Magnetic resonance in the ordered state
by Stuart Lynn Hutton
A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in
Physics
Montana State University
© Copyright by Stuart Lynn Hutton (1988)
Abstract:
Traditional ferromagnetic resonance and antiferromagnetic resonance are reviewed and their limits and
shortcomings are examined. A new approach to the problem of ordered state resonance is presented.
This so called local coordinate method permits direct substitution of magnetic field angles for
magnetization angles in high field limits. This method is then generalized to multi-sublattice systems. It
is shown that in ordered systems, the usual torque equations can be generalized to provide resonance
equations based upon a Hessian matrix of a free energy expansion. This formulation is applied to a
proposed two sublattice ferromagnetic system in order to obtain a measurement of interplanar exchange
fields which are in approximate agreement with previous susceptibility work. MAGNETIC RESONANCE IN THE ORDERED STATE
by
Stumt Lynn Hutton
A thesis submitted in partial fulfillment
of the requirements for the degree
of
Doctor of Philosophy
in
Physics
MONTANA STATE UNIVERSITY
Bozeman, Montana
December 1988
£57*
n
APPROVAL
of a thesis submitted by
Stuart Lynn Hutton
This thesis has been read by each member of the thesis committee
and has been found to be satisfactory regarding content, English usage,
format, citations, bibliographic style, and consistency, and is ready for
submission to the College of Graduate Studies.
Il / A
Date
hChAAAM4 Jub^
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VV
Approved for the Major Department
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Date
Graduate Dean
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^ ^ / J/dV //fcPP
IV
ACKNOWLEDGEMENTS
It is
difficult to
encouragement
foremost,
I
guidance
in
Dick
would
like
complicated
Other
Franz
Robiscoe,
to
greatly
thank
affairs
people,
of
who
Hugo
Jack
life
and
Waplak,
Schmidt,
Drumheller
deserve
Waldner, Stefan
Bender and my family.
have
all the people who
have contributed
to me in orderto make this thesis a reality.
appreciated.
Rubins,
acknowledge
a
whose
physics
special
George
First and
has
continual
always
mention
are
Tuthill, Sachiko
Jerry Rubenacker,
Yadollah
been
Roy
Tsuruta,
Hassani,
Paul
There are, of course, others whose influence I
appreciated,
most
notably
Jennifer
Tuthill
and
Patty
Drumheller.
I
would
and contributed
especially like
to
to
form.
the
final
thank the people
These people
who
read this thesis
are Jack Drumheller,
Dick Robiscoe, George Tuthill, and Hugo Schmidt.
This acknowledgment would be incomplete without also thanking my
very good friends in life.
three have
three
times
been provides
by
my
three
To
have even one friend as enduring as these
a fortunate life.
very
good
My life
friends
Mitzi
has been enriched
Hundley,
Benaquista and Karin Hochli.
Financial support for work was from NSF grant DMR-8702933.
Matthew
TABLE OF CONTENTS
Page
I.
INTRODUCTION..................
II.
THE DEVELOPMENT OF A NEW APPROACH TO
FERROMAGNETIC RESONANCE ....................................................................... 5
I
The Problem as Formulated by Kittel................................................................ 5
Free Energy Approaches to the Problem of
Large Anisotropy Fields...................................................................................... 7
The Failure o f Standard Free Energy Approaches
in High-field L im its....................................................
17
The Local Coordinate Approach to Ferromagnetic
Resonance........... ................................................................................................... 21
Equilibrium Conditions in the Local Coordinate
Formalism............................................................................................................... 27
Relating the Free Energy to Local Coordinates................................................29
Systems with Cubic Anisotropy and the Local
Coordinate Formulation.........................................................................*.......... 35
Smit and Beljers Revisited (and Revised)...........................................................38
A General Solution to the Ferromagnetic
Resonance Problem................................................................................................40
The Eigenvalue Approach to Ferromagnetic
Resonance Problems.................
44
III.
A SURVEY OF MAGNETIC RESONANCE IN SYSTEMS OF
SEVERAL SUBLATTICES..................; . . . ...........................................................49
IV.
MAGNETIC RESONANCE IN MULTIPLE SUBLATTICE SYSTEMS . . . . 54
Ordered State R esonance.....................................................................................54
Effective Fields in Multi-sublattice System s....................................................60
Obtaining Effective Fields in Ordered State
Systems ................................................................................................................. 61
A Hessian Matrix for the General Resonance
Equation...........................................................
64
Examples of the Application of the Generalized
Equation................................................................................................................. 67
The Two Sublattice Ferromagnet........................................................................69
Examples Where Hessians Might Appear in
Classical M echanics...............
76
vi
TABLE OF CONTENTS-Continued
page
V.
MAGNETIC RESONANCE IN A TWO SUBLATTICE
FERROMAGNET.........................................................................................................81
The Two Sublattice Ferromagnet................ * ; ..................................................81
Limits to Interplanar Exchange Fields Which
Can be Measured....................................................... ...........................................92
The Real Model for [NH3(CH2)7NH 3ICuBr4. ..............................................95
VI.
CONCLUSIONS AND FUTURE EXTENSIONS OF THE THEORY............ 96
Results of This W ork .....................
96
Future Extensions of the Theory and Experiment .......................................... 97
VII.
LITERATURE CITED........................
98
vii
LIST OF FIGURES
Figure
1.
page
The coordinate system used
in the Smit and Beljers
formulation...............................................................................
8
2.
The free energy as a function of magnetic field for a
system with uniaxial anisotropy when the magnetic field is
directed
perpendicular
to
the
easy
axis.
The
magnetization
orientation
will
be
the
curve
which
minimizes the free energy.............................................................................................. 14
3.
Behavior of the resonancefrequency as a function of
magnetic field in a uniaxial system...............................................................................16
4.
The coordinate system used in the
development of
local coordinate m ethod....................................................................
5.
the
The geometry used to illustrate that in a more general
formulation, the equations of Smit and Beljers do contain
first derivative tenns...............
23
39
6.
The frequency dependence upon field o f a two sublattice
ferromagnet with a free energy model described by Eq.
[4.42] for two values of the inter-sublattice exchange field................................... 75
7.
A classical coupled harmonic oscillator......................................................................77
8.
The angular dependence of
the two resonance
peaks
observed in [NH3(CH2)7NH 3]GuBf4. .........................................................
9.
87
The antiferromagnetic circle is shown
centered about the
x-axis.
The presence of a second resonance peak
indicates [NH3(CH2)4NH 3JCuBr4 is a four sublattice system ................................ 88
Viii
LIST OF FIGURES-Continued
Figure
10.
The temperature dependence of
the two resonance peaks
with magnetic field slightly
off the y-axis.
Data
indicated with a star are uncertain..................................................
page
89
11.
The temperature dependence o f resonances observed in
[NH3(CH2)7NH3ICuBr4 ..............................................................................................91
12.
The theoretical dependence o f
resonance fields for a
uniaxial
two-sublattice
ferromagnetic
mode.
Modes
corresponding to resonance curves are indicated..............................
93
Theoretical range of interplanar exchange which can be
observed with an x-band EPR apparatus................................................
94
13.
IX
ABSTRACT
Traditional ferromagnetic resonance and antiferromagnetic resonance
are reviewed and their limits and shortcomings are examined.
A new
approach to the problem of ordered state resonance is presented. This so
called local coordinate method permits direct substitution o f magnetic
field angles for magnetization angles in high field limits.
This method is
then generalized to multi-sublattice systems.
It is shown that in ordered
systems, the usual torque equations can be generalized to provide
resonance equations based upon, a Hessian matrix o f a free energy
expansion.
This formulation is applied to a proposed two sublattice
ferromagnetic system in order to obtain a measurement of interplanar
exchange fields which are in approximate agreement with previous
susceptibility work.
I
CHAPTER I
INTRODUCTION
Magnetism
thousand
has
years
as
been
is
known
to
documented by
civilization
the
fust
for
record
more
than
four
of the use o f a
compass during the reign o f emperor Hoang-ti in 2637
SC.
During a
pursuit of an enemy,
the emperor’s troops, who were pursuing the rebellious prince
Tcheyeou, lost their way, as well as the course o f the wind, and
likewise the sight of their enemy, during the heavy fogs
prevailing in the plains of Tchou-Iou.
Seeing which, Hoang-ti
constructed a chariot upon which stood erect a prominent
female figure which indicated the four cardinal points, and
which always turned to the south whatever might be the
direction taken by the chariot.
Thus he succeeded in capturing
the rebellious prince, who was put to death. [1]
This
thesis
properties
of
began
an
as
exchange
interest
estimate
in
matter
about
when
assignment
a layered
stemmed
of
is
from
interplanar
high-temperature
measurements.
the
theoretical
subject
to
to
measure
ferromagnetic
work
by
exchange
series
the
Rubenacker
was
possible
data these
to
The
et.
al.[3]
based
exchange
the
high-temperature
but
series
since
magnetic
The
work
ferromagnetic
reason for this
where
upon
powder
measurements
ferromagnetic
small,
interplanar
compound.
is
so
radiation.
weak
(NH 3(CH2) 7NH 3)CuBr4
is
experimental
microwave
expansions
With powder
and
the
only
an
fit
of
susceptibility
clearly
show
that
the
interplanar
expansion
convergence
2
is
not
good
enough
to
determine
an
accurate
value
of
the
interplanar
exchange.
Ferromagnetic
have
several
Kittel
of
resonance
confusing
resonance
sophisticated
totally
peaks.
anisotropy
can
is
be
Next,
Smit
be incorrect
and Beljers1113 to
relative
the
to
the
magnetization
therefore
has
vector
been
proceeding.
crystal
method.1123
axis
is
led
to
to
no
that
provide
way
assumed
at
a
that
and keep
certain
as well
for
found
turned
to
us
what
the
to
external
correct
we
will
historical
clear
analysis
reasonably
those
equations
formalism of
call
the
applied
assumption that
magnetic
these
field.
formalisms
the
to
the
angles of
as under the
out
any
found the free energy
parallel
necessary
This
we
compound
we
inadequate
There
fields
this
First,
mathematically tractable.
field
of
elements.
equations14"103 were
the
spectra
local
It
before
coordinate
Finally, we have generalized this local coordinate method to
be applicable to systems of many sublattices and any free energy model
with terms up to fourth order in magnetization.
In
from
Chapter
thefirst
enhancement,
II,
the
theory
observation
the
local
of
coordinate
of
ferromagnetic
ferromagnetic
method
is
generalized
to
include
tomany-sublattiee magnets
Finally,
V
Chapter
we present
our
In
Chapter
reviewed
the
III,
latest
systems
In Chapter IV, the local
whichapplications
in
is
resonance to
method.1123
with more than one sublattice are examined.
coordinate
resonance
multi-sublattice
data
are
on
still
systems
for
waiting.
the layered
system
[NH 3(CH2) 7NH 3JCuBr4. We will show that when this system is considered
to
of
be
just
a different type
one,
of ferromagnet,
additional
resonance
one with two
peaks
are
sublattices instead
predicted
in
the
3
ferromagnetic
interplanar
resonance
exchange.
spectrum with
Measurements
a
separation
from
of
twice
[NH3(CH2 )4NH 3JCuBr4
the
agree
with estimates from the work by Rubenaeker et. a l.^
In
many
respects,
equation
draws
from
Hesse
the
formulationof
the mathematical
a
generalized
foundations laid
o f the last century and it is only
resonance
by
Ludwig
Otto
fitting that a short biography of
him be presented.
Hesse is little known for the Hessian matrix, which is a matrix of
second derivatives of functions.
student
of
Jacobi
whose
It is worth noting that Hesse was
determinant
gives
approximation to coordinate transformations.
Konigsberg.
university
Hesse’s
In
in
life,
facilitated
was
through
and
showed
Konigsberg
he
to reduce a
terms
1840,Hesse
received
then
began
to
show
how
determinants
and
by
that
"his
the
introduction of
doctorate
at
a
linear
determinant
gives
there.
elimination
substitution,
Hessian
for every
degree
professorship
algebraic
his
(linear)
1811
form of curves in third degree to one
hence,
order
Hesse was bom in
his
and
able
thefirst
the
the
In
could
he
at
was
be
able
involving only three
determinant.
curve
Hesse
another curve, such
that the double points of the first are points on the second".
In
1946
ferromagnetic
J.H.E.
resonance.
others
introduced
shape
effects
peaks
frequency
in
were necessary.
Later,
torque equations
for
resonance
GrifftIis^14-* reported
high fields.
could
zero
be
external
from
1947
with
magnetic
first
to 1949,
it
which
field,
These modifications were
was
Kittelt4"101
fields
found
gave
and
observation
C.
demagnetizing
Later,
observed
the
a
to
of
and
explain
that
additional
finite
resonance
additional
modifications
obtained in the years
o f 1955
4
to 1956 by J. Sniit and H G , Beljers^^, H. Suhl^^-*, P. Tannenwald and
B.
Laxfl6j,
success
TL.
Gilbertci7j,
was based
ferromagnet’s
upon
and
the
magnetization
vector
expansion in terms
vector,
Rf,
equations
it
which
Within
this
results
is
are
the
free energy
numerically
the
formulation
approaches
fails
to
the key
that for low magnetic
does
not
necessarily
assumed by Kittel.
to
determine
magnetization
approach,
correct,
approximation that Ivf and i f
Thus,
and
By
of laboratory coordinates o f
possible
yield
Artmanci8j
concept
magnetic field direction as was
energy
J.O.
it
the
is
explicit,
for
shown
later
equations
fail
are parallel with i f
o f FMR
reproduce
which
the
the
along
assuming
the
a free
the magnetization
any
complicated,
applied
field.
that although
in
the
very
the
simple
not in the x-z plane.
was based upon
equations
their
fields,
lie
though
direction
to
of
Kittel
the
free
when
energy
the
same
approximations are used.
This discrepancy was clarified in 1988.
As a portion o f this work,
the standard equation o f FMR is revised to give a form which is valid
in the
high-field limit and although the form is
morecomplicated than
that
Smit
able
of
and
physical
pictures
previous
free
Beljers,
of
energy
the
the
new
resonance
formulations.
theory
process
It
is
is
which
with
this
is
to yield
important
lacking
from
the
that
we
background
now trace the development of the new ferromagnetic resonance equation.
5
CHAPTER II
DEVELOPMENT OF A NEW APPROACH
TO FERROMAGNETIC RESONANCE
In
this
presented.
Kittel
chapter,
The
to
the
evolution
development
enhancements
by
of
ferromagnetic
is
traced
from
an
Smit
and
Beljers
initial
and
resonance
is
fonnulation
by
later
to
the
final
addition which we call the "local coordinate" method.
The Problem as Formulated by Kittel
The
fundamental
magnetically
ordered
equation
systems
describing
is
the
torque
sublattice
equation.
motion
The
in
torque
equation was due to Kittel and is simply given by
dM/dt = YhtxIfleff
Efeff
where
by
the
large
as
of
one
magnetic field,
between
the
effective
magnetization
group
appear
is
Ef
spins
entity.
ht,
field
where
so
In
acting
the
strongly
the
upon
sublattice
coupled
limit
[2.1]
that
by
Efcff
the
sublattice
is
considered
exchange
is
simply
designated
to
fields
the
be
as
a
to
external
this equation will yield resonance with the proportionality
and G), the microwave frequency, given by y, which is the
gyromagnetic ratio and is equal to ge/2me.
With certain approximations,
6
it
was
shown
that
mechanics.1-6’19"211
this
torque
Anisotropy
could be
and
obtained
demagnetizing
fields
from
were
so that H^eff no longer represented the lab magnetic field.
field
approximation,
these
anisotropy
fields
were
quantum
introduced
In the mean
assumed
to
be
of the
form
I t a= KxMxx+KyMyy+K Mzz
[2.2]
where Kx, Ky and Kz were first order anisotropy constants and Mx, My
and Mz were
laboratory
components
effective
was
by
field
fields.
