introduction-weak-interaction-volume-one

u = e 3 P 'r 1 + r + (J p'r)2 . . . (3 .4 .28 )
e J 2 '- 2 :
It may be shown that the first tern in this expansion corresponds to th e
form of the electron spinor occuring in the matrix elements of decay s
involving no change in orbital angular momentum, the second term to thos e
in which it is altered by one unit, and so on . These transitions are know n
as S-wave ('bt = 0), P-wave (o( = 1), D-wave, F-wave, G-wave, etc .
In allowed transitions, the S-wave term will dominate the decay rate, an d
hence the matrix element will be of order unity. However, in 'first-forbidden '
transitions, the first term vanishes, and hence the major contributio n
comes from the second term of the multipole expansion . When r is aroun d
the nuclear radiu sl , this will be --1 0-1 . Taking into account the fact
that a first-forbidden decay necessitates a change in nuclear configuratio n
and parity, which decreases the matrix element, we find that ou r
rough approximation yields values for ht of the correct order of magnitude .
'Second-forbidden' transitions effectively consist of two consecutiv e
first-forbidden ones, and have corresponding small matrix elements .
3 .5 The Beta Decay of Unpolarized Nuclei .
The matrix element for neutron decay may be calculated directl y
from (3 .3 .6) : (38 )
:E i J up (i) (a) Oi un(+)(9n) (c i ue (+) (c,) O i X
x uv(-)(_1v) + C . ue(+)(1e) Oi f'5 uv(-)(_a, ) ) X
x exp (j x (gn - qp - q e - qv) , (3 .5 .1 )
assuming that all particles involved in the interaction may be describe d
by plane waves (i .e . they are nonrelativietic) . However, in true nuclear
beta decay, the decaying nucleon is bound within a nucleus, and henc e
it may not be described by a simple plane *save solution of the Dira c
equation . Furthermore, the nucleus will usually contain many nucleons ,
any of which may undergo beta decay. The best approximation (38) i s
probably obtained by Fourier analysing the plane wave nucleon spinor ,
1 . Since this is the domain of the integral appearing in the matrix element .