RFSS: Lecture 5 Beta Decay
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Readings: Nuclear and Radiochemistry: Chapter 3, Modern
Nuclear Chemistry: Chapter 8
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•
•
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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
131
spin
53
Radioactive decay process in which A remains unchanged,
but Z changes
26
 β- decay, electron capture, β+ decay
13
 energetic conditions for decay:
 β decay: MZ  MZ+1
 Electron capture: MZMZ-1,
22
 β+ decay: MZ  MZ-1+2me
11
Beta decay half-life

few milliseconds to ~ 1016 years

How does this compare to alpha decay?
•
−
I →131
Xe
+
β
+ ν + Energy
54
26
Al + e − →12
Mg + ν + Energy
•
Na →1022Ne + β + + ν + Energy
5-1
Q value calculation (Review)
•
•
•
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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
Beta decay
Z → ( Z + 1) + β − + ν + Q
Qβ− = M ( Z) − M ( Z + 1)
Positron decay
Z → ( Z − 1) + β + + ν + Q
Qβ+ = M ( Z) − (M ( Z − 1) + 2e)
Electron Capture
+
Z → ( Z − 1) + ν + Q
Q EC = M ( Z) − M ( Z − 1)
Qβ− are ~0.5 – 2 MeV, Q β+ ~2-4 MeV and QEC ~ 0.2 – 2 MeV
What about positron capture instead of EC?
5-2
β-Decay
• Decay energies of β -unstable
nuclei vary systematically with
distance from stability

Shown by mass parabolas

Energy-lifetime relations
are not nearly so simple as
alpha decay

β -decay half lives depend
strongly on spin and parity
changes as well as energy
• For odd A, one β-stable nuclide;
for even A, at most three β-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
5-3
The Neutrino
• Solved problems associated with βdecay
 Continuum of electron emission
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
5-4
Neutrino
• Carries away appropriate amount of energy and
momentum in each β process for conservation
• Nearly undetectable due to small rest mass and magnetic
moment
 observed by inverse β processes
37Cl+ν37Ar+e-: Detection of 37Ar
71Ga+ν71Ge+e-: Detection of 71Ge
• Antineutrinos emitted in β- decay, neutrinos emitted in β+
decay
 indistinguishable properties, except in capture
reactions
• Neutrinos created at moment of emission
 n → p + β- + ν
 p → n + β+ + ν
• Spin of created particles are involved in assigning decay
 Spin up and spin down
5-5
Spin in Beta Decay
• Spins of created particles can be combined in
two ways
 Electron and neutrino spin both 1/2
Sβ=1 in a parallel alignment
 Sβ= 0 in an anti-parallel alignment
• two possible relative alignments of "created"
spins
 Fermi (F) (Sβ=0)
Low A
 Gamow-Teller (GT) (Sβ =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
• Can be used to define decay
5-6
Spin in Beta Decay
• Decay of even-even nuclei with N=Z (mirror nuclei)
 neutron and protons are in the same orbitals
 shell model, Nuclear Structure and Models lecture
 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
 Basis of model to calculate decay constant
5-7
* Fermi golden rule (slide 15)
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•
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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

Conversion of mass to energy
when positron and electron
interact

simultaneous emission of
corresponding amount of
energy in form of radiation

Responsible for short lifetime
of positrons
 No positron capture decay
Positrons
• Annihilation radiation
 energy carried off by two γ
quanta of opposite
momentum
 Annihilation conserves
momentum
 Exploited in Positron
Emission Tomography
5-8
Weak Interaction: Model of Beta Decay
• Fermi's theory of beta decay based on
electromagnetic theory for light emission
 Fermions interact during reaction
 Degree of interaction from 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
4π 2
2
2
P ( pe )dpe =
ψ e (0) ψ ν (0) M if g
h
dE0
5-9
Weak Interaction
2 2 dn
4π
2
2
ψ e (0) ψ ν (0) M if g
P( pe )dpe =
h
dE0
2
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•
•
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P(pe)dpe probability electron with momentum pe+dpe
ψe electron wave function
ψν neutrino wave function
|ψe(0)|2 and |ψν(0)|2 probability of finding electron and
neutrino at nucleus
• Mif matrix element
 characterizes transition from initial to final nuclear state
• |Mif|2 a measure of overlap amount between wave functions of
initial and final nuclear states
• dn/dEo is density of final states with electron in specified
momentum interval
 number of states of final system per unit decay energy
5-10
Weak Interaction
• Integration over all electron momenta from zero to maximum
should provide transition probabilities or lifetimes
 Variations in number of electrons at a given energy
 Derivation of emission spectrum
 Calculation of decay constant
• Classically allowed transitions both have electron and neutrino
emitted with zero orbital angular momentum
 Allowed have s orbital angular momentum
 Relatively high probabilities for locating electron and
neutrino at nucleus for s wave compared to higher l
 p,d,f, etc.
 |ψ|2 of allowed transitions > |ψ|2 of forbidden transitions
• Magnitudes of |ψν(0)| and |Mif| are independent of energy division
between electron and neutrino
2 2 dn
4π 2
2
2
P ( pe )dpe =
ψ e (0) ψ ν (0) M if g
h
dE0
5-11
Weak Interaction
• Spectrum shape determined entirely
by |ψe(0)| and dn/dEo
 dn/dEo density of final states with
electron momentum
Coulomb interaction between
nucleus and emitted electron
(|ψe(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 16π 2mo5c 4
2
1/ 2
2
=
W
(
W
−
1
)
(
W
−
W
)
dW
o
6
dEo
h
5-12
Coulomb Correction
• Agreement of experiment and modeling at low Z

