The magnon-magnon interactions in easy-plane antiferromagnets

Physica A 184.992) 523-557 North-Holland

The magnon-magnon interactions in easy-plane antiferromagnets D.A.

G a r a n i n , V.S. L u t o v i n o v a n d L.V. P a n i n a

Moscow Institute of Radioengineering, Electronics and Automation, 117454, Prospect Vernadskogo, 78, Moscow, Russia

Received 14 November 1991 In the framework of the spin-operator diagram technique, the processes of relaxation of lower branch magnons in a two-sublattice easy-plane antiferromagnet are investigated. The procedure of the spin Hamiltonian diagonalization allowing to obtain compact expressions for magnon interaction amplitudes is proposed. The relaxation frequencies due to three- and four-magnon processes, processes of elastic scattering and interbranch transformation by thermal and concentrational magnetization fluctuations as well as the three-magnon confluence processes with rescattering of thermal magnons by fluctuations are calculated. The results obtained are consistent with the experimental data on parametric excitation of spin waves in the low-temperature antiferromagnets MnCO3 and CsMnF3.

1. Introduction It is undoubtedly that, in general, the most consistent description of static and dynamic properties of magnetic systems m a y be achieved in terms of variables which are inherent in the problem, i.e. the spin operators. H o w e v e r , since the algebra of the spin operators is rather complex, n u m e r o u s rounda b o u t methods are used in practice. A m o n g t h e m the most widespread ones are those based on the boson representation of the spin operators (e.g. the H o l s t e i n - P r i m a k o f f and the D y s o n - M a l e e v representations) in the f r a m e w o r k of which the main part of the results of the spin wave theory was obtained. Although the boson methods are handy for calculations, their applicability is confined to the low-temperature region where the excitations of a magnetically o r d e r e d s y s t e m - spin waves ( S W ) - may be considered as small deviations f r o m the completely ordered state. A t higher t e m p e r a t u r e s where the magnetization substantially differs from its nominal value and, especially, in the vicinity of the phase transition point, the boson representations of the spin o p e r a t o r s b e c o m e inconvenient. In fact, in this region the spin wave must be considered as a collective motion against the thermally activated, strongly 0378-4371/92/$05.00 ~) 1992- Elsevier Science Publishers B.V. All rights reserved

524

D . A . Garanin et al. / M a g n o n - m a g n o n interactions

fluctuating background. One of the dominant SW scattering processes in this region is elastic scattering by thermal fluctuations of magnetization. The mathematical framework supporting the picture outlined above was originated by Vaks, Larkin and Pikin [1] who proposed the spin-operator diagram technique allowing, in principle, to calculate physical quantities of magnetic systems at all temperatures, including the paramagnetic state. However, since the graphical definitions proposed by Vaks, Larkin and Pikin and further developed by Izyumov, Kassan-Ogly and Scryabin [2] are still far from those commonly accepted in the diagrammatic technique for boson and fermion systems, the spin-operator diagram technique was reputed complicated and was not practiced on a large scale. The more serious difficulty which till recently hampered the application of the spin diagram technique to real (anisotropic and multi-sublattice) magnetic systems, was the absence of a spin Hamiltonian diagonalization procedure similar to the well-known u - v transformation. As for the application of the matrix (normal and "abnormal") Green's functions to non-diagonal spin hamiltonians, this leads to drastic complication of the calculations and to disguising of the physical sense of elementary excitations and their interaction processes [3]. Not long ago the spin Hamiltonian diagonalization procedure was proposed by the authors [4-6]. This has allowed to describe magnetic systems in terms of diagonal Green's functions and to clarify the nature of spin wave processes evolving against the strongly fluctuating background. As an extreme example, we considered the dynamical properties of a nuclear subsystem of antiferromagnets [6] in which the nuclear polarization ( I ) constitutes only a few percents of its nominal value, I. With the use of the spin-operator diagram technique it was shown that unphysically large boson occupation numbers of thermally excited nuclear spin waves are, in fact, cancelled because of the finiteness of the spin-operator spectrum, and the leading role among the scattering processes passes to the fluctuational scattering. Besides this, we proposed graphical representations which are maximally close to those used in the diagram technique for boson and fermion systems. However, the spin Hamiltonian diagonalization procedure mentioned above is all the same substantially less handy than the u - v transformation - instead of the linear substitution of the operator variables in the Hamiltonian one has to solve systems of inhomogeneous linear first-order differential equations for the coefficients of the Hamiltonian under transformation. These complications result from the fact that the originally proposed transformation virtually coincides with the u - o one only for some essentially non-linear transformation not affecting the quadratic part of the Hamiltonian and the interaction amplitudes on the energy shell. But as for the latter beyond the energy shell, the corresponding expressions turn out to be rather cumbersome which may be

