P461 - decays II 1

Beta Decays

• Beta decays are proton neutrons or neutron proton transitions

• involve W exchange and are weak interaction

• the last reaction is electron capture where one of the atomic electrons overlaps the nuclei. Same matrix element (essentially) bit different kinematics

• the semi-empirical mass formula gives a minimum for any A. If mass difference between neighbors is large enough, decay will occur

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P461 - decays II 2

Beta Decays - Q Values

• Determine Q of reactions by looking at mass difference (careful about electron mass)

• 1 MeV more Q in EC than beta+ emission. More phase space BUT need electron wavefunction overlap with nucleus..

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P461 - decays II 3

Beta+ vs Electron Capture

• Fewer beta+ emitters than beta- in “natural” nuclei (but many in “artificial” important in Positron Emission Tomography - PET)

• sometimes both beta+ and EC for same nuclei. Different widths• sometimes only EC allowed

• monoenergetic neutrino. E=.87 MeV. Important reaction in the Sun. Note EC rate different in Sun as it is a plasma and not atoms

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P461 - decays II 4

Beta+ vs Electron Capture

• from Particle Data Group

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P461 - decays II 5

Beta Decay - 3 Body• The neutrino is needed to conserve angular

momentum

• (Z,A) (Z+1,A) for A=even have either Z,N even-even odd-odd or odd-oddeven-even

• p,n both spin 1/2 and so for even-even or odd-odd nuclei I=0,1,2,3…….

• But electron has spin 1/2 I(integer) I(integer) + 1/2(electron) doesn’t conserve J

• need spin 1/2 neutrino. Also observed that electron spectrum is continuous indicative of >2 body decay

• Pauli/Fermi understood this in 1930s electron neutrino discovered 1953 (Reines and Cowan) muon neutrino discovered 1962 (Schwartz +Lederman/Steinberger) tau neutrino discovered 2000 at Fermilab

P461 - decays II 6

3 Body Kinematics• While 3 body the nuclei are very heavy and easy

approximation is that electron and neutrino split available Q (nuclei has similar momentum)

• maximum electron energy when E(nu)=0

• example

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P461 - decays II 7

Beta decay rate• Start from Fermi Golden Rule

• first approximation (Fermi). Beta=constant=strength of weak force

• Rule 1: parity of nucleus can’t change (integral of odd*even=0)

• Rule 2: as antineutrino and electron are spin 1/2 they add to either 0 or 1. Gives either

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P461 - decays II 8

Beta decay rate II• Orbital angular momentum suppression of 0.001

for each value of (in matrix element calculation)

• look at density of states factor. Want # quantum states per energy interval

• we know from quantum statistics that each particle (actually each spin state) has

• 3 body decay but recoil nucleus is so heavy it doesn’t contribute

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P461 - decays II 9

Beta decay rate III• Conservation of energy allows one to integrate over

the neutrino (there is a delta function)

• this gives a distribution in electron momentum/energy which one then integrates over. (end point depends on neutrino mass)

• F is a function which depends on Q. It is almost loqrithmic

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P461 - decays II 10

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P461 - decays II 11

Beta decay rate IV• FT is “just kinematics”

• measuring FT can study nuclear wavefunctions M’ and strength of the weak force at low energies

• lower values of FT are when M’ approaches 1

• beta decays also occur for particles

• electron is now relativistic and E=pc. The integral is now easy to do. For massive particles (with decay masses small), Emax = M/2 and so rate goes as fifth power of mass

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P461 - decays II 12

Beta decay rate V• M=M’ is strength of weak interaction. Can

measure from lifetimes of different decays

• characteristic energy

• strong energy levels ~ 1 MeV

• for similar Q, lifetimes are about

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P461 - decays II 13

Parity Violation in Beta Decays

• The Parity operator is the mirror image and is NOT conserved in Weak decays (is conserved in EM and strong)

• non-conservation is on the lepton side, not the nuclear wave function side

• spin 1/2 electrons and neutrinos are (nominally) either right-handed (spin and momentum in same direction) or left-handed (opposite)

• Parity changes LH to RH

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P461 - decays II 14

“Handedness” of Neutrinos

• “handedness” is call chirality. If the mass of a neutrino = 0 then:

• all neutrinos are left-handed all antineutrinos are right-handed

• Parity is maximally violated

• As the mass of an electron is > 0 can have both LH and RH. But RH is suppressed for large energy (as electron speed approaches c)

• fraction RH vs LH can be determined by solving the Dirac equation which naturally incorporates spin

P461 - decays II 15

Polarized Beta Decays

• Some nuclei have non-zero spin and can be polarized by placing in a magnetic field

• magnetic moments of nuclei are small (1/M factor) and so need low temperature to have a high polarization (see Eq 14-4 and 14-5)

• Gamow-Teller transition with S(e-nu) = 1

• if Co polarized, look at angular distribution of electrons. Find preferential hemisphere (down)

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P461 - decays II 16

Discovery of Parity Violation in

Beta Decay by C.S. Wu et al. • Test parity conservation by observing a

dependence of a decay rate (or cross section) on a term that changes sign under the parity operation. If decay rate or cross section changes under parity operation, then the parity is not conserved.

• Parity reverses momenta and positions but not angular momenta (or spins). Spin is an axial vector and does not change sign under parity operation.

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Beta decay of a neutron in a real andmirror worlds:If parity is conserved, then the probability of electron emission at is equal to that at 180o-.Selected orientation of neutron spins - polarisation.

