ElectrodynaDlics of Continua II Fluids and Complex Media
With 56 Illustrations
Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong
Kong
A.C. Eringen Princeton University Princeton, N.J. 08544
U.S.A.
G.A. Maugin Laboratoire de Modelisation
en Mecanique Universite Pierre et Marie
Curie et C.N.R.S. 75252 Paris 05 France
Library of Congress Cataloging in Publication Data Eringen, A.
Cerna!.
Electrodynamics of continua / A.C. Eringen, G.A. Maugin. p.
cm.
Includes bibliographical references. Contents: I. Foundations and
solid media - 2. Fluids and complex media.
I. Fluid mechanics. 2. Electrodynamics. 3. Magnetohydrodynamics.
mechanics. I. Maugin, G. A. (Gerard A.), 1944-
4. Continuum
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987 6 543 2 1
ISBN-13: 978-1-4612-7928-0
DOl: 10.1007/978-1-4612-3236-0
Preface to Volume II
The first volume of Electrodynamics of Continua was devoted mainly
to the development of the theory of, and applications to,
deformable solid media. In the present volume we present
discussions on fluid media, magneto hydro dynamics (MHD) (Chapter
10), electrohydrodynamics (Chapter 11), and media with more
complicated structures. Elastic ferromagnets (Chapter 9) and
ferrofluids (Chapter 12) require the inclusion of additional
degrees of free dom, arising frQm spin-lattice interactions and
supplementary balance equations.
With the discussion of memory-dependent materials (Chapter 13) and
nonlocal electromagnetic theory (Chapter 14), we account for the
nonlocal effects arising from motions and fields of material
points, at past times and at spatially distant points. Thus, the
damping of electromagnetic elastic waves, photoelasticity, and
streaming birefringence are the subjects of Chapter 13. Nonlinear
constitutive equations developed here, and in Chapter 14, are
fundamental to the field of nonlinear optics and nonlinear
magnetism. The content of these chapters is mostly new and is
presently in the development stage. However, they are included here
in the hope that they will stimulate further research in these
important fields.
Volume II is self-contained and can be studied without the help of
Volume I. However, certain prerequisites are necessary. In order to
provide quick access to the basic equations and the underlying
physical ideas, we have included a section (Section 9.0) in Chapter
9, where the constitutive equations for electromagnetic fluids are
also presented. This section serves as a founda tion for the fluid
media discussed in Chapters 10 and 11. Basic equations and the
underlying physical ideas, necessary for each chapter, are
presented at the beginning of each chapter.
TIle second volume extends the development of the first volume to
richer and newer grounds. Because of space limitations, and the
logical development and continuity of the text, recent developments
in mixtures, semiconductors, superconductivity, nonlinear optics,
and electronic theories could not be included.
It can be said that the electrodynamics of continua touches every
aspect of
VI Preface to Volume II
the world of physics. In this regard, the present volume hopes to
stimulate certain aspects. This volume may be used as a basis for
several graduate courses in engineering schools, applied
mathematics, and physics departments. It also contains fresh ideas
and directions for further research. Ferromag netism and
plasticity, memory-dependent materials as applied to polymers,
nonlinear optics, and the nonlocal theory developed in Chapter 14
are can didates for deeper research, penetrating into microscopic
and atomic scale phenomena.
Nonlocal theory (Chapter 14) is still in its infancy. However, it
is a parallel discipline to the well-developed field oflattice
dynamics. It has the advantage that it can be used to discuss
physical phenomena in intermediate scales between microscopic and
atomic dimensions.
Electromagnetic theory properly falls into the domain of the theory
of relativity. Consequently, we have included a chapter (Chapter
15) on this subject to close this volume.
Contents (Volume II)
CHAPTER 9
Elastic Ferromagnets. 437
9.0. An Overview of Basic Equations 437 9.l. Scope of the Chapter.
443 9.2. Model of Interactions 444
A. Gyroscopic Nature of the Spin Density. 445 B. Spin-Lattice Model
ofInteractions. 446
9.3. Balance Equations 449 A. Global Balance Equations 449 B. Local
Balance Equations 450 C. The Clausius-Duhem (C-D) Inequality 452 D.
Boundary Conditions 453
9.4. Constitutive Theory . 453 A. Saturated Ferromagnetic Elastic
Insulators 453 B. Free Energy 456 C. Correspondence Between the
Microscopic Model and the
Continuous Representation . 458 D. Infinitesimal Strains . 460 E.
Centro symmetric Cubic Crystals 461 F. Uniaxial Crystals. 463 G.
Elementary Dissipative Processes 464 H. Small Fields Superposed on
a Constant Bias Magnetic Field 466
9.5. Resume of Basic Equations . 469 9.6. Coupled Magnetoelastic
Waves in Ferromagnets . 472
A. Preliminary Remarks 472 B. Plane Harmonic Waves. 474 C. Damping
of Magnetoelastic Waves. 482 D. Magnetoelastic Faraday Effect .
484
9.7. Applications of the Magnon-Phonon Coupling 487 A. Pumping and
Temporal Magnon-Phonon Conversion 487 B. Drift-Type Amplification
of Magnetoelastic Waves 490
viii Contents (Volume II)
9.8. Other Works. 490 A. Continuum Descriptions of Ferromagnetic
Deformable Bodies. 490 B. Wave Propagation . 491 C. Ferrimagnetic
Deformable Bodies. 492 Problems. 497
CHAPTER 10
Magnetohydrodynamics .
10.1. Scope of the Chapter 10.2. Basic Equations of Electromagnetic
Fluids 10.3. Magnetohydrodynamic Approximation 10.4. Perfect
Magnetohydrodynamics
A. Field Equations. B. "Frozen-In" Fields . C. Bernoulli's Equation
in Magnetohydrodynamics. D. Kelvin's Circulation Theorem in
Magnetohydrodynamics E. Alfven Waves F. Generalized Hugoniot
Condition .
10.5. Incompressible Viscous Magnetohydrodynamic Flow A.
Magnetohydrodynamic Poiseuille Flow B. Magnetohydrodynamic Couette
Flow.
10.6. One-Dimensional Compressible Flow. 10.7. Shock Waves in
Magnetohydrodynamics.
A. Classification of Magnetohydrodynamic Shock Waves B. Shock
Structure .
10.8. Magnetohydrodynamic Equilibria. 10.9. Equilibrium of Magnetic
Stars. 10.10. Magnetohydrodynamic Stability .
A. The Energy Method B. Equilibrium States and Perturbations. C.
Quantities Conserved in the Perturbation D. Elementary
Perturbations . E. Change in the Energy Integrals F. Application to
the Linear Pinch Problems.
CHAPTER II
502
502 503 507 512 512 513 514 515 515 516 518 518 520 521 525 526 530
530 533 537 537 539 540 540 543 545 547
Electrohydrodynamics 551
11.1. Scope of the Chapter 551 11.2. Field Equations. 552 11.3.
Charge Relaxation . 554 1 i.4. Stability Condition . 554 11.5.
Helmholtz and Bernoulli Equations 555
A. Generalization of the Helmholtz Equation 555 B. Vorticity
Generation in a Space-Charge-Loaded Electric Field. 556 C.
Generalization of Bernoulli's Equations 556
11.6. Equilibrium of a Free Interface. 557
Contents (Volume II) IX
11.7. Effect of Free Charges at an Interface. 11.8.
Electrohydrodynamic Stability. 11.9. Electrohydrodynamic Flow in a
Circular Cylindrical Conduit 11.10. Electrogasdynamic Energy
Converter.
Problems.
Ferrofluids .
12.1. Scope of the Chapter 12.2. Constitutive Equations of
Ferromagnetic Fluids. 12.3. Theory of Ferrofluids
A. Equilibrium Constitutive Equations . B. Nonequilibrium
Constitutive Equations C. Balance Laws
12.4. Existence and Stability of a Constant Magnetization in
aMoving
558 561 567 569 573
574
Ferrofluid 582 12.5. Ferrohydrodynamic Approximation . 585 12.6.
Some General Theorems in Ferrohydrodynamics 587
A. Generalization of the Helmholtz Equation 587 B. Generalization
of the Bernoulli Equation. 588
12.7. Ferrohydrostatics 589 A. Equilibrium of a Free Surface . 589
B. Energy Conversion . 590
12.8. Ferrohydrodynamic Flow of Nonviscous Fluids. 591 A.
Preliminary Remarks 591 B. Steady Two-Dimensional Source Flow
593
12.9. Simple Shear of a Viscous Ferrofluid . 596 12.10.
Stagnation-Point Flow ofa Viscous Ferrofluid 598 12.11. Interfacial
Stability of Ferrofluids 603 12.12. Other Problems in Ferrofluids .
608
Problems. 609
CHAPTER 13
Memory-Dependent Electromagnetic Continua.
13.1. Scope of the Chapter 13.2. Constitutive Equations. 13.3.
Thermodynamics of Materials with Continuous Memory 13.4.
Quasi-Linear and Linear Theories.
A. Quadratic Memory Dependence B. Finite-Linear Theory C. Linear
Theory D. Linear Isotropic Materials . E. General Polynomial
Constitutive Equations .
13.5. Rigid Bodies. A. Continuous Memory B. Polynomial Constitutive
Equations
611
611 612 613 620 621 622 624 627 629 630 630 631
x Contents (Volume II)
13.6. Dispersion and Absorption. 13.7. A Simple Atomic Model 13.8.
Free Motion of an Electron Under Magnetic Field 13.9.
Electromagnetic Waves in Memory-Dependent Solids 13.10.
Electromagnetic Waves in Isotropic Viscoelastic Materials. 13.11.
Nonlinear Atomic Models for Polarization . 13.12. COI}~titutive
Equations of Birefringent Viscoelastic Materials
A. Rate-Dependent Materials. B. Linear, Continuous Memory of
Strains
13.13. Propagation of Waves in Birefringent Viscoelastic Materials
13.14. Photoviscoelasticity.
Problems.
CHAPTER 14
632 634 637 641 647 652 657 659 660 661 666 673
Nonlocal Electrodynamics of Elastic Solids 675
14.1. Scope of the Chapter 675 14.2. Constitutive Equations. 677
14.3. Thermodynamics 679 14.4. Linear Theory . 682 14.5. Material
Symmetry. 686 14.6. Nature of Nonlocal Moduli 688 14.7. Nonlocal
Rigid Solids . 693 14.8. Electromagnetic Waves. 694 14.9. Point
Charge. 696 14.10. Rigid Magnetic Solids 696 14.11.
Superconductivity . 699 14.12. Piezoelectric Waves. 702 14.13.
Infrared Dispersion and Lattice Vibrations 704 14.14.
Memory-Dependent Nonlocal Electromagnetic Elastic Continua 707
14.15. Linear Nonlocal Theory for Electromagnetic Elastic Solids.
710 14.16. Natural Optical Activity 712 14.17. Anomalous Skin
Effects. 713
Problems. 715
CHAPTER 15
15.1. Scope ofthe Chapter 15.2. Space-Time, Notation
A. Space-Time. B. Special Relativity C. General Relativity D.
Inertial Frames and Rest Frame E. Proper Time, Timelikeness . F.
