Page 129 - 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

340 Electron Spin Resonance

of isotropic hyperfine values to obtain the amount of spin lated and compared with experimental data. The difficulty
density in s orbitals. is that information about the excited-state energy levels
needs to be known to properly calculate the g tensor and
this is generally known only for simple molecular systems.
−
VII. g ANISOTROPY In a few cases, such as for the CO radical ion, detailed
2
calculations have been carried out and the experimental
In the general spin Hamiltonian given by Eq. (2), the g fac- g anisotropy has given information about the molecular
tor given in the electron Zeeman energy term is written as wave function.
a tensor connecting the electron spin angular momentum Organic radicals generally have the unpaired electron
operator S and the magnetic field vector H. A free elec- in a p orbital, which has orbital angular momentum. How-
tron has only spin angular momentum, and its orientation ever, the weak molecular electrostatic field splits the M L
in a magnetic field is determined only by this physical components and gives M L = 0 as the lowest state. In this
property. However, in general, in atomic and molecular case g g e . Nevertheless, small deviations from g e do
systems there will be some contribution from orbital an- occur and can be readily measured, particularly in single
gular momentum to the total unpaired electron wave func- crystals, since at a typical 3300-G field, g = 0.0006 for a
tion. In this case the orbital and spin angular momentum 1-G shift.
vectors interact, and by convention this interaction is in- When an organic radical contains an atom with a large
corporated into an “effective” anisotropy in the g factor. In spin–orbit coupling constant, such as oxygen, sulfur, or
this representation the spin angular momentum vector S halogens, the g anisotropy becomes significantly greater
no longer represents “the true spin” because the true spin and the average g value generally shifts to larger values.
has only spin angular momentum and is associated with This serves as a diagnostic tool for radicals in solution and
an isotropic g value. Instead, when g is written as a tensor in solids. One particular example involves peroxy radicals
the spin angular momentum vector represents an effective in which the unpaired spin is localized largely on the oxy-
spin, which instead of being oriented along the magnetic gen and for which the average g equals 2.015 in a wide
field direction is oriented along the vector H · g. For most variety of environments. Sulfur-containing radicals also
purposes this nuance will not affect our utilization of the often show large g anisotropy. This can be used as a diag-
g tensor formulation. nostic test for the localization of the radical site in some
The experimental determination of the g tensor is car- biological molecules that contain sulfur.
ried out by a procedure completely analogous to that for The largest g anisotropy occurs for transition-metal
determination of the anisotropic hyperfine tensor. Mea- ions, where the g anisotropy is very useful for discrim-
surements are required as a function of angle in three inating between transition-metal ions in different types of
mutually perpendicular planes. From this data, a general environments. The range of g anisotropy can be rather
g tensor is obtained, which is diagonalized to find the prin- large. Typical values for axial g anisotropy range from
cipalvalues.Theprincipalaxesofthe g tensorareoftenthe g ⊥ = 2.04 and g = 2.17 for copper complexes to g ⊥ = 6
same as for the hyperfine tensor, but they do not have to be. and g = 2 for some ferric complexes.
The interpretation of the principal value of the g tensor
can be conveniently discussed by Eq. (17):
VIII. SPIN RELAXATION
Cλ
g obs = g e + . (17)

E The energy between the magnetic energy levels at 3000 G,
In this expression, g e is the g factor for an isolated spin gβH, is only 10 −3 of kT at 300 K. At thermal equilibrium
(2.0023), λ is the spin–orbit coupling constant, C is a pro- the Boltzmann factor, exp(−gβH/kt), gives the popula-
portionality constant calculated from the electronic wave tion ratio of the two levels, so the levels are almost equally
functions, and
E is the energy difference between the populated. The application of microwave energy causes
ground state and the first excited state. Values of λ have transitions between the magnetic levels. The microwave
been obtained for a number of atoms and ions from atomic field stimulates transitions in both directions with a prob-
spectra, but the particular value to be used in a molecular ability that depends on the microwave power and on the
system can only be approximated by this. In general, λ number of spins in each level. Transitions from the lower
values increase with atomic number. The values of
E to upper levels absorb energy, while upper- to lower-level
can sometimes be deduced from electronic spectra. Thus transitions emit energy. Since the population is slightly
the g anisotropy is related to the electronic wave function, greater in the lower level there will be a net absorption of
and if sufficient information is known about the electronic microwave energy; this provides the observed ESR signal.
wave function the principal g components can be calcu- Under steady application of the microwave field with no