Minimized charge on nucleus
• At higher Z need a correction factor to account for coulomb interaction

Coulomb interaction between nucleus and 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 ψe(0)

Called Fermi function

Defined as ratio of |ψe(0)|2Coul /|ψe(0)|2free

perturbation on ψe(0) and spectrum multiplied by Fermi function
 Z daughter nucleus
 v beta velocity
 + for electrons
 - for positron
Ze 2
2πx
F ( Z ,W ) =
;x = ±
v
1 − exp(−2πx)
5-13
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 plotted against beta-particle energy (W)

x axis intercept is Q value
• Linear relationship designates allowed transition
5-14
Fermi Golden Rule
• Used for transition probability
• Treat beta decay as transition that depends upon strength of
coupling between initial and final states
• Decay constant given by Fermi's Golden Rule


2π
2
λβ =
M ρf

matrix element couples initial and final states
density of states that are available to system after transition
M = ∫ψ f Vψ i dv
 Wave function of initial and final state
 Operator which coupled initial and final state
• Rate proportional to strength of coupling between initial and final
states factored by density of final states available to system
 final state can be composed of several states with the same
energy
 Degenerate states
5-15
Comparative Half Lives
• Based on probability of electron energy emission coupled with
spectrum and Coulomb correction fot1/2
 comparative half life of a transition
ln 2
= K M if
λ=
t1/ 2
2
fo
K = 64π 4 mo5c 4 g 2 / h 7
Wo
f o = ∫ F ( Z , W )W (W 2 − 1)1/ 2 (Wo − W ) 2 dW
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)
5-16
• Log ft = log f + log t1/2
Comparative
 t1/2 in seconds
• Z is daughter
• Eo is maximum energy in MeV (Q value)
half-lives
log f β − = 4.0 log Eo + 0.78 + 0.02 Z − 0.005( Z − 1) log Eo
log f β +
log f EC
E 

= 4.0 log Eo + 0.79 + 0.007 Z − 0.009( Z + 1) log o 
3 

= 2.0 log Eo − 5.6 + 3.5 log( Z + 1)
•
to 14N

positron decay

Q=4.121 MeV

T1/2 =70.6 s
Log fβ+ = 3.30, log t = 1.85
•
Log ft=5.15
•
2
14 O
log f β
+
4.121 

= 4.0 log 4.121 + 0.79 + 0.007( 7) − 0.009( 7 + 1) log

3 

2
5-17
Log ft calculation
• 212Bi beta decay
• Q = 2.254 MeV
• T1/2 = 3600 seconds
 64 % beta branch
 λβ =1.22E-4 s-1
 T1/2Beta =5625 seconds
log f β − = 4.0 log Eo + 0.78 + 0.02 Z − 0.005( Z − 1) log Eo
log f β − = 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
5-18
Log ft data
•
What drives changes in log ft values for 205Hg?

Examine spin and parity changes between parent and daughter state
5-19
Selection Rules
• Allowed transitions are ones in which 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/2π and parities must be same
 0 or 1
Fermi or Gamow-Teller transitions
• 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
5-20
Selection Rules
• All transitions between states of ∆I=0 or 1 with no
change in parity have allowed spectrum shape
 I is nuclear spin
• Not all these transitions have similar fot values
 transitions with low fot values are “favored” or
“superallowed”
β emitters of low Z
between mirror nuclei
* one contains n neutrons and n+1 protons, 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
5-21
•
When 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
Forbidden
Transitions
 ft values, spectrum shapes
•
•
•
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If l is odd, initial and final nucleus have opposite
parities
If l is even, parities are same
Emission of electron and nucleus in singlet state
requires ∆I ≤ l
Triple-state emission allows ∆I ≤ l+1
5-22
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 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
5-23
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
5-24
Topic Review
• Fundamentals of beta decay
 Electron, positron, electron capture
• Neutrino Hypothesis
 What are 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
5-25
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 βdecay to the 1- state of 144Pr?
• Why is there no β decay to the 2+ level?
• Calculate and compare the logft values for
EC, positron and electron decay for Sm
isotopes.
5-26
Question
• Respond to PDF Quiz 5
 Submit quiz when complete
• Comment on the blog
 http://rfssunlv.blogspot.com
5-27