Page 128 - Academic Press Encyclopedia of Physical Science and Technology 3rd Analytical Chemistry

P1: GLQ/LSK P2: FQP Final Pages
Encyclopedia of Physical Science and Technology EN005E-212 June 15, 2001 20:32

Electron Spin Resonance 339

functions are mixed, the normally forbidden transitions diagonal elements of a diagonalized hyperfine tensor are
become weakly allowed, due to components of the oscil- called the principal values.
lating microwave magnetic field parallel to the direction of To determine the anisotropic hyperfine tensor experi-
the external magnetic field, instead of perpendicular to the mentally, one must usually use a single crystal, although
external magnetic field as is the case for ordinary allowed if there is only one interacting nucleus or possibly more
ESR transitions. In the practical situation there is usually with sufficiently large anisotropy, it is sometimes possible
a small component of the applied microwave magnetic to determine the tensor from powder spectra. For single
field in the direction parallel to the external magnetic field, crystals it is necessary to measure the angularly dependent
which can bring about such nominally forbidden transi- hyperfine splitting in three mutually perpendicular planes
tions. These types of transitions are seldom seen in liquids with respect to the external magnetic field. From this data
but are relatively common in solids. there are well-known procedures to obtain the hyperfine
tensor in the axis system chosen for measurement. Then,
as outlined above, this tensor may be diagonalized and the
VI. ANISOTROPIC HYPERFINE
INTERACTION principal values with their associated direction cosines
may be determined.
The physical interpretation of the anisotropic hyperfine
The general spin Hamiltonian was given by Eq. (2), in
principal values is given by the classical magnetic dipolar
which the interaction parameters were written in the gen-
interaction between the electron and nuclear spin angular
eral tensor form. The total hyperfine tensor A can be rep-
momenta. This interaction energy is given by
resented by a 3 × 3 matrix that connects the three com-
ponents each of the electron spin angular momentum and 2
1 − 3(cos φ)
the nuclear spin angular momentum. The hyperfine tensor aniso =−gβg N β N 3 I · S, (15)
r
is a real matrix and can always be diagonalized. Thus a
general tensor referenced to an x–y–z axis system can be where r is the vector between the unpaired electron and
written as the nucleus with which the interaction occurs and φ is the
angle between r and the electron spin angular momentum

A xx A xy
A xz
vector S, which is in the direction of the external magnetic
A ≡ A yx A yy A yz . (12)
field. The A principal values are given by

A zx A zy A zz

2
This general tensor can be transformed to another axis −1 1 − 3(cos φ)
A =−gβg N β N h , (16)
system in which the new tensor is diagonal. This requires r 3
av
finding the proper transformation matrix L, which diago-
where av denotes a spatial average over the electronic or-
nalizes the A tensor as
bital of the unpaired electron. The three components of A
T d
L · A xyz · L = A αβγ , (13) are given by three different values of cos φ corresponding
T
d
where L is the transpose of L. The diagonalized tensor A to rotation in three mutually perpendicular planes of the
is diagonal in the α–β–γ axis system, and the components principal axis system. 2 3
of L are the three sets of direction cosines that relate the Thedipolarfunction 1 − 3(cos φ)/r ) av canbeevalu-
x–y–z axis system to the α–β–γ axis system. ated from known wave functions of electrons in s, p, d, and
The components of the diagonalized hyperfine tensor other orbitals on different atoms. For s orbitals the dipolar
consist of an isotropic part A 0 and a purely anisotropic function is zero because of spherical symmetry. The cylin-

part A , whose orientational average is zero. This decom- drical symmetry of p orbitals gives three components, A ,

position is shown in Eq. (14): A , and A , which are related by A =−2A . Note that
⊥
⊥

⊥
◦
the dipolar angular function changes sign at φ = 54 44 .

A αα A 0 0
0 0
0 Thus, the space around a nucleus can be divided into four
d
A = 0 A ββ 0 = 0 A 0 0 regions alternating in sign. In the determination of the hy-

0 0 A γγ 0 0 A 0 perfine tensor, a set of signs of the components will be

(14) obtained so that the sum of the diagonal principal values

A αα 0 0 is zero.

= 0 A ββ 0 . The A components from Eq. (16) have been evaluated

0 0 A
theoretically for unit spin density in atomic radial wave
γγ
functions for p orbitals. Thus the experimental anisotropic
d
Thus the sum of the diagonal elements of A gives 3A 0 , hyperfine components can be used to estimate the amount

and the sum of the diagonal elements of A is zero. The of spin density in p orbitals. This complements the use