Electric and magnetic moments

In Section 6.2.1 we considered a general sinusoidal perturbation of the Hamiltonian of a particle introduced by interaction with an electromagnetic wave. To be more specific about spectral transitions we must consider the detailed interaction of an atom or molecule with an electromagnetic wave.

An electromagnetic wave has an electric field whose strength varies with time and position as E = Eq cos(ut — k-r), where u is the angular frequency and k is the wavevector (|k| = 2^/A). The wave also has an associated magnetic field B = (Eq/c) cos(ut — k-r) and thus the total force acting on each electron and nucleus in the molecule is given by F; = qt (E + v; x B), where qt is the electric charge of the i th particle and is its velocity. The energy of interaction between the wave and the molecule is then defined as u(r, t) = ^ i F; -ri.

Since the electric field strength is greater than the magnetic field strength by a factor of c, and since particle velocities are in general small compared with c, the electric field terms in the perturbation potential tend to completely dominate the magnetic ones. However, since the electric field varies with position, there are different ways in which the electric interaction can take place. The electric field of the wave may be expanded as the following power series

E = E0 cos(ut - k-r) = E0[cos ut + (k-r) sin ut - 2 (k-r)2 cos ut H----] (6.10)

and the molecule may be considered to interact with different terms independently. Interaction with the first term, where the amplitude of the field is constant over the molecule, leads to electric dipole transitions. Interaction with the second, much weaker term leads to electric quadrupole transitions, interaction with the third even weaker term leads to electric octopole transitions, and so on. Although small, magnetic field interactions may be similarly resolved as magnetic dipole transitions, magnetic quadrupole transitions, etc. For most molecules, electric dipole transitions completely dominate everything else and thus it is these transitions that are usually considered. However, some molecules such as H2 do not have a dipole moment and thus may not engage in electric dipole transitions although they do have an electric quadrupole moment and hence may engage in electric quadrupole transitions. Such transitions are clearly observable in giant planet spectra. Magnetic transitions are very weak and difficult to observe in giant planet spectra and may be neglected except for the effect of magnetic dipole 02 absorption in Earth's atmosphere discussed in Section 7.3.1.