Page 123 - 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

334 Electron Spin Resonance

In Eq. (2) the summations are taken over all the nuclei
in the molecular species. The new symbols in Eq. (2) are
defined as follows: g n is the nuclear g factor, which is di-
mensionless; β n is the nuclear magneton, having units of
joules per gauss or per tesla; the nuclear spin angular mo-
mentum operator I n ; the electron–nucleus hyperfine ten-
sor A n ; the quadrupole interaction tensor Q n ; and Planck’s
constant h.
Note that the nuclear Zeeman term involving the inter-
action of the magnetic field with the nuclear spin angular
momentum operator has a negative sign. This is essen-
tially due to the difference in charge between electrons
and nuclei. Also note that the nuclear g factor can be either
FIGURE 3 Schematic of the first-order spin energy levels of a hy-
positive or negative, while the electron g factor is taken
drogen atom, showing successive interactions in the spin Hamil-
as intrinsically positive. In Eq. (2) the electron g factor tonian, the allowed ESR transitions, and the spin wave functions.
has been written in tensor form involving a 3 × 3 matrix
that connects the magnetic field vector and the electron energy levels given in Eq. (5), where m S and m I refer to
spin angular momentum vector. Similarly, the hyperfine the electron and nuclear spin angular momentum quantum
interaction is written in tensor form connecting the elec- numbers:
tron spin and nuclear spin angular momentum vectors. The
E n = gβHm S − g n β n Hm In + m S hA n m In .
quadrupole interaction is also written in tensor form.
n n
First consider the special case of isotropic hyperfine (5)
interaction in which the hyperfine interaction becomes a
The wave functions correct to first-order perturbation the-
scalar and can be written in front of the dot product of the
ory are just the product functions of the respective electron
nuclear and electron spin angular momentum vectors. For
and nuclear spin combinations. The hyperfine interaction
simplicity the electron g factor will also be considered to
term involves only the z-components of the electron and
be isotropic and to be a scalar. This simplification typically
nuclear spin angular momentum operators when treated
applies to most organic and some inorganic free radicals
by first-order perturbation theory.
in liquids and also to a few cases in solids. This occurs
This simplified treatment can be applied exactly to a
because rapid tumbling of the molecular species averages 1 1
hydrogen atom with S = 2 and I = 2 where the corre-
outtheanisotropicinteractions.Also,sincethequadrupole 1
sponding m S and m I values are both ± . The spin wave
2
interaction is typically small and can only be experimen-
functions such as α e β n and the spin energy levels for a
tally resolved in special cases, it will be left out of the
hydrogen atom in a magnetic field are shown in Fig. 3.
simplified spin Hamiltonian. The resulting simplified spin
The transition probabilities between the first-order spin
Hamiltonian becomes
energy levels can be calculated from time-dependent per-

spin = gβH · S − g n β n H · I n + hA n S · I n . (3) turbation theory by standard methods in which the mag-
n n netic dipole moment operator is used. It is found that the
The external magnetic field is unidirectional, and by con- transition moment is only finite for a magnetic dipole mo-
vention it is taken to be in the z-direction. ment operator oriented perpendicular to the magnetic field
Then the energy levels of the spin system are given by direction. This means that in the design of an electron spin
resonance cavity into which the sample is placed, the mi-
∗
E n = spin n dτ, (4) crowave magnetic field must be arranged to be perpendic-
n
ular to the external static magnetic field. The second point
where representsthespinwavefunctions.Thespinwave
of interest is that transition moments that determine the se-
functions can be taken as products of the electronic and
lection rules for magnetic resonance are finite for electron
nuclear spin wave functions. These product functions are
spin transitions corresponding to a change in the electron
not exact wave functions for the spin Hamiltonian includ-
spin orientation but are zero for nuclear spin transitions.
ing the hyperfine interaction term, but they serve as good
These selection rules can be compactly written as
first approximations. The correct treatment of the prob-
lem usually involves some approximation method, and a
m S =±1 (6)
common one is perturbation theory, which is a standard
and
quantum-mechanical method. Applying first-order pertur-
bation theory to the spin Hamiltonian in Eq. (3) gives the
m I = 0, (7)