Dionne G.F. Magnetic Oxides
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1.4 Gyromagnetism 29
moved. Because the amplitudes of M
x
and M
y
must also decay at a rate proportional
to
1
1
, as analyzed in Appendix 1A, the phenomenological damping rate for these
components from (1.93) can be expressed as
x;y
1
Š
2
1
C f./
1
1
; (1.76)
where f./D
jcos j
1Cjcos j
,
which varies between 0 and 1/2. The relation for the limiting case of f.0/D 1=2
was stated without proof in [18].
We can therefore express
dM
x;y
dt
D
M
x;y
x;y
DM
x;y
1
2
C f./
1
1
Š
M
x;y
2
; (1.77)
under the usual paramagnetic condition that
2
1
and the influence of spin–
lattice relaxation is moot. In terms of the Heisenberg uncertainty principle, the de-
coherence rate 1=
2
! H of the resonance line, that is, the half-linewidth
at half maximum. Thus, greater dipolar interactions cause broader linewidths. For
paramagnetic systems
2
10
10
s and produces line broadening of a few kHz at
microwave frequencies of several GHz.
By contrast,
1
is usually not short enough to affect the linewidth unless fast-
relaxing impurities are present. However, the spin–lattice interaction is a sensitive
function of temperature, with
1
being shorter
10
6
s
at room temperature, and
lengthening into the millisecond range to produce line narrowing for easy measure-
ment only at liquid helium temperatures of 4 K and below. In the paramagnetic limit,
line broadening due to spin precession decoherence from dipole–dipole interactions
can therefore be observed in systems where broadening effects are determined by
2
1
. Where concentrations of spins are large enough, exchange coupling produces
an effective field that can align the spins into a coherent collective magnetic moment
that renders
2
1
! 0 under ideal gyromagnetic resonance conditions.
Spatial decoherence of the M
x;y
component can also occur in ferromagnetic sys-
tems through spin waves where conditions permit degeneracy of rf signal and spin
wave frequencies. Line broadening by a
2
1
effect can reappear and even deterio-
rate into irreversible nonlinear transfer of energy from signal to spin wave system.
1.4.3 Complex Susceptibility Theory
There are facets to the mysteries of magnetic resonance that have challenged and
fascinated researchers from the time of the phenomenon’s discovery. One topic that
has received attention involves the rapid rotation of the m vector when the ampli-
tude H
rf
is very large, often administered in the form of a high-power pulse. From
the perspective of the rotating frame of H
rf
(x
0
, y
0
, z
0
,wherez
0
D z if the now-
static H
rf
is along the x axis), we can visualize the polar rotation of m as confined
30 1 Introductory Magnetism
to the fixed z
0
-y
0
plane, for which the corresponding angular precession frequency
becomes H
rf
,assketchedinFig.1.15.IfH
rf
1
1
,
2
1
, the two relaxation
rates, the reversal of m through an angle is termed “adiabatic fast passage” be-
cause the damping effects associated with the relaxation do not have time to take
effect. In the sense of the previous description of dispersion with H
rf
parallel to
H , it would mean that ! 1. This regime of rf drive fields leads to the clas-
sical studies of resonance-line “hole burning” and “spin-echo” effects from which
estimates of precession-phase decoherence influenced by dipolar spin interactions
and spin-flip damping by spin–lattice relaxation can be obtained directly from ex-
periment [19]. In a medium with an inhomogeneously broadened resonance line,
spin diffusion can be investigated by spin-echo methods with H
rf
1
1
,
2
1
[20]. However, these examples are esoteric enough to fall outside of the scope of our
immediate interests. For an insightful treatment of this general topic, the reader is
referred to standard texts, for example, by Pake [21] and Abragam [22]. In Chap.6,
the subject of spin–lattice relaxation is examined further in relation to the general
microwave properties of magnetic oxides.
To explain high frequency and microwave signal propagation as functions of
frequency, magnetic field, and temperature, the behavior of the complex rf suscep-
tibility must be understood. In very high dc magnetic fields, transmission properties
at optical wavelengths can also be influenced by magnetic resonance. If the rf field
is small enough to be treated as a perturbation, so that fast-passage effects are not
present, that is, H
rf
1
1
,
2
1
, it is convenient to work within the stationary
frame x, y, z. Here, we can call the regime “slow passage” and include all of the
damping effects, and then derive the complex susceptibility relations for a circularly
polarized rf signal confined to the x-y plane, as indicated in Fig. 1.15.