D . A . Garanin et al. / M a g n o n - m a g n o n interactions

525

considered as evidence of some incompleteness of the suggested approach (see ref. [7]). In this article we propose an improved version of the spin Hamiltonian diagonalization procedure being the direct spin analog of the u - v transformation and allowing to achieve large simplifications. In particular, the results of the diagonalization for the physically observed quantities- the interaction amplitudes- may be written compactly as some linear transformations and retain their functional form beyond the energy shell. As a concrete physical system we consider two-sublattice antiferromagnets with easy-plane anisotropy (AFEP) (MnCO3, CsMnF3, FeBO 3, c~-FeeO3, etc.) characterized by two spin wave branches with widely separated activation energies: el0 ~ e20. The lower of the them may be investigated by parallel pumping and AFMR techniques [8, 9]. Although experimental and theoretical investigations of antiferromagnets have been carried out for a long time, the nature of relaxational processes in AFEP has not been completely clarified up to now. In the limiting case of the isotropic AF the spin wave relaxation frequencies due to the four-magnon scattering processes were correctly calculated only in 1970 by Harris et al. [10] after numerous unsuccessful attempts made by other groups. The main part of the mistakes was due to utilization of erroneous scattering amplitudes which did not satisfy the Adler principle. This principle is used in the theory of magnetism for checking the amplitudes starting from the work of Baryakhtar, Sobolev and Kvirikadze [11] where the four-magnon processes in AFEP were considered in the case of zero magnetic field (at H = 0 the lower spin wave branch is a Goldstone one: el0 = 0, elk ~-sk). The four-magnon processes in another particular case: k = 0, H ~ 0 were treated by Baryakhtar and Sobolev [12]. At the same time the corresponding results for the relaxation frequency of the lower-branch magnons in the general case s k ~ e l o , realized in most of the parallel pumping experiments, are absent. The spin wave damping in AFEP due to the three-magnon confluence process for k = 0 was calculated by Ozhogin [13] (see also ref. [14]). It is worth noting that the calculation of the lower-branch magnon damping, due to the three-magnon confluence process in the lowest order of the perturbation theory, may be insufficient. This problem arises from the fact that because of the energy conservation law this process is only allowed in a narrow region of the phase space. As a consequence, the result may be unstable with respect to taking into account the damping of magnons involved in the process and with respect to the vertex corrections. Furthermore, under certain conditions, namely when the allowed energy interval falls into the "underthermal" region (T O= e20/2ex0 2 ~ T) or into the "superthermal" region (T O>> T), the contributions of higher-order diagrams into the magnon damping may exceed that of the starting 2nd-order one. If the energy of the "probe" magnon ek

526

D . A . Garanin et al. / M a g n o n - m a g n o n interactions

exceeds the damping of those interacting with it (e k ~> yq), then it is sufficient to take into account the next order of the perturbation theory, i.e. the insertions into the three-magnon loop corresponding to concrete mechanisms of rescattering of intermediate excitations. In the opposite case (e k <~ yq), the rescattering must be taken into account in all orders of the perturbation theory (the hydrodynamic situation). The crossover from one region to the other was traced qualitatively (neglecting vertex corrections) by Lutovinov and Semin [15], suggesting that the relaxation of thermal magnons is governed by fourmagnon scattering. It was shown that if the inequality To ~ T1, where T I tOE(el0/tOE)1/5 and o)E is the exchange frequency, is satisfied, then the sharp rise of the damping of lower-branch magnons (Yk ~ T 7 ) in the range T o ~ T ~ T 1 (where e k >> y q ) is followed by a slow decrease (Yk ~ T-1/2) for T >> T 1 (e k ~ yq). The earliest approach to the problem of damping of quasiparticles in the hydrodynamic regime was developed by Akhiezer [16] who investigated the attenuation of sound waves in dielectrics at low frequencies. Akhiezer's method consists in the solution of the kinetic equation for a phonon gas subjected to external perturbation by a "probe" phonon with subsequent calculation of dissipated energy. As was shown by Lutovinov, Olkhov and Semin [17], in the diagrammatic language Akhiezer's method corresponds to the ladder approximation for the quasiparticle's self-energy part, the kinetic equation being equivalent to the integral equation for the vertex. The solution of the kinetic equation in the ~--approximation is equivalent to neglecting the vertex corrections in the diagrammatic approach. As can be seen above, the transition to the hydrodynamic region in AFEP occurs at rather high temperatures where the damping of thermal magnons caused by the elastic fluctuational scattering is of the order of that caused by the four-magnon one. In real crystals with defect concentration c' = 10-2-10 -3, the damping of spin waves with ek ~ e20 due to the joint influence of concentrational and thermal magnetization fluctuations is large in the whole region T < T N and must be taken into account first of all considering the threemagnon confluence process in A F E E It must be mentioned that due to the simplicity of the fluctuational amplitudes, the system of integral equations for the vertices is reduced to the algebraic system and it becomes possible to write down the expression for the spin wave damping as a quadrature for all values of the parameter ek/'y q (see eqs. (5.13), (5.14)). The composition of the subsequent part of the article is as follows. In section 2 the procedure of the spin Hamiltonian diagonalization is described. In section 3 the amplitudes of various interaction processes between the quasiparticles of AFEP are constructed. In section 4 we calculate in the lowest orders of the perturbation theory, the relaxation frequencies of the lower-branch magnons due to the fluctuational, three- and four-magnon scattering processes and also =

