Lesson 9

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 β-, β+ and electron capture (EC).

•  β- 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 β- decay is

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14C→14N + β− + ν e

(animation)

Why do we “need” neutrinos?

•  Conservation of energy •  Conservation of angular momentum

Beta decay and the weak interaction

•  e- created at the instant of emission by weak interaction

•  Weak interaction force carriers are W± and Z0. Masses of these particles large (81, 93 GeV/c) and forces are short range (10-3 fm)

•  n(udd)→p(duu) + β-+

•  (Animation) €

νe

A fundamental view of beta decay

Beta decay (cont) •  In β- decay, ΔZ = +1, ΔN =-1, ΔA =0 •  Most of the energy emitted in the decay appears in the rest

and kinetic energy of the emitted electron (β- ) and the emitted anti-electron neutrino,

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

•  β- decay is seen in all neutron-rich nuclei •  The emitted β- are easily stopped by a thin sheet of Al €

ν e

Beta decay (cont) •  The second type of beta decay is β+ (positron) decay. •  In this decay, ΔZ = -1, ΔN =+1, ΔA =0, i.e., a nuclear proton changes

into a nuclear neutron with the emission of a positron, β+ , and an electron neutrino, νe

•  An example of this decay is

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

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

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

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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

•  β- decay

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14C→14N + β− + ν e

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

−)]c 2

Energy = [M(14C) −M(14N)]c 2

• β+ decay

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64Cu→64Ni− + β + + ν e

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

+)]c 2

Energy = [M(64Cu) −M(64Ni) − 2melectron ]c2

Mass Changes in Beta Decay •  EC decay

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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.

Fermi theory of beta decay

•  Fermi assumed β-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 Fermi’s golden rule

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λ =2πhM 2 dn

dE0

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M ≡ matrix⋅ element = ψ f*Mψ idτ∫

ψ f =ψRψeψν

M reflects the probability of going from state I to state f (nuclear structure information)

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

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ΔpxΔx = hΔpyΔy = hΔpzΔz = h

ΔpxΔxΔpyΔyΔpzΔz = h3

This volume is the unit cell in phase space

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

•  Not all ways are equally probable

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

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λ =g2 Mif

2

2π 3h7c3F(ZD ,pe )pe

2(Q − Te )2dp

0

pmax

∫

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λ =g2 M 2me

5c4

2π 3h7f (ZD ,Q )

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ft1/ 2 = ln 2 2π 3h7

g2 M 2me5c4

∝1

g2 M 2

f=Fermi integral

Use of log ft1/2 •  Consider the β+ decay of 25Al. t1/2=7.6 s, Eβ+=3.24 MeV

log f0t=3.7, log(C)=-0.2 log ft=3.5

Allowed vs Superallowed Transitions

mirror nuclei

Superallowed

Allowed non-mirror nuclei

Transition types

•  Fermi vs Gamow-Teller

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Ii = I f + l

Ii = I f + l +1

Fermi

Gamow-Teller

• Allowed transitions

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l = 0Δπ = no

What is ΔI?

Transition types(cont.)

•  First forbidden

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l =1Δπ = yes

What is ΔI?

Electron capture decay

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λEC =g2 Mif

2Tν2

2π 2c3h3ϕK (0)

2

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ϕK (0) =1π

Zmee2

4πε0h2

⎛

⎝ ⎜

⎞

⎠ ⎟

3 / 2

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λK −EC =g2Z 3 Mif

2Tν2

cons tan ts

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λKλβ +

= cons tan ts Z 3Tν2

f (ZD ,Q )

Electron capture decay

Extranuclear effects after EC

•  X-rays vs Auger emission (animation)

•  Fluorescence yield

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ω =λX −ray

λX −ray + λAuger

β-delayed radioactivity

•  β-decay followed by another decay •  fission product examples •  β-delayed neutron emitters •  β-delayed fission

Double beta decay