Transcript Decay Kinetics

Alpha Decay
•
•
•
•
•
•
•
•
Readings

Nuclear and Radiochemistry: Chapter 3

Modern Nuclear Chemistry: Chapter 7
Energetics of Alpha Decay
Theory of Alpha Decay
Hindrance Factors
Heavy Particle Radioactivity
Proton Radioactivity
Identified at positively charged particle by Rutherford

Helium nucleus (4He2+) based on observed emission bands

Energetics
 Alpha decay energies 4-9 MeV
 Originally thought to be monoenergetic, fine structure discovered
AZ(A-4)(Z-2) + 4He + Q
a
3-1
Alpha Decay Energetics
• Q value positive for alpha decay

Q value exceeds alpha decay energy

maTa = mdTd

md and Td represent daughter
• From semiempirical mass equation
ma Ta
Q

T


emission of an α-particle lowers Coulomb
a
md
energy of nucleus

increases stability of heavy nuclei while
ma
Q

T
(
1

)
not affecting overall binding energy per
a
md
nucleon
 tightly bound α-particle has
md
Q

T

Q
(
)
a
approximately same binding
m
ma  md
energy/nucleon as original nucleus
(1  a )
md
* Emitted particle must have
reasonable energy/nucleon
* Energetic reason for alpha rather
than proton
• Energies of alpha particles generally increase
3-2
with atomic number of parent
Energetics
• Calculation of Q value from mass excess
238U234Th + a + Q

 Isotope
Δ (MeV)
238U
47.3070
234Th
40.612
4He
2.4249

Qa=47.3070 – (40.612 + 2.4249) = 4.270 MeV

Q energy divided between α particle and heavy recoiling
daughter
 kinetic energy of alpha particle will be slightly less than Q
value
• Conservation of momentum in decay, daughter and alpha are equal
rd=ra
 recoil momentum and a-particle momentum are equal in
magnitude and opposite in direction
 p2=2mT where m= mass and T=kinetic energy
• 238U alpha decay energy
m
234
Ta  4.720 (
)  4.198 MeV
4  234
Ta  Q(
d
ma 3-3 md
)
Energetics
• Kinetic energy of emitted particle is less than Coulomb barrier
α-particle and daughter nucleus
 Equation specific of alpha
2
2
Z
e
2Z
 Particles touching
Vc 

1.44MeV fm

For 238 U decay
R 4 o
R
2(90)
259MeV fm
Vc 
1.44MeV fm 
 28MeV
1/ 3
1/ 3
1.2(234  4 ) fm
9.3 fm
• Alpha decay energies are small compared to required energy for
reverse reaction
• Alpha particle carries as much energy as possible from Q value,
• For even-even nuclei, alpha decay leads to ground state of
daughter nucleus
 as little angular momentum as possible
 ground state spins of even-even parents, daughters and
alpha particle are l=0
3-4
•
•
•
•
Distance of closest approach for
scattering of a 4.2 MeV alpha
particle is ~62 fm

Distance at which alpha
particle stops moving
towards daughter

Repulsion from Coulomb
barrier
Alpha particle should not get
near nucleus

should be trapped behind
a potential energy barrier
Wave functions are only
completely confined by infinitely
highpotential energy barriers

With finite size barrier
wave function has
different behavior

main component inside
barrier

finite piece outside
barrier
Tunneling

trapped particle has
component of wave
function outside potential
barrier

Some probability to go
through barrier
 Related to decay
probability

Higher energy has higher
tunneling probability
Alpha decay theory
Vc
Alpha decay energy
3-5
Alpha Decay Theory
•
•
Closer particle energy to barrier
maximum more likely particle will
penetrate barrier
More energetic alpha will
encounter barrier more often
T

•
Increase probability of
barrier penetration due to
Geiger Nuttall law of alpha decay
log t1 / 2  A 

•
1 2
mv
2
B
Qa
constants A and B have Z
dependence.
simple relationship describes data
on α-decay

over 20 orders of magnitude
in decay constant or half-life

1 MeV change in a-decay
energy results in a change of
105 in half-life
3-6
Alpha Decay Calculations
• Alpha particle barrier penetration
from Gamow
 T=e-2G
• Determination of decay constant
from potential information
R2
 4

h
1/ 2
1/ 2

exp
(2 )  (U (r )  T ) dr
2R12
h


R1

• Using square-well potential,
integrating and substituting
2
Zze
1
 Z daughter, z alpha T 
 v 2
R2
2
1/ 2
1/ 2
1/ 2

h
 8Zze 2 

T 
 T   T  

exp

arccos

1











2R12
hv
B
B
B








 


