Nuclear Decays
• Unstable nuclei can change N,Z.A to a nuclei at a
lower energy (mass)
 : Z N A  Z 2N n A4  2He 4
 : N  N  e   /
Z
A
Z 1
A
n

• If there is a mass difference such that energy is
released, pretty much all decays occur but with
very different lifetimes.
• have band of stable particles and band of “natural”
radioactive particles (mostly means long lifetimes).
Nuclei outside these bands are produced in labs and
in Supernovas
• nuclei can be formed in excited states and emit a
gamma while cascading down.
P461 - nuclear decays
1
General Comments on Decays
• Use Fermi Golden rule (from perturbation theory)
2
rate 
| Vif |2  f

Vif   i*V f dVolume
• rate proportional to cross section or 1/lifetime
• the matrix element connects initial and final states
where V contains the “physics” (EM vs strong vs
weak coupling and selection rules)
• the density of states factor depends on the amount
of energy available. Need to conserve momentum
and energy “kinematics”. If large energy available
then higher density factor and higher rate.
• Nonrelativistic (relativistic has 1/E also. PHYS684)
 f  p dpi dEi each particle
2
i
P461 - nuclear decays
2
Simplified Phase Space
• Decay: A  a + b + c …..
• Q = available kinetic energy
Q  Mass A   mi ( final state)
• large Q  large phase space  higher rate
• larger number of final state products possibly
means more phase space and higher rate as more
variation in momentums. Except if all the mass of
A is in the mass of final state particles
B   D 0   2  body
Q  5279  1865  770  2644MeV
B   D 0  0 
3  body
Q  5279  1865  770  139  2505MeV
• 3 body has little less Q but has 4 times the rate of
the 2 body (with essentially identical matrix
elements)
P461 - nuclear decays
3
Phase Space:Channels
• If there are multiple decay channels, each adds to
“phase space”. That is one calculates the rate to
each and then adds all of them up
• single nuclei can have an alpha decay and both
beta+ and beta- decay. A particle can have hundreds
of possible channels
• often one dominates
• or an underlying virtual particle dominates and then
just dealing with its “decays”
c  s W
W  ud , e ,  ,
s  K mesons
• still need to do phase space for each….
P461 - nuclear decays
4
Lifetimes
• just one channel with N(t) = total number at time t
1
dN
 t
 Rate    gamma  width
   N  N ( t )  N ( 0) e

dt
1
t  
half  life  t1 / 2   ln 2

• multiple possible decays. Calculate each (the
“partial” widths) and then add up
1
   1  2  3 


branching fractioni  i

• Measure lifetime.
long-lived (>10-8sec). Have a certain number and
count the decays
dN / dt
1

N

P461 - nuclear decays
5
Lifetimes
• Measure lifetime.
medium-lived (>10-13sec). Decay point separated
from production point. Measure path length. Slope
gives lifetime
Dx  cDt
100
10
1
Dx
• short-lived (10-23 <  <10-16 sec). Measure invariant
mass of decay products. If have all  mass of
initial. Width of mass distributions (its width)
related to lifetime by Heisenberg uncertainty.
 ( x, t )   ( x )e  t / 2 eit 
   ( x ) e t / 
2
Dt  
2
M
 DE 
 

Dt 
  1020 sec  DE 
P461 - nuclear decays

 100 MeV
1020
6
Alpha decay
• Alpha particle is the He nucleus (2p+2n)
• ~all nuclei Z > 82 alpha decay. Pb(82,208) is
doubly magic with Z=82 and N=126
X  X  
ZX  ZX  2 NX  NX  2
• the kinematics are simple as non-relativistic and
alpha so much lighter than heavy nuclei
Q  mX  mX   mHe
p X   p
p2
TX  
 small
2m X 
A4
KE 
Q 4  9MeV
A
• really nuclear masses but can use atomic as number
of electrons do not change
P461 - nuclear decays
7
Alpha decay-Barrier penetration
• One of the first applications of QM was by Gamow
who modeled alpha decay by assuming the alpha
was moving inside the nucleus and had a
probability to tunnel through the Coulomb barrier
• from 1D thin barrier (460) for particle with energy
E hitting a barrier potential V and thickness gives
E
E  2 ka
Transmission = T
T  16 (1  )e
V
V
2m(V  E )
k