Consider the a wave of amplitude H
rf
linearly polarized along the x axis of the
laboratory frame of reference given by
H
x
D H
rf
cos !t D
1
2
H
rf
Œexp .i!t / C exp .i!t/;
H
y
D iH
rf
sin !t D
1
2
H
rf
Œexp .i!t / exp .i!t/; (1.78)
where the two exponentialfunctions represent the counter-rotating modes of circular
polarization. For the resonance condition only the positive exponent has influence
when ! ! !
L
.If(1.78) is combined with (1.71), (1.74), and (1.77), the following
general relations can be constructed:
dM
z
dt
D
M
y
H
rf
cos !t M
x
H
rf
sin !t
C
M M
z
1
;
dM
x
dt
D
M
z
H
rf
sin !t HM
y
M
x
2
; (1.79)
dM
y
dt
D .HM
x
M
z
H
rf
cos !t/
M
y
2
:
1.4 Gyromagnetism 31
A complex rf susceptibility can then be expressed in the standard form of
rf
D
rf
0
i
rf
00
(1.80)
and the components of M are expressed in terms of
rf
by the relations
M
x
D
rf
0
H
rf
cos !t C
rf
00
H
rf
sin !t
M
y
D i
rf
0
H
rf
sin !t C
rf
00
H
rf
cos !t
: (1.81)
For the remainder of this derivation, we leave the manipulations of the relations as an
exercise or to be found in standard texts on magnetic resonance [21–23]. Solutions
for the z component of M under the resonance conditions reduce to
M
z
D
m
H
1 C
2
2
.H !/
2
1 C
2
2
.H !/
2
C
2
H
rf
2
1
2
; (1.82)
where
m
is the dc susceptibility that would apply far from resonance. For the x and
y components we obtain
M
x;y
D .M /
2
2
.H !/ H
rf
cos !t C H
rf
sin !t
1 C
2
2
.H !/
2
C
2
H
rf
2
1
2
; (1.83)
recalling that M D
m
H . Combination with (1.80)and(1.81) enables the identifi-
cation of
rf
0
and
rf
00
according to
rf
0
D
1
2
.M /
2
2
.H !/
1 C
2
2
.H !/
2
C
2
H
rf
2
1
2
;
rf
00
D
1
2
.M /
2
1
1 C
2
2
.H !/
2
C
2
H
rf
2
1
2
: (1.84)
Note that the factor 1/2 from (1.78) reappears because only the right-hand circu-
lar polarization component (RHCP) of the linearly polarized signal H
rf
enters the
susceptibility resonance expressions of (1.84). In Fig. 1.16, rough sketches of
rf
0
and
rf
00
about the resonance frequency illustrate two important features: (1) the
Lorentzian tails of the derivative (dispersion) curve representing phase shift extend
further than those of the absorption curve that represents loss, and (2) the corre-
sponding linewidth is defined at the half-power point of the absorption curve, which
is approximately equal to the separation of the two peaks of the dispersion curve.
The design parameter in the selection of a gyromagnetic property for a microwave
application is critically dependent on the effective intrinsic linewidth. For the discus-
sions of ferromagnetic resonance in Chap.6 where both circular polarization modes
are important, the factor of 1/2 is not included.
32 1 Introductory Magnetism
Fig. 1.16 Conceptual illustration of the real
rf
0
and imaginary
rf
00
components of the complex
rf susceptibility. Note the definition of linewidth at the 0.5 point and the longer extension of the
dispersion tails, which are important for high efficiency in signal transmission
If the H .dM =dt/ is integrated over one cycle an expression for the power
absorbed form the out-of-phase component is obtained:
A D .!=2/
2=!
Z
0
H .dM =dt/dt D 2!
rf
00
H
2
rf
D
m
H
2
rf
.H /
2
!
.H !/
2
C .!/
2
C .H
rf
/
2
1
!