Ma M R
Ma  M R
Zze2
B
R1
3-7
Gamow calculations
t1 / 2
ln 2
ln 2




fP
• From Gamow
logt1/ 2
ln 2
e 2G
( 2(Vo  Qa )

B
 A
Qa
• Calculated emission rate typically one order of
magnitude larger than observed rate
 observed half-lives are longer than
predicted
 Observation suggest a route to evaluate
alpha particle pre-formation factor
3-8
Alpha Decay
• Even-even nuclei undergoing l=0 decay

average preformation factor is ~ 10-2
Theory

neglects effects of angular momentum
 Assumes α-particle carries off no orbital angular momentum (ℓ
= 0)

If α decay takes place to or from excited state some angular
momentum may be carried off by α-particle

Results in change in decay constant when compared to calculated
3-9
Hindered a-Decay
• Previous derivation only holds for even-even nuclei
 odd-odd, even-odd, and odd-even nuclei have longer half-lives than
predicted due to hindrance factors
• Assumes existence of pre-formed a-particles
 Ground-state transition from nucleus containing odd nucleon in highest
filled state can take place only if that nucleon becomes part of a-particle
 therefore another nucleon pair is broken
less favorable situation than formation of an a-particle from already
existing pairs in an even-even nucleus
* may give rise to observed hindrance
 a-particle is assembled from existing pairs in such a nucleus, product
nucleus will be in an excited state
this may explain higher probability transitions to excited states
• Hindrance from difference between calculation and measured half-life
 Hindrance factors between 1 and 3E4
 Hindrance factors determine by
ratio of measured alpha decay half life over calculated alpha decay
half life
ratio of calculated alpha decay constant over measured alpha decay
constant
t1 / 2a measured a calculated

 Hindrance factor
t1 / 2a calculated a measured
3-10
•
Hindrance Factors
Transition of 241Am (5/2-) to 237Np

states of 237Np (5/2+) ground state and
(7/2+) 1st excited state have hindrance
factors of about 500 (red circle)

Main transition to 60 keV above
ground state is 5/2-, almost unhindered
3-11
Hindrance Factors
• 5 classes of hindrance factors based on hindrance values
 Between 1 and 4, transition is called a “favored”
 emitted alpha particle is assembled from two low
lying pairs of nucleons in parent nucleus, leaving
odd nucleon in its initial orbital
 Hindrance factor of 4-10 indicates a mixing or
favorable overlap between initial and final nuclear
states involved in transition
 Factors of 10-100 indicate that spin projections of
initial and final states are parallel, but wave function
overlap is not favorable
 Factors of 100-1000 indicate transitions with a change
in parity but with projections of initial and final states
being parallel
 Hindrance factors of >1000 indicate that transition
involves a parity change and a spin flip
3-12
Topic Review
• Understand and utilize systematics and energetics
involved in alpha decay
• Calculate Q values for alpha decay
 Relate to alpha energy and fine structure
• Correlate Q value and half-life
• Models for alpha decay constant
 Tunneling and potentials
• Hindered of alpha decay
• Understand proton and other charged particle
emission
3-13
Homework Questions
• Calculate alpha decay Q value and Coulomb barrier potential
for following, compare values
 212Bi, 210Po, 238Pu, 239Pu, 240Am, 241Am
• What is basis for daughter recoil during alpha decay?
• What is relationship between Qa and alpha decay energy (Ta)
• What are some general trends observed in alpha decay?
• Compare calculated and experimental alpha decay half life for
following isotopes
 238Pu, 239Pu, 241Pu, 245Pu
 Determine hindrance values for odd A Pu isotopes above
• What are hindrance factor trends?
• How would one predict half-life of an alpha decay from
experimental data?
3-14
Pop Quiz
• Calculate alpha decay energy for 252Cf and 254Cf from
mass excess data below.
• Which is expected to have shorter alpha decay half-life
and why?
• Calculate alpha decay half-life for 252Cf and 254Cf from
data below. (use % alpha decay)
3-15
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?
•
131
53
26
13

