Alpha Decay
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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
α
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Fine Structure and
Energetics
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Different alpha decay energies
for same isotope

Relative intensities vary

Coupled with gamma
decay
Over 350 artificially produced
alpha emitting nuclei

Alpha energy variations
used to develop decay
schemes
All nuclei with mass numbers
greater than A of 150 are
thermodynamically unstable
against alpha emission (Qα is
positive)

However alpha emission is
dominant decay process
only for heaviest nuclei,
A≥210

Energy
ranges 1.8 MeV
Nd)
to
11.6 MeV
(144
212m
Po)
(
 half-life
of 144Nd is
29
times longer
5x10 212m
Po
then
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Energetics
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Q values generally increase with
A

variation due to shell
effects can impact trend
increase

Peaks at N=126 shell
For isotopes decay energy
generally decreases with
increasing mass
82 neutron closed shell in the
rare earth region

increase in Qα

α-decay for nuclei with
N=84 as it decays to N=82
daughter
short-lived α-emitters near
doubly magic 100Sn
107Te, 108Te, 111Xe

alpha emitters have been
identified by proton dripline
above A=100
3
Alpha Decay Energetics
• Q value positive for alpha decay

Q value exceeds alpha decay energy

mαTα = mdTd

md and Td represent daughter
• From semiempirical mass equation
mα Tα
Q
=
T
+

emission of an α-particle lowers Coulomb
α
md
energy of nucleus

increases stability of heavy nuclei while
mα
T
Q
(
1
)
=
+
not affecting the overall binding energy
α
md
per nucleon
 tightly bound α-particle has
md
Q
(
)
=
T
=
Q
α
approximately same binding
m
mα + md
energy/nucleon as the original
(1 + α )
md
nucleus
* Emitted particle must have
reasonable energy/nucleon
* Energetic reason for alpha rather
than proton
4
• Energies of alpha particles generally increase
with atomic number of parent
Energetics
• Calculation of Q value from mass excess
238U234Th + α + Q

 Isotope
Δ (MeV)
238U
47.3070
234Th
40.612
4He
2.4249

Qα=47.3070 – (40.612 + 2.4249) = 4.270 MeV

Q energy divided between the α particle and the heavy recoiling
daughter
 kinetic energy of the alpha particle will be slightly less than
Q value
• Conservation of momentum in decay, daughter and alpha are equal
ρd=ρα
 recoil momentum and the α-particle momentum are equal in
magnitude and opposite in direction
 p2=2mT where m= mass and T=kinetic energy
• 238U alpha decay energy
Tα = 4.270(
234
) = 4.198MeV
4 + 234
md
)
Tα = Q( 5
mα + md
Energetics
• Kinetic energy of emitted particle is less than Coulomb barrier
α-particle and daughter nucleus
 Equation specific of alpha
2Z e 2
2Z
Vc =

For 238 U decay
R 4πε o
=
R
1.44 MeV fm
259MeV fm
2(90)
Vc =
1.44 MeV fm ≈
= 28MeV
1/ 3
1/ 3
1.2(234 + 4 ) fm
9.3 fm
• Alpha decay energies are small compared to the required energy
for reverse reaction
• Alpha particle carries as much energy as possible from Q value
• For even-even nuclei, alpha decay leads to the ground state of
the daughter nucleus
 as little angular momentum as possible
 ground state spins of even-even parents, daughters and
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alpha particle are l=0
Energetics
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Some decays of odd-A heavy nuclei populate
low-lying daughter excited states that match
spin of parent

Leads to fine structure of alpha
decay energy
Orbital angular momentum of α particle can
be zero

83%
of alpha decay245of 249Cf goes to
th
9 excited state of Cm

lowest lying state with same spin and
parity as parent
Long range alpha decay

Decay from excited state of parent
nucleus to ground state of the
daughter
212mPo

 2.922 MeV above 212Po ground
state
 Decays to ground state of 208Pb
with emission of 11.65 MeV
alpha particle
Systematics result from

