Beta Decay
•
Readings: Nuclear and Radiochemistry:
Chapter 3, Modern Nuclear Chemistry:
Chapter 8
•
•
•
•
•
•
Neutrino Hypothesis
Derivation of Spectral Shape
Kurie Plots
Beta Decay Rate Constant
Selection Rules
Transitions
•
Majority of radioactive nuclei are outside
range of alpha decay

Beta decay
 Second particle found from U decay
* Negative particle
* Distribution of energies
* Need another particle to balance
spin
 Parent, daughter, and
electron
 Need to account for half
integer spin
Beta decay half-life

few milliseconds to ~ 1016 years

How does this compare to alpha decay?
•
5-1
•
•
•
•
•
-Decay
Class includes any radioactive decay process in which
A remains unchanged, but Z changes
 - decay, electron capture, + decay
131
131
I

Xe  
 energetic conditions for decay:
53
54
  decay: MZ  MZ+1
26

26
 Electron capture: MZMZ-1,
Al

e

Mg
 + decay: MZ  MZ-1+2me
13
12
* + decay needs to exceed 1.02 MeV
* Below 1.02 MeV EC dominates
22
22
+
Na

Ne  
*  increases with increasing energy
11
10
Decay energies of  -unstable nuclei rather
systematically with distance from stability

Predicted by mass parabolas

Energy-lifetime relations are not nearly so
simple as alpha decay

 -decay half lives depend strongly on spin and
parity changes as well as energy
For odd A, one -stable nuclide; for even A, at most
three -stable nuclides
 Information available from mass parabolas
Odd-odd nuclei near the stability valley (e.g., 64Cu) can
decay in both directions

Form even-even nuclei
Beta particle energy not discrete

Continuous energy to maximum

   Energy
   Energy

   Energy
5-2
The Neutrino
• Solved problems associated with -decay
 Range of particle energies
• Zero charge
 neutron -> proton + electron
• Small mass
 Electron goes up to Q value
• Anti-particle
 Account for creation of electron particle
• spin of ½ and obeys Fermi statistics
 couple the total final angular momentum to initial
spin of ½ ħ,
 np+ + e- is not spin balanced, need another
fermion
5-3
Neutrino
• Carries away appropriate amount of energy and
momentum in each  process for conservation
• Nearly undetectable due to small rest mass and
magnetic moment
 observed by inverse  processes
• Antineutrinos emitted in - decay, neutrinos emitted in
+ decay
 indistinguishable properties, except in capture
reactions
• Neutrinos created at moment of emission
 n  p + - + 
 p  n + + + 
• Spin of created particles are key in assigning decay
 Spin up and spin down
5-4
Spin in Beta Decay
• Spins of created particles can be combined
in two ways
 S=1 in a parallel alignment
 S= 0 in an anti-parallel alignment
• two possible relative alignments of
"created" spins
 Fermi (F) (S=0)
Low A
 Gamow-Teller (GT) (S =1)
High A
*Spin change since neutron number
tends to be larger than proton
• A source can produce a mixture of F and GT
spins
5-5
Spin in Beta Decay
• Decay of even-even nuclei with N=Z (mirror nuclei)
 neutron and protons are in the same orbitals
 0+ to 0+ decay can only take place by a Fermi
transition
• Heavy nuclei with protons and neutrons in very different
orbitals (from shell model)
 GT is main mode, need to account for spin difference
• Complex nuclei
 rate of decay depends on overlap of wave functions of
ground state of parent and state of the daughter
 final state in daughter depends on decay mode
 spin and parity state changes from parent to
daughter
• Half life information can be used to understand nuclear
states
 Decay constant can be calculated if wave functions are
known
 Observed rate indicates quantum mechanical overlap
of initial and final state wave functions
5-6
Energetics (Review)
•
Beta decay

•
•

Z  ( Z  1)  e    Q
electron can be combined with the positive ion to create a neutral
atom
 release of very small binding energy
 use neutral atoms to calculate the Q value
* assuming that the mass of the antineutrino is very small
Consider beta decay of 14C
14C14N+ + β- +antineutrino + energy

