Dionne G.F. Magnetic Oxides
Подождите немного. Документ загружается.
Appendix 5A 261
Appendix 5A Four-Level Degenerate Perturbation Solution
for d
1
The following calculation has been excerpted from the analysis of the param-
agnetic resonance spectrum of Ti
3C
d
1
in the hydrated rubidium alum salt
Rb
1C
Al
3C
.SO
4
/
2
2
12H
2
O[3]. The energy level diagram for this model is
showninFig.5.4 with orbital eigenstates
j
0
i
through
j
4
i
corresponding to the set
listed in (2.12)and(5.3). It begins with the computation of the 10 10 matrix for
the elements of the combined crystal field and spin–orbit coupling operators, i.e.,
hk
j
H
cf
C L S
j
ni, for which no restrictions are placed on the relative strengths
of the two contributions. The introduction of the Zeeman perturbation elements can
be postponed here because we assume that they are small by comparison.
We begin the process by setting up the full Hermitian matrix:
j
0˛
ij
1ˇ
ij
2ˇ
ij
3˛
ij
4ˇ
i j
0ˇ
ij
1˛
ij
2˛
ij
3ˇ
ij
4˛
i
h
0˛
j
0 i
2
2
i0 0000 0
h
1ˇ
j
i
2
ı
1
i
2
2
0 0000
p
3
2
h
2ˇ
j
2
i
2
ı
2
i
2
0 0000i
p
3
2
h
3˛
j
i
2
i
2
0 0000 0
h
4ˇ
j
00 00
0
0
p
3
2
i
p
3
2
00
h
0ˇ
j
00 00 0 0 i
2
2
i0
h
1˛
j
00 00
p
3
2
i
2
ı
1
i
2
2
0
h
2˛
j
00 00i
p
3
2
2
i
2
ı
2
i
2
0
h
3ˇ
j
00 00 0i
2
i
2
0
h
4˛
j
0
p
3
2
i
p
3
2
000000
0
(5.70)
where
j
˛
i
and
j
ˇ
i
are the S D˙1=2 eigenfunctions. By inspecting (5.70) we rec-
ognize that the full 1010 matrix can be reduced to two identical 44 matrices with
only a small sacrifice in accuracy. The equation formed from the secular determinant
P
k
.
h
k
j
H
cf
C L S
j
n
i
Eı
kn
/ D 0 of these smaller matrices is expressed as
E
4
C AE
3
C BE
2
C CE C D D 0; (5.71)
where
7
A D.ı
1
C ı
2
C /,
B D ı
1
ı
2
C ı
1
C ı
2
.9=4/
2
C D .1=4/
2
.6ı
1
C 6ı
2
C 3/ ı
1
ı
2
C .5=8/
3
D D .1=8/
3
.2 4ı
1
4ı
2
/
.1=4/
2
.ı
1
C ı
2
C 4ı
1
ı
2
/ C .3=4/
4
:
7
These relations have been corrected for small errors that appeared in the original computation [3].
262 5 Anisotropy and Magnetoelastic Properties
For the ground doublet (the solution with lowest value of E), the eigenfunctions for
the two states can be shown to be
j
e
0
C
i
D a
j
0˛
i
C b
j
1ˇ
i
C c
j
2ˇ
i
C d
j
3˛
i
(5.72)
j
e
0
i
D a
j
0ˇ
i
C b
j
1˛
i
c
j
2˛
i
d
j
3ˇ
i
;
where a
a Cb
b Cc
c Cd
d D 1 for normalization. Similar relations for
j
e
1
˙
i
,
j
e
2
˙
i
, and the other eigenfunctions can be derived for the higher energy solutions.