; (1.85)
which simplifies to
A Š
M!
.H !/
2
C .!/
2
H
rf
2
(1.86)
for small H
rf
. For large values of H
rf
at resonance with ! D H , we obtain a
limiting relation that exposes the dependence on the spin–lattice relaxation time and
provides the “saturation method” for its measurement [24]:
A D
m
H
2
1
1
D MH
1
1
: (1.87)
In these relations, ! or H
D
2
1
is the half-linewidth at half-maximum.
Note that the previous condition for “fast passage” can now be expressed as H
rf
H .When
x
y
due to fast spin–lattice relaxation,
2
should be replaced by
x;y
from (1.76) to account for
1
influence on H broadening.
1.4 Gyromagnetism 33
1.4.4 Resonance Line Shapes
Although the limitations of the scope of this text will not permit an extensive
description of the intricacies of magnetic resonance, some general statements about
line shapes and their causes will prove useful in later topics. Line broadeningmecha-
nisms may be divided into homogeneous (from broadening processes inherent in the
line itself) and inhomogeneous (from overlapping resonances offset in frequency by
an irregular arrangement of anisotropic resonance centers). For the basic resonance
phenomenon under discussion, only the homogeneous case applies. The effect of
inhomogeneities will be introduced in a review of the resonance behavior of porous
ferrimagnetic ceramics.
In general, there are two basic lineshape functions that are used as reference
extremes: Lorentzian and Gaussian. In the sketches of Fig. 1.17, the Lorentzian
shape is identified by a sharper center and sweeping tails; the Gaussian has the
opposite characteristics. For analytical convenience, the Lorentzian curve is charac-
teristic of homogeneous broadening and is generally applicable for widely separated
or uniformly spaced moments. Paramagnetic resonance line broadening due to dipo-
lar interactions was examined by Van Vleck [25], and the dependence of the shape
function on the concentration of individual magnetic moments was studied by Kittel
and Abrahams [26], who reported that increased concentrationsof like paramagnetic
ions tend to drive the lineshape toward a Gaussian function, suggesting an inhomo-
geneous distribution. When the concentration is sufficiently high to allow exchange
forces to narrow the line as mentioned earlier, the lineshape tends to be Lorentzian.
This latter observation is important in the analysis of microwave and optical effects
in ferrimagnets to be examined in later chapters.
Fig. 1.17 Normalized Lorentzian and Gaussian resonance lines emphasizing their contrasting
shapes. Homogeneous systems feature Lorentzian contours with sharper peaks and longer tails
of a single resonance frequency; inhomogeneous systems have the opposite characteristics due to
multiple resonance frequencies and are often described by a Gaussian function
34 1 Introductory Magnetism
Appendix 1A Spin–Lattice Contribution to Linewidth
The basic phenomenological analysis of magnetic relaxation considered the phase
coherence of M
x
and M
y
, but neglected the decrease in M
z
amplitude, which arises
from the canting angle between the vector M and the magnetic field direction
(z axis). The standard relations in spherical polar coordinates are
M
z
D M cos ;
M
x
D M sin cos ' D
M
2
M
z
2
1=2
cos '; (1.88)
M
y
D M sin sin ' D
M
2
M
2
z
1=2
sin ':
By definition, the spin–lattice damping time constant
1
of longitudinal relaxation
follows from
dM
z
dt
D
M M
z
1
: (1.89)
If the x component is used as the example of transverse relaxation
dM
x
dt
D
M
2
M
z
2
1=2
d.cos '/
dt
C cos '
d
M
2
M
z
2
1=2
dt
: (1.90)
The first term represents the phase decoherence of the precessing spins and the sec-
ond is the effect of the amplitude relaxation. Therefore, (1.90) can be expressed as
dM
x
dt
D
M
x
2
M
x
.M
2
M
z
2
/
1=2
M
z
.M
2
M
z
2
/
1=2
dM
z
dt
: (1.91)
After substituting from (1.89)fordM
z
=dt and simplifying,
dM
x
dt
D
M
x
2
M
x
1
M
z
M M
z
DM
x
1
2
C f./
1
1
; (1.92)
where
f./D
j
cos
j
1 C
j
cos
j
:
Note that 0 f./ 1=2, for which the extremes represent the case of a wide-
open spin cone, sometimes used to measure
2
in spin-echo experiments conducted
with D =2,andthe D 0 case for the complete spin alignment of ferro- or
ferrimagnetism. In general, where
1
2
, the linewidth would then be related by
H
1
2
C f./
1
1
; (1.93)
so that spin–lattice relaxation broadening could become dominant, as in the
cases where rare-earth or other fast-relaxing ions are present in large enough
concentrations.