Q  = M (Z) 
 Energy = mass 14C – mass 14N
Positron decay

2 extra electrons (daughter less Z, emission of positron)

Z  ( Z  1)  e    Q
•
Electron Capture
M ( Z  1)
Q   = M ( Z )  ( M ( Z  1)  2 e )

Z  ( Z  1)    Q
Q EC = M ( Z )  M ( Z  1)
5-7
Q value calculation (Review)
•
•
•
•
Find Q value for the Beta decay of 24Na

1 amu = 931.5 MeV

M (24Na)-M(24Mg)
 23.990962782-23.985041699
 0.005921 amu
* 5.5154 MeV

From mass excess
 -8.4181 - -13.9336
 5.5155 MeV
Q value for the EC of 22Na

M (22Na)-M(22Ne)

21.994436425-21.991385113

0.003051 amu
 2.842297 MeV

From mass excess
 -5.1824 - -8.0247
 2.8432 MeV
Q are ~0.5 – 2 MeV, Q + ~2-4 MeV and QEC ~ 0.2 – 2 MeV
What about positron capture instead of EC?
5-8
Positrons
•
•
•
•
Postulated in 1931

Relativistic equations could be solved for
electrons with positive energy states

Require energies greater than electron mass

Creation of positive hole with electron
properties
Pair production process involves creation of a
positron-electron pair by a photon in nuclear field
 Nucleus carries off some momentum and energy
Positron-electron annihilation

Interaction of electron into a whole in sea of
electrons of negative energy
 simultaneous emission of corresponding
amount of energy in form of radiation
 Responsible for short lifetime of
positrons
* No positron capture decay
Annihilation radiation

energy carried off by two  quanta of opposite
momentum

Annihilation conserves momentum

Exploited in Positron Emission Tomography
5-9
Weak Interaction: Model of Beta Decay
• Fermi's theory of beta decay based on electromagnetic
theory for light emission
 Electromagnetic interaction characterized by
electron charge
 Needs to be replaced for beta decay
 Fermi constant (g)
* Value determined by experiment
* 10-3 of the electromagnetic force constant
• Used to determine emitted electron momentum range
per unit time P(pe) dpe;
2 2 dn
4
2
2
P( pe )dpe =
 e (0)   (0) M if g
h
dE0
2
5-10
Weak Interaction
2 2 dn
4
2
2
P ( pe )dpe =
 e (0)   (0) M if g
h
dE 0
2
•
•
•
•
•
P(pe)dpe probability electron with momentum pe+dpe
e electron wave function
 neutrino wave function
e(0)2 and (0)2 probability of finding electron and neutrino at nucleus
Mif is matrix element characterizing the transition from the initial to the
final nuclear state
• Mif2 a measure of the amount of overlap between the wave functions of
initial and final nuclear states
• dn/dEo is the density of final states with the electron in the specified
momentum interval

number of states of the final system per unit decay energy
• Fermi constant (g) governs other interactions in addition to beta decay

m-meson decay, -meson decay, neutrino-electron scattering
 Weak interactions
5-11
Weak Interaction
• Integration over all electron momenta from zero to
maximum possible to evaluate spectrum should provide
transition probabilities or lifetimes
 Variations in number of electrons at a given energy
• Classically allowed transitions both have electron and
neutrino emitted with zero orbital angular momentum
 Allowed have s orbital angular momentum
 Relatively high probabilities for location of electron and
neutrino at nuclear for s wave compared to higher l
 p,d,f, etc.
 2 of allowed transitions  2 of forbidden
transitions
• Magnitudes of (0) and Mif are independent of division of
energy between electron and neutrino
2 2 dn
4 2
2
2
P ( pe )dpe =
 e (0)   (0) M if g
5-12
h
dE0
Weak Interaction
• Spectrum shape determined
entirely by e(0) and dn/dEo
 dn/dEo density of final states
with electron momentum
Coulomb interaction
between nucleus and
emitted electron (e(0))
neglected
* Reasonable for low Z
• Density of final states determined
from total energy W
 W is total (kinetic plus rest)
electron energy
 Wo is maximum W value
• Dn/dEo goes to zero at W = 1 and
W = Wo
 Yields characteristic bell shape
beta spectra
dn 16 2 mo5c 4
2
1/ 2
2
=
W
(
W