The coefficients expressed relative to d that can apply to any of the E solutions and
are given by
a D i
1
2
3
.•
1
E/.•
2
E/
1
4
2
Œ.•
1
E/ C .•
2
E/
1
4
2
Œ. E/ C .•
1
E/C.•
2
E/ E.•
1
E/.•
2
E/
d;
b D
1
4
2
. E/
1
2
. E/.•
2
E/
1
4
2
Œ. E/ C .•
1
E/C.•
2
E/ E.•
1
E/.•
2
E/
d; (5.73)
c D i
1
4
2
. E/
1
2
. E/.•
1
E/
1
4
2
Œ. E/ C .•
1
E/C.•
2
E/ E.•
1
E/.•
2
E/
d:
With the eigenfunctions determined for each solution for E, we can now compute
the Zeeman splittings by applying the method used above to each of the Kramers
doublets. In cases where the ground state is a degenerate group of Kramers doublets,
all of the states must be included in the calculation. Where the ground state
j
e
0
i
is
a singlet created by lower symmetry crystal fields, the ground doublet alone may be
all that is required. If magnetic fields of average strength .H < 1 T/ are applied, the
Zeeman energy term is negligible compared with the constant and the upper levels
j
e
1
i
and
j
e
2
i
have little influence except possibly for paramagnetic susceptibilities
at higher temperatures. The general procedure is to repeat the above process by
forming secular equations from the Zeeman perturbation H
h
D m
B
.L C 2S / H
utilizing the new sets of eigenfunctions obtained from (5.72)and(5.73). If all three
doublets are required, we compute the elements to form a 6 6 matrix; if the lowest
doublet is needed, the problem is simplified to the 22 matrix of the ground doublet
j
e
0
C
i
and
j
e
0
i
, as indicated by
j
e
0
C
ij
e
0
i
j
e
1
C
ij
e
1
i
j
e
2
C
ij
e
2
i
h
e
0
C
j
E
0
C H
11
H
12
H
13
H
14
H
15
H
16
h
e
0
j
E
0
C H
22
H
23
H
24
H
25
H
26
h
e
1
C
j
E
1
C H
33
H
34
H
35
H
36
h
e
1
j
E
1
C H
44
H
45
H
46
h
e
2
C
j
E
2
C H
55
H
56
h
e
2
j
E
2
C H
66
:
(5.74)
For the ground
j
e
0
˙i doublet submatrix, we can drop the E
0
energy from the
diagonal elements and determine that H
22
DH
11
,andH
12
D H
21
D 0, with the
Appendix 5B 263
result that we obtain the following relations for the g factors for the three H field
directions:
g
z
D 2
a
a b
b c
c C d
d
C4i
a
d d
a
2i
b
c c
b
g
x
D 2
a
a C b
b c
c d
d
C 2i
a
b b
a
C2i
c
d d
c
(5.75)
g
y
D 2
a
a b
b C c
c d
d
C2
a
c C c
a
2
b
d C d
b
:
Calculations for parameter values D 154, ı
1
D 1;050, ı
2
D 1;320,and
D 22;000 cm
1
yield g
z
D 1:895, g
x
D 1:715,andg
y
D 1:767, in exact
agreement with the measured results [4].
Appendix 5B T
2g
Solution for d
1
in an Exchange Field
Where spin–orbit coupling is stronger than any likely lower symmetry component
of the crystal field, it is advantageous to use a solution that begins with the eigen-
functions of the L S operator. The details of the analysis for the d
1
case can
be found in Ballhausen [74]orSchl¨afer and Gliemann (Table 5.10)[93]. This ver-
sion of the d
1
solution is particularly useful in understanding the behavior of ions
with triplet ground states i.e., d
1
and d
6
, d
2
and d
7
in octahedral fields, or d
3
and
d
4
, d
8
and d
9
in tetrahedral fields. For our purposes, we expand this solution to
include a tetragonal field component by constructing matrices that include the V
cf
T
operator, which produces off-diagonal elements that mix the starting wavefunctions.
