This simplified version ofgJis off by only about 0.1% from the expression in
equation 16.12. The selection rules in terms ofJ,L,ML, and Sin equation 16.8
above apply. The Landé gfactor is named after Alfred Landé, a German scien-
tist who in 1923 (before the development of quantum mechanics) derived
equation 16.13 from a visual inspection of a lot of atomic spectra.
Figure 16.7 shows a simple electronic spectrum with and without a magnetic
field as an example of the anomalous Zeeman effect. Notice that the splitting of
the lines when the sample is exposed to a magnetic field isn’t as simple as with
the normal Zeeman effect. This is one reason it was considered anomalous.16.4 Electron Spin Resonance
Normally, molecules composed of main-group elements have a ground state
where all of the electrons are spin-paired.* Compounds having all-spin-paired
electrons show no noteworthy magnetic effects in the electronic spectrum due
to spin magnetic dipoles.
In molecules that have unpaired electrons, such as d-block and f-block com-
pounds as well as free radical species, there is a net spin magnetic dipole. In the
presence of a magnetic field, the presence of a spin magnetic dipole creates a
potential energy of interaction, given by equation 16.11. When a single electron
is involved (as is usually the case for main-group free radicals), the total spin
vector Sis simply ^12 , and the potential energy of interaction can be rewritten in
terms of the msquantum number, which is either ^12 or ^12 . In terms ofms,
EmaggeBmsB (16.14)
where the gfactor for a free electron,ge, is used instead ofgJ, and mscan be
either ^12 or ^12 . This implies a relatively simple two-level system where the
ms ^12 state goes up in energy and the ms^12 state goes down in energy.
Such a system is illustrated in Figure 16.8. The difference between the two
energy levels is equal to geBB.
What is the difference between equations 16.14 and 16.11? In equation
16.14, we are considering the effect on one electron, whereas equation 16.11 is
a more general case with more than one electron. For a single electron, the
magnetic field effects are determined by ms, whereas for multiple electrons the
effects are better described by Jand MJ. Therefore, for multiple electrons gJand
MJare the appropriate variables, and for a single electron geand msare the rel-
evant variables.
The splitting of the two spin states for the unpaired electron is not a lot in
terms of energy: using equation 16.14, one can see that the splitting between
the two is equal to 1.855 10 ^23 J/tesla, or about 0.934 cm^1 per tesla. If one
wanted to irradiate a sample having an unpaired electron and exposed to a
magnetic field, then radiation of the proper wavelength would cause an elec-
tron lying in a lower,ms^12 state to absorb radiation and move to the up-
per,ms ^12 state. When the magnetic-interaction-induced difference in en-
ergy of the two states equals the energy of the photon, absorption of a photon
can occur and we say that a state ofresonanceis established. The relevant equa-
tion is
̃resge
hcBB (16.15)
16.4 Electron Spin Resonance 5671
MJ 2
1
MJ 2B 0Spectrum:Energy(^2) P1/2
(^2) S1/2
1
MJ 2
1
MJ 2
B 0
Spectrum:
Energy
(^2) P1/2
(^2) S1/2
Figure 16.7 When JL(that is, when S0),
the splitting due to a magnetic field is more com-
plicated and is called the anomalous Zeeman ef-
fect. The figure shows a transition in the absence
and in the presence of a magnetic field.
*NO and NO 2 are two of the rare exceptions to this statement, each having an odd num-
ber of electrons. The ground state of O 2 has two unpaired electrons in degenerate molecu-
lar orbitals, as explained in Chapter 12.