527

D . A . Garanin et al. / M a g n o n - m a g n o n interactions

due to the confluence process with rescattering by fluctuations. In section 5 the calculation of the lower-branch magnon damping due to the confluence process with rescattering by fluctuations in all orders of the perturbation theory is presented. Finally, in section 6 a comparison of theoretical results with existing experimental data is carried out.

2. The spin Hamiltonian diagonalization The Hamiltonian of a two-sublattice antiferromagnet with easy-plane type anisotropy is of the form

"

- ~t, S t.Sg - dzg[S¢, Sgly) ,

g

.

(2.1)

f,g

Here H is an external magnetic field lying in the basis plane (x, z), o~rg is the intersublattice exchange interaction, f and g are the indices of magnetic ions belonging to different sublattices, dig is the Dzjaloshinsky interaction,/3rg > 0 is the anisotropy of the exchange interaction. From now the sublattice spins will be viewed in coordinate systems with the axes " z l , z2" along the equilibrium directions of the magnetization of each sublattice, which constitute a small angle 0 with the axis " z " : x )= Si(g

szl(z2)

sin 0 + s x l ( x 2 ) cos 0 ,

z )= Sr(g

+_SZl(Z2)

SYf(g) = +_S yl(y2) ,

(2.2) cos 0 w- s x l ( x 2 ) sin 0 .

The canting angle O is determined in the mean field approximation from the condition of nullification of the coefficient in the term of the Hamiltonian which is linear with respect to transverse spin components sxl'x2: H c o s 0 + H D cos 2 0 -

(2.3)

H E sin 2 0 = 0 .

Here H E - toE = B~0 , H o = Bdo, ~o and d o are the corresponding zero Fourier components, B is the sublattice polarization,

B=SBs(SI2I/T),

Bs(x ) =

1+

1

cotanh 1 +

x-~-~cotanh2--S, (2.4)

S is the spin value of a magnetic ion, and /4 = H sin 0 + HE cos 2 0 + HD sin 2 0

(2.5)

D . A . Garanin et al. / Magnon-magnon interactions

528

is the molecular field. In practice the inequality H , H D ~ H E is usually satisfied, and one gets H D ( H + HD)

H + HD

O ~

2H~

~ 1,

/-) ~ H E +

(2.6)

2H E

It must be mentioned that taking into account only the first term in the above expression for H , one obtains incorrect three-magnon interaction amplitudes (see ref. [11]). A f t e r Fourier transformation while taking into account relation (2.2), the Hamiltonian of A F E P may be written in the form 1 --- -/--)(S 01 --b 302) - ~ E 6(1 + 2 ) [ v ] ~ AS{' AS22 1,2 rl]xx~xl~x2 rl]xz(.~xl SXl2 AS21)] -~- --1 ~1 ~2 -1- rI/ryY~yl~y2 --1 ~1 ~2 q- --1 \°1 AS22 ~-

(2.7)

where A S ~ ' = S~i - N B 6 ( k , ) ,

1 (~,+I -1 S~' = i W k , + S - k l ) '

V] ~ = - V ~ ~ = J k C O S 2 0

+ dksin20,

etc.,

VYly = J , , - / 3 k l ,

(2.8)

H H V~ ~ = - J k , sin 2 0 + dk~ cos 2 0 = - ~ Ak. cos 2 0 ~ -- ~ , and Ak ----J J J o = d J d o . For the diagonalization of the spin Hamiltonian (2.8) it is convenient to introduce a matrix notation, zm

~L° = - / ~ ( S o 1 q- 302) - ~ -[- V l m f l n s ~ l m S 2

zn

E 8(1 q- 2 ) ( v 7 " A S 1 A S 2 1,2

,8n -.~ V ~,~;, I S am 1

AS;")