Coulomb potential
 Higher mass accelerates
products

larger mass
 daughter and alpha particle
start further apart
mass parabolas from semiempirical mass
equation
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cut through the nuclear mass surface
at constant A
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Explains beta decay in decay chain
Beta Decay to Energy minimum,
then Alpha decay to different A
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Alpha decay theory
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Distance of closest approach for
scattering of a 4.2 MeV alpha particle
is ~62 fm
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Distance at which alpha
particle stops moving towards
daughter
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Repulsion from Coulomb
barrier

Alpha particle should not get
near the nucleus from outside
Alpha particle should be trapped
behind a potential energy barrier
Wave functions are only completely
confined by potential energy barriers
that are infinitely high
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With finite size barrier wave
function has different behavior

main component inside the
barrier

finite piece outside barrier
Tunneling
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classically trapped particle has
component of wave function
outside the potential barrier
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Some probability to go
through barrier
 Related to decay
probability
Vc
Alpha decay energy
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Alpha Decay Theory
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Closer the energy of the particle
to the top of the barrier more
likely the particle will penetrate
barrier
More energetic the particle is
relative to a given barrier height,
more frequently the particle will
encounter barrier
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Increase probability of
barrier penetration
Geiger Nuttall law of alpha
decay

Log t1/2=A+B/(Qα)0.5

constants A and B have a
Z dependence.
simple relationship describes the
data on α-decay

over 20 orders of
magnitude in decay
constant or half-life

1 MeV change in α-decay
energy results in a change
of 105 in the half-life
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Expanded Alpha Half Life Calculation
• More accurate determination of half life from Hatsukawa,
Nakahara and Hoffman
log10 (t1/ 2 ) = A( Z )(
Ad 1/ 2
) [arccos X − X (1 − X ] − 20.446 + C ( Z , N )
Ap Qα
C ( Z , N ) = 0 Outside of closed shells
C ( Z , N ) = [1.94 − 0.020(82 − Z ) − 0.070(126 − N ) 78≤Z≤82; 100≤N≤126
C ( Z , N ) = [1.42 − 0.105( Z − 82) − 0.067(126 − N ) 82≤Z≤90; 100≤N≤126
Q
X = 1.2249( A1/ 3 + 41/ 3 )( α 2 )
2Z d e
• Theoretical description of alpha emission based on calculating
rate in terms of two factors
 rate at which an alpha particle appears at the inside wall of
the nucleus
 probability that the alpha particle tunnels through the
barrier
• λα=P*f
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 f is frequency factor
 P is transmission coefficient
Alpha Decay Theory
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Alternate expression includes additional factor that describes probability of
preformation of alpha particle inside parent nucleus
No clear way to calculate such a factor
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empirical estimates have been made
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theoretical estimates of the emission rates are higher than observed rates
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preformation factor can be estimated for each measured case
 uncertainties in the theoretical estimates that contribute to the
differences
Frequency for alpha particle to reach edge of a nucleus
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estimated as velocity divided by the distance across the nucleus
 twice the radius
 lower limit for velocity could be obtained from the kinetic energy of
emitted alpha particle
 However particle is moving inside a potential energy well and its velocity
should be larger and correspond to the well depth plus the external
energy
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On the order of 1021 s-1
2(Vo + Q) / µ
v
f =
≈
2R
2R
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Alpha Decay Calculations
• Alpha particle barrier penetration
from Gamow
 T=e-2G
• Determination of decay constant
from potential information
R2

 4π
h
1/ 2
1/ 2
exp −
(2 µ ) ∫ (U (r ) − T ) dr 
λ=
2µR12
h


R1
Mα M R
µ=
Mα + M R
• Using the square-well potential,
integrating and substituting
2
Zze
1
 Z daughter, z alpha T =
= µv 2
R2
2
1/ 2
1/ 2
1/ 2
 8πZze 2 
h
 T   T   
T 
λ=
exp−
arccos  −   1 −   
2 µR12
hv
 B   B   
B