If we assume that the cubic Dq parameter is large enough to allow the upper E
g
state
to be ignored, we can reduce the problem to the solution of 2 2 matrices gener-
ated by applying the .L C gm
B
H
ex
/ S operator to the basis vectors of spin–orbit
coupling. In order to retain the single-ion nature of the Hamiltonian, the source of
the exchange interaction with spins of adjacent ions is treated phenomenologically
as a quasimagnetic field H
ex
that forms a scalar product with S , analogous in cer-
tain respects to the point-charge crystal-field concept. Stabilization of only the spin
Table 5.10 Eigenfunction limits and orbital angular momentum for axial crys-
tal field and spin–orbit coupling energy levels of a d
1
electron
a
ı= '
0
'
1
'
2
1 t
2g
˛t
2g
o
ˇt
2g
C
˛
m
`
D1m
`
D 0m
`
DC1
0 t
2g
˛
q
2
3
t
2g
o
ˇ C
q
1
3
t
2g
C
˛
q
1
3
t
2g
o
ˇ C
q
2
3
t
2g
C
˛
m
`
D1m
`
DC
1
3
m
`
DC
2
3
C1 t
2g
˛t
2g
C
˛t
2g
o
ˇ
m
`
D1m
`
DC1m
`
D 0
a
For the second set of wavefunctions '
0
0
, '
0
1
,and'
0
2
, the corresponding l
z
values
will reverse sign
264 5 Anisotropy and Magnetoelastic Properties
momentum state can be considered in the exchange contribution to the perturba-
tion energy. The magnetic resonance implications of this abbreviated solution in the
presence of a “real” magnetic field are discussed further in connection with “ex-
change” resonance in Chap. 7. As a result, H
ex
is assumed to be directed along the
z-axis and g D 2 for the spin-only value. The complete matrix of the perturbed T
2g
term with the crystal field axial splitting parameter ı included is expressed as:
c
8
d
8
a
8
a
7
b
8
b
7
c
8
1
2
C
1
3
ı
Cm
B
H
ex
0 0 0 0 0
d
8
0
1
2
C
1
3
ı
m
B
H
ex
0 0 0 0
a
8
0 0
1
2
1
3
ı
1
3
m
B
H
ex
2
p
2
3
m
B
H
ex
p
2
3
ı
0 0
a
7
0 0
2
p
2
3
m
B
H
ex
p
2
3
ı
C
1
3
m
B
H
ex
0 0
b
8
0 0 0 0
1
2
1
3
ı
C
1
3
m
B
H
ex
2
p
2
3
m
B
H
ex
p
2
3
ı
b
7
0 0 0 0
2
p
2
3
m
B
H
ex
p
2
3
ı
1
3
m
B
H
ex
; (5.76)
where the eigenfunctions are based on the set listed in (2.11) according to:
c
8
D t
2g
˛
a
8
D
r
2
3
t
o
2g
ˇ C
r
1
3
t
C
2g
˛
a
7
D
r
1
3
t
o
2g
ˇ C
r
2
3
t
C
2g
˛
And
c
8
D t
2g
˛
a
8
D
r
2
3
t
o
2g
ˇ C
r
1
3
t
C
2g
˛ (5.77)
a
7
D
r
1
3
t
o
2g
ˇ C
r
2
3
t
C
2g
˛:
For this exercise, the eigenfunction nomenclature used by Ballhausen is adopted
for consistency. The secular equation formed from (5.76) provides the following
solutions for the energy states:
E
c
8
D
1
2
C
1
3
ı
C m
B
H
ex
E
d
8
D
1
2
C
1
3
ı
m
B
H
ex
Appendix 5C 265
E
a
8;7
˙
D
1
2
1
2
C
1
3
ı
˙
1
2
r
9
4
2
C ı
2
ı C 2m
B
H
ex
.
2ı
/
C4
.
m
B
H
ex
/
2
(5.78)
E
b
8;7
˙
D
1
2
1
2
DC
1
3
ı
˙
1
2
r
9
4
2
Cı
2
ı 2m
B
H
ex
.
2ı
/
C 4
.
m
B
H
ex
/
2
:
Computations of E as a function of m
B
H
ex
for < 2ı=3, D 2ı=3,and
> 2ı=3 indicate six distinct levels and the removal of all degeneracy. Of partic-
ular interest is the comparison between the lowest energy states of the spin–orbit
stabilized case corresponding to ı D 3E
JT
=2 and the Jahn–Teller case with
ı D –3E
JT
=2,whereE
JT
is the maximum available spontaneous ligand strain
energy. For H
ex
>0, the respective ground states are the E
d
8
and E
a
8;7
solutions of (5.78).
8
Additional discussion of this solution can be found in [16]
By equating these two expressions, the threshold condition for the stabilization of
the E
d
8
level (for ı>0) can be derived by algebraic manipulation, according to
ı
2
3
1 C
2
3
ı
gm
B
H
ex
; (5.79)
which is introduced as (5.16) in the text.