The
given
the
quantity
of
the
magnetization
sum
of
anisotropy
was
calculated
vector.
fields
using
the
The
and external
high
field
assumption that h f and t f are parallel:
Y h lxtfeff = y{ [My( H +K Mz)-Mz(H + K yMy)]x+
[Mz(Hx+KxMx)-Mx(H + K M z)]y+
[2.3]
[Mx(Hy+KyMy)-My(Hx+ K M x)]z }.
The
usual
assumption which yields
is that which permits both h i
the
condition for magnetic resonance
and t f
to have large steady components
and small time dependent components.
Eq.
[2.3] was then expanded to
first order and the resulting set of equations were solved for t f
parallel
A
to the z axis. The result, known as the Kittel equation, is given by
to = -y{[H+(Ky-K)M ][H+(Kx-Kz)M ]}1/2.
[2.4]
I
Fxee Energy Approaches to the Problem of Large Anisotropy Fields
The
equation
introduced
free
took
energy
formulations
place in
1955
what
has
since
and
resonance.118,22"26’121
advantage
large
show
the
derivation
details
of
of
this
anisotropy
this
very
the
1956
could
formulation,
important
the
This
fields
ferromagnetic
when Smit
become
ferromagnetic
that
of
it
resonance
and
standard
Beljers[11]
equation
formulation
be
included.
is
helpful
resonance
had
In
to
equation.
of
the
order to
present
the
Also,
this
derivation will serve as a model to which the newer formulations can be
compared.
It
begins with the same
torque equation
(Eq. [2.1]) as used
by Kittel.
Since the definition of torque is given by the cross product
of a distance and a force, it was necessary to define an effective force
acting upon the magnetic sublattice which is assumed to be the gradient
of some free
energy, F.
It is fairly straightforward
to show then that
in spherical coordinates, the torque is given by
r>= -[(dF/d9)ee -(l/sin9)OF/d<l>)ee]
A
[2.5]
A
where e^, and ee are the unit vectors associated with the usual spherical
coordinates
as
equation can be
shown inFigure
expressed
I.
in terms
The
left hand
of the spherical
side of the torque
components
magnetization vector to obtain the time variation of the angles 9 and <E>:
of the
8
X
Figure I.
The coordinate
formulation.
system
used
in
the
Smit
and
Beljers
9
(l/y)dlvf/dt
For
non-zero
unit
= M{ (d0/dt)e0 +. (d<D/dt)sin 0 e^}
vectors,
a
set
of
coupled
first
[2.6]
order
differential
equations which govern the time dependence of 0 and <E> is obtained by
equating components of Eq. [2.5] and Eq. [2.6]. The result is
dO/dt = -(y/M sin0)(dF/d0)
[2.7]
d0/dt
[2.8]
= +(y/M sin0)(dF/dO).
The next step is to expand the free energy to lowest order in a Taylor
series about equilibrium values of 0 and <E>:
F =const.+A0 F0 + A 0 F^ +(l/2)A 02F00.+(l/2)AO2Fo o + A0AC>FO0
where
assume
subscripts
the
denote
orientation
differentiation.
The
which minimizes
the
+ e«,e
magnetization
free
energy
[2.9]
vector
will
and thus it is
required that
F0 = 0
Later,
it
will
equivalent
direction
to
of
unimportant;
expansion,
be
the
the
shown
= 0.
that
requirement
effective
here we
only
and
this
that
field.
[2.10]
minimization
the
The
magnetization
constant
assume it to be zero.
second
order
derivatives
condition
vector
term
in
is
lie
Eq.
almost
in
[2.9]
the
is
Thus, in the free energy
survive.
Substitution
of
Eq.
10
[2.9]
into
Eqs.
[2.7]
and
[2.8]
then yield
equations
of motion
for the
angles 9 and C> in terms o f 0 and O themselves:
d0/dt = -(y/Msin0)(A0 F00
[2.11]
d9/dt = +(YZMsmO)(Ad)
where it
their
is now
understood
equilibrium
derivatives
variation
of
of
positions
the
the
angles
angles
that the derivatives
given
are
from
[2.12]
+ Ae f 6$)
by
equal
Eq.
to
their
are to
[2.10].
the
be evaluated at
Since
time
the
derivatives
tune
of
equilibrium positions,replacement
the
of
d0/dt by d(A0)/dt and dO/dt by d(A<D)/dt is possible.
The final step is
to
<3> with
seek
normal
modes
for
the
variation
of
0
and
harmonic
variation. Thus, one employs the rotating wave approximation,
d(A0)/dt=zcoA0
When
this
is
used
and
with
Eqs.
d(AO)/dt=zcoA<$>.
[2.11]
and
[2.12],
[2.13]
it
gives
the
set
of
equations:
A0[ (Y/Msin0)Fee ]
+ A # [ (y/M sin0)F^ + zw ]
A0[ (Y/Msin9)Fe(I) - m ] +
This system
may
now
be
=0
[2.14]
A $[ (y/M sin9)F ^ ] = 0 .
[2.15]
solved
by
setting thedeterminate
of
the
11
coefficients
equal
to
zero.
The
result
is
the
standard
equation
for
ferromagnetic resonance which is
CO2
It
= (Y/Msin6)2{Fe6F ^ - F 2$ }.
is now important to
this
it
will
Kittel
be
equation
discuss simple applications of Eq. [2.16].
explicitly
results
[2.16]
seen
for
that
some
this
very
form
does
simple
not
From
reproduce
geometries
when
the
one
assumes high field !units.
The first and simplest example for the application o f Eq.
a
uniaxial ferromagnet.
For
this
system,
one
assumes
a
[2.16] is
free energy
expansion of the form
F = (l/2)K zM2- l M
[2.17]
where Kz is proportional to the uniaxial anisotropy field
and
is the
sublattice
axis
anisotropy
field,
magnetization.
Kz= - 1Kz
A
along the z
problem
to lie
is
I
so
when
Kz to
H^=O,
represent
the
an
easy
magnetization
vector
will
lie
AA
axis.
When the magnetic field lies in the x-y plane, this
exactly
strictly
that
For
solvable.
If the magnetic field
is
further restricted
along the x axis, then there is no loss
of generality in
the problem. With these simplifications, Eq. [2.17] reduces to
F = ( I ^ K zM2Cos2B1-HMsinQ1Costh1
[2.18]
12
where the magnetization polar angles are by 0^ and
field angles are 0 and <E>,
and the magnetic
The next step is to obtain first and second
derivatives of Eq. [2.18]. Explicitly, these derivatives are given by
F
= H M sin0,sin0
[2.19]
= -KzM2sin01eos01-HMcos01cos<I>1,
[2.20]
I
F0
and
F0 0
must be
[2.21]
F0 0
I I
[2.22]
= HMcosO1SinO1,
= -KzM2Ccos2O1-Sin2O1!+HMsinO1C o s O 1 .
According to Eq.
zero
F^ ^ = HMsinO1CosO1,
I I
[2.23]
[2.10], the solutions to the first derivatives equated to
obtained in
order to
determine the
equilibrium orientation
of the magnetization vector. When SinO1 is non-zero, Eq. [2.19] implies
SinO1 = 0.
[2.24]
Clearly, then the solution for O1 is
O1 = 0.
[2.25]
13
Wlien Eq. [2.25] is substituted into Eq. [2.20], a simplified version o f the
second minimization equation is obtained,
-KzM 2SinO1CosG1-HMcosG1 = 0.
[2.26]
The two solutions to Eq. [2.26] are now easily obtained
and
There
are
therefore
and
theseare
CosG1 = 0,
[2.27]
SinG1 = -(H/KM ).
[2.28]
two
dependent
distinct
upon
behaviors
the
for
magnetic
the
field.
magnetization
Which solution
chosen in the high-field regions is easily understood since if
Eq.
[2.28] would
solution
imply imaginary values
G=7t/2 would be correct here.
choose at
for the
vector
angle
is
| h /K zM |> 1,
Gr
Only
the
Which of the two solutions we
a lower field,however, is not so clear.
In Figure
2, the free
energy is shown as a function of magnetic field for each of these two
possible solutions and it is clear that for
by Eq.
[2.28]
bothsolutions
| h /K zM |< 1, the solution given
always yields the lower free energy.
apply
so
that
there
is no
When
| h /K zM |=1,
discontinuity
in
the
magnetization angle as a function of field.
The
next
step
is
to
evaluate
the
[2.23]) for each of these two solutions.
6=7t/2, the three derivatives are given by
second
derivatives
(Eqs.
[2.21]-
In the high-field region, where
14
=TT
/2
FREE ENERGY
I?
MAGNETIC FIELD
Figure 2.
The free energy as a function of magnetic field for a system
with uniaxial anisotropy when the magnetic field is directed
perpendicular to the easy axis.
The magnetization orientation
will be the curve which minimizes the free energy.
15
[2.29]
[2.30]
and
F
KzM2H-HM.
[2.31]
The resonance frequency for this region is then given from Eq. [2.16] as:
In
the
other
region,
(to/y)2
= H(Hh-KzM). { IH / K M I>1}
where
SinO1=-(HZKzM),
(co/y)2
= K2M2-H2.
{ HZKzM < 1
regions,
additional
the
[2.32]
resonance
frequency
is
given by
Note
that
for
one
of
the
an
IHZKzM I
predicted to lie below the field
I
I }
[2.33]
resonance
general
behavior
external
field.
become
the
This
standard
of
the
simple
resonance
formulation
approach to
frequency
due
for
approach in
as a
Smit
and
function
of
Beljers
had
until in
1988,
The failure of the Smit and
certain high-field limits has been discussed by us[12]
a system of cubic
which is presented next.
to
Figure 3 shows
ferromagnetic resonance
when a serious discrepancy was uncovered.
Beljers
is
and indeed it is also predicted
that one can expect to observe resonance at zero field.
the
curve
symmetry and it is
a discussion
of this failure
MICROWAVE FREQUENCY
16
H /K M =1
EXTERNAL MAGNETIC FIELD
Figure 3.
Behavior of the resonance frequency
magnetic field in a uniaxial system.
as
a
function
of
17
The FMure of Standaid Free Energy Approaches in High-field Limits
The Smit and Beljers formulation has been enormously successful for the
simple systems in which no approximations need be
more
complicated
magnetization
is
systems
parallel
where
to
the
the
made.
approximation
magnetic
field,
is
the
However, in
used
Smit
that
and
the
Beljers
formulation fails for most models as well as for most symmetries unless
the external magnetic field is applied strictly in the x-y plane where the
A
z
axis
is
determined
demonstration
of
this
by
the
failure
direction
is
o f the
easy
presented here for
axis.
the
An
case
explicit
of
cubic
anisotropy.
The free energy model is assumed to be
F = Fz + K[L M2M2
+ M2M2
+ M2M2
1
x y
x z
y zJ
where no
first order anisotropy terms are present and the Zeeman term
( Fz=-H^lvf ) is represented by Fz.
formulation,
[2.34]
second
derivatives
with
According to the Smit and Beljers
respect
to
<D and
0
are
required.
For this free energy form, these derivatives are given by
+ ZKM4Sin4(G1) Cos^fc1),
+ K M 4I l 2 Sin2(G1)COS2(G1)Cos2(O 1)Sin2(O 1)-
[2.35]
18
4sin4(61)cos2(C>1)sm2( 0 1)+
[2.36]
2(cos4(01)-6sin2(91)cos2(61)+sin4(91)],
and
F0 0
=
+ 8KM4sin3(01)cos(01)cos(4<l)1).
[2.37]
With a system of cubic anisotropy, when a resonance experiment obtains
angular dependent data in rotating the field from the <1,0,0> axis to the
<0,1,0>
axis, the results are expected to be the same if the field were
rotated from the <0,0,1>
applies
to
magnetic
this symmetry.
expects
to
magnetization
be
axis to the <1,0,0>
resonance
in
cubic
axis.
crystals
Any
must
theory which
therefore
exhibit
In the high-field limit, when Et and Ef are parallel, one
able
to
make
angles
in
the
a
simple
second
replacement
derivatives
given
of
field
above.
angles
for
With this
approximation, the derivatives of the free energy become
?$*
= ^
Fe 0 = F00 + KM4[12
+ 2KM4sin4(9 )co s(4 0 ),
[2.38]
S in 2(O)COS2(O)COS2( O ) S in 2( O ) -
4sin4(0)cos2(O)sin2(O)+
[2.39]
2(cos4(0)-6sin2(0)cos2(0)+sin4(0)],
and
F
tpI6I
F00 + 8KM4sin3(0)cos(0)cos(4O).
[2.40]
19
In rotating from the <1,0,0> axis to the <0,1,0> axis,
0= tu/2 and 0
is
represented by a. The derivatives then become
f <d o
[2.41]
= F $e> + 2KM4cos(4a),
0IeI = Fe0 + 2KM4[(3/4) + (l/4)cos(4a)],
[2.42]
and
fV
The
angular
dependence
of
[2.43]
1=0 '
the
resonance
frequency
in
this
plane
then
reduces to
((O/Y)2 =
[Fq0ZM + 2KM3((3/4) + (l/4)cos(4a) ]x
[2.44]
[ F ^ /M + 2KM3cos(4a) ].
In rotating
from the <0,0,1>
axis
to the
<1,0,0>
axis,
0=0
and 0
is
represented by a. For this rotation, the derivatives then become
F*!*! = F ^ +
2KM4[( l/2)-(l/2)cos(2a)]sin2(a),
F0 e = F00+
i I
and
The
[2.45]
2KM4cos(4a),
[2.46]
[2.47]
Fv r 0'
angular
dependence
<1,0,0> plane then reduces to
of
the
resonance
frequency
in
the
<0,0,1>-
20
(co/y)2 =
[Fq9ZM + 2K M 3cos(4 cc) ]
[2.48]
x
[ F ^ (M s in 2Ca)) + 2KM3((l/2)-(l/2)cos(2a)) ].
A
comparison of Eqs.
[2.44]
not
behave
shows
in the
that
the tenns explicitly
proportional
to
[2.48]
the required symmetry with respect to the two-fold axis
show
K do
and [2.48]
same manner
nor
does
Eq.
at
a=7t/2.
There
are
formulation
[2.9]),
actually
two
shows this
it
behavior.
was observed
orientation,the
first
reasons
that
derivatives
that the
In the
in
(given
order
by
zero,
and thus the equilibrium orientation is
field
approximation
usually
is,
is
violated.
assumed,
If the
this
free
free energy
to achieve
and
Beljers
expansion
an
(Eq.
equilibrium
Eq. [2.10])
must
obtained.
When the high-
minimization
energy
Smit
is not
equate
condition
can,
expanded
about
to
and
an
equilibrium orientation, then one obtains a system of equations analogous
to Eqs. [2.14] and [2.15], namely
A0[ (y/Msin0)F00 ] + A 0 [ (y/MsinO)?^ + i d ) ] = -(yZMsinO)F0
[2.49]
A0[ (YZMsinO)F63l -
[2.50]
/CO ]+
The problem with the
first
seeks
CO
A<$>[ (y/Msin0)F$3) ] = -(y/Msin0)F$ .
solution to such a system
and obtains the
angles
of equations is that one
of deviation later.
Note that in
21
equilibrium, this problem does not arise.
Smit
andBeljers
formulation
transformation
to
F0^ /(M sin 2(9))
shows
does
azimuthal
the
The second reason is that the
not
properly
coordinates,
since
inconsistency.
It
we
system
so
resonance
reformulate
that
the
problem
in
and
correct
form
a new
equation
is
obtained
a
which
only
was
failures that a new approach to the problem was
section,
accomplish
the
because
sought.
local
the
term
of
these
In the next
cartesian
coordinate
Smitand
Beljers
of
the
is
applicable
under
the
attempt
to
approximation that M and H are parallel.
