Nuclear Shell Model amp Alpha Decay Stanley Yen TRIUMF
Nuclear Shell Model
Electrons in atoms occupy well-defined shells of discrete well-separatedenergy Do nucleons inside a nucleus do the same or not
Evidence for electron shells in atoms sudden jumps in atomicproperties as a shell gets filled up eg atomic radius ionization energy chemical properties
from Krane IntroductoryNuclear Physics
atomicradius
atomicionizationenergy
chemical reactivity
Not evident a priori that nuclei should show shell structure Why not
1 Success of liquid drop model in predicting nuclear binding energies Liquid drops have smooth behaviour with increasing size and do not exhibit jumps
2 No obvious center for nucleons to orbit around unlike electronsin an atom
one pionexchangetwo pion
andheavy mesonexchange
overlap of 3-quark bagscomplicatedshort-rangebehaviour
3rd objection to a shell model Strong repulsivecore in the nucleon-nucleonpotential should scatterthe nucleons farout of their orbits --nucleons should nothave a well-definedenergy but shouldbehave more likemolecules in a gascolliding and exchanging energywith each other
before
after
Many theoretical reasons were given why nuclei shouldnot show any shell structure
but experiment says otherwise
2-neutron separationenergy in nuclei-- analogousto ionization energyin atoms
Note jumps atnucleon numbers8 20 28 50 82 126
from KraneIntroductoryNuclear Physics
from Gadioli ampGadioliIntroNuclearPhysics
abundancesof even-evennuclides
neutron capturecross section--analogous tochemicalreactivity
dips at N=20 50 82 126
local maximaat 50 82 126
from KraneIntroductoryNuclear Physics
alpha particleemissionenergy vs neutron numberof parent nucleusmax for N=128 parent (N=126 daughter)
neutroncapturecross section(repeat)
nuclearchargeradius
minima atN=20 28 50 82 126
Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo
2 8 20 28 50 82 126
These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms
Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from
It must be related to the shape of the potential well that the nucleons sit in
Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise
Greg Goebel wwwvectorsitenettpqm_02html
wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)
wavelength λ = 2Ln n=1 2 3
De Broglie momentum p = hλ = nh2L
Potential energy V(x)=0 inside well
E = p22M = n2h2 (8ML2)
A different shape for the potential well would result in a different set of level spacings eg
Krane IntroductoryNuclear Physics
infinite squarewell potential
harmonic oscillatorpotential
Notations means L=0p means L=1d L=2f L=3
p state can have2L+1 =3 substates
each substate canhave spin up orspin down
Thus 6 protons and6 neutrons in 1plevel
From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this
from KraneIntroductoryNuclear Physics
But this shape of potential doesnt give the right energy level spacings either
No reasonable shape of nuclear potential seems to work
Empirical magic numbers2 8 20 28 50 82 126
Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo
rarr rarr rarrV(r) = V
0(r) + V
LS LS L = orbital angular momentum
rarr S = spin angular momentum
where the potentials V0 and V
LS are negative (ie attractive)
L and S paralleldeeper attractive potential
L and S anti-parallelshallower attractive potential
So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting
J=32 and J=12
J=12
J=32
In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps
Hydrogen atom
Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P
32 and 2P
12 is
only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)
With a nuclear spin-orbit force the levels now match what is observed experimentally
1g levelis split somuch thatthe twocomponentsnow liein differentshells
Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Nuclear Shell Model
Electrons in atoms occupy well-defined shells of discrete well-separatedenergy Do nucleons inside a nucleus do the same or not
Evidence for electron shells in atoms sudden jumps in atomicproperties as a shell gets filled up eg atomic radius ionization energy chemical properties
from Krane IntroductoryNuclear Physics
atomicradius
atomicionizationenergy
chemical reactivity
Not evident a priori that nuclei should show shell structure Why not
1 Success of liquid drop model in predicting nuclear binding energies Liquid drops have smooth behaviour with increasing size and do not exhibit jumps
2 No obvious center for nucleons to orbit around unlike electronsin an atom
one pionexchangetwo pion
andheavy mesonexchange
overlap of 3-quark bagscomplicatedshort-rangebehaviour
3rd objection to a shell model Strong repulsivecore in the nucleon-nucleonpotential should scatterthe nucleons farout of their orbits --nucleons should nothave a well-definedenergy but shouldbehave more likemolecules in a gascolliding and exchanging energywith each other
before
after
Many theoretical reasons were given why nuclei shouldnot show any shell structure
but experiment says otherwise
2-neutron separationenergy in nuclei-- analogousto ionization energyin atoms
Note jumps atnucleon numbers8 20 28 50 82 126
from KraneIntroductoryNuclear Physics
from Gadioli ampGadioliIntroNuclearPhysics
abundancesof even-evennuclides
neutron capturecross section--analogous tochemicalreactivity
dips at N=20 50 82 126
local maximaat 50 82 126
from KraneIntroductoryNuclear Physics
alpha particleemissionenergy vs neutron numberof parent nucleusmax for N=128 parent (N=126 daughter)
