Nonlinear dynamics of coupled systems near magnetic phase transitions of the “order-order” type

Journal of Magnetism and Magnetic Materials 100 (1991) 544-571 North-Holland

Nonlinear dynamics of coupled systems near magnetic phase transitions of the "order-order" type V.I. O z h o g i n Kurchatov Institute of Atomic Energy, 123182 Moscow, USSR

and V.L. P r e o b r a z h e n s k i i Moscow Institute of Radioengineering, Electronics and Automation, Moscow 117454, USSR

The results are reviewed of the 15-yeardevelopment of the concept of effective anharmonicity of the normal modes that describe the vibrations of a continuous medium having two or more mutually coupled subsystems. Magnetoelastie interaction near a spin reorientation phase transition leads to a giant acoustic nonlinearity and its magnetic field sensitivity in antiferromagnets such as a-Fe203, FeBO3, TmFeO3, etc.

I.

Introduction

The time period which has passed since the publication of the 1st volume of the Journal of Magnetism and Magnetic Materials (JMMM) turned out to be the period of an intensive development of the physics of nonlinear p h e n o m e n a in continuous media (plasma, solid state, liquids, polymers and biological media). This general tendency reverberated in the physics of magnetoordered substances as well. Strong nonlinearity and complex dispersion inherent to spin systems have attracted and are still attracting attention to magnets not only as to objects of self-dependent scientific interest but also as to model media for a study of general problems of nonlinear dynamics of a continuous medium. The nonlinearity of an electron spin system manifests itself most vividly (even under relatively weak excitations) in the vicinity of the second kind phase transitions of the " o r d e r - o r d e r " type (so-called spin-reorientation transitions, SRT). At the phase transition a spin system looses its stability, while on nearing it the activation energy of

one of the branches ("soft m o d e " ) of the spin spectrum anomalously shrinks. At the same time the extraordinary increase of the contribution of nonlinearity into the energy of long-wave "soft" excitations of the finite amplitude can be noted. A principal distinction of SRT in magnets is a rather narrow fluctuation region for thermally activated fluctuations of the order p a r a m e t e r (see below), which gives a good opportunity to study the anomalies of linear and nonlinear properties of magnetic crystals outside the region, the close proximity to the critical point included. In real crystals the excitations of an electron spin system are to a certain extent coupled with excitations of a different physical nature, such as crystal lattice oscillations, nuclear spin precession, electromagnetic waves or electric polarization oscillations in ferroelectromagnetic substances. Excitations of a small amplitude in this case exist in the form of normal mixed (hybridized) modes. Generally speaking all the degrees of freedom of interacting "partial" subsystems are excited simultaneously in any normal mode of coupled vibrations. If even one of the

0304-8853/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved

EL Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems

subsystems or a coupling between them is nonlinear in amplitude the normal modes will also be nonlinear. At resonance conditions, i.e. at conditions of intersecting the partial subsystem spectra (dispersion curves to(k)), the excitation energy is equally distributed between them. Obviously, the main contribution to nonlinearity of coupled waves is expected here from the subsystem where strong nonlinearity is realized with less energy densities. At a distance off the resonance each of the normal modes retains to a great extent the physical individuality of the subsystem which is the closest to it in frequency. Nevertheless the contribution of the interaction of vibrations to the nonlinearity of a specific mixed mode may turn out to be decisive. A glaring example of such a situation is the interaction of a magnetic and an elastic subsystem in magnets near SRT. A detailed discussion of this example is still more important because in many cases or SRT it is not a spin (or rather a quasispin) mode but an activationless acoustic (quasisound) magnetoelastic mode that plays the role of a soft mode with the critical behavior, while in the quasispin branch spectrum there remains an energy gap [1-4] resulting from the magnetostriction coupling. In ref. [3] it was noted that this situation is similar to spontaneous symmetry breaking [43]. Atom displacements of a relatively small amplitude in a wave of the acoustic branch may cause substantial deviations of magnetic moments from the equilibrium direction in the region of the spin system stability loss. Magnetostriction stresses turn out to be nonlinearly dependent on the strains amplitude in an acoustic wave. The nonlinearity of the stress dependence on deformations is nothing else but elastic anharmonicity which is introduced in this case into acoustic oscillations by the interaction of a sound with a spin subsystem of a crystal. Such "effective" anharmonicity which is a reflection of the nonlinearity of mixed modes is often of a giant value and hence is a main mechanism of a number of nonlinear wave processes. The concept of the effective anharmonicity of mixed modes, first formulated in ref. [5] and analyzed in detail in ref. [6], appeared to be very fruitful not only for the nonlinear magnetoacous-

545

tics and the kinetics of mixed excitations near SRT [8-10] but also for acousto-optics of magnets [11-13]. The admixture of spin excitations in acoustic oscillations near SRT was found to be capable to increase considerably the amplitudes of the light-on-sound scattering (acoustomagnetooptical interaction [11]) as compared to conventional photoelasticity - similarly to the increase of the Raman sound-on-sound scattering amplitudes near SRT [6,14]. The increase of nonlinearity near the point of the spin system stability loss reduces considerably the applicability of anharmonic expansions of the magnetic energy density by dynamic variables of the critical soft mode. Thus, for a quasisound wave the anharmonic approximations turn invalid near SRT already for dynamic deformations (t~ ~) of the order of magnitude of spontaneous magnetostriction I d01--10 -5, while outside the SRT region a wave with such deformations remains very weakly nonlinear. It suggests that magnets near SRT are objects of special interest for experimental and theoretical investigations of strongly nonlinear wave processes in a mixed wave system (in particular, of the processes of formation and propagation of mixed solitons of a topological and a dynamic type [16-22] and nonlinear periodic mixed waves [12,23]. They are also interesting for a study of the stability problems in strongly nonlinear mixed excitations [20,12]. A particular place in the investigations of nonlinear dynamics of coupled waves near SRT belongs to the phenomena of strong phase nonlinear effects of double bistability over the threshold of the parametric instability of acoustic waves [24]. Taking into account a specific character of the jubilee edition of the journal this article is aimed at summarizing the results of a 15-year development period of the physics of mixed nonlinear excitations of the acoustic (hydrodynamic) type in a magnet near SRT. We have dwelt on a number of original papers initially published in editions almost inaccessible for JMMM readers. We also wanted to draw the readers' attention to some results in the field of nonlinearity of coupled waves obtained within the last three years which passed since the publication of our previous review [7].

V.L Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems

546

2. Nonlinear magnetoelastic dynamics of an antiferromagnet with the "Easy Plane" type Anisotropy (EPAF) Crystals of EPAF are the most convenient and interesting objects for studying nonlinear dynamics of coupled modes. In crystals with a negligibly small in-plane anisotropy the situation, which is completely similar to the region of the second kind SRT, is realized at weak magnetic fields (required only for rendering a single-domain in the sample). The antiferromagnetic character of spin ordering weakens the long-range dipole-dipole interactions which only complicate the nonlinear dynamics pattern near SRT. The decrease of the Zeeman energy due to the AFM exchange widens the field interval for the observation of critical anomalies. Finally, such EPAFs as ctF e 2 0 3 and FeBO 3 have a high temperature of magnetic ordering (T N = 960 and 348 K, respectively) which not only facilitates the experimental conditions but also enables one to hope for practical applications of the phenomena under investigation in solid-state electronics. The magnetoelastic vibrations in an antiferromagnet are described phenomenologically in the two-suitable model by a system of the coupled nonlinear equations for the sublattice magnetization vectors M 1 and M 2 and the equations for elasticity:

-y-'M,,=[M,,×H.]

( n = 1, 2),

O~= O°'i~ ( i , y = 1 , 2 , 3 ) . 3xj

(1) (2)

Here the effective fields are

Ho

- a f F dVf~M.,

the displacement vector is U~, and the tensor of the mechanical stresses is

In its magnetic component F m we shall take account of the energy of intersublattice exchange interaction, the Dzyaloshinskii interaction, the uniaxial anisotropy energy (with the effective fields respectively of H E, H D and HA), and the energy of interaction of the magnetic moments with the external field H:

[ Fm = 2Mo]IHE m 2 - HD[m × 1]z

2/axj

"

Here m - (M l + M2)/2M o is the ferromagnetic vector, ! - (M 1 - M z ) / 2 M o is the antiferromagnetic vector, a - vZ/4HEY 2, and u m is the socalled "limiting" velocity of the spin waves (magnons), which is proportional to the exchange field and the square of the lattice constant. The effective exchange field H E ~ kTN/p. B in crystals having a high enough N6el temperature T N considerably exceeds the fields of the relativistic ( H A) and exchange-relativistic ( H D) interactions, just as it does the external fields ( H ) used in experiments. For example, in crystals of ~-Fe20 3 the characteristic values of the fields are the following: H E -~ 1 0 7 Oe, H o = 2 x 1 0 4 0 e and H A = 2 × 10 2 Oe. In many cases, taking account of the actually existing hierarchy of interactions in the spin system enables one substantially to simplify the description of the nonlinear dynamics of antiferromagnets. In the first approximation in the parameter I:I/H E << 1, where /1 = HD, H A or H, it proves possible to reduce the system of precession equations to the equation of motion for the antiferromagnetic vector ! alone, since 12 = 1 - m 2 = 1 and m << l [6,25,26]: [IXL] =0,

= aF/a(aE/axj). L- y-2(f-L'Zv2/)-

T - ' { 2 [ H ×/'] + [ H × i ] }

The energy density of the crystal is

(

OM~ OM2 0Ui) F M~,M2, Ox~' ax~ ' Ox/ "

+ H(H'I) + HD[H×z ] + ( 2 H E H A + Hl~)lzz - 2mEHme.

(3)

V.L Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems

In this approximation it suffices to restrict the treatment in the expression for the magnetoelastic energy density to taking account of invariants of the type rra,

(4)

Here/~t is the tensor of magnetoelastic constants corresponding to the antiferromagnetic vector. The spin-wave vector of an EPAF, which can be found from the linearized eq. (3) or, as is usual, directly from the linearized system (1), contains two branches. Their frequencies without allowance for the magnetoelastic interaction are given by the relationships -2 =T2[2HEHA C.Oak

+HD(H+HD)] +v2k2

2 2 W2k = y H ( H + H D ) + vmk .

(5) (6)

One of the branches, the "antiferromagnetic" one, 03ak has the relatively high activation energy y(2HEHA) 1/2. For example, for cx-Fe20 2 03a0 lies in the millimeter UHF range [27]. The other ("quasiferromagnetic") branch 03fk a m o u n t s to the "soft mode" of the spin system, whose activation energy is small, in line with the smallness of the external field intensity. The activation branch of the spectrum 03ak corresponds to vibrations with departure of the vector l from the basis plane and with change of the angle inclination of the magnetic sublattices. The soft mode 03fk corresponds to rocking of 1 in the base plane and precession of the ferromagnetic vector m about the equilibrium vector ms, as in a ferromagnet. The interaction of the elastic subsystem with the soft spin-wave mode gives rise to very strong coupling. It determines the fundamental features of the magnetoacoustic properties of EPAFs. One can show that, under ordinary conditions of not too high frequencies 03 << C.0ak and weak magnetostrictive fields nme << HA, where OFm¢ -2 M o Ol 1

-/me --

we can neglect the departure of the antiferromagnetic vector from the base p l a n e i n describing the

547

magnetoelastic dynamics (l z --- 0). Then the equation of motion for 1 is reduced to the form Y-2[ l × ( [ - v2V2/]~

=(n'l)([n×l]~+HD) + 2He[ l × Hme]z + y - 'I4z.

