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abg Decay Theory. Previously looked at kinematics now study dynamics (interesting bit). QM tunnelling and a decays Fermi theory of b decay and e.c. g decays. a Decay Theory. Consider 232 Th Z=90 R=7.6 fm  E=34 MeV Energy of a E a =4.08 MeV Question: How does the a escape?

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Abg decay theory
abg Decay Theory

  • Previously looked at kinematics now study dynamics (interesting bit).

  • QM tunnelling and a decays

  • Fermi theory of b decay and e.c.

  • g decays

Nuclear Physics Lectures

A decay theory
a Decay Theory

  • Consider 232Th Z=90 R=7.6 fm  E=34 MeV

  • Energy of a Ea=4.08 MeV

  • Question: How does the a escape?

  • Answer: QM tunnelling

Nuclear Physics Lectures

Abg decay theory

radial wave function in alpha decay



barrier (negative KE)

small flux of real α




Exponential decay of y

Nuclear Physics Lectures

Qm tunnelling
QM Tunnelling

  • B.C. at x=0 and x=t for Kt>>1 and k~K gives for 1D rectangular barrier thickness t gives T=|D|2=exp(-2Kt)

  • Integrate over Coulomb barrier from r=R to r=t





Nuclear Physics Lectures

A decay

DEsep≈6MeV per nucleon for heavy nuclei

DEbind(42a)=28.3 MeV > 4*6MeV





Nuclear Physics Lectures

Alpha decay rates
Alpha Decay Rates

  • Gamow factor

  • Number of hits, on surface of nucleus radius R ~ v/2R.Decay rate

Nuclear Physics Lectures

Experimental tests
Experimental Tests

  • Predict log decay rate proportional to (Ea)1/2

  • Agrees ~ with data for e-e nuclei.

  • Angular momentum effects:

    • Additional barrier

    • Small compared to Coulomb but still generates large extra exponential suppression. Eg l=1, R=15 fm El~0.05 MeV cf for Z-90  Ec~17 MeV.

  • Spin/parity

    DJ=L parity change=(-)L

Nuclear Physics Lectures

Experimental tests1
Experimental Tests






EnergyE (MeV)

Nuclear Physics Lectures

Fermi b decaytheory











( )


Fermi b DecayTheory

  • Consider simplest case: n decay.

  • At quark level: du+W followed by decay of virtual W.

Nuclear Physics Lectures

Fermi theory
Fermi Theory

  • 4 point interaction (low energy approximation).

Nuclear Physics Lectures

Fermi theory1
Fermi Theory

  • e distribution determined by phase space (neglect nuclear recoil energy)

  • Use FGR : phase space & M.E. decay rate

Nuclear Physics Lectures

Kurie plot
Kurie Plot

Tritium b decay


Coulomb correction  Fermi function K(Z,p)

Continuous spectrum neutrino

End point gives limit on neutrino mass



Electron energy


Electron energy (keV)

Nuclear Physics Lectures

Selection rules
Selection Rules

  • Fermi Transitions:

    • en couple to give 0 spin: DS=0

    • “Allowed transitions” DL=0  DJ=0.

  • Gamow-Teller transitions:

    • en couple to give 1 unit of spin: DS=0 or ± 1.

    • “Allowed transitions” DL=0  DJ=0 or ± 1.

  • “Forbidden” transitions:

    • Higher order terms correspond to non-zero DL. Therefore suppressed depending on (q.r)2L

    • Usual QM rules give: J=L+S

Nuclear Physics Lectures

Electron capture
Electron Capture

  • Can compete with b+ decay.

  • For “allowed” transitions.

  • Only l=0. n=1 largest.

Nuclear Physics Lectures

Electron capture 2
Electron Capture (2)

  • Density of states:

  • Fermi’s Golden Rule:

Nuclear Physics Lectures

Anti neutrino discovery
Anti-neutrino Discovery

  • Inverse Beta Decay

  • Same matrix elements.

  • Fermi Golden Rule:

Nuclear Physics Lectures

Anti neutrino discovery 2
Anti-neutrino Discovery (2)

  • Phase space factor

  • Neglect nuclear recoil.

  • Combine with FGR

Nuclear Physics Lectures

The experiment
The Experiment

  • For E~ 1MeV s~10-47 cm2

  • Pauli prediction and Cowan and Reines.