• now go to a Coulomb barrier V= A/r from the edge
of the nucleus to edge of barrier and integrate- each
dr is a thin barrier
rc
T  exp( 2 
rn
2m 2 Ze 2
(
 E ) dr
2
 40 r
P461 - nuclear decays
2 Ze 2
rc 
40 K
8
Alpha decay-Barrier penetration
• this integral isn’t easy, need approximations
rc
T  exp( 2 
rn
2m 2 Ze 2
(
 E ) dr
2
 40 r
2 Ze 2
rc 
40 K
• see nuclear physics textbook (see square) Get
T  exp( 2zZe2 / v )
 exp( 2zZe2 M /  2 K )
• where K = kinetic energy of alpha. Plug in some
numbers
2  2  90  (1.4 MeVF )
4  931MeV
T  exp( 

)
197 MeVF
2  6 MeV
T  exp( 70)  4  1031
• see
www.haverford.edu/physics-astro/songs/alpha.htm
P461 - nuclear decays
9
Alpha decay-Barrier penetration
• Then have the alpha bouncing around inside the
nucleus. It “strikes” the barrier with frequency
•
•
•
•
•
velocity
f 
2rN
the decay rate depends on barrier height and barrier
thickness (both reduced for larger energy alpha)
and the rate the alpha strikes the barrier
larger the Q larger kinetic energy and very strong
(exponential) dependence on this
as alpha has A=4, one gets 4 different chains (4n,
4n+1, 4n+2, 4n+3). The nuclei in each chain are
similar (odd/even, even/even, etc) but can have spin
and parity changes at shell boundaries
if angular momentum changes, then a suppression
of about 0.002 for each change in L (increases
potential barrier)
2
 l (l  1)
2mr 2
P461 - nuclear decays
10
Alpha decay-Decay chains
4n+2
4n
P461 - nuclear decays
11
Alpha decay-Energy levels
• may need to have orbital angular momentum if subshell changes (for odd n/p nuclei)
• Z= 83-92 1h(9/2)
N=127-136 2g(9/2)
Z=93-100 2f(7/2)
N=137-142 3d(5/2)
• so if
f(7/2)  h(9/2) need L>0 but parity change if
L=1  L=2,4
• or d(5/2)  g(9/2) need L>1. No parity change
L=2,4
• not for even-even nuclei (I=0). suppression of
about 0.002 for each change in L (increases
potential barrier)
s 0
p 1
d 2
f 3
g 4
h 5
P461 - nuclear decays
12
Parity + Angular Momentum
Conservation in Alpha decay
• X  Y + . The spin of the alpha = 0 but it can
have non-zero angular momentum. Look at Parity P
PX  PY P Porbit  P  1, Porbit  ( 1)l
• if parity X=Y then L=0,2…. If not equal L=1,3…
U 235  Th 231  
# n  143  1i11/ 2  112
# n  141  3d 5 / 2 