(2.9)

in which the summation over Latin (m, n = 1, 2) and G r e e k ( a , / 3 = -+) indices is implied. The composite indices of the type a m take their values in the order: + 1 , - 1 , + 2 , - 2 . The matrix coefficients of the Hamiltonian (2.9) are of the form V mn = ½ V ~ 3 ( m + n - 3 ) , g am'Bn = 1 [I~8(a/3 -- 1) + ik = !4,-k t,,.xx

+

),

V "m;" = ½VX~8(m + n - 3 ) ,

]8(~/3 +

1)]8(m + n - 3 ) ,

(2.10)

L = 4 ,!(,,.xx -k-GY),

where the values ---1 are attributed to the G r e e k indices a, /3. To get rid of

529

D.A. Garanin et al. / Magnon-magnon interactions

non-diagonal elements of the matrix V am~n, transformation

one

has to carry out a unitary

(2.11)

~*(n)=exp(n~) ~exp(-n~), which will be used later in its differential form: 2_ Or/

=

(2.12)

The parameter of the transformation 7/ changes from 0 to 1; at 77= 1 the hamiltonian ~* must be diagonal. Unlike the preceding papers [4-6] where the operator ~ was chosen bilinear with respect to the spin operators S +-, here we will consider a transformation of a general form: 1

1

= ~ Z 6(1 + 2) ~

l~Ctm~n~ctm~ -~n

--1

°1

""2

1,2

+~

1

1

Z

l~amjSn;p~otm~n AS;P

6(1+2+3)~-~--,2

~1

u2

1,2,3

1

1

E

l~o~ml3nylKp¢2o~rn.l~On~yl~Kp

+ .

(2.13)

1,2,3,4

The terms of higher orders in the spin operators (the counter-terms) in this expression has no influence on the quadratic part of the transformed Hamiltonian. Their concrete form is determined by the requirement of a direct analogy of the present transformation (2.11) with the boson u - v transformation-in this case a large simplification of multi-spin terms, l

N2 ~

1,2,3

6(1 + 2 +

-~- --231"rotm;npcam°l

1

N3 Z

,.,4.(l/rotml3n;p.~otm~13n AS;P ~- varn[Jnyl~otrn~[3n~.yl ~)",--12

~I

~2

--123

~1

~2

~3

zn AS2 AS;p)

6(1+2+3+4)

1,2,3,4

(Vl/rotm~nyl;P~°tm ~ ~n ~ yl A S 4P -~- rL,rctm~nylKP.~otm Q~n ~ yl ~Kp ~ . . . . X k--123

°1

~'2

~3

--1234

~1

u2

~3 ~4

)'

(2.14)

generated in the Hamiltonian ~*(r/) with the increasing of the parameter ~7, is achieved. The operator differential equation (2.11) is equivalent to the infinite set of ordinary differential equations for the tensor coefficients of the Hamiltonian:

D.A.

530

Garanin

+ &

et al.

I Magnon-magnon

[PRY”-‘“V;”

interactions

+ ~YR;~-~~V;~]

etc. It is seen from eqs. (2.15) and (2.16) that coefficients corresponding to spin-operator combinations of lower orders (here V”“‘“(q)) enter inhomogeneous terms of higher-order differential equations. Hence, these equations connected with each other may be solved step by step up to the arbitrary order in spin operators, though inhomogeneous terms result in difficulties in analytical calculations rising with each order. Fortunately, the inhomogeneous terms in differential equations may be eliminated by a proper choice of the counterterms in eq. (2.13), amPn;p

RI2

-&j (R;m~pn~np + R;n-am8,,,p)

_ -

,

(2.17) R amPvl+v _ -

1234

~R;“-pnSnlpS(y

-&

(9 is the symmetrization

operator),

+

K)

and passing to new variables:

1 ampn _ @, - V~““” - 4B Ha,, a,,,,, ,

@umPnYlKP 1234

_-

Va;yyY’KP

-

“;2”p”‘p6,6(y

+

K)

1 -2

SB

v:m-p”&,p%’

olm;n= Va”;” @I 1 CYRZ;lZp =v @23

7

+ K) + &

@ amOnvl_ 123

ampnyl -

VI,,

v%%,&p6(~

+ p)a(?’ +

K),

D.A. Garanin et al. / Magnon-magnon interactions ~ r n f l n ~ / l ; P = v~ml3nyl;P ~-

vaml3nylx

3

- - 1+4,2,3 ~ m p

123

123

531

1

2 am;n

16B 2 6 m 6 ( ~ + y)[ V2+3 6rap +

vam;PS ] --4 ~mnl

(2.18)

where the symmetrization corresponding to the symmetry of the Hamiltonian with respect to spin variables must be carried out. The new variables qOsatisfy homogeneous differential equations, __

O~ c~m/gr~. l~$Jrfln

<3r/ (2.19) <3~ ~alTI3n'P

*'1

'~=12

*'2

"~12

'

etc., which are solved easily. In particular, the solution of the first equation may be written in the form ~}1 = exp(--'r//~l ) tPl (0) exp(r//~ 1),

(2.20)

where ~_)ctrn~n l/2¢~)?mfln 1/2 1 =Ol j~ , g l rnon = o ~ l f 2 R l m # n ~

1/2 ,

(-1)

-+I/2 =

±i.

