Pp 2012 Problems

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Subatomic problem set 1

Radioactivity and nuclear stability

Three vectors are written in bold e.g. x.Four vectors are written sans-serif, e.g. P.The metric is diag(1,1,1,1) so that with X X= c2t2 x x,time-like intervals have positive signs, and propagating particles sat-isfy P P= +m2.

Core questions

1.1. a) What assumptions underlie the radioactivity law

dN

dt = N ?

b) IfN0 nuclei are present at time t = 0 how many are present at time t?

c) Calculate the mean life of the species.

d) Relate the half-life t 12

to .

e) How do things change when the particle moves relativistically?

1.2. Consider Compton scattering + e + e of a photon of energyEfrom a stationary electron.

a) Show that the energy of the scattered photon is

E0= meE

me+ E(1 cos ) ,

where is the angle through which the photon is scattered.

b) If an incoming photon is scattered through an angle 180 at the surface ofa material, how much of the original photons energy would you expect not tobe deposited in the material?

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c) A high energy photon can create an electron-positron pair within the material.

When a positron comes to rest it will annihilate against an electron from themateriale+ + e +.

What will be the energy of these secondary photons?

d) The figure shows the energy spectrum from the gamma decay of 24Na asmeasured in a small Ge(Li) detector. Explain the origins of the peaks A, B, Cand the edge D. For such a detector describe the stages by which gamma rayenergy is converted into a measurable voltage pulse.

1.3. A sample consists originally of nucleus A only, but subsequently decaysaccording to

A A B B C.

Write down differential expressions for dAdt

, dBdt

and dCdt

. Solve for, and thensketch, the fractions of, A(t),B (t)andC(t). At what time is the decay rate of

B maximum?

1.4. Briefly explain the origin of each of the terms in the semi-empirical massformula (SEMF)

M(N, Z) =Z mp+ Nmn A +A23 +(N Z)2

A +

Z2

A1

3

+(N, Z)

and obtain a value for .

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Show that we can include the gravitational interaction between the nucleons by

adding a term to the SEMF of the form

A5/3

and find the value of.

Use this modified SEMF to obtain a lower bound on the mass of a gravitationally-bound nucleus consisting only of neutrons (a neutron star).

1.5. An analysis of a chart showing all stable (t 12

>109 years) nuclei shows thatthere are 177 even-even, 121 even-odd and 8 odd-odd stable nuclei and, for each

A, only on, two or three stable isobars. Explain these observations qualitativelyusing the SEMF. Energetically 10648Cdcould decay to

10646Pdwith an energy release

of greater than 2 MeV. Why does 10648Cd occur naturally?

1.6. The radiusr of a nucleus with mass number A is given by r = r0A1

3 withr0 = 1.2 fm. What does this tell us about the nuclear force?

a) Use the Fermi gas model (assuming N Z) to show that the energy F ofthe Fermi level is given by

F = ~2

2mr20

9

8 23

.

b) Estimate the total kinetic energy of the nucleons in an 16Onucleus.

c) For a nucleus with neutron number Nand proton number Zthe asymmetryterm in the semi-empirical mass formula is

(N Z)2A

.

Assuming that (N Z) A use the Fermi gas model to justify this form andto estimate the value of. Comment on the value obtained.

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1.7. Alpha-decay rates are determined by the probability of tunnelling through

the Coulomb barrier. Draw a diagram of the potential energyV(r)as a functionof the distance r between the daughter nucleus and the particle, and of thewave functionhr|i.

The decay rate can be expressed as =f P, wherefis the frequency of attemptsby the alpha particle to escape and P = exp(2G) is the probability for thealpha particle to escape on any given attempt.

By using a one-dimensional approximation to the Hamiltonian

H= p2x

2m

+ V,

and by representing the wave function by

hx|i = exp[(x)],show that

G=

2m

~

Z ba

pV(r)Qdr.

Integrate (a substitutionr= rbcos2 helps) to give

G=

2Zz

s2mc2

Q F(ra/rb)

where the dimensionless function

F(x) = 2

cos1

x

px(1 x)

lies in the range between 0 and 1, and for small Q approaches 1.

[Hint: can you convince yourself that00 (0)2?]

