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Advances in CHEMICAL PHYSICS
Edited by
I. PRIGOGINE
Center for Studies in Statistical Mechanics and Complex Systems
The University of Texas Austin, Texas
and International Solvay Institutes, Universite Libre de Bruxelles
Brussels, Belgium
and
STUART A. RICE
Department of Chemistry and
The James Franck Institute The University of Chicago
Chicago, Illinois
VOLUME 109
AN INTERSCIENCE” PUBLICATION JOHN WILEY & SONS, INC.
NEW YORK CHICHESTER WEINHEIM BRISBANE SINGAPORE TORONTO
ADVANCES IN CHEMICAL PHYSICS VOLUME 109
EDITORIAL BOARD
BRUCE J. BERNE, Department of Chemistry, Columbia University, New York, New York, U.S.A.
KURT BINDER, Institut fur Physik, Johannes Gutenberg-Universitat Mainz, Mainz, Germany
A. WELFORD CASTLEMAN, JR., Department of Chemistry, The Pennsylvania State University, University Park, Pennsylvania, U.S.A.
DAVID CHANDLER, Department of Chemistry, University of California, Berkeley, California, U.S.A.
M. S. CHILD, Department of Theoretical Chemistry, University of Oxford, Oxford, U.K.
WILLIAM T. COFFEY, Department of Microelectronics and Electrical Engineering, Trinity College, University of Dublin, Dublin, Ireland
F. FLEMING CRIM, Department of Chemistry, University of Wisconsin, Madison, Wisconsin, U.S.A.
ERNEST R . DAVIDSON, Department of Chemistry, Indiana University, Bloomington, Indiana, U.S.A.
GRAHAM R. FLEMING, Department of Chemistry, The University of Chicago, Chicago, Illinois, U.S.A.
KARL F. FREED, The James Franck Institute, The University of Chicago, Chicago, Illinois, U.S.A.
PIERRE GASPARD, Center for Nonlinear Phenomena and Complex Systems, Brussels, Belgium
ERIC J. HELLER, Institute for Theoretical Atomic and Molecular Physics, Harvard- Smithsonian Center for Astrophysics, Cambridge, Massachusetts, U.S.A.
ROBIN M. HOCHSTRASSER, Department of Chemistry, The University of Pennsylvania, Philadelphia, Pennsylvania, U.S.A.
R. KOSLOFF, The Fritz Haber Research Center for Molecular Dynamics and Department of Physical Chemistry, The Hebrew University of Jerusalem, Jerusalem, Israel
RUDOLPH A. MARCUS, Department of Chemistry, California Institute of Technology, Pasadena, California, U.S.A.
G. NICOLIS, Center for Nonlinear Phenomena and Complex Systems, UniversitC Libre de Bruxelles, Brussels, Belgium
THOMAS P. RUSSELL, Department of Polymer Science, University of Massachusetts, Amherst, Massachusetts
DONALD G. T~UHLAR, Department of Chemistry, University of Minnesota, Minneapolis, Minnesota, U.S.A.
JOHN D. WEEKS, Institute for Physical Science and Technology and Department of Chemistry, University of Maryland, College Park, Maryland, U.S.A.
PETER G. WOLYNES, Department of Chemistry, School of Chemical Sciences, University of Illinois, Urbana, Illinois, U.S.A.
Advances in CHEMICAL PHYSICS
Edited by
I. PRIGOGINE
Center for Studies in Statistical Mechanics and Complex Systems
The University of Texas Austin, Texas
and International Solvay Institutes, Universite Libre de Bruxelles
Brussels, Belgium
and
STUART A. RICE
Department of Chemistry and
The James Franck Institute The University of Chicago
Chicago, Illinois
VOLUME 109
AN INTERSCIENCE” PUBLICATION JOHN WILEY & SONS, INC.
NEW YORK CHICHESTER WEINHEIM BRISBANE SINGAPORE TORONTO
This book is printed on acid-free paper @
Copyright 0 1999 by John Wiley & Sons, Inc. All rights reserved.
Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4744. Requests to the Publisher for permission should be addressed to the Permissions Department, JohnWiley & Sons, Inc., 605 Third Avenue, NewYork, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: PERMREQ @ WILEY.COM.
Library of Congress Catalog Number: 58-9935
ISBN 0-471-32920-7
10 9 8 7 6 5 4 3 2 1
CONTRIBUTORS TO VOLUME 109
BIMAN BAGCHI, Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560 012, India
RANJIT BISWAS, Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560 012, India
ARIEL A. CHIALVO, Department of Chemical Engineering, University of Tennessee, Knoxville, TN 37996-2200
J. CRAIN, Department of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 352, Scotland, United Kingdom
PETER T. CUMMINGS, Chemical Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6181
A. DE WIT, Service de Chemie Physique, Centre for Nonlinear Phenomena and Complex Systems, CP 231, Universite Libre de Bruxelles, Campus Plaine, 1050 Brussels, Belgium
V. KOMOLKIN, Institute of Physics, Saint Petersburg State University, Saint Petersburg 198904, Russia
K. P. SCAIFE, School of Engineering, Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland
V
INTRODUCTION
Few of us can any longer keep up with the flood of scientific literature, even in specialized subfields. Any attempt to do more and be broadly educated with respect to a large domain of science has the appearance of tilting at windmills. Yet the synthesis of ideas drawn from different subjects into new, powerful, general concepts is as valuable as ever, and the desire to remain educated persists in all scientists. This series, Advances in Chemical Physics, is devoted to helping the reader obtain general information about a wide variety of topics in chemical physics, a field that we interpret very broadly. Our intent is to have experts present comprehensive analyses of subjects of interest and to encour- age the expression of individual points of view We hope that this approach to the presentation of an overview of a subject will both stimulate new research and serve as a personalized learning text for beginners in a field.
I. PRIGOGINE STUART A. RICE
vii
CONTENTS
O N THE THEORY OF THE COMPLEX, FREQUENCY-DEPENDENT SUSCEPTIBILITY OF MAGNETIC FLUIDS 1
By B. K. I? Scaife
SIMULATING MOLECULAR PROPERTIES OF LIQUID CRYSTALS 39
By J. Crain and A. K Komolkin
MOLECULAR-BASED MODELING OF WATER AND AQUEOUS SOLUTIONS AT SUPERCRITICAL CONDITIONS 115
By Ariel A. Chialvo and Peter T. Cummings
POLAR AND NONPOLAR SOLVATION DYNAMICS, ION DIFFUSION, AND VIBRATION RELAXATION: ROLE OF BIPHASIC SOLVENT RESPONSE I N CHEMICAL DYNAMICS 207
By Biman Bagchi and Ranjit Biswas
SPATIAL PATTERNS AND SPATIOTEMPORAL DYNAMICS IN
CHEMICAL SYSTEMS 435
By A. De Wil
AUTHOR INDEX 515
SUBJECT INDEX 549
ix
ON THE THEORY OF THE COMPLEX,
OF MAGNETIC FLUIDS FREQUENCY-DEPENDENT SUSCEPTIBILITY
B. K. P. SCAIFE
School of Engineering, Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland
CONTENTS
I. Introduction 11. Resonance absorption
A. Lorentz (Type I) Absorption B. Van Vleck-Weisskopf-Frohlich (Type 11) Absorption C. Comparison of the Two Types of Absorption
111. Ferromagnetic Resonance: Theory of Landau and Lifshitz IV. Some Results from the Theory of Linear Systems V. Some Results from Fluctuation Theory
A. Basic Relations B. Particular Results
1. The Kubo Relation 2. Correlation Function for a Random Telegraph Signal 3. Correlation for a Sinusoid with Random Abrupt Changes in Phase
VI. Longitudinal and Transverse Polarizabilities for a Fixed, Spherical, Single-Domain Particle A. Zero-Frequency Polarizabilities B. Frequency-Dependent Longitudinal Polarizability C. Frequency-Dependent Transverse Polarizability
VII. Calculation of the Frequency-Dependent, Complex Susceptibility of a Magnetic Fluid Appendix A. Comparison of the Functions YD, YI, and YII Appendix B. Different Forms of the Type I1 Resonance Equation Acknowledgments References
Advances in Chemical Physics, Volume 109, Edited by I. Prigogine and Stuart A. Rice ISBN 0-471-32920-7 0 1999 John Wiley & Sons, Inc.