Space and Time Decomposition G. Antisymmetric Tensors and Axial
Four-Vectors.
716
Contents (Volume II) Xl
15.3. Relativistic Kinematics of Continua 725 A. Motion, Strain
Tensors. 725 B. Relativistic Rate of Strain . 727 C. Contravariant
Convective Time Derivative 728
15.4. Covariant Formulation of Maxwell's Equations in Matter 729 A.
Electromagnetic Fields . 729 B. Integral Formulation of Maxwell's
Equations 731 C. Four-Vector Formulation of Maxwell's Equations
733
15.5. Relativistically Invariant Balance Laws 734 15.6.
Electromagnetic Interactions with Matter 738 15.7. Thermoelastic
Electromagnetic Insulators 741 15.8. Electromagnetic Fluids.
743
A. General Nondissipative Case . . 743 B. Linear Electromagnetic
Constitutive Equations 744 C. Elementary Dissipative Processes .
745 D. Relativistic Perfect Magnetohydrodynamics . 746
15.9. Further Problems in the Relativistic Electrodynamics of
Continua. 747 Problems. 748
References
Index
753
II
CHAPTER 2
CHAPTER 5 Constitutive Equations
9.0. An Overview of Basic Equations
Basic equations of electrodynamics of continuous media were
developed in Chapters 3 and 5. Here we give a summary of these
equations, with a supple mentary discussion regarding their
extensions to some more complex media, which will be elaborated in
this volume.
Macroscopic electromagnetic theory is based on two sets of
equations:
(I) Balance Laws: These consist of Maxwell's equations and
mechanical balance laws. These equations are valid irrespective of
material con stitution.
(II) Constitutive Equations: These equations characterize the
nature of the material media. They express the response of the
medium to external stimuli. Consequently, they have different forms
depending on the nature and constitution ofthe bodies. Elastic
solids, viscous fluids, ferromagnetic materials, memory-dependent
electromagnetic elastic solids, and electro magnetic fluids all
have different constitutive equations.
For simple materials balance laws are the same, irrespective of the
material constitution, but, constitutive equations change from one
type of material body to the next. As discussed in Chapter 5,
electromagnetic elastic solids have different constitutive
equations from those of electromagnetic fluids. However, for some
complex media, e.g., ferromagnetic solids, additional internal
degrees of freedom are brought into play. In such cases both
balance laws and constitutive equations will have to be
supplemented by additional equations. An example of such media is
the subject of the present chapter. Ferrofluids, discussed in
Chapter 12, is another example of such complex media, where the new
degree offreedom arising from the spin-lattice interaction is
brought into play. Among many other important fields requiring the
considera tion of internal degrees of freedom, we mention briefly
ferroelectric media, semi-conductors, liquid crystals (DeGennes
[1974], Eringen [1979a, b]), and magnon-phonon interactions
(Matthews and Lecraw [1962]).
Here we give a summary of basic equations for simple media as
discussed
438 9. Elastic Ferromagnets
in Chapters 3 and 5. Ferromagnetic media and ferrofluids are
discussed in Chapters 9 and 12, respectively. Chapters 13 and 14
take up memory effects and nonlocality. In Chapter 13 the effects
of past deformations and electro magnetic fields are brought into
play, and in Chapter 14 those occurring at points distant from the
reference point are introduced. In all these theories the balance
laws, given below, remain valid, possibly with supplementary terms
and/or equations.
I. Balance Laws
Balance laws are the local field equations consisting of Maxwell's
equations and the mechanical balance equations. These are valid in
the body, with volume 1/, excluding the discontinuity surface (J,
which may be sweeping the body with its own velocity v. On the
discontinuity surface, we have the jump conditions which provide
the boundary condition on the surface a1/ of the body, when (J
coincides with a"r.
A. Maxwell's Equations (in 1/ - (J)
V·O - qe = 0,
1 aB VxE+--=O
1 ao 1 v x H---=-J c at c'
aa~e + v . J = 0.
B. Mechanical Balance Equations (in 1/ - (J)
Po = plII~j2 or p + pV·v = 0,
(9.0.1)
(9.0.2)
(9.0.3)
(9.0.4)
(9.0.5)
(9.0.6)
(9.0.7)
t[kll = t&'[kPll + B[kAIl' (9.0.8)
p(1' + e1] + ery) + tklVl.k - V· q - ph + PkJk + AJ3k - ~kt&'k
= 0, (9.0.9)
py == pry - V· (q/e) - (ph/e) ?: 0.
Accompanying these equations, we have the jump conditions.
C. Jump Conditions (on (J) 0·[0] = We'
n x [ E + ~ V X B ] = 0,
o· [B] = 0,
nX[H-~vXDJ=O, (9.0.14)
n'[J - qev] = 0, (9.0.15)
where surface polarization and surface currents have been
discarded. In this regard, see Chapter 3.
[p(v - v)]n = 0,
[(p('I' + el1) + ~. P + 1PV2 + 1(E2 + B2)} (Vk - vd
(9.0.16)
(9.0.17)
[PI1(V - v) -~qln ~ O. (9.0.19)
There is no jump condition corresponding to (9.0.8).
E. Mechanical Surface Traction (on a1/') In the absence of the
moving discontinuity surface, the mechanical surface
traction is given by (9.0.20)
F. Definitions of Electromagnetic Field and Loads The
electromagnetic fields in the fixed laboratory frame RG are denoted
by
D, E, B, H, P, M. In the frame Rc, co-moving with the reference
point, they are denoted by script majuscule letters,~,~, PlJ, Yt',
fYJ, and At. Cauchy's stress tensor is denoted by tk/ and the
electromagnetic stress tensor by tf,. A part of tkl is the
symmetric stress tensor Etkl' The electromagnetic body force is
given by FE, the electromagnetic momentum by G, and the Poynting
vector by 9':
1 ~ = E + -v x B,
c
c
c
'I' = e - el1 = 6 - el1 - p-l~kPk'
t~l = Pk~l - BkAl + EkEl + BkBl - 1(E2 + B2 - 2.${· B)(jkl'
440 9. Elastic Ferromagnets
II. Constitutive Equations
Constitutive equations were discussed thoroughly in Chapter 5.
According to the axioms of constitutive theory, the fields are
divided into two distinct classes:
(a) the dependent variables; (b) the independent variables.
The dependent variables are considered to be functionals of the
indepen dent variables.
The dependent variables, at a reference point X, at time t,
are:
'P = free energy,
q = heat vector,
P = polarization vector,
The independent variables are:
8 = electric field vector,
They are given at all points X' of the body, at all past times,
including the present time, -00 < t ' ::;; t.
<W'(X/, t'} = {e, ve, C, 8, B}, X' E V, 00 < t ' ::;; t.
(9.0.23)
For more complex media, such as ferromagnetic elastic solids and
viscous fluids, other additional variables are brought into play,
as discussed in this chapter and in Chapter 12.
1. Nonlocal Media
The general theory of constitutive equations for memory-dependent
nonlocal media begins with the formal constitutive equation
~(X, t) = ~[<W'(X/, t')J, (9.0.24)
where ~ is a functional of <W'(X/, t'} over space and tjme. This
is the basis of the nonlocal elasticity discussed in Chapter
14.
9.0. An Overview of Basic Equations 441
2. Memory-Dependent Local Media
For the local theory, only the values of the fields <??I(X', t')
at the reference point X are considered. Thus, 1Z is a functional
of only the time histories <??I(t') at the reference point
X
1Z(X, t) = ff[<??I(t')], -00 < t' :::;; t. (9.0.25)
For simple memory-dependent materials, this covers all sorts of
viscoelastic solids and fluids presented in Chapter 13.
3. Electromagnetic Elastic Solids (Section 5.8)
For local elastic solids, nonlocality and memory effects are
discarded. In ,this case then, 1Z is a function of <??I at (X,
t).
1Z = F(<??I). (9.0.26)
4. Electromagnetic Viscous Fluids
For fluids, C, in (9.0.23), is replaced by the mass density p and a
new variable dkl is included (Section 5.12), e.g.,
(9.0.27)
Similar equations are written for 1], Pk, Jltk, tkl , qk' and Jk'
Constitutive equations are subject to various invariance
requirements as
elaborated in Chapter 5. Among these, the restrictions arising from
the second law of thermodynamics (9.0.10) are prominent. These
restrictions are explored separately in each of the following
chapters, except Chapter 10, on electro magnetic fluids which was
discussed in Volume 1, Section 5.12. In order to make Volume 2
self-contained we present here a brief discussion.
Constitutive equations for electromagnetic viscous fluids begin
with (9.0.27). Similar equations, involving the same set of
independent variables, are assumed to be valid for 1], q, t, P, Jt,
and J.
Eliminating ph/() between (9.0.9) and (9.0.10) we obtain
., 1 . . py == -p('¥ + 1]() + tk1vl,k + eqk8,k - PeCk - JltkBk +
ASk ~ O. (9.0.28)
Substituting 'i' from (9.0.27) into this inequality we have
where we used (9.0.6) to replace p and iI)troduced the spin tensor
Wkl =
V[k,lj' This inequality is linear in 0, Wkl , dkl , li.k' $k' and
13k, The necessary and
442 9. Elastic Ferromagnets
sufficient conditions, for (9.0.29) to remain in one sign,
are
and
o\{l of) = 0,
t[klJ = 0,
(9.0.30)
(9.0.31)
where Dt is the dissipative stress tensor and 11: is the
thermodynamic pressure, defined by
o\{l 11:= -Op-l. (9.0.32)
From the first three equations of (9.0.30), it is clear that \{I is
independent of d and Vf) and the stress tensor is symmetric.
Thus,
P[krffl] + vIt[kBl] = O. (9.0.33)
If Dt, q, and ,$ are continuous in d, Vf), and iff, from (9.0.31)
it follows that
Dt = 0, q = 0, ,$ = 0, when d = 0, Vf) = 0, iff = O. (9.0.34)
Thus, we have proved the following theorem (Eringen [1980, Sect.
10.24]):
Theorem. The constitutive equations of electromagnetic fluids do
not violate the second law of thermodynamics, if they are of the
form (9.0.30) subject to (9.0.31)-(9.0.34).
Equation (9.0.33) implies that, in fluids the electromagnetic
couple vanishes. Since the free energy function \{I must be an
objective scalar function of iff
and B, it can depend on these variables only through their
invariants.
(9.0.35)
where 13 is selected so as to satisfy the invariance under the
time-reversal. Hence
\{I = \{I(1 1, 12, 13, f), p-l),
o\{l rJ = -ai)'
3 (g'B)B ,
At= -2P[:~ B + :~ (g. B)g J (9.0.36)
Constitutive equations for ot, q, and f can be constructed by using
Table E2 (Vol. I) to obtain the generators of these quantities as
functions of the joint invariants of d, ve, g, and B. Below we give
the linear theory.
Linear Constitution Equations
The free energy for the linear theory is a quadratic isotropic
function, but P, At, ot, q, and f are linear in independent
variables. Hence
1 1 'P='P _-XEg·g_-XBB·B
o 2p 2p ,
a'Po 1 aXE 1 axB 11 = -Te + 2p ae g · g + 2pae B ' B,
P = xEg,
q = KVe + KEg,
f = (Jg + (Jove,
where the material moduli 'Po, XE, XB ; the viscosities Av, .uv;
the heat conduction coefficients K, KE; and the electric conduction
coefficients (J and (Jo are func tIOns of p-l and e. Coefficients
of KE and (Jo are known as Peltier and Seebeck coefficients.
