Atomic Structure Theory

176 6 Radiative Transitions
2zpxɛx = Ly + im
¯h
Similarly, we find
2zpyɛy = −Lx + im
¯h
[h, zx] ɛx.
[h, zy] ɛy.
These terms can be recombined in vector form to give
2 ˆ k · rp· ˆɛ = L · [ ˆ k × ˆɛ]+ im
ˆkiˆɛj [h, Qij] , (6.89)
3¯h
where Qij =3xixj − r 2 δij is the quadrupole-moment operator. Using the
identity (6.89), we find
T (1)
ba
= ik
c 〈b|M|a〉·[ˆ k × ˆɛ] − kωba
6c
ij
〈b|Qij|a〉 ˆ kiˆɛj , (6.90)
where M is (up to a factor e) the magnetic moment operator
M = 1
[L +2S] , (6.91)
2m
with S = 1
2σ. As in the definition of the electric-dipole moment, we have
factored the electric charge e in our definition of the magnetic-dipole moment.
The magnetic moment divided by c has the dimension of a length. Indeed,
¯h/mc = αa0 is the electron Compton wavelength.
The first of the contributions in (6.90) is referred to as the magneticdipole
amplitude and the second as the electric quadrupole amplitude. As we
will prove later, in the general discussion of multipole radiation, these two
amplitudes contribute to the decay rate incoherently. That is to say, we may
square each amplitude independently, sum over the photon polarization, and
integrate over photon angles to determine the corresponding contribution to
the transition rate, without concern for possible interference terms.
Magnetic Dipole
Let us consider first the spontaneous magnetic-dipole decay
dΩk |〈b|M|a〉·[ ˆ k × ˆɛλ]| 2 .
w sp α k2
a→b = ω
2π c2 λ
The sum over photon polarization states can easily be carried out to give
|〈b|M|a〉·[
λ
ˆ k × ˆɛλ]| 2 = |〈b|M|a〉| 2 sin 2 θ,
ij