1
2 B. K. P. SCAIFE
I. INTRODUCTION
Magnetic fluids are stable, colloidal suspensions of ferromagnetic materials [l]. The small size of the roughly spherical particles (diameter on the order of 10 nm) ensures that they each consist of a single magnetic domain. Extensive theoretical and experimental studies have been made of the mag- netization of magnetic fluids in alternating magnetic fields of low intensity
The behavior of magnetic fluids in such fields is characterized by a dimen- ~ 3 1 .
sionless, isotropic, complex, frequency-dependent susceptibility:
Thus for a sufficiently small, alternating magnetizing force
h(t) = RH(w) exp(zwt) (1 4 (R denotes “real part of”) of frequencyf = w/2n will induce, along the axis of a long, thin, needle-shaped sample, a magnetization
Prnag(t) = mrnag(U) e x ~ ( l o t ) (1 *3)
such that
Prnag(w) = PoXrnag(w>H(w> (1 *4)
po(= 4n x lop7 H/m) being the absolute permeability, For simplicity it will be assumed that the colloid particles do not conduct electricity.
The purpose of this review is to derive, in as direct a manner as possible, and discuss an equation to describe the dependence of ~,,,(w) on the angular frequency w.
In this review we shall assume that all the particles of a magnetic fluid have the same size and are spherical in shape. Therefore, the magnitude of the magnetic moment of a particle of radius ap is
(1.5) 4 3 3 p
m = vdibfs = -na MS
where Ms (Wb m-*) is the saturation magnetization of the material and Vd is the magnetic-domain volume, which we shall take to be the same as the actual volume of the particle.
In a uniformly magnetized, crystalline ferromagnetic material the energy associated with the magnetization depends on the direction of the magnetic moment vector m with respect to the crystallographic axes. This phenomenon is known as magnetic anisotropy [4, $401. In a single-domain particle there
FREQUENCY-DEPENDENT SUSCEPTIBILITY OF MAGNETIC FLUIDS 3
will be preferred orientations of m corresponding to minima of the anisotropy energy. In this article attention is confined to materials with uniaxial magnetic anisotropy; this means that at low temperatures the magnetic moment vector will lie along a particular axis through the particle. Consequently m can have two antiparallel equilibrium orientations. At finite temperatures in any par- ticular particle, we must expect that thermal fluctuations will cause m to reverse direction in an abrupt and random manner.
If the energy of magnetic anisotropy is large, the direction of m with respect to the particle will not change for long periods of time, and we can assume that m is fixed rigidly in the particle. In this situation, we speak of the magnetic moment being “blocked,” and all changes in orientation of m are determined by the rotational motion of the particle.
The calculation of x,,,(w) for a magnetic colloid in which the magnetic moments are blocked is a simple matter: We simply take over the Debye [5] theory of dielectric polarization in an assembly of weakly interacting, rigid electric dipoles. According to this theory, in which each spherical molecule carries a permanent electric dipole moment p, the dielectric sus- ceptibility
where P-’ = kBT (kp, is Boltzmann’s constant, and Tis the absolute tempera- ture), NIV is the molecule number density, EO (% 8.854 x FIm) is the absolute permittivity of free space, and TD is the relaxation time. Debye, in adapting Einstein’s (1926) theory of translational Brownian motion [6] to rotational Brownian motion [7], treated each dipolar molecule as a small sphere, of radius up, performing rotational Brownian motion. The mol- ecular interactions, which give rise to the random changes in orientation, exert a damping effect on the rotational motion of a molecule; this damping may be accounted for in terms of the dynamic viscosity y. Debye showed that
(1.7)
Applying the same analysis to a magnetic fluid, with each particle carrying a blocked magnetic moment m, and a particle number density N/K we find that
TD = 47ca,yj 3
with
4 B. K. P. SCAIFE
In Eq. (leg), the thickness of the surfactant coating of the colloidal particles is ignored and the effective hydrodynamic radius is assumed to coincide with the particle radius.