(7)

When we take account of the condition of conservation of the modulus I11= 1, the only dynamical magnetic variable in eq. (7) proves to be the angle ~p of rotation of the vector 1 in the base plane. Transforming to this variable allows us to write the energy density of spin excitations wm and the magnetoelastic excitation Free in the form M° [,)/-2((~) 2 + ')/-2U2m(~D)2 wm - 2 H E -(H

cos ~ +HD)2],

Free = ( / ~ l ( a )

COS 2¢ "1-/~2(0~ ) sin 2q)fi.

(s) (9)

The explicit form of the components of the tensor of magnetoelastic constants Bn(a) (n = 1, 2) depends on the specific symmetry of the crystal and the angle a of the external field orientation with respect to the marked crystal axis in the basis plane. For rhombohedral symmetry, which many of the experimentally studied EPAFs possess (~xFe203, FeBO 3, MnCO 3, COCO3, having the symmetry D36d and the marked axis x II U2), the following relationships hold: /~,(a) =/~,(0) cos 2a +/~2(0) sin 2a, /~2(0~) = - n l ( 0 ) sin 2a +/~2(0) cos 2a,

(lo)

/~l(0)t~ = -- ½(BI1 - B l 2 ) ( U x x - Uyy) -- 2B14uyz,

/~2(0)fi = - (B,, - B12)uxy - 2B14Ux~. Thus it proves possible to describe phenomenologically the magnetoelastic dynamics of EPAFs by using the four-vector magnetoelastic

II..I. Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems

548

displacement (U, ~0), which satisfies the system of equations of motion [16]

considerably enrich the pattern of possible wave processes.

O/J/= - - ( d ( z ) a ~ + / ~ l ( a ) cos 2q~ ~Xj \

3. Coupled magnetoacoustic modes in EPAFs in the SRT-like region

0

+ Bz( a) sin 2q~)ij =-- OXj~.j,

=

(11)

2t ' y[ l --20)f02__ HHD) sin 2q~ + HH D sin q~ 2HE +--

Let us take up in greated detail in properties of coupled magnetoelastic excitations of small amplitude. Their spectrum O k is determined by the well-known dispersion equation, which can be easily derived from the system (11), (12) linearized over small deviations ~0 << 1:

Mo

3

xl/~2(a)a

cos

(w2,-a2)

2q~,-/~,(a)d~ sin 2 1,

~'2

2

2

~Sk O)fk O)Sk

a2_OJs2, = 0 .

(13)

S=I

(12) here tl _ = t1 - t~0 and fi 0 = - [~(2)]- l/~l(a) is the tensor of spontaneous magnetostrictive deformations, °~f0 = T [ H ( H

+ E

+ HD) + 2HEHms]l/2

is the frequency of the ferromode of the antiferromagnetic resonance (AFMR), Hms = - ( 2 / Mo)Bl(a)Ro is the magnitude of the effective field of the spontaneous striction; and we use

(~- 2 ~ Oxy + Oxi " We should note that using eqs. (11) and (12) does not presuppose any restrictions on the amplitude of the spin oscillations. They are suitable for describing both weakly and strongly nonlinear effects to which anharmonic approximations are inapplicable. We note also that one can derive eq. (12) directly from the relationships (8) and (9) by variation methods. It has the form of the well known double sine-Gordon equation [28] supplemented by nonlinear dynamical couplings. It is encountered in the theory of nonlinear spin excitations in liquid 3He and in the theory of self-induced optical transparency. Antiferromagnets are another example of physical systems whose description can be reduced to an equation of this type. Here the nonlinear dynamical couplings

Here ~Os, is the partial frequency of the concrete (S = 1, 2, 3) elastic mode having the wave vector k and the polarization e (s), and o)fk is the partial frequency of the ferromode. The adjective "partial" is essential - it means that the frequencies O)sk and o~fk that enter into eq. (13) are calculated for a fixed partner subsystem. For example, wfk is calculated for a "frozen" lattice (for more details on this, see ref. [3]) and hence it contains a so-called magnetoelastic gap: O)ek= -2 + 2TZHEHms )1/2. Recurring to eq. (13), now (wkf we have: ^

HE

(2B: s)

2

~s2k= M0 (~ofJT)gcO)~sUs '

(14)

1

(Us)ij=

~-£(e}S)kj + e}S)ki).

The asymptotic linearity in k (i.e, when k >> k m - tof0/Vm, fig. 1) o f the spin-wave spectrum in

an antiferromagnet determines the qualitative differences of the spectra of magnetoelastic excitations in crystals having a high and a low N6el temperature T N, to which the "limiting" magnon velocity u m is related by direct proportionality. In low temperature EPAFs (such as MnCO 3, CoCO 3 and CsMnF3), T N is lower than the Debye temperature T D and the velocity of spin waves is smaller than the velocity of sound (see fig. lb). Here the spectra of the " p u r e " (partial) spin and

V.L Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems

elastic excitations O.)fk and tOsk intersect. The conditions for strongest interactions are realized in the resonance region near the intersection. In high-temperature EPAFs (T N > To) , for which um > Us, intersection of the "partial" spectra is not realized. In this case the spin waves for any k are unambiguously separated into acoustic (more exactly - soundlike) and spin (magnonlike) waves, while the strongest coupling corresponds to the long-wavelength region of the spectrum (k << tOfo/Vm). We note for the discussion below that, in an EPAF of the type of ct-FezO 3 or FeBO3, the relation holds that Vs2 << v2, and the frequencies of the acoustic waves satisfy the condition O 2 << to2k for any wave vectors k. Magnetoelastic coupling renormalizes the velocities of the acoustic waves and causes them to depend on the field intensity. The acoustic waves contain a substantial admixture of "nonresonance" spin excitations caused by the deformations. Their amplitude can be found from the linearized equation (11) with allowance for the condition a/at << tofk:

q~k

2HE ~ Mo O)fk

/~2ak.

(15)

549

that in weak magnetic fields, and namely when

H ( H + H D) <<.2HEHms, the amplitudes of the nonresonance excitations prove to be large (~o -= 1) even at deformations of the order of the spontaneous deformations (u k = u0). The nonlinearity of the acoustic modes under these conditions

cannot be considered to be weak. When u k <
(16)

= ~(2) + A f ( 2 ) ( k ) ,

At~(2)(k)

HE 2 . '(2Y/~2) mO')fk o

(17)

The spectrum and polarization of the magnetoelastic waves of the acoustic branch are found here, as in ordinary elasticity theory, from an equation like the Green-Christoffel equation:

[Paz[-Paf(k)]ek=O,

(18)

where we have

It is precisely these excitations, participating in nonlinear interactions inherent in the spin system, that introduce anharmonicity into the acoustic vibrations. We can easily convince ourselves

l~eff

(~(2)b ~..,eff ~, • k.

The field dependence of the moduli ~<2)teff~r~),,and of the corresponding velocities of sound bear

63t0

k=

"~ a

0

Jr,,,

"~ b

Fig. 1. Spectra of coupled magnetoelastic waves in crystals of an EPAF (solid lines) - high-temperature (T N > TD) (a) and low-temperature (T N < TD) (b).

550

V.I. Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics o f coupled systems

direct information on the magnitude of the linear coupling of the elastic and spin waves. In the simplest case of interaction of one elastic mode (S) with the spin system, the field dependence of the velocity of sound is described by the relationship

Vs,( H ) = USk(1 - ~S2k(H ) ) 1/2.

(19)

The specific participation of the exchange reaction, which is characteristic specifically of antiferromagnets, in the formation of the amplitudes of the interaction of the excitations and the magnetoelastic activation of the spectrum of spin waves has the result that the coupling coefficient proves to be of the order of unity over a broad interval of magnetic fields 0 < H ( H + H D) <_ 2HEHms that considerably exceeds the monodomainization field of the crystal. For hematite the characteristic field is H * = 2HEHms/HD = 0.5 kOe. The acoustic modes that satisfy the condition of limiting strong coupling ~'s,k~ 0( H --* 0) ~ 1 are

of fundamental physical and practical interest. The existence of such modes in a crystal that is isotropic in its magnetic properties (though not necessarily in its magnetoelastic properties) involves losses of stability of the equilibrium state with respect to slow rotation of the magnetization on the xy plane by an arbitrary angle (with a corresponding change in the spontaneous deformation). Measurement of the field dependence of the velocity of sound is one of the fundamental methods of experimental study of the coupling of acoustic with spin waves. Early experiments on standing waves in a - F e 2 0 3 and FeBO 3 crystals were carried out in refs. [29,30]. Fig. 2 shows the data of measurements of the velocities of certain types of running bulk and surface acoustic waves in hematite. Also shown there are the results of measurements of the intrinsic frequency of acoustic vibrations of the "contour-shear" mode for a resonator made of ot-Fe20 3 in the form of a disk cut in the basis plane [31]. The results of calculations performed by the methods of elasticity the-

vs/o~, ~(~)/~ro~) I

2 rr

0.,8 /

e'

II 0

!

2

3 ~I, kOc

Fig. 2. Dependence of the normalized velocities (curves 1 and 2) and acoustic-resonancefrequencies (curve 3) on the magnetic field i n t e n s i t y in c t - F e 2 0 4 . 1 - v o l u m e t r a n s v e r s e w a v e s (1 - e ' IIx IIH , 1" - e " ± x II//[15]); 2 - s u r f a c e w a v e s ( 2 ' - H , x = . ~ / 4 , 2" H ]1x [60]; 3 - r e s o n a n c e o f t h e s h e a r m o d e o v e r t h e c o n t o u r o f t h e t h i n disk. D o t s - e x p e r i m e n t [31], lines - c a l c u l a t i o n , i n s e t s geometry of excitation and detection of magnetoelastic waves.

