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
14
(animation)
€
C→14 N + β − + ν e
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-+ ν e
• (Animation)
€
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, ν e
• The decay energy is shared between the emitted electron
and neutrino.
• β- decay is seen in all neutron-rich nuclei
€ stopped by a thin sheet of Al
• The emitted β- are easily
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
22
22
+
An example of this decay is
Na→ Ne + β + ν e
Like β- decay, in β+ decay, the decay energy is shared between the
residual nucleus, the emitted positron and the electron neutrino.
€ larger than normal p/n ratios. It is
β+ decay occurs in nuclei with
€ elements
restricted to the lighter
β+ particles annihilate when they contact ordinary matter with the
emission of two 0.511 MeV photons.
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.
e− + 209Bi→209 Pb + ν e
• An example of this type of decay is
• The occurrence of this decay is detected by the emitted Xray (from the vacancy in the electron shell).
€ for proton-rich heavy nuclei.
• It is the preferred decay mode
Mass Changes in Beta Decay
• β- decay
14
C→14 N + β − + ν e
Energy = [(m(14 C) + 6melectron ) − (m(14 N) + 6melectron ) − m(β − )]c 2
Energy = [M(14 C) − M(14 N)]c 2
•β+ decay
€
64
Cu→64 Ni− + β + + ν e
Energy = [(m( 64 Cu) + 29melectron ) − (m( 64 Ni) + 28melectron ) − melectron − m(β + )]c 2
Energy = [M( 64 Cu) − M( 64 Ni) − 2melectron ]c 2
€
Mass Changes in Beta Decay
• EC decay
207
Bi + + e− →207 Pb + 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, GamowTeller 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
2π
2 dn
λ=
M
h
dE 0
M ≡ matrix⋅ element = ∫ ψ *f Mψ i dτ
€
ψ f = ψ Rψ eψν
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
Δpx Δx = h
Δpy Δy = h
ΔpzΔz = h
Δpx ΔxΔpy ΔyΔpzΔz = h 3
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
2
λ=
2
€
g M if
2 p
max
3 7 3
2π h c
∫ F(Z
2
e
2
,pe )p (Q − Te ) dp
D
0
2
5 4
e
3 7
g M mc
λ=
f (ZD ,Q )
2π h
ft1/ 2 = ln 2
2π 3 h7
2
2
5 4
e
g M mc
f=Fermi integral
∝
1
2
g M
2
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
Superallowed
Allowed
mirror
nuclei
non-mirror
nuclei
Transition types
• Fermi vs Gamow-Teller
Ii = I f + l
Ii = I f + l + 1
•Allowed transitions
€
l=0
Δπ = no
What is ΔI?
€
Fermi
Gamow-Teller
Transition types(cont.)
• First forbidden
l =1
Δπ = yes
What is ΔI?
€
Electron capture decay
2
2
λEC =
g M if Tν2
2 3 3
2π c h
ϕ K (0)
2 ⎞ 3 / 2
⎛
1 Zme e
ϕ K (0) =
⎜
2 ⎟
π ⎝ 4 πε 0 h ⎠
€
2
€
€
λK −EC =
3
2
g Z M if Tν2
cons tan ts
λK
Z 3Tν2
= cons tan ts
λβ +
f (ZD ,Q )
2
Electron capture decay
Extranuclear effects after EC
• X-rays vs Auger emission (animation)
• Fluorescence yield
λX −ray
ω=
λX −ray + λAuger
€
β-delayed radioactivity
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•
•
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β-decay followed by another decay
fission product examples
β-delayed neutron emitters
β-delayed fission
Double beta decay