Eq. (1.8) can account quite well for the frequency dependence of ~,,,(co) at low frequencies; however, such a simple theory neglects entirely the internal dynamics of the magnetic moment. When the magnetic moment vector m deviates from the magnetic axis of the particle, it is subjected to a mechanical torque that tends to return m to its equilibrium orientation; the result of this torque is that m undergoes Larmor precession [8, p. 2341 about the magnetic axis. In a study of the frequency dependence of the magnetic polarizability of a fixed, single ferromagnetic domain, Landau and Lifshitz [9] proposed the following equation of motion for the magnetic moment m:
(1.10)
where y denotes the gyromagnetic ratio, A is a damping constant, and r(t) is the total torque exerted on m both by magnetic anisotropy and by any external magnetic field. Pending a discussion of Eq. (1.10) in Section 111, we remark that for a uniaxial domain, with its axis parallel to the [-axis (Fig. l), and a vanishingly small, external, perpendicular magnetizing force hl(t), we obtain the following equation for mt(t):
(1.11)
The constant K takes account of the magnetic anisotropy, the characteristic angular frequency
Upr = P o F m
and the time constant for the damping of precessional motion is
Tpr = ( P ~ Y J J C - '
For
and
h<(t) = RHt(w)exp(zcot)
mt(t) = RMl(co) exp(zwt)
(1.12)
(1.13)
(1.14a)
(1.14b)
FREQUENCY-DEPENDENT SUSCEPTIBILITY O F MAGNETIC FLUIDS 5
Figure 1. The geometry of the magnetic moment m of a spherical, single-domain, uniaxial ferromagnetic particle undergoing Larmor precession about the magnetic axis 05. The coordi- nate system Ogql is fixed in the particle; the polar coordinates of m are 0 and 4. The effective magnetizing force owing to magnetic anisotropy is ha [Eq. (3.7)].
we find, from Eq. (Lll), that the magnetic polarizability
Eq. (1.15), first derived by Landau and Lifshitz, describes what is sometimes referred to as ferromagnetic resonance, which we shall refer to as Type I1 resonance absorption. Although this type of resonance has many similarities with the resonance of a damped harmonic oscillator (which we shall call Type I), there are important differences, which will be discussed in Section 1I.C.
Unrelated to the work of Landau and Lifshitz were the studies by VanVleck and Weisskopf [lo] and Frohlich [11,12], which were concerned with attempts to improve on the efforts of Lorentz [13] to take proper account of the effect of collisions on the shape of spectral lines. Our interest in these matters is twofold: first, the equations derived by Van Vleck and Weisskopf and by Frohlich for the electrical susceptibility of a damped harmonic oscillator coincide with the Landau-Lifshitz relation, Eq. (1.15); second, one of the
6 B . K . P. SCAIFE
methods used by Frohlich has a direct relevance in our derivation of an equation for x,,,(o).
Our task is to combine the results of Landau and Lifshitz with those of Debye on rotational Brownian motion and, with the minimum of mathemat- ical complexity, to obtain an expression for x,,,(o). We briefly review and discuss the essential results of Lorentz, of Van Vleck and Weisskopf, and of Frohlich in Section I1 and those of Landau and Lifshitz in Section 111. In Section IV a short account is given of some important results from the theory of linear systems. Section V lists some relevant topics in the theory of fluctuations, the calculation of the transverse and longitudinal fre- quency-dependent polarizabilities of a spherical particle is described in Section VI, and the derivation of a formula for x,,,(o) is given in Section VII.