neutroncapturecross section(repeat)
nuclearchargeradius
minima atN=20 28 50 82 126
Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo
2 8 20 28 50 82 126
These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms
Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from
It must be related to the shape of the potential well that the nucleons sit in
Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise
Greg Goebel wwwvectorsitenettpqm_02html
wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)
wavelength λ = 2Ln n=1 2 3
De Broglie momentum p = hλ = nh2L
Potential energy V(x)=0 inside well
E = p22M = n2h2 (8ML2)
A different shape for the potential well would result in a different set of level spacings eg
Krane IntroductoryNuclear Physics
infinite squarewell potential
harmonic oscillatorpotential
Notations means L=0p means L=1d L=2f L=3
p state can have2L+1 =3 substates
each substate canhave spin up orspin down
Thus 6 protons and6 neutrons in 1plevel
From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this
from KraneIntroductoryNuclear Physics
But this shape of potential doesnt give the right energy level spacings either
No reasonable shape of nuclear potential seems to work
Empirical magic numbers2 8 20 28 50 82 126
Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo
rarr rarr rarrV(r) = V
0(r) + V
LS LS L = orbital angular momentum
rarr S = spin angular momentum
where the potentials V0 and V
LS are negative (ie attractive)
L and S paralleldeeper attractive potential
L and S anti-parallelshallower attractive potential
So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting
J=32 and J=12
J=12
J=32
In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps
Hydrogen atom
Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P
32 and 2P
12 is
only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)
With a nuclear spin-orbit force the levels now match what is observed experimentally
1g levelis split somuch thatthe twocomponentsnow liein differentshells
Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
from Krane IntroductoryNuclear Physics
atomicradius
atomicionizationenergy
chemical reactivity
Not evident a priori that nuclei should show shell structure Why not
1 Success of liquid drop model in predicting nuclear binding energies Liquid drops have smooth behaviour with increasing size and do not exhibit jumps
2 No obvious center for nucleons to orbit around unlike electronsin an atom
one pionexchangetwo pion
andheavy mesonexchange
overlap of 3-quark bagscomplicatedshort-rangebehaviour
3rd objection to a shell model Strong repulsivecore in the nucleon-nucleonpotential should scatterthe nucleons farout of their orbits --nucleons should nothave a well-definedenergy but shouldbehave more likemolecules in a gascolliding and exchanging energywith each other
before
after
Many theoretical reasons were given why nuclei shouldnot show any shell structure
but experiment says otherwise
2-neutron separationenergy in nuclei-- analogousto ionization energyin atoms
Note jumps atnucleon numbers8 20 28 50 82 126
from KraneIntroductoryNuclear Physics
from Gadioli ampGadioliIntroNuclearPhysics
abundancesof even-evennuclides
neutron capturecross section--analogous tochemicalreactivity
dips at N=20 50 82 126
local maximaat 50 82 126
from KraneIntroductoryNuclear Physics
alpha particleemissionenergy vs neutron numberof parent nucleusmax for N=128 parent (N=126 daughter)
neutroncapturecross section(repeat)
nuclearchargeradius
minima atN=20 28 50 82 126
Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo
2 8 20 28 50 82 126
These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms
Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from
It must be related to the shape of the potential well that the nucleons sit in
Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise
Greg Goebel wwwvectorsitenettpqm_02html
wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)
wavelength λ = 2Ln n=1 2 3
De Broglie momentum p = hλ = nh2L
Potential energy V(x)=0 inside well
E = p22M = n2h2 (8ML2)
A different shape for the potential well would result in a different set of level spacings eg
Krane IntroductoryNuclear Physics
infinite squarewell potential
harmonic oscillatorpotential
Notations means L=0p means L=1d L=2f L=3
p state can have2L+1 =3 substates
each substate canhave spin up orspin down
Thus 6 protons and6 neutrons in 1plevel
From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this
from KraneIntroductoryNuclear Physics
But this shape of potential doesnt give the right energy level spacings either
No reasonable shape of nuclear potential seems to work
Empirical magic numbers2 8 20 28 50 82 126
Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo
rarr rarr rarrV(r) = V
0(r) + V
LS LS L = orbital angular momentum
rarr S = spin angular momentum
where the potentials V0 and V
LS are negative (ie attractive)
L and S paralleldeeper attractive potential
L and S anti-parallelshallower attractive potential
So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting
J=32 and J=12
J=12
J=32
In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps
Hydrogen atom
Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P
32 and 2P
12 is
only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)
With a nuclear spin-orbit force the levels now match what is observed experimentally
1g levelis split somuch thatthe twocomponentsnow liein differentshells
Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
chemical reactivity
Not evident a priori that nuclei should show shell structure Why not
1 Success of liquid drop model in predicting nuclear binding energies Liquid drops have smooth behaviour with increasing size and do not exhibit jumps
2 No obvious center for nucleons to orbit around unlike electronsin an atom
one pionexchangetwo pion
andheavy mesonexchange
overlap of 3-quark bagscomplicatedshort-rangebehaviour
3rd objection to a shell model Strong repulsivecore in the nucleon-nucleonpotential should scatterthe nucleons farout of their orbits --nucleons should nothave a well-definedenergy but shouldbehave more likemolecules in a gascolliding and exchanging energywith each other
before
after
Many theoretical reasons were given why nuclei shouldnot show any shell structure
but experiment says otherwise
2-neutron separationenergy in nuclei-- analogousto ionization energyin atoms
Note jumps atnucleon numbers8 20 28 50 82 126
from KraneIntroductoryNuclear Physics
from Gadioli ampGadioliIntroNuclearPhysics
abundancesof even-evennuclides
neutron capturecross section--analogous tochemicalreactivity
dips at N=20 50 82 126
local maximaat 50 82 126
from KraneIntroductoryNuclear Physics
alpha particleemissionenergy vs neutron numberof parent nucleusmax for N=128 parent (N=126 daughter)
neutroncapturecross section(repeat)
nuclearchargeradius
minima atN=20 28 50 82 126
Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo
2 8 20 28 50 82 126
These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms
Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from
It must be related to the shape of the potential well that the nucleons sit in
Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise
Greg Goebel wwwvectorsitenettpqm_02html
wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)
wavelength λ = 2Ln n=1 2 3
De Broglie momentum p = hλ = nh2L
Potential energy V(x)=0 inside well
E = p22M = n2h2 (8ML2)
A different shape for the potential well would result in a different set of level spacings eg
Krane IntroductoryNuclear Physics
infinite squarewell potential
harmonic oscillatorpotential
Notations means L=0p means L=1d L=2f L=3
p state can have2L+1 =3 substates
each substate canhave spin up orspin down
Thus 6 protons and6 neutrons in 1plevel
From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this
from KraneIntroductoryNuclear Physics
But this shape of potential doesnt give the right energy level spacings either
No reasonable shape of nuclear potential seems to work
Empirical magic numbers2 8 20 28 50 82 126
Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo
rarr rarr rarrV(r) = V
0(r) + V
LS LS L = orbital angular momentum
rarr S = spin angular momentum
where the potentials V0 and V
LS are negative (ie attractive)
L and S paralleldeeper attractive potential
L and S anti-parallelshallower attractive potential
So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting
J=32 and J=12
J=12
J=32
In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps
Hydrogen atom
Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P
32 and 2P
12 is
only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)
With a nuclear spin-orbit force the levels now match what is observed experimentally
1g levelis split somuch thatthe twocomponentsnow liein differentshells
Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Not evident a priori that nuclei should show shell structure Why not
1 Success of liquid drop model in predicting nuclear binding energies Liquid drops have smooth behaviour with increasing size and do not exhibit jumps
2 No obvious center for nucleons to orbit around unlike electronsin an atom
one pionexchangetwo pion
andheavy mesonexchange
overlap of 3-quark bagscomplicatedshort-rangebehaviour
3rd objection to a shell model Strong repulsivecore in the nucleon-nucleonpotential should scatterthe nucleons farout of their orbits --nucleons should nothave a well-definedenergy but shouldbehave more likemolecules in a gascolliding and exchanging energywith each other
before
after
Many theoretical reasons were given why nuclei shouldnot show any shell structure
but experiment says otherwise
2-neutron separationenergy in nuclei-- analogousto ionization energyin atoms
Note jumps atnucleon numbers8 20 28 50 82 126
from KraneIntroductoryNuclear Physics
from Gadioli ampGadioliIntroNuclearPhysics
abundancesof even-evennuclides
neutron capturecross section--analogous tochemicalreactivity
dips at N=20 50 82 126
local maximaat 50 82 126
from KraneIntroductoryNuclear Physics
alpha particleemissionenergy vs neutron numberof parent nucleusmax for N=128 parent (N=126 daughter)
neutroncapturecross section(repeat)
nuclearchargeradius
minima atN=20 28 50 82 126
Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo
2 8 20 28 50 82 126
These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms
Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from
It must be related to the shape of the potential well that the nucleons sit in
Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise
Greg Goebel wwwvectorsitenettpqm_02html
wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)
wavelength λ = 2Ln n=1 2 3
De Broglie momentum p = hλ = nh2L
Potential energy V(x)=0 inside well
E = p22M = n2h2 (8ML2)
A different shape for the potential well would result in a different set of level spacings eg
Krane IntroductoryNuclear Physics
