Gamma and Beta DecayBasics
[Secs: 9.1 to 9.4, 10.1to 10.3 Dunlap]
Gamma and Beta decays are similar
W W
0Z
A Family of Four Force Carriers
β decay
γ decay
Unlike α decay, β and γ decays are closely related (e.g. like cousins).
• They often occur together as in the typical decay scheme (i.e. 198Au)
• They just involved changes in nucleon states (p n, n p, p p)
• They involve the same basic force (γ, W± ) carrier but in different state
• But β decays are generally much slower (~100,000) than γ decays (produced by EM force) because the Ws are heavy particles (which makes force weaker)
N
N
Gamma and Beta decays are very similarGamma and Beta decays are very similar
i f
i atom f free
atom e
i f
e
e
p p
p e p e
p e n
p p e e
p n e
n p e
Gamma Decay
Internal Conversion
Electron Capture
Pair Internal Conversion
β+ Decay
β- Decay
EM
EM
weak
EM
weak
weak
Nucleon + One Lepton
Nucleon + Two Leptons
Nucleon + Zero Leptons
Decay Name of process Interaction Out Channel
p
n
BETA PLUS DECAYBETA MINUS DECAY
n
p
PAIR INTERNAL CONVERSION
p’
p
Feynman Diagrams - SimilarityFeynman Diagrams - Similarity
e
ee
e
TIME
ee
All these decay types are similar in structure
They all have a 4 point vertex
They all have 3 particles in the final state
The fact that the Q of the decay is shared between 3 particles means that the outgoing observed particle [ie. electron or positron] has a spectrum of energies in the range (0 to Q).
OUT CHANNEL ---- One nucleon + 2 leptons
n
p
INTERNAL CONVERSION ELECTRON CAPTURE
[Mono-energetic electrons ] [Mono-energetic neutrinos]
Feynman Diagrams - Similarity
TIME
p
GAMMA DECAY
p’
p
[Mono-energetic photons ]
p’
eK eK
e e
All these decays have only two particles in their output state.
The Q of the decay is shared between only 2 particles
Conservation of Energy: The emitted particle (γ , e-, νe) is monoenergetic.
OUT CHANNEL ---- One nucleon + 1 lepton
BETA PLUS DECAYBETA MINUS DECAY PAIR INTERNAL CONVERSION
Quark level Feynman Diagrams - SimilarityQuark level Feynman Diagrams - Similarity
e
e
ee
TIME
ee
-W
d u d
d u u
d u u
d u d
+W
d u u
d u u
The proton is made of 3 quarks – uud (up, up, down)
The neutron is made also of 3 quarks - udd (up, down, down)
We see the very close similarity of pattern between reactions through W and γ particles. NOTE: only vertices of 3 particles are now seen (makes sense)
INTERNAL CONVERSION ELECTRON CAPTURE
Quark level Feynman Diagrams - SimilarityQuark level Feynman Diagrams - Similarity
TIME
GAMMA DECAY
eK eK
e e
d u d
d u u
d u u +Wγ
d u u
d u u
d u u
Again we see that there are ONLY 3 PARTICLE – VERTICES
We see the similarity of the decays are propogated through the intermedicate “Force” particles (W and γ).
Remember in INTERNAL CONV. And ELECTRON CAPTURE the electron comes from the core electron orbitals of THE ATOM.
Beta and Gamma similarities
PAIR INTERNAL CONVERSIONBETA PLUS DECAY
Note how similar the spectral shapes are for positron emission even though the BETA PLUS is via the WEAK force, while PAIR INTERNAL is via the EM force. This is because in the final state there are 3 PARTICLES (Daughter nucleus + 2 Leptons).
64 64Cu Ni ee 24 24Na Mg e e
Beta and Gamma similarities
GAMMA DECAY INTERNAL CONVERSION
60 601 2Ni Ni 0214 214Po PoE e
1.17MeV
1.33MeV
4+
2+
0+
0+
0+
1.42 MeV
GAMMA decay and INTERNAL CONVERSION decay both show discrete lines – WHY because these are 2 body decays. What about data for ELECTRON CAPTURE – well that would require looking at the energy spectrum of emitted neutrinos – something not yet achieved (Why?).
Gamma and Beta decays are similarGamma and Beta decays are similar
W W
0Z
β
γ
Because the force carriers all have Jπ =1- the basic decays are similar in terms of angular momentum:.. NO MORE THAN ONE UNIT OF ANG. MOMENTUM.