11. RESONANCE ABSORPTION
A. Lorentz (Type I) Absorption
The study of collision-broadened spectra [I41 has been a subject of continuing interest since the pioneering work of Lorentz [13, Note 571 who studied the effect of random collisions on the resonant absorption, in an alternating elec- tric field, of an electric dipole capable, when isolated, of continuous harmonic oscillations with a particular frequency (c0~/2n) . Between collisions the equation of motion of the vibrating charge q of mass m4 is
where x( t ) is the coordinate of q, kx(t) is the restoring force, and e(t) is the external electric field. If we introduce the electric dipole moment
A t ) = qx(0 (2.2)
we have
Lorentz envisaged a system made up of a large number N of such oscil- lators (all with their axes parallel to one another) in thermal equilibrium with a heat bath. The thermal agitation interrupts the oscillatory motion of the oscillators; electrical coupling between the oscillators is neglected. The collisions are assumed to be distributed in time according to a
FREQUENCY-DEPENDENT SUSCEPTIBILITY OF MAGNETIC FLUIDS 7
Poisson distribution [ 151, with time constant zcoll. By statistical arguments, Lorentz showed that the total electric dipole-moment Np(t) induced by an external electric field
e(t) = RE(w) exp(iwt) (2.4)
could be expressed by the equation
Np(t ) = NlRa~(w)E(o) exp(imt)
in which the complex electric polarizability
with
and
It is most important to appreciate that ~ ( w ) describes the average behavior of a large group of oscillators. An individual oscillator, as described above, does not have a polarizability.
In the limit of very light damping w ~ z ~ ~ l l >> 1, Eq. (2.6a) reduces to
where g = (2k/w~z,,11). Lorentz pointed out that one could also derive Eq. (2.9) by assuming that a moving charged particle in the oscillator was con- tinuously subject to viscous damping, so that in place of Eq. (2.3), which holds between collisions, one has the following equation, valid for all times:
(2.10)
For convenience we shall describe the resonance described by the complex hnction YI(u, OL, Z,,II) [Eq. (2.6a)], as Type I resonance.
8 B . K. P . SCAIFE
B, Van Vleck-Weisskopf-Frohlich (Type 11) Absorption
VanVleck and Weisskopf [lo], in an attempt to remove from the Lorentz treat- ment of collision broadening features which they regarded as deficiencies, proposed, in effect, that Eq. (2.6) should be replaced by
where
(2.12)
The various forms that Eq. (2.11) can take are found in Appendix B. The differencebetween the analyses ofcollisionbroadening by Lorentz and by
VanVleck and Weisskopflies in their treatment ofthe mechanics ofthe collisions. As we have already mentioned, Frohlich [ll] obtained the same equation; the analysis for the harmonic oscillator given in his book [12, p. 1721 is based on the Boltzmann equation. In his first publication [ll], he points out that the method was originally developed for the case ofa rigid dipole performing angu- lar oscillations about an equilibrium position.
Frohlich [12, p. 991 assumes the form of a time-dependent decay function and by linear-system theory obtains Eq. (2.1) for YII(O, (UL, z,,I~) [see Eq. (4.11)].
C. Comparison of the livo Types of Absorption
From the discussion and figures in Appendix A, it is clear that, near the fre- quency of maximum absorption, Type I absorption is similar to Type 11. At low values of w it follows from Eqs. (2.6) and (2.11) that
FREQUENCY-DEPENDENT SUSCEPTIBILITY OF MAGNETIC FLUIDS 9
where, for completeness, we have included the expansion for YD(w, ZD) from Eq. (1.6). In the limit of large values of o we find that
UZD >> 1 (2.14~)
It is clear from these results that it is at very high frequencies that the two types of resonance differ. For example, when o is large, the real part of 'PI is pro- portional to o-2 and is negative; whereas although the real part of YII is also proportional to it is negative only if u ) L T , , ~ ~ > 1. The imaginary parts of YI and "11 have the same sign, but while Y; is proportional to
YyI is proportional to o-'. It is this last difference that is at the heart of the physical difference betweenType I and Type I1 resonance absorp- tions. As will be discussed in more detail when we deal with the spectra of polarization fluctuations in Section V, the fact that the frequency dependence of Y~I(cco-l) falls off so much more slowly than that of Y;(ccw-~) stems from the different treatments of the collision processes by Lorentz (Type I), on the one hand, and by Van Vleck, Weisskopf, and Frohlich (Type II), on the other. In Type I1 collision broadening, the particle position can change abruptly; in Type I collision broadening, only the velocity changes abruptly Since abrupt changes in position can occur only in the absence of inertia, Type I collision broadening is preferred as being more in harmony with basic principles.
