Download Lesson 8 - Oregon State University

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
14
C14 N      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
e
electron and neutrino.
• - decay is seen in all neutron-rich nuclei
• The emitted - are easily stopped by a thin sheet of Al

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, e
• An example of this decay is
22
Na22Ne      e
• 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.
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
e  209Bi209Pb   e
• 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.
Mass Changes in Beta Decay
• - decay
14
C14 N      e
Energy  [(m(14C)  6melectron )  (m(14 N)  6melectron )  m(  )]c 2
Energy  [M(14 C)  M(14N)]c 2
•+ decay

64
Cu64 Ni      e
Energy  [(m( 64Cu)  29melectron )  (m( 64 Ni)  28melectron )  melectron  m(  )]c 2
Energy  [M( 64 Cu)  M( 64Ni)  2melectron ]c 2
Mass Changes in Beta Decay
• EC decay
207
Bi   e 207Pb   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.
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.
H  Eˆ 
H 
2
2m
2  V

Eˆ  i
t
(x, y,z.t)   (x, y,z) (t)
1  2 2
i 
 V 
 2m
 t
 2 2
   V  E
2m
i

 E
t
 (t)  eiE t /
n
  a11  a2 2 
   an n
n

 an n
a*nan is the probability that the system will be
in 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   1*H '2d  1 H '2
Matrix element describes the probability that H’ will
transform 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 H
• Beta decay energy E0 divided between electron

and neutrino
• Not all divisions are equally probable (would
mean flat beta spectrum)
f
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
pxx  h
pyy  h
pzz  h
pxxpyypzz  h 3
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
2

g M if
2
2 p
max
3 7 3
c
 F(Z
,pe )p (Q  Te ) dp
2
e
D
2
0
2
2
5 4
e
3 7
g M mc

f (ZD ,Q )
2
ft1/ 2  ln 2

2 3
2
2
7
5 4
e
g M mc

1
g2 M
2
Electron capture decay
2
2
EC 
g M if T2
2 c
2 3 3
 K (0)
2 3 / 2

1 Zme e
 K (0) 

2 
 4  0 

2

K EC 
2
g Z M if T2
constan ts
K
Z 3T2
 cons tan ts

f (ZD ,Q )


3
2
Electron capture decay
-delayed radioactivity
•
•
•
•
-decay followed by another decay
fission product examples
-delayed neutron emitters
-delayed fission
Double beta decay