1
)
(
W

W
)
dW
o
6
dEo
h
5-13
Coulomb Correction
•
•
•
Agreement of experiment and modeling at low Z
At higher Z need a correction factor to account for coulomb interaction

Coulomb interaction between nucleus and the emitted electron

decelerate electrons and accelerate positrons
 Electron spectra has more low-energy particles
 Positron spectra has fewer low-energy particles
Treat as perturbation on electron wave function e(0)

Called Fermi function

Defined as ratio of e(0)2Coul /e(0)2free

perturbation on e(0) and spectrum multiplied by the Fermi function
 Z daughter nucleus
 v beta velocity
 + for electrons
 - for positron
F ( Z ,W ) =
2 x
1  exp(  2  x )
;x = 
Ze
2
v
5-14
Kurie Plot
• Comparison of theory and experiment for momentum
measurements
 Square root of number of beta particles within a
certain range divided by Fermi function against
beta-particle energy
• Linear relationship designates allowed transition
5-15
Fermi Golden Rule
• Treat beta decay as transition that depends upon strength of
coupling between the initial and final states
• Decay constant given by Fermi's Golden Rule
2
2
 =
M  ( E o ); M =   f V  i dv

 matrix element couples initial and final states
 phase space factor which describes volume of phase
space available for the outgoing leptons
 Electron is charged lepton
* electron, muon, and tau
 Neutral lepton is neutrino
 Small system perturbation
 Contained within M
• E is Q value
• Rate proportional to strength of coupling between initial and
final states factored by the density of final states available to
5-16
system
Comparative Half Lives
• Based on probability of electron energy emission coupled with
spectrum and Coulomb correction fot1/2
 comparative half life of a transition
K = 64  m o c g / h
4
5
4
2
7
 =
Wo
fo =
 F ( Z , W )W (W
2
 1)
1/ 2
(W o  W ) dW
2
ln 2
t1 / 2
= K M if
2
fo
1
• Assumes matrix element is independent of energy
 true for allowed transitions
• Yields ft (or fot1/2), comparative half-life
 may be thought of as half life corrected for differences in Z and
W
W is total kinetic energy
• fo can be determine when Fermi function is 1 (low Z)
• Rapid estimation connecting ft and energy
 Simplified route to determine ft (comparative half-life)5-17
Comparative half-lives
log
•
•
f

= 4 . 0 log E o  0 . 78  0 . 02 Z  0 . 005 ( Z  1) log E o
= 4 . 0 log E o
log
f
log
f EC = 2 . 0 log E o  5 . 6  3 . 5 log( Z  1)