5
2
(i, l  6)
(d , l  2)
 Lorbital  ( 112  52 )  3
 Lorbital  4,6
• to conserve both Parity and angular momentum
P461 - nuclear decays
13
Energy vs A Alpha decay
P461 - nuclear decays
14
Lifetime vs Energy in Alpha
Decays
Perlman, Ghiorso, Seaborg, Physics
Review 75, 1096 (1949)
10
log10 half-life
in years
0
-10
5
7
Alpha Energy MeV
P461 - nuclear decays
15
Beta Decays
• Beta decays are proton  neutrons or neutron 
proton transitions
• involve W exchange and are weak interaction
M Z , A  M Z 1, A  e    e
( p  ne )
M Z , A  M Z 1, A  e    e
(n  pe )
e   M Z , A  M Z 1, A   e
(ep  n )
• the last reaction is electron capture where one of
the atomic electrons overlaps the nuclei. Same
matrix element (essentially) bit different kinematics
• the semi-empirical mass formula gives a minimum
for any A. If mass difference between neighbors is
large enough, decay will occur
P461 - nuclear decays
16
Beta Decays - Q Values
• Determine Q of reactions by looking at mass
difference (careful about electron mass)
  : X Z , A  YZ 1, A  e    e
m X ( Zme )  mY  ( Zme )  KY  me  K e  K
Q  AtomicMass X  AM Y  KY  K e  K
  : X Z , A  YZ 1, A  e    e
m X ( Zme )  mY  ( Zme )  KY  me  K e  K
Q  AM X  AM Y  2me
EC : e   X Z , A  YZ 1, A   e
me  m X ( Zme )  mY  ( Zme )  K Y  K e  K
Q  AM X  AM Y
• 1 MeV more Q in EC than beta+ emission. More
phase space BUT need electron wavefunction
overlap with nucleus.....
P461 - nuclear decays
17
Beta+ vs Electron Capture
• Fewer beta+ emitters than beta- in “natural” nuclei
(but many in “artificial” important in Positron
Emission Tomography - PET)
• sometimes both beta+ and EC for same nuclei.
Different widths
• sometimes only EC allowed
3
Li 7
M  7.01600u
Be 7 M  7.01693u
DM  .00093u < 2  me  2  .00055u
4
4
Be 7  e  3Li 7  
• monoenergetic neutrino. E=.87 MeV. Important
reaction in the Sun. Note EC rate different in Sun
as it is a plasma and not atoms
P461 - nuclear decays
18
Beta+ vs Electron Capture
• from Particle Data Group
p  p2H  e  
8
7
B8Be  e   
Be  e7Li  
P461 - nuclear decays
19
Beta Decay - 3 Body
• The neutrino is needed to conserve angular
momentum
• (Z,A)  (Z+1,A) for A=even have either
Z,N even-even  odd-odd or odd-oddeven-even
• p,n both spin 1/2 and so for even-even or odd-odd
nuclei I=0,1,2,3…….
• But electron has spin 1/2
I(integer)  I(integer) + 1/2(electron) doesn’t
conserve J
• need spin 1/2 neutrino. Also observed that electron
spectrum is continuous indicative of >2 body decay
• Pauli/Fermi understood this in 1930s
electron neutrino discovered 1953 (Reines and
Cowan)
muon neutrino discovered 1962 (Schwartz
+Lederman/Steinberger)
tau neutrino discovered 2000 at Fermilab
P461 - nuclear decays
20
3 Body Kinematics
• While 3 body the nuclei are very heavy and easy
approximation is that electron and neutrino split
available Q (nuclei has similar momentum)
• maximum electron energy when E(nu)=0
X  Y  e 
p y  pe
let E  0
conserve momentum
(mx  Ee  m ) 2  E y2
Ee max 
mx2  m y2  me2
2m x
K e  Ee  me 
• example
conserve energy
( m )
(mx  m y  me )( mx  m y  me )
2mx
Q
Mg 12  27Al13  e   
m27,12  26.9843, m27,13  26.981
27
me  .00055  Q  2.8MeV
pe  Ee2  me2  2.75MeV ,  e 
E
 5 .5
m
p2
K Al 
 0.2keV  small
2m
P461 - nuclear decays
21
Beta decay rate
• Start from Fermi Golden Rule
2
2
Rates 

| M |  Final
M   F* d
• first approximation (Fermi).
Beta=constant=strength of weak force
M  M  M    Z* 1 Z d
• Rule 1: parity of nucleus can’t change (integral of
odd*even=0)
• Rule 2: as antineutrino and electron are spin 1/2
they add to either 0 or 1. Gives either
Fermi : Di  iZA  iZ 1 A  0
42
Sc 21 42Ca 20  e   
0  0
Gamow  Teller : Di  1 (not 0  1  0)
32
P15 22S 16  e   
1  0 
P461 - nuclear decays
22
Beta decay rate II
• Orbital angular momentum suppression of 0.001
for each value of L (in matrix element calculation)
36
Sc17 36Ca18  e    2   0
Di  1  L  1
• look at density of states factor. Want # quantum
states per energy interval
dN n
2
2
Rates 
| M |  Final  n 

dEn
• we know from quantum statistics that each particle
(actually each spin state) has
p2
dN  4 3 dp
h
• 3 body decay but recoil nucleus is so heavy it
doesn’t contribute
2
2
pe
p
dN  4 3 dpe 4 3 dp
h
h
p  (Q  K e ) / c
P461 - nuclear decays
23
Beta decay rate III
• Conservation of energy allows one to integrate over
the neutrino (there is a delta function)
Rates 
dN 2

