Appendix C: Magnetic

dipole transitions

C

The electric dipole transitions that lead to the emission and absorption
of light by atoms are discussed in many places in this book, and the
selection rules for this type of transition are summarised in Table 5.1.
These rules are not obeyed in radio-frequency transitions between Zee-
man sub-levels where only the magnetic quantum number changes, or
in transitions between different hyperfine levels whereF andMF can
change—these transitions in radio-frequency spectroscopy are magnetic
dipole or M1 transitions induced by the oscillating magnetic field of the
radiation^11 An important exception is the Lamb
shift transition, where two levels of op-
posite parity have a very small energy
separation (see Section 2.3.4). This is
an electric dipole or E1 transition.

Brf=B 0 cosωt. (C.1)

The transition matrix element between hyperfine levels is

μ 21 ∝〈 2 |μ·Brf| 1 〉, (C.2)

whereμis the magnetic dipole operator in eqn 5.9. This gives transitions
for which ∆l=∆L=∆S=0.
The selection rules for magnetic dipole transitions between hyperfine
states are:

∆F=0,±1(butnot0→0) ,

∆MF=0,± 1.

These are as expected from angular momentum conservation and a
dipole operator that can change angular momentum by one unit (as
discussed for electric dipole transitions in Section 2.3.5).
The spontaneous decay rate for radio-frequency transitions is propor-
tional to
A 21 ∝ω^3 |μ 21 |^2 , (C.3)

whereωis the angular frequency, which is small relative to optical tran-
sitions. The matrix elementμ 12 is also much smaller than that for E1
transitions:
|〈 2 |μ·B| 1 〉|^2
|〈 3 |er·E| 1 〉|^2

∼

(

μB/c

ea 0 /Z

) 2

∼(Zα)^2 , (C.4)

where the factorcarises because the ratio of the magnetic to the electric
field,|B|/|E|=1/cin an electromagnetic wave; the atomic size scales
as 1/Z, see Section 1.9. Therefore spontaneous emission is negligible in
radio-frequency and microwave spectroscopy. In outer space, however,