Eo 

3 
Z is daughter and Eo is maximum energy in MeV (Q value)
Log ft = log f + log t1/2

t1/2 in seconds
•
14 O
•
•

positron decay

Q=1.81 MeV

T1/2 =70.6 s
Log f = 1.83, log t = 1.84
Log ft=3.67
to

 0 . 79  0 . 007 Z  0 . 009 ( Z  1)  log

14N
log
log
f   = 4 . 0 log E o
f 
Eo 

 0 . 79  0 . 007 Z  0 . 009 ( Z  1)  log

3 

2
1 . 81 

= 4 . 0 log 1 . 81  0 . 79  0 . 007 ( 7 )  0 . 009 ( 7  1)  log

3 

5-18
2
2
Log ft calculation
• 212Bi beta decay
• Q = 2.254 MeV
• T1/2 = 3600 seconds
 64 % beta branch
  =1.22E-4 s-1
 T1/2Beta =5625 seconds
log f   = 4 . 0 log E o  0 . 78  0 . 02 Z  0 . 005 ( Z  1) log E o
log f   = 4 . 0 log 2 . 254  0 . 78  0 . 02 ( 84 )  0 . 005 ( 84  1) log 2 . 254
• Log f=3.73; log t=3.75
• Log ft=7.48
5-19
Log ft data
• What drives the changes in the log ft values for 24Na and 205Hg?
5-20
Extranuclear Effects of EC
• If K-shell vacancy is filled by
L electron, difference in
binding energies emitted as xray or used in internal
photoelectric process
 Auger electrons are
additional extranuclear
electrons from atomic
shells emitted with kinetic
energy equal to
characteristic x-ray
energy minus its own
binding energy
• Fluorescence yield is fraction
of vacancies in shell that is
filled with accompanying xray emission
 important in measuring
disintegration rates of EC
nuclides
radiations most
frequently detected
are x-rays
5-21
Selection Rules
• Allowed transitions are ones in which the electron and
neutrino carry away no orbital angular momentum
 largest transition probability for given energy release
• If electron and neutrino do not carry off angular
momentum, spins of initial and final nucleus differ by no
more than h/2 and parities must be the same
• If electron and neutrino emitted with intrinsic spins
antiparallel, nuclear spin change (I )is zero
 singlet
• If electron and neutrino spins are parallel, I may be +1,
0, -1
 triplet
5-22
Selection Rules
• All transitions between states of I=0 or 1 with no
change in parity have the allowed spectrum shape
• Not all these transitions have similar fot values
 transitions with low fot values are “favored” or
“superallowed”
found among  emitters of low Z and between
mirror nuclei (one contains n neutrons and n+1
protons, the other n+1 neutrons and n protons)
 Assumption of approximately equal Mif2 values
for all transitions with I=0, 1 without parity
change was erroneous
5-23
Forbidden Transitions
•
When the transition from initial to final nucleus cannot take place by emission
of s-wave electron and neutrino
 orbital angular momenta other than zero
•
l value associated with given transition deduced from indirect evidence
 ft values, spectrum shapes
•
•
•
•
If l is odd, initial and final nucleus have opposite parities
If l is even, parities must be the same
Emission of electron and nucleus in singlet state requires I  l
Triple-state emission allows I  l+1
5-24
Other Beta Decay
• Double beta decay

Very long half-life
 130Te and 82Se as
examples

Can occur through beta
stable isotope
76Ge to 76Se by double beta

 76Ge to 76As
 Q= -73.2130- (-72.2895) •
 Q= -0.9235 MeV

Possible to have
neutrinoless double beta
decay
 two neutrinos
annihilate each other
 Neutrino absorbed by
nucleon
Beta delayed decay

Nuclei far from stability can populate
unbound states and lead to direct nucleon
emission

First recognized during fission
 1 % of neutrons delayed
* 87Br is produced in nuclear fission
and decays to 87Kr

decay populates some high energy states in
Kr daughter
 51 neutrons, neutron emission to form
86Kr
5-25
Topic Review
• Fundamentals of beta decay
 Electron, positron, electron capture
• Neutrino Hypothesis
 What are the trends and data leading to
neutrino hypothesis
• Derivation of Spectral Shape
 What influences shape
Particles, potentials
• Kurie Plots
• Beta Decay Rate Constant
 Calculations
 Selection rules
Log ft
* How do values compare and relate to
spin and parity
• Other types of beta decay
5-26
Homework questions
• For beta decay, what is the correlation
between decay energy and half life?
• What is the basis for the theory of the
neutrino emission in beta decay.
• In beta decay what are the two possible
arrangements of spin?
• What is the basis for the difference in positron
and electron emission spectra?
• What log ft value should we expect for the decay to the 1- state of 144Pr?
• Why is there no  decay to the 2+ level?
• Calculate and compare the logft values for
EC, positron and electron decay for Sm
isotopes.
5-27
Pop Quiz
• Calculate the logft for the decay of 241Pu, 162Eu,
44Ti, and 45Ti. Provide the transition for each?
5-28