With a proper choice of a diagonalization matrix ~ = e x p ( - ~ ) , the "energy matrix" o~m~n becomes diagonal, 1 tP~mt3"(1) = --~-~ e , , 6 , ~ , a ~ .

(2.21)

Here el,2k ~ - [ ( / - I + 2Bik) 2 - (2BJ,)2] 1/2 are the magnon energies of AFEP, which in the long-wavelength region have the form 2 2 s2k 2 ~nk ~ ~nO +

S2 =

2HEa2/z, (2.22)

2 = EIO

H(H +

HD)

e202 = 2HAH E + H D ( H + HD )

where H A = B[3o ~ H E is the anisotropy field, s is the spin wave velocity, a is the lattice spacing, z is the number of nearest neighbours. With the use of eq. (2.21) and the first of the formulas (2.18) one can derive the coefficients of the quadratic part of the transformed Hamiltonian ~*(1):

D.A. Garanin et al. / Magnon-magnon interactions

532

v""~" -k

1 (ft 4B

e,~)6,,,6,~

(2.23)

Now by the usual procedure [4-6] one may introduce Green's functions (GF's) for the upper and the lower branches of the elementary excitation spectrum of AFEP. In the mean field approximation, as it is seen from (2.9), these Green's functions have degenerate poles at the frequency w =/4. Taking into account the chain correlations generated by the quadratic part of the Hamiltonian associated with eq. (2.23), one obtains magnon GF's having the poles at the true magnon energies enk (n = 1, 2). The solution of other equations (2.19) is quite similar and may be represented in the tensor convolution form. At r/= 1 one gets • ~";P(1)

ti~2nn~'/(1)

k'ama'm'k'~nO'n'(ba'm"~'n';P[(I]

=

~1 =

*~2

~12

\v)

k-ama 'm' l,cflnfl 'n'/CT/T '/' (]~a 'm 73 'n 'Y '/' "'1

*'2

"'3

~

123

(2.24) (0),

etc., where K '~m3n= Ol-1/20°tm3n[31/2. In explicit form, K=

M

L =

,

Vl

L/1

M=

,

__/32

U2

,

(2.25)

where /unk/= 1 tv,,,

j

(

Y,,k 1 +

•nk ~1/2

Y, 2e = / ~ + 2Bi~ '

,

-

(2.26) "

Thus the problem of the spin Hamiltonian diagonalization is solved. With the quantities 4(1) known one may obtain the coefficients of the diagonalized Hamiltonian V(1) with the use of formulas (2.18) and construct the amplitudes of various scattering processes. Moreover, it turns out that on the energy shell, the scattering amplitudes coincide within numerical factors and symmetrization with appropriate components of the quantities 4. This circumstance clarifies the physical sense of the latter ("the amplitude tensors"). As is seen above, the amplitude tensors are formed automatically by the diagonalization procedure as invariant (i.e. transformed through themselves) variables. So, despite that the Hamiltonian of the system undergoes intricate metamorphoses when being diagonalized, the physically observed quantities- the scattering amplitudesare transformed, as in the case of boson and fermion systems, in a simple manner with the use of u - v coefficients (see eq. (2.26)). Note also that if, the counter-terms being specified by eq. (2.17), we express the operator ~ (see eq. (2.13)) in terms of boson variables through the Holstein-Primakoff or the

D . A . Garanin et al. / Magnon-magnon interactions

533

D y s o n - M a l e e v representations, then we will see that ~ is a quadratic form in boson operators. In this sense the unitary transformation of the spin Hamiltonian suggested in this paper is the direct spin analog of the u - v transformation. Nevertheless, this "boson analogy" alone is insufficient for the determination of both matrices: Rarn~n;P and ~RamI3nylKp *'12 *1234 " The explicit form of the quantities ~ ( 0 ) being given by eqs. (2.9) and (2.18), one may fulfill the transformation (2.24) and obtain final expressions for the quantities of interest: ~Tm;P(1 ) = -- ~H ( - 1 ) m P p l m , am;np

~23

H 4B 2 6 ( n + p - 3) 6 ( m - 2 ) P 7 m

(1) =

H

dl)aml3nyl(1. ~ = 123

\'1

t

l"~m+n+lllgam() Onyl

[1 + k - * ,

~

J--I

~-'23

'

(2.27) ~pamfln~.l;P(l] 123

=

\*)

~

) =

H

[2 + ,

(__l~mlt__l](m+n+l)(p+l)pamQt3ny ! ,

J\

)'

1234

23

'

1 4B

(f)amflnylKp(l]

1

=

Y*]

'

1 [1 + ( - 1 ) m+"+'+p] ~m°"nv'KP 32B 2 ~12 434

•

Here t~m _ _

Pk

m

,,

1,m-1Vml,

= P k = u,,~: + t-- )

.