References 35
References
1. R.M. Bozorth, Ferromagnetism, (D. Van Nostrand, New York, 1951)
2. J.A. Osborn, Phys. Rev. 67, 351 (1945)
3. W.E. Henry, Phys. Rev. 88, 559 (1952)
4. N. Cusack, The Electrical and Magnetic Properties of Solids, (Longmans, Green and Co., New
York, 1958)
5. A.H. Morrish, The Physical Principles of Magnetism, (Wiley, New York, 1965), Chapter 2
6. G.F. Dionne, J. Appl. Phys. 41, 4874 (1970)
7. W. Heisenberg, Z. Phys. 49, 619 (1928)
8. L.D. Landau and E.D. Lifshitz, Quantum Mechanics, (Addison-Wesley, Reading, MA, 1958),
Chapter IX
9. W. Heitler and F. London, Z. Physik 44, 455 (1927)
10. K. Yosida, Theory of Magnetism, (Springer-Verlag, New York, 1996)
11. R.M. White, Quantum Theory of Magnetism, (Springer-Verlag, New York, 1985)
12. E.C. Stoner, Proc. Leeds Phil. Soc. 2, 391 (1933)
13. J.C. Slater, Phys. Rev. 49, 537 (1936)
14. A.H. Morrish, The Physical Principles of Magnetism, (Wiley, New York, 1965)
15. B. Lax and K.J. Button, Microwave Ferrites and Ferrimagnetics, (McGraw Hill, New York,
1962)
16. F. Bloch, Phys. Rev. 70, 460 (1946)
17. N. Bloembergen, Phys. Rev. 78, 572 (1950); also N. Bloembergen, Proc. IRE 44, 1259 (1956)
18. A.H. Morrish, The Physical Principles of Magnetism, (Wiley, New York, 1965), p. 198
19. E.L. Hahn, Phys. Rev. 80
, 580 (1950)
20. J.C. Dyment, Can J. Phys. 44, 637 (1966)
21. G.E. Pake, Paramagnetic Resonance, (W.A. Benjamin, New York, 1962)
22. A. Abragam, The Principles of Nuclear Magnetism, (Clarendon Press, Oxford, 1962)
23. C.P. Slichter, Principles of Magnetic Resonance, (Springer, New York, 1996)
24. A.L. Kipling, P.W. Smith, J. Vanier, and G.A. Woonton, Can. J. Phys. 39, 1859 (1961)
25. J.H. Van Vleck, Phys. Rev. 74, 1168 (1948)
26. C. Kittel and E. Abrahams, Phys. Rev. 90, 238 (1953)
Chapter 2
Magnetic Ions in Oxides
To establish the pattern of this book, a logical first step is to review the Periodic
table of chemical elements, to identify the transition groups within it, and to explain
the local influences of the chemical bonding environment in which the various ions
reside in a crystal lattice of an oxide. By the terminology of transition groups is
meant those elements for which the inner shells remain unfilled while electrons oc-
cupying outer shells participate in chemical bonding. Consequently, the electrons of
the unfilled inner shells are responsible for a variety of magnetic properties because
of the magnetic moments carried by their unpaired spins.
From the theory of atomic spectra, the angular momentum of the electron spin
is coupled to the angular momentum that is derived from the motion of the electron
in its orbit about the nucleus, that is, the orbital angular momentum. The strength
of spin–orbit coupling is a key factor in determining the extent to which the orbital
moment contributes to the magnetic properties and conversely, to what extent the
spins interact with the lattice. When placed in a crystal lattice, the magnetic ion is
subjected to two effective fields that separately influence the spin and orbital mo-
menta – the crystal electric field of the lattice site that captures or “quenches” the
orbital moment by a Stark effect, and the exchange interaction that orders the spin
into a collective ferromagnetic or antiferromagnetic state. The origin of the crystal
and exchange fields is reviewed first. The role of spin–orbit coupling is examined
later in relation to magnetocrystalline anisotropy and magnetostriction.