| M |2  Final 
dpe

2
2
2
2 4pe 4 (Q  K e )
|M |

h3
(hc)3
K e  ( pe2  me2 )1/ 2  me
• this gives a distribution in electron
momentum/energy which one then integrates over.
(end point depends on neutrino mass)
1 me5c 4
2
Rate  
|
M
|
F ( Ee max )
3 7
T 2 
• F is a function which depends on Q. It is almost
loqrithmic
log F  A log K e max
P461 - nuclear decays
24
actual. not “linear” due to
electron mass
log F  A log K e max  4.4 log K  .5
 F  3K 4.4
P461 - nuclear decays
25
Beta decay rate IV
• FT is “just kinematics”
• measuring FT can study nuclear wavefunctions M’
and strength of the weak force at low energies
• lower values of FT are when M’ approaches 1
• beta decays also occur for particles
    0  e  e
K    0  e  e
• electron is now relativistic and E=pc. The integral
is now easier to do. For massive particles (with
decay masses small), Emax = M/2 and so rate goes
as fifth power of mass
p max
2 2
5
(
Q

K
)
p
dp

E
e
e
e
max / 30

0
P461 - nuclear decays
26
Beta decay rate V
• M=M’   is strength of weak interaction. Can
measure from lifetimes of different decays
  1062 joule  m3  100 eV  F 3
• characteristic energy

100eV * F 3

 0.1eV
3
vol
(10F )
• strong energy levels ~ 1 MeV
 weak
 107  relative strength  1014
 strong
• for similar Q, lifetimes are about
 strong  10 s
23
 EM  1016 s
 weak  1010 s
P461 - nuclear decays
27
Parity Violation in Beta Decays
• The Parity operator is the mirror image and is NOT
conserved in Weak decays (is conserved in EM and
strong) P ( x, y, z )  ( x, y, z )
P ( r , ,  )  ( r ,    ,    )
• non-conservation is on the lepton side, not the
nuclear wave function side
• spin 1/2 electrons and neutrinos are (nominally)
either right-handed (spin and momentum in same
direction) or left-handed (opposite)
• Parity changes LH to RH
•


RH
P( p)   p
  

P( L  r  p)  L
LH
P461 - nuclear decays
28
“Handedness” of Neutrinos
• “handedness” is call chirality. If the mass of a
neutrino = 0 then:
• all neutrinos are left-handed
all antineutrinos are right-handed
• Parity is maximally violated
• As the mass of an electron is > 0 can have both LH
and RH. But RH is suppressed for large energy (as
electron speed approaches c)
• fraction RH vs LH can be determined by solving
the Dirac equation which naturally incorporates
spin
P461 - nuclear decays
29
Polarized Beta Decays
• Some nuclei have non-zero spin and can be
polarized by placing in a magnetic field
• magnetic moments of nuclei are small (1/M factor)
and so need low temperature to have a high
polarization (see Eq 14-4 and 14-5)
Co60Ni  e   
60
i5
i  4 s  12 , 12
• Gamow-Teller transition with S(e-nu) = 1
• if Co polarized, look at angular distribution of
electrons. Find preferential hemisphere (down)
Pnu
pe
Spin antinu-RH
Spin e - LH
Co
P461 - nuclear decays
30
Discovery of Parity Violation in
Beta Decay by C.S. Wu et al.
• Test parity conservation by observing a
dependence of a decay rate (or cross section) on
a term that changes sign under the parity
operation. If decay rate or cross section changes
under parity operation, then the parity is not
conserved.
• Parity reverses momenta and positions but not
angular momenta (or spins). Spin is an axial
vector and does not change sign under parity
operation.
180o
Pe
neutron