V.L Ozhogin, V.L Preobrazhenskii / Nonlinear dynamics of coupled systems

ory using the effective moduli t'~(2)(b ---+ 0, H ) are presented at the same time. The differing symmetries of the tensors C (2) and "c~(2) e f t lead to removal of the degeneracy of t h e spectrum of transverse waves propagating along the trigonal axis of rhombohedral EPAFs. One of the normal modes is characterized by strong coupling, whereas the other one does not interact linearly with the spin system (see the curves la and b in fig. 2). The interest in the properties of the contourshear mode arises from the fact that, according to the calculations, specifically it satisfies the criterion of limiting strong coupling. The field dependence of its frequency is described by a relationship analogous to that derived in refs. [4,6] for running waves: 1/2

a(H)

=a(o0)(1

2HEH's~20y_2 )

Here the frequency o3f0 =

T[ H( H + HD) + 2HEH, s],/2

differs from the experimentally measured antiferromagnetic resonance frequency by the amount of the magnetoelastic "gap". The data presented in fig. 2 imply that the magnetoelastic interaction leads to a variation in the frequencies and velocities of the bulk acoustic waves by a factor of practically two (i.e., fourfold for the corresponding dynamical moduli), while the variation in the velocities of surface acoustic waves reaches 35%. Such a substantial renormalization of the acoustic parameters is direct experimental proof of the strong coupling of the elastic and spin waves in ot-Fe20 3 crystals near SRT. Magnetoelastic waves in FeBO 3 possess analogous properties. Similar strongly coupled modes are the fundamental ob-" jects of experimental studies in the nonlinear magnetoelastics of high-temperature EPAFs. The orientation phase transition in EPAFs can be caused by a combined action of a magnetic field and of applied mechanical stresses. Relatively weak elastic deformations comparable in value with spontaneous magnetostriction induce a magnetoelastic anisotropy in the basis plane of an EPAF. Its energy may exceed the

551

energy of the crystallomagnetic in-plane anisotropy as it takes place in a hematite. This induced anisotropy manifests itself, in particular, in the variation of the antiferromagnetic resonance (AFMR) frequency. When the "hard axis" of the induced anisotropy is oriented in parallel to the applied magnetic field in the basis plane the deformation lowers the A F M R frequency down to a complete compensation of the contribution of the Zeeman interaction to the spectrum activation. The point of compensation is correlated with the reorientation phase transition of the second kind [4]. Further build-up of deformations results in the deviation of the magnetic moment m off the field direction and in the corresponding reorientation of the AFM vector 1. The A F M R frequency begins to increase. Such nonmonotonic dependence of the A F M R frequency on the deformation stresses in e~-FezO 3 was experimentally observed in refs. [30,32] and the corresponding reorientation of vector 1 was studied using the magneto-optical method in ref. [33]. A detailed experimental study of the behavior of an acoustic wave spectrum near SRT when a field and external stresses (~) are applied to an c~-Fe203 crystal was first carried out in ref. [34]. On compressing the crystal (~r** > 0) along the magnetic field in the basis plane the effective field of the external pressure is directed in antiparallel to the applied magnetic field and compensates the latter on reaching the critical value of Hpc =H(H + HD)/2H E. The coupled magnetoacoustic wave velocity in this case, similarly to the case of H--+ 0, has its minimal value. The results of measuring the dependence of shear coupled waves propagating along the crystal threefold axis on elastic stresses are plotted in fig. 3 for different values of the magnetizing field. (The experiment geometry is shown in the insert). What draws our attention is a noticeable limitation of the sound velocity in the proximity to the SRT. It should be noted that the acoustic mode under investigation in a rhombohedral E P A F is not critical and the wave velocity should not turn to zero within the limit of k--+ 0 in the critical point. However the observed limitation of the sound velocity can be ascribed quantitatively neither to the specific elastic and magnetoelastic

552

V.L Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems in a w i d e r a n g e o f fields A H = 2 H E H m s / H o ( = 103 O e for h e m a t i t e ) a r o u n d t h e S R T field.

a n i s o t r o p y in c t - F e 2 0 3 n o r to t h e s o u n d velocity d i s p e r s i o n in the t r a n s i t i o n point. T h e c o m p a r i son o f e x p e r i m e n t a l d a t a with t h e c a l c u l a t e d curve (see fig. 3, curve 1') suggests a s u b s t a n t i a l differe n c e b e t w e e n t h e m n e a r t h e transition. In ref. [34] an e x p l a n a t i o n o f t h e e x p e r i m e n t was p r o p o s e d t h a t t h e S R T is a f f e c t e d by t h e defects o f the " r a n d o m f i e l d " - t y p e [35]. T h e t h e r m a l fluctuations a r e k n o w n to b e insignificant at r e o r i e n t a tion transitions, on t h e c o n t r a r y the d e f e c t s dist o r t i n g the t r a n s l a t i o n s y m m e t r y o f a crystal s e e m to play t h e decisive role in critical b e h a v i o r at S R T . T h e c o r r e l a t i o n r a d i u s of spin fluctuations r c = Vm//(tO2o - 2 T 2 H E H m s )1/2 for h i g h - t e m p e r a ture E P A F s t u r n s o u t to b e m a c r o s c o p i c a l l y large

4. Mixed m o d e a n h a r m o n i c i t y near SRT In t h e crystal lattice d y n a m i c s t h e w e a k nonline a r i t y is d e s c r i b e d by a n h a r m o n i c t e r m s in the expansion of the potential energy density Fe of a crystal in a p o w e r series in small d e f o r m a t i o n s : 1 1 ^ F e = __ ~(2)t~ a + --C(3)/~/~/~ + . - . . 2! 3! T h e feasibility of this e x p a n s i o n is j u s t i f i e d by usually small d i s p l a c e m e n t s o f the lattice a t o m s

~v v 0,06

N •

.

1

-0.3

,2

2

3

3

4

Fig. 3. Relative variation of the magnetoelastic wave velocity vs. elongation-compression stresses at the field H, kOe: 1 - 0.5; 2 - 0.8; 3 - 1.0; 4 - 1.5, Solid lines - calculation with allowance for the influence of the "random field"-type defects [34], 1' - calculation without allowance for defects.

553

V.L Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems

as compared to interatomic distances: l a I<< 1, and by relatively weak variations of the magnitude of anharmonic moduli with the increase of their order [36]: C (" +1)/C('° = 1-10. This ratio of constants is in agreement with widely spread ideas of the mechanism of interionic interactions in crystals with different types of chemical bond. In the acoustic experiment conditions the deformations are, as a rule, sufficiently small for the nonlinearity of elastic waves to be considered weak ( [ ~(3)/~1//1 ~(2) I (< 1). In the lowest order of perturbation theory, the anharmonicity of the acoustic branch excitations arises from the nonlinearity of the magnetoelastic interaction, as well as from pure elastic nonlinearity [6]. The expansion of the magnetoelastic energy density of (9) in a power series in the amplitudes ~o of the spin oscillations contains the anharmonic term F(_ m32= - 2/~,a¢, 2.

(20)

Eq. (20) describes the processes of interaction of one sound wave and two spin waves. Upon taking account of the spin oscillations of (15) that accompany the sound wave, we can easily pick out the contribution to the energy density of the acoustic excitation, which is proportional to the cube of the deformation. One can correlate this contribution with the tensor of the anharmonic elastic moduli A(~ (3) [6]: 1

M° ['l f°t2

W(4)

~'-E[~tT] - 4&a

-¼HHD

]

~o4

3.

(22)

Taking into account all the contributions cited above, the anharmonic terms of the expansion of the energy density of interest to us are reduced to the standard form [12,37]: 1

A W (4) = - - Af(4)/~a/,~/~.

4~

Here we have

mc (4) =

12

(~))3

(2/~2)4 ( ((.Of0//r) 6

1+

l yZHHD) 4

tOf0

^

Aw(3) = --AC(3)~t~/~, 3[ AC 0 ' =

higher-order effective nonlinearity to describe them. The magnetic and magnetoelastic energies of the crystal contain anharmonic terms of all orders in the amplitudes of the magnetoelastic excitations. The effective fourth-order elastic moduli are formed by three fundamental mechanisms: four-wave interaction of non-resonanceexcited spin waves, magnetoelastic interaction with participation of sound wave and three nonresonance spin waves, and the interaction of (20) in second-order perturbation theory. The first two mechanisms corresponds to the following terms in the energy density:

( H E ) 3 (2/~2)2(2/~')2 - 4 8 M00 (O)f0//~/) 6

6[ HE ]2 (2/~,1(2/~2) 2

-- ~~'-'~0]

(23)

(21)

( (.0f 0 / " ~ ) 4

For simplicity we have restricted the treatment to the wavelength region of the spectrum, i.e., o~ << we0 = oJfk. Using the characteristic parameters B 107 erg/cm 3, H E l M 0 ...m10 4, and (¢Ofo/T)2 107 e r g / c m 3, we obtain for a-Fe203 at H = 1 kOe the estimate presented above of AC (3)= 104C (2) ~ 1016 e r g / e m 3. Eq. (21) describes the processes of interaction of three acoustic (soundlike) waves. A number of experimentally observable nonlinear acoustic phenomena requires account to be taken of a

For a - F e 2 0 3 with H ~- 0.5 kOe we have AC (4) = 10 20 e r g / c m 3. A steep increase of effective moduli with the growth of their order necessitates the discussion of the problem of the anharmonic expansion parameter. It is easy to notice that it is X--~2[U/Uo[ (where u is the dynamic deformation amplitude and u 0 is the spontaneous magnetostriction deformation) that plays the role of this parameter. As it has already been mentioned, a distinctive feature of the behaviour of the coupling coefficient of a critical mode near SRT is its tendency

554

V.L Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems

to the limit ~'2._> 1 which is independent of the crystal constants and environment (fields, pressure, temperature, etc.). As a result, near SRT the anharmonic expansion parameter turns out to be not the amplitude of dynamic deformation proper but its ratio to the magnitude of spontaneous striction. Thus, at the reorientation point the magnetostriction-introduced elastic nonlinearity increases with the decrease of the magnetostriction constants of a crystal (it is true only if Hms > H a, where H a is the in-plane anisotropy field). This, on the face of it, paradoxal result conforms to experimental data which demonstrate a much stronger effective acoustic nonlinearity near "in-plane" SRT of tx-Fe20 3 crystals with a conventional weak striction (u 0 - 10 -5) in comparison with recently investigated [38] effective nonlinearity of the rare-earth compound Tb0.3Dy0.7Fe 2 with the compensated magnetic anisotropy and the giant magnetostriction (u 0 = 10 -3 ) [39]. The peculiarity under consideration, i.e. the increase of nonlinearity of activationless (hydrodynamic) modes near the critical point with the decrease of the constants of their interaction with the order parameter, is, in all probability, of general character for the second kind phase transitions of the " o r d e r - o r d e r " type.

"turns off" the magnetoelastic coupling (a k is the decay coefficient for the intensity of the wave). The nonlinear moduli were determined from the change in the velocities of the acoustic waves under a relatively weak static deformation of the crystal. We note that the anomalously high tensosensitivity of the velocities of sound in EPAFs had been prediced also in ref. [30b] - on the basis of analysis of the linear effects of static elastic stresses on the antiferromagnetic resonance frequency and the magnetoelastic coupling. In ref. [40] the geometry of experiment was chosen taking into account the strong magnetoelastic anisotropy of hematite, owing to which the anharmonic moduli AC455 and AC155 have the largest magnitude (with H parallel to the twofold axis Uz[I x). In determining these components of the tensor AC ~3), one can use a transverse wave with polarization e II x and wave vector k II z. Here one must create static stresses of two types that are homogeneous throughout the crystal: tensile (~ryy) and shear (¢ryz). An acoustic wave at the frequency 30 MHz was excited and detected with piezotransducers. The variation of the velocity of this transverse wave ~v t upon deforming the crystal was measured from the change in the phase of

,,4£,~ .10 -t6

5. Experimental nonlinear magnetoacoustics near SRT The anomalously large magnitude, specific symmetry and strong dependence of the effective elastic moduli on the magnitude and direction of the magnetic field facilitate the experimental detection and identification of the magnetoelastic machanisms of many nonlinear acoustic processes. Direct experimental measurements [40] of the effective third-order anharmonic moduli have been performed on synthetic single crystals e~F e 2 0 3 of dimensions 5 × 4 mm z in the basis plane and 13 mm in length along the C 3 axis. The quality of the crystal was sufficiently high: for the soundlike wave being studied having ozk = 600 c m - 1, the quality factor Qk = ook/At°k = k / a k = 3 × 103 for a field H---3 kOe which practically

I

0

i

0.5"

t /,Q

I /.5 H, kOe

Fig. 4. Dependences of the effective third-order elastic moduli on the magnetic field intensity (solid line - calculation by eqs. (21)) [40].