From the Clausius-Duhem (C-D) inequality (9.0.31), it follows
that
K ;:0: 0, (J ;:0: 0, 3Av + 21Lv ;:0: 0, .uv ;:0: o. 4K(Je-1 -
(KEe-1 + (J°f ;:0: O.
(9.0.38)
This completes the constitutive theory for the electromagnetic
theory of viscous fluids.
9.1. Scope of the Chapter
Chapter 8 was devoted to the magnetoelasticity of solids which
present no magnetic ordering (e.g., paramagnetic bodies), or bodies
in which this ordering does not manifest itself (e.g., the
so-called soft ferromagnetic bodies). Here we examine the case of
magnetic solids in which the ordered arrangement of magnetic spins,
be it of the ferromagnetic or antiferromagnetic type, has
444 9. Elastic Ferromagnets
important consequences for magnetoelastic couplings. Most of the
chapter, however, is devoted to the case of hard ferromagnetic
bodies. As was briefly recalled in Section 4.5, the most important
coupling phenomenon is the so-called phonon-magnon coupling with
the allied magneto acoustic resonance effect. On a microscopic
scale, this phenomenon follows from the fact that both phonons and
magnons obey the Bose-Einstein statistics in quantum physics, and
are therefore expected to interact in a sufficiently strong manner.
Here, however, all discrete details are avoided so that, of
necessity, a heuristic model of interactions must be considered;
this is the object of Section 9.2. The continuum formulation of a
theory of elastic ferromagnets, apart from the few new ingredients
introduced by the model of interactions, then follows the same
development as other continuum theories (see Chapters 3 and 5).
Local balance equations are given in Section 9.3, and the
constitutive theory for thermoelastic ferromagnetic insulators is
developed in Section 9.4. Special attention is given to the cases
of quadratic free energy, infinitesimal strains, the correspondence
between microscopic and macroscopic representations of exchange
forces, cubic and uniaxial crystals, and elementary dissipative
processes such as viscosity and spin-lattice relaxation.
Section 9.5 presents a resume of the basic equations. Section 9.6
introduces the reader to the study of coupled magnetoelastic waves
in ferromagnetic insulators. The effects of magnetoacoustic
resonance, damping of magneto elastic waves, and the
magnetoelastic Faraday effect are exhibited analytically and
illustrated by many curves. Applications of magnon-phonon couplings
are examined in Section 9.7. These include pumping and temporal
magnon phonon conversion. Section 9.8 discusses briefly more
general problems such as the case of deformable
antiferromagnets.
Dynamic magnetoelastic effects in magnetically ordered solid bodies
occur in the ultrasonic (hypersound) region, and are particularly
important from the viewpoint oftechnological applications which
include hypersound generators, high-frequency magnetostrictive
transducers, the amplification of waves by using nonlinear
interactions, the design of wave filters and delay lines with fixed
delay or electronically variable delay, analysis of the internal
magnetic field, internal-magnetic field synthesis, etc. A great
variety of physical effects, which these materials can be made to
exhibit, to greater or lesser degree, has been utilized in the
ever-broadening search for new components. The strong links between
solid state physics on the one hand, and the electronics industry
on the other, have resulted in an area of research scrutinized
simultaneously by applied physisists, electrical engineers, and
scientists in the field of mechanics.
9.2. Model of Interactions
In previous chapters we have dealt with paramagnetic elastic
bodies. Here we focus our interest on the case of ferromagnetic
elastic bodies. Since ferro magnetism is, by its very nature, a
quantum mechanical effect, and we intend
9.2. Model of Interactions 445
to consider a phenomenological approach dealing with continuous
fields, there is a need to introduce a heuristic model of
interactions. This model will serve to describe, in the language of
continuum physics, the interactions which take place between the
lattice continuum, i.e., the usual substrate of elastic defor
mations and the magnetization field.!
In the quantum-mechanical sense, individual particles have
associated with them a magnetic moment and an internal angular
momentum called spin. Electrons are also believed to provide a
predominant contribution to the magnetic moment of an atom (see
Section 4.4). In continuum theory it is convenient to refer to the
continuum, which smoothly represents the discrete distribution of
individual spins over the material body, as the "electronic" spin
continuum.2 S\nce the field equations governing the lattice
continuum-the equations of motion-have already been examined in
detail in Chapter 3, the main purpose of this section is to
elaborate upon the ingredients which allow the formulation of the
field equations that govern the electronic spin continuum.
A. Gyroscopic Nature of the Spin Density
In Section 9.0 it was mentioned that, in ferromagnetic materials,
spin-lattice interactions play an important role in this magnetic
behavior. The field primarily associated with the electronic spin
continuum is the intrinsic spin density per unit mass in the
co-moving frame. This is introduced by the isotropic gyromagnetic
relationship (compare Section 2.2).
S(x, t) = y-! p(x, t}. (9.2.1)
The isotropy of the effect follows from the fact that only one type
of particle, the electron, provides a predominant contribution.
Accordingly, the gyro magnetic ratio y is not very different from
the constant y. = - e/moc (here only spin angular momentum is
accounted for, see (2.2.10», and p and S
depend on the position x at time t in the present configuration .Yt
of the material body.
In order to treat phenomenologically the magnon-phonon coupling
(thus long-wave magnons) we consider low levels of energy, i.e.,
low temperatures. Consequently, the present discussion is limited
to the case of temperatures B which are much smaller than the Curie
temperature Be of the material: B « Be. In these conditions the
local magnetization may be considered saturated with a
time-independent magnitude. It follows from these assumptions
that
p' p = /1; = const. (9.2.2)
1 Such a model has been devised by H.F. Tiersten [1964]. See also
Maugin [1976b] for general electromagnetic continua. 2 As usual,
not too much attention should be paid to the wording, which only
provides a modus vivendi; and it must be noticed that there is not
necessarily any one-to-one correspondence between microscopic
concepts and macroscopic ones.
446 9. Elastic Ferromagnets
and
But via the equation of motion (2.2.1) we can also write
Ji = Ji(X, t),
XE V,
/1i,K/1i =0.
(9.2.3)
(9.2.4)
(9.2.5)
(9.2.6)
Direct consequences of (9.2.3) are as follows: since I Jil =
const., if Dn denotes the co-rotational derivative (see Section
1.12) with respect to a frame rigidly attached to Ji in Rc(x, t),
we will have DnJi = ° where, symbolically, Dn = d/dt - n x . Thus
it is necessarily of the form
s=flxs.
By taking the vector product of this with Ji we find that
n = /1;2 [Ji X ft + (w n)Ji],
(9.2.7)
(9.2.8)
where n is called the precessional velocity vector of the
magnetization density or the spin density. Equation (9.2.7) reminds
us of the equation governing a gyroscope in a frame attached to it.
Indeed, the gyroscopic character of the spin s is asserted from
(9.2.7')2:
n· s = y-1 g. it = 0, (9.2.9)
where s is a couple per unit mass. Equation (9.2.9) states that, in
a real precessional velocity field fl, the couple s = y -1 it does
not produce any power. Such a couple is said to be a gyroscopic or
d'Alembertian inertia couple. Consequently, the inertia associated
with the field Ji will not participate in the statement of the
first principle of thermodynamics.
B. Spin-Lattice Model of Interactions
For the sake of simplicity we consider nonelectrically polarized
ferro magnets. 3 Thus,
P =0, 1t = 0, (9.2.10)
everywhere in the body at any time. Then the usual motion of the
deformable body
x = x(X, t) (9.2.11)
and the magnetization function (9.2.5) may be considered as
defining a kind
3 However, there is no difficulty in accounting for nonzero P. In
this regard, see Tiersten and Tsai [1972J and Collet and Maugin
[1974].
9.2. Model of Interactions 447
of generalized motion for the total lattice-plus-spin continuum.
With each material point in the body there is associated, in a
unique way, a magnetization field or a spin field, so that the
electronic spin continuum cannot translate with respect to the
lattice continuum at that point. Therefore, it is clear that the
spin continuum expands and contracts with the lattice continuum and
must occupy the same volume with its volumetric behavior being
governed by the usual continuity equation. As usual, the lattice
continuum responds to volume and surface forces (thus, to stresses)
and to volume couples. We suppose here that it does not possess any
mechanism to respond to surface couples.4
Based on this observation, the conservation oflinear momentum
states that whatever force of magnetic origin (e.g., the
ponderomotive force) is applied to a point on the spin continuum,
it is transferred directly to the lattice con tinuum at that
point. Indeed, by its very nature, the spin continuum can respond
only to couples, which may be either volume or surface couples. In
view of (9.2.10) the ponderomotive couple (see (3.5.44)) applied to
the spin continuum is reduced to
cE = .H x B = PI1 x B. (9.2.12)
As far as the interactions between the lattice continuum and the
spin con tinuum are concerned, they must necessarily be of the
couple type (see Figure 9.2.1). We assume that this couple results
from the existence ofa local magnetic induction field LB, so that
the couple exerted by the lattice continuum on the spin continuum
is given by the "recipe" (compare (9.2.12)):
(9.2.13)
Since the angular momentum is conserved between the two continua,
an equal and opposite couple
C(S/L) = -e(L/S) = LB x .H= P LB x P (9.2.14)
is exerted on the unit volume of the lattice continuum. Finally, in
order to account for ferromagnetic exchange effects, we must
consider that each spin of the electronic spin continuum
experiences from its nearest neighbors an action caused by the
exchange forces of a quantum mechanical nature (see Section 4.5).
Given the rapid fall with distance of this type of force, it will
be assumed that they give rise to contact (i.e., surface) actions
in much the same manner as the stress vector for the lattice con
tinuum. More specifically, a surface exchange contact force !!T is
introduced which produces a couple per unit area equal to .H x !!T
on the spin con tinuum. After the discussion given in Section 4.5,
it is clear that !!T will be related in some way to the spatial
nonuniformities in the magnetization field. !!T is an axial vector
and has the dimension of a magnetic field times length,
4 This possibility, however, is envisaged in Collet and Maugin
[1975] and Collet [1978].
448 9. Elastic Ferromagnets
LATTICE CONTINUUM (LC) (I ne rt i a : p v)
Volume c (SCI LC) couples
Maxwell's SPIN CONTINUUM (SC) equations (Inertia: p)-1 ~)
force f
--..""."", ,....";,,,. B~
Surface coup! ~
Figure 9.2.1. Interactions in deformable ferromagnets (from Maugin
[1979c, p. 253]). Reprinted with permission of Elsevier Science
Publishers.
or of a surface distribution of magnetic dipoles. Similar to the
generalized stress principle (3.8.3), it is assumed that the value
of :!I on 0"Y depends only on the normal vector n of the surface,
i.e., fT = 5(0) = fT(x, t; n), x E 0"Y. This implies a first-order
gradient theory as far as magnetization effects are con cerned.