1.8. a) What are the basic assumptions of the Fermi theory of beta decay?

b) The Fermi theory predicts that in a beta decay the rate of electrons emittedwith momentum betweenpandp + dpis given by

d

dpe=

2

~G2|Mfi|

2 1

44~6c3(EQ)2p2.

Justify this result.

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c) Show that for Q mec2 the total rate is proportional to Q5

d) What spin states are allowed for the combined system of the electron +neutrino?

e) Why are transitions between initial and final nuclei with angular momentadiffering by more than ~suppressed?

Optional questions

1.9. The figure shows the-decay scheme of 244

96Cm and 240

94Pu.

Show either by (a) using a suitable approximation ofFcalculated in a previousquestion or (b) by redoing the corresponding integral neglecting Q that wewould expect the rates to satisfy and equation of the form

log =A BZQ

.

The Q value for the ground state to ground state transition is 5.902 MeV and

for this transition A = 132.8 and B = 3.97( MeV)1/2 when is in s1. Thebranching ratio for this transition is given in the figure. Calculate the mean lifeof 244Cm.

Estimate the transition rate from the ground state of 244Cm to the 6+ level of240Pu using the same AandB and compare to the branching ratio given in thefigure.

Suggest a reason for any discrepancy.

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[Hint: what form does the Schrodinger equation take for angular momentum quantum

numberl 6= 0?]1.10. Discuss the evidence for shell structure in the atomic nucleus. Indicatehow closed shells for proton and neutron numbers of 2, 8, 20, 28, 50 can beexplained. What are the other magic numbers?

Deduce from the shell model the spins and parities of the ground states of thefollowing nuclei, stating any assumptions you make: 73Li,

178O,

2010Ne,

2713Al,

147N,

3919K,

4121Sc.

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Quarks and deep inelastic scattering

Core questions

2.1. TheJP = 32

+decuplet contains the following baryons:

++

+

0

+0

+

What is the quark content of each of these baryons?

The constituent quarks have no relative orbital angular momentum. How doesthe baryon state behave under exchange of any pair of quarks? Explain howthis is achieved in terms of the space, spin, flavour and colour parts of the state

vector.

Account for the absence ofsss,ddd anduuustates in the JP = 12

+octet.

What is meant by quark confinement?

2.2. Write down the valence quark content for each of the different particlesin the reactions below and check that the conservation laws of electric charge,flavour, strangeness and baryon number are satisfied throught.

(1) +p

K0 +

(2) K

+p K0 + 0(3) +p + (4) K +p K+ + K0 +

Draw a quark flow diagram for the last reaction.

2.3. Consider the decay of the 0 meson (JP = 1) in the following decaymodes:

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a) 0 0

b) 0 +

c) 0 00

In each case, draw an appropriate Feynman diagram.

Consider the symmetry of the wave-function required for 00 explain why thisdecay mode is forbidden.

From consideration of the relative strength of the different fundamental forces,

determine which of the other two decay modes will dominate.

2.4. What is meant by the cross section and the differential cross section?

Consider classical Rutherford scattering of a particle with mass m and initialspeedv0 from a potential

V(r) =

r

a) Show from geometry that the change in momentum is given by

|p|= 2p sin(/2).

b) Considering the symmetry of the problem, show that

bv0=r2 d

dt

where b is the impact parameter, ris the location of the particle from the originand is the angle (r, r) where r is the point of closest approach.

c) Starting from Newtons second law show that

|p|= 2v0b

cos

2

.

d) Show that the scattering angle is given by

tan(/2) =

2bT (1)

where b is the impact parameter (the closest distance of the projectile to the

nucleus if it were to be undeflected) and T = p2

2mis the initial kinetic energy.

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e) Calculate the Rutherford scattering cross section for scattering of projectiles

by angles greater than

min.

f) Show that the differential cross section

d

d=

1

16

T

2 1sin4(/2)

where T is the kinetic energy of the particle.

Why is it not possible to calculate the total cross section for this reaction?