Orbital angular momentum contributions to magnetoelastic effects are there-
fore dependent on the strength of the exchange field. Based on this simple model,
the threshold ratio =ı reaches the value of 2/3 suggested by (2.34) only when
H
ex
!1. Because exchange energy is dependent on the degree of spin-ordering,
the collective gm
B
H
ex
S , and hence the stabilization itself, are temperature depen-
dent. Consequently, a transition from S–O to J–T stabilization as the temperature
increases would be theoretically tied to the decline in strength of the molecular field
that controls the shape of the Brillouin function dependence of the sublattice mag-
netization.
Appendix 5C Orbital States of d
5
in a Cubic Field
Two computations were undertaken to determine the energy ladder of orbital states
of the various d
n
configurations in a cubic crystal field. Orgel began by choosing
the free-ion orbital wavefunctions and applying the cubic field potential [94]fol-
lowing the method of Finklestein and Van Vleck [95]. For the d
5
case, the energy
8
Note that the Hamiltonian for a second ion, with an opposing spin direction needed to comply
with the Pauli principle, would be set up with z !z. The basis wavefunctions would be similar
to those of (
5.77), but with each orbital and spin component reversed in sign. The corresponding
ground states would therefore be E
c
8
and E
a
8;7
C
with ˛ and ˇ spin states that complement
those of the first ion to provide the spin degeneracy in a two-ion molecular-orbital bonding state.
Such a degeneracy would then appear spectrally as a Kramers doublet, removable only by an
external magnetic field.
266 5 Anisotropy and Magnetoelastic Properties
relations of the starting unperturbed states following Racah’s terminology defined
in Sect. 2.3 are:
E
F
4
F
D 10A 13B C 7C
E
D
4
D
D 10A 18B C 5C
E
P
4
P
D 10A 28B C 7C (5.80)
E
G
4
G
D 10A 25B C 5C
E
S
6
S
D 10A 35B:
The calculation proceeds by solution of secular equations obtained from matrices
with nonzero elements formed between states of common symmetry group repre-
sentations. In addition to three orbital singlets, there are six triplets and two doublets
that are sorted out according to the following matrices formed with
6
S as the ground
state reference energy:
T
1g
4
P
4
F
4
G
4
P 7B C 7C 0 4
p
5Dq
4
F 0 22B C 7C 2
p
5Dq
4
G 4
p
5Dq 2
p
5Dq 10B C 5C
; (5.81)
T
2g
4
D
4
F
4
G
4
D 17B C 5C 20
p
7Dq 0
4
F 20
p
7Dq 22B C 7C 10
p
3=7Dq
4
G 010
p
3=7Dq 10B C 5C
; (5.82)
E
4
D
4
G
4
D 17B C 5C 0
4
G 0 10B C 5C
: (5.83)
The alternative approach that was carried out earlier in a more comprehensive scope
was reported by Tanabe and Sugano [96]. In their work the starting functions were
formed from the strong field limit, i.e., the condition obtained when Dq !1,the
exact reverse of the Orgel method. For details of the computation of the nonzero
matrix elements, the reader is directed to the original paper. The matrices for deter-
mining the nonsinglet states under the action of a field of strength Dq are as follows:
T
1g
4
G;
4
P;
4
F
t
2g
4
"""# e
g
"
t
2g
3
""#
e
g
2
""
t
2g
2
""
e
g
3
""#
t
2g
4
e
g
25B C 6C 10Dq 3
p
2B C
t
2g
3
e
g
2
3
p
2B 16B C7C 3
p
2B
t
2g
2
e
g
3
C 3
p
2B 25B C 6C C10Dq
; (
5.84)
Appendix 5D 267
T
2g
4
G;
4
D;
4
F
t
2g
4
"""#
e
g
"
t
2g
3
""#
e
g
2
""
t
2g
2
""
e
g
3
""#
t
2g
2
e
g
3
17B C 6C 10Dq
p
6Dq 4B C C
t
2g
2
e
g
3
p
6Dq 22B C 5C
p
6Dq
t
2g
2
e
g
3
4B C C
p
6Dq 17B C 6C C 10Dq
;
(
5.85)
E
4
G;
4
D
t
2g
3
""#
e
g
2
""
t
2g
3
"""
e
g
2
"#
t
2g
3
e
g
2
22B C 5C 2
p
3Dq
t
2g
3
e
g
2
2
p
3Dq 22B C 5C
: (5.86)
For both the Orgel and Tanabe-Sugano analyses the energy states as a function of
Dq can be obtained by analytical solution of secular equations that involve nothing
more complex than cubic equations. An example of such a calculation for 3d
5
is
shown in Fig. 2.15, which is the identical result for either method using the same
values of B D 1; 133 cm
1
and C D 3; 867 cm
1
.