The Local Coordinate Approach to Ferromagnetic Resonance
The
understand
resonance
when
the
local
why
coordinate
the
apparently
high
development
Smit
and
failed
to
field limits
was
Beljers
correctly
were
begun
as
formulation
reproduce
assumed
and a
an
of
ferromagnetic
expected symmetries
direct
substitution
of
magnetic field angles for magnetization angles was performed.
The work
was
group
first
started
in direct
collaboration
with
the
Waldner
in
Zurich, and eventually expanded to include the Wigen group in Ohio as
well as Marysko in
Prague.
Henceforth, this work
is either referred to
as the "local coordinate" or the Baselgia formulation.
This method begins, as does the Smit and Beljers method, with the
torque
equations
but now
a
solution
is sought
ina
coordinate
frame
along tyi, and derivatives are now taken with respect to local components
of
magnetization
(MpM 2 M 3).
Here
M 3 is
parallel
to
Mt,
the
steady
22
state
magnetization.
As
a
result,
we
obtain
a
new
form
for
the
ferromagnetic resonance equation which is given by
«o/y)2
= [MF
11
The
torque
equation
is
- F m HMFm2m2- F M3]
;
expressed
in
the
local
-
[2.51]
I 2
(body)
coordinate
system
shown in Figure 4. In local coordinates, the torque equation appears as
dM/dt = /IvixH^eff
[2.52]
but now Ni refers to the local Cartesian coordinates (MpM25M3) with M 3
parallel to Ivf, the steady state magnetization.
between
the
local
coordinate
method
and
the torque equation is intimately attached to
Thus, the first difference
previous
in
a
Taylor
series
to first
is
that
a coordinate system defined
by the magnetization vector and not the laboratory.
expanded
formulations
order
The torque is now
in
components
of
magnetization. If we identify the torque as
Fr- Ylvtxlfeff
[2.53]
then the Taylor expansion appears as
3
r>= r>°+ X
/=I
where the
sum over j
is
am .
(a ry d M .)l
< Mj=O1M^=O5Mg=M >
7
J
[2.54]
over the three local Cartesian components
of
magnetization and
AM. =M 1-M?.
[2.55]
23
Z
Figure 4.
The coordinate system used in the development of the local
coordinate method.
The
two
components
relevant
components
of
magnetization
are
the
I
and
N»
24
2
since
all precession is assumed to occur around the 3 axis.
A A
Thus, we require the I and 2 components of the Taylor expansion (Eq.
JS
[2.57]
uF
JX
I
[2.56]
Il
and
r I = M2H3- M3H2
Jti
[2.54]):
Since it is from the derivatives of these two torque components that the
first derivative with respect to M 3 comes into Eq. [2.51], it is helpful to
show these derivatives. They are, for the F1 component,
and
CdF1ZdM1) = Mi CdH3ZdM1) - M 3CdH2ZdM1),
[2.58]
CdF1ZdM2) = M 2CdH3ZdM2) - M 3CdH2ZdM2) + H3,
[2.59]
CdF1ZdM3) = M 2CdH3ZdM3) - M 3CdH2ZdM3) - Hr
[2.60]
For the F2 component, they are
and
CdF2ZdM1) = M 3CdH1ZdM1) - M 1CdH3ZdM1) - H3,
[2.61]
CdF2ZdM2) = M 3CdH1ZdM2) - M1CdH3ZdM2),
[2.62]
CdF2ZdM3) = M 3CdH1ZdM3) - M 1CdH3ZdM3) - Hr
[2.63]
25
Eqs.
[2.58]-[2.63]
aie
substituted
into
tlie
first
order
Taylor
expansion (Eq. [2.54]), we obtain approximate expressions for the I and
N»
When
components of torque which are
r , - r > MllM2OH3ZdM1) - M3(SH2ZdM1)11
M2-0 M3_M>
+ M2CM2(SH3ZdM2) -M 3OH2ZdM2) + H3JI
^
[2.64]
+ (M3 - M )[M2(dH3/dM3) - M3CdH2ZdM3) - H7]!
2 <Mj=0,M^=O,M^=M>
and
T2 = q + M 1EM3CdH1ZdM1) - M 1CdH3ZdM1) - H3])
<M1=0,M2=0,M^=M >
+ M2EM3CdH1ZdM2) -M 1CdH3ZdM2)] I
<M1=0,M^=O,M^=M>
+ (Ms - M m 3OH 1ZdM3) - M 1OH 3ZdM3) + H1] I
For
a
system
to
assume
an equilibrium
[2.65]
M3^ M 3=M >
orientation,
the
■
constant torque
terms must vanish which thus means that the magnetization vector is in
the
direction
evaluated
first
at
order
simplify.
of
the
the
effective
equilibrium
in
M1
As
a
and
result
M2,
to
field.
orientation,
so
these
that
Furthermore,
retaining
Eqs.
only
[2.64]
assumptions,
we
the
system
is
terms
which
are
and
now
[2.65]
have
greatly
the
two
expressions:
E1 = M 1E -MCdH2ZdM1)] + M2[ -MCdH2ZdM2) + H3]+
(M3 - M)[ - MCdH2ZdM3) - H2]
[2.66]
26
and
F2 = M1[ M(dH1/8M1)-H3] + M2[ MCdH1ZdM2)] +
[2.67]
(M3 - M)[ MCdH1ZdM3) - H1].
The term (M 3 - M) is actually second order since
M3-M= M3-[Mj+M2+M2]1/2= M3-M3[1+(M2+M2)ZM2]= -(M2+M2)Z2M,
and
so
the
final
terms
of
Eqs.
[2.66]
and
[2.67]
are
ignored.
[2.68]
At
equilibrium the fields H1 and. H2 vanish which later will be shown to be
almost equivalent to requiring the free energy to be at a minima.
With
these simplifications, the new expressions for the I and 2 components of
torque become:
F1 =
-M M 1CdH2ZdM1)+ M2[ H3 - MCdH2ZdM2) ]
[2.69]
and
F2 = M 1C-H3+ MCdH1ZdM1I-H3 ] + MM3MCdH1ZdM2).
A
[2.70]
A
These expressions for the I and 2 components of torque are now equated
A
A
to the I and 2 components of the time rate of change of magnetization
(by Eq. [2.52]) to yield
(IZyXdM1Zdt)
=: -M M 1CdH2ZdM1)+ M2[ H 3 - MCdH3ZdM3) ]
[2.71]
(IZy)CdM2Zdt)
=
[2.72]
and
M1C-H3+ MCdH1ZdM1) ] + MM 3CdH1ZdM3).
27
The I and 2 components of magnetization are assumed to have harmonic
time dependence according to the rotating wave approximation, i.e.
M1= M ^ ztot) and M2=M"e(z'tot).
[2.73]
The two equations o f motion then become
(ZtoZy)M1
=
-M M 1CdH2ZdM1)+ M2I H3 - MCdH2ZdM2) ]
[2.74]
and
(ZtoZy)M2
We
can now
=
M1R I 3+ MCdH1ZdM1) ] + MM2MCdH1ZdM2).
obtain non-trivial solutions which thus gives
[2.75]
the resonance
equation,
W rr
= IM F h
-F
1
1
][MF
3
-F m ]
2 2
- [MFm
3
J2
[2.51]
1 2
Equilibrium Conditions Jn the Local Coordinate Formalism
The
two
local
components
assumed to vanish in equilibrium.
of
effective
field,
An investigation
H1
however,
a
discussion
formalism is essential.
of
effective
fields
in
H2
were
of this assumption
which will eventually lead to the equilibrium conditions.
done,
and
the
Before this is
local
coordinate
If we look at only the Zeeman term in the free
energy expansion, then we have
28
Fz = - E W .
[2.76]
It is clear that if we take the local derivative,
V^m = -[ (d/dMx)x + (d/dMy)y + (d/9Mz)z
then
the
general
result
gives
manner,
we
simply
have,
the operator in Eq.
the
in
external
fact,
[2.77].
A
],
magnetic
defined
our
[2.77]
field.
In
a more
magnetic
fields
through
similar definition of the effective fields
was used by Hernnannp7j in a four-sublattice resonance problem.
also
introduced
related
the
to
the
direction
an
effective
polar
angle
and
field
for
of
the
0
magnitude
are
a
specific
orientation
magnetization
ambiguous.
to
For the
Kittel
which
he
demonstrate
that
Kittel
this
case,
definition agrees with the definition:
I f 6ff =- V L
F
M
[2.78]
Thus, the assumption that H1 and H2 vanish in equilibrium is simply a
statement
of
vector
equilibrium
in
however,
is
the
in
assumption
is
contrast
in
to
that
as anisotropy
A
direction
the direction
the
of the
assumption
the direction to the external field
fields
the
that
of
the
effective
magnetization
field.
the magnetization
This,
is
in
since the effective field includes such
and demagnetizing fields.
Thus, it is
clear that if
A
we require the I and 2 components of effective fields to vanish, this is
equivalent to the requirement that the free energy is
A
respect
to
the
I
at a minima with
A
and
2
directions.
Note
that
there
is
no
such
29
requirement
placed
upon
the
3
direction
of
magnetization
would in effect require the absence of an effective field.
worth noting
that this minimization
condition is
since
this
Finally, it is
equivalent to
Fq=O and
(IZsinO)F4 =O from the Smit and Beljers formulation.
Relating the Free Energy to Local Coordinates
A
the
free energy expression for any system is ultimately connected to
laboratory
crystalline
the
torque
local
coordinate
axes
coordinates,
for
the
the
local
problem
coordinates.
AAA
1,2,3
since
the
external
magnetic
field
are defined relative to the laboratory coordinates.
equation
specified local
system
is
We
coordinate
to
transform
choose
the
A
completely,
letting
the
free
system
is
written
energy
to be
Since
to
in
the
defined by
AA
2
between the y and 2 axis.
formulation
and
be within
the
x-y plane
with
the
angle
O
With the angle 6 between z and the 3 axis,
the orthogonal transformation, B with elements b becomes:
■
-
0
0
X
COS COS
y
cosGsinO
-sin0
Z
-
-sin(&
sin0cos0
0
sin0sm<E>
COS
0
COS
I
6
0
2
2 79]
[ .
3
-
Each component M. in the free energy expansion can be replaced by
3
%
k—1 hJk Mk
[2.80]
30
with
is
the
definition
possible
to
of the transformation matrix
connect
the
two
coordinate
given
systems.
above.
An
Thus,
example
it
will
serve to simplify the actual mechanics of this process.
Returning
uniaxial
to
system
the
could
free
be
energy
expression,
described by
a free
it
was
energy
assumed
that
a
expansion of the
form
F = (1/2)K zM£ - E M .
For the magnetic field directed along the x-axis, in the
[2.17J
1,2,3 coordinate
system, using the transformation given by Eq. [2.80], the free energy can
be represented as
F = (1/2)K [
UzlM 1 + Uz2M2 + Uz3M 3 ] -
[2.81]
H, [ bX1w I + I»x2M2 + bX3M3 IIf we then use the transformation matrix defined by Eq.
[2.79], then the
transformed free energy becomes
F = (l/2)Kz[
- M1 SinO1 + M3 CosO1 ] [2.82]
Hx [ M1CosO1Costh1 - M2Sinth1 + M3SinO1Costh1 ]
where the magnetization angles are indicated by the subscript I .
31
Now
that
appropriate to
the
local
consider
coordinate
method
an example.
has
been
presented
Our problem will
begin
it
is
with the
familiar uniaxial anisotropy given by Eq. [2.17] with H* applied parallel to
A
the x axis. Then, the free energy appears as
F = (1/2)K
- HxMx.
[2.83]
The first step to obtaining the resonance frequency is,
according to Eq.
[2.51],
the
to
obtain
the
first
and
second
derivatives
of
free
energy
expression. Explicitly, these are given by
f M3
= Kz[
\M . =
fm
and
'MzbZ3 I
K,[ (b,if
M =
I 2
FMM
2 2
-
bzlbz2
Hbx3
[2.84]
,
]
,
[2.85]
]
,
[2.86]
[2.87]
= K[ (b^f
The resonance frequency is then given from Eq. [2.79] as
= ( M K ,[ ( b j :
] - [ K ,[
] - H b x, ]) x
l2.ooJ
I M K IbzA
From the
2
] - I t y MzX
transformation
matrix,
] - H b x3 ]) - I MKzIbzlUz2 ]
Eq.
[2.80],
elements b . The necessary elements are given by
we
are
able
to
)2
obtain the
\
32
b Zl
and
When Eqs. [2.89]-[2.92]
[ 2 .8 9 ]
= -sin 9 V
b z2 = ° ’
[2 .9 0 ]
b z3
" CosOj ,
[2 .9 1 ]
b x3 = SinQ1CostDj .
[2 .9 2 ]
are substituted into the resonance equation,
(Eq.
[2.88]), the result is given by
(fo/y)2
= { M Kz[ Sin2Oj]
- [ K J M zCosOj] -
x
SinOj COsOj ]]
[2.93]
{ M K z[ 0 ] - [ Kz[ M zCosO1] - HxSmO1Cos^j] }
The
next
step
magnetization
to
is
to
the
transform
local
all
coordinate
- { M KJO ] }2.
the
laboratory
components
of
system.
For
[2.93],
is
Eq.
it
necessary only to transform the component Mz which is given by
M z = -M j SinO1 + M 3CosOj .
In
the
local
magnetization
have
coordinate
no
system,
steady-state
~ [2.94]
however,
components
the
in
the
components
I
or
2
direction
and the only steady-state component of magnetization is M" = M33.
the transformed magnetization simply becomes
of
Thus
33
Mz = McosGr
[2.95]
Thus, the resonance frequency is seen to become
(a)/y)2
[ MKz[ S in 2G1-
Cos2G1]
+ Hx SinG1CosO1}
x
[2.96]
{- MKz[ Cos^G1I
At
this
point,
no
condition
+ Hx SinG1CosO1}.
has
been
placed
upon
the
magnetization
angles O1 and G1 but this will come from the solutions to the equilibrium
conditions
SFZdM1
and
SFZdM21<
M1=0,M^=O,M^=M>
= O
[2.97]
M1=OjM^=O1M2=M >
0.
[2.98]
The two equations are thus given by
^M1
and
K ZM A l
fm2 = W
'
12 -
H xb xl
hA
[2.99]
< M1=OjM2=O1M3=M >
2 l<
Once again, Mz is transformed according to Eq.
b are obtained from Eqs. [2.89]-[2.92]. The result is
= o[2.80]
pjoo]
and the elements
34
- MKzCosG1SinG1. -HxCosG1CostD1 = 0 ,
and
[2.101]
-HxSinO1 = 0.
[2.102],
The second o f these (Eq. [2.102]) implies that for Hx non-zero,
SinO1
which is
Smit
and
With this
= 0,
[2.103]
identical to the equilibrium equation obtained for O under
Beljers formulation
when
SinG1 is non-zero
(see Eq.
the
[2.24]).
solution for Op the solution to Eq. [2.96] becomes simpler.
It
is given by
CosG1[
M KzSinG1-
Hx] = 0.
[2.104]
The solution thus proceeds as before. Namely, one obtains
CosG1=
and
SinG1=
Substitution of Eqs.
O
if IHxZMKzI>1
-(HxKM )
[2.105]
if
and [2.106]
IHxZMKzI<1.
[2.105]
[2.106]
into the resonance equation
for
this model (Eq. [2.96]) then yields
(co/y)2 = Hx(H + MKz) if
IHxZMKzI>1
[2.107]
35
and
A
(co/y)2
comparison
= (KzM)2- H2 if
with
the
Smit
and
IIy M K z I<1.
Beljers
[2.108]
formulation
shows
identical
results (see Eqs. [2.32] and [2.33]).
