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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
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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
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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.
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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.
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e−+209Bi→209Pb + ν e
Mass Changes in Beta Decay
- 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
+ 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 ]c 2
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.
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.
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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
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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)
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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
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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
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λ=g2 M if
2
2π 3h7c3F(ZD ,pe )pe
2(Q − Te )2 dp0
pmax
∫
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λ=g2 M
2me
5c4
2π 3h7f (ZD ,Q )
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ft1/ 2 = ln 22π 3h7
g2 M2me
5c4∝
1
g2 M2
Electron capture decay
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λEC =g2 M if
2Tν
2
2π 2c3h3ϕ K (0)
2
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ϕ K (0) =1
π
Zmee2
4πε 0h2
⎛
⎝ ⎜
⎞
⎠ ⎟
3 / 2
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λK −EC =g2Z 3 M if
2Tν
2
cons tan ts
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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