10 B. K . P. SCAIFE
Last, we draw attention to the fact that when the characteristic frequency wL = 0, Type I1 absorption reduces to Debye nonresonant absorption, i.e.,
y I I ( W , 0, Tcoll) YD(W, Tcoll) (2.15)
Since it is well known that the Debye theory of dielectric absorption does not take full account of the effects of inertia [16], it is not surprising that Type I1 absorption should reduce to Debye-type absorption. Notice that in Type I absorption, when WL = 0, there is still resonance-formally at least-because Eq. (2.6a) leads to the relation:
(2.16)
which still has the same frequency behavior as the full Lorentz equation, except that now the characteristic angular frequency 1/zcOll is determined by the frequency of the collisions.
111. FERROMAGNETIC RESONANCE: THEORY OF LANDAU AND LIFSHITZ
The equation of undamped motion of the magnetic moment m(t) of a spherical, single-domain ferromagnet is [ 17; 4, p. 2701
dm(t) dt
m(t) = - = -,uoym(t) x ha
in which the gyromagnetic factor is denoted by
where g, the Land& splitting factor for electrons, has been set equal to 2, and thus ,uoy = 2.210197915 x lo5 m/As, ha is an effective magnetizing force caused by magnetic anisotropy [ 181. For uniaxial magnetic anisotropy (Fig. 11,
ha =z t ~ h a ~ = tha (3.3)
where t is a unit vector along the O i axis fixed in the particle. The magnitude of ha is determined by equating the torque that such a magnetizing force would exert on m, namely m x ha, with the actual torque being created by
FREQUENCY-DEPENDENT SUSCEPTIBILITY OF MAGNETIC FLUIDS 11
magnetic anisotropy Thus
au(o) ae m x ha = -n-
where
(3.4)
is the anisotropy energy of the domain when m makes an angle 0 with the symmetry axis 06 Ku is the magnetic-anisotropy constant, and fi is a unit vector parallel to the torque m x ha (Fig. 2). Consequently,
mha sin Q = 2Kuvd cos 0 sin 0 (3.6)
or
2Ku
MS h, = c---cos 0 = clcm cos 6
Eq. (3.1), which may now be written in the form
0 -
!
(3.7)
Figure 2. The variation with polar angle B of the uniaxial magnetic-anisotropy energy U(0) = -vdKu cos2 0 [Eq. (3.5)]. There are two energy minima: one at 0 = 0 and one at 0 = n.
12 B. K. P. SCAIFE
indicates that when the direction of m deviates from the OC-axis, m precesses about this axis with Larmor angular velocity
Wpr cos 0 = 6Opr cos 0 (3.9a)
with
(3.9b)
which is independent of the size of the spherical particle. For the materials of interest here [18-201, Ku x lo5 J/m3 and Ms x 1 Wb/m2, so that oP,/2n x 10 GHz.
It is clear from Figure 2, and from Eq. (3.5) that there are two antiparallel equilibrium orientations for m. Therefore, at finite temperatures it is to be expected that m will reverse direction in a random manner and at a rate deter- mined by the height &Vd of the energy barrier that separates the two orien- tations. The significance of these fluctuations is that the domain will have a finite magnetic susceptibility along the Or-axis. This fact will be considered in more detail in Section VI.
Any uniformly magnetized body experiences a demagnetizing force arising from the uncompensated magnetic poles on the surface of the body. In the special case of a sphere, the demagnetizing force is
l m
3PO vd *
Such ademagnetizing force cannot exert a torque on m, since m x m = 0. In the case of an ellipsoid, however, the demagnetizing force is no longer antiparallel to the magnetic moment and will, therefore, exert a torque on m [21,22].