V.L Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems the signal in the receiving transducer. T h e results were processed by using the relationships AC155 = 1 [C44 ( Cl I _ C, 2) Sil( H )

"[- C14C445 ± ( O ) ] ,

(24)

AC455 = 1[C42S.1_ ( H ) + 2C,4C44Sll ( H ) ] . H e r e we have

p(av2),, Sll

_ p(Sv2)3-

C440ryy

S 3_

C440.yz

T h e subscripts II and ± corresponds to tensile and shear deformations. T h e results of the

555

m e a s u r e m e n t s and of calculation of the modulus ACn55(H) are presented in fig. 4. T h e use of static deformation for determining the dynamical fourth-order elastic moduli (e.g., from its influence on frequency doubling of sound) involves a f u n d a m e n t a l difficulty. T h e static deformations that cause the change in the velocity of the magnetoelastic wave owing to fourth-order anharmonicity alter the direction of the equilibrium magnetization. Since the magnetostrictive field that determines the gap in the spectrum of spin waves does no hinder static remagnetization, in contrast to high-frequency remagnetization, the deviation of the magnetic m o m e n t s (and the

AV V ' 10"2 1.0

y/

l

xlIHg~

z

0.5

!

0

7

14

~xz " 106-dYn¢

Fig. 5. Dependence of the relative variation of the magnetoelastic wave velocity on shear stresses [34] H, kOe: 1 - 0.5; 2 - 0.7; 3 1.0; 4 - 1.3; 5 - 1.7; 6 - 2.5.

V.L Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems

556

changes in the velocity of sound associated with them) in static and dynamic deformation can differ substantially. Accordingly the need arises of distinguishing the dynamic and quasistatic moduli. As H ~ 0 and near orientational phase transitions, the latter diverge proportionally to 1 / ( o ~ 0 - 2T2HEHms )n, where n is the order of the expansion of the energy density in the components of the static deformation tensor that are associated with the change in the direction of the vector 1 in the linear approximation. Measurements of the quasistatic fourth-order moduli in et-Fe203 using static deformations were performed in ref. [34]. Fig. 5 shows the measurement results of the mixed shear acoustic mode velocity, propagating along the threefold axis (z)

of a crystal, vs. the magnitude of shear stresses O-xz. In this figure the result of the square approximation of experimental dependences is plotted in solid lines. While processing the experimental data presented in the figure, an allowance was made for the fact that shear stresses O'zx (for n II x) give rise to the variation not only of velocities but also of polarizations of transversal waves propagating in the direction of the threefold axis. Rotation of polarization vectors of normal waves results from the rotation of vector 1 in the basis plane under the o-xz stresses, the effective field of anisotropy of the latter being oriented at an angle of -rr/4 to the x-axis in the basis plane. Transverse wave polarization degeneracy peculiar to rhombohedral crystals in the basis plane is elimi-

C44(H) ; A~4~H), 1021erg/tt:In 2

c44(c~) 1.0 r

t

C44

c~4(oz)

0.5

l 0

•

1.0 '

2.0 '

3.'0 H, kOe

Fig. 6. Second- and fourth-order elasticity moduli vs. magnetic field intensity (dots - experiment, lines - calculation) 134l.

V.L Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems nated by the magnetoelastic interaction. A linear magnetoelastic coupled and incoupled waves have different velocities and their polarizations are directed at angles 2~00 and (290 + at/2), respectively, to the twofold axis (x), where ~o0 is the angle of the vector l deviation from axis y. The polarization rotation appears in the sound propagation description in terms of effective elasticity moduli because of the existence of nondiagonal (in variable deformation indexes) components of quasistatic fourth-order moduli r(3) ~(5)45 where indexes in brackets relate to the static component of deformation, U~z. Allowing for their contribution to the renormalization of wave velocities in the second order of perturbation theory it can be proved that it is the combination of effective m o d u l i AC (55)44 (4) + -~(55)55 Ag'(4) which is responsible for the transverse wave velocity variation (Av t) following the square law in relation to static deformations Uxz.'~H e r e we have

p(av:)-- 2[ aq55(4) 4 + ac('4;)5,] (u:z) 2.

where ton = y ~ H ( H + H D ) . The solid line in fig, 6 indicates the result of the calculation of elasticity moduli vs. field dependences using eq. (26). The dots in the same figure show the results of processing the m e a s u r e m e n t results according to eq. (25). Point of reduction is chosen at the field strength H = 1 kOe. It is easy to notice the qualitative conformity between calculated and experimental curves of the field dependence at H > 0.5 kOe. The quantitative calculation of moduli in the point of reduction with allowance for independently determined magnetoelastic parameters of a crystal provides the value very close to that in fig. 6. These results verify theoretically predicted anomalous rise of nonlinear moduli with the growth of the nonlinearity order as well as a giant value (101°-1021 e r g / c m 2) and strong field sensitivity of effective fourth-order elast.icity moduli of EPAFs near SRT. Measurements of the field dependence of the dynamic moduli AC5555 and AC6666 have been performed while using the effect of nonlinear frequency shift (NFS) of the magnetoelastie oscillations of a thin plate. In the e x p e r i m e n t the acoustic vibrations of a monocrystalline resonator in the form of a disk 0135 m m thick and 5.5 m m in diameter cut in the basis plane were studied. The NFS was measured from the distortion of the shape of the resonance line upon increasing the

(25)

For the combination of moduli in eq. (25) the following expression can be written: m c (4) -+- m c (4) (55)55 (55)44

(7HHD+H2),

= ( H E ] 3 ( 2BI4"y2 Mo ] t tOfotOH

C26)

2 I

•

I

557

I

,~ I

I

6 H, kOe |

I

\vl-2

1

¢.a

i

,;042.4

1B-

£,~-I_,

404Z8 (f?,/2zJ, kHz Q

Fig. 7. Nonlinear frequency shift and hysteresis of the amplitude-frequency characteristics of acoustic resonance of the thickness-shear mode observed for different values of the amplitude of the alternating field (h z/h 1= 2, H =2 kOe).:(b) Field dependences of the.effective fourth-order elastiCmoduli (curves - calculation by eqs. (27)) [37].

558

F..L Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems

strong field dependence of the anharmonic elastic moduli of EPAFs. One of the first confirmations of the concept of effective elastic anharmonicity was the strong field dependence discovered in a - F e 2 0 3 of the amplitude of the second acoustic harmonic generated by a running elastic wave [41]. A wave at the frequency of 37 MHz was excited at one end of the specimen. At the opposite end the acoustic signal was received with a resonance piezotransducer tuned to the doubled frequency. Fig. 8 shows the results of measuring the dependence of the power of the second-harmonic signal on the power of the pump wave and the field dependence of the efficiency of conversion, which is proportional to oj~8(H), in agreement with the theory. The decline in the efficiency of conversion with decreasing field in weak fields is explained by the increase in damping of the acoustic waves as the homogeneity of the magnetization distribution over the volume of the specimen breaks down - primarily owing to crystal-structure defects, since the disorientation of the magnetization near a defect increases with decreasing field. At the same time an effect was discovered in hematite of acoustic detection by running waves.

amplitude of the vibrations (a recording of a characteristic shape of an acoustic resonance line is shown in fig. 7a). The magnitude of the NFS of the acoustic vibrations is proportional to the fourth-order elastic moduli: ACssss for the thickness shear mode and AC6666 for the contour shear mode. Fig. 7b compares the results of measurements and calculations of the field dependence of the parameters Ac(a)(H). The calculations were performed by using the relationships ACsss5

= 1 2 ( H E t 3 (2BI4) 4 (I + T 2 H H D t M00] (oJf0/y) 6

4°~20 I ' ( 2 7 a )

3 [ ( E l l - C12)C44- 26124] 2 AC6666 = 2 MoHmsC244 ×

2HEHmsT213[1

( °20

T2HHD (27b)

In agreement with the theoretical ideas on the features of the effective elastic anharmonicity, the experiment performed on hematite demonstrates the anomalously large magnitude and

%9 ~ ~

,

J

l

10 2 _ P2 ,10-9 W 1.5 i IJ "G l,O 0,8 / kOc

,

,

i o

1# l

o

o

o

O.5 O.5

0

i 0.1

0.2

0,~

a

l ~ 4 p~, W 2

0

1

2

3

,~

H, kOe

b

Fig. 8. (a) Dependence of the power of the second acoustic harmonic on the square of the power of the pump wave in a-Fe203. (b) Field dependence of the efficienw of conversion of sound into the second harmonic (line - calculation) [41].

559

V.L Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems

The essence of the effect consists in generation of a sound wave at the frequency of the envelope of the amplitude-modulated acoustic signal. The field dependence of the power of the detected signal that was found was also proportional to fo~8(H). Generation of the second acoustic harmonic of a surface magnetoelastic wave was found later - also for tx-Fe20 3 [42]. A convincing demonstration of the strong acoustic nonlinearity of hematite was the observation of the effect predicted in ref. [6] of stimulated combination scattering (SCS) of running acoustic waves [14]. U p o n propagation of a " p u r e " transverse p u m p wave of frequency tOp and wave vector k parallel to the trigonal axis, a threshold process was observed of generation of backwardrunning magnetoelastic waves at the combination frequencies tOt and tO2, namely such that tOl + tO2 =fOp.

The conditions for s p a c e - t i m e synchronization for this process are illustrated in fig. 9a. Since the velocities of magnetoelastic waves (with polarization el, 211H 11x) substantially depend on the field

intensity (see fig. 2), the combination frequencies corresponding to the synchronization condition also depend on the field. Fig. 9b shows the data of the m e a s u r e m e n t s and the results of calculation of the field dependences of the generation frequencies. Calculation [6] of the magnitude of the threshold deformation within the framework of the theory of effective anharmonicity for the process being discussed yields the following relationships: c __ U p - - "IT

21B14 ] (1 - ~2) 3/2

kpL~ 5

C44

,

(28) ~-2 = HE (2B14y)2 M0

fo20C44

H e r e L is the length of the specimen. A calculated estimate of the threshold deformation (Up) --- (3.5 + 1.5) × 10 -7 for H = 0.5 kOe agrees in order of magnitude with the result of m e a s u r e m e n t of (Up) = (8 + 4) × 10 -7. In the same geometry and also for tx-Fe20 3 under syn-

!

l

=~tr~MH

P

35

10

.

s s

~!

5

|

I

O

0.2

1

I

O.4 H, kOe

a b Fig. 9. (a) Diagram of the conditions of synchronization for acoustic stimulated combination scattering [6]. Field dependence of the combination frequencies of transverse magnetoelastic waves in c ~ - F e 2 0 4 ( p - 60p, 1 - ~Ol,2 - w2, lines - calculation) [14].

560

V.L Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems

chronization conditions, a threshold-free process conditions, a threshold-free process was observed of merger of magnetoelastic waves into a running pure-sound wave that was the reverse of the SCS [44]. When a high-frequency magnetic field h(t) homogeneous throughout the specimen acts on a crystal of a-Fe203, a number of parametric acoustic phenomena can occur [45]. The parametric coupling of sound with an alternating field, just like the effective interactions of elastic waves, is mediated by the excitation of the spins. In an alternating field parallel to the magnetizing field (the so-called parallel pumping (hll << H), the following term in the energy density is responsible for the parametric coupling:

M0

- --(2H WplI -- 2 H E

+ H o ) hllq~2.