The next example is a system with cubic anisotropy and will show
the utility
of the local
coordinate
formulation.
Here,
the
correction to
the Smit and Beljers formulation can be seen in the high-field limits.
Systems with Cubic Anisotropy and the Local Coordinate Formulation
In the following example, the solution for systems having first and
second
order
coordinate
cubic
anisotropy
method.
approximation,
It
unlike
will
the
terms
be
Smit
are
obtained
shown
and
that
Beljers
using
the
local
the
high
field
in
formulation,
a
simple
replacement o f field angles for magnetization angles does not lead to an
inconsistent solution. We assume a free energy model of the form
F = - F M + (I/2) [KxM2 + KyM2+ KzM2] +
[2.109]
y I ub(M2M2
+ M2M2+
M2M2)+
K=ubM2M2M2.
x x y
x z
y zz
2
x y z
In
first
order
to
and
second
complicated,
only
the
obtain
an
final
the
resonance
derivatives
explicit
results
AA
are
frequency,
required.
derivation
are
according
presented.
is
not
Since
this
particularly
The
when evaluated in the x y plane are given (for <E>=a) by
relevant
to
Eq.
model
[2.51],
is
so
illuminating,
so
second
derivatives,
36
f
M i M1
f M2M2
Iicz-1
=
+ 2K=ub+2K=ubcos2asin2a,
- Ek zI + 2K™b[sin4a
and
f Mi M2
[2.110]
- 4cos2asin2a
+ eos4a],
[2.111]
= °-
12.112]
The first derivative term is simply
Fjvi3 = -Hx(Cosa)-Hy(since)
+ M[Kxcos2a + Kysin2a]
+2MK™b(cos2asin2a
[2.113]
+ cos2asin2a).
The resonance frequency can then be expressed as
(co/y)2
where
the
anisotropy
other than
with
Eq.
contribution
the
Smit
coordinate
rotation
= [F1+K1M ((3/4)+(l/4)cos(4a))][F2+K1M cos(4a)],
F1 and
term.
F2
terms
For
the
are
due
purpose
of
the first cubic term have
[2.44]
shows
behaves
and
in
Beljers
formulation
direction,
that
from
the
in
give
the z
In
a
axis
than
set to zero.
as
the
to
expression
to the x
axis, the
cubic
anisotropy
tenns
first
show
different
the
A
these tenns
order
are required only now we have 0 = a and 0 = 0 .
the effective field is given by
other
demonstration,
expression,
fashion
formulation.
does
been
this
same
to tenns
[2.114]
comparison
order
cubic
behaved under
that
for
same
the
the
local
second
derivatives
The derivative which yields
37
Fm = -Hxsina-Hzcosa
+ [Kxsin2a + Kzcos2a]
[2.115]
+4K^ub(sin2acos2a ) .
The
other derivatives, which
are the
second derivatives
appearing in the
resonance equation, are given by
Fm M
= [K co s2a+K sin2a]+
1 i
[2.116]
2K'ub[cos4a-4sin2acos2a+sin4a ] ,
Fm m = [Kxcos2a+Kzsin2a]+2K^'b+2K™b[sin2acos2a ] ,
[2 J 17]
Fm M = 0.
[2.118]
and
Then the resonance frequency becomes
(to/y)2
where F 3
= [F3+K1M ((3/4)+(l/4)eos(4a))][F4+K1M cos(4a)],
and F4
again refer to terms
[2.119]
other than the cubic
anisotropy.
A comparison of Eq. [2.119] with Eq. [2.114] shows that the cubic terms
behave in
the same manner.
Eq. [2.48] which was obtained by the Smit
and Beljers formulation does not agree with Eq.
demonstration
that
the
local
coordinate
[2.119]
formulation
and this is the
correctly
reflects
the
38
symmetry of the problem where as the Smit and Beljers formalism does
not.
As was mentioned earlier, the Smit and Beljers formulation can be
changed to imply the local coordinate formulation, but this has not been
a
part
of
this
standard
description
for
ferromagnetic
resonance
until
local coordinate formulation.
Smit and Beljers Revisited (and Revised)
That
implied
the
by
local
coordinate
fonnulation
of ferromagnetic
a more general form o f the Smit and Beljers
included for completeness.[2?]
resonance
is
equations, is
In a more general form, the equations of
Smit and Beljers can be written as
(co/y)2
= (1/M2){ F ^ F 11ti - (F ^ 1)2)
where E, and T) are two orthogonal angular directions.
this:
Can the first derivative term in Eq.
into Eq. [2.120]?
F^
[2.120]
The question is
[2.51] , p.22 be incorporated
The geometry necessary for this is shown in Figure 5.
describes a change in the free energy by a rotation through a small
angle % in the ^-T) plane.
The variation of the free energy is given by
ClF1^ 3= (1/2)F^%2
Since the Fj^
= (1/2)F ^ M |/M 2.
[2 . 121]
term will only describe a . rotation from point I to point
3 in Figure 5, an additional term must be ,added to go from point 3 to
point 2. This term is given by
39
Figure 5.
The geom etry
used to illustrate that in a more
formulation, the equations o f Smit and Beljers do
first derivative terms.
general
contain
40
[ 2 . 122]
where 8M^~-(1/2)M^ /M.
Thus, the total variation in the free energy in
executing this rotation is given by
8F i ^ 3 ^ 2=(1/2)(Fm ^ -
Fm ^/M )M |.
['2.123]
It is not clear- from this more general equation, however, how one makes
the
Smit
transformation
and
derivative
to
Beljers
terms
the
usual
formulation
should
not
be
axes.
can
too
In
be
addition,
modified
the
to
surprising when
fact
include
one
that
the
the
first
considers
Eqs.
[2.49] and [2.50] in which it was shown that by not evaluating the free
energy at equilibrium, one could obtain first derivative terms.
A General Solution to the Ferromagnetic Resonance Problem
In
this
resonance in
section,
we
show
that
the
general
problem of
a one sublattice system can be solved. Though
proves to be fairly involved, the principles involved
magnetic
the
are the same
solution
as we
used for the two previous examples.
It is verifiable that in the mean
field
ferromagnet
approximation, any
expression
one
sublattice
has
the free
energy
41
P
E
P.
O
P -P x
E
CP
MPx MPy MP;
PxPy *
y
=
Il
o'
E
I
O
CO
[2.124]
where the index pz is defined by
Pz = P - Px - Py•
According
obtained
Since
die
to
by
I
Eq.
[2.78],
derivatives
and
2
the
of
components
Eq. [2.124]
components
of
the
[2.125]
of
with
the
effective
respect
effective
field
to
field
are
magnetization.
yield
equilibrium
conditions, we obtain them first. The two equations are
F
I
<Mj =0,M^=O,M^=M>
=
00
P
P- Px eye
E
E
E
E •
P= I Px= 0 py= 0 n=x,y,z
[2.126]
P
CPxPy
and
F
00
E
p=l
I
<Mj=0,M^=O,M^=M>
=
P
P - Px eye
E
E
E
px= 0 p = 0 n=x,y,z
[2.127]
P
0 PxPy
42
These two derivatives are then set equal to zero in order to obtain the
equilibrium orientation
for
equilibrium
should
equations
ferromagnetic
resonance
the
magnetization.
describe
experiments
In
not
only
principle,
the
these
orientation
in
but
also
in
susceptibility
p
experiments once the expansion parameters C
have been determined
^xPy
for a given ferromagnetic system.
The next derivative needed is that of
the one non-zero component of the effective field. This is given by
[2.128]
<Mj=0,M^=OjMg=M >
X
P=I
The
second
P
P-Px
E
E
p
=
0
p
=
0
rx
ry
derivatives
Fm^ ,
P
C
PxPy
P-I Px
Fm^m ,
and
x,3
Fm m
PZ
b z,3*
are
also
required.
These are
M1M1 <M1=0,M^=O,M^=M
P
E
p= 2
E
P - Px
E
O
I-
oo .
Py = 0
[2.129]
>
eye
E
P
C
PxPy
P-2
M
x
n = x ,y ,z
r 0
.
. Pn - 1
P ( n + 1 ) " 11, P (n + 2)
L P n P ( n + l ) f , n , l b n, S b Cn + ! ) , ! 15 (n + l ) , 3 b (n + 2),3
+
43
and
[2.130]
M M <M =O1M =0,M =M >
^ ^I
I
2
3
00
P
Z
P“ PX eye
Z
Z
P-2
M
x
Z
p= 2 px= 0 py= 0
n=x,y,z
t 2 P n P ( n + l ) l , n , 2 l)
Pn 'I
n, 3 b ( n + l ) „ 2 b
(n + 1 ) ' 1. P ( n + 2)
(n + l ) , 3 b . ( n + 2),3
+
3bPi-l!L bP(«ti),3]and
FM^M^ i <M1=0,M2=0,M3=M >
OO
P
Z '
Z
P -P x
Z
[2.131]
eye
Z
p = 2 Px= O Py= O
P-2
M
x
P
cP A
n=x,y,z
1h P(n + 1)"1|iP$l+2)
[-P n P (n + l ) ( , , n , l b (n + l ) , 2 + b n , 2 b (n + l ) , l
3U
(n + l ) , 3 °
+
(n+2),3
Pn "2 P,
Substitution
orientation
of
yields
feiTomagnetic
involved,
the
these
number
the
system .
the
three
resonance
Though
sym m etries
of
terms
derivatives
of
w hich
the
go
evaluated
frequency
the
general
system s
into
the
at
for
any
problem
involved
free
the
one
appears
w ill
energy
equilibrium
to
sublattice
be
drastically
expansion
quite
reduce
given
by
44
Eq.
[2.124].
calculated
in
In
addition, since
these general
computerized.
As
the
derivatives
expressions,
a final part to
the
this
have
problem
chapter,
an
already
can
be
been
readily
alternative view
of
the solution to ferromagnetic resonance problems is presented.
The Eigenvalue Approach to Ferromagnetic Resonance Problems
In
the
general,
form
effective
of
any
an
fields.
ferromagnetic
eigenvalue
The
resonance
problem
importance
of
problem
where
this
the
can
be
cast
into
eigenvalues
are
the
process
for
ferromagnetic
resonance is not so great but in the next chapter, we shall see that the
ability to formulate eigenvalue problems in the high field limit is useful
in order to provide information concerning resonance modes in a system.
Here,
we
present
the
simplest
case,
ferromagnet with uniaxial symmetry.
that
there
components
were
of
two
coupled
magnetization.
the
one
familiar
Recall from Eqs.
equations
These
for
the
rate
equations
problem
of
a
[2.71] and [2.72]
of
could
change
be
of
expressed
the
in
matrix notation as
[2.1.32]
MCdH1ZdM1)
MCdH1ZdM2) - zcoZy
M1
H3
0
M1
9
MCdH2ZdM1) + zcoZy
MCdH2ZdM2)
M2
0
It is clear that this problem is now in the form of an eigenvalue
H3
M2
45
problem.
If
Cramer’s
rule
is
applied
to
this
system,
then
the
eigenvalues are obtained as
H3 = (l/2 )[
MOH1ZdM1) + MOH2ZdM2)] ± [[
MOH1ZdM1) - M OH2ZdM2)]2 +
[2.133]
[MCdH1ZdM2) - z'cgZyH
From
the definition
MCdH2ZdM1) + ztoZy]]1/2
of the effective field
(Eq.
[2.78]),
it is
also clear
that for conservative free energies,
CdH1ZdM2)=
CdH2ZdM1).
[2.134]
Thus, the eigenvalue equation reduces to
H3 = (1Z2)[
MCdH1ZdM1) + MCdH2ZdM2)] ± [ [ MCdH1ZdM1) - MCdH2ZdM2)]2 +
[
From
field
Eq. [2.135],
are
obtainable
termproportional
Zeeman
terms
term
since
to
the
from
H3,
of magnetic
step is to
in the
field.
obtain modes.
MCdH2ZdM1)]2 + (toZy)2
high field region,
the
free
energy
Once
expansion
eigenvalues
the
[2.135]
the eigenvalues of magnetic
magnetic field.
then the
] 1/2
If
one
are
eigenvalues
Then it is possible to
always
separates
expressible
are
obtained,
contains
out
directly
a
this
in
the next
continue the problem to
46
obtain the modal matrix which will
then diagonalize the first matrix in
Eq. [2.132],
An
Our
example
model
magnetic
is
field
of
the
the
actual
standard
applied
process
problem
perpendicular
should
of
to
a
the
simplify
uniaxial
easy
things
system
direction.
greatly.
with
the
The
free
energy for this is then given by Eq. [2.83]:
F = (1/2)KM2 - HxMx.
[2.83]
The first matrix in Eq. [2.131] then becomes
-Ksin2G
-z'co/Y
[2,136]
zcq/ y
O
Thus, the eigenvalues o f effective field are given by
H3 = (1/2) {(-Ksin26)±[K2sin4e+(co/Y)2] 1/2].
Since
that
this
a
formulation
clear
separation
is
most helpful
between
[2.137]
in the high field
magnetic
field
and
fields is possible, we look at the solution for 0=7t/2.
limit in order
anisotropy
effective
From Eq. [2.137],
it is
H3 = (1/2) {-K±[K2+(to/Y)2] 1/2}.
[2.138]
47
Separation
o f the
magnetic
field
is now
facilitated.
Similar to before,
the effective field is given by
H3=-dF/dM3.
[2.139]
Thus, the external magnetic field for this model is simply H3=H and we
have the eigenvalues of the magnetic field which are given by
H = (1/2) {-K+[K2+(co/y)2] 1/2}.
[2.140]
Now that the eigenvalues for the magnetic field have been obtained, the
next step is to determine normal mode eigenvectors.
Substitution of the
eigenvalues into the top equation in [2.132] gives the eigenvectors as
-/( k +( 1/2) {-K±[K2+( oo/y)2] 1/2})y/to
M1
[2.141]
I
M2
Now
we
eigenvectors
matrix
in
define
and
order
a
modal
then
to
matrix
multiplying
obtain
a
Thus, the modal matrix is defined by
whose
the
columns
energy
diagonalized
form
are
matrix
of
the
made
by
from
the
this
modal
energy
matrix.
48
-*(K+( 1/2) {-K+[K2+( co/ y)2] 1/2})y/to
-z(K +(l/2) {-K-[K2+(co/y)2] 1/2) )y/to
I
I
[2,142]
The
final
eigenvalues
step
have
would
be
already
to
been
does not lend additional insight.
diagonalize
obtained,
the
the
energy
matrix
diagonalized
but
energy
since
matrix
In the next chapter, it will be shown
that the formulation of the general resonance problem will predict many
resonance
modes
and
the
question
of
upon the types of modes present in the system.
observability
depends
somewhat
49
CHAPTER HI
A SURVEY OF MAGNETIC RESONANCE IN SYSTEMS OF
SEVERAL SUBLATTICES
In this
chapter,
an overview
several sublattices is presented.
defined
in
terms
experiences.
magnetic
which
For
ions
the
the
in
with
to orient
The concept of a
local
example,
interact
serves
compounds,
of
environment
"classical"
each
these
exchange
o f magnetic resonance in
other
ions
in
is three
a
ferromagnetic
a
an
magnetic
ion
compounds,
all
exchange
interaction,
common direction.
dimensional
of
magnetic sublattice is
that
through
systems
and
the
In
entire
these
set
of
magnetic ions would be considered to be a single sublattice.
In the layered magnetic compounds, however, this exchange is not
three
dimensional.
Instead,
super-exchange
between
planes
much
interact
these
ions in
less
compounds
the
have
planes
strongly.In
the
strong
while
layered
ions
ferromagnetic
on
compounds,
differing
each
plane would be a magnetic sublattice while successive planes may not be
part of the
are
same sublattice.
antiferromagnetic
sublattice.