Appendix 5D Angular Dependence of Cubic Anisotropy Fields
The effective fields created by the stiffness of the anisotropy torques that affect the
rotation of M can be determined analytically for any particular lattice plane by a
transformation of coordinates from x, y, z (or r, , )tox
0
, y
0
, z
0
(or r,
0
,
0
)that
places the crystallographic plane of interest in the x
0
z
0
plane .
0
D 0/,wherez
0
is
set to the direction of vector M . To establish the location of M in the new plane, the
parameter ı is introduced as the angle M makes with a crystallographic reference
axis, e.g., [111], [110]. Thus, while ı fixes the position of z
0
in the crystallographic
plane,
0
remains the variable for the determining the stiffness torque (
0
, ı)asM
is deflected from z
0
by the rotation of H . With an relation for E
K
(
0
, ı) developed
from the discussion in Sect. 5.3.1 expressed in terms of these new coordinates, we
can then obtain the relation for the in-plane anisotropy field as a function of rotation
angle ı. By choosing the y
0
z
0
plane .
0
D =2/, the out-of-plane stiffness torque
and anisotropy field can also be determined. Knowledge of these orthogonal H
K
relations is necessary for analyzing ferromagnetic resonance spectra described in
Chap. 6.
The relations for H
K
were used to obtain the results listed in Table 5.7 and are
based on the diagrams of 5.39.
Rotation between axis and orthogonal plane in uniaxial crystal: ı D 0 referenced
to the axis.
H
K
==
D
2K
1
M
s
.cos 2ı/ .in plane/
H
K
?
D
2K
1
M
s
.cos ı/ .out of plane/: (5.87)
268 5 Anisotropy and Magnetoelastic Properties
Fig. 5.39 Vector diagrams for M as functions of rotation angle ı in the four planes of interest for
a cubic crystal lattice: [100], [110], [111], and [112]
Rotation in f100g planes: ı D 0 referenced to a h100iaxis.
H
K
==
D
2K
1
M
s
.cos 4ı/ .in plane/
H
K
?
D
2K
1
M
s
1
1
2
sin
2
2ı
C
K
2
2M
s
sin
2
2ı
.out of plane/ (5.88)
Rotation in f110g planes: ı D 0 referenced to a h100iaxis.
H
K
==
D
K
1
M
s
2 sin
2
ı 3 sin
2
2ı
C
K
2
2M
s
sin
2
ı
6 cos
4
ı 11 sin
2
ı cos
2
ı C sin
4
ı
.in plane/ (5.89)
H
K
?
D
K
1
M
s
2 4 sin
2
ı
3
4
sin
2
2ı
K
2
2M
s
sin
2
ı cos
2
ı
3 sin
2
ı C 2
.out of plane/
References 269
Rotation in f111g planes: ı D 0 referenced to a h112iaxis.
H
K
==
D
K
2
3M
s
cos
3
2ı 3 sin
2
2ı cos 2ı
.in plane/
H
K
?
D
K
1
M
s
C
K
2
6M
s
1
2
3
3 sin
2
ı cos ı cos
3
ı
2
.out of plane/ (5.90)
Rotation in f112g planes: ı D 0 referenced to a h111iaxis.
H
K
==
D
4K
1
3M
s
cos
4
ı C
3
4
sin
4
ı
21
4
sin
2
ı cos
2
ı
4K
2
9M
s
cos
6
ı
3
8
sin
6
ı C
57
8
sin
4
ı cos
2
ı
41
4
sin
2
ı cos
4
ı
.in plane/
H
K
?