infinite squarewell potential
harmonic oscillatorpotential
Notations means L=0p means L=1d L=2f L=3
p state can have2L+1 =3 substates
each substate canhave spin up orspin down
Thus 6 protons and6 neutrons in 1plevel
From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this
from KraneIntroductoryNuclear Physics
But this shape of potential doesnt give the right energy level spacings either
No reasonable shape of nuclear potential seems to work
Empirical magic numbers2 8 20 28 50 82 126
Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo
rarr rarr rarrV(r) = V
0(r) + V
LS LS L = orbital angular momentum
rarr S = spin angular momentum
where the potentials V0 and V
LS are negative (ie attractive)
L and S paralleldeeper attractive potential
L and S anti-parallelshallower attractive potential
So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting
J=32 and J=12
J=12
J=32
In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps
Hydrogen atom
Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P
32 and 2P
12 is
only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)
With a nuclear spin-orbit force the levels now match what is observed experimentally
1g levelis split somuch thatthe twocomponentsnow liein differentshells
Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
one pionexchangetwo pion
andheavy mesonexchange
overlap of 3-quark bagscomplicatedshort-rangebehaviour
3rd objection to a shell model Strong repulsivecore in the nucleon-nucleonpotential should scatterthe nucleons farout of their orbits --nucleons should nothave a well-definedenergy but shouldbehave more likemolecules in a gascolliding and exchanging energywith each other
before
after
Many theoretical reasons were given why nuclei shouldnot show any shell structure
but experiment says otherwise
2-neutron separationenergy in nuclei-- analogousto ionization energyin atoms
Note jumps atnucleon numbers8 20 28 50 82 126
from KraneIntroductoryNuclear Physics
from Gadioli ampGadioliIntroNuclearPhysics
abundancesof even-evennuclides
neutron capturecross section--analogous tochemicalreactivity
dips at N=20 50 82 126
local maximaat 50 82 126
from KraneIntroductoryNuclear Physics
alpha particleemissionenergy vs neutron numberof parent nucleusmax for N=128 parent (N=126 daughter)
neutroncapturecross section(repeat)
nuclearchargeradius
minima atN=20 28 50 82 126
Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo
2 8 20 28 50 82 126
These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms
Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from
It must be related to the shape of the potential well that the nucleons sit in
Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise
Greg Goebel wwwvectorsitenettpqm_02html
wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)
wavelength λ = 2Ln n=1 2 3
De Broglie momentum p = hλ = nh2L
Potential energy V(x)=0 inside well
E = p22M = n2h2 (8ML2)
A different shape for the potential well would result in a different set of level spacings eg
Krane IntroductoryNuclear Physics
infinite squarewell potential
harmonic oscillatorpotential
Notations means L=0p means L=1d L=2f L=3
p state can have2L+1 =3 substates
each substate canhave spin up orspin down
Thus 6 protons and6 neutrons in 1plevel
From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this
from KraneIntroductoryNuclear Physics
But this shape of potential doesnt give the right energy level spacings either
No reasonable shape of nuclear potential seems to work
Empirical magic numbers2 8 20 28 50 82 126
Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo
rarr rarr rarrV(r) = V
0(r) + V
LS LS L = orbital angular momentum
rarr S = spin angular momentum
where the potentials V0 and V
LS are negative (ie attractive)
L and S paralleldeeper attractive potential
L and S anti-parallelshallower attractive potential
So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting
J=32 and J=12
J=12
J=32
In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps
Hydrogen atom
Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P
32 and 2P
12 is
only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)
With a nuclear spin-orbit force the levels now match what is observed experimentally
1g levelis split somuch thatthe twocomponentsnow liein differentshells
Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Many theoretical reasons were given why nuclei shouldnot show any shell structure
but experiment says otherwise
2-neutron separationenergy in nuclei-- analogousto ionization energyin atoms
Note jumps atnucleon numbers8 20 28 50 82 126
from KraneIntroductoryNuclear Physics
from Gadioli ampGadioliIntroNuclearPhysics
abundancesof even-evennuclides
neutron capturecross section--analogous tochemicalreactivity
dips at N=20 50 82 126
local maximaat 50 82 126
from KraneIntroductoryNuclear Physics
alpha particleemissionenergy vs neutron numberof parent nucleusmax for N=128 parent (N=126 daughter)
neutroncapturecross section(repeat)
nuclearchargeradius
minima atN=20 28 50 82 126
Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo
2 8 20 28 50 82 126
These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms
Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from
It must be related to the shape of the potential well that the nucleons sit in
Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise
Greg Goebel wwwvectorsitenettpqm_02html
wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)
