Physical Chemistry , 1st ed.

Example 16.6

What magnetic field is necessary for an ms^12 →ms^12 transition to

be in resonance with microwave radiation having a wavelength of 11.8 cm?

(This is the approximate wavelength of the microwaves that are used in some

microwave ovens.)

Solution

A of 11.8 cm implies a wavenumber  ̃ of 1/11.8 cm 0.0847 cm^1. Using

equation 16.15, we get

0.0847 cm^1 

where we have used the value for the speed of light in units of cm/s and the

magnetic field strength,B, is the only unknown. Solving for B:

B0.0906 T 906 G

This is a relatively weak magnetic field, although it is more than 1000 times

Earth’s magnetic field.

ESR spectra are usually collected by radiating a sample with monochro-

matic microwave radiation and then varying the magnetic field. It is easier to

do this than to hold the magnetic field constant and vary the frequency of the

microwave radiation. In either case, a resonance condition can be established.

Many (but not all) ESR spectra are a conglomeration of closely spaced, unre-

solved lines; see Figure 16.10a. To emphasize the exact positions of the differ-

ent absorptions, it is conventional to plot an ESR spectrum as a derivativeof

the absorption with respect to the varying magnetic field. In this way, the dif-

ferent absorptions are enhanced. Figure 16.10b shows the derivative spectrum

of the absorption spectrum in Figure 16.10a. It is much easier to interpret a

spectrum like Figure 16.10b than one like Figure 16.10a (although they con-

tain the same information).

The above discussion may lead one to think that all electrons absorb the

same microwave radiation at a particular magnetic field. If this were so, then

ESR spectroscopy would have limited use. However, such is not the case.

The exact value of the gfactor gedepends strongly on the local environment

of the unpaired electron. This means that the exact frequency of resonant

absorption depends on the specific molecule of interest. In particular, because

nuclei themselves also have a spin, there is an interaction, or a coupling,be-

tween the unpaired electron’s spin and the spin angular momentum of the in-

dividual nucleus, which is labeled I.

In a molecule having several nuclei with nonzero spin, the nuclear spins can

couple to give a total molecular nuclear spin MI,molgiven by

MI,mol

0

nuclei

MI

where MIis the spin of an individual nucleus. The molecule has 2MI,mol 1

possible orientations of the total nuclear spin in the zdimension,MI,z.Each

orientation couples differently with an unpaired electron. This type of cou-

pling is called hyperfine coupling.Because nuclear spin states are quantized (as

with any angular momentum), the interaction energies are also quantized and

(2.002)(9.274 ^10 ^24 J/T)B

(6.626  10 ^34 Js)(2.9979  1010 cm/s)

16.4 Electron Spin Resonance 569

(a)

Absorption

Magnetic field

(b)

Derivative of absorption

Magnetic field

Figure 16.10 (a) ESR spectra plotted as an
absorption spectrum show multiple, unresolved
absorptions. Such spectra are difficult to inter-
pret. (b) ESR spectra plotted as derivative spectra
are easier to interpret, because the individual
peaks are more easily resolved.