J =1
J =1
RIGHT HANDED PHOTON
LEFT HANDED PHOTON
FAMILY FAMILY CHARACTERISTICCHARACTERISTIC
J =1 J = 0
j=1/2
j=1/2
j=1/2
j=1/2
GAMOW-TELLER transition
FERMI transition
LEFT LEFT HANDED HANDED ELECTRONELECTRON
RIGHT RIGHT HANDED HANDED ANTI - ANTI - NEUTRINONEUTRINO
e e
e
e
Jπ = 1-
Gamma and Beta decays are similarGamma and Beta decays are similarJ
yes no no yes yes no no yes
±1
±2
±3
±4
Electric Dipole (E1) RadiationElectric Dipole (E1) Radiation
L=1
Magnetic Dipole (M1) RadiationMagnetic Dipole (M1) RadiationBut an oscillating magnetic dipole gives exactly the same radiation pattern.
Both E1 and M1 radiations have L=1 which means that this sort of radiation carries with it ONE unit of angular momentum. The distribution on the right is the probability of photons being emitted.
1st Forbidden Transitions are L=1 and have this same emission pattern.
L=1
Electric Quadrupole (E2) Radiation
L=2
Magnetic Quadrupole (M2) Radiation
L=2
Both E2 and M2 are L=2 radiations – i.e. the photons carry away with them 2 units of angular momentum. The diagram on the right shows the directional probability distribution of photons. This same distribution applies to 2nd forbidden transitions. How is L= 2 possible? Doesn’t the photon only have only one unit of angular momentum? In Beta decay isn’t J=1 for the lepton pair the maximum?
EMISSION FROM HIGHER MULTIPOLES IS DIFFICULT BUT EMISSION FROM HIGHER MULTIPOLES IS DIFFICULT BUT NOT IMPOSSIBLE. QUANTUM MECHANICS ALLOWS IT.NOT IMPOSSIBLE. QUANTUM MECHANICS ALLOWS IT.
222)(
c
ERR
pkR
(8)where p and E are the momentum and energy of the outgoing lepton pair wave. Putting E=1MeV (typical decay energy) and R=5F (typical nuclear radius) we get
2 22 4( ) ~ 5x10
p ERkR R
c
R
Imagine that the nucleus wants to get rid of 1 unit of ang. mom. by emitting a photon from its surface. The maximum ang. mom that can be transferred is:
Ep
c
. .ER ER
J p Rc c
1 .5~ 0.025
197 .
E R MeV F
c MeV F
i.e. classically one could only get 1/40 of an ang.mom unit. Quantum mechanics allows tunneling to larger distances – but the transition rates are reduced by factor:
22 fdnf H i
dE
The FERMI GOLDEN RULEBeta and Gamma decay are also alike in that the transition (decay) rate for their transitions can be calculated by the FERMI GOLDEN RULE. You can read about its derivation from the Time Dependent Schrodinger equation in most QM books
It gives the rate of any decay process. It finds easy application to BETA and GAMMA decay. Fermi developed it and used it in 1934 in his THEORY OF BETA DECAY that is still the basic theory used today.
It looks as follows:
Enrico Fermi (1901-1954)
= the decay rate:
f H i = the “matrix element” or the overlap between initial state and final state via the Force interaction H
= the “density of states”. The more available final states the faster the decay will go.
fdn
dE
THEORY OF GAMMA DECAY
3
. 3
ˆ* *
2ˆ = . .( )( ). * ( . )
EM if D EM P
ik rD P
f H i H H d r
c e d rV
ε r
(i) MATRIX ELEMENT This is difficult to obtain without a full knowledge of quantum field theory but we can quote the result.
Parent wavefunction
Photon wavefunction
Daughter wavefunction
(ii) DENSITY OF STATES. The density in phase space is a constant so that:
photon momentum space
p
dkkV
kdV
dN 23
33
4.)2()2(
3
2
32
3 )2()2(
dppV
dkkV
d
dN
Where V is the volume of a hypothetical box containing the nucleus. Because emission is directional we must work per unit solid angle:
THEORY OF GAMMA DECAYNow lets write down the no of states per solid angle and per unit energy (remember E=pc, or
photon momentum space
p
2 22
3 3 3 3. (2 ) (2 ) ( )
p dp Ed N V V
d dE dE c
2
22
3 3
3
2
2
( )2 2 . .( )( ) .
(2 ) ( )
1 d . .
(2 )
f
if
if
dnd f H i
dE
EVc M d
V c
Ec M d
c
(iii) APPLY THE FERMI GOLDEN RULE
rdeM Pi
Dif3. ).ˆ( rεrk
Where:
This is the PARTIAL DECAY RATE: I.E. the decay rate into an solid angle dΩ in a certain direction as determined by Mif
1dp
dE c
THEORY OF GAMMA DECAY(iv) INTEGRATE OVER ANGLE TO GET THE TOTAL DECAY RATE
3
21 . . .