P461 - decays II 17

Wu’s experiment• Beta-decay of 60Co to 60Ni*. The

excited 60Ni* decays to the ground state through two successive emissions.

• Nuclei polarised through spin alignment in a large magnetic field at 0.01oK. At low temperature thermal motion does not destroy the alignment. Polarisation was transferred from 60Co to 60Ni nuclei. Degree of polarisation was measured through the anisotropy of gamma-rays.

• Beta particles from 60Co decay were detected by a thin anthracene crystal (scintillator) placed above the 60Co source. Scintillations were transmitted to the photomultiplier tube (PMT) on top of the cryostat.

P461 - decays II 18

Wu’s results

• Graphs: top and middle - gamma anisotropy (difference in counting rate between two NaI crystals) - control of polarisation; bottom - asymmetry - counting rate in the anthracene crystal relative to the rate without polarisation (after the set up was warmed up) for two orientations of magnetic field.

• Similar behaviour of gamma anisotropy and beta asymmetry.

• Rate was different for the two magnetic field orientations.

• Asymmetry disappeared when the crystal was warmed up (the magnetic field was still present): connection of beta asymmetry with spin orientation (not with magnetic field).

• Beta asymmetry - Parity not conserved

P461 - decays II 19

Gamma Decays

• If something (beta/alpha decay or a reaction) places a nucleus in an excited state, it drops to the lowest energy through gamma emission

• excited states and decays similar to atoms

• conserve angular momentum and parity

• photon has spin =1 and parity = -1

• for orbital P= (-1)L

• first order is electric dipole moment (edm). Easier to have higher order terms in nuclei than atoms

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P461 - decays II 20

Gamma Decays

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P461 - decays II 21

Mossbauer Effect

• Gamma decays typically have lifetimes of around 10-10 sec (large range). Gives width:

• very precise

• if free nuclei decays, need to conserve momentum. Shifts gamma energy to slightly lower value

• example. Very small shift but greater than natural width

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P461 - decays II 22

Mossbauer Effect II

• Energy shift means an emitted gamma won’t be reabsorbed

• but if nucleus is in a crystal lattic, then entire lattice recoils against photon. Mas(lattice)infinity and Egamma=deltaM. Recoiless emission (or Mossbauer)

• will have “wings” on photon energy due to lattice vibrations

• Mossbauer effect can be used to study lattice enregies. Very precise. Use as emitter or absorber. Vary energy by moving source/target (Doppler shift) (use Iron. developed by R. Preston, NIU)

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P461 - decays II 23

Nuclear Reactions, Fission and Fusion

• 2 Body reaction A+BC+D

• elastic if C/D=A/B

• inelastic if mass(C+D)>mass(A+B)

• threshold energy for inelastic (B at rest)

• for nuclei nonrelativistic usually OK

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P461 - decays II 24

Nuclear Reactions (SKIP)

• A+BC+D

• measurement of kinematic quantities allows masses of final states to be determined

• (p,E) initial A,B known

• 8 unknowns in final state (E,px,py,pz for C+D)

• but E,p conserved. 4 constraints4 unknowns measure E,p (or mass) of D OR C gives rest or measure pc and pd gives masses of both

• often easiest to look at angular distribution in C.M. but can always convert

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P461 - decays II 25

Fission

• AB+C A heavy, B/C medium nuclei• releases energy as binding energy/nucleon = 8.5 MeV for Fe and

7.3 MeV for Uranium• spontaneous fission is like alpha decay but with different mass,

radii and Coulomb (Z/2)2 vs 2(Z-2). Very low rate for U, higher for larger A

• induced fission n+AB+C. The neutron adds its binding energy (~7 MeV) and can put nuclei in excited state leading to fission

• even-even U(92,238). Adding n goes to even-odd and less binding energy (about 1 MeV)

• even-odd U(92,235), U(92,233), Pu(94,239) adding n goes to even-even and so more binding energy (about 1 MeV) 2 MeV difference between U235 and U238

• fission in U235 can occur even if slow neutron

P461 - decays II 26

Spontaneous Fission

•

P461 - decays II 27

Induced Fission

•

P461 - decays II 28

Neutron absorption

•

P461 - decays II 29

Fusion

• “nature” would like to convert lighter elements into heavier. But:

• no free neutrons

• need to overcome electromagnetic repulsion high temperatures

• mass Be > twice mass He. Suppresses fusion into Carbon

• Ideally use Deuterium and Tritium, =1 barn, but little Tritium in Sun (ideal for fusion reactor)

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P461 - decays II 30

Fusion in Sun

• rate limited by first reaction which has to convert a p to a n and so is Weak

(pp) ~ 10-15 barn

• partially determines lifetime of stars

• can model interaction rate using tunneling – very similar to Alpha decay (also done by Gamow)

• tunneling probability increases with Energy (Temperature) but particle probability decreases with E (Boltzman). Have most probable (Gamow Energy). About 15,000,000 K for Sun but Gamow energy higher (50,000,000??)

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P461 - decays II 31

Fusion in Sun II

• need He nuclei to have energy in order to make Be. (there is a resonance in the if have invariant mass(He-He)=mass(Be))

• if the fusion window peak (the Gamow energy weighted for different Z,mass) is near that resonance that will enhance the Be production

• turns out they aren’t quite. But fusion to C start at about T=100,000,000 K with <kT> about 10 KeV each He. Gamow energy is higher then this.

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P461 - decays II 32

Fusion in Sun III

• Be+HeC also enhanced if there is a resonance. Turns out there is one at almost exactly the right energy --- 7.65 MeV

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