Since :!I acts through the couple oR x fT, only the portion of 5(0)
which is orthogonal to oR is effectively defined, so that, without
loss of generality, we can set forth the condition
5(o)'oR= 0
on 0"Y. A similar condition may be imposed on LB. That is,
LB'oR= 0 at all points in "Y.
(9.2.15)
(9.2.16)
We are now in a position to set forth the global balance laws that
govern non polarized moving ferromagnetic bodies.
9.3. Balance Equations 449
A. Global Balance Equations
Using the notation of Section 3.10, and taking into account the
model just constructed, we have the following global balance laws,
in addition to Maxwell's equations «9.0.1) (9.0.5)),
Conservation of Mass:
dd f P dv = o. t "Y"-a
Balance of Momentum for the Lattice Continuum:
dd f PVi dv = f (Ph + Fn dv + r t(O)i da t "Y" -a "Y" -a J
a"Y"-a
Balance of Moment of Momentum for the Lattice Continuum:
dd f (x x PV)i dv t "Y"-a
Balance of Angular Momentum for the Electronic Spin
Continuum:
d f -1 f E f -d PY fl dv = [c + C(L/S)] dv + Pfl X 9(0) da, t "Y"
-a "Y" -a a"Y"-a
Conservation of Energy for the Combined Continuum:
dd f p(!v2 + e) dv = f (pf·v + ph + WE) dv t "Y" -a "Y" -a
+ f [w;]np da, a(l)
(9.3.1)
(9.3.2)
(9.3.3)
(9.3.4)
(9.3.5)
As noted in Section 9.0, for the first time, here we encounter a
new balance law (9.3.4), in addition to modifications introduced by
the spin degrees of freedom, in other balance laws.
450 9. Elastic Ferromagnets
Principle of Entropy
dd Ie PY/dv~I p~edv+ r. e-1q·nda. t ,t -(1 "//_(1 JaY'-(1
(9.3.6)
All symbols bear the same significance as in Chapter 3. 5(0) is the
only new field which contributes to the expression of the first
principle of thermo dynamics, since the inertia associated with J!
yields a zero contribution (see (9.2.9)) and LB cannot contribute
because of the action-reaction principle between the two continua.
The power developed by 5(0) is estimated in accordance with the
rule that a magnetic field A, acting on a magnetization field At =
PJ!, produces a power pA . Ji. As before O'(t) is a moving
discontinuity surface having velocity v with respect to the
laboratory frame RG •
B. Local Balance Equations
The localization of equations (9.3.1)-(9.3.3) is made, as modeled
by (3.10.3):
Conservation of Mass: jJ + pVk,k = 0 in "r - 0',
[P(Vi - v;)]ni = 0 on O'(t),
Balance of Momentum for the Lattice Continuum:
PVi = pj; + N + tji,j in "f/' - 0',
[pvi(vj - Vj) - tji - (t}f + vpJ]nj = 0 on O'(t),
with t(D)i = njtji on o"f/' - 0';
Balance of Moment of Momentum for the Lattice Continuum:
eijk(tjk + P LBj!1k) = 0 in "f/' - 0'.
Applying the alternation symbol to the free index this reads
tUiJ = - P LB[j!1iJ in "f/' - 0'.
(9.3.7)
(9.3.8)
(9.3.9)
(9.3.10)
(9.3.11)
(9.3.12)
(9.3.13)
Applying the tetrahedron argument (3.8.4) to (9.3.4) we can show
that
(9.3.14)
where £!lJj; is a new (general second-order) tensor which plays a
role similar to the stress tensor tji . Since 5(0) is supposed to
represent the effects of exchange forces, we shall refer to flj]j;
as the spin-spin interaction or exchange-force tensor. Taking into
account (9.3.14), the localization of equation (9.3.4) yields at
once the following results:
Balance of Angular Momentum for the Electronic Spin
Continuum:
y-lp; = e;jk!1j(Bk + LBk + P-1£!lJzk,Z) + p-1e;jk£!lJzk!1j,Z in"f/'
- 0', (9.3.15)
[py-l !1;(vj - v) - e;pQ!1pflj]jQ]nj = 0 on O'(t). (9.3.16)
9.3. Balance Equations 451
If we note that p. is ofthe form (9.2.7), and compare it with
(9.3.15), we deduce
p. = n x J.1 in "Y - (J, (9.3.17)
where now n = _yBeff (9.3. IS)
with (9.3.19)
and or (9.3.20)
The latter is a direct consequence of the conditions oflow
temperature (J « (Jc
and saturation of the magnetization. Equations (9.3.18) and
(9.3.17) reminds us ofthe Larmor spin precession of an isolated
electron (om/ot = -YeB x m). However, the precession ofJ.1 here is
caused by an effective induction Beff which results from the
combined action of the Maxwellian magnetic induction, the local
magnetic induction, and the exchange forces via (fiji'
Conservation of Energy: Employing the equations obtained above and
(3.10.3), the localization of
(9.3.5) yields - p(q; + (J1j + (11) + tjiVi,j + (fIji(iti),j - P
LB' P.
+ V·q + f·1t + ph = 0 in"Y - (J (9.3.21)
and
[{!pv2 + p'li + P11(J + !(E2 + B2 - 2Jt· B)}{vj - v)
- (tji + tIf + VjGi)Vi - (fIjiiti - (qj - Yj)]nj = 0 on (J(t),
(9.3.22)
where 'Ii = e - 11(J = /:: - B· J.1 - 11(J. (9.3.23)
According to the Legendre transformation (9.3.21h, e and 'Ii depend
func tionally on the magnetization density. Consequently, by using
(9.2.9), (9.3.1S), and (9.3.19), we have
(fIji,j{t; = - pB' P. - p LB· p.. Finally, we have from (9.3.6)
the
Local Entropy Inequality:
p1j ~ (J-l ph + (J-l V . q + q' V G) in "Y - (J,
(9.3.24)
(9.3.25)
Together with Maxwell's equations, eqs. (9.3.7), (9.3.S),
(9.3.9)-(9.3.11), (9.3.13), (9.3.14), (9.3.16), (9.3.17), (9.3.22),
(9.3.24), and (9.3.25) form the complete set of local field
equations, boundary conditions, jump conditions, and thermo
dynamic constraints for the present theory. To close the theory,
these must be supplemented with initial conditions and constitutive
equations for the
452 9. Elastic Ferromagnets
dependent variables e, 1], t ji , LB, !?4ji , $, and q. The
Clausius-Duhem (C-D) inequality is central to the development of
the constitutive theory. Note finally, that the above-obtained
equations are independent of the exact mechanical behavior which
may be that of elastic bodies, fluids, or media with an inter
mediary behavior.
C. The Clausius-Duhem (C-D) Inequality
Eliminating h between (9.3.21) and (9.3.25), we are led to the
following Clausius-Duhem inequality:
~ . 1 - p('¥ + 1]8) + tjiVi•j - P LB' Ii + [!8ji(itJ,j + $ . iff +
(j q . V8 ~ O. (9.3.27)
It is important to express this inequality in terms of objective
time rates. To fulfill this requirement, we note that 11 is
objective since it is the magnetization in the co-moving frame and
we introduce the objective time rates
mi == iti - V[i.jJi1j = (DJIl)i'
@ij == (itJ. j - V[i.k]i1k,j = (DJ VIl)ij + i1i,k d kj'
(9.3.28)
(9.3.29)
where (VIl)ij = i1i,j and D J indicate the Jaumann derivative. The
second part of (9.3.29) is readily checked from (1.12.6) and
(1.9.7). Then, using (9.3.28), (9.3.29), (9.3.13), and (9.3.20),
(9.3.27) takes the form 5
- p(4 + 1]0) + (Jjidij - P LB' ill + !?4ji@ij + $ . iff + 8-1 q .
VB ~ 0, (9.3.30)
so that (9.3.31)
(9.3.32)
The last equation is none other than the canonical decomposition of
the Cauchy stress in its symmetric and skew-symmetric parts. This
decomposition is fundamental in that it shows that if the field LB
possesses a thermo dynamically irreversible contribution (which
will be shown to represent the so-called spin-lattice relaxation
phenomenon), then this contribution partici pates in the Cauchy
stress. As a consequence, it will follow that the damping of
elastic waves can be partly due to spin-lattice relaxation.
Another comment is in order concerning (9.3.30) and the duality
inherent in thermodynamics. If the field 11 is frozen in the
deformable matter (i.e., if its precessional velocity n equals the
vorticity w), then LB no longer contributes to (9.3.30). This fact
upholds the interpretation already granted to LB. How ever, !?4ji
is clearly associated with the gradient of 11.
5 The inequality (9.3.30) was first obtained by Maugin and Eringen
[1972aJ and Maugin [1974aJ for insulators. Its extension to the
case of electrically polarized conductors is due to Maugin and
Collet [1974].
9.4. Constitutive Theory 453
D. Boundary Conditions
If (T is a material surface, v = v, then (9.3.10), (9.3.16),
(9.3.22), and (9.3.26) reduce to
[tji + tj~ + vjG;]nj = 0,
[eipqJ.lpgHjq]nj = 0,
[0-1 qj] nj ::; O.
(9.3.33)
Letting (T coincide with o"f/ then reproduces the boundary
conditions on o"f/. Now consider (9.3.14). Jt x 9(0) is a surface
couple acting on the electronic
spin continuum. If angular momentum is conserved for the whole
continuum on o"f/ - (T, then a surface couple 9(0) x Jt should be
applied to the lattice continuum. But the latter is not supposed to
be able to respond to such couples. Therefore, in general, for any
J1 different from zero, we must have
9(0) = 0 on o"f/ - (T. (9.3.34)
If follows from (9.3.14) and (9.3.34) that the vectors of
components J.li and nj86ji must be parallel on o"f/ - (T.
Introducing a multiplier A. (an unknown to be found in the process
of problem solving), the spin boundary condition takes the
restricted form6
(9.3.35)
A. Saturated Ferromagnetic Elastic Insulators
Numerous ferromagnetic materials can be considered to be
insulators. Here we consider the case of elastic ferromagnetic
insulators that are magnetically saturated. Therefore, in addition
to (9.2.10), we set
,$ =0.
FE = (VB)·M.
(9.4.1)
(9.4.2)
6 This boundary condition was established by one of the authors in
the more general ferri~agnetic case (Maugin [1976c]). In cubic
crystals where Blj; = all;,}, (9.3.35) takes the form e(8,,/8n) + "
= 0 with 0 ~ e = a/).. ~ 00. This is the boundary condition
obtained by Soohoo [1963] (see also Gurevich [1973, p. 140]). In
rigid bodies only conditions of the type (9.3.35) can be imposed
with, for instance, ).. = 0 -+ njBlj; = 0 or ).. = 00 -+ " = O. In
deformable bodies Collet and Maugin [1975] have considered a finer
mechanical description to obtain a broader class than (9.3.35) for
the spin boundary conditions. Hyperstresses and double normal
forces must be considered there along with the usual
stresses.
454 9. Elastic Ferromagnets
Fundamental to the development of the constitutive equations is the
second law of thermodynamics (9.3.27) which reads
- p(qJ + ;,8) + tjiVi,j - P LB' it + ~ji(itJj + 8-1 q . V8 2: O.