,

(2.28)

the matrices G ~m°" and Q amt~n a r e composed of blocks G m" and Qmn having the diagonal (a/3 = 1) and off-diagonal ( a f t = - 1 ) elements equal pairwise: Qo, rnt~n = Qmn;,~o; G,~rnt~n = Gmn;e,t~. In explicit form Q am2;+ = (--1) m

1

U,,kV,,k2 + (--

l)n-1 VmklUnk2, (2.29)

0 7 ; ; - = ( - 1 ) ,.+n U,,
where v = + 1.

(2.30)

534

D . A . Garanin et al. / M a g n o n - m a g n o n interactions

3. The spin wave scattering amplitudes in AFEP

In the present section we describe all basic magnon-magnon interactions in AFEP. These interactions are divided into the direct ones (the scattering amplitudes are linear in the coefficients of the Hamiltonian) and the indirect ones, which are constructed in higher orders of a perturbation theory taking into account virtual magnon states. At first, we will consider the amplitudes of direct interactions and then with the use of the latter we will find additions to them due to indirect ones. As it is known [4-6], in the approach based on the spin-operator diagram technique, the amplitudes (including direct ones) are the sums of terms corresponding to various spin-operator combinations in the Hamiltonian. In particular, the amplitude of the scattering of lower-branch magnons by fluctuations of the longitudinal component of spins belonging to the 1st sublattice is given by the graphs represented in fig. la and equal to (i) 1;1(1;2) 1 = 2B(V~2'-1;' + V21 ' + ' ; 1 ) + 2 ( V 2 ' - ' _--2B(qb ~1-';1 + (/)21l+l;1) -+- ~

1

+V~'+')

( E l ' -- El2 ) •

(3.1)

For the similar amplitude associated with fluctuations of the spins of the 2nd sublattice (see fig. lb) one gets (b 21;1(1,2)= 2B(V~2J- 1;2 + V211 + 1;2) -- V l + 2 • = 2n(qb?21-1;2 + (/)211+1;2).

(3.2)

As follows from eqs. (3.1) and (3.2), the scattering amplitudes are practically determined completely by the tensors t~(1) introduced in the previous section.

Fig. 1. The amplitude of the lower-branch magnon scattering by fluctuations of the longitudinal spin components of the 1st (a) and the 2nd (b) sublattices.

D.A. Garanin et al. / Magnon-magnon interactions

535

An exclusion is the last term of eq. (3.1) vanishing on the energy shell. This term arises from the asymmetry of the first graph in fig. la with respect to the wavevectors k 1 and k 2. Just the same asymmetry arises also in the four-magnon scattering amplitudes. Note also that according to adopted cipher notation of wavevectors (see eq. (2.7)) i =- ki for the incoming arrow (the magnon annihilation) and i =- - k i for the outgoing one (the magnon creation). For example, in eq. (3.1) 1 =- k I and 2 --- - k 2. Similarly, with the use of the "diagonalized" version of the spin-operator diagram technique [4-6], one may construct the amplitudes of all other direct scattering processes. For the fluctuational processes of magnon scattering and interbranch transformation of magnons by fluctuations (e~k-+ 82q and vice versa) with the use of eq. (2.27) one gets 1 qOPm;,,(1;2) = (-1)(m+n)P+ l Gn~2 ,- + - ~ (~°ml -- En2)~mn p .

(3.3)

The "explosion" fluctuational amplitudes corresponding to creation of two magnons by a fluctuation are written in the form p 1 q~;m.(;12) = ~-~ (-1)(m+")P+IG~2 n'+ , (3.4) qOPmn;(12;) -__ ( 2 B ) 2• p;m~(;12) . The quantities G are written as ll,v

G12

....

entering eqs. (3.3) and (3.4) in the longwavelength region

1

__

[H2(/~1+2-

2 B ~

1/~.1- 1/~.2) -- 1 V e l l e 1 2 1 ,

1

[-vH2(AI+2G22;v 12 2 B e2X/-k-S]-~1822 1 _

2 .(E20

2 _

ElO ) +

o12; = & ( < 2 - . 8 , , ) 12

4X/--el I 822

1 E21 822]

½A 1 -

(3.5) ,

c21; '

½A2)

~ 12 =

0(8 1 - .81 ) 4X/'-e'21812

It is seen from eq. (3.5) that among the fluctuational amplitudes the interbranch ones having the order of the exchange energy J0 are dominant. In particular, on the energy shell (elk = e2q ) one gets 1;2 = ½ ( - - 1 ) P - 1 ~ 0

•

(3.6)

536

D.A.