Zze 2
B=
R1
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Gamow calculations
t1/ 2
ln 2
ln 2
ln 2
− 2G
e
=
=
=
fP (2(Vo + Qα ) / µ )1/ 2
λ
• From Gamow
 Log t1/2=A+B/(Qα)0.5
• Calculated emission rate typically one order of
magnitude larger than observed rate
 observed half-lives are longer than
predicted
 Observation suggest probability to find a
‘preformed’ alpha particle on order of 10-1
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Alpha Decay Theory
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Even-even nuclei undergoing l=0 decay
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average preformation factor is ~ 10-2
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neglects effects of angular momentum
 Assumes α-particle carries off no orbital angular momentum (ℓ = 0)
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If α decay takes place to or from excited state some angular momentum
may be carried off by the α-particle
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Results in change in the decay constant when compared to calculated
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Hindered α-Decay
• Previous derivation only holds for even-even nuclei
 odd-odd, even-odd, and odd-even nuclei have longer halflives than predicted due to hindrance factors
• Assumes existence of pre-formed α-particles
 ground-state transition from a nucleus containing an odd
nucleon in highest filled state can take place only if that
nucleon becomes part of the α-particle
another nucleon pair is broken
less favorable situation than formation of an α-particle
from existing pairs in an even-even nucleus
* observed hindrance.
if α-particle is assembled from existing pairs in such a
nucleus, the product nucleus will be in an excited state,
* explain the “favored” transitions to excited states
• Hindrance factor determine by ratio of measured alpha decay
half life over calculated alpha decay half life
 Calculations underpredict half life
 Hindrance factors between 1 and 3E4
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Hindrance Factors
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Transition of 241Am (5/2-) to 237Np
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states of 237Np (5/2+) ground state and (7/2+) 1st excited state have
hindrance factors of about 500
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Main transition to 60 keV above ground state is 5/2-, almost
unhindered
5 classes of hindrance factors (half live measure/half life calculated)
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Between 1 and 4, the transition is called a “favored”
 emitted alpha particle is assembled from two low lying pairs of
nucleons in the parent nucleus, leaving the odd nucleon in its
initial orbital
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Hindrance factor of 4-10 indicates a mixing or favorable overlap
between the initial and final nuclear states involved in the transition
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Factors of 10-100 indicate that spin projections of the initial and
final states are parallel, but the wave function overlap is not
favorable
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Factors of 100-1000 indicate transitions with a change in parity but
with projections of initial and final states being parallel
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Hindrance factors of >1000 indicate that the transition involves a
parity change and a spin flip
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Heavy Particle Decay
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Possible to calculate Q values for the
emission of heavier nuclei
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Is energetically possible for a
large range of heavy nuclei to
emit other light nuclei.
Q-values for carbon ion emission by
a large range of nuclei
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calculated with the smooth
liquid drop mass equation
without shell corrections
Decay to doubly magic 208Pb from
220Ra for 12C emission
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Actually found 14C from 223Ra
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large neutron excess favors the
emission of neutron-rich light
products
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emission probability is much
smaller than the alpha decay
simple barrier penetration estimate
can be attributed to the very small
probability to preform 14C residue
inside the heavy nucleus
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Proton Decay
• For proton-rich nuclei, the Q value
for proton emission can be positive
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Line where Qp is positive,
proton drip line

Describes forces holding nuclei
together
• Similar theory to alpha decay
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no preformation factor for the
proton
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proton energies, even for the
heavier nuclei, are low (Ep~1
to 2 MeV)
• barriers are large (80 fm)
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Long half life
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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
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Homework Questions
• Calculate the alpha decay Q value and Coulomb barrier
potential for the following, compare the values
 212Bi, 210Po, 238Pu, 239Pu, 240Am, 241Am
• What is the basis for daughter recoil during alpha decay?
• What is the relationship between Qa and the alpha decay
energy (Ta)
• What are some general trends observed in alpha decay?
• Compare the calculated and experimental alpha decay half life
for the following isotopes
 238Pu, 239Pu, 241Pu, 245Pu
 Determine the hindrance values for the odd A Pu isotopes
above
• What are the hindrance factor trends?
• How would one predict the half-life of an alpha decay from
experimental data?
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Pop Quiz
• Calculate the alpha decay energy for 252Cf and 254Cf
from the mass excess data below.
• Which is expected to have the shorter alpha decay halflife and why?
• Calculate the alpha decay half-life for 252Cf and 254Cf
from the data below.
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