(9.4.3)
In accordance with the axioms of constitutive theory enunciated in
Chapter 5, we assume that the dependent constitutive
variables
are functions of the set of independent variables
{Xi,K' l1i, l1i,K, 8,K' 8}.
(9.4.4)
(9.4.5)
More convenient is a set of equivalent objective variables that can
be con structed from (9.4.5), namely,
(9.4.6)
However, for magnetically saturated insulators, a more useful set
is
(9.4.7)
where 8,K = 8,iX i,K,
(9.4.8) .ffKL = l1i,Kl1i,L = .ffLK ·
This contains three members less than those contained in (9.4.6),
since the number of independent components of mKL is nine, while
that of .ff KL is six. We have four constraints (9.2.2) and (9.2.6)
restricting 11 and 11;,K' Therefore, if we use (9.4.7), the three
constraints (9.2.6) can be disregarded and we will be left with a
single constraint, (9.2.2), or equivalently,
(9.4.9) -1
where C KL is the Piola strain measure defined by (1.4.11). Hence
we set -1
qJ = 'P(EK£> mK, .ffKL , 8. K, 8) - gli( CKLmKmL - 11;)
(9.4.10)
where gli is the Lagrange multiplier. Employing (1.9.11), we
compute the following time rates:
EKL = d· ·x· KX, L lJ l, ).'
rYtK = (itj + l1i Vi,j)xj ,K,
AKL = 2(itJ,jXj ,Kl1i,K'
(9.4.11 )
+ (tii - P :: J.liXj,K) Wij + ~q. V() ~ 0, (9.4.12)
where we used vi,i = dij + wij' We note that th<? coefficient of
the Lagrange multiplier & in (9.4.10) does not contribute to
'P. The inequality is linear in the quantities 0, O,K' dij, {Ii'
and ({Ii),j which can be varied independently and arbitrarily.
Moreover, wij = -Wji' Consequently,
Theorem. Constitutive equations of saturated ferromagnetic elastic
insulators are thermodynamically admissible if and only if
8'1' 8'1' 1] = -aij' 8() = 0,
,K
8'1' t·· = p--X· KX, L - P LB.J.l. J' 8EKL " J, J "
0'1' LB.= --X' K
J' 8..HKL J, ','
tUiJ = - P LBuJ.lil'
In view of(9.4.13), 'I' is independent of (),K so that
'P = 'I'(EKL' mK, ..H KL, ().
(9.4.13)
(9.4.14)
(9.4.15)
(9.4.16)
(9.4.17)
(9.4.18)
(9.4.19)
From (9.4.18) and the continuity of q with respect to V() it
follows that
q = 0 when V() = O. (9.4.20)
The constitutive equation for q is of the form
(9.4.21)
subject to (9.4.18) and (9.4.20). For. some purposes it is useful
to express the stress tensor in the forms
of which the second follows from the definition (9.3.31) and
0'1' Etji = P OEKL Xj,LXi,K = Etij (9.4.23)
456 9. Elastic Ferromagnets
is the elastic part of the stress tensor. Note, however, that Et
not only contains elastic effects, but also effects such as magneto
stricti on, piezomagnetism, and exchange-strictive effects.
We also note that the saturation condition (9.3.20) is satisfied
automati cally, since from the symmetry of A KL and (9.4.16) it
follows that
o\{l BUk[iflJl.k = 2p~ fl[i.KflJl.L == O.
U~tKL (9.4.24)
Finally, since mK essentially represents the components of the
magnetiza tion in the initial configuration, the local magnetic
field LB is the field which accounts for the dependence of the
energy of the ferromagnet upon the direction of the magnetization.
In accordance with the discussion presented in Section 4.5, LB can
also be referred to as the magnetic anisotropy field or
magnetocrystalline magnetic field.
The free energy function \{I is further restricted by the material
and mag netic symmetry regulations. To this end, we employ
representation theorems as discussed in Appendix C (Vol. I).
However, to exhibit various physical phenomena, we resort to a
quadratic expansion of the free energy which is valid for the
low-energy levels. 7
B. Free Energy
Let To > 0 be the uniform temperature field and Po the mass
density in the reference configuration
(9.4.25)
A polynomial approximation for the free energy, irrespective of any
material symmetry, is of the form
Po q.; = LEI + LEx + La + Lpyro + LPiezo + LMs + L ES +
higher-order terms, (9.4.26)
where
LEI = 0'0 - Porto T - (Po'Y/2To)T2 + (AKL - BKL T)EKL +
~LKLMNEKLEMN (Thermoelastic energy),
LEx = !aKLAKL + !AKLMNAKLAMN + fKLMmKALM (Exchange energy),
(Magneto-crystalline energy), (9.4.27)
(Pyromagnetic energy),
(Piezomagnetic energy),
7 Reduced forms of constitutive equations for hemitropy were given
by Maugin and Eringen [1972b] and Maugin [1976a].
9.4. Constitutive Theory 457
I:MS = !AKLMNEKLmMmN (Magneto-Strictive energy),
I:ES = 'YKLMNEKLJ( MN (Exchange-Strictive energy).
The symmetry regulations of various tensorial moduli (which depend
on To and Po) are deduced easily from the symmetry of E KL , J(KL'
and the expressions in (9.4.27). (10 is a constant which can be
discarded without loss of generality.
Several special cases are of importance:
(a) For centrosymmetric materials there exist no material tensors
of odd rank, so that fKLM' NK , and EKLM are nil. Discarding the
effect represented by NKL which is of second order in J1.i,K-see
(9.4.8)-we see that the phenomena of pyromagnetism and
piezomagnetism do not show up In centro symmetric materials.
(b) If we consider isothermal processes, aT/at = 0 and VT = 0, then
all terms containing T in factor can be discarded.
(c) The term linear in EKL in the thermoelastic energy can be shown
to yield, in the linearized theory, a homogeneous state of stresses
(see Section 5.11), which can be formally removed from the
formulation by considering that it defines a new reference
configuration.
(d) The term AKLMNJ( KLJ( MN is of fourth order in J1.i,K and can
be neglected, compared with the term aKLJ( KL, in most cases where
the spatial non uniformities in the spin field are small.
Thus, for a noncentrosymmetric material in isothermal situations
and in the absence of homogeneous stresses, we can consider the
following simplified expression for the free energy:
Po 'I' = !I:KLMNEKLEMN + !AKLMNEKLmMmN
+ EKLMmKELM + h~LmKmL + !aKLJ( KL + 'YKLMNEKLJ( MN' (9.4.28)
This expression still contains contributions of elastic,
magnetostrictive, piezo magnetic, magnetocrystalline, exchange,
and exchange-strictive origins. We have gathered the last two
contributions into a single expression for the following reason:
these effects arise simultaneously when we approximate (by a
continuous expression) the microscopic exchange energy in solid
materials subject to large elastic deformations.
(e) As far as magnetomechanical effects are concerned,
magnetostriction is of second order in p (or mK ), whereas
piezomagnetism is of first order. For sufficiently small p, the
first effect may be neglected compared with the second. The
remaining piezomagnetic effect is sufficient to produce the
coupling of phonons and magnons (see Section 9.6). Similarly, the
exchange-strictive effect is of second order in J1.i,j and can be
neglected for small spatial nonuniformities of p. In these
conditions (9.4.28) reduces to
458 9. Elastic Ferromagnets
the simple expression
Po'¥ = 1L.KLMNEKLEMN + h~LmKmL + 1 a KL A KL + EKLMmKELM'
(9.4.29)
This expression may be thought of as an expansion in E KL and jl KL
followed by an expansion of the tensorial coefficients in terms
ofmK • Finally, only terms of second order at most are kept, so
that along with the magnetocrystalline term only the piezomagnetic
term appears as a new term during the process. Constitutive
equations (9.4.14)-(9.4.16) read
(9.4.31)
(9.4.32)
C. Correspondence Between the Microscopic Model and the Continuous
RepresentationS
Here we consider elastic ferromagnets subject to large
deformations, and try to find an expression for the exchange energy
in terms of its microscopic representation given in Chapter 4. To
that end, consider Heisenberg's expres sion (4.5.7). The exchange
integral J is a function of the distance r(ap) between the atoms
IY. and [3. Now assume that the angle cpaP between sa and sP is
very small, so that, with ISal = ISPI == s,
where n" is the unit vector along S", i.e., along the associated
magnetic moment. Then (4.5.7) yields
(9.4.34)
Suppose that, at fixed time, n" can be represented by a continuous
function n(X) of the position in the reference configuration K,
i.e., in the undeformed lattice. Then, to a sufficient
approximation,
nP - n" = n,K(XK - Xa
where XK are the coordinates of the atom IY. in K. Equivalently,
using the director cosines d", we can write
(9.4.35)
9.4. Constitutive Theory 459
For the sake of simplicity consider the origin of coordinates at
the undeformed site IX. Then XK = O. Substituting from (9.4.35)
into (9.4.34) we obtain
(9.4.36)
We then sum this expression over nearest neighbors /3, multiply the
whole expression by ! to avoid counting each term twice, and
multiply by the number of atoms no per unit of undeformal volume.
The first contribution in (9.4.36) can only depend on d and EKL
through J(r(aP»). Therefore it can be absorbed in other
contributions to the whole free energy that also contain E KL and
d. Hence, the contribution of microscopic exchange forces to the
phenomenological (free) exchange energy in the undeformed state is
obtained by summing only over the second contribution of (9.4.36).
That is,
pot/lex = ! noS2 L J(r(aP»)x~Xfdi,Kdi,L' (9.4.37) P
The estimation of this expression is as follows: Let R be the
distance between an atom and its nearest neighbors in the
configuration K. Since Rand r(ap) are obviously very small on the
continuum scale, to a sufficient approximation we can write [see
(1.4.2hJ
r(aP)2 = x\aP)x\aP) = cKLx~xf
= R2 + 2EKLX~xf, (9.4.38)
since R = «(jKLX~Xf)1/2. Set rl == XVR, the director cosines of the
line that join the sites IX and /3 in K, then (9.4.38) reads
r(ap)2 = R2(1 + 2EKLrlrf),
so that, to the first order in EKL, we have
r(ap) = R(1 + EKLrlrf). (9.4.39)
J(r(aP») can be evaluated by considering its expansion in function
of the small distance variation, i.e.,
J(r(aP») ::::::: J(R) + J'(R)(r(aP) - R),
J(r(aP») = J(R) + J'(R)REKLrlrl. (9.4.40)
aKL == n~ S2 R2 J(R) L rlrl, fl. P
YKLMN == 2no2 S2 R3 J'(R) L rlrlrftrA, fl. P
(9.4.41)
460 9. Elastic Ferromagnets
Taking into account (9.4.27h, we can rewrite (9.4.37) in the
form
porjJex = }aKLAKL + 'YKLMNEKLAMN. (9.4.42)
This is nothing other than the last contribution in (9.4.28). The
phenomeno logical tensorial coefficients are related to
microscopic parameters through equations (9.4.38). a KL is
obviously symmetric, but (9.4.38) tells us that YKLMN
is also a completely symmetric tensor.9 We see that a continuous
approxima tion of the Heisenberg exchange energy here yields both
pure exchange and exchange-strictive contributions to the whole
free energy. Strictly considered as a formula issueing from
microscopic considerations, (9.4.42) is valid only at e = O.