D
4K
1
3M
s
cos
4
ı C
3
4
sin
4
ı
3
4
sin
2
ı
(5.91)
4K
2
9M
s
0
@
1
2
cos
6
ı C
1
2
cos
4
ı
3
4
sin
4
ı
C
9
8
sin
4
ı cos
2
ı
3
4
sin
2
ı cos
2
ı
3
2
sin
2
ı cos
4
ı
1
A
.out of plane/
References
1. J.H. Van Vleck, The Theory of Electric and Magnetic Susceptibilities, (Oxford University
Press, London, 1932)
2. J.S. Griffith, The Theory of Transition-Metal Ions, (Cambridge University Press, London,
1961)
3. G.F. Dionne, Phys. Rev. 137, A743 (1965)
4. G.F. Dionne, Can. J. Phys. 42, 2419 (1964)
5. J.A. MacKinnon and G.F. Dionne, Can. J. Phys. 44, 2329 (1966)
6. G.F. Dionne and J.A. MacKinnon, Phys. Rev. 172, 325 (1968)
7. B. Bleaney, G.S. Bogle, A.H. Cooke, R.J. Duffus, M.C.M. O’Brien, and K.W.H. Stevens, Proc.
Phys. Soc. A68, 57 (1955)
8. G.A. Woonton and J.A. MacKinnon, Can. J. Phys. 46, 59 (1968)
9. M.H.L. Pryce, Proc. Phys. Soc. A63, 25 (1950); A. Abragam and M.H.L. Pryce, Proc. Roy.
Soc. A205, 135 (1951)
10. H. Hartmann and H.L. Schlafer, Z. Physik Chem. (Leipzig) 197, 116, (1951)
11. A.J. Freeman and R.E. Watson, Phys. Rev. 127, 2058 (1962)
12. K.W.H. Stevens, Proc. Phys. Soc. A65, 209 (1952)
13. M.T. Hutchings, Solid State Phys. 16, 227 (1964)
14. W. Low, Paramagnetic Resonance in Solids, (Academic, New York, 1960)
15. H.A. Kramers, Proc. Amsterdam Acad. Sci. 33, 959 (1930); W. Low, Paramagnetic Resonance
in Solids, (Academic, New York, 1960), p. 34
16. G.F. Dionne, J. Appl. Phys. 91, 7367 (2002)
17. L. Dubicki and M.J. Riley, J. Chem. Phys. 106, 1669 (1997)
18. L.W. Tragenna-Pigott, S.P. Best, M.C.M. O’Brien, K.S. Knight, J.B. Forsyth, and J.R. Pillbrow,
J. Am. Chem. Soc. 119, 3324 (1997)
19. L.S. Kornienko and A.M. Prokorov,
Sov. Phys. JETP 11, 1189 (1960)
270 5 Anisotropy and Magnetoelastic Properties
20. F.S. Ham, Phys. Rev. 138, 1727 (1965)
21. R.M. MacFarlane, J.Y. Wong, and M.D. Sturge, Phys. Rev. 166, 250 (1968)
22. C.A. Bates and J.P. Bentley, J. Phys. C: Solid St. Phys. 2, 1947 (1969)
23. K.W.H. Stevens, J. Phys. C: Solid St. Phys. 2, 1934 (1969)
24. M. Abou-Ghantous, C.A. Bates, and K.W.H. Stevens, J. Phys. C: Solid St. Phys. 7, 325 (1974)
25. W. Low, Phys. Rev. 101, 1827 (1956)
26. G.A. Slack, Phys. Rev. 134, A1268 (1964)
27. B. Bleaney, K.D. Bowers, and R.J. Trenam, Proc. Roy. Soc. A228, 157 (1955)
28. M.H.L. Pryce, Il. Nuovo Cimento 6 (Suppl.) 817 (1957)
29. C.A. Bates and P.E. Chandler, J. Phys. C: Solid St. Phys. 4, 2713 (1971)
30. G.M. Zverev and A.M. Prokorov, J. Exptl. Theoret. Phys. (U.S.S.R.) 34, 1023 (1958)
31. W. Low, Paramagnetic Resonance in Solids, (Academic, New York, 1960), p.91
32. A. Abragam and M.H.L. Pryce, Proc. Roy. Soc. A205, 135 (1951); A206, 173 (1951)