This
antiferromagnet
interaction
determines
antiferromagnetic.
successive
is
since
Inthe examples of these compounds which
in
in
planes
contrast
those
whether
are
to
systems,
the
not
the
the
part
of
the
usual
concept
nearest
neighbor
crystal
is
same
of
an
exchange
ferromagnetic
or
50
The
through
terminology
a
functions
super-exchange
non-magnetic
is
ion
so
non-vanishing.
investigated by Snively
that
The
et.
refers
the
to
magnetic
overlap
interplanar
al.[29] for
ions
integral
interacting
of
super-exchange
the
wave
has
been
[NH 3(CH2)nNH3JCuX for n=2-5
and
X=Cl4 and X=Cl2Br2 and it was determined that for two halide bridges,
the
super-exchange
susceptibility
was
clearly
decreases
studies
shown
on
as
the
the
IOth
magnetic
series
that as the distance
for
n=2-6
and n=8,
compounds
for
n=7,9
n=7,9
and
10
the
plane
and weakly
the
and
compounds
are
are
distance.
In
(NH 3(CH2)nNH 3)CuBr4,
it
In that work, it is also shown
compounds
10
of
between the planes is increased,
the interplanar superexchange decreases.
that
power
are
antiferromagnets
ferromagnets.^
The
ferromagnetically
coupled between the
while the
fact
coupled
plane leads
one
to
that
for
within
the
suspect that
they could be interpreted as two-sublattice systems.
There
are
references
literature,*-31,32^ but
the
to
two
systems
being
sublattice
referred
antiferromagnets.*33,34*
The
term
canted
sublattices
away
from
a
are
canted
an asymmetric exchange
Hamiltonian
as
E^-(S1XS2).
introduce
weak
many
a
It
ferromagnetic
compounds.*35'38*
There
has
are
been
which
however,
an
P.
Bloembergen*39* on
considered
planar
which
interactions.
has
(CnH2n^1NH 3)2CuBr4,
an anisotropy
One
also
finds
this
been
orientation
of
a
in
two
In the work by
two
sublattice
introduced
for
inter-sublattice,
possibility
does
measured
a
the
the
term
example
sublattice ferromagnet which has appeared in literature.
canted
appears in the
that
has
the
means
antiferromagnetic
shown
in
actually
interaction which
moment
is,
to
antiferromagnets
true
due usually to
ferromagnets
of
a
system
is
inter­
intra-planar
51
symmetric
exchange
term
in
such
sy ste m s^
which
forces
the
cant away from a common
direction thus giving a weaker
moment.
In
another
the
interaction
in
ferromagnetic
work,
possibility
systems
is
of
the
spins
to
ferromagnetic
Dzyaloshinski-Moriya
presented,
In
this
work,
the
compound (C6H5 (CH2)nNHg)CuX4 was investigated for X=Cl (n=l,2 and 3)
and X=Br
the
(n=l).
emphasis
alone,
as
conclude
One o f
that on
the more important points from
this work is
basis
measurements
the
in
the
work
by
that
the
presence
of powder
susceptibility
Rubenacker et.
al.,
of ferromagnetic
it
is not
possible
to
moment
is not
due to
an
anti-symmetric exchange tenn.
Magnetic
years
been
resonance
well
antiferromagnetic
problem
with
in
understood.
resonance
a
antiferromagnetic
is
treated
intersublattice
magnetic
field
is
applied
In
the
in
terms
molecular
similar to that o f a ferromagnet.
systems
has
founding
field
of
a
defined
for
many
work^9,41"48^
two
in
sublattice
a
manner
This early work showed that when the
along
the
easydirection,
two
resonance
(oVy)—[H+(Hmb-Hma)/2]±[-H^+Hk(Hmb+Hma)+(Hmb-Hma)2/4]1/2
[3.1]
frequencies are obtained which are given by
with
Hma
and
representing
the
Hmb
uniaxial
magnetic
field.
field
sufficiently
is
representing
The
the
anisotropy
two
large
field and
resonance
to
result
intersublattice
modes
in
the
H
exchange
is the external
precess
spin
field,
flop
until
the
transition,
Hfc
applied
magnetic
that
is
until
H-CZHt Hm)1".
[3.2]
52
The
other
easily
solved
case
is
perpendicular to the easy axis.
when
the
magnetic
field
is
applied
In this case, the resonance frequency is
given by
(to/Y)=(H2+2HkHm)1/2.
The
more
was
also
general
case
treated
experimental
in
of magnetic
theliterature
observations
extensive.
of
[3.3]
field
not
strictly
along
mentioned above.
antiferromagnetic
either
axis
The literature
on
resonance
is
fairly
Observations of the resonance field as a function of magnetic
field angle have been seen to result in the well-known antiferromagnetic
circle. ^49"51-1
This
antiferromagnetic
straightforward
circle
is
obtained
from
free
energy
resonance by
manner
by
expanding
the
the
original
formulations
considerations
in
free
about
energy
of
avery
the
equilibrium orientation for small angles o f deviation.
There
but which
are
examples
of
possibilities
Henmannt27-* has
from
a
Dzyaloshinski-Moriya
free
multi-sublattice
yielding
net
An
canting
antiferromagnetic
antiferromagnetic
the
approach
including
terms
motion
In
this
because
the
from
introduces a
coupling
modes.
which
The
paper,
four
sublattice
such
the
as
the
modes
are
in which the magnetization vectors process
important point
modes
elegant
systems
For example in
analyzed
interaction.
exchange modes
hidden
are
completely
energy
analyzed for
directions.
which
are not the usual antiferromagnetic systems.
orthoferrites,
no
systems
modes
this
always
work
between
is
move
the
in
opposite
conclusion
that
exchange
modes
of
the
also
been
results
in
observability
importance
of
modes
has
and
53
observed by
necessary
Tanaka et.
to
fit
antiferromagnets.
alJ52-* where
a six
resonance
data
their
In particular,
it was
sublattice system was found
on
hexagonal
assumed that only
ABX3-Iype
modes which
Correspond to a precession of the total magnetization are observable.
their
six
similar
sublattice
to
that
of
systems,
effective
Baselgia
et.
al.
fields
and,
are
as
obtained
is
in
described
a
in
In
manner
the
next
chapter, by the gradient of some free energy expression with respect of
components
of
magnetization.
There
are
also
other
magnetic
systems
which are thought to consist o f more than two sublattices, for example,
LiCuCl3o2H20 . [53"55]
In
of
the
several
next
chapter,
sublattices
is
the
theory
developed.
of magnetic
From
resonance
the previous
in
systems
discussion,
the
need for a coherent treatment of magnetic resonance in these systems is
apparent
and
it
will
be
shown
that magnetic
resonance
in
comprised o f sublatttices can be expressed in terms of a single equation.
all
systems
54
CHAPTER IV
MAGNETIC RESONANCE IN MULTIPLE SUBLATTICE SYSTEMS
Ordered State Resonance
Magnetic
described
possible
resonance
by
to
the theory
discuss
development
of
a
general
coordinate
method is
result
equations
are shown
although,
have
in
have been
which
many
shown
a
et.
strictly
below
very
equation.
resonance".
resonance
theory
in
is
is
is
not
so
easily
Nevertheless,
such
systems
more
and
it
complicated
it
is
is
the
systems
describing
magnetic resonance
sophisticated
literature
of several sublattices.
for
in
some
alJ56^ developed
the
application.
specific
ordered
a
state
Similar equations
compounds
but
number o f sublattices.
a form of the
sublattices
straightforward,
nevertheless
it
in
As
which is simple in appearance
applicable to two
general
It
II.
for these
only forsystems of a specified
the general problem
for
Chapter
obtained for systems
the
systems
In this chapter, the generalization of the local
cases,
in
Besser
is
in
to follow a resonance equation
appeared
example,
multi-sublattice
presented
magnetic
which we now present.
the
in
resonance
. While
has
these
As an
equation
the generalization
importance
in
that
o f resonance in a system of several sublattices and
free
this
energy
model can
general solution
be
that
described
we
call
by
a
"ordered
single
state
55
The
starting point
again is
the torque equations.
For the
simple
one sublattice systems, the torque equation appeared in Chapter II as
divt/dt = yjv£xH*eff
where
all
the
multi-sublattice
for
the
acting
system,
entire
on
relevant
the
it is
system.
a given
straight-forward.
of
terms
have been
sublattice.
vector
we
The
Instead o f the
magnetization
previously
inconvenient
Instead,
[2.1]
to
consider
will
consider
In
the
the effective
field
the
effective
generalization o f Eq.
torqueequations
for
defined.
the
entire
[2.1] will be
describing
system,
the
field
the
torque
motion
equations
refer to the motion or perturbation o f the magnetization vector of each
of
the sublattices.
Thus,
the
generalized
torque
equations
are
expressible as
dM^/dt = yivfcxH*0
Ivta
where
a
represents
magnetization
vector
and H*° represents the effective field
relevant
coordinates
defined
by
are
local
Ma
with
sublattice
acting upon sublattice <j.
Cartesian
with
associated
coordinates
parallel
to
of
M °,
The
magnetization,
the
sublattice
As with the ferromagnetic case, the components M° and
considered
magnetization
additional
are
<Ma,Ma Ma>
magnetization.
M°
the
[4 .1]
away
sublattice
problem by three.
to
be
small
from the
steady
is
seen
to
perturbations
state
of
direction
increase
the
Ma
the
sublattice
Thus,
dimensionality
of
each
the
In the final resonance equation, however, it will be
seen that the dimensionality is only increased by a factor of two since
the
3°
component
is
assumed
to
not
precess
being
always
in
the
56
direction
these
o f the relevant effective
torque
equations
then
field.
As
describe
the
in the
ferromagnetic case,
motions
of
magnetization
vectors when placed in the presence o f an external magnetic field.
order
to
solve
for
resonance
frequencies,
one
now
defines
the
In
torque
acting upon sublattice <y through
F^ct= yivfcxH^
[4 ,2 ]
where Eq. [4.2] is simply the right hand side of Eq. [4.1].
obtain
resonance
acting
upon
about
the
contrast
equations
sublattice
direction
to
the
a
of
one
for
is
the
the
expanded to
effective
sublattice
system
since
the
effective
field
in
the
sublattice
it
is
at sublattice
the
torque
a Taylor series
upon
account the effects
all
<y,
order in
acting
however,
general case,
experienced
of
first
field
case,
this Taylor expansion take into
the
motion
In order to
sublattice.
now
essential
By
that
of all
sublattices in
other sublattices
contribute to
<j.
For the
ith component
o f the torque appearing in Eq. [4.2], the Taylor expansion appears as
I
O p0
rT
j
where
+
3
Z Z
AM" (dlf/dM "),
CX=I 7=1
the
ellipsis
J
reflects
the
s
8
s
s
+
[4.3]
OOO
I < m ^= o,m ^= o,m ^= m 8>
fact
that
present in future formulations^571 and a
higher
order
terms
may
be
refers to a sum over sublattices.
The unusual notation of combining Greek and Roman indices is useful in
order
the
to
clearly
constant
separate
components
sublattices
of
torque
from
not
in
coordinates.
the
In
35 direction
thus for the 1° and 2° components of torque, we are left with
equilibrium,
vanish
and
57
r? ~
Z
Z
am “
(dJ7 /dMa)i
C H 7=1
J
s
s
.
[4.4]
I<M*=OAjf=OJM*=M >
It is now useful to break Eq. [4.4] into two parts, one part due only to
contributions
from
sublattice
<j and
the
other part
due
to
contributions
from all other sublattices. Thus,
rT
Z AM? (d n /d M ”),
7=1
J
s
[4.5]
J l<M?
■j 0 ,M^=0 ,M^=M^>
Z Z AM" (dl^/dM")
Ote^cr7=1
In a one sublattice system, of course, only the first term in Eq.
would be present.
direction
of
the
[4.5]
It is this first term which we now evaluate in the
effective
field.
The
notation
used
to
represent
the
derivatives of the effective fields will be
A"-5 = -CdH0VdM*).
It
is
necessary
to look
at each
[4.6]
component o f torque
appearing in Eq.
[4.5] and, owing to the complexity of Eq. [4.5], it is more useful to look
at the first portion and then the second portion.
For the
I component
of torque, the first portion of the expansion simplifies to become (using
I to denote this portion)
MCT{M"A"’"+ M"[A"J + (H"/M°)]}
[4.7]
58
where
evaluation
in
the
direction
of
the
effective
fields
is
understood.
For the 2 component, this first portion becomes
h=2
By
pulling
Eq.
[4.4],
appear in
= + (H3ZMa)] + MaA aJ ).
out
the
specific dependence upon
the first derivative terms, i.e.
the second portion of Eq.
[4.8]
sublattice
o
that
was
the effective field terms, do
[4.5].
in
not
Thus, these derivatives
are
seen to be given by (using II to denote this term)
Hm
=
2 [MaAaJ + MaA aJ]
c#a
[4.9]
for the Ia components. For the 2a components, these are given by
H.=2
Thus,
the
=
E [MaAaJ + MaAaJ].
Ia and 2a components
[4.10]
o f torque can be represented to first
order as
I J = Ma (MaAaJ + Ma[AaJ + (Ha/Ma)] } + Ma E (MaA aJ+ MaAaJ )
I J ~-Ma {Ma[AaJ + (HaZMa)] + MaA J a ) - Ma E (MaA J a+ MaA J a).
a#?
At
this
point,
represent
many
matrix
notation.
it
is
worth noting
equations
Now
and
that
these
that
the
are
Eqs.
[4.11]
capable
expression
of
for
and
being
the
[4.12]
[4.11]
[4.12]
actually
expressed
first
in
order
59
approximation
obtained,
to
the
normal
magnetization
manner
mode
have
similar
torques
solutions
harmonic
to
acting
the
time
upon
in
which
dependence
ferromagnetic
a
sublattice
the
are
been
components
sought.
resonance
have
case
Thus,
of
in
discussed
a
in
Chapter II, one uses the rotating wave approximation
MF=
where,
as
was
component
also
the
o f magnetization
[4.13]
case
is
in
the
ferromagnetic
constant
in
time
problem,
and along
the
3°
the effective
field, but now this field is not the effective field for the entire system
but
rather
proceeds
that
as
in
which
the
is
acting
ferromagnetic
upon
the
case,
sublattice
for
<y.
sublattice
If
a,
one
two
then
coupled
equations o f motion result. These equations o f motion are given by
[4.14]
0 = M "(A^ + Hf/M0) +
+ zto/yMCT) +
X [MfAf’f + MfAf’f]
0 #G
and
[4.15]
0 = M f(A f° - ZtoZyM0) + M f(Af° + HfZM0) +
We
shall
leave
these
two
equations
fields in the general resonance problem.
for
X
a time
[MfAf0 + MfAf0].
and
look
at effective
60
Effective Fields in Multi-sublattice Systems
In
the
one
sublattice
problem,
the
spins
were
assumed
to
be
interacting with an effective field which could be expressed in terms of
the
gradient
one
can
general
crystal
of
not
a
free
speak
experiences
and
of
energy.
a
single
different
magnetic
For
effective
effective
symmetries
sublattice
systems,
it
would
is possible
field
fields.
would be the same in each sublattice.
one
multi-sublattice
predict
systems,
since
It
that
each
is
sublattice
likely
the
however,
that
effective
in
the
fields
In any event, by analogy to the
to
define
effective
fields for each
sublattice by definition o f a sublattice-specific gradient operator:
= -t OZdM^l0 + (d/dM£)2° + (d/dM£)3c ],
[4.16]
where now, in a magnetic system comprised of I sublattices, one would
also in general expect / such distinct operators.
