Page 127 - Academic Press Encyclopedia of Physical Science and Technology 3rd Analytical Chemistry

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Encyclopedia of Physical Science and Technology EN005E-212 June 15, 2001 20:32

338 Electron Spin Resonance

H
where Q is a constant with a value of about −25 G. Equa- The difference between the allowed transitions shown
tion (10) applies to protons bonded to any atom, except in Fig. 8 is not changed from the first-order situation at
H
that Q is dependent on the atom. It has been most exten- constant magnetic field. However, in an actual ESR spec-
sively tested for protons bonded to carbon and to a lesser trometer the magnetic field is swept, and since the second-
extent for protons bonded to nitrogen. order correction term depends on the magnetic field, the
Although proton hyperfine interactions are by far the “apparent” splitting will change in second order. This ob-
most common in aromatic radicals, a great deal of in- served apparent splitting will be larger than the actual hy-
formation about spin densities can also be derived from perfine splitting, and for the case of hydrogen atoms this
13
14
17
19
hyperfine interactions with C, N, F, O, and so forth. difference is about 2 G.
The simple relation of Eq. (10) for protons does not gen- The second-order effect on the hydrogen-atom spec-
erally apply to these nuclei because the interactions of the trum is a subtle one, in the sense that the nominal appear-
unpaired spin with spin density on adjacent nuclei and ance of the spectrum does not change. Generally in more
with lone-pair π electrons must be considered. complex paramagnetic systems additional lines in the ESR
spectrum appear that are due to second-order effects in the
energy-level diagram. These types of transitions must be
V. SECOND-ORDER HYPERFINE EFFECTS identified in order to interpret a spectrum well enough to
assign the structure of a radical. The “extra” lines that
When the hyperfine coupling constants are large and the commonly occur due to second-order hyperfine effects
linewidth is small, second-order hyperfine effects must be can be divided into two classes. One class arises from the
considered to explain the observed spectra in many cases. splitting of some of the degeneracies of the inner lines
A simple example is the spectrum of a hydrogen atom. of a hyperfine pattern involving several equivalent nuclei.
The second-order effects in the hydrogen spectrum cause The second class involves forbidden transitions where the
a shift of the center of the spectrum to lower field and nuclear spin selection rule is violated and transitions cor-
cause the observed hyperfine splitting to be larger than responding to
m I =±1 are observed.
the actual value for a field-swept spectrum. Second-order In general, extra spectral lines due to the splitting of
analysis is required because the hyperfine splitting of 507 the degeneracies of equivalent nuclei occur when the
G is a significant fraction of the typical 3300-G magnetic second-order hyperfine correction magnitude is greater
field used to observe the spectrum. The second-order per- than the linewidth. This has been observed only in liquid-
turbation theory correction factor to the energy levels is phase spectra. To second order, the transition energies for
proportional to the square of the hyperfine coupling con- a system with one unpaired electron and one type of de-
stant divided by the observing field. This causes a second- generate nuclei of nuclear spin I is given by
order shift in two of the four hydrogen-atom energy levels, 2
2
as shown in Fig. 8. The two transition energies are in- E n = gβH + AM I + 1 A I(I + 1) − M , (11)

I
creased, so the entire spectrum is shifted to lower field due 2 gβH
to this effect. For hydrogen atoms the shift is about 18 G, where M I is the total nuclear spin quantum number for the
which corresponds to a change in the apparent g factor of set of equivalent nuclei. For n equivalent nuclei in general
2
2
0.0108. I = M , so from Eq. (11) one can see that some splitting
I
ofthefirst-orderdegeneracywillresult.Anexampleofthis
situation occurs for the trifluoromethyl radical in solution,
where the first-order spectrum would predict four equally
spaced lines with relative intensities of 1 : 3 : 3 : 1. Since
A iso for 19 F is 145 G, which is a large value, second-
ordereffects are observable. In the actual spectrumat suffi-
ciently high resolution six lines are observed with relative
intensities 1 : (1 : 2) : (1 : 2) : 1, where the (1 : 2) lines are
two closely spaced lines whose splitting is due to second-
order shifts. All lines are shifted slightly to low field, but
different energy levels are shifted by different amounts,
which leads to the additional splitting of about 9 G in this
case.
The other result of second-order hyperfine effects is to
FIGURE 8 Schematic of the first- and second-order energy lev-
els of a hydrogen atom, the first-order allowed ESR transitions, mix the first-order wave functions as shown in Fig. 8 so
and the first- and second-order spin wave functions. The magni- as to partially allow some forbidden transitions that in-
2
tude of the second-order shift is A [4(gβH + g N β N H)] −1 . volve flipping of nuclear spins. When the first-order wave