Beta decay of a neutron in a
real and
mirror worlds:
If parity is conserved, then the
probability of electron
emission at  is equal to that at
180o-.
Selected orientation of neutron
spins - polarisation.
Pe
P461 - nuclear decays
31
Wu’s experiment
• Beta-decay of 60Co to 60Ni*. The
excited 60Ni* decays to the
ground state through two
successive  emissions.
• Nuclei polarised through spin
alignment in a large magnetic
field at 0.01oK. At low
temperature thermal motion does
not destroy the alignment.
Polarisation was transferred from
60Co to 60Ni nuclei. Degree of
polarisation was measured
through the anisotropy of
gamma-rays.
• Beta particles from 60Co decay
were detected by a thin
anthracene crystal (scintillator)
placed above the 60Co source.
Scintillations were transmitted to
the photomultiplier tube (PMT)
on top of the cryostat.
P461 - nuclear decays
32
Wu’s results
• Graphs: top and middle - gamma
anisotropy (difference in counting
rate between two NaI crystals) control of polarisation; bottom - 
asymmetry - counting rate in the
anthracene crystal relative to the
rate without polarisation (after the
set up was warmed up) for two
orientations of magnetic field.
• Similar behaviour of gamma
anisotropy and beta asymmetry.
• Rate was different for the two
magnetic field orientations.
• Asymmetry disappeared when the
crystal was warmed up (the
magnetic field was still present):
connection of beta asymmetry with
spin orientation (not with magnetic
field).
• Beta asymmetry - Parity not
conserved
P461 - nuclear decays
33
Gamma Decays
• If something (beta/alpha decay or a reaction) places
a nucleus in an excited state, it drops to the lowest
energy through gamma emission
• excited states and decays similar to atoms
• conserve angular momentum and parity
• photon has spin =1 and parity = -1
• for orbital P= (-1)L
• first order is electric dipole moment (edm). Easier
to have higher order terms in nuclei than atoms
N*  N 
3  2   
L  0, edm
2  0  
L  1, e.quad .mom.
Pfinal  P PN (1) L  (1)( 1)( 1)  
P461 - nuclear decays
34
Gamma Decays
2
5

17
Cl 38
18
Ar38
26%
E
MeV
gamma
11%
3

2
53%
gamma

0
0
N  N 
*


2 3
GT
3  2   
L  0; edm
2   0  
L  1; eqm

2  2
GT  L  1 ( P change) L  1
2   0
Di  2; GT  L  1
P
conserve angular momentum and parity. lowest order is electric dipole
moment. then quadrapole and magnetic dipole
P461 - nuclear decays
35
Mossbauer Effect
• Gamma decays typically have lifetimes of around
10-10 sec (large range). Gives width:
1015 eVs
  DE   10
 105 eV
 10 sec