The usual method for
obtaining the effective fields appearing in the torque equation then is to
allow this operator to act upon the free energy.
quantity
can
be
expressed
as
the
second
It is apparent that the
derivatives
of
this
free
energy expression, so, a revised definition for the A“^ is given by
AaJ
SE
Cd2FZdMicW j5)
[4.17]
where, for a conservative free energy,
Cd2FZdMi0W j ) = Cd2FZdMjWa).
[4.18]
61
One
notational
ferromagnetic
simplification
case,
is
appropriate
derivatives
will
be
at this point.
denoted
by
As
with the
subscripts
as,
for
example,
FaJ5 = S2FZdMicW j5
[4.19]
and
Fa'5 I s S2FZdMaSM^I
ij
5
8
S
<M^=0,M^=0,M^=M >
[4.20]
Obtaining Effective Fields in Ordered State Systems
The
which
sublattice-specific
by,
effective
analogy
fields
energy
models
Under
the
sublattice
to
is
to
the
must
be
assumption
explicit
operator
case
defined
these
complicated
but
also
simultaneously solved
Nevertheless,
regard
direction
solutions
to
the
resonance
experiments
mean field approach.
and
for
by
of
two
the equilibrium
equilibrium
also
coordinates
to
the
are typically
the number
in principle,
[4.16]
while
laboratory
vector
yields
angles
free
coordinates.
associated
with
able
O0 and
G0
by
In ordered state resonance problems,
conditions
increases
by Eq.
o f this effective field, oneis
equilibrium conditions
equilibrium
local
magnetization
solving the equilibrium conditions.
however
to
with
that the
defined
o f ferromagneticresonance,
defined with regard
<j orients in the
obtain
gradient
conditions
to
not
equations
for
only much
which
must
each additional
conditions
should
magnetization
can be
apply
more
be
sublattice.
solved.
to
measurements
The
magnetic
in
the
Thus, we show explicitly how these conditions are
62
obtained
and
then
present
what
should
demonstrate the calculation involved.
be
a
simple
example
to
If one applies Eq. [4.16] to a free
energy, it is seen that the effective field is given by
I f a= - (dF/dMa) l 5
The
that
-(dF/dMa)25
- (dF/dMa)35.
[4.21]
yv*?
sublattice magnetization is parallel to the 3 direction and we see
As?
A<?
the I and 2 components o f magnetization must vanish.
Hence,
for sublattice <y, the equilibrium conditions are given by
OFZdMp I
0
< M ^= 0,M ^=O ,M ^=M >
[4.22]
and
(dF/dMp I<M5=0)M5=0M5=M >
These two equations,
a
system
comprised
which
usually
permits
direct
must
= o.
[4.23]
though sunple, become much more
of
I sublattices
be
solved.
substitution
of
In
magnetic
since
there
are
high
field
limits
field
directions
complicated for
2/
such
this
for
equations
formulation
magnetization
directions and thus the solution to these equations is not needed in this
limit.
local
Before we proceed, however, a connection between laboratory and
coordinate
systems
must
be
obtained.
In the
ferromagnetic
case,
this connection was provided by means of the transformation matrix B (
Eq. [2.79]) which gave the transformation as
63
X
cosOcosO
y
cos8sin<D
COS0
sin0sind>
Z
-sin0
0
COS0
resonance
problems,
In
ordered
state
-sinO
sin0cos<X>
I
°
2
[4.24]
3
the
generalization
•
of
this
transformation matrix is obvious and for sublattice a , it is given by
[4.25]
XCT
yG
COS0<7COS0<7
=T
cos0 a sin® a
z°
Thus,
-sin® O
sin0 a cos® a
Ia
cos® a
sin0<7sin® a
2C
COS0<7
3a
rsinG<7
as before,
each
0
component
of the
sublattice
magnetization M? in
the free energy expansion is replaceable by
3
M?=
Z
J
h=l
b?, M?
Jlc
K
with the elements bj^. given by the transformation matrix above.
[4.26]
64
A Hessian Matrix for the General Resonance Equation
If
the
effective
fields,
the
sublattice-specific
model,
are
substituted
which
operator
into
are
(Eq.
the
obtained
[4.16])
equations
to
of
through
a
motion
application
general
(Eqs.
free
of
energy
[4.14]
and
[4.15] p. 59), then these equations reappear as
[4.27]
0 = MjrCFJJ + FJZMct) +
- /co/yM0) +
E [MjFJ-J + MjFJ-J]
and
[4.28]
E [MJFJ-J + MJFJ-J],
0 = MJCFJJ + zco/ yMct) + MJCFJ-J + FJZMct) +
Ot^G
where
both
equations
have
been multiplied by
formulation is practically complete.
equations in the form of a matrix.
-I.
At
this point,
the
The next step is to express these I
Thus, for a system comprised of I
sublattices, one would have a matrix which is block-diagonal, termed the
frequency matrix and a second matrix which is the Hessian matrix.
first matrix is given by
The
65
[4.29]
I
-F^ZM1
/(o/yM1
-zto/yM1
-F1ZM1
0
0
0
o
o
o
0
0
0
0
0
0
z'to/yM2
0
0
0
0
. -FyM 2
-zto/yM2
-F2ZM2
O
O
0
0
0
0
0
0
matrix
diagonal matrix.
to
the
local
is
,
designated by
0
' -Fz3 /Mz
0
-ZtoZyMz
0 (1 ,2 ," .,/)
and it
is
ZtoZyMz
-FzZMz.
a 2/x2/ block
The second matrix is the Hessian matrix with respect
coordinates
IVTjr and M"
and is
Thus, the Hessian matrix of the system is given by
designated by
H (l,2,•••,/).
66
[4.30]
I
f !:2
F 1’1
2,1
FW
F:::
f I:!
F%
fS
fU
fIz
■
Fi:l
F2-1
U
m
F2-2
U
F%
fU
F%
F2’1
2,1
F%
F%
F%
Fzf1
Fz:',
fW
F'l2
FU
F'i2
■ F'i'z
f SJ
fI !
Fg
F21
F%
FIZ
f z'i
If we now define the column vector whose elements are the I5 and
N3>
F 1-I
U
components of the magnetization vectors for each sublattice as
Mj
Ml
[4.31]
67
then
it
is
easy
to
see
that
the
general
resonance
equation
can
be
written in the form
H(1,2,««,/)»]v£
-
= 0.
[4.32]
The solutions for frequency from Eq. [4.32] are given by
det{
H (l,2,•••,/)
-
£2(1,2,»••,/)
} = 0,
[4.33]
which is the general resonance equation.
Examples o f the Application of the General i zed Equation
There
are many simple
examples for the application o f Eq.
[4.33]
and the simplest example would be that of a one-sublattice system, i.e.
When
Eq.
I
0
-zco/yM1
2,1
[4.34]
is
zco/yM1
t
fS
F1-1
_____ I
fI:!
'Tl
ferromagnetic resonance. In this case, Eq. [433] can be written as
expanded,
the
[4.34]
-F^M1
result
is
the
single-sublattice
ferromagnetic resonance equation,
[MF>
„1„
M
1M11-F m1][MFm11„‘- F m1]
(Otf)
*3
Application
facilitated
by
of
some
Eq.
[4.33]
general
M2M2
to
expansions
systems
for
- [MFmV
‘“ 3
i2
* T* ~2
of
the
several
case
of
sublattices
two
is
sublattice
68
magnets.
Such
expansions
application
of
numerically
evaluate
much
Eq.
information
for two
[4,33]
Eq.
is
sublattice
are
since
not
computer
[4.33].
obtainable
systems
necessary
For
without
requires
for
the
algorithms
exist
two-sublattice
which
can
systems,
however,
The
expansion
computerization.
definition
computerized
of several matrices.
First
the reduced Hessian matrix, Hj., is defined by
Hr(l,2,...,/)
where
Qd is
= H (l,2,...,/)-f2d(l,2,...,Z)
the matrix with only
[4.35]
the diagonal elements
o f Q (l, 2 ,...,/).
The other matrix which is useful to define is termed the two sublattice
interaction matrix and it is given by
p
a
P a 8
M1M2
8
M1M1
X (a,5)
[4.36]
p
a
8
p
a
8
i M2M1
If one also defines the quantity %by
[4.37]
Sg = (i/yM5)
then
the
expansion
of
the
resonance
equation
for
a
two
sublattice
system is expressible as
[4-38]
2
- to2
E
IH (5+1)1
+
IX ( 8 ,5+1)I
} + IHr( 1 ,2 ) I
= 0
8= 1
where
the
index
5
is
cyclic
in
that
8=3
means
5=1.
Under
the
69
assumption
that
all
sublattice
magnetizations
are
the
same,
this
expansion reduces to
2
(toO4 - (roO2
[4.39]
2
{ IH (8+1)I + |X (5 ,5+1)1
} + IH (1 ,2 )I
=0.
The solution to Eq. [4.68] is given by
2
(toO2 = (1/2) 2
5=1
{ IHr(5 + l)|
+ |X (5 ,8+1)1
}
[4.40]
2
± (l/4 )[
E { IH (8+1)1
5=1
For
case
the
of
an
+ |X (5 ,8+1)1
}2 - 4 | H (1,2)1
antiferromagnet
in
the
] 1/2
antiferromagnetic
state,
evaluation of Eq. [4.40] is required.
The Two Sublattice Ferromagnet
A
particularly
fruitful
is on layered ferromagnets.
alternate plane
the
weak
sublattices
is
application
the
multi-sublattice
formulation
Here the crystal structure is such that each
assumed to
interplanar
of
be
associated with
exchange
and appears explicitly
becomes
one
the
sublattice
interaction
in the free energy.
so
that
between
As will be seen
below, the theory predicts two modes, that is, two resonances such that
the
difference
provides
a
between
them
theoretical
basis
is
for
twice
the
the
interplanar
sought
interplanar exchange for the case of ferromagnets.
after
exchange.
measurement
This
of
70
The
starting point for this example is
a free energy model for a
uniaxial ferromagnet. This is given by
F = (l/2 )K [(M i ) 2
where
in
interaction
term.
the
mean
between
+(M z2) 2 ]+ e S M 2- I fo tM 1+ M 2]
field
the
two
approximation,
sublattices
is
the
[4.41]
simple
represented
ferromagnetic
by
the
second
In Eq. [4.41], if one wishes to have the z axis represent an easy
axis,
then
K z= - I K z I
so
that
in
zero
applied
field,
the
magnetization
vectors Kf1 and Kf2 will lie parallel to the z direction.
Furthennore, if
e is to represent a ferromagnetic interaction, then e = -|e |
so that if all
other fields vanished,
energy
state.
applied
in
M 1 would
For this problem,
the
x-y
plane
and
be parallel to M 2 in thelowest free
we
assume that the magnetic
without
loss
of
generality,
it
field is
may
assumed that the magnetic field is applied parallel to the x axis.
be
Thus,
the free energy is seen to simplify to
F = (1/2)K £i[(M 1)
it 2
The
next
step
toward
+(M 2)Z2 ] + EM1-M 2- HX(MX
1 + M2).
X
obtaining
the
resonance
[4.42]
frequencies
determine the elements o f the Hessian matrix given by Eq.
[4.30].
is
to
The
explicit derivatives are given by
Fjj = Ktsin2G1]
[4.43]
F22 = K[sin2G2]
[4.44]
71
=
Fg' = Fg - Fg - Fg = Fg - 0,
[4.45]
Fn = E[cos01cos0 2 (cosC>1cos<D»2+sinO 1sinC>2)+sin0 1sine2] =F^,
[4.46]
F22 = £( SinO1SinO2 + CosO1CosO2 ) = F22,
and
Since
the
sublattices
the
F22 = EcosO1CsinO1CosO2
■■ CosO1SinO2),
F1I = EcosO2 CsinO2CosO1
• CosO2SinO1)
interaction
e
is
are parallel
in
their
derivatives
above
are
assumed
static
to
he
or
they
[4.48]
.
[4.49]
ferromagnetic,
orientations.
duplicated
[4.47]
Therefore,
vanish.
The
the
many
two
of
non-zero
elements o f the Hessian matrix are thus
F|J = Ktsin2O1]
and
Fn =
fS
=
F 22
=
FU
[4.50]
=
Fg =E.
[4.51]
With these simplifications, the Hessian matrix appears as
72
F}}
O
e
O
O
O
O
e
[4.52]
The next
step
is
to
e
O
O
8
obtain the
O
O
elements
o f the
frequency
matrix
(Eq.
[4.29]). The derivatives which give effective fields are
F3 = Fg = KzM 1(CosG1) 2
+ EM1 - HxSinG1Cosdi1.
[4.53]
The equilibrium conditions are also required for this problem if we wish
to look at solutions outside of the high-field limiting cases.
first
approximation
external
field
is
to
the
solutions,
sufficiently
large
let
that
lie in the direction of the external field.
in
the
above
expressions
angles 0 and <D.
0 = 7t/ 2
can be
simply
us nevertheless
the
effective
To obtain a
assume that the
field
does
indeed
Then, the angles G1 and (D1
replaced by
the
external field
Since the magnetic field is applied along the x axis,
and <D=0. With this simplification, the derivatives reduce to
Fjj = K
and
F 31 = eM 1 - Hx .
[4.54]
73
The
of
resonance
the
frequency
difference
is
between
then
the
obtained
by
Hessian matrix
evaluating
and the
the
determinant
frequency matrix
from Eq. [4.29]. Thus,
(to/y)
2
= [ H + KMg- (e ± e)Mg ][ H - (e + e)Mg ].
[4.55]
where Mg is the magnetization associated with a sublattice.
that
two
show
resonance
that
these
modes
two
are
modes
obtained
are
and
separated
it
by
is
It is clear
straight-forward
twice
the
to
inter-sublattice
exchange field.
This
to Eq.
For the lower sign solution, the resonance frequency is
[4.55].
result can be seen by looking at the two solutions
<2
given by (to/y)
= [ H + KMg][ H].
Hf=-(KMs/2)+[(KMg/2) 2-(to/y)^] 1^2.
solution is now chosen,
H=Hf-2eMg.
the
the
Inversion
exchange
is
Inversion o f Eq. [4.55] then yields
From
field
Eq.
in
Eq.
[4.55],
[4.55]
if
can
the
upper
be
replaced
o f Eq. [4.55] then shows that Haf=2eMg+Hf.
ferromagnetic
here,
Haf will
be
at
a
lower
sign
by
Since
magnetic
field than Hf.
The
two
modes
correspond
to an in-phase
resonance
mode
where
M 1 and M 2 precess in-phase about their effective field and to an out-of
phase
mode
where
phase
about
their
modes,
it
is
the
two
effective
necessary
to
magnetization
field.
In
substitute
the
vectors
order
to
precess
180°
determine
frequencies
obtained
out
of
resonance
in
Eq.
[4.55] into the system of equations given by Eq. [4.32] and then to solve
for the components o f
M 1 and M2.
For the frequency designated
lower sign, the resonance frequency is given by
by the
74
(oo/y) 2
similar to the value
[2.32]).
[4.56],
solve
it
for
order to
is
necessaryto
M|+M^
precession
about
which
frequency
the
The
M^+M^
corresponds
designated
simple ferromagnetic case (see
this
result
direction
mode
corresponding
frequency
yields
which
of the
to
[4.56]
resonance
substitute
and
the
in the
obtain
and M"
quantities
the
obtained
In
M"
= H(H+KMg)
into
solutions
Eq.
in
corresponds
effective
field
by
the upper
sign
is
and
of the
anin-phase
and
the sublattices precessing
to Eq.
[4.32]
terms
to
Eq.
Mj-M^
and
out-of-phase.
If
desired,
the
result
is
given by
(co/y) 2
which
is
unlike
a
= (H-2eMs)(H -2eM +KM )
ferromagnetic
resonance
[4.57]
frequency
in
several regards.