The

Garanin

direct four-magnon

et al.

/ Magnon-magnon

scattering

interactions

amplitudes

are due to the terms

A S z" A S z~, S+~S -~, s~mst3" A S ~p and S~"S~nSvlS~P of the diagonalized Hamil-

tonian and are given by the general expression

C19mn;tp(12;34) =

½11 +

[

l~m+n+l+p](l,-~mn,+[~tlp,+

,-~,

1\~12

i,-~ ml, - [ ~ np , -

+ ~13

-~-24

[.~ np , - i , ~ m l , -

"~ ~ 2 4

¢,~lp,+

~34 ~13

+ ~34 +

nl. -

ran,+

Q12 -

G23 Q~np, +

G14P'-Qnl'-'~23 ,

1 2B

(eml

"~- ~ n 2 -

el3 -

Ep4)~mnlp

'

(3.7)

which reflects in an explicit form the symmetry of the Hamiltonian which forbids the four-magnon processes with one or three lower-branch magnons involved. The decay amplitude q~m;,tp(1;234) may be obtained from eq. (3.7) by throwing off the last term, dividing by 2B and changing the sign in the index where the index n occurs (i.e. GI~'----~ GI~'+). The "explosion" amplitudes ~);mnlp may be obtained from ~m;,lp by changing the sign in the index where the index m occurs and dividing by 2B. Note also that the relations qbntp;m(234;1) = (2B) 2@m;,lp(1;234),

(3.8) ~mnlp;(1234; ) = (2B) 4~;mntp(;1234)

are valid. The quantities QI2"'" (see eq. (2.29)) entering eq. (3.7) in the longwavelength region (e k ~ tOE) have the form Qmn,v

12

onn,•

12

z

_

mn;v

-G12

m~n,

/Jo,

(3.9)

(-- V)n - 1HE 2 ~ / e ~ 1 en 2

With the use of eqs. (3.9), (3.5) and (3.6) and the long-wavelength expansion 2 2 A k ~ 1 - ( s k ) / 2 t o E (s is the magnon velocity) one gets

P0

q~lml(12;34) = 4x/--g1e2e3e 4 [e~0 - a(Ae) 2] - Ae/2n,

P0 q~22;22(12;34) = 4V e I e 2 e 3 e 4 [4e20 -- 3e~0 -- ~(Ae) 2] -- Ae/2B, (3.10)

P0

q~a2;12(12;34) = 4X/-gl ez e 3 e4 × [e~0 + s2(k~ "k3

-{- k 2 " k 4 ) -

E.1E 3 - - E 2 E 4 - - I ( A E ) 2 ]

537

D . A . Garanin et al. / M a g n o n - m a g n o n interactions

~11;22(12;34) =

4V"-gl e2 e3 e4 x

+

s2(kl

•

k 2 d- k 3 - k 4 ) -

e le 2

-

•

Here Ae = 61 d-62 - - e 3 - - e 4 , and the evident indices corresponding to SW branches at the right-hand side of eq. (3.10) are omitted. It is seen that in the longwavelength region the asymmetry terms (the last terms in the first two equations) are small. Note also that in the limit e20-->0 (i.e. HA--->0) the last two amplitudes tend to those obtained previously for the isotropic case. The four-magnon amplitudes corresponding to the processes which do not conserve the number of quasiparticles involved and which are needed for the construction of indirect interactions may be calculated in a similar way. The three-magnon processes in AFEP arise via two different terms in the Hamiltonian: S am A S zp and samsflns 71. In particular, the diagrams for the three-magnon confluence vertex (1 + 1--->2) are given in fig. 2. The general expressions for decay and confluence amplitudes are of the form: H

t/~m;nt(1;23 ) = ~-~ [1 + (--1)

m+n+l

m

hi,+

pnl~rnl,-

](P1 Q23 -k- - - 2 ~ -

13

ioll.')mn,-'~ q- - - 3 ~ - 12

l,

(3.11) q~ t;m(23;1) = 2Bq0m;~t(1;23) . The "explosion" amplitude (~);mnl may be obtained from @,~;nt by changing the sign in the index where the index m occurs and dividing by 2B. In particular, in the long-wavelength region the amplitude of the decay of an upper-branch magnon into two lower-branch ones (2--~ 1 + 1) has the form ~2;11(1;23)=

H

" HE

xl/2

4B ( el--~2~ )

(el + e2 + e3);

(3.12)

the amplitude of the decay of a lower-branch magnon into upper and lower branch ones reads ~v21(1;23) = '

H( HE 4B ~ )

\1/2

(~1 + ~ ' 2 - E3)"

(3.13)

One more type of magnon-magnon interaction processes in AFEP are the

,( Fig. 2. The amplitude of the three-magnon confluence process.