However, it may be reinterpreted as a formula valid at an arbi
trary temperature (of course, such that e « eJ, by assuming that
the material coefficients a KL and YKLMN are temperature
dependent.
D. Infinitesimal Strains
In the case of infinitesimal strains we no longer need to
distinguish between lowercase and capital Latin indices. For
instance, considering (1.6.7), (1.6.8), and (9.4.8), and neglecting
exchange-strictive effects, we have
~ _ HI 1 - - + 1 M + - + 1 1 - + 1 £.. = Po T = Z(Jijkleijekl zXij
Ilillj eijklliejk zAijkleijllkll1 zaijllk,illk,j'
(9.4.43)
This expression is valid for noncentrosymmetric linear elastic
bodies, when there are no initial stresses, and for isothermal
situations. To the same degree of approximation, the constitutive
equations (9.4.14)-(9.4.16) yield, with p ~ Po,
(9.4.44)
(9.4.45)
(9.4.46)
and
9 There are other cases in physics where a microscopic approach
yields a stronger symmetry than the corresponding macroscopic one.
This results from the fact that certain special forms of
interactions and potentials must necessarily be introduced in a
microscopic theory to make it tractable (e.g., central forces,
two-body interactions). This happens in the determination of
elastic moduli from lattice theory where the so-called Cauchy
relation results between the elastic moduli of cubic crystals (see
Musgrave [1970, p. 234]).
9.4. Constitutive Theory 461
This last expression shows that the magnetocrystalline effect can
contribute to the Cauchy stress at the second order in J.1 in the
same way as the magneto striction. This follows from the
decomposition of (9.4.22). The relative impor tance of the two
effects depends on the strength of the fields. 10 In a fully linear
theory we keep only the quadratic terms jointly in eij and fli in
the expression (9.4.44). This amounts to setting A ijkl = 0.
The coupling between stresses and the magnetic anisotropy field-or
lattice and spin continua-subsists only in the form of the
piezomagnetic effect. The local balance law of the moment of
momentum is no longer satisfied, for tji is now symmetric. The
simplified theory obtained is thus similar to Voigt's theory of
piezoelectricity discussed in Chapter 7. Finally, by virtue of
Euler's identity for quadratic forms, L can be rewritten in the
form
(9.4.48)
which proves useful in studying theorems of existence and
uniqueness for this simplified theory.
Returning to the more general case (9.4.43) we note that, with the
help of representation theorems,11 expressions for the various
tensorial material coefficients for many of the crystallographic
classes and magnetic groups, referred to in Chapter 5, can be
obtained. However, numerous ferromagnetic materials are either
centrosymmetric cubic or unixial (i.e., transversely isto ropic).
Consequently, we focus our attention on these two classes.
E. Centrosymmetric Cubic Crystals
Iron and nickel, and some of their compounds such as
yttrium-iron-garnet (Y.I.G.), are ferromagnetic materials with a
cubic structure. Iron garnets are typical elastic ferro magnets of
which the general formula is M 3 Fe1S 0 12 ,
where M is a trivalent magnetic ion and Fe appears in its trivalent
form Fe3+ . Y.I.G. is the best known iron garnet with the formula
Y3 Fes0 12 where the ion y3+ is diamagnetic.
As an example, we consider a cubic crystal of the magnetic class
1Jl31Jl (which belongs to the subgroup m3 ), the generators of
which have been given in Appendix B (Vol. I). This material is
centrosymmetric. The tensorial coefficients appearing in (9.4.43)
have the representations
(Jijkl = C(jijkl + c 12 (jij(jkl + C44 (Oik Oji + 0UOjk)'
eijk = 0, (9.4.49)
A ijkl = AOijkl + A 12 (jij Okl + A44 (Oik Oji + OUOjk),
where the symbol Oijkl has components equal to one when all indices
are alike and zero otherwise. The second equation of (9.4.49) tells
us that the magnetic
10 See Maugin and Eringen [1972b] and Maugin [1976a, pp. 299-300].
11 See Sirotin [1960], [1961], Mason [1966], and Appendices Band E
(Vol. I).
462 9. Elastic Ferromagnets
anisotropy will not have any effect on the spin precession if it is
only of second order in p. Therefore, we must consider the next
contribution, which will be of fourth order in p, in the expression
(9.4.43). This contribution reads
with bijkl = Mijkl + b12 (c5ijc5kl + c5ik c5jl + c5jkc5i/)'
Setting Kl == -bj2,
(9.4.50)
(9.4.51)
(9.4.52)
introducing the Cartesian system (x, y, z) of coordinates,
discarding terms which contain only Jl == Ipl and using (9.4.48), a
short computation leads to the following expression for L: '
L = tCll (e~x + e;y + e;z) + c12(exXeyy + eyyeZZ + ezzexx)
+ tC44(e~y + e;z + e;J + taJli,jJli,j
+ (A44 + Aj2)(exxJl~ + eyyJl; + ezzJl;)
+ K 1 (Jl~ Jl; + Jl; Jl; + Jl; Jl~), (9.4.53)
in which there appear seven material constants, Cll' C12 , C44' A,
A44' a, and K l' The fact that there are two magnetostriction
constants in cubic crystals was recognized by Akulov (1936). For
the purpose of illustration, typical values of these constants are
given in Table 9.4.1 for y'I.G. It must be remarked that the
deviation from elastic isotropy, which may be measured by the
parameter
Table 9.4.1. Room temperature constants ofY.I.G. (after Strauss
[1968] from different sources).
Constant Symbol Value
C12 10.77 X 1011 dyn x cm-2
C44 7.64 x 1011 dyn x cm-2
Density P 5.17 gr x cm- 3
Gyromagnetic ratio l' 1.76 x 107 (Oersted x secr1
Saturation moment 4nMs = 4npJls 1750 Oersted Reduced exchange
~ == 2aJls 5.2 x 10-9 Oersted x cm2 constant
K' Anisotropy constant -2_1Jl3K -45 Oersted M -2 s 1
s Magnetoelastic AJl; 3.48 x 106 erg x cm- 3
constants 2A44Jl; 6.96 x 106 erg x cm-3
(9.4.54)
9.4. Constitutive Theory 463
is weak, for ~ ~ 0.06 for the values given in Table 9.4.1.12 This
means that Y.I.G. is elastically isotropic for all practical
purposes. It must be remarked that for certain materials such as
iron, it may be necessary to include terms of the sixth order in Ji
in the anisotropy energy. The additional term then has the form K
zJ1;J1;J1;.
F. Uniaxial Crystals
Uniaxial crystals are crystals, such as hexagonal cobalt, which
exhibit a preferred direction in their magnetic properties. The
representation of the elastic, piezomagnetic, and magnetostrictive
energies corresponding to this symmetry are to be found elsewhere.
13 If di is a unit vector pointing in the preferred direction, we
only recall that, due to the symmetry of eijk in its indices j and
k, the piezomagnetic tensor has the following useful representation
for uniaxial symmetry:
(9.4.55)
where e1 , ez, and e3 are piezomagnetic constants. This follows
directly from the application of a single representation theorem
(already used in Chapter 4) to describe optically uniaxial
crystals. Similarly, the tensorial coefficients bij and aij of
(9.4.55) admit the representations
bij = XMbij - X~didj'
aij = abij + aZdidj , (9.4.56)
where XM and X~ are magnetic anisotropy constants and a and a2 are
exchange constants (dependent on () for nonisothermal regimes).
Discarding contri butions which contain only IJiI, the expressions
(9.4.56) yield the contributions
La = -h~(Ji' d)2, (9.4.57)
and
1 1 (aJi)2 Lex = zaJi.i· Ji,i + za2 ad ' (9.4,58)
where a/ad == d· V in (9.4.58). Upon examining the expression
(9.4.57) we see that, if x~ < 0, L2 reaches
its minimum when Ji and d are orthogonal. For instance, if d is
along the x3-axis, at equilibrium Jilies in the (Xl' x2 )-plane. If
X~ > 0, then La reaches its minimum when Ji and d are aligned.
Crystals in which X~ < 0 are called crystals of the easy-plane
type. Crystals in which X~ > 0 are called crystals of the
easy-axis type. In hexagonal cobalt (in which the preferred
direction d is the hexagonal axis), X~ > 0 below approximately
200 DC and X~ < 0 above
12 Clark and Strakna [1960]. 13 For example, in Maugin and Eringen
[1972b, Sect. 9].
464 9. Elastic Ferromagnets
that temperature. At room temperature X"tjp5 ~ 4.1. However, it
must be noted that the series expansion of ~a in p converges slowly
and that in many cases terms of high order (such as the fourth
order) must also be included. In fact, since (p' d)2 = p2 - (p X
d)2 and terms in p2 can be discarded, (9.4.57) can be rewritten as
~a = K' sin2 <p, where <p denotes the angle between p and d
and K' = xrp2j2. An expansion, up to fourth-order terms in p,
yields
~a = K~ sin2 <p + K~ sin4 <po (9.4.59)
In hexagonal cobalt at room temperature, K'l = 4.1 X 106 erg x cm -
3 and K~ = 1.0 X 106 erg x cm-3•
As for the expression (9.4.58), the deviation from isotropy or
cubic sym metry is usually weak and the second contribution
involving a2 may ,often be discarded.
G. Elementary Dissipative Processes
An inspection of the C-D inequality (9.3.30), and the duality
inherent in continuum theromodynamics, indicates that in a
sufficiently general case we may have thermodynamically
irreversible processes associated with uji
(viscosity), LB (dissipative contribution to the anisotropy field
due to the fact that p is not frozen in the material; lit =F 0),
flJj ! (dissipative contribution associated with spin-spin
interactions according to the significance granted to flJji ), f
(electrical conduction), and q (heat conduction). We shall examine
only linear irreversible processes and, in fact, discard the last
two effects as well as the dissipative part for flJji for which the
evidence is weak. We index the thermodynamically recoverable parts
introduced previously by a left superscript R, and the new
irreversible contributions by a left superscript D. In
particular,
(9.4.60)
(9.4.61)
By virtue of (9.3.32), and considering the elastic case for the
parts indexed R,
tji = Rtji + Dtji ,
The D parts satisfy the remaining dissipation inequality (for
non-heat conducting insulators)
py == DUji dij - p rB·1it ~ 0 (9.4.65)
if the R parts have been derived from the potential 'P. For the
linear anisotropic insulators, constitutive equations for DO" and
rB read
DUji = '1jikl d kl + '1jik 1Ylk'
'iBi = Lijkdjk + Lij1Ylj'
9.4. Constitutive Theory 465
subject to the restrictions of the entropy inequality (9.4.65). In
the case of isotropic insulators Da and ~B are uncoupled
DUji = Avdkkbji + 2Jlvdji>
?Bi = - prrYti'
where viscosities Av, Jlv, and the relaxation constant r are
subject to
Jlv ~ 0, r ~O.