When the modes are obtained for this frequency, it is seen that the two
sublattices
precess
difference
of
resonance
frequency
and thus
implies
through
the
about
180°.
the
effective
field
The most obvious
explicitly
contains
the possibility
resonance
to
formulation
but
they
have a
difference, however,
a reference
measure
to
an
phase
is that this
exchange
field
this inter-sublattice exchange
developed
in
this
chapter.
In
Figure 6 , an example o f a frequency curve for a system such as this is
presented.
frequency
The
and the
solid
line
straight line
is
is
the
normal
ferromagnetic
an experimental
microwave
The dashed lines then correspond to this second mode with the curve
resonance
frequency.
Microwave Frequency (U/7)
75
External Magnetic Field
Figure 6.
The frequency dependence upon field of a two sublattice
ferromagnet with a free energy model described by Eq. [4.42]
for two values of the inter-sublattice exchange field.
76
shown
for
exchange
two
values
field
is
of
the
increased,
inter-sublattice
one
would
exchange
observe
field.
one
As
resonance
this
peak
moving to higher and higher frequencies until it vanished because all the
resonance
frequencies
Thus,
any experiment performed
in
that there is
lie
beyond
experimental
microwave
frequency.
at a constant frequency,
it is
clear
a certain maximum value of the exchange field for which
this resonance can be seen.
the
the
exchange
field,
the
On the other hand, for very small values of
two
resonance
lines
are close
together
and the
linewidth o f the more intense resonance line prevents resolution.
antiferromagnetic
ferromagnetic
case,
the
resonance
second
line.
mode
Inthe
lies
on
the
other
antiferromagnetic
In the
side
of the
systems,
an
increase in the exchange field would only shift the field position o f the
resonance
Thus,
an
but
it is
it
would
be
observable
at
a
constant
frequency.
clear that observation o f the ferromagnetic resonance due to
inter-sublattice
requirements
than
antiferromagnetic
slightly
always
more
exchange
the
field
observation
exchange.
In
complicated
the
version
would
o f the
next
have
resonance due
chapter,
o f this
more
we
model
stringent
to
shall
and
consider
discuss
an
a
its
application to ONH3(CH2) 7NH 3JCuBr4.
Examples Where Hessians Might Appear in Classical Mechanics
As
Hessian
suggested
formulation
by
the
to be
development above
useful in
one
might
any problem o f coupled
expect
the
oscillators.
Consider a system consisting of two point masses of mass M which are
connected, as shown in Figure 7. It is well known that the potential
Figure 7. A classical coupled harmonic oscillator.
78
energy
for
the
system
when
the
masses
are
displaced
from
their
equilibrium positions by distances X1 and X2 appears as
U = (IZZ)Kx1
In
contrast
to
the
+ (1/2 )K x 2
magnetic
resonance
+(VZ)E(X1-X2)2.
problem
one-dimensional problem but with two elements.
[4.58]
this
system
is
only
a
The Hessian matrix for
this system would appear as
K+£
-8
[4.59]
H ( X p X2)
-E
Determination
If
one
the frequency
recalls
frequency
elements
of
the
matrix
(in the
which
components
accelerations
of
ordered
magnetization.
matrix would be
matrix,
state
absence
corresponded
and
K+E
not velocities
to
however,is
formulation, it
o f effective
the
time
In
this
so
that the
straightforward.
was
seen
fields)
derivatives
system
we
had
of
are
elements
in terms of frequency squared.
frequency matrix is given by
less
of
the
that
off-diagonal
A
I
and
concerned
the
the
A
Z
with
frequency
For this problem, the
79
Mco2
0
[4.60]
Q(Xr X2):
0
Mco2
Thus, the eigenfrequenci.es are given by
det(H(xv x2)
Explicitly,
then,
one
-
obtains
= 0.
Q ( X p X2) )
the well
know
[4.61]
solution
for
two
coupled
harmonic oscillators as
co = [(l/M)(K+e ± e)](1/2).
we
see then
important
indeed
the
in
physics,
problem
different
from
imaginary
matrix.
that
In
that
elements
the
in
do
the specific
not
form
matrix
however,
magnetic
can
the
appear
resonance
problem
matrix
may
of
have
of this matrixmust
come
this
problem.
many
many
in
frequencymatrix
appear and the matrix was
moregeneral
oscillators, thefrequency
and
the
Hessian
[4.62]
only
from
is
Here,
the
a diagonal
coupled
off
very
diagonal
harmonic
elements
the problem
at
hand.
It is
interesting
to note
associated with the Hessian.
some
of
the
mathematical
properties
One o f the most obvious places to find the
80
Hessian matrix is in the study o f inflection points o f curves.
is
a
mathematical
theorem
that
given
an
algebraic
curve,
In fact, it
F,
then
inflection points o f
F are given by the intersections o f F and the
det(H)=0 where H
is the
mathematical
property
worth
Hessian matrix associated with F.
mentioning
is
the
fact
that
the
One
curve
final
nature
critical points o f a
curve, F, can be determined by looking at the
o f the determinant
of
H
the
of
value
at those critical points. If H(Xq) is positive
definite, then the critical point is a local maximum J58,59^
81
CHAPTER V
MAGNETIC RESONANCE IN A TWO SUBLATTICE FERROMAGNET
The Two Sublattice Ferromagnet
Earlier
we
compounds,
planes,
those
pointing
which
planes
discussed
out
defines
(sublattices).
with
that
the
they
have
suggests
In
this
strong
and
that
interplanar
systems.
sublattice structure
sublattices,
This
ferromagnetic
two-sublattice
the special
chapter,
coupling
the
exchange,
can
we
the layered
coupling
weak
even
of
within
the
between
the
ferromagnets,
be
that
considered to
present
the
is
be
experimental
evidence in support of the consequences of this assumption.
It
system
was
shown
in
with
only
an
sublattices,
equations
a second
which
Chapter
that
exists (see
permit
a
collapsed
observed
twice
antiferromagnet
to
the
be
separated
interplanar
a
in
which
from
the
ferromagnetic
acting
[4.55])
direct
to the
measurement
between
resonance
of
the
This is similar to the case of
such
normal
exchange field.[60]
uniaxial
term
Eq.
possible
interplanar exchange in high field limits.
a
in
inter-planarexchange
solution
would
IV
an
excited
resonance
ferromagnetic
Also, in
the
resonance
was
by
antiferromagnetic
compound [NH 3(CH2)4NH 3]CuC14, the same mode which corresponds to an
out-of-phase
Additional
coupled
precession
resonance
chains
by
lines
Phaff
of
the
have
two
been
et. aV61]and
sublattices
observed
they
in
were
was
observed.^51-1
antiferromagnetically
able
to
interpret
82
observed
weak
resonance
spectra
antiferromagnetic
canted,
line
magnetic
from
which
additional
o f canting
could be
angle
line
is
which
moment
[NH3(CH2) 7NH 3JCiiBr4.
work
by
defined
with
a feature
deposited over
two
et.
of
the
chains
were
additional
in
compound
the
system
two
films
These
and
sublattice
in
thus
having two
experiments
the
on
this
ferromagnetic
exchange
measurements
which
ferromagnetic
a
substrate as
systems,
an
because it shows a
that observed in
thin
resonance
Also,
resonance
a ferromagnetic
origins
two
determined.
their
Ferromagnetic
with
al.[63]
sublattice
models
similar to
obtained
Cochran
attributed
and magnetic
compound
is
with
This system is also interesting
show
material
system
an
observed
is
sublattice
Since
the
compound
been
interaction.
two
showed
sites.[62]
also
a
experiments
ferromagnetic
have
of
resonance
[(CH3) 3NHJNiCl3<>2H20
weak
terms
interchain
resonance
inequivalent
in
however,
in
the
all
have
the
treatment
of
these
ferromagnetic
resonance
equations
elegant
well-
systems
is
not
surprising.
Initial
experiments
were performed
on
crystals o f
the
compound
PNH3(CH2 )7NH 3JCuBr4 which were approximately .013 mm on a side and
approximately
.005 mm in width.
The fact that these crystals were so
thin permitted polarization with a light microscope in order to determine
the
crystalline
resonance
axes.
peaks
appearing
experiments
showed
could
be
not
Due
the
attributed
to
in
the
the
existence
to
small
size
spectrum
were
of
a
second
twinning.
For
of these
quite
crystals,
small
resonance
but initial
peak
resonance
all
which
experiments
A A
performed
allows
strictly
in
determination
the
of
z-y
plane,
exchange
a
simplified
fields
assuming
free
that
energy
model
fourth
order
83
anisotropy
terms
magnetic
to
fit
fields
the
includes
such
as
(MxMy) 2
are sufficiently
data
two
from
(MxMz)2
are
unimportant
high that Mx vanishes.
these
uniaxial
and
experiments,
anisotropy
a
terms
free
and
Then,
energy
an
when
in order
model
which
anisotropy
in
the
interplanar exchange is postulated. The form o f this model is given by
F=(l/2)K y[(My) 2 +(M 2) 2 ]+(l/2)K z[(Mz) 2 +(M 2) 2 ]+(l/2)K a[(Mz)4 +(M2)4]
[5.1]
- Hx[Mx+M2] -Hy[My+M2] -HJM z+M2] +E1[Mx«M2+MyoM 2]+£2 [Mz°M2].
In
this
model,
a
possible
demagnetizing
term
is
combined
with
the
uniaxial anisotropy fields Ky and Kz since a single resonance experiment
cannot distinguish between the two types of fields.^
Hx, Hy and Hz
are components o f the external magnetic field, Ka is proportional to the
inter-planar
exchange
field,
and
M?
is
the
the magnetization associated with sublattice a
represent
ferromagnetic
fields,
and M 2 will be parallel.
orientation
of
sublattice
the
sublattice
of
energy.
From the
free
static
(a= 1,2).
equilibrium
component
of
Since E1 and E2
orientations
of
M1
Thus, we need only to find the equilibrium
one
direction
the
iUl cartesian
which
can be
magnetization
energy
in
Eq.
obtained by
which
[5.1],
determining the
minimizes
the two
the
free
equations which
must be solved in order to obtain the polar magnetization angles 9 and O
which
make
a minima in the free
energy
are obtained by
taking first
derivatives o f the free energy with respect to local coordinates Mp
(with M 3 parallel to M) and then setting the derivatives, F ^
M2
and F ^
84
equal to zero.
For the model in Eq. [5.1], this yields the following two
equations for O 1 and S1:
K■yM sSinSi1CosSJi1Sin2O i -K
M SinSx1CosSx1-2KaMs 3Cos-Sx1SinSx1-Hx CosSx1CosO1l
z s
[5.2]
-HyCosS1SinO1+HzSinS1+(E1-e2)MscosS 1SinS1=O,
and
KyMsSinS1CosO1SinO1+HxSinO1-HyCosO1=S
[5.3]
where Mg is the magnetization o f a sublattice (Mg= IM*I = IM2 I).
equations must be
resonance
this
type,
second
solved numerically
frequencies
for
specifically
derivative
a
model
excluding
such
for the angles
with
interactions
as
for
O 1 and S1-
inter-sublattice
which
example,
These
yield
is
interactions
The
of
a non-vanishing
given
from
the
formulation of the previous chapter as
[5.4]
2
(to/y)=
iT
DVIsFM^M!"FMiii:MsFM^M?]rMsFM
lM r FM li:MsFM^M2]"[MsFM!Ml
2 2
i i
1 1
'IxyxI
where
F ^ i ^ i =O2FZdM^dMj evaluated at equilibrium for example
i i
the model given by Eq. [5.1], the relevant derivatives are given by
FM 1M 1= F yCOs2®Sin 2<F+KzSin2®+6 KaM 2sm26 cos2 S,
FM jM ^ Kyeos2$’
FM |M j=elCOs2 0 +e2sin20’
■
and for
[5.5]
[5.6]
[5.7]
85
f M1
2M
^ eV
[5.8]
FM }M ^ KyCOs6sin<I>cos<I)’
[5.9]
Fm i =K M sin26 sin2C>+K M cos 29+2K M 3Cos4 GMs y 5
" '
* ’
[5.10]
Hx sinGcosO-Hy sinGsinO-Hz CosG-He1M
sin2G+e.M
cos 2G,
x
s
Z s
with 6
and
equilibrium
O the
magnetization
conditions.
two
resonances
field,
one
In
are
would
angles
order
separated
obtained
by
solving
to obtain theapproximation
by
twice
the intersublattice
set Ky=O and Ka=O with
E1=E2=E while
for
the
that
the
exchange
assuming the
high field limit so the M 1 and M 2 are parallel to the magnetic field:
the
case
of
collapsed
antiferromagnets
one
employs
a
In
similar
approximation.
Magnetic
resonance
[NH 3(CH2 )7NH 3]CuBr4
experiments
using
the
wereperformed
at
EPRspectrometer
9.2
GHZ
on
previously described
which was fitted with an Oxford cryostat in order to provide a range of
temperatures
Magnetic
for
above
resonance
several
complicated
and
deformations. I-65"68!
the
measurements
reasons.
because
below
First,
of
In
in
these
precise
internal
addition,
critical
temperature
of
10.2K.
layered compounds is
difficult
alignment
stresses
due
the samples
to
are
of
the
plastic
and
crystal
is
structural
quite fragile and break
easily when subjected to temperature changes1-66"681
Angular
dependencies
of
the
resonance
using Eq. [5.4] with the free energy
peaks
at
4.5+.5K
model given by Eq. [5.1].
were
fit
A best
86
fit
(shown
in
Figure
8)
is
given
by
the
parameters
K Mg=703
Oe,
KzMg= 1952 Oe, K M s3=-76 Oe, e^Mg=-554 Oe (J/k=.039 K), £2M =-917 Oe
(J/k=.062K) where an average g value o f 2.097 has been assumed.
When
E1 and E2 are not equal ie., when there is anisotropy in the weak interplanar
exchange,
as
function
a
interplanar
of
and
interaction.
In
resonances
as
resonance
was
in
was
fact
see
this
is
in
initially
rotated in
this
of
to
resonance
The
field
x-y
a
angle
only
plane
travels
rapidly
separation
to
the
y-axis
high
field
and
data
shows
that
the
direction
of
the
two
high
field
when
the
showed that
such
easy
type
the
behavior
could be explained in terms of a symmetric exchange interaction.
experimental
the
exchange
of
The
but later experiments
to
in
dipole-dipole
evident.
close
separation
symmetric
the
is
their
anisotropy
by
possible
in
vary
large
explained
variation
observed
the
resonances
plane.
due
8,
function
two
completely
be
Figure
the
the
not
may
a
that
angle
exchange
interactions
crystal
we
axis
Later
in
this
compound is in the x direction.
This is in agreement with data in the
antiferromagnetic
[NH 3 (CH 2 ) 4 NH 3ICuBr4
antiferromagnetic
of-plane
direction
compound
circle
is
located to
within
as Shown in Figure 9.
a few
where
degrees
The presence
of the
the
out-
o f the second
resonance in Figure 9 indicates that [NH 3(CH2)4NH 3]CuBr4 has an
additional process occurring such as spin canting and is a candidate for
a
four
sublattice
system.
In
the
compound
[NH3(CH2)4NH 3ICuCl4, we
have observed the analogous resonance for antiferromagnetic systems.
The
rough
temperature
dependencies
for
the
was also obtained and this is shown in Figure 10.
two
resonance
peaks
With better methods
o f temperature measurement and control, the possibility exists to obtain
RESONANCE FIELD (kO e)
87
x
y
z
FIELD ANGLE (DEGREES)
Figure 8 .
The angular dependence of the two resonance peaks observed
in [NH 3(CH2)7NH 3ICuBr4.
88
4 DA CuBn
M
[easy axis]
-4 0 -20 0 20 4 0 6 0 8 0 100 120 140 160 180 2 0 0
0 (degrees)
Figure 9.
The antiferromagnetic circle is shown centered about the xaxis.