2.5. a) A projectile travels through a medium of thicknessx withn targets perunit volume. Show that the fraction absorbed or deflected by the medium is

Pabsorb(x) = 1 enx,

where is the absorption cross section.

b) Estimate, stating any assumptions you make, the thickness of lead that wouldbe required to have a 50% chance of stopping a 2.3 MeV neutrino coming froma solar nuclear fusion reaction.

The cross section for the scattering of a neutrino from a stationary target isapproximately

tot= 2G2F

4p2CM(2)2

dpCMdECM

where ECMis the centre-of-mass energy of the system, andpCMis the momentumof the neutrino in the centre-of-mass frame.

c) Justify the form of this expression.

d) Explain how Figure1supports a model in which the proton contains point-likeconstituents?

[The density of lead is about11.3 g cm3. Some data for the cross section of neutrinos

scattering from nucleons meaning protons or neutrons are shown in Figure1.]

2.6. A wave function is modelled as the sum of the incoming plane wave andan outgoing (scattered) spherical wave,

hx|(+)i =A

eikx +eikr

r f(k,k0)

.

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Figure 1: tot/E for neutrino-nucleon N (and anti-neutrinonucleon N)interactions as a function of neutrino energy. From [PDG(2008)].

Calculate the flux associated with the plane wave and the spherical wave sepa-rately. Hence show that the cross section into solid angle d is

d

d=|f(k,k0)|

2.

justifying any assumptions you make.

[Hint: Remember that the flux is given by ~2mi( ).]

2.7. In the Born approximation, the scattering amplitude is given by

f(1)(k0,k) = 14

(2m)(2)3hk0|V|ki

Explain the terms in this equation, and state the conditions for which it is valid.

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b) The Yukawa potential is given by

V(r) = g2

4

er

r .

Show that for this potential

hk0|V|ki = g2

4

1

(2)3

Z d3x eikx

er

r

where k= k0 k.

c) Hence show that

hk0

|V|ki = g2

(2)31

q2 + 2

where q= |k|.

d) Show that within the Born approximation

d

d=

g4

(4)24m2

[2k2(1 cos) + 2]2 (2)

where is the scattering angle.

e) Find the total cross section for the case when

6= 0.

f) When 0, V(r) 1/r. Compare the differential cross section (2) to theclassical Rutherford scattering cross section

d

d=

1

16

T

2 1sin4(/2)

.

What must be the relationship between g and for Born approximation toreproduce the classical Rutherford formula for electronproton scattering?

2.8. The Klein-Gordon equation,2

t2 2 + m2

(r, t) = 0. (3)

is the relativistic wave equation for spin-0 particles.

a) Show that

=er

ris a valid static-field solution.

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b) Show that another possible solution to the Klein-Gordon equation is:

(X) =A exp[iP X]

Where P and X are the momentum and position four-vectors respectively. Whatrestrictions does (3) place on the components of P?

What are the physical interpretation of these solutions?

Optional questions

2.9. A proton is travelling through a material and scattering the electrons inthe material.

a) Express the scattering angle in terms of the impact parameter b, the reducedmass , the relative speed v, and the scattering angle in the ZMF. Hence showthat the momentum transfer is

q=2v1 + z2

.

where z=bv2

/.

b) Write down the energy given to an electron for a collision for a given impactparameterb. Integrate this up with area element2b dbto show that the averageenergy lost by the projectile per distance travelled is

dE

dx

=

4ne2

mev2

Z zmaxzmin

z dz

1 + z2,

where ne is the number density of electrons.

[Hint: recycle results from the Rutherford scattering question.]

2.10. The figure shows the fractionN/N0 transmitted when protons of kineticenergy E = 140 MeV impinge as a collimated beam on sheets of copper ofvarious thicknesses x. By considering the 2-body kinematics of protonelectroncollisions and protonnucleus collisions, account for the attenuation for values ofx between 0 and 20 mm, and for the sudden change in behaviour around thevalue ofxmarkedR(E).

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Results similar to the figure were obtained for protons ofE= 100 MeV, exceptthat in this case a value ofR(E) = 14 mm was obtained. Offer a brief explana-tion for the change in R(E). What is the relative size of the nuclear scatteringcross section Nucl in copper compared to the geometric cross section?