Introducing the relaxation term R and the viscous force FV by
R = yp x ~B = -pyrp x rit,
FV = div Da,
and noting that
we can rewrite the Euler-Cauchy equation (9.3.9) and the
spin-precession equation (9.3.17) in the form
where
Ii = yp x RHeff + R,
RHeff_H +RB + -1"" i - i LiP ~ji,j'
(9.4.73)
(9.4.74)
(9.4.75)
H replaces B in the last expression without loss of generality and,
con sequently, the resulting combined field is renamed the
effective magnetic field. The relaxation R, which results from
spin-lattice interactions according to the significance granted to
LB, is shown to participate in the Euler-Cauchy equations of
motion.
Note that no hypothesis has been made concerning the magnitude of
r. The expression (9.4.69), which serves to describe the relaxation
of the spin density toward its equilibrium position (see Figure
9.4.1), and the subsequent equation (9.4.74) are valid for
relatively strong damping of the spin precession. However, if r is
small and considered as an infinitesimally small quantity of the
first order, then the term R in (9.4.74) may be considered as a
perturbation. Then Ii can be evaluated from the spin equation
(9.4.74) in the absence of R. Carrying out this perturbation
procedure yields, in lieu of (9.4.74),
(9.4.76)
where
it = __ 1 p x [p x (RHeff + ~)J. 2r'p2 y
(9.4.77)
Here we took into account the definition of rit, and introduced the
vorticity vector wand the new relaxation time
(9.4.78)
Figure 9.4.1. Relaxation ofthe magnetic spin.
To the same degree of approximation, R can be replaced by R in
(9.4.73). It must be understood that (9.4.76) and the corresponding
Euler-Cauchy equa tion must be used only for sufficiently large
relaxation times 1:'. Thus, only for slight deviations from the
adiabatic processes, R is the objective relaxation term which
generalizes, to the case of deformable bodies, the term deduced
from a Rayleigh dissipation function (by Gilbert [1955J) for rigid
bodies. R is the relaxation term which generalizes to deformable
bodies, the term intro duced heuristically in rigid bodies by
Landau and Liftshitz in their pioneering work [1935].14
H. Small Fields Superposed on a Constant Bias Magnetic Field
In many applications, materials are acted upon by a constant
magnetic field Ho and magnetization Po- The solid is then subjected
to nonuniform small displacement u and fields ji. In this case,
constitutive equations can be
14 In rigid bodies the experimental evidence for a Landau-Lifshitz
damping of the spin was given by Robdell [1964]. The superiority of
Gilbert's term for strong damping is'shown in Kambersky and Patton
[1975] (see also Anderson [1968, p. 180]). Akhiezer et al. [1968]
deduced (9.4.77) on an unsound thermodynamical basis. The present
thermodynamical derivation of both (9.4.71) and (9.4.77), and the
establishment of the restricted conditions involving (9.4.77), are
due to Maugin [1975]. Previous attempts along the same lines in
elastic ferromagnets are due to Alblas [1968] and Maugin [1972a].
The antiferromagnetic and ferrimagnetic cases are established in
Maugin [1976c]. Numerical values of 1: can be found in Gilbert
[1956].
9.4. Constitutive Theory 467
H = Ho + h,
fl = flo + p,
(9.4.79)
and dropping products of h, p, 15, eij' and their rates and
gradients. The underlying assumption is that
Ihl« IHol,
Ipi « Iflol.
to replace flo.
(9.4.80)
(9.4.81)
The linearization is to be performed on the initial state (Po, flo,
Ho) where the body carries no initial stress or internal field LBi
• Consequently, from the expression of m K , we drop a constant
term m~ = J1.0 dj <>jK which causes stress at the initial
state. With this, the linearization gives
m K ~ mAk = (lii + J1.0 d i Ui)<>jK,
vi{ KL = iii,kiii,l<>kK<>'L'
~tji = UijklUk,1 + ekijiik + ekijJ1.od p u p ,k,
~Bi = - POl [ei/mU"m + x~'(lii + J1.0d kUk)J,
!!4ji = ajliii,I'
~ = Po'P = tUijkleijekl + tAijkleijmkm, + eijkmiejk + tx~mimj +
taijvl{ij + Yijkleijvl{kl,
where the spatial moduli Uijkl, ••• , Yijkl are related to their
material counter parts, expressed in the material frame X K
by
(Uijkl, Aijkl, Yijkl) = (~KLMN' AKLMN ,
rKLMN)<>iK<>jL<>kM<>,N'
e ijk = EKLM<>iK<>jL<>kM'
x~ = X~L<>iK<>jL'
(9.4.83)
The constitutive equations read
468 9. Elastic Ferromagnets
!J6ji = a 1Jii,j'
L = (!O"ijkl + 1l0 diejkl + !1l6diXj~ddUi,jUk" + (eijk +
1l0Xitdj)JiiVj,k
+ 1 M- - 1 - - -rXij Ilillj + -ra l Ili.klli,b
where the material moduli O"ijkl' ... , Yklmn are functions of To,
Po and d.
(9.4.85)
(9.4.86)
(9.4.87)
We consider the special case of induced anisotropy due to the
orientation d of the magnetic field. The material moduli are
assumed to be isotropic functions of d. By means of the tables
given in Appendix E (Vol. I) we can construct these functions
X~ = Xl (jij + X2 dA, aij = a 1 (jij + a2dA,
eijk = e11l0 di(jjk + e21l0(dAk + dk(jij) + e31l'fAdjdk,
O"ijkl = O"O(jij(jkl + 0"1 ((jik(jjk + (ji/(jjk) + 0"2(dkd,(jij +
dA(jkl) (9.4.88)
+ 0"3(didk(jjl + did,(jkj + djdk(ji/ + djdl(jik) + 0"4
didikd"
where Xa' ea, and O"a are functions of To, Po, and 110' Here Xl and
X2 are the magnetic anisotropy constants, ea are the piezomagnetic
constants, O"a are the elastic constants, and aa are the exchange
constants of which we set a2 = 0 and, as discussed before, we
neglect higher-order terms represented by Aijkl and Yijkl' Then
dissipative parts of the constitutive equations (9.4.63) and
(9.4.64) retain their forms with dkl and mk, having the linear
forms
1 [(aUk) (aul) ] dkl = "2 at ,I + at ,k ; 1 [(aUk) (aul) ]
Wkl = "2 at ,I - at ,k ' (9.4.89)
The dissipation moduli '1ijkl, ... , 'ij have the same general
forms as in (9.4.88) under the assumption of Onsager reciprocity.
However, the effect of uniaxial anisotropy on these moduli are
generally small, so that expressions (9.4.66) and (9.4.67) are used
in practical applications.
For some purposes, it is convenient to express the constitutive
equations (9.4.84)-(9.4.86), for the uniaxial case, in the
form
where
~Bi = - POl (X 1 Jii + X2 did' Ji + bijkUj.k)'
!J6ji = aJii,j'
cij = 1l0el (jij + 1l0(X2 + 1l6 e3)didj,
(9.4.90)
(9.4.91)
(9.4.92)
b;jk = Jio[eld;bjk + (el + Xl)dAk + eldkbij + (Ji6 e3 +
Xl)dAdkJ,
O';jk/ = (J;jk/ + Ji6[e l (dAbk/ + dkd/bij) + el(d;dkbj/ + d;d/bjk
+ djd/b;k + djdkb/i)
+ (Xl + 2Ji6 e3)d;diA· (9.4.93)
The linearization process applied to the balance laws results
in
op ou at + poV' at = 0,
( r OlU;) E tj;,j + Po J; - otl + F; = 0,
- oJ! Y" d x Beff - - = 0
/"'0 ot'
V x h = O.
P = - POuk,k; P = Po(1 - uk,d·
For the magnetization, we have
M = PJ! = Mod(l - ekk ) + PoJ!,
B = H + PoJ! = Ho + Mod + h - Moekkd + PoJ!,
(9.4.94)
(9.4.95)
(9.4.96)
(9.4.97)
(9.4.98)
(9.4.99)
(9.4.100)
where Mo = PoJio. Consequently, the magnetic forces FE and B~ff are
given by
(9.4.101)
B~ff = h; + Mo(1 - ur,r)d; - Po (Z: bij + pi/xt! ))Ij + polal
Vl)I;
- POl (eijk + JioX~dj)Uj,k' (9.4.102)
For the uniaxial anisotropy induced by the initial field d,
(9.4.102) takes the form
Btff = h; + Mod; + - POl [(pJ Z: + Xl) bij + Xld;dj] ~ +
polaVl)I;
- PolJiO[(e l + pJ)d;b;k + (e 2 + XddAk + e2 dkbij
(9.4.103)
9.5. Resume of Basic Equations
The theory of ferromagnetic insulators is based on the following
equations.
Maxwell's Equations (in 1'") For the treatment of the interactions
oflong-wave magnons and phonons,
only quasi-magnetostatic equations are kept.
470 9. Elastic Ferromagnets
v x H = 0,
tji,j + p(J; - 1\) + FiE = 0,
Boundary Conditions (across or) n x [H] = 0,
n' [B] = 0,
njPAji + AJl.i, = 0,
Constitutive Equations (nonlinear)
Ii = Yfl x Beff•
R P o~ t·· = ---x· KX, L - P LB.Jl.· J' Po oEKL ',J, J "
P o~ PAji = 2- :'l ~~ Xj,KJl.i,L'
Po Uv'nKL
Po 'I' = ~ = tUijkleijekl + h~Jl.iJl.j + eijkJl.iejk +
taijJl.k,iJl.k,j'
R -Etji = Uijklekl + ekijJl.k,
Dissipative parts of the constitutive equations are
Dtji = l1jikldkl + l1jikmk'
PBi = 'rijkdjk + 'rimj.
Definitions
Rtj; = Etj; - P ~BjJ1.;, D D DR __ tj; = CTj; - P L~UJ1.;I'
CTj; = tu;) = CTij = RCTj; + DCTj;'
FE = (VB)' M = (VH)' M + tVM2,
M = Pfl,
H=Ho+h,
fl = J1.oil + p, P = Po(l - ekk),
Mo = PoJ1.o, d = flo/J1.o, J1.0 == 1J1.01,
where BO = Ho + Mod satisfies
d x Bo = O.
a- -.l!: = "" d x Beff at 11"'0 ,
V . (h + PoP - ModUk,k) = 0,
V x h = O.
FE = Mo(Vh)·d - M5Vur,r + PoMo(Vp)·d,
Biff = h; + Mod; - POl [ (P5 ~: + Xl) <>;j + X2d;dj] Jlj +
polaV2Jl;
- polJ1.o[(e l + P5)d;Ojk + (e2 + xddAk + e2 dA j
+ (X2 + e3J1.~)d;djdk]Uj,k> \; = [aijkl + J1.~(e2 +
Xdd;dk<>jl - J1.~e2dA<>;k]Uk,1
+ J1.0[(e2 + Xdd;<>jk + e2dAk]Jlk
+ J1.o[el <>ij + (X2 + J1.~e3)d;dj]d· p,
(9.5.22)
(9.5.23)
(9.5.24)
(9.5.25)
(9.5.26)
(9.5.27)
(9.5.28)
(9.5.29)
(9.5.30)
(9.5.31)
fIlji = aJii,j'
where iiijkl is given by (9.4.93)5' For the free energy, see
(9.4.87).