33. C.J. Ballhausen, Introduction to Ligand Field Theory, (McGraw-Hill, New York, 1962) p. 124
34. A.L. Kipling, P.W. Smith, J. Vanier, and G.A. Woonton, Can. J. Phys. 39, 1859 (1961)
35. J.S. Thorp, Masers and Lasers: Physics and Design, (MacMillan, London, 1967) Chapter 4
36. J.P. Gordon, H.J. Zeiger, and C.H. Townes, Phys. Rev. 95, 282 (1954)
37. W. Low, Paramagnetic Resonance in Solids, (Academic, New York, 1960), p.92
38. J. Gielessen, Ann. Physik. 22, 537 (1935)
39. D.S. McClure, J. Phys. Chem. Solids 3, 311 (1957)
40. W.P. Doyle and G.A. Lonergan, Discuss. Faraday Soc. 26, 27 (1958)
41. H. Hartmann and H. M¨uller, Discuss. Faraday Soc. 26, 49 (1958)
42. H. Watanabe, Prog. Theoret. Phys. (Kyoto), 18, 405 (1957)
43. W. Low, Paramagnetic Resonance in Solids, (Academic, New York, 1960), p.120
44. J.H. Van Vleck and W.G. Penney, Phil. Mag. 17, 961 (1934)
45. B. Bleaney and K.W. H. Stevens, Rep. Prog. Phys. 16, 108 (1953)
46. V. J . F o l e n , Paramagnetic Resonance Vol. 1, (Proceed. First Intl. Conf., ed. W. Low (Academic
Press, New York, 1962), p. 68
47. R.J. Elliott and K.W.H. Stevens, Proc. Roy. Soc. A219, 387 (1953)
48. J.W. Orton, Rep. Prog. Phys. 22, 204 (1959)
49. K.D. Bowers and J. Owen, Rep. Prog. Phys. 18, 304 (1955)
50. S. Chikazumi, Physics of Magnetism, (Wiley, New York, 1964)
51. J. Smit and H.P.J. Wijn, Ferrites, (Wiley, New York, 1959)
52. R.M. Bozorth, Ferromagnetism, (D. Van Nostrand, New York, 1951)
53. A.H. Morrish, The Physical Principles of Magnetism, (Wiley, New York, 1965)
54. H.J. Williams, Phys. Rev. 52, 747 (1937); A.H. Morrish, The Physical Principles of Magnetism,
(Wiley, New York, 1965), p. 310
55. L.R. Bickford, Phys. Rev. 78, 449 (1950)
56. E.W. Lee, Rep. Prog. Phys. 18, 184 (1955)
57. E. du Tr´emolet de Lacheisserie, Magnetostriction, (CRC Press, Boca Raton, FL, 1993)
58. C. Kittel, Rev. Mod. Phys. 21, 541 (1949)
59. G.F. Dionne, IEEE Trans. Magn. 5, 596 (1969)
60. G.F. Dionne, Mater. Res. Bull. 6, 80 (1971)
61. J.H. Van Vleck, Phys. Rev. 52, 1178 (1937)
62. C. Zener, Phys. Rev. 96, 1335 (1954)
63. S. Chikazumi, Physics of Magnetism, (Wiley, New York, 1964), Section 7.2
64. K. Yosida and M. Tachiki, Progr. Theoret. Phys. (Kyoto) 17, 331 (1957)
65. W.P. Wolf, Phys. Rev. 108, 1152 (1957)
66. G.P. Rodrigue, H. Meyer, and R.V. Jones, J. Appl. Phys. 31, 376S (1960)
67. S. Geschwind, Phys. Rev. 121, 363 (1961)
68. B. Luthi and T. Henningsen, Proceed. Intl. Conf. Magn. (Nottingham) 1965, p. 668
69. P. H a ns e n, J. Appl. Phys. 45, 3638 (1974)
70. J.R. Cunningham, Jr., J. Appl. Phys. 36, 2491 (1965)
71. G.F. Dionne, J. Appl. Phys. 40, 1839 (1969)
72. A.H. Morrish, The Physical Principles of Magnetism, (Wiley, New York, 1965), p. 529