[The density of copper is8.9 g cm3, and it has relative atomic mass63.5. You may

assume that the nuclear radius is given byr = r0A1/3 with r0= 1.25 fm ]

2.11. Consider the HamiltonianH = H0+ V, where H0 is the free-particleHamiltonian, and V is some localised potential. Let the ket |i represent aneigenstate ofH0, and the ket|i represent an eigenstate ofHwhich shares thesame energy eigenvalue as the |i in the limit 0.

Show that the Lippmann-Schwinger equation

|()i =|i+ 1EH0 iV|

()i.

is consistent with the states defined above by multiplying it by the operator(E

H0 i) and taking the limit

0.

2.12. Show that the Greens function

G(x,x0) ~

2

2m

Dx

1EH0 i

x0Ecan be written

G(x,x0) =

i

42

Z

0

dq qeiq eiqq2 k2 i (4)

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where E= ~2k2

2m and =|x x0|.

[Hint: start by inserting identity operatorsRd3p0|p0ihp0| and Rd3p00|p00ihp00| on each

side of the operator, and changingH0p2/2m.]

2.13. If you have done the course on functions of a complex variable, finish theintegral in (4) using appropriate contour integrals to obtain

G(x,x0) = 1

4

eik

.

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Scattering & reactions

Core questions

3.1. Consider the scattering of an electron from a nucleus with extendedspherical charge densityN(|x0|)which is normalised such that Rd3x0N(|x0|) = 1.The potential at any point is then

V(x) =zZ

Z d3x0

N(|x0|)

|x x0| ,

where Zandzare the charges of the nucleus and the projectile respectively.

a) By expanding in the position basis, and defining a new variable X= x x0,show that the Born approximation to the scattering amplitude can be expressedin the form

f(k0,k) =f(k0,k)point Fnucl(k)

wheref(k0

,k)pointis the scattering amplitude from a point charge with potentialZz/|x|, the nuclear form factor

Fnucl(k) =

Z d3x eikxN(|x|)

is the 3D Fourier transform of the charge density distribution, and k is thechange in momentum of the projectile.

Hence show thatd

d=

d

d

Rutherford

|Fnucl(k)|2.

b) Consider the example form factors in Figure2(i) and (ii). How would theseform factors scale along the k-axis if the radius r of the corresponding sphereof charge was doubled? By relating the momentum transfer to the scatteringangle, use the data to estimate the size of the silver nucleus. Compare to theexpectation for an incompressible nucleus, r= r0A

1

3 withr0= 1.25 fm.

c) How might one accelerate protons to kinetic energy of17 MeV, and subse-quently detect the scattered protons experimentally?

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2 4 6 8 10k

0.01

0.1

1

10

Fk21 r

(i)

2 4 6 8 10k

0.01

0.1

1

10

Fk2

1

5.r1

1

(ii)

(iii)

Figure 2: (LHS)Scattering cross section for K.E. = 17 MeV protons normalizedto the Rutherford scattering cross section (from [Glassgold(1958)]).(RHS)Examples of nuclear form factors|F(|k|)|2 for different charge densityfunctions: (i) uniform unit sphere; (ii) Saxon-Woods (r) [1 + exp((rR)/a)]1 withR = 1,a = 0.2. The corresponding Saxon-Woods charge density[Woods and Saxon(1954)] is shown in (iii).

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3.2. The cross section for the production of-rays by neutrons incident on a

certain nucleus (N, Z) is dominated by a resonance and given by the Breit-Wignerformula,

(n, ) =

k2n

(E E0)2 + 2/4 . (5)

a) Define the symbols in this formula and explain the physical principles thatunderlie it, and the conditions under which it applies.

b) What are the mass and lifetime of the resonant state?

c) All spin effects have been ignored in (5). How would the formula differ if spinsare included?

d) On the same plot draw how the inelastic (n, ) and elastic (n, n) cross sec-tions would behave close to the resonance, when = 4n, labeling importantquantities including the peak cross section values.

[In (d) you may assume that resonant scattering dominates both cross-sections, and

that decays other than ton andare negligible.]

