Subatomic Physics

302 The Electromagnetic Interaction
Figure 10.9: A few examples of the possible values of angular momentum and parity emitted in a
given transition. The vector diagrams for the transition 1 − → 1 + are shown at the right.
As an example, the electric dipole radiation carries an angular momentum j =1
and, according to Eq. (10.80), a negative parity; it is written as E1. More
generally, an electric (magnetic) radiation with quantum number j is written as
Ej(Mj). [We remind the reader that the quantum number j is defined by Eq. (5.4):
If J is the photon angular momentum operator, j(j +1)� 2 is the eigenvalue
of J 2 .]
The values of j and ηP of the photons emitted in a transition α → β are limited
by the conservation of angular momentum and parity
J α = J β + J, ηP (α) =ηP (β)ηP . (10.81)
A few examples of possible values of j and ηP are given in Fig. 10.9. Note that
initial and final spins are vectors. The various values of the angular momentum
of the emitted radiation are obtained by vector addition, as also shown in
Fig. 10.9.
The selection rules equation (10.81) state which transitions are allowed in a given
decay, but they do not give information about the rate with which they occur. To
find the rate, dynamical computations must be performed. In the previous section,
the transition rate for E1 radiation was found, and Eq. (10.77) expresses this rate
in terms of the matrix element 〈β|x|α〉. Expressions similar to Eq. (10.77) can
be found for all multipole orders Ej and Mj. The real problem then begins: The
relevant matrix elements must be evaluated, and this step requires a knowledge of
the wave functions ψα and ψβ. Finding the correct wave functions for a particular
subatomic system is usually a long and tedious process, and only in a few cases has
it come to a satisfactory conclusion. For an estimate of the transition rate, a crude
model is therefore a necessity; it will provide at least an approximate value with
which observed half-lives can be compared.