wavelength λ = 2Ln n=1 2 3
De Broglie momentum p = hλ = nh2L
Potential energy V(x)=0 inside well
E = p22M = n2h2 (8ML2)
A different shape for the potential well would result in a different set of level spacings eg
Krane IntroductoryNuclear Physics
infinite squarewell potential
harmonic oscillatorpotential
Notations means L=0p means L=1d L=2f L=3
p state can have2L+1 =3 substates
each substate canhave spin up orspin down
Thus 6 protons and6 neutrons in 1plevel
From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this
from KraneIntroductoryNuclear Physics
But this shape of potential doesnt give the right energy level spacings either
No reasonable shape of nuclear potential seems to work
Empirical magic numbers2 8 20 28 50 82 126
Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo
rarr rarr rarrV(r) = V
0(r) + V
LS LS L = orbital angular momentum
rarr S = spin angular momentum
where the potentials V0 and V
LS are negative (ie attractive)
L and S paralleldeeper attractive potential
L and S anti-parallelshallower attractive potential
So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting
J=32 and J=12
J=12
J=32
In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps
Hydrogen atom
Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P
32 and 2P
12 is
only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)
With a nuclear spin-orbit force the levels now match what is observed experimentally
1g levelis split somuch thatthe twocomponentsnow liein differentshells
Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
2-neutron separationenergy in nuclei-- analogousto ionization energyin atoms
Note jumps atnucleon numbers8 20 28 50 82 126
from KraneIntroductoryNuclear Physics
from Gadioli ampGadioliIntroNuclearPhysics
abundancesof even-evennuclides
neutron capturecross section--analogous tochemicalreactivity
dips at N=20 50 82 126
local maximaat 50 82 126
from KraneIntroductoryNuclear Physics
alpha particleemissionenergy vs neutron numberof parent nucleusmax for N=128 parent (N=126 daughter)
neutroncapturecross section(repeat)
nuclearchargeradius
minima atN=20 28 50 82 126
Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo
2 8 20 28 50 82 126
These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms
Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from
It must be related to the shape of the potential well that the nucleons sit in
Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise
Greg Goebel wwwvectorsitenettpqm_02html
wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)
wavelength λ = 2Ln n=1 2 3
De Broglie momentum p = hλ = nh2L
Potential energy V(x)=0 inside well
E = p22M = n2h2 (8ML2)
A different shape for the potential well would result in a different set of level spacings eg
Krane IntroductoryNuclear Physics
infinite squarewell potential
harmonic oscillatorpotential
Notations means L=0p means L=1d L=2f L=3
p state can have2L+1 =3 substates
each substate canhave spin up orspin down
Thus 6 protons and6 neutrons in 1plevel
From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this
from KraneIntroductoryNuclear Physics
But this shape of potential doesnt give the right energy level spacings either
No reasonable shape of nuclear potential seems to work
Empirical magic numbers2 8 20 28 50 82 126
Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo
rarr rarr rarrV(r) = V
0(r) + V
LS LS L = orbital angular momentum
rarr S = spin angular momentum
where the potentials V0 and V
LS are negative (ie attractive)
L and S paralleldeeper attractive potential
L and S anti-parallelshallower attractive potential
So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting
J=32 and J=12
J=12
J=32
In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps
Hydrogen atom
Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P
32 and 2P
12 is
only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)
With a nuclear spin-orbit force the levels now match what is observed experimentally
1g levelis split somuch thatthe twocomponentsnow liein differentshells
Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
from Gadioli ampGadioliIntroNuclearPhysics
abundancesof even-evennuclides
neutron capturecross section--analogous tochemicalreactivity
dips at N=20 50 82 126
local maximaat 50 82 126
from KraneIntroductoryNuclear Physics
alpha particleemissionenergy vs neutron numberof parent nucleusmax for N=128 parent (N=126 daughter)
neutroncapturecross section(repeat)
nuclearchargeradius
minima atN=20 28 50 82 126
Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo
2 8 20 28 50 82 126
These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms
Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from
It must be related to the shape of the potential well that the nucleons sit in
Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise
Greg Goebel wwwvectorsitenettpqm_02html
wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)
wavelength λ = 2Ln n=1 2 3
De Broglie momentum p = hλ = nh2L
Potential energy V(x)=0 inside well
E = p22M = n2h2 (8ML2)
A different shape for the potential well would result in a different set of level spacings eg
Krane IntroductoryNuclear Physics
infinite squarewell potential
harmonic oscillatorpotential
Notations means L=0p means L=1d L=2f L=3
p state can have2L+1 =3 substates
each substate canhave spin up orspin down
Thus 6 protons and6 neutrons in 1plevel
From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this