I 131
Xe


  Energy
54
26
Al  e 12
Mg   Energy
22
11
22
Na10
Ne      Energy
3-16
-Decay
•
•
•
•
•
Class includes any radioactive decay process
in which A remains unchanged, but Z
changes
 - decay, electron capture, + decay
 energetic conditions for decay:
 - decay: MZ  MZ+1
 Electron capture: MZMZ-1,
 + decay: MZ  MZ-1+2me
* + decay needs to exceed 1.02
MeV
* Below 1.02 MeV EC dominates
* + increases with increasing
energy
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
3-17
The Neutrino
• Solved problems associated with decay
 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
3-18
Spin in Beta Decay
• Spins of created particles can be combined in
two ways
 Electron and neutrino spin both 1/2
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
3-19
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
Beta decay
Z  ( Z  1)       Q
Q  M(Z)  M(Z  1)
Positron decay
Z  ( Z  1)       Q
Q  M(Z)  (M(Z 1)  2e)
Electron Capture

Z  (Z 1)    Q
QEC  M(Z)  M(Z 1)
Q are ~0.5 – 2 MeV, Q + ~2-4 MeV and QEC ~ 0.2 – 2 MeV
What about positron capture instead of EC?
3-20
•
•
•
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
Positrons
• Annihilation radiation
 energy carried off by two 
quanta of opposite
momentum
 Annihilation conserves
momentum
 Exploited in Positron3-21
Emission Tomography
Weak Interaction
2 2 dn
4
2
2
P( pe )dpe 
 e (0)  (0) M if g
h
dE0
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 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
• Fermi constant (g) governs other interactions in addition to beta decay

-meson decay, -meson decay, neutrino-electron scattering
 Weak interactions
3-22
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
• 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 nucleas 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 energy
division between electron and neutrino
2 2 dn
4
2
2
P( pe )dpe 
 e (0)  (0) M if g
h
dE0
2
3-23
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 2mo5c 4
2
1/ 2
2

W
(
W

1
)
(
W

W
)
dW
o
6
dEo
h
3-24
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 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 Fermi
function
 Z daughter nucleus
 v beta velocity
 + for electrons
 - for positron
2x
Ze2
F ( Z ,W ) 
;x  
1  exp(2x)
v
3-25
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
3-26
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


2
2
 
M rf

matrix element couples initial and final states
density of states that are available to system after transition
M   f V 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
3-27
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
t1/ 2
2
fo
K  64 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)3-28
Comparative half-lives
lo g f    4.0 lo g Eo  0.7 8  0.0 2Z  0.0 05( Z  1) lo g Eo
lo g f  
lo g f EC
•
•
E 

 4.0 lo g Eo  0.7 9  0.0 07Z  0.0 09( Z  1) lo g o 
3 

 2.0 lo g Eo  5.6  3.5 lo g(Z  1)
Z is daughter and Eo is maximum energy in MeV (Q value)
Log ft = log f + log t1/2

t1/2 in seconds
2
•
•
•

positron decay
2

Q=1.81 MeV
1.81

log f   4.0 log1.81 0.79  0.007(7)  0.009(7  1) log


T1/2 =70.6 s
3 

3-29
Log f = 1.83, log t = 1.84
Log ft=3.67
to
14N
E 

 4.0 log Eo  0.79  0.007Z  0.009( Z  1) log o 
3 

14 O
log f  

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 Eo  0.78  0.02Z  0.005( Z  1) log Eo
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
3-30
Log ft data
•
What drives changes in log ft values for 24Na and 205Hg?

Examine spin and parity changes between parent and daughter state
3-31
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
3-32
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/2 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
3-33
 triplet
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”
 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
3-34
Forbidden Transitions
•
When transition from initial to final nucleus cannot take place by emission of swave 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 same
Emission of electron and nucleus in singlet state requires I  l
Triple-state emission allows I  l+1
3-35
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
3-36
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
3-37
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.
3-38
Pop Quiz
• Calculate the logft for the decay of 241Pu, 162Eu,
44Ti, and 45Ti. Provide the transition for each?
3-39