Lesson 8

Beta Decay

Beta -decay

• Beta decay is a term used to describe three types of decay in which a nuclear neutron (proton) changes into a nuclear proton (neutron). The decay modes are -, + and electron capture (EC).

- decay involves the change of a nuclear neutron into a proton and is found in nuclei with a larger than stable number of neutrons relative to protons, such as fission fragments.

• An example of - decay is

€

14C→14N + β − + ν e

Beta decay (cont)

• In - decay, Z = +1, N =-1, A =0• Most of the energy emitted in the decay appears in the rest and kinetic energy of the emitted electron (- ) and the emitted anti-electron neutrino,

• The decay energy is shared between the emitted electron and neutrino.

- decay is seen in all neutron-rich nuclei

• The emitted - are easily stopped by a thin sheet of Al

€

ν e

Beta decay (cont)• The second type of beta decay is + (positron) decay.

• In this decay, Z = -1, N =+1, A =0, i.e., a nuclear proton changes into a nuclear neutron with the emission of a positron, + , and an electron neutrino, νe

• An example of this decay is

• Like - decay, in + decay, the decay energy is shared between the residual nucleus, the emitted positron and the electron neutrino.

+ decay occurs in nuclei with larger than normal p/n ratios. It is restricted to the lighter elements

+ particles annihilate when they contact ordinary matter with the emission of two 0.511 MeV photons.

€

€

22Na→22Ne + β + + ν e

Beta decay (cont)• The third type of beta decay is electron capture (EC) decay. In EC decay an orbital electron is captured by a nuclear proton changing it into a nuclear neutron with the emission of a electron neutrino.

• An example of this type of decay is

• The occurrence of this decay is detected by the emitted X-ray (from the vacancy in the electron shell).

• It is the preferred decay mode for proton-rich heavy nuclei.

€

e−+209Bi→209Pb + ν e

Mass Changes in Beta Decay

- decay

€

14C→14N + β − + ν e

Energy = [(m(14C) + 6melectron ) − (m(14N) + 6melectron ) − m(β −)]c 2

Energy = [M(14C) − M(14N)]c 2

+ decay

€

64Cu→64Ni− + β + + ν e

Energy = [(m(64Cu) + 29melectron ) − (m(64Ni) + 28melectron ) − melectron − m(β +)]c 2

Energy = [M(64Cu) − M(64Ni) − 2melectron ]c 2

Mass Changes in Beta Decay

• EC decay

€

207Bi+ + e−→207Pb + ν e

Energy = [(m(207Bi) + 83melectron ) − (m(207Pb) + 82melectron )]c 2

Energy = [M(207Bi) − M(207Pb)]c 2

Conclusion: All calculations can be done with atomic masses

Spins in Beta Decay

• The electron spin and the neutrino spin can either be parallel or anti-parallel.

• These are called, respectively, Gamow-Teller and Fermi decay modes.

• In heavy nuclei, G-T decay dominates

• In mirror nuclei, Fermi decay is the only possible decay mode.

A fundamental view of beta decay

Perturbation Theory

• Up to now, we have restricted our attention primarily to the solution of problems where things were not changing as a fucntion of time, ie, nuclear structure calculations. Now we shall take up the issue of transitions from one state to another.

• To do so, we need to introduce an additional concept in quantum mechanics, perturbation theory. A full accounting can be found in any quantum mechanics textbook.

€

HΨ = ˆ E Ψ

H = −h2

2m∇ 2 + V

ˆ E = ih∂

∂tΨ(x,y,z.t) =ψ (x,y,z)τ (t)

1

Ψ

−h2

2m∇ 2Ψ + V =

ih

τ

∂τ

∂t

−h2

2m∇ 2ψ + Vψ = Eψ

ih∂τ

∂t= Eτ

τ (t) = e−iEnt / h

Ψ = a1Ψ1 + a2Ψ2 +L + anΨn

Ψ = anΨn

n

∑

a*nan is the probability that the system will bein state n corresponding to the wave function n

Now consider a two state system

How do we handle this in the Schrodinger equation?

Make an’s time dependent

Modify the Hamiltonian

H=H0+H’

For two state system

Weak perturbation, neglect term 1

€

Matrix element = ψ1*H 'ψ 2dτ = ψ1 H 'ψ 2∫

Matrix element describes the probability that H’ willtransform state 2 into state 1

Fermi theory of beta decay

• Fermi assumed -decay results from some sort of interaction between the nucleons, the electron and the neutrino.

• This interaction is different from all other forces and will be called the weak interaction. Its strength will be expressed by a constant like e or G. Call this constant g. (g~10-6 strong interaction)

Fermi theory of beta decay(cont)

• Interaction between nucleons, electron and neutrino will be expressed as a perturbation to the total Hamiltonian.

• Decay probability expressed by matrix element

• Beta decay energy E0 divided between electron and neutrino

• Not all divisions are equally probable (would mean flat beta spectrum)

€

f ′ H ψ i

Fermi theory of beta decay(cont)

• How do we do the counting? First guess is 50-50 split between electron and neutrino.

• Define dn/dE0 as the number of ways the total energy can be divided between electron and neutrino

Fermi theory of beta decay(cont)

• Probability for emission of electron of momentum pe

Fermi theory of beta decay(cont)

Calculating dn/dE0

• Consider the electron at position (x,y,z) with momentum components (px,py,pz)

• Heisenberg tells us that

€

pxΔx = h

ΔpyΔy = h

ΔpzΔz = h

ΔpxΔxΔpyΔyΔpzΔz = h3

This volume is the unit cell in phase space

Calculating dn/dE0

(cont.)• The probability of having an electron with momentum pe (between pe and pe+dpe) is proportional to the number of unit cells in phase space occupied.

Calculating dn/dE0

(cont.)

Calculating dn/dE0

(cont.)• Have neglected the effect of the nuclear charge on the electron energy

Calculating dn/dE0

(cont.)• Add a factor the Fermi function F(Z,Ee)

Kurie Plots

log ft

€

λ=g2 M if

2

2π 3h7c3F(ZD ,pe )pe

2(Q − Te )2 dp0

pmax

∫

€

λ=g2 M

2me

5c4

2π 3h7f (ZD ,Q )

€

ft1/ 2 = ln 22π 3h7

g2 M2me

5c4∝

1

g2 M2

Electron capture decay

€

λEC =g2 M if

2Tν

2

2π 2c3h3ϕ K (0)

2

€

ϕ K (0) =1

π

Zmee2

4πε 0h2

⎛

⎝ ⎜

⎞

⎠ ⎟

3 / 2

€

λK −EC =g2Z 3 M if

2Tν

2

cons tan ts

€

λK

λβ +

= cons tan tsZ 3Tν

2

f (ZD ,Q )

Electron capture decay

-delayed radioactivity

-decay followed by another decay

• fission product examples-delayed neutron emitters-delayed fission

Double beta decay