from KraneIntroductoryNuclear Physics
But this shape of potential doesnt give the right energy level spacings either
No reasonable shape of nuclear potential seems to work
Empirical magic numbers2 8 20 28 50 82 126
Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo
rarr rarr rarrV(r) = V
0(r) + V
LS LS L = orbital angular momentum
rarr S = spin angular momentum
where the potentials V0 and V
LS are negative (ie attractive)
L and S paralleldeeper attractive potential
L and S anti-parallelshallower attractive potential
So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting
J=32 and J=12
J=12
J=32
In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps
Hydrogen atom
Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P
32 and 2P
12 is
only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)
With a nuclear spin-orbit force the levels now match what is observed experimentally
1g levelis split somuch thatthe twocomponentsnow liein differentshells
Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
from KraneIntroductoryNuclear Physics
alpha particleemissionenergy vs neutron numberof parent nucleusmax for N=128 parent (N=126 daughter)
neutroncapturecross section(repeat)
nuclearchargeradius
minima atN=20 28 50 82 126
Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo
2 8 20 28 50 82 126
These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms
Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from
It must be related to the shape of the potential well that the nucleons sit in
Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise
Greg Goebel wwwvectorsitenettpqm_02html
wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)
wavelength λ = 2Ln n=1 2 3
De Broglie momentum p = hλ = nh2L
Potential energy V(x)=0 inside well
E = p22M = n2h2 (8ML2)
A different shape for the potential well would result in a different set of level spacings eg
Krane IntroductoryNuclear Physics
infinite squarewell potential
harmonic oscillatorpotential
Notations means L=0p means L=1d L=2f L=3
p state can have2L+1 =3 substates
each substate canhave spin up orspin down
Thus 6 protons and6 neutrons in 1plevel
From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this
from KraneIntroductoryNuclear Physics
But this shape of potential doesnt give the right energy level spacings either
No reasonable shape of nuclear potential seems to work
Empirical magic numbers2 8 20 28 50 82 126
Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo
rarr rarr rarrV(r) = V
0(r) + V
LS LS L = orbital angular momentum
rarr S = spin angular momentum
where the potentials V0 and V
LS are negative (ie attractive)
L and S paralleldeeper attractive potential
L and S anti-parallelshallower attractive potential
So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting
J=32 and J=12
J=12
J=32
In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps
Hydrogen atom
Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P
32 and 2P
12 is
only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)
With a nuclear spin-orbit force the levels now match what is observed experimentally
1g levelis split somuch thatthe twocomponentsnow liein differentshells
Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo
2 8 20 28 50 82 126
These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms
Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from
It must be related to the shape of the potential well that the nucleons sit in
Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise
Greg Goebel wwwvectorsitenettpqm_02html
wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)
wavelength λ = 2Ln n=1 2 3
De Broglie momentum p = hλ = nh2L
Potential energy V(x)=0 inside well
E = p22M = n2h2 (8ML2)
A different shape for the potential well would result in a different set of level spacings eg
Krane IntroductoryNuclear Physics
infinite squarewell potential
harmonic oscillatorpotential
Notations means L=0p means L=1d L=2f L=3
p state can have2L+1 =3 substates
each substate canhave spin up orspin down
Thus 6 protons and6 neutrons in 1plevel
From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this
from KraneIntroductoryNuclear Physics
But this shape of potential doesnt give the right energy level spacings either
No reasonable shape of nuclear potential seems to work
Empirical magic numbers2 8 20 28 50 82 126
Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo
rarr rarr rarrV(r) = V
0(r) + V
LS LS L = orbital angular momentum
rarr S = spin angular momentum
where the potentials V0 and V
LS are negative (ie attractive)
L and S paralleldeeper attractive potential
L and S anti-parallelshallower attractive potential
So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting
J=32 and J=12
J=12
J=32
In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps
Hydrogen atom
Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P
32 and 2P
12 is
only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)
With a nuclear spin-orbit force the levels now match what is observed experimentally
1g levelis split somuch thatthe twocomponentsnow liein differentshells
Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise
Greg Goebel wwwvectorsitenettpqm_02html
wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)
wavelength λ = 2Ln n=1 2 3
De Broglie momentum p = hλ = nh2L
Potential energy V(x)=0 inside well
E = p22M = n2h2 (8ML2)
A different shape for the potential well would result in a different set of level spacings eg
