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