1. 16.451 Lecture 18: Total Rate for Beta Decay (etc...) 6/11/2003
Our formalism determines if, which is the rate (s-1) to a particular final state electron
(or positron) momentum p:
electron
neutrino
recoil
( refers to  decay modes)
The total decay rate is obtained by integrating if over all allowed e momenta p:
“Fermi integral”, f (Z’,Q)
Key point: by integrating over the momentum distribution, which is specific only to
the kinematics of the case under consideration, we can relate rates for various
decays that differ only in the nuclear matrix element ...

2. undistorted spectrum – i.e. no Coulomb effects.2
Krane, Figure 9.8: Dimensionless Fermi integral
By convention:
log10 f(Z’,Eo)
Note: Z’ = 0 gives
the “phase space”
integral for the
undistorted spectrum –
i.e. no Coulomb effects.

3. The Fermi integral is not symmetric for + and - decay !3
(+ get shifted to higher
momentum, and the
integral is momentum –
weighted, so this accounts
for the difference) Z’ - +
(Points read from previous graph at Q = 1 MeV )

4. Comparative half lives:4
By convention, the half-life, is used as a comparison standard for
different nuclear beta decays:
Using our formalism and , we can write:
Notice: The only difference in the “ft” value between different nuclear beta
decays is the value of the nuclear matrix element.
If |Mnucl|2 = 1 (“superallowed” case in nuclei), the ft values can
be used to determine the weak coupling constants G = (GV, GA)
Special case: “superallowed” decays in nuclei with initial and final nuclear states
0+  0+, e.g.
must have S = 0 for the leptons  pure Fermi decay...

5. Superallowed Fermi 0+  0+ decays: world’s best data for light nuclei5
all have the same ft value ~ sec
determines the weak coupling constant for Fermi decays:
(And GA/GV = -1.25, more later....)

6. Can we understand beta decay rates in general?6
first page of Krane, Appendix C: (symbol  stands for electron capture/ + decay)
 27 isotopes: 8 - decays, 6 + decays, spanning 16 orders of magnitude in rate!
fastest
slowest

7. 1. According to our theory, the very slow decay: (1.6 x 106 yrs)
Some anomalies: 7
1. According to our theory, the very slow decay: (1.6 x 106 yrs)
should not occur at all, because angular momentum does not add up, i.e.:
2. Another example: (16.1 hr)
This should not occur because the wavefunctions in the nuclear matrix element
have opposite parity, so the integrand is odd and should vanish:

8. 8
Forbidden Decays:
These are two examples of forbidden decays – they cannot proceed under the allowed
approximation, ie.
electron
neutrino
recoil
Reconsider the electron and antineutrino wave function as a multipole expansion:

9. Effect on beta transitions:9
Effect on beta transitions:
angular momentum coupling for the multipole order L, together with S and nuclear
angular momentum allows previously impossible reactions to proceed
multipole term has parity (-1)L, which allows nuclear states of opposite parity to
be “connected” by the beta decay operator
momentum dependence of the matrix element varies as
since this is small, the lowest multipole order L that satisfies the conservation
laws will dominate the transition
rate ~ |M|2 ~ (pR r/ħ) 2L  (0.01) 2L  dramatically smaller for large L
momentum dependence also affects the shape of the spectrum; Kurie plots
are not linear unless “shape factors” are taken into account....
naming convention:

10. Classification: all known decays....10
variation within a
class is due to the
nuclear matrix element

11. Classification of decays: general11
Nuclear case: -
electron
neutrino
recoil
Conservation laws:
with S = 0 (Fermi) or S = 1 (Gamow-Teller)
Smallest value of L that is consistent with conservation laws will dominate the transition.

12. Example: + decay of 18Ne 12
92% 8%
0.002% 1 2 3
Branching ratio (BR): the fraction of decays that go to a particular final state.
In this case, total = 1/ = sec-1 ;  = 1 + 2 + 3, with i = BR(i) total
Transition 1: 0+  This is a first forbidden GT decay, with the slowest partial rate:
Transition 2: 0+  This is an allowed Fermi decay:
Transition 3: 0+ 1+ This is an allowed Gamow-Teller decay

13. ??? Other weak interaction decays – perhaps a surprise....13
We have, so far two coupling constants for (nuclear) beta decay: GV (S=0 decays)
and GA (S=1). These set the overall scale of the interaction, with GV determined from
the transition rates for “superallowed” 0+  0+ nuclear decays, and GA from Gamow-
Teller decays (0+  1+ and vice versa).
Other related processes:
1. muon decay:
a “purely leptonic” weak decay -- no quarks before or after!
no change of electric charge; must be “mediated” by the neutral Z0 boson
no “Fermi function” needed, since no Coulomb effects in the final state.
analogous to neutron decay, so we can try the same formalism, assuming
weak interactions for quarks and leptons are the same W- d u
neutron Z0
muon
???

14. Muon lifetime implications:14
Muon lifetime implications:
measured lifetime:  =  s
theoretical prediction:
Muon decay gives a weak coupling constant G that is about 2.5% larger than in nuclear
beta decays.... (assuming GV = GA = G here and comparing results), or alternatively
the coupling constant for the d  u quark weak transition is about 2.5% smaller
than that for the   e lepton weak transition.
2. Pion decay:
Another d  u quark transition; rate is consistent with the same coupling constants
as nuclear beta decay
3. K meson decay:
This is an s  u quark transition; rate is much smaller than the equivalent d  u rate;
coupling constants are reduced by about 20% compared to nuclear beta decay....

15. Weak interactions of quarks:15
Weak interactions of quarks:
There are hundreds of examples of weak decays in nuclear and particle physics.
Purely leptonic rates are all consistent with a single weak coupling constant G
Hadronic rates, involving quark transitions, occur at a comparable scale but with
consistent differences that depend on the type of quarks involved.
A simple pattern emerges if we assume that the quarks that participate in weak
interactions are linear combinations of the strong interaction eigenstates,
represented by a unitary matrix called the CKM (Cabbibo-Kobayashi-Maskawa)
matrix:
weak eigenstates
strong eigenstates
Unitary matrix, like a rotation
matrix – preserves “length”

16. “mapped out” the CKM matrix as follows: (Particle Data Group , 2002)
How does this work? 16
Instead of a d  u transition in neutron beta decay, only the contribution from the
weak eigenstate d’ plays a role, and the weak coupling constant is effectively reduced
by a factor Vud =
Similarly, instead of an s  u transition in kaon decay, we have an s’  u transition,
effectively reducing the weak coupling constant by a factor Vus =
Studies of a large number of particle decays and beta transitions have effectively
“mapped out” the CKM matrix as follows: (Particle Data Group , 2002)
2 error
limits
Diagonal terms dominate the CKM matrix
All “large” terms are real; small imaginary component in lower right 2x2 submatrix
allows for time reversal, or alternatively “CP violation” -- a hot research topic!

17. CKM Unitarity test: are there more than three generations of quarks?17
CKM Unitarity test: are there more than three generations of quarks?
(the best-tested row of Vij)
world data, 2001:
2 discrepancy:
contentious issue...
H. Abele, NIM A 440
(2000), p. 499
 = |GA/GV|
 asymmetry: the
correlation
Answer soon, perhaps!