Krane IntroductoryNuclear Physics
infinite squarewell potential
harmonic oscillatorpotential
Notations means L=0p means L=1d L=2f L=3
p state can have2L+1 =3 substates
each substate canhave spin up orspin down
Thus 6 protons and6 neutrons in 1plevel
From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this
from KraneIntroductoryNuclear Physics
But this shape of potential doesnt give the right energy level spacings either
No reasonable shape of nuclear potential seems to work
Empirical magic numbers2 8 20 28 50 82 126
Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo
rarr rarr rarrV(r) = V
0(r) + V
LS LS L = orbital angular momentum
rarr S = spin angular momentum
where the potentials V0 and V
LS are negative (ie attractive)
L and S paralleldeeper attractive potential
L and S anti-parallelshallower attractive potential
So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting
J=32 and J=12
J=12
J=32
In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps
Hydrogen atom
Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P
32 and 2P
12 is
only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)
With a nuclear spin-orbit force the levels now match what is observed experimentally
1g levelis split somuch thatthe twocomponentsnow liein differentshells
Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
A different shape for the potential well would result in a different set of level spacings eg
Krane IntroductoryNuclear Physics
infinite squarewell potential
harmonic oscillatorpotential
Notations means L=0p means L=1d L=2f L=3
p state can have2L+1 =3 substates
each substate canhave spin up orspin down
Thus 6 protons and6 neutrons in 1plevel
From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this
from KraneIntroductoryNuclear Physics
But this shape of potential doesnt give the right energy level spacings either
No reasonable shape of nuclear potential seems to work
Empirical magic numbers2 8 20 28 50 82 126
Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo
rarr rarr rarrV(r) = V
0(r) + V
LS LS L = orbital angular momentum
rarr S = spin angular momentum
where the potentials V0 and V
LS are negative (ie attractive)
L and S paralleldeeper attractive potential
L and S anti-parallelshallower attractive potential
So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting
J=32 and J=12
J=12
J=32
In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps
Hydrogen atom
Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P
32 and 2P
12 is
only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)
With a nuclear spin-orbit force the levels now match what is observed experimentally
1g levelis split somuch thatthe twocomponentsnow liein differentshells
Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this
from KraneIntroductoryNuclear Physics
But this shape of potential doesnt give the right energy level spacings either
No reasonable shape of nuclear potential seems to work
Empirical magic numbers2 8 20 28 50 82 126
Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo
rarr rarr rarrV(r) = V
0(r) + V
LS LS L = orbital angular momentum
rarr S = spin angular momentum
where the potentials V0 and V
LS are negative (ie attractive)
L and S paralleldeeper attractive potential
L and S anti-parallelshallower attractive potential
So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting
J=32 and J=12
J=12
J=32
In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps
Hydrogen atom
Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P
32 and 2P
12 is
only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)
With a nuclear spin-orbit force the levels now match what is observed experimentally
1g levelis split somuch thatthe twocomponentsnow liein differentshells
Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
But this shape of potential doesnt give the right energy level spacings either
No reasonable shape of nuclear potential seems to work
Empirical magic numbers2 8 20 28 50 82 126
Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo
rarr rarr rarrV(r) = V
0(r) + V
LS LS L = orbital angular momentum
rarr S = spin angular momentum
where the potentials V0 and V
LS are negative (ie attractive)
L and S paralleldeeper attractive potential
L and S anti-parallelshallower attractive potential
So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting
J=32 and J=12
J=12
J=32
In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps
Hydrogen atom
Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P
32 and 2P
12 is
only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)
With a nuclear spin-orbit force the levels now match what is observed experimentally
1g levelis split somuch thatthe twocomponentsnow liein differentshells
Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo
rarr rarr rarrV(r) = V
0(r) + V
LS LS L = orbital angular momentum
rarr S = spin angular momentum
where the potentials V0 and V
LS are negative (ie attractive)
L and S paralleldeeper attractive potential
L and S anti-parallelshallower attractive potential
So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting
J=32 and J=12
J=12
J=32
In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps
Hydrogen atom
Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P
32 and 2P
12 is
only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)
With a nuclear spin-orbit force the levels now match what is observed experimentally
1g levelis split somuch thatthe twocomponentsnow liein differentshells
Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting
J=32 and J=12
J=12
J=32
In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps
Hydrogen atom
Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P
32 and 2P
12 is
only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)
With a nuclear spin-orbit force the levels now match what is observed experimentally
1g levelis split somuch thatthe twocomponentsnow liein differentshells
Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Hydrogen atom
Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P
32 and 2P
12 is
only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)
With a nuclear spin-orbit force the levels now match what is observed experimentally
1g levelis split somuch thatthe twocomponentsnow liein differentshells
Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
With a nuclear spin-orbit force the levels now match what is observed experimentally
1g levelis split somuch thatthe twocomponentsnow liein differentshells
Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells
J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell
J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell
In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
filled 1S shellcorrespondingto magic no 2
4He
so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
16O
filled 1S shell
filled 1P shell
Oxygen-16 is another double closed-shell nucleus
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+
eg 12C N=6 Z=6 has a 0+ ground state
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus
Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability
As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes
This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Alpha Decay
α-particles are 4He nuclei
Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A
eg 238U rarr 234Th + α
Z 92 90 A 238 234Z2A 3556 3461
Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then
The α particle is especially tightly bound (especially small mass)so that in the decay
238U rarr 234Th + α
this gives an especially large amount of available energy to the decayproduct
It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate
Rate ~ p where p=momentum of the particle
and for non-relativistic particles E = p22M
so Rate ~ E12
ie doubling the energy results in radic2 increase in the decay rate
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Q = energy released
compare 232Th and218Th
roughly double theenergy but 24 ordersof magnitude changein decay rate
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
This defied explanation until George Gamow realized it was the result of quantummechanical tunneling
Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics
Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave
But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection
The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this
E
The bigger the difference in refractive index the faster theexponential drops off
E
rarr
rarr
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection
The decaying exponentialwave in the gap regionis called an evanescentwave
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Initial glass blockStrong light beam(large amplitude sinusoidal E-field)
Air gapAmplitude ofE-field dropsexponentiallywith distance
Second glass blockWeak light beam(small amplitudesinusoidal E-field)
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Other examples of classical evanescent waves
submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave
The better the conductingmedium the faster theexponential drops off
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V
E
E
V
Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall
But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off
And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Heres what the wavefunction looks like
Initial sinusoidal wavewith large amplitude A
Barrier regiondecayingexponential
Final sinusoidal wavewith small amplitude B
Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2
It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier
ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier
So a small change in energy E or thickness d makes a huge change in penetration
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier
Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
For a hyperbolic-shaped barrier it can be shown that
Transmission prob T = exp(-2G) = exp(-2Z E-12)
The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)
Half-life t12 ~ 1T = exp(+2Z E-12)
ln(t12 ) ~ 2 Z E-12
from Segregrave Nuclei and Particles
Confirms linear relationship between
ln(t12 ) and E-12
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier
The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Putting in some typical numbers
For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)
those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second
and yet it takes ~14x1010 years to leak out
It is evident that the tunneling probability is very low
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Next week Nuclear reactions
The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus
accelerator
projectiletarget
The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that
Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus
I hope you find learning nuclear physics easier than that