2.1 The Transition Metals
In the introduction to orbital angular momentum in Sect. 1.2, each electron has an
orbital quantum state encoded by n and L.Then label identifies the basic Coulomb
state energies of the Bohr atomic model and L specifies the particular orbital angular
momentum state. For an atom without perturbation, including that from spin–orbit
coupling, the array of orbital angular momentum states are initially quantum me-
chanically degenerate (of same energy), but are identified according to the scheme
described in Sect. 1.2.1, with 2L C 1 states for the sequence labeled s.LD 0/,
p.LD 1/, d.LD 2/, f.LD 3/, g.LD 4/, etc. Because there are two spin states
G.F. Dionne, Magnetic Oxides, DOI 10.1007/978-1-4419-0054-8 2,
c
Springer Science+Business Media, LLC 2009
37
38 2 Magnetic Ions in Oxides
for each orbital state, in filled orbital shells there are correspondingly 2 s electrons,
6p electrons, 10 d electrons, 14 f electrons, and 17 g electrons. As a general rule,
all inner orbital shells are complete, that is, the only unfilled orbital states are in the
outermost shell. Certain important exceptions exist, however, and it is from these
metal elements that most magnetic effects originate. They are called the transition-
metal groups.
2.1.1 The Periodic Table
Figure2.1 is modeled from the Mendelief Periodic table that has been tailored to
highlight the transition element groups. The definition of a transition group is one in
which inner electron shells are unfilled while outer shells are occupied by electrons.
Characteristically, in the building of the table there is an orderly progression of
state occupations from hydrogen with a single 1s electron all the way out to the
first transition series that begins with a single electron in the ten-state 3d shell of
scandium Sc (atomic number 21) outside a completed argon (Ar) core, but inside
a completed two-state 4s shell, as sketched in Fig. 2.2a. The series ends at number
29 with copper (Cu) when the ten electrons fill the 3d shell (designated 3d
10
/.This
series is commonly referred to as the “iron group” and will be the central feature of
this book.
In a similar manner, the second series is built on a krypton core (Kr) with a
filled outer 4p
6
. It begins at number 39 with yttrium (Y) and ends with number 47
silver (Ag), but notably features a completely unfilled 4f shell, shown in Fig. 2.2b.
Next in importance to the 3d shell, the “rare-earth” or lanthanide transition series
Fig. 2.1 Periodic table configured to highlight the transition element groups
2.1 The Transition Metals 39
Fig. 2.2 Electron shell diagrams of the transition elements: (a) 3d
n
,(b) 4d
n
,(c) 4f
n
,(d) 5d
n
defined by the filling of the 4f shell begins with number 57 lanthanum (La) and
continues to number 70 ytterbium (Yb) with a complete 4f
14
.InFig.2.2c, it is
shown schematically that the rare-earth series is built on an xenon (Xe) core, even
though the 4f shell is shown pictorially inside the Xe 5s
2
and 5p
6
filled shells
(not shown). Beyond the Xe core there are also empty 5d, 5f ,and5g shells before
the outermost 6s
2
shell completes the electronic configuration of these elements.
The transition series shown in Fig. 2.2d begins with number 71 lutetium (Lu) and
ends with number 79 gold (Au) and is fashioned from the completion of the 5d
shell outside of the Xe core (now with the 4f shell filled), still inside the 6s
2
shell.
A final transition series may be considered as beginning with number 89 actinium
(Ac) and ending with number 101 mendelivium (Md), formed on a radon (Rn), but
with an incomplete 5f shell inside the radon core, analogous to the configurations
of the rare earth series. This group comprises many synthetic elements and is not of
any practical importance in magnetism.
Among the four groups of interest, only certain members become interesting for
their magnetic properties in the ionic states, and only the iron group and rare-earth
group have so far made a significant impact on the practical properties of oxide
compounds.