Wpll=

(30)

The magnetoelastic interaction of (20) also determines the coupling of the sound with the transverse pump field h ±(t). Taking into account the nonresonance excitations of the spin system caused by the transverse field, e.g. (¢h = ( H + HD)H±(tofo/y) -2, and by the sound (15), we find Wpz = 8 ~ - ~ ( n

-1- n D ) ( t o f 0 / ~ ) -4

X (/~,Q)(/~zfi)h ± .

Y..t7

2.5

0

/

2

3

H, kOe Fig. 10. Dependence of the amplitude of the reduced threshold field on the constant magnetic bias field; the reduction point is H = 1.6 kOe [45].

(29)

Upon substituting into this the amplitudes of the nonresonance spin oscillations of (15), we obtain the following expression for the coupling of the acoustic waves with the field: 2H E ] 2H+ H D ^ 2 M0 / (tof0/,y)4(B2/~) hll.

ZS"-

(31)

The parametric interactions (30) and (31) allow a graphic physical interpretation. The effective elastic moduli AC~2)(H) (and this implies also the frequencies of the acoustic spectrum and velocities of sound) depend on the magnitude and direction of the external magnetic field. Modulation of the external field in magnitude or in direction leads to modulation of the elastic pa-

rameters of the crystal and parametric Couplings with the energy Wp = wplI + wp ±, where Wp =

( O A(~(2) t ^^ OH ]h uu2

(32)

When the amplitude of the ac field with the frequence to exceeds a critical value, e.g., hit > h c = Q-tJ2/(OJ2/OH) (Q is the Q-factor of the quasisound mode), a parametric instability arises in a - F e 2 0 3 crystals of the magnetoelastic acoustic modes with the frequency J2 = to/2. An analogous effect has been observed earlier in the ferrite garnet Eu3FesOl2 [46]. Under the conditions of the experiment the amplitude of the parametric interactions characterized by the quantity 0 AC(2)/OH is substantially larger for a - F e 2 0 3 than for the ferrite. Moreover, in hematite the effect is observed over a considerably broader range of magnetic fields. Fig. 10 shows the field dependence of the threshold field for the thickness shear mode of a disc acoustic resonator made of a - F e 2 0 3 (the plane of the disk is parallel to the basis plane). The pumping and recording of the instability were performed by an induction method. The line in fig. 10 shows the result of calculation of the field dependence of hc from the data of independent measurement of Q(H) and J2(H) = J~(ooXl - ~2(H))1/2 of the studied specimen. For

V.L Ozhogin, V.L. Preobrazhenskii / Nonlinear dynam&s of coupled systems

2

g'f2 7 9~~ 0 9.',-g.9, RHz Fig. 11. Dependence of the amplitude of parametric magnetoelastic vibrations on the pump frequency for different degrees of supercriticality hp/hpc = 1.25 (1); 2 (2).

t h e c h a r a c t e r i s t i c v a l u e s Q = 10 4, H = 0.5 k O e , a n d O / ( O O / O H ) = 0.2 k O e , t h e t h r e s h o l d a m p l i t u d e has the m a g n i t u d e h c = 0 . 5 0 e .

561

W h e n t h e r e s o n a n c e c o n d i t i o n s a r e satisfied a n d with an i n t e n s e e n o u g h e x t e r n a l a g e n t ( p u m p ) , o n e observes t h e so-called n o n d e g e n e r ate t h r e s h o l d p r o c e s s e s o f p a r a m e t r i c g e n e r a t i o n o f m a g n e t o e l a s t i c m o d e s - at the c o m b i n a t i o n f r e q u e n c i e s O n + ~'Qm = tOp [45]). Past the t h r e s h o l d of p a r a m e t r i c excitation, the effects o f n o n l i n e a r - s e l f - a c t i o n o f acoustic m o d e s m a n i f e s t themselves. In p a r t i c u l a r t h e y a r e f o u n d from t h e d i f f e r e n c e of t h e values of the f r e q u e n c y d e t u n i n g (Ato = t o p - 2 g ~ n ) with respect to the p a r a m e t r i c r e s o n a n c e at which s o u n d g e n e r a t i o n arises a n d d i s a p p e a r s . A c h a r a c t e r i s t i c r e c o r d i n g of the hysteresis d e p e n d e n c e of t h e a m p l i t u d e o f a p a r a m e t r i c a l l y excited oscillation on the d e t u n i n g is shown in fig. 11. T h e states with d e t u n i n g Ato n > Q~-l[(hp/hc)2 2 --1] 1/2 a r e bistable. Small f l u c t u a t i o n s of the q u a s i s o u n d intensity relax t o w a r d s to s t a t i o n a r y m e t a s t a b l e state while oscillating with the characteristic f r e q u e n c y (g~M) d e p e n d on t h e s t a t i o n a r y

Fig. 12. Oscillogram of sound intensity dependence on pumping frequency tOp. Line widening corresponds to "collective oscillations". Arrow a indicates the frequency at which the "primary" parametric quasi-sound vibrations arise while ~Or,decreases; arrow b - the same for collective oscillations. Arrows c and d show the quenching points for primary vibrations and collective oscillations while tOp increases.

V.L Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems

562

state parameters. These oscillations can be excited by the parametric parallel low frequency (LF) pumping h M cos toMt II H that modulates the quasisound spectrum with frequency tom = 2g2 M. The effects observed here [24] are to a certain extent similar to earlier investigated processes of a double parametric resonance under collective oscillation excitations of parametric magnons in ferrites and antiferromagnets [47,48]. The most interesting result of the acoustic experiment [24] proved to be the discovered bistability of oscillations of the quasi-sound intensity against the background of the bistability of an overthreshold stationary state (a double over threshold bistability). Fig. 12 shows an oscillogram of an observed dependence of the quasisound intensity (associated with the critical contour-shift mode with frequency tOn/2"n" = 0.5 MHz) on the frequency detuning of H F pumping AtOn = tOp -- 2tO n under LF modulation tOM = 1 kHz. Beam trace spreading corresponds to the intensity oscillations. The oscillation bistability can be clearly seen in the dependence of their intensity on the modulation frequency (fig. 13). The double bistability effect can be explained in terms of the theory of the parametric excitation of quasisound modes with allowance for the effective elastic fourth-order anharmonicity which is responsible for the nonlinear frequency shift of the acoustic mode. Complex amplitude b n of a parametric quasi-sound is described by equation 1 ]" bn + 7tonQil ibn + 7'

Atonbn

= ihoV, b*,

--

i~bnlbnlebn (33)

I bn [E/ton :

tOn

+ Q~-' ,_-/5- - 1

hoe

,

O,5

~AM,. k H z 2~ ,

,

,

(34)

The linear dependence of intensity on detuning

t

I. 2 2.2 Fig. 13. Collective oscillations intensity vs. modulation field frequency. The frequency ~M/2"rr of collective oscillations is 1 kHz. 1.6

agrees well with the experiment (fig. 11). The oscillation frequency of small perturbations of the stationary state is deduced from eq. (33) and has the form of the following relationship:

riM=to

n Nil

Nil-

to---~-

As it follows from eq. (33) the parametric quasisound phase On = 2 arg(bil) is described by the expression: On + tonQnl0n + (o~nQ~-' - 2hpVn s i n On) × (Aw, - 2 h p V . cos 0,) = 0,

where Vn - ½0ton/OH is the parametric coupling amplitude, ~On is an amplitude of four quasiphonon interaction and Qn is the Q-factor of a mode. A stationary overthreshold state possesses the following normalized sound intensity Nil - 2~0n

gn=

N M rel. uo~t5

(36)

where detuning (under a slow modulation of the quasisound spectrum) is equal to 0to n

AWn = tOp -- 2tOn -- 2 - ~ - h M cos tOMt. The phase equation is strongly nonlinear and for small deviations from the stationary state it allows the expansion (to an accuracy of cubic m e m bers) into phase deviation from the stationary value On - On0 _-- dexP(½itOMt) + c.c. The equation being formed for complex amplitudes d(t) is completely identical to the initial one. Detuning, a relaxation rate, a threshold modulation field,

V.L Ozhogin, V.L.Preobrazhenskii / Nonlineardynamicsof coupledsystems amplitudes of parametric coupling and of nonlinear self-action in this case have the forms of

AtoM = toM _ 2 ~ M ,

1 --1 = 2tonQn 1 --1 , 2OMQM

hMc=J"2M - ~ [ ( h p / h p c ) 2 - 1]

VM

Oh

ton

-1 h

h

, 2

111/2

and

4'M

a M f Aton + Q n l 2_ I ) Nn t 4t°n [(hp/hpc) 1-1/2 '

respectively. In conformity with the experimental data (fig. 13) the nonlinear shift of oscillation frequency appears to be negative (~OM < 0) in contrast to the nonlinear shift of the initial quasisound frequency (On > 0). Detunings

AtO M > f2MQMI[(hM/hMc) 2 - 1] 1 / 2 - A M correlate with metastable excited states. After the quenching of oscillations they restore on increasing the pumping frequency to M in the region of absolute instability I AtoM[ < A M and this is an explanation of the experimentally observed hysteresis of the oscillation amplitude. Let us note that the above approach towards the double bistability description can be applied to a newly formed metastable state, etc. It gives good ground to believe that in case of sufficiently high Q-factor and strong nonlinearity it is possible to realize not only double but also cascaded bistabilities near SRT. A manifestation of self-action and nonlinear intermode interactions proves to be the automodulation of the over-threshold amplitude of deformations observed in ct-Fe20 3 in the parametric generation of sound. The effect arises at values of the magnetic field, supercriticality, and frequency detuning defined for the given mode [45]. The parametric effects in running acoustic waves also include the frequency shift discovered in hematite of a wave in a non-steady-state, monotonically varying magnetic field. Phenomena of parametric amplification and

563

front reversal of running magnetoelastic waves in a high frequency magnetic field were observed in ~-~e203 [49]. The experimentally observable phenomena of generation of long-wavelength nonresonance excitations of the spin system in a field of nonlinearly interacting acoustic waves are associated with the interactions of (30), (31) and (32) [50]. The acoustic waves give rise in the crystal to oscillations of the magnetization Ix(t) that depend nonlinearly on the elastic deformations. The quadratic components of the ac magnetizations can be calculated by using (32): Ix = -Owo/Oh. When harmonic sound waves with waves vectors identical in magnitude propagate in opposite directions (k2 = - k l), spatially homogeneous oscillations of the magnetization are excited in the crystal at the sum frequency. In the case of amplitude-modulated waves, the time envelope of the integral of the magnetization over the length L of the interaction region amounts to the convolution function of the envelopes of the interacting waves: Ix(t) ~x f

u,( Ou2( e - 2vt ) d~.