9.6. Coupled Magnetoelastic Waves in Ferromagnets
A. Preliminary Remarks
(9.5.32)
(9.5.33)
In the study of spin waves in rigid ferromagnets the coupling
between the spins and the motion of ions in the crystal lattice is
discarded. However; in real systems, this coupling, in spite of its
weakness, is present and it gives rise to displacements and
oscillations in spin. More precisely, this results in the
propagation of coupled magnetoelastic waves rather than purely
magnetic (spin) waves or purely elastic waves in magnetically
ordered crystals. Ac cording to the terminology introduced in
Chapter 4, we must then account for the magnon-phonon interactions.
Pioneering work in this field of research is due to Kittel [1958a,
b] and Akhiezer et al. [1958],15 Since that time many physicists,
electronic engineers, and theorists of the continuum have focused
their attention on this problem because of its physical
implications and potential engineering applications.
In practice, the coupling between magnons and phonons is quite
weak, but it is especially important under certain definite
resonance condictions. When these conditions are not fulfilled,
magnons and phonons can be separated and regarded as independent to
a high degree of accuracy. Nevertheless, they do interact with each
other. The importance of this interaction is judged from the
following argument. From dimensional analysis we can form the
following coupling parameter
(9.6.1)
where j, Mo, Po, and CT are, respectively, a typical piezomagnetic
constant, a typical saturation moment, the matter density, and a
typical elastic-wave velocity. The order of magnitudes of these
quantities are
j ~ 3-10, Mo ~ 1,000 Gauss,
Po ~ 10 gr/cm3, cT ~ 3 X 105 cm/sec.
15 The work of Akhiezer and his coworkers is synthesized in the
monograph by Akhiezer et al. [1968]. Other prominent contributors
in this field are, among others, Comstock [1964], [1965], Eshback
[1963], Kaliski [1969a, b], Matthews and Lecraw [1962], Motogi
[1979], Morgenthaler [1966], [1968a, b], Rezende and Morgenthaler
[1969], Schlomann [1960], [1961], [1964], Strauss [1965], [1968],
and Tiersten [1965b]. The treatment presented in this section is
based in part on Maugin [1979a, b] and Maugin and Pouget [1981].
Other recent work is found in Alblas [1974] and Van de Ven
[1975].
9.6. Coupled Magnetoe1astic Waves in Ferromagnets 473
Then' ~ 10-5-10-4, which shows the extreme weakness of the
coupling. The conditions of resonance, whose numerical values
indicate the regime for which the interaction is of practical
interest, can be estimated in a simple fashion as follows: Let
ws(k) and wp(k) be typical wave-number dependences of the frequency
for pure magnons and pure phonons, respectively. We consider the
excitation of magnons by phonons or vice versa. At resonance, ws(k)
= wp(k) and the amplitude of magnons excited by phonons will be
particularly large. This condition defines specific values ko of
the wave number which, in turn, allows us to compute the
corresponding values of w. The same reasoning holds true for
phonons excited by magnons, so that we can speak of a definite
resonance that may be referred to as the magnetoelastic (or
magnetoacoustic) resonance. A typical dispersion relation for pure
magnons, in the absence of an externally applied field, is given
by16
(9.6.2)
where (J( and P are reduced exchange and magnetic anistropy
constants. For pure transverse elastic waves in linear elasticity,
we have the linear dispersion curve
(9.6.3)
Equating (9.6.2) and (9.6.3) we obtain the resonance condition
which provides the roots of ko
kOl = ~ ± [~(~)2 _ fiJ1 /2. 2 2(J(wM 4 (J(WM (J(
(9.6.4)
In ad hoc units, the following approximation can be introduced from
micro physical arguments:
(9.6.5)
where eo is Debye's temperature17 and v. (~ O.lc) is the order of
magnitude of the velocity of electrons in the lattice. Considering
(9.6.5), the approximate roots of (9.6.4) are kOl ~ PWM/CT and k02
~ CT/(J(WM. Corresponding to these, we have the resonance
frequencies in accordance with (9.6.3)
(9.6.6)
The first of these does not contain any acoustic parameter but
contains the reduced anisotropy constant p. The second involves CT
and the reduced exchange constant (J(. In ad hoc units, assuming
that Mo ~ 1,000 Gauss, CT ~ 3 x 105 cm/sec, P ~ 1, (J( ~ 10-12 cm2,
then
(9.6.7)
For Y.I.G., we get w~) ~ 1010 sec-1 (see Table 9.4.1). These
numerical values
16 See Problem 9.10. 17 Debye's temperature is defined by eD =
hwmax/ks where W max is the maximum circular frequency of the
lattice vibrations (see Musgrave [1970, p. 231].
474 9. Elastic Ferromagnets
show that the frequencies of magnetoelastic resonance lie in the
high-frequency ultrasonic (hypersound) region. This fact is
important for engineering applica tions. In particular,
magnetoelastic resonance can be used in hypersound
generators.
Of course, a situation similar to that described above will also
prevail for all the branches of elastic oscillations. In the case
of elastically anisotropic ferro magnets, separation between
transverse and longitudinal oscillations is not, in general,
meaningful. In general, the interactions between magnons and the
branches of elastic oscillations will lead to a resonance
excitation by magnons of all the elastic waves and vice versa. In
particular, magnetoelastic resonance can occur through both
transverse and longitudinal elastic waves.
In the sequel to this section the continuum theory, developed in
previous sections, will be used to examine a particular case of
magnetoelastic resonance in an infinite ferromagnet. Propagation of
waves in an arbitrary direction, the damping of magnetoelastic
waves, the magnetoelastic Faraday effect, and possible engineering
applications will be discussed briefly.
B. Plane Harmonic Waves
A ferromagnet carrying a constant bias field is disturbed from its
static equilibrium position. We wish to determine the frequency of
oscillations.
Basic equations of perturbations are listed in (9.5.25)-(9.5.33)
where we substitute
(9.6.8)
Here p:o, uo, and hO are constant vectors, k is the wave vector,
and w is the circular frequency to be determined.
Substituting (9.6.8) into (9.5.29)-(9.5.33), and the result into
field equations (9.5.25)-(9.5.28), we obtain the following
expressions:
{[PoWZ + Il~ez(d· k)ZJOkl - akjlmkjkm -1l~kZ(ez + Xl)dkd1
+ Mlikkk1}uf + illo[(e l + pli)kk + (Xz + ll~e3)d·kdkJd·p: +
illo(ez + Xl)dk(P:°·~) + illoez(k· d)j1f + iMo(d· hO)kk = 0,
(9.6.9)
-iYIl~P01(ez + Xd(d·uO)d x k - iYIl~polez(d·k)d x UO
- YIlOPOl [akZ + (Xl + pli Ho/Mo)]d x p:o + iwp:o + Yllod x hO = 0,
(9.6.10)
Mo(d·k)k·uo + iPok·p:o + ik·ho = 0, (9.6.11)
k x hO = O. (9.6.12)
Equation (9.6.12) shows that hO = exk where ex is independent of k.
Since electrodynamic effects are not present in Maxwell's
equations, we consider the important case ex = 0, i.e., hO = O.
From (9.6.10) it is clear that
(9.6.13)
9.6. Coupled Magnetoelastic Waves in Ferromagnets 475
Employing this and (9.6.11), (9.6.9) and (9.6.10) may be reduced
to
([Poco2 + ll~e2(d' k)2](jkl - Ukjlmkjkm - 1l~(e2 + Xl)
. [d' kdkk, + k2dkd,] + M5kkk,}U? + illoe2k' dJi~ = 0,
(9.6.14)
- iYIl~Pol (e2 + Xd(d' uO)d x k - iYIl~pole2(d' k)d x UO
- Yllopol(aP + Xl + P5Ho/Mo)d x jio + icojio = O. (9.6.15)
These equations constitute a system of six equations for the six
unknowns u~ and ji~. For the solution to exist the coefficient
determinant must vanish. This leads to a polynomial equation for co
whose roots give the dispersion relations co = co(k), expressing
the frequency as a function of the wave vector. However, the
analysis is rather involved. To simplify the matter, we consider
the fol lowing working hypothesis: Following Fedorovl8 we consider
that the ma terial with Uijkl elastic moduli is "weakly
anisotropic". According to Fedorov's criterion, effective Lame's
constants or, equivalently, effective velocities of longitudinal
and transverse elastic waves, cL and cT , can be introduced to
express Uijkl to a sufficient degree of approximation by
(9.6.16)
where
ci == ii/Po, (9.6.17)
where 1 and ii are effective Lame's constants, so that the
right-hand side of (9.6.16) "differs the least" from the expression
present in the right-hand side of (9.4.93).
Using (9.6.12) and (9.6.16) in (9.6.9) and (9.6.10), we
obtain
[Poco2 + M5e2(d' k)2 - PociP]uO - Po(ci- ci - POl M5)(k' uO)k
- M5(e2 + Xl)[(d· k)k' UO + ko(d· uO)]d + illoe2(d' k)jio = 0,
(9.6.18)
iYIl~Pol [(e2 + Xl)(d· uO)d x k + e2(d· k)d x UO]
+ Yllopo l (ak2 + Xl + P5Ho/Mo)d x jio - icojio = O. (9.6.19)
Below we will discuss several special cases.
(i) Longitudinal Waves For the longitudinal waves, k and UO are in
the same direction. Without
loss of generality, we take this common direction to be along the
z-axis. k cannot be parallel to d, since (9.6.11) and (9.6.13) give
k· UO = O. From (9.6.18) and (9.6.11) it follows that
(9.6.20)
476 9. Elastic Ferromagnets
in which (9.6.22)
Consequently, longitudinal waves are uncoupled from other waves and
they can propagate in a direction not parallel to the anisotropy
axis. In general, M6;Poct « 1 so that the phase velocity is very
nearly the same as cL . Neverthe less, a real change in the
velocity of propagation oflongitudinal waves can be observed if a
bias external field H~ is introduced (see Section 8.8).
(ii) Transverse Waves For the waves propagating in the direction of
the anisotropy axis, k is
parallel to d. From (9.6.11) it follows that
or d·uo = 0. (9.6.23)
Without loss of generality we select the z-axis in the direction of
d so that
jI~ = u~ = 0.
It is convenient to introduce the circular components of UO and Po
by
U± = u~ ± iu~, Ji± = jI~ ± ijI~. With these (and since d = 1),
(9.6.9) becomes
where
[w + ws(k)]Ji± = ±ikfwMJiou±,
(9.6.24)
(9.6.25)
(9.6.26)
(9.6.27)
(9.6.28)
(9.6.29)
From (9.6.26) with Ji± = 0, it is clear that W T is the transverse
elastic wave frequency modified by the bias magnetic field (compare
with Section 8.8). Similarly, for u± = 0, (9.6.27) shows that Ws is
none other than the pure magnon frequency. The coupling of these
waves is through f.
Now consider the case oftransverse magnetoelastic waves whose
dispersion relation is given by the coupled equations (9.6.26) and
(9.6.27). Upon elimi nating the amplitudes u± and jI± between
these equations, we obtain the dispersion relations
D+(w, k) == [w 2 - wj.(k)] [w - ws(k)] - (wMwj.(k) = 0,
(9.6.30)
D-(w, k) == [w 2 - wj.(k)] [w