Liquid Scint.

1 GW Nuclear Reactor




Nuclear Physics Lectures

Parity definitions
Parity Definitions

  • Eigenvalues of parity are +/- 1.

  • If parity is conserved: [H,P]=0  eigenstates of H are eigenstates of parity. If parity violated can have states with mixed parity.

  • If Parity is conserved result of an experiment should be unchanged by parity operation.

Nuclear Physics Lectures

Parity conservation
Parity Conservation

  • If parity is conserved for reaction a+b c+d.

  • Nb absolute parity of states that can be produced from vacuum (e.g. photons) can be defined. For other particles we can define relative parity. e.g. define hp=+1, hn=+1 then can determine parity of other nuclei.

  • If parity is conserved <pseudo-scalar>=0 (see next transparency).

Nuclear Physics Lectures

Abg decay theory

<Op> = 0 QED

Nuclear Physics Lectures

Is parity conserved in nature
Is Parity Conserved In Nature?

  • Feynman’s bet.

  • Yes in electromagnetic and strong interactions.

  • Big surprise was that parity is violated in weak interactions.

Nuclear Physics Lectures

Mme wu s cool experiment
Mme. Wu’s Cool Experiment

  • Align spins of 60Co with magnetic field.

  • Adiabatic demagnetisation to get T ~ 10 mK

  • Measure angular distribution of electrons and photons relative to B field.

  • Clear forward-backward asymmetry  Parity violation.

Nuclear Physics Lectures

The experiment1
The Experiment

Nuclear Physics Lectures

Improved experiment
Improved Experiment

q is angle wrt spin of 60Co.

Nuclear Physics Lectures

G decays
g decays

  • When do they occur?

    • Nuclei have excited states cf atoms. Don’t worry about details E,JP (need shell model to understand).

    • EM interaction << strong interaction

    • Low energy states E < 6 MeV above ground state can’t decay by strong interaction  EM.

  • Important in cascade decays a and b.

  • Practical consequences

    • Fission. Significant energy released in g decays.

    • Radiotherapy: g from Co60 decays.

    • Medical imaging eg Tc.

Nuclear Physics Lectures

Energy levels for mo and tc
Energy Levels for Mo and Tc

b decay leaves Tc in excited state.

Useful for medical imaging

Nuclear Physics Lectures

G decay theory beyond syllabus
g Decay Theory (Beyond Syllabus)

  • Most common decay mode for nuclear excited states (below threshold for break-up) is g decay.

  • Lifetimes vary from years to 10-16s. nb long lifetimes can easily be observed unlike in atomic. Why?

  • Angular momentum conservation in g decays.

    • intrinsic spin of g is1 and orbital angular momentum integer  J is integer.

    • Only integer changes in J of nucleus allowed.

    • QM addition of J:

    • Absolutely forbidden (why?): 00

Nuclear Physics Lectures

G decays1
g Decays

  • Electric transitions

  • Typically k~1 MeV/c r~ 1 fm k.r~1/200  use multipole expansion. Lowest term is electric dipole transitions, L=1.

  • Parity change for electric dipole.

Nuclear Physics Lectures

Forbidden transitions
Forbidden Transitions

  • If electric dipole transitions forbidden by angular momentum or parity can have “forbidden” transitions, eg electric quadropole.

  • Rate suppressed cf dipole by ~ (k.r)2

  • Magnetic transitions also possible:

  • Classically: E=-m.B

  • M1 transition rate smaller than E1 by ~ 10-3.

  • Higher order magnetic transitions also possible.

  • Parity selection rules:

    • Electric: Dp=(-1)L

    • Magnetic: Dp=(-1)L+1

Nuclear Physics Lectures

Internal conversion
Internal Conversion

  • 00 absolutely forbidden:

  • What happens to a 0+ excited state?

  • Decays by either:

    • Internal conversion: nucleus emits a virtual photon which kicks out an atomic electron. Requires overlap of the electron with the nucleus only l=0. Probability of electron overlap with nucleus increases as Z3. For high Z can compete with other g decays.

    • Internal pair conversion: nucleus emits a virtual photon which converts to e+e- pair.

Nuclear Physics Lectures