(2 ) if
Ec M d
c
2cos dependence3
2if
1. . D
3
Ec
c
2 .sin d
3Dif D Pz d r Nuclear dipole matrix element
More generally this result can be extended to higher angular momentum (L) radiation components:
2 12
2
2( 1)( ) . . ( )
(2 1)!!
L
if
ELEL c Q L
cL L
is the matrix element of the multipole operator Q(L).
3( ) ( )if f iQ L Q L d r
Eqn. 10.16
- because the decay rate varies with angle
The Weisskopf EstimatesThe lines show the half-life against gamma decay 1/ 2
0.692
For what are known as the Weisskopf Estimates named after Victor Weisskopf who showed using the single particle shell model (1963) that (Eqns 10:20, 10 :22):
23
3L
ifQ RL
2 122 2 /302
2/3 2
2( 1) 3( ) . .
[(2 1)!!] 3
( ) 0.308 . ( ) 10 . ( )
L
L LELEL c R A
L L L c
ML A EL EL
Giving:
One can work out transition speeds approximately by scaling the result that
5 201/ 2 ( , 100, 1 ) 10 LEL A E MeV s
and noting that the rate scales as
and as
2 /3LA2 1LE
- speeding up with both size and energy.
THEORY OF BETA DECAY(i) MATRIX ELEMENT As with EM decay we write:
3ˆ* * *eweak if D e Weak Pf H i H H d r
3* * *.eD e PG d r
This follows because the 4 point vertex is close to being a “point” interaction. G is a constant representing the strength of the weak interaction. The outgoing electron and neutrino waves into imaginary volume V are written
. /1( ) ip r
e r eV
and
. /1( ) iq rr e
V
( ).3*. .
i p q rif
if D P
GMGH e d r
V V
where ( ).
3 3*. . * .i p q r
if D P D PM e d r d r
and: 5 38.8 10 . decay coupling constantG x MeV F
D
-e
e
THEORY OF BETA DECAY(ii) DENSITY OF STATES. The density is more complex in beta decay because there are both the electron and neutrino momentum space to consider:
Electron mom. space Anti-neutrino mom. .space
23 3 2 2
3 3 4 6
V. .
(2 ) (2 ) 2f e e
V Vdn dN dN d k d k p q dpdq
The number of final states is a product of both phase space probabilities.
22 2
4 62f
f f
dn V dqp q dp
dE dE
22 2
4 6 3.( )
4f
ef
dn VQ T p dp
dE c
Note that: eT qc Q
i.e. that eQ Tq
c
So that with Te fixed 1
f
dq
dE c
THEORY OF BETA DECAY(iii) APPLY THE FERMI GOLDEN RULE
2
2 2 22 2
2 4 6 3
2 22 2
3 7 3
2
2 . . .( )
4
G ( )
2
f
ife
ife
dnd f H i
dE
G M VQ T p dp
V c
MQ T p dp
c
Which explains the rise here which goes as p2
And explains the drop here which goes as (Q-Te)2
THE BETA SPECTRUM SHAPE
Electron mom. p
N(p) ~ dλ
The Fermi-Kurie Plot
Te (keV)
The Fermi-Kurie plot is very famous in β decay. It relies upon the shape of the β spectrum – i.e. that
F(Z,Te) is the “Fermi factor “ that deals with the distortion of the wavefunctions due to repulsion and attraction of the e+ or e- on leaving the nucleus.
2 2( ) Const. ( ) ( , )e eN p p Q T F Z T
1/ 2
2
( )
( , )e
N p
p F Z T
3 3 -H He + e e
18.62Q keV
THEORY OF BETA DECAY(iv) INTEGRATE OVER ENERGY TO GET THE TOTAL DECAY RATEUnlike gamma decay where we needed to integrate over angle – in BETA DECAY we have to INTEGRATE OVER ENERGY in order to get the total decay rate. [For L=0 the there is no directional emission pattern]
max2 2 5
3 70
( )( ) . .f( , )
2 ( )
peG m c
N p dp c Z Qc
where f(Z,Q) is the Fermi integral
max 2 22 5 0
1f ( , ) ( , )( ) ( ) ( )
( )
p
D D e ee
Z Q F Z T pc Q T d pcm c
which is a very complex integral – needing calculation by computer. The values can be found in nuclear tables. But there is one solution if then -
2 2 2 2( ) ( )e e cT pc m c m c
2eQ m c
5
2
1f ( , )
30De
QZ Q
m c
22
53 7
. .60 ( )
ifG Mc Q
c
Sargent’s Law
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