Notice that it follows from Eq. (3.1) that the magnitude of m is a constant because
dm dt
m- = m . m = -yoym. (m x h,) 3 0 (3.10)
The presence of a small, time-dependent external magnetizing force h(t) may be taken into account by adding h(t) to h, in Eq. (3.1) with the result that
(3.11)
with
f(t) = ha + h(t) (3.12)
FREQUENCY-DEPENDENT SUSCEPTIBILITY OF MAGNETIC FLUIDS 13
Thermal agitation will tend to damp out any precessional motion. Landau and Lifshitz [9] proposed to take account of such damping by the empirical equation:
(3.13) m(t) ’ = - dm(t) - - ,uoyf(t) x m(t> - ,uoy2m(t) x [m(t> x f(t) l
The term in Eq. (3.13) with the damping constant i is so arranged as to be perpendicular to m and to the vector m x f. Notice that the condition m . m = 0 holds for Eq. (3.13) and, therefore
i dt
m x (m x m) = m(m.m) - m 2 ’ m = -m2m (3.14)
By means of repeated use of this result Gilbert and Kelly [38] have shown that, provided 2 < m, the Landau-Lifshitz Eq. (3.13) may be transformed to the form
On the basis of this result, Gilbert and Kelly proposed the following equation:
(3.16)
in which A is an empirical damping constant. We shall confine our attention to solving Eq. (3.13) for the case of a small
alternating external magnetizing force transverse to the magnetic axis of the particle. With this in mind we shall write
f(t) = ha + h(t) = i h a + ?hg(t) = i ~ m [ ( t ) + tht(t> (3.17)
with which we obtain from Eq. (3.13):
he(t> = PO? -~mr(t)m~(t> + Ahdt ) - G[ht(t)m[(t) + ~mi(r)]m-(i)] (3.18a) /z
3, { 1 hdt) = Po? KmdOmdt) - ht(t)mg(t) - --&ht(t)mr(t) + Km;(t)lm,(t)}
{ (3.18b)
h;zi(t> = Po? he(t)m,(t) + i~m;zi(t) - ~ [ h t ( f ) m t ( l ) + ~m:(t)]mr(t) (3.18~)
As already noted, the basic equation of motion proposed by Landau and Lifshitz ensures that m(t) is a vector of constant magnitude. Consequently,
I A
14 B. K . P. SCAIFE
since
(3.19)
and if mt(t) and m,(t) are small quantities, we may use the approximation:
(3.20)
The damping ensures that, in the absence of external fields, the vector m lies parallel to the magnetic OC-axis (Fig. 1). Thus to first order in hg(t), Eq. (3.18) reduces to:
M t ) = PoY[-Kmm,(t) + wo - ~7Q%(t>l (3.2 1 a)
f i , ( t) = PoY[Kmmdt) - mht(t) - /2Km,(Ol (3.2 1 b) f i ( ( t ) = 0 (3.21~)
Introducing the precession relaxation time z~~ defined by the equation
Z p r = ( P L ~ Y A K I - ’ (3.22)
and using the characteristic angular frequency upr, defined by Eq. (3.9b), we may write Eq. (3.21) in the form:
(3.23 b)
h[(t) = 0 (3.23~)
If we eliminate m,(t) from these equations we obtain Eq. (1.11). Suppose that
h(t) = &t(t) = R&-lg(w) exp(iot) (3.24)
and writing
(3.25) m(t) = Zmt(t) + ilm,(t> + imi(t>
= P S [ ~ M C ( ~ ) ) + +M,(w) + ~ M [ ( W ) I exp(iwt)
we find that
= “@do> + +@,,t(w> + i U [ , 5 ( 4 1 H t ( 4 e x P ( l 4 (3.26)
FREQUENCY-DEPENDENT SUSCEPTIBILITY O F MAGNETIC FLUIDS 15
in which the complex magnetic polarizabilities are
a d o ) = 0 (3.27~)
These equationswere first obtained by Landau and Lifshitz (1935). By chang- ing from h(t) = &zt(t) in Eq. (3.24) to h(t) = $z,(t), it may also be shown that [23-261
and that
also, by symmetry,
According to Eq. (3.21c), a ~ , ~ ( o ) vanishes; we postpone a discussion of a[,e(o) until Section VI.
