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
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) + -+e
A fundamental view of beta
decay
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.
e
• - 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
+ decay,Ne
• Like - decay, in Na
the
is shared between
 decay
  energy
e
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.
22
22

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
209
209
• The occurrence of
this
decay
by the emitted
e 
Bi
Pbisdetected
e
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.
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
Allowed vs Superallowed
Transitions
Superallowed
Allowed
mirror
nuclei
non-mirror
nuclei
Transition types
• Fermi vs Gamow -Teller
Ii  I f 
Ii  I f   1
•Allowed transitions

0
  no
What is I?
Fermi
Gamow-Teller
Transition types(cont.)
• First forbidden
1
  yes
What is I?

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
Extranuclear effects after EC
• X-rays vs Auger emission
• Fluorescence yield
X ray

X ray  Auger

-delayed radioactivity
•
•
•
•
-decay followed by another decay
fission product examples
-delayed neutron emitters
-delayed fission
Double beta decay