3.3. The cross section for the reaction p 0n shows a prominent peakwhen measured as a function of the energy. The peak corresponds to the

resonance which has a mass of 1232 MeV, with

= 120 MeV. The partialwidths for the incoming and outgoing states are i = 40 MeV, and f= 80 MeVrespectively for this reaction.

At what pion beam energy will the cross section be maximal for a stationaryproton?

Describe and explain the similarities and differences you would expect betweenthe cross section forp 0nand the one for p p, for centre-of-massenergies not far from 1.2 GeV. Giving values for the variables in the Breit-Wignerformula where possible. Use quark-flow diagrams to explain what is happening.

By considering the quark content of the intermediate states, discuss whether youwould expect similar peaks in the cross sections for the reactions (a) Kpproductsand (b)K+p products.

3.4. Draw all the lowest order Feynman diagram(s) for each of the followingprocesses:

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a) e+e in the presence of matterb)e

+ e

e

+ e

c) + +

3.5. Why does the ratio

(e+ + e + + )(e+ + e + +)

tend to unity at high energies? Would you expect the same to be true for

(e+

+ e

+

+

)(e+ + e e+ + e) ?

3.6. Draw leading order QED Feynman diagrams for the processes

e+e + and e+e qq.

How do the vertex and propagator factors compare?

In the figure below you can find the ratio of the cross sections for the process of

electronpositron annihilation to hadrons, and the corresponding cross section tothe muonantimuon final state as a function of

s, the centre-of-mass energy.

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10-1

1

10

10 2

10 3

1 10 10 2

R

s[GeV]

R=(e+e hadrons)(e+e+) .

Considering the number of quarks that can be created at particular centre-of-mass energy, what values ofR would you expect for centre-of-mass energy inthe range 2 GeV m2)for (i)e+ e+ and for (ii)e+e + +.3.9. Draw Feynman diagrams showing a significant decay mode of each of thefollowing particles:a) 0 mesonb) + mesonc)

d) to a final state containing hadronse) K0

f) top quark

3.10. The pions can be represented in an isospin triplet (I = 1) while thenucleons form an isospin doublet (I= 1

2),

+0

and p

n

while the series of resonances have I= 32

.

By assuming that the isospin operators I, I3, I obey the same alegbra as thequantum mechanical angular momentum operators J, Jz, J, explain why theratio ofi/f 12 was found in Question 3.3.

[Hint: you will need the Clebsch-Gordon coefficients forhj1j2m1m2|j1j2JMiforJ=32 , j1= 1, j2=

12 , M= 12 .]

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The Standard Model

Core questions

4.1. a) Draw Feynman diagrams for the the production of W bosons beproduced in a pp collider, and the various possible decays of the W+ boson.

(Which final states are kinematically accessible?)

b) What fraction of decays would you expect to produce electrons in their decay?

c) Suggest why theWwas discovered in the leptonic rather than hadronic decaychannels.

d) How could the outgoing electron momentum be determined? How might thecomponents of the neutrino momentum perpendicular to the beam be deter-mined?

4.2. Consider the rate of the neutron decay in the Fermi theory. Justify theform of the the three-body density of states

dN= E2(2)3

ddE E2e

(2)3dedEe

when the electron is highly relativistic.

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Hence show that the rate for a particular electron energy is

d

dEe=

G2F23

(Q Ee)2 E2e .

and that the total rate is

=G2FQ

5

603 .

Write down Feynman diagrams for the decays of the muon and the tau lepton.Are hadronic decays possible? By considering the propagator factor in each caseexplain why one might expect on dimensional grounds that lifetimes should be

in the ratio ( e)( e)=

mm

5

.

Using the following data

m= 1777.0 MeV = 2.91 1013 sm= 105.66 MeV = 2.197 106 s

BR( e) = 17.8%

test this prediction.

4.3. Which of the Standard Model fermions couple to theZ0 boson? To whichfinal states may aZ0 boson decay?

Explain why for the Z0 the sum of the partial widths to the observed states(e+e, +, +, hadrons) does not equal the FWHM of the Breit-Wigner.

By referring to the properties of the Breit-Wigner forumla, suggest how theLEP e+e collider operating at centre-of-mass energies in the range 80 GeV to100 GeV could have inferred that there are three neutrino species with m

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Pp 2012 Problems