An interesting result is obtained when the external magnetizing force is a rotating field of constant magnitude; consider, therefore, that in place of Eq. (3.24) we have
h*(t) = H(o)[icos ot k ijsin ot] = R H ( w ) ( i F I$) exp(rwt) (3.29)
The -t superscript indicates a right-handed rotation about the Ol-axis, and the - superscript a left-handed rotation, From Eqs. (3.27) and (3.29) we find that
16
With the notation:
B . K. P. SCAIFE
it follows that
(3.31)
(3.32a)
(3.32b)
It is sometimes convenient to introduce the dimensionless parameter 0, which is the ratio of the barrier height to the thermal energy, i.e.,
so that
1 1 1 m2 _ - _ - pm2 = -- K 20 20 k B T
(3.3 3a)
(3.3 3 b)
and hence Eq. (3.27a) can be written
IX SOME RESULTS FROM THE THEORY OF LINEAR SYSTEMS
According to linear system theory [16,27], to each of the normalized polar- izability functions - YD(W, ZD), [Eq. (1.6)], YI(w, UL, Z,,II) [Eq. (2.6)], and YII(O, Wpr , z ~ ~ ) [Eq. (2.11)] - there corresponds a function of time, which is the response to a large, but very brief, pulse. More precisely, this hnction of time is the pulse, or delta-function, response. The pulse response is the time derivative of the unit step-function response.
We shall denote the pulse response corresponding to the dispersion, or frequency response, function Y(o) by $(t). For the moment we shall use a simplified notation that omits the subscripts (D, I, and 11) and that omits reference to the constants ZD, WL, zcOll, Wpr , and 7pr). The pulse response
FREQUENCY-DEPENDENT SUSCEPTIBILITY OF MAGNETIC FLUIDS 17
and the dispersion fimction form a Fourier transform pair, i.e.,
Y(o) = dt $(t)exp(-zwt) 1: (4. la)
(4.1 b) l o o
$( t ) = -1 dw Y(co)exp(zwt) 271 -m
The lower limit in the first integral is set to zero, because to satisfy causality, $(t) must vanish for t < 0. We shall denote the unit step-function response by s(t), which is related to $(t) as follows:
Consequently,
"(0) = dt-exp(-zcot) s," d:tt) (4.3)
When we come to discuss fluctuation theory in Section V we shall need expressions for the decay, or aftereffect, function. The decay hnction, denoted by b(t), is related to the unit step-function by the relation:
b(t) = s(o0) - s(t) ( t p 0) (4.4)
From Eqs. (4.1) and (4.2), we deduce that
Because no physical system can respond instantly to an external stimulus,
s(0) = 0 (4.6a)
and since, by definition, "(0) = 1 we see that
b(0) = 1 (4.6b)
Hence we see from Eq. (4.5) that
s(o0) = 1 (4.7)
18 B. K . P. SCAIFE
Using Eq. (4.4) and carrying out an integration by parts in Eq. (4.3), we obtain
Y ( w ) = 1; dtd[-bolexp(-mt) dt
w
= [-b(t)]r - EO 1 dt b(t) exp(-lot)
= 1 - 10
= [ 1 - o 1; dt b(t) sin cot] - E [O /: dt b(t) cos ot
0
dt b(t)exp(-lot) I 1
We now list expressions for the step, aftereffect, and pulse functions for the three types of absorption considered in this Chapter.
Nonresonant Absorption [Eq. (1.6)]
(4.9b)
(4.9c)
(4.9d)
Type I Resonance Absorption [Eq. (2.6)]
(4.1 OC)