(We assume that L is larger than the spatial extent of the acoustic pulses.) The effect of acoustic convolution was observed in a crystal of et-Fe20 3 [50] upon interaction of transverse bulk waves propagating along the trigonal axis. Sound waves of frequencies = 30 MHz were excited at opposite ends of the crystal by piezotransducers made of LiNbO 3. The spatially integrating detection of the signal of magnetization oscillations was performed by an induction method. In agreement with the theoretical concepts of mediated interactions of the type of (30) and (31), intramode interactions of coupled waves (e I II e2 II x II H) were observed in the experiment with excitation of a longitudinal component of the magne2 and intermode interactions of tization IXll~ Uxz the coupled and the pure sound waves (e I I[ x [[ H; e 2 2_ e 1) with excitation of a transverse component IX± ot UxzUy z. The efficiency of conversion (for fixed polarizations of the emitters) shows a sharp dependence on the magnitude of the exter-

V.I. Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems

564

nal field and on its direction (see ref. [50]). H e r e the character of the interaction is described by the discussed mechanisms of the interactions. Effects of acoustic convolution in tx-Fe20 3 have been observed also in the nonlinear interaction of surface magnetoelastic waves [51,52]. The internal bilinear factor of the processes, i.e., the ratio of output electromagnetic power to the product of the acoustic powers of the interacting signals, in fields H = 0.5 kOe amounted to about - 3 0 dB m ( - 1 0 dB m corresponds to 0.1 m W at the output for 1 m W at each of the two inputs). In closing this section we note the experimental results in the field of nonlinear magnetoacoustics obtained with another high-temperature E P A F - FeBO 3. Under transverse U H F pumping

I

T ~ r ' - - "

• • ° olnl' olb~ • • •

Tm Fe 0 3 , H=0

e

0

." oo% •

•

~Cd

6

•

5

2

;

° 93,5 S¢,5

O0

• I eo~eo

80

eD

;

oe.........o..

85

.....

~qo

°ql o

•

ooe

%lOeo

95

I00 T,K

Fig. ]4. Temperature dependence of the amplitudes of the first and second harmonics of sounde at the output of the TmFeO3 crystal near the SRT points T~ and T2.

I

J

I

I M

od oo ° 0

•-1.7

+

O0 0

o-0,2

,roD++

* O

~I I

0•

•

"4"

+

_oo ° 8 • 0 w •

•

"1-

•

44"

•

•

•

-•

+ O

•

0 0

QI

4"

0

3 •

+ + + + +

o

•

0 0 0

H =4,SkOe

40°

t

2o

I

:

1

,J r,K +-NO

0

o•

I

f0-6

+

1+ + l

t

i

I

I

I

lO - 5

Fig. 15. D e pe nde nc e of the amplitude of the second acoustic harmonic on the amplitude of the pump wave in T m F e O 3 (AT --- T - T]).

under conditions of antiferromagnetic resonance in iron borate, parametric instability of sound was observed [53,54]. The effect was treated theoretically in ref. [55]. The excitation of a parametric acoustic echo by a high-frequency magnetic field was discovered in F e B O 3 [56]. It was noted in an excellent review on iron borate [57], and this viewpoint cannot but be shared, that F e B O 3, just like t~-Fe203, will serve in many regards as an extremely convenient object for studying numerous effects engendered by the very strong dynamic coupling of the elastic and magnetic subsystems of this EPAF. Experimentally the a p p e a r a n c e of strong acoustic nonlinearity near an orientational phase transition (of second-order) has been demonstrated by observing an effect of generation of the second acoustic harmonic in the orthoferrite T m F e O 3 [58]. The T m F e O 3 crystal has an orthorhombic structure (the b axis is the " h a r d axis" for the magnetic moments of the sublattices), and the temperature of antiferromagnetic ordering is T N = 630 K. In the absence of an

V.L Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems

external magnetic field, changing temperature leads to reorientation of the spins from the a axis at T > T ~ = 9 4 K t o t h e c axis at T < T z = 8 2 K . At T 1 < T < T2 the vector l rotates smoothly in the ac plane from one axis to the other. The temperatures T 1 and T2 are second-order orientational-transition points, near which the activation energy of the spin-wave modes is depressed (the mode becomes "soft"), as is confirmed experimentally by measuring the antiferromagnetic resonance frequency. In the vicinity of the temperatures T 1 and T 2 a sharp increase has been observed in the efficiency of generation of the second acoustic harmonic (figs. 14, 15). With increasing power of the pump wave, the output of the second harmonic is saturated and even declines owing to conversion of the energy of the incident wave, not only into the second harmonic, but also into a multitude of higher harmonics.

565

account the coupling of the subsystems [59]. This has the form /~(2, = • [OJskbs+kbsk + tofkc;c~ k,s 3

+ (Gskb;kck + H.c.)] = E aa,da+kdak • A=O

(37) Here b+sk, bsk, c; and c k are the operators for creation and annihilation of phonons and magnons, and we have Gsk = ~Sk(tOSktOfk)1/2. The operators for creation and annihilation of mixed excitations d ~ and dak are connected to the operators for the " p u r e " excitations by the unitary transformation: 3

c; + c _ k =

E Pxk(d~k+da.-k),

(38)

A=0 3

bsk -+ b+s,-k = E Rak(d-~k + dx,-k) .

6. On the quantum theory of nonlinear interactions of coupled excitations The interaction of elementary excitations determines many thermodynamics and kinetic properties of crystals. As a rule, description of these properties requires one to apply quantumstatistical methods. The coupling of excitations gives shape to the effective nonlinear interactions, whose amplitudes can have an anomalously large magnitude and unusual dispersion properties. The selection rules for such interactions are often controlled by the external conditions, e.g., the orientation of the external field with respect to the crystallographic axes. All this introduces certain specifics into a number of phenomena that arise in crystals in the presence of coupling of subsystems. The concept of the effective nonlinearity as being the nonlinearity of mixed elementary excitations is the basis for constructing its quantum theory [8-10]. In a second-quantization scheme the transition to a representation of mixed modes is performed by diagonalizing the bilinear component of the Hamiltonian operator taking into

(39) x=0 Here Pak and R~k are the transformation coefficients given in ref. [8]. Far from the intersection point of the " p u r e " spectra, the coupling of the terms with A = 1, 2, 3 with the operators c~- and c_ k in eq. (38), which is the quantum analog of eq. (15), governs the nonresonance (virtual) excitation of magnons by quasiphonons. Therefore every nonlinear process in the system of magnons is also a source of nonlinearity in the system of quasiphonons. Thus, an interaction of the type of (20) with participation of one phonon and two magnons studied phenomenologically is described by a contribution to the Hamiltonian having the form [8] n ( 3 ) = N - l ~ 2 E t'/fm(3)-ph(k, q ) ( b p k + b~,,-k)

k,q

X ( c ; +c_,)(c+_k_q+Ck+q).

(40)

When we take account of the transformation (38), (39), this interaction is' the source of threeparticle processes in the system of quasiphonons: AH(3) = N - ' / a

E ~(t3)~(k, q)(d-~k + da-k) +

k,q l,h,u

'+

X (d., + d~._q)(d,( k +q) + dt(_k_q) ) . (41)

V.L Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems

566

J a

-,-,_

J

J

>< + - < - o

z--X

b

Fig. 16. Diagrams for vertices arising from effective anharmonicity of third (a) and fourth (b) order [8].

Here l, A and u are the polarization indices of the quasiphonons, while the amplitude qt~3) of the interaction is expressed in terms of the parameters of the quasiphonons and the effective elastic moduli, just as in ordinary lattice dynamics:

q) 1 23/2

X

[ 12xk12.qOt(k+q) ) 1/2 1 MvV V V,

J(~ ^ ^ ^^ ^^ ~(fl2UAk)(fl2Ui, q)(fllUt,k+q)*

interaction of quasiphonons mediated by spin interactions is constructed analogously. The construction of the amplitude of the interaction of four quasiphonons corresponding to the effective fourth-order anharmonicity of (23) is illustrated in fig. 16b. A quantum theory of the EPAF acoustic nonlinearity based on the diagram technique for spin operators [61-63] has been developed in refs. [9-10]. A specific spatial dispersion of the interaction amplitudes (i.e., their dependence on k) caused b the dispersion of magnons in characteristic for effective anharmonicity. Dispersion restricts the phase volume of the interacting excitations to the region of small wave v e c t o r s k m < O)fO/Urn , since phonons with wave vectors of the order of the Debye value practically do not interact linearly with magnons (~'k-+kD~ 0). In this regard the effective anharmonicity of the elastic subsystem, which is giant in the long-wavelength region of the spectrum, is weak in the short-wavelength region and hence gives rise to relatively small energy losses of acoustic waves in processes of their scattering by short-wavelength (thermal) excitations.

(42)

Here J0 is the exchange energy; V~ =12xk/k; ill,2 = B1,2U0; and v 0 and M v are the volume and mass of the unit cell. We can make the construction perspicuous by using the graphic representation for the vertex parts of the Feynman diagrams (fig. 16). The diagram of fig. 16a illustrates the interaction (40). The straight and wavy lines correspond to magnons and phonons. We can conveniently treat the transformation to the representation of the coupled waves, with formation of an effective vertex of interaction of quasiphonons (which corresponds to the double line), as the result of joining the phonon and the magnon lines. This merger is put into correspondence with the factor (Gsk/@Oyk) in the analytic expression for the vertex with subsequent replacement of the phonon parameters (frequencies and polarizations) by the quasiphonon parameters. The description of any

[,., ~ - [ I

× -1,5

5,5

+' k

o

-4,~

-I- -- 4,g

4.5

~'~ 0

"-'==-=~

I

I

J

1

2

H, kOe

Fig. 17. Field dependence of the damping of magnetoelastic vibrations at the frequency 0.5 MHz in crystals of a-(Fe 1-xAlx)203.

V.L Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems

The contributions of effective anharmonicity to the damping Yk of sound is determined by the relationship [8] "Yk

TTN~ 6

g]t,

0);o(Mvvs2) 3"

For hematite an estimate of the acoustic Q-factor yields Q ~-- 105-106 for k << k m. Here it is pertinent to note that the experimentally measurable acoustic losses in crystals of aFe203 are substantially higher [12] (fig. 17). Apparently this is explained by the defect content of real crystals, which corresponds to the results of comparing the losses upon passage of sound through a specimen of a-Fe203 and relaxation times of the sonic field in experiments of reversal of the wave front of sound. We should note that crystals of hematite containing an admixture of AI 3+, which is isomorphous to trivalent iron, have a relatively higher Q-factor, owing to the decreased concentration of Fe z+ ions usually present in hematite in an amount of about 1%. Q =- g 2 k / 2 y k

7. Outside the anharmonic approximations The arguments presented in section 1 imply that, for magnetics having near SRT a strong enough m a g n o n - p h o n o n coupling (,~ = 1), the region of applicability of anharmonic expansions is limited to elastic deformations close in magnitude to the deformations of spontaneous magnetostriction. Such deformations are usually far from the limits for breakdown of real crystals and are relatively easily attained under the conditions of acoustic experimentation. In this regard it is of interest to study magnetoelastic excitations whose description lies outside the framework of anharmonic approximation and which involves the need for exact solution of strongly nonlinear systems of the dynamic equations (11) and (12). Under certain conditions the nonlinearity and dispersion introduced by the spin system into the magnetoelastic excitations can compensate one another. Consequently the possibility arises of formation of isolated coupled magnetoelastic waves - magnetoacoustic solitons. Such waves as applied to

567

EPAFs were first studied in ref. [16]. Their different modifications problems of stability and the character of evolution of soliton solutions have been studied in refs. [17-21]. We shall analyze the conditions of formation of magnetoacoustic solitons by using eq. (7) with the example of waves propagating in rhombohedral EPAFs along the trigonal axis. If the wave is a steady-state one, i.e., U = U(~) and 9 = 9(~:), where ~ = z - V t , then we can eliminate the dynamic deformations from the system of equations (11), (12) and reduce it to the "double sineG o r d o n " steady-state equation: 229

mp--~ =A

sin 9 + ½D sin 29.