• very precise
• if free nuclei decays, need to conserve momentum.
Shifts gamma energy to slightly lower value
A*  A  
p A  p  E 
M A2*  M A2
2M A*
DM
 DM (1 
)
2M
• example. Very small shift but greater than natural
width
DM  .13MeV , M  191* 931.5
 E  .13MeV  .005eV
P461 - nuclear decays
36
Mossbauer Effect II
• Energy shift means an emitted gamma won’t be
reabsorbed
A*  A   E  .13  .000000005MeV
A    A* E  .13  .000000005MeV
• but if nucleus is in a crystal lattic, then entire lattice
recoils against photon. Mass(lattice)infinity and
Egamma=deltaM. Recoiless emission (or
Mossbauer)
• will have “wings” on photon energy due to lattice
vibrations
• Mossbauer effect can be used to study lattice energies. Very
precise. Use as emitter or absorber. Vary energy by moving
source/target (Doppler shift) (use Iron. developed by R.
Preston, NIU)
P461 - nuclear decays
37
Nuclear Reactions, Fission and Fusion
•
•
•
•
2 Body reaction A+BC+D
elastic if C/D=A/B
inelastic if mass(C+D)>mass(A+B)
threshold energy for inelastic (B at rest)
2
2
M 2  Etot
 ptot
 (mC  mD ) 2
mA  mB  mC  mD
 K th  Q
2 mB
K th  Q
Q  DM
m A  mB
(non  relativistic )
mB
• for nuclei nonrelativistic usually OK
p  3H 2H  2H
Q  (1.007825  3.016049  2  2.014102)u  4.03MeV
K th  4(1  13 )  5.38MeV (non  rel )
K th  5.47 MeV (rel )
P461 - nuclear decays
38
Nuclear Reactions (SKIP)
• A+BC+D
• measurement of kinematic quantities allows masses
of final states to be determined
• (p,E) initial A,B known
• 8 unknowns in final state (E,px,py,pz for C+D)
• but E,p conserved. 4 constraints4 unknowns
measure E,p (or mass) of D OR C gives rest
or measure pc and pd gives masses of both
• often easiest to look at angular distribution in C.M.
but can always convert
d
d
CM
P461 - nuclear decays
39
Fission
• AB+C A heavy, B/C medium nuclei
• releases energy as binding energy/nucleon = 8.5
MeV for Fe and 7.3 MeV for Uranium
• spontaneous fission is like alpha decay but with
different mass, radii and Coulomb (Z/2)2 vs 2(Z-2).
Very low rate for U, higher for larger A
• induced fission n+AB+C. The neutron adds its
binding energy (~7 MeV) and can put nuclei in
excited state leading to fission
• even-even U(92,238). Adding n goes to even-odd
and less binding energy (about 1 MeV)
• even-odd U(92,235), U(92,233), Pu(94,239) adding
n goes to even-even and so more binding energy
(about 1 MeV)  2 MeV difference between U235
and U238
• fission in U235 can occur even if slow neutron
P461 - nuclear decays
40
Spontaneous Fission
•
P461 - nuclear decays
41
Induced Fission
•
P461 - nuclear decays
42
Neutron absorption
•
P461 - nuclear decays
43
Fusion
m(1H )  1.007825u
m( H )  2.014102u
m( 4He ) < 4  m(1H )
m( He )  4.002603u
m(12C ) < 3  m( 4He )
2
4
m(8Be )  8.005305u
m(12C )  12.00000u
•
•
•
•
•
“nature” would like to convert lighter elements
into heavier. But:
no free neutrons
need to overcome electromagnetic repulsion 
high temperatures
mass Be > twice mass He. Suppresses fusion into
Carbon
Ideally use Deuterium and Tritium, =1 barn, but
little Tritium in Sun (ideal for fusion reactor)
2
H 3H 4He  n Q  17 MeV
P461 - nuclear decays
44
Fusion in Sun
m(1H )  1.007825u
m( H )  2.014102u
2
m( He )  4.002603u
4
m( Be )  8.005305u
8
p  p  2H  e   
p  2H 3He  
3
He  3He  4He  p  p
m(12C )  12.00000u
•

•
•
•
rate limited by first reaction which has to convert a
p to a n and so is Weak
(pp) ~ 10-15 barn
partially determines lifetime of stars
can model interaction rate using tunneling – very
similar to Alpha decay (also done by Gamow)
tunneling probability increases with Energy
(Temperature) but particle probability decreases
with E (Boltzman). Have most probable (Gamow
Energy). About 15,000,000 K for Sun but Gamow
energy higher (50,000,000??)
P461 - nuclear decays
45
Fusion in Sun II
m(1H )  1.007825u
4
He  4He 8Be
m( 2H )  2.014102u
8
Be  4He 12C  
m( 4He )  4.002603u
m(8Be )  8.005305u
m(12C )  12.00000u
mBe  2mHe  92 KeV
 Be  1012 sec
•
need He nuclei to have energy in order to make
Be. (there is a resonance in the  if have invariant
mass(He-He)=mass(Be))
• if the fusion window peak (the Gamow energy
weighted for different Z,mass) is near that
resonance that will enhance the Be production
• turns out they aren’t quite. But fusion to C start at
about T=100,000,000 K with <kT> about 10 KeV
each He. Gamow energy is higher then this.
P461 - nuclear decays
46
Fusion in Sun III
m(1H )  1.007825u
4
He  4He 8Be
m( 2H )  2.014102u
8
Be  4He 12C  
m( 4He )  4.002603u
m(8Be )  8.005305u
m(12C )  12.00000u
•
mBe  2mHe  92 KeV
 Be  1012 sec
Be+HeC also enhanced if there is a resonance.
Turns out there is one at almost exactly the right
energy --- 7.65 MeV
Dm  0.28MeV
C * 0 11,185.65 MeV
11,185.37  mBe  mHe
12
7.65 MeV
11,185.27  3mHe
2
4.44 MeV
0
11,178 MeV
P461 - nuclear decays
47