The presence of a second resonance peak indicates
[NH3(CH2)4NH3]CuBr4 is a four sublattice system.
R E S O N A N C E FIE L D (K O e )
89
T E M P E R A T U R E (K )
Figure 10.
The temperature dependence of the two resonance peaks
with magnetic field slightly off the y axis.
Data indicated
with a star are uncertain.
90
the
critical
this
exponents
variation
function
of
critical
is
/3 and 5
due
to
temperature.
exponents
c o m p a r iso n ,
through
th e
from
such
the
sublattice
The
possibility
line-width
temperature
dependence
magnetization
changing
also
exists
to
measurements.[65]
tem p eratu re
as
obtain
By
d e p e n d en ce
since
way
o b se rv e d
a
these
of
in
[NH3(CH2 )4NH 3JCuBr4, shown in Figure 11, also shows multiple resonance
modes
with
shown
in
two
Figure
strong
9.
and
This
two
is
weak
also
modes
evident
indicative
of
a
which
multiple
are
not
sublattice
behavior.
We
the
have
layered
presented
compound
we measured agrees
high
temperature
with
the
our evidence
for
a two
[NH 3(CH2)7NH 3JCuBr4.
sublattice behavior in
The
interplanar
fairly well with the estimate o f <1K
series
estimates
expansions on
obtainedfrom
the n=2-6
powder
exchange
obtained from
compounds
susceptibility
and
also
measurements
which were obtained from molecular field theory on the two neighboring
antiferromagnetic
compounds,
[NH3(CH2)6NH3] CuBr4
[NH 3(CH2 )8NH 3]CuBr4 with J=-.05 K.
compounds
obtained
with
from
small
high
values
of
values of
J=-.01
K
and
Susceptibility measurements on the
J have been
temperature series
was uncertain for small
with
J.
based
expansions, the
upon
validity
estimates
of which
The large anisotropy measured in
the interplanar exchange is more than would be expected from a dipoledipole interaction since this would lead only to a few percent difference
in values measured.It can be shown, however
the
inter-sublattice
exchange
field
will
tend to
the frequency vs. field curves o f the two resonances.
that this
change
the
anisotropy in
minimum
in
91
T ( 0 K)
Figure 11.
The temperature dependence
[NH3(CH2)7NH3ICuBr4.
of
resonances
observed
in
92
Limits to Interplanar Exchange Fields Which Can Be Measured
An
what
important
conditions
constant
the easy axis.
intersection
frequency.
system
of
about
the
one might expect
microwave
simpleuniaxial
function
question
with
for
o f the
basis
magnetic
for
with
a
at a
this discussion
is the
field applied perpendicular
of
co=constant with
small value
of
inter-planar
to
as a
inter-planar exchange field.
The
one of these curves yields
resonance position as a function of frequency.
that
isunder
12, resonance frequency has been plotted
two values
line
sublattice concept
observe two resonance peaks
The
the
In Figure
field
to
two
the
From Figure 12, one sees
exchange,
two resonance
peaks
are observable where as with a higher value of this exchange field, only
one
resonance
line
is
seen to
occur
and this would correspond
normal ferromagnetic resonance line.
It is possible
upon
resonance
the
range
including
in
which
theexperimental
have line-widths on
resonance
is
multiple
to place rough
curves
observation that these
may
lower in
intensity
by
a
factor
of
10.
Thus, when
to
the
smaller peak.
On
the
which for an
roughly
1500
expects
peaks.
The
gauss,
viable
sketched
by
in
one
range
Figure
of
to
In
the exchange
x-band EPR spectrometer is
observe no
observability
13.
the
could not expect
other hand, when
field is more than co/2y,
is
seen
the order of 500 gauss and also that the higher-field
peaks are closer than roughly 600 gauss, one
peak
be
limits
resonance lines tend to
resonance
resolve
to the
for the
addition
to
additional
resonance
additional
resonance
these
limits,
one
additional limitation would seem to be that fields should not be so high
that the system is forced to cross a phase boundary.
93
MAGNETIC FIELD (kOe)
Figure 12.
The theoretical dependence of resonance fields
uniaxial
two-sublattice
ferromagnetic
mode.
corresponding to resonance curves are indicated.
for a
Modes
94
£
Figure 13.
m
mi
Theoretical range of interplanar exchange
observed with an x-band EPR apparatus.
which
can
be
95
The Real Model For FNHJCH2I7NHJCuBr 4
Earlier, it was pointed out that [NH 3(CH2)7NH 3ICuBr4 seemed to be
a
two
sublattice
fields.
system
owing
Laterexperimental
only
to
evidence
on
the
weak
inter-planar exchange
larger
crystals
showed
that
A A
indeed the intensity of the higher field resonance line in the y-x plane
did
not
simply
within 45°
vanish
o f either axis
but rather
with
rapidly
a slight
travelled
to
a
higher
field
asymmetry towards the x
axis.
This behavior can be interpreted as being owed to a symmetric exchange
term
of
the
form
S [M^M^+M^M^]
which
thus
implies
that this
system
may in fact be a two sublattice system due to this exchange rather than
due to the simplified model described above.
that
the
susceptibility
[NH3(CH2)7NH 3ICuBr4
was
measurements
ferromagnetic
It should be pointed out
which
were
determined
done
on
since a large single crystal was not easily obtained.
Data collected from
presence
of a weak ferromagnetic moment in one direction due to spin
system
magnetic
[(CH3) 3NHjNiCl3eZH2O
not
samples
susceptibility
Indeed,
can
powder
powder
cantingl37-*
measurements
that
resonance
^
antiferromagnetic system and in
was
necessarily
experiments
determined
appear
axis
which
towards
cannot
be
the hard
attributed
axis,
to
a
second
crystal
represent
on
the
a canted
When rotating from
resonance
twinning
of twinning would show up in another orientation.
was offered for this resonance peak.
performed
the
Their work does not refer to the
presence of a second resonance peak in this plane.
easy
out
one plane, their angular dependent data
shows the behavior described above.
the
to
rule
since
peak
does
the
effect
No clear explanation
96
CHAPTER VI
CONCLUSIONS AND FUTURE EXTENSIONS OF THE THEORY
Results of This Work
The
early
formalisms
intractable
or
fields.
this
thesis,
local
coordinate
method.
to
cover
systems
but
we
In
inadequate
of ferromagnetic resonance were
term
the
then
generalized
discussed
above
formulation
and
when
these
used
in
equations
had
were
This
of
earlier
recognized
systems
the
accurate
and usable
approach to
large
reformulated
local
several
sublattices.
problems
the
anisotropy
into
coordinate
generalized
development o f the local coordinate method.
more
with
found to be
Smit
inherent
what
we
method
was
It
not
was
and
Beljers
it
before
in
In an attempt to present a
ordered state resonance, the more
complete generalization had to wait for a year until the development o f
the local coordinate method.
In an attempt to use magnetic resonance techniques to measure the
inter-planar
o f a two
was
shown
permit
taken
ferromagnetic
in
the
layered
sublattice ferromagnet was assumed.
that
an
additional resonance
a measurement
on
exchange
the
layered
of
this
magnetic
elusive
compounds,
is
predicted which
interplanar exchange.
the predicted additional resonance line was observed.
model
From this assumption, it
mode
compound
a
could
Data were
[NH3(CH2)7NH3]CuBr4
where
97
Future Extensions of the Theory and Experiment
There are many paths this work can now take.
additional
systems
which
sublattice
behavior
could
sublattice
ferromagnets
in order to
work
by
Rubenacker
magnetometer
could
which
et.
have
no
al.*-2-* one
of
the
and
of spin
performed
more
useful
qualitative
have
two
additional
two
search
for
spin
canting
should
at
such
be
be
and
contributions
done
to
eliminate
which
a
quantitative
the
scientific
presentation
If
can be
Additional
frequencies
to
candidates
[NH 3(CH2)^N H 3ICuBr4.
arrangement.
various
initiated
From the
possible
[NH 3(CH2)2NH 3ICuBr4
should
canting
two
the
larger
obtained,
possibility
experimental work
would
certain data on free energy models for the system.
the
yet
The
finds
and
compound
measurements
type
be
ferromagnets
pursued.
[NH 3(CH2) 9NH3] CuBr4
crystals
o f some
be
be
see if the second resonance can be confirmed.
immediately:
single
might
An examination of
yield
more
In addition, one of
literature
of
would
magnetic
be
a
resonance
observations in the ordered state for this entire series.
It
was
observability
generalized
pointed
that
of resonance lines
formulation
important
future
investigate
more
compounds.
out
If
yielding
modes
properties
large
[NH3(CH2) 1NH 3ICuBr4
can
single
be
be
important
due to higher sublattice
modes
for
It
would
contribution.
the
may
of
all
crystals
obtained,
of
would provide clues about the multi-sublattice behavior.
interactions.
A
be
fields
the
crystal
the
models
also
anisotropy
to
in
series
analysis
would
be
interesting
these
of
for
an j
to
layered
compounds
this
series
98
LITERATURE CITED
[1]
RF. Mottelay,
Bibliographical
Magnetism, (Amo,New York: 1975), I.
[2]
LJ. de Jongh and A.R. Miedema, Adv. Phys. 23,1(1974).
[3]
G.V. Rubenacker, D.N. Haines,
Magn. Magn. Mat. 43, 238(1984).
[4]
C. Kittel, Phys. Rev. T l, 270(1947).
[5]
C. Kittel, Phys. Rev. 73, 155(1948),
[6]
J.M. Luttinger and C. Kittel, Helv. Phys. Acta. 21,480(1948).
[7]
C. Kittel, Phys. Rev. 76, 743(1949).
[8]
L.R. Blickford, Jr., Phys. Rev. 7 6 , 137(1949);78, 743(1950).
[9]
C. Kittel, Phys. Rev. 82, 565(1951).
[10]
F. Keffer and C. Kittel, Phys. Rev. 85, 329(1952).
[11]
J. Smit and H G. Beljers, Philips Res. Rep. 10, 113(1955).
[12]
L. Baselgia, M. Warden, F. Waldner, S.L. Hutton, J.E. Dmmheller,
Y.Q. He, P.E. Wigen and M. Marysko, Phys. Rev. B38, 2237(1988);M.
Marysko, S.L. Hutton, J.E. Dmmheller, Y.Q. He, P.E. Wigen, L.
Baselgia, M. Warden and F. Waldner, J. de Physique(1988).
[13]
C.
Cajori,
A
York:1893),371.
[14]
J.H.E. Griffths, Nature 158, 670(1946).
[15]
H. Suhl, Phys. Rev. 97, 555(1955).
[16]
P. Tannenwald and B. Lax, Private communication to ref. [15].
[17]
T.L. Gilbert, Phys. Rev. 100, 1243(1955).
[18]
J O. Artman, Proc. IRE 44, 1248(1956).
History
of
History
J.E.
of
Electricity
Dmmheller,
Mathematics,
K.
and
Emerson,
(Macmillan,
J.
New
99
[19]
D. Polder, Phil. Mag. 40, 99(1948).
[20]
J.M. Richardson, Phys. Rev. 75, 1630(1949).
[21]
J.H. Van Vleck, Phys. Rev. 78, 266(1950).
[22]
A.H. Morrish, The Physical
Principles
New York: 1965), and references therein.
[23]
P.E. Wigen, Thin Solid Films 144, 135(1984).
[24]
Y.Q. He and P.E. Wigen, J. Magn. Magn. Mater. 53,115(1985).
[25]
C. Chappert, K. Le Dang, P.
Renard, Phys. Rev. B34, 3192(1986).
[26]
P.E. Wigen, in "Physics of Magnetic Garnets," Proceedings of the
International School of Physics, Varenna, Course LXX, edited by
A. Paoletti North-Holland (1978), 196.
[27]
G.F. Herrmann, Phys. Rev. 133, A1334(1964).
[28]
Referee report for Phys. Rev. BA3702 manuscript (1987).
[29]
L O. Snively, P.L. Seifert, K. Emerson, and
J.E. Drumheller, Phys.
Rev. B20,
2101(1979);L.O.
Snively,
K.
Emerson,
and J.E.
DrumheEer^ Phys. Rev. 823,6013(1981); L.O. Snively, G.F. Tuthill,
and J.E. Drumheller, Phys. Rev. B24, 5349(1981). L.O. Snively, D.N.
Haines, K. Emerson and J.E. Drumheller, Phys. Rev.B26, 5245(1982).
[30]
L.J. de Jongh, A.C. Botterman, F R . de Boer and A.R. Miedema, J.
App. Phys. 10,1363(1969).
[31]
H. van Kempen, F.H. Mischgofsky, and P. Wyder, Phys. Rev. B 15,
386(1977); H. von Kanel, Physiea 96B, 167(1979).
[32]
W.E. Estes, D B .
630(1980).
[33]
I. Dzyaloshinsky, J. Chem. Solids 4, 241(1958).
[34]
T. Moriya, RhyS. Rev. 120, 91(1960).
[35]
L.J. de Jongh, Physica, 49, 305(1972).
[36]
G, Srinivasan and M S. Seehra, Phys. Rev. B28, 1(1983).
[37]
A.S. Borvik-Rominov
18(1961).
of Magnetism
Beauvillain,H.
Losee and W.E. Hatfield,
and
V I.
Ozhogin,Sov.
(John Wiley,
Hurdequint,
and
D.
J. Chem. Phys. 72(1),
Phys.
JETP
12,
100
[38]
D B . Losee, J.N. McEleamey, G E . Shankle, C.L.
Cresswell and W.T. robinson, Phys. Rev. B5, 2185(1973).
[39]
P. Bloembergen, Physica 8 IB , 205(1976).
[40]
P. Bloembergen, PJ. Berkhout and J.J.M.Franse, Int.
4, 219(1973).
[41]
Von Wemer Doring, Z. fur Nat., 7, 373(1948);T. Nagamiya, Prog.
Theor. Phys. (Kyoto) 6, 342(1951).
[42]
T. Nagamiya, Prog. Theor. Phys., 11, 309(1954).
[43]
T. Nagamiya, K. Yosida, R. Kubo, Adv. Phys. 13, 1(1955).
[44]
M. Date and M. Motokawa, J. Phys. Soc. Jpn. 22, 165(1967).
[45]
K. Yosida, Prog. Theor. Phys. 6, 691(1951).
[46]
K. Yosida, Prog. Theor. Phys. 7, 25(1952).
[47]
K. Yosida, Prog. Theor. Phys. 7,1178(1952).
[48]
J. Ubbink, Physica XIX, 9(1953).
[49]
C. Coulon,
6235(1986).
[50]
M. Tanimoto, T. Kato and Y. Yokozawa, Phys. Lett. 58A, 66(1976).
[51]
S.L. Hutton, S. Waplak and J.E. Drumheller, Bull.
31(3), 338(1986).
[52]
H. Tanaka,
published).
[53]
J.W. Metsellar and D. de Klerk, Physica 69,499(1973).
[54]
R.L. Carlin
231(1980).
[55]
G.V. Rubenacker, S. Waplak, S.L. Hutton,
Drumheller, J. AppL Phys. 57, 3341(1985).
[56]
P.J. Besser, A.H. Morrish and C.W. Searle, Phys.
632(1967);B.R. Morrison, Phys. Stat. Sol. 59B, 581(1973).
[57]
F. Waldner, J. Phys. C. 21,1243(1988).
[58]
R.J. Walker, "Algebraic Curves", (Dover: New York,1950), 117-119.
J.C.
S.
Scott
Teraoka,
and
A.J.
and
E.
van
R.
Laversanne,
Kakehashi,
Jour,
Duyneveldt,
Carlin,
J.Magnetism
Phys.
de
Ace.
D.N.
P.J.
Rev
B33,
A. Phys. Soc.
Physique
Chem.
(to
Res.
Haines
be
13,
and J.E.
Rev.
153,
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