(43)

Here we have A = HHD,

mp =

v D = ,~-2

0920 u2t

_v 2 V 2

HHDT2),

VS t = Vst(1 _ ~ 2 ) 1 / 2 is the velocity of the soundlike wave. We can easily establish the intervals of velocities V of waves corresponding to soliton solutions and the qualitative features of the motion of the magnetic moments by using the evident analogy of eq. (32) with the equation of motion of a particle of mass mp and momentum p = mp~9/O ~ in a force field having the potential

f(9) =A(cos 9-

1) + ¼D(cos 2 9 -

1).

In crystals of EPAfs having a high N6el temperature (a-Fe203, FeBO3), for which u m > USt , the attainable velocities of motion of solitons in the z direction must satisfy the conditions: V 2 < 2 > V 2 > US2t"In these cases the effective VS2t o r u m potential has the characteristic form shown in fig. 18a. An isolated wave on the background of the equilibrium state of the crystal (9 -= 0) corresponds to motion of the "particle" in the potential well F ( 9 ) from the point ( 9 [ ~ - = = 0 for p [ ~ _ ~ _ ~ = 0 ) to the point ( 9 [ ~ + ~ = 2 w for p I ~_~ +~ = 0), i.e., continuous rotation of the magnetic moment in the wave by the angle 2"rr. A solution of the given type amounts to the well

V.I. Ozhogin, ICL. Preobrazhenskii / Nonlinear dynamics of coupled systems

568

F

Vm k

i/ b

a

k'

Fig. 18. (a) Dependence of the effective potential on the angle ¢ of deviation of the vector I from the equilibrium direction. (b) Regions of existence for the velocities of solitons in a high-temperature EPAF (v M > r'st)-

known " 2 r r k i n k " of the " d o u b l e s i n e - G o r d o n " e q u a t i o n [16]:

q~

tg 2 =

(D+A)l/2(sh ~

~ 1-'

~5~

(44)

H e r e the c h a r a c t e r i s t i c d i m e n s i o n of a soliton is

~o = [mo/(D + X ) ] 1/2. F o r the a - F e 2 0 3 crystal with H = 0.5 k O e a n d V << Vst the m a g n i t u d e o f so0 is ~ 1 0 - 3 - 1 0 -4 cm, while the m a x i m u m def o r m a t i o n s in the wave a r e of t h e o r d e r o f 10 -5 . W e should p a y a t t e n t i o n to the fact that the velocities of solitons t a k e on values t h a t do not coincide with the p h a s e velocities o f l i n e a r magn e t o e l a s t i c waves. In the wk p l a n e each value c,ph - ¢o/k o f the h a r m o n i c wave can be p u t into

c o r r e s p o n d e n c e with a straight line passing t h r o u g h the c o o r d i n a t e origin with a slope e q u a l to tan a = %h" T h e regions in which t h e s e straight lines can lie a r e left u n c r o s s h a t c h e d in fig. 18b. I n turn, the c r o s s h a t c h e d r e g i o n s c o r r e s p o n d to t h e a t t a i n a b l e velocities o f solitons ( V = t a n a ) . Such a s e g m e n t a t i o n o f the tok p l a n e is valid for any directions o f p r o p a g a t i o n o f m a g n e t o e l a s t i c waves, if we d o not t a k e into a c c o u n t relaxation, which a m o u n t s to a g e n e r a l i z e d result o f the a c t i o n o f the n o n l i n e a r i n t e r a c t i o n s n o t t a k e n into a c c o u n t in (11), (12). In l o w - t e m p e r a t u r e E P A F s for which u m < USt ( M n C O 3, COCO3), solitons c o r r e s p o n d to velocities Cst > V > Vst o r V < u m (fig. 19b). In the f o r m e r case the solitons also have the f o r m of a

~sr ]r

F

/ I jm.~

G.)~ 0

O

"5

k

a

Fig. 19. (a) Dependence of the effective potential on the angle q~ of deviation of the vector 1 from the equilibrium direction. (b) Region of existence for the velocities of solitons in a low temperature EPAF (c m < Ust).

V.l. Ozhogin, V.L. Preobrazhenskii / Nonlineardynamics of coupled systems 2"rr kink, and in the latter case their structure proves qualitatively different. The characteristic form of the effective potential a of "particle" of mass Imp I for the range of near-sonic velocities Vst > V > Vst is shown in fig. 19a. A soliton corresponds to motion of the "particle" from the point q~ = 0 (for p = 0) to the point @max with subsequent return to the point ~0 = 0. Here the solution of eq. (32) has the form [16] q~ ( I D + A I tg~A

~1/2[ ~ ~--1 ) [ch~0) "

(45)

In contrast to a 2rr kink, the given type of solitons can be realized for any deviations from equilibrium (~max << 1). In this case the result goes over into the known soliton solution of the modified Korteweg-de Vries equation. From the experimental standpoint it seems interesting to estimate the time ~" during which such a soliton evolves from the initial steplike magnetoelastic perturbation moving with the velocity Vst [17]. For the typical low-temperature E P A F MnCO 3 we find ~"= ~0/I V - Vst I -~ 10 -6, and the length of specimen necessary for a pulse experiment does not exceed 1 cm. Estimates of z for an elastic soliton with a nonlinear elastic lattice or for a magnetoelastic soliton in a ferromagnet having a moderate magnetoelastic interaction yield r --- 0.1 s, thus demonstrating the advantages of nonconductive EPAFs for direct observation of solitary magnetoelastic waves. We must note that soliton solutions of the original system (11), (12) for magnetoelastic waves of small amplitude with near-sonic velocities also exist in high-temperature EPAFs, but for other directions of propagation [17]. However, in these crystals such excitations prove to be unstable with respect to transverse perturbations of their front [18,20]. One of the variants of the development of this instability turns out to be self-focusing of the magnetoelastic excitations [18]. Its physical cause is the decrease in velocity of the excitation with increase in its amplitude, which causes an accumulating deflection of the wave front in a direction opposite to the direction of propagation. A detailed analysis of the initial distributions of the ampli-

569

tude from which the self-focusing wave is formed has been performed in ref. [20]. The evident qualitative differences between solitons of the types (44) and (45) (see figs. 18 and 19) allow a certain topological treatment. The solitons of (44) have a nonzero topological charge (1/2"rr)fV~ • d l = 1, where the integration is performed over a contour closed at infinity and penetrating the basis plane. By analogy with the vortex states in extended Josephson structures, such excitations can be classified as vortex excitations. For the solitons of (45) the topological charge is zero. In the case of the excitations (45) a crystal with fixed boundary conditions ( ~ l z = +L = 0) can be converted by continuous transformation to the equilibrium state. Under the same conditions for excitations of the form of (44) the transition to equilibrium involves overcoming a finite energy barrier due to the exchange interaction. Accordingly, the solitons (44) are topologically stable in contrast to the solitons (45). A calculation [12] of the spectra of their localized excitations shows that the given type of solitons is stable not only topologically, but also dynamically, which allows us to expect a possible experimental observation. We note that strongly nonlinear spin excitations are substantially magnetoelastic even in crystals with relatively weak m a g n o n - p h o n o n coupling if their velocity of propagation is close the sound velocity. In particular, such a situation arises in the orthoferrites in the motion of domain boundaries with near-sonic velocities [21,22]. Recently experimental results have been obtained [64,23,74] on FeBO 3 which confirm the conclusions of the theory on magnetoelastic gaps [65,66] in the spectrum of the stationary motion velocity of domain boundaries in EPAFs. Also it was obtained the evidence of dynamic SRTs in the sound wave field [67].

8. Conclusion

As we see it, the idea presented in the review on the nonlinearity of mixed modes are highly general in character. Not only magnetoelastic, but also electron-nuclear-spin and electron-

570

V.L Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems

nuclear-magnetoelastic waves, ferroelectromagnetic, a n d f e r r o e l e c t r i c - m a g n e t o e l a s t i c w a v e s a n d other types of coupled oscillations can play the r o l e o f m i x e d e x c i t a t i o n s . N o n l i n e a r p r o c e s s e s in such systems have been intensively studied, but m a i n l y t h e o r e t i c a l l y [ 6 8 - 7 3 ] a n d all o f t h e m a r e p a r t i c u l a r l y p r o m i n e n t in t h e vicinity o f s p i n reo r i e n t a t i o n t r a n s i t i o n s - f r o m a c o r r u g a t e d instability o f a m a g n e t o e l a s t i c w a v e f r o n t [74] u p to s h o c k w a v e s [73c] a n d e v e n s t r a i n s - s p o u t s (to b e published). An expansion of the experimental s t u d i e s o f s t r o n g d y n a m i c i n t e r a c t i o n s in c o u p l e d systems in t h e vicinity o f S R T w o u l d f a c i l i t a t e t h e further development of our views of the mechanisms of formation of the dynamical properties of solids a n d t h e i r f u n c t i o n a l p o t e n t i a l i t i e s in t e c h nical a p p l i c a t i o n s .

[13] [14]

[15] [16] [17] [18] [19]

[20] [21]

[22]

References [1] A.S. Borovik-Romanov and E.G. Rudashevskii, Zh. Eksp. Teor. Fiz. 47 (1964) 2095 [Sov. Phys. JETP 20 (1965) 1407]. [2] E.A. Turov and V.G. Shavrov, Fiz. Tverd. Tela (Leningrad) 7 (1965) 217 [Sov. Phys. Solid State 7 (1965) 166]. [3] E.A. Turov and V.G. Shavrov, Usp. Fiz. Nauk 140 (1983) 429 [Soy. Phys. Usp. 26 (1983) 593]. [4] I.E. Dikshtein, V.V. Tarasenko and V.G. Shavrov, Zh. Eksp. Teor. Fiz. 67 (1974) 816 [Sov. Phys. JETP 40 (1975) 404]. [5] V.I. Ozhogin and V.L. Preobrazhenskii, Physica B 86-88 (1977) 979. [6] V.I. Ozhogin and V.L. Preobrazhenskii, Zh. Eksp. Teor. Fiz. 73 (1977) 988 [Sov. Phys. JETP 46 (1977) 523]. [7] V.I. Ozhogin and V.L. Preobrazhenskii, Sov. Phys. Usp. 31 (1988) 713. [8] V.S. Lutovinov, V.L. Preobrazhenskii, and S.P. Semin, Zh. Eksp. Teor. Fiz. 74 (1978) 1159 [Sov. Phys. JETP 47 (1978) 609]. [9] L.V. Panina and V.L. Preobrazhenskii, Fiz. Met. MetaUoved. 57 (1984)39 [Phys. Met. Metallogr. (USSR) 57 (1984)30]. [10] L.V. Panina, V.L. Preobrazhenskii, V.N. Shumilov and N.A. Ekonomov, Akust. Zh. 30 (1984) 566 [Sov. Phys. Acoust. 30 (1984) 335]. [11] N.N. Evitkhiev, V.V. Moshkin, V.L. Probrazhenskii and N.A. Ekonomov, Pis'ma Zh, Eksp. Teor. Fiz. 35 (1982) 31 [JETP Lett. 35 (1982) 38]. [12] V.L. Preobrazhenskii, Abstract of dissertation for doctorate in physical-mathematical sciences, Institute of Radio

[23] [24] [25]

[26] [27]

[28]

[29] [30]

[31]

[32] [33] [34] [35] [36]

Engineering, Electronics, and Automation, Moscow, (1986) in Russian. E.A. Turov, JETP 98 (1990) 655. A.Yu. Lebedev, V.I. Ozhogin and A.Yu. Yakuhovskii, Pis'ma Zh. Eksp. Teor. Fiz. 34 (1981) 22 [JETP Lett. 34 (1981) 19]. V.V. Berezhnov, N.N. Evtikhiev, V.L. Preobrazhenskii and N.A. Ekonomov, Radiotekh. Electron 28 (1983)]. V.L. Preobrazhenskii and M.A. Savchenko, Proc. 20th Congr. AMPERE. Tallinn (1978) p. 410. V.I. Ozhogin, and A.Yu. Lebedev, J. Magn. Magn. Mater. 15-18 (1980) 617. V.I. Ozhogin, D.Yu. Manin, V.I. Petviashvili and A.Yu. Lebedev, IEEE Trans. Magn. MAG-19 (1983) 1977. V.D. Buchel'nikov and V.G. Shavrov, Fiz. Tverd. Tela (Leningrad) 25 (1983) 90 [Soy. Phys. Solid State 25 (1983) 49]. S.K. Turitsyn and G.E. Fal'kovich, Zh. Eksp. Teor. Fiz. 89 (1985) 258 [Soy. Phys. JETP 62 (1985) 146]. A.K. Zvezdin and A.F. Popkov, Fiz. Tverd. Tela (Leningrad) 21 (1979) 1334 [Sov. Phys. Solid State 21 (1979) 771]. V.G. Bar'yakhtar, B.A. Ivanov and V.V. Chetkin, Usp. Fiz. Nauk 146 (1985) 417 [Sov. Phys. Usp. 28 (1985) 563]. M.V. Chetkin and V.V. Lykov, JETP Lett. 52 (1990) 869. A.O. Kiselev, V.I. Ozhogin and V.L. Preobrazhenskii, JETP Lett. 51 (1990) 82. V.G. Bar'yakhtar, B.A. Ivanov and A.L. Sukstanckii, Zh. Eksp. Teor. Fiz. 78 (1980) 1509 [Sov. Phys. JETP 51 (1980) 757]. A.F. Andreev and V.I. Marchenko, Usp. Fiz. Nauk 130 (1980) 39 [Sov. Phys. Usp. 23 (1980) 21]. L.V. Velikov, S.V. Mironov and E.G. Rudashevskii, Zh. Eksp. Teor. Fiz. 75 (1987) 1110 [Sov. Phys. JETP 48 (1978) 559]. R.K. Bullough, P.J. Caudrey and H.M. Gidds, in: Solitons, eds. R.K. Bullough and P.J. Caudrey (SpringerVerlag, Berlin, 1980) p. 107 [Russ. transl. (Mir, Moscow, 1983) p. 122]. M.H. Seavey, Solid State Commun. 10 (1972) 219. a) V.I. Ozhogin and P.P. Maksimenkov, Digests of Intermag Conf., Kyoto, (1972) 49-4; b) IEEE Trans. Magn. MAG-8 (1972) 130; c) Zh. Eksp. Teor. Fiz. 65 (1973) 657 [Sov. Phys. JETP 38 (1974) 342]. E.F. Andruschak, N.N. Evtikhiev, S.A. Pogozhev, V.L. Preobrazhenskii and N.A. Ekonomov, Akoust. Zh. 27 (1981) 170 [Sov. Phys. Acoust. 27 (1981)]. A. Tasaki and S. Iida, J. Phys. Soc. Japan 18 (1963) 1148. V.S. Merkulov, E.G. Rudashevskii, H. LeGall and C. Leycuras, Zh. Eksp. Teor. Fiz. 75 (1978) 628. A.F. Ermolov, V.V. Moshkin, V.L. Preobrazhenskii and N.A. Ekonomov, Fiz. Tverd. Tela 30 (1988) 65. A.P. Levanyuk, V.V. Osipov, A.S. Sigov and A.A. Sobyanin, Zh. Eksp. Teor. Fiz. 76 (1979) 345. L.K. Zarembo and V.A. Krasil'nikov, Usp. Fiz. Nauk 102 (1970) 549 [Sov. Phys. Usp. 13 (1971) 778].

V.L Ozhogin, V.L. Preobrazhenskii / Nonlinear dynamics of coupled systems [37] V.L. Preobrazhenskii, M.A. Savchenko and N.E. Ekonomov, Pis'ma Zh. Eksp. Teor. Fiz. 28 (1978) 93 [JETP Lett. 28 (1978) 87]. [38] A.M. Zaikin and V.L. Preobrazhenskii, 5th All-Union Conf. on Precisive Alloys, Rostov-Velikii (September 1991) Abstracts. [39] A.E. Clark, H.S. Belson, N. Tamagave and E. Callen, Proc. ICM '73, 4 (1973) 335. [40] V.V. Berezhnov, N.N. Evitkhiev, V.L. Probrazhenskii and N.A. Ekonomov, Fiz. Tverd. Tela (Leningrad) 25 (1982) 1870 [Soy. Phys. Solid State 25 (1982) 1079]. [41] V.I. Ozhogin, A.Yu. Lebedev and A.Yu. Yakubovskii, Pis'ma Zh. Ekskp. Teor. Fiz. 27 (1978) 333 [JETP Lett. 27 (1978) 313]. [42] V.A. Krasil'nikov, T.A. Mamatova and V.G. Prokoshev, Pis'ma Zh. Tekh. Fiz. 10 (1984) 1196 [Sov. Tech. Phys.. Lett. 10 (1984) 506]. [43] D.A. Kirzhnits, Usp. Fiz. Nauk 124 (1978) 487. [44] V.V. Berezhnov, V.L. Preobrazhenskii, N.A. Ekonomov and D.E. El'yashev, Abstracts of reports at the 16th All-Union Conf. on Physics of Magnetic Phenomena, Tula (1983) p. 23. [45] N.N. Evtikhiev, V.L. Preobrazhenskii, M.A. Savchenko and N.A. Ekonomov, Vopr. Radioelektron. Ser. Obshchetekhn. 5 (1978) 124. [46] R.L. Comstock and R.C. LeCraw, Phys. Rev. Lett 10 (1963) 219. [47] V.V. Zautkin, V.S. L'vov, B.I. Orel and S.S. Starobinets, Zh. Eksp. Teor. Fiz. 72 (1977) 272. [48] V.I. Ozhogin, A.Yu. Yakubovskii, A.V. Abryutin, S.M. Sileimanov, J. Magn. Magn. Mater. 15-18 (1980) 757. [49] V.A. Krasil'nikov, T.A. Mamatova and V.G. Prokoshev, Fiz. Tverd. Tela (Leningrad) 28 (1986) 615 [Sov. Phys. Solid State 28 (1986) 346]. [50] V.V. Berezhnov, N.N. Evtikhiev, V.L. Preobrazhenskii and N.A. Ekonomov, Akust. Zh. 26 (1980) 328 [Sov. Phys. Acoust. 26 (1980) 180]. [51] M.K. Gubkin, T.A. Mamatova and V.G. Prokoshev, ibid. 31 (1985) 678 [Sov. Phys. Acoust. 31 (1985) 410]. [52] V.A. Ermolov, A.I. Alekseev and V.G. Pankratov, Pis'ma Zh. Tekh. Fiz. 11 (1985) 377 [Sov. Tech. Phys. Lett. 11 (1985) 156]. [53] W. Wettling, W. Jantz and C.E. Patton, J. Appl. Phys. 50 (1979) 2030. [54] B.Yu. Kotyuzhanskii and L.A. Prozorova, Zh. Eksp. Teor. Fiz. 83 (1982) 1567 [Soy. Phys. JETP 56 (1982) 903]. [55] V.S. Lutovinov and M.A. Savchenko, Abstracts of reports at the All-Union Conf. on the Physics of Magnetic Phenomena, Khar'kov (1979) p. 85.

571

[56] M.V. Petrov, A.P. Paugurt, I.V. Pleshakov and A.V. Ivanov, Pis'ma Zh. Tekh. Fiz. 11 (1985) 1204 [Sov. Tech. Phys. Lett. 11 (1985) 498]. [57] R. Dichl, W. Jantz, B.I. Nolang and W. Wettling, Current Topics in Materials Science, vol. II, ed. E. Kaldis, (1986) p. 242. [58] A.Yu. Lebedev, V.I. Ozhogin, V.L. Safonov and A.Yu. Yakubovskii, Zh. Eksp. Teor. Fiz. 85 (1983) 1059 [Sov. Phys. JETP 58 (1983) 616]. [59] M.A. Savchenko, Fiz. Tverd. Tela (Leningrad) 6 (1964) 864 [Soy. Phys. Solid State 6 (1964) 666]. [60] R.I. Kukhtin, V.L. Preobrazhenskii and N.A. Ekonomov, Fiz. Tverd. Tela (Leningrad) 26 (1984) 884 [Sov. Phys. Solid State 26 (1984) 356]. [61] V.G. Vaks, A.I. Larkin and S.A. Pikin, Zh. Eksp. Teor. Fiz. 53 (1967) 1089 [Sov. Phys. JETP 28 (1969) 734]. [62] E.M. Pikalev, M.A. Savchenko and J. Shoyom, Zh. Eksp. Teor. Fiz. 55 (1968) 1404 [Soy. Phys. JETP 28 (1969) 734]. [63] Yu.A. Izyumov, S.A. Kasan-Ogly and Yu.N. Skryabin, Field Methods in the Theory of Ferromagnetism (in Russian), (Nauka, Moscow, 1984) p. 224. [64] M.V. Chetkin, V.V. L'vov and V.D. Tereshchenko, Fiz. Tverd. Tela (Leningrad) 32 (1990) 939. [65] A.K. Zvezdin, V.V. Kostuchencko and A.A. Mukhin, Preprint No. 209, Lebedev Phys. Inst. of Ac. Sci (1983). [66] L.V. Panina and V.L. Preobrazhenskii, Fiz. Tverd. Tela (Leningrad) 60 (1985) 449. [67] A.F. Kabychenkov and V.G. Shavrov, Acta Phys. Polon. 4 (1988) 531. [68] A.V. Andrienko, V.I. Ozhogin, V.L. Safonov and A.Yu. Yakubovskii, Soy. Phys. Usp. (October 1991). [69] M.A. Savchenko and V.P. Sobolev, Abstracts of reports at the All-Union Conf. on the Physics of Magnetic Phenomena, Donetsk (1985) p. 318. [70] M.A. Savchenko and M.A. Khabakhpashev, Fiz. Tverd. Tela (Leningrad) 18 (1976) 2699; 20 (1978) 1845 [Sov. Phys. Solid State 18 (1976) 1573, 20 (1978) 1065]. [71] G.A. Farias and A.A. Maradudin, Phys. Rev. Ser. B 28 (1983) 1870. [72] V.V. Men'shenin, I.F. Mirsaev and G.G. Taluts, Fiz. Met. Metalloved. 56 (1983) 1078 [Phys. Met. Metallogr. 56 (1983) 1983]. [73] A.F. Kabychenkov, V.G. Shavrov and A.L. Shevchenko, Fiz Tverd. Tela (Leningrad) a) 31 (1989) 193, b) 32 (1990) 1182, c) 32 (1990) 2010. [74] M.V. Chetkin and V.V. Lykov, J. Magn. Magn. Mater. (1991) to be published.