538

D.A.

G a r a n i n et al. / M a g n o n - r n a g n o n

interactions

three-magnon fluctuational ones which are due to the terms S am A S zp, s'~mst3"S:'t and s"mst3"S~'t A S ~p in the Hamiltonian. In particular, the vertex

corresponding to the confluence of two magnons into the third one when interacting with a fluctuation is represented in fig. 3. As in the other cases, the majority of terms corresponding to six graphs represented in fig. 3 cancel each other and the result turns out to be expressed through the amplitude tensor ~otm~n'gl;p (see eq. (2.27)). Finally, p

.

n

m

m

nl,-

q~m,;,(12,3) = ~-~ (--1)(m+"+t)(P-'){[2+ (--1) ]PI Q23

+ [ 2 + ( - 1 ,v q,P nzQ mr',3 + [ 2 + ( - 1 )]t p t3Qm,~ ,z' + } .

(3.14)

Herefrom with the use of eqs. (2.28) and (3.9) in the long-wavelength region one obtains

q b f l ; , ( 1 2 ; 3 ) = ( - - 1 ) P -1

H H E/2 8 B ~ HH

q)P2;2(12;3)

= ( - 1 ) p-1

'

3/2 E

(3.15)

8 Bv'-ff 1 E"2 E 3

(the rest amplitudes of this group are small a s e l , 2 , 3 / n E in comparison with eq. (3.15)). Now we will proceed to the determination of indirect contributions to the scattering amplitudes. In particular, in the longwavelength region the contribution to the four-magnon scattering amplitudes in the second order of a perturbation theory with respect to the three-magnon ones is accounted for by the addition of the term 3H 2 into the brackets of the first two equations of (3.10) (the analogous contributions into the cross-scattering amplitudes ~,2;12

Fig. 3. The amplitude of the three-magnon fluctuational process.

539

D . A . Garanin et al. / Magnon-magnon interactions

4

2"

~

2

Y

Z

Fig. 4. The indirect contributions into the amplitudes of fluctuational scattering processes.

and q~11;22may be neglected). The additions to the fluctuational amplitudes are combined from the three-magnon amplitude and the amplitude of the creation/ annihilation of a magnon by a fluctuation (see fig. 4). These additions to the scattering amplitudes ~.P;. given by eqs. (3.3) and (3.5) are equal to H ( - - 1 ) n-1 4 n V , enlSn2

(3.16)

As a result, at zero Dzjaloshinsky field (HD = 0) the fluctuational scattering amplitudes are independent of a magnetic field. On the energy shell (e k = eq) ~P:l(k;q) =

1 (2sZk • q + HHD). 4B eX/-g~k eq

(3.17)

The corresponding additions to the interbranch fluctuational amplitudes (see eqs. (3.3) and (3.6)) are small. The indirect contributions to the scattering amplitudes are of the greatest significance for the three-magnon fluctuational processes. These contributions for the three-magnon confluence process inside the lower branch (1 + 1 ~ 1) are shown in fig. 5. The first graph in fig. 5 contains the full four-magnon amplitude which includes the three-magnon one in the second order. In the rest of the graphs fluctuational amplitudes are taken bare because the indirect additionals are taken into account by the first graph. The resulting expression for the amplitude of the process under consideration reads

HH3/2

q~fl"(12;3) = (-1)P 2 B ~

[

1

2

e'° +23He E4

el +2

e2

2 __ 2 -}- 2 __ 2 + E2+3 E1 El+3 E2

e3

(3.18)

2-E l + 2 -- E23 P

P

KH

II

K,~

4

K1

K2

-~

3

I(2.

~..~pK L ;¢'t

Fig. 5. The indirect contributions into the amplitudes of three-magnon fluctuational processes.

D . A . Garanin et al. / M a g n o n - m a g n o n interactions

540

where E4 corresponds to the lower branch, and the energies e1+2, el+ 3 and e2+ 3 to the upper one. At small wavevectors (el,2, 3 - e~o) the last three terms between round brackets have the order (e~o/e20)2~ 1 but must be retained because the first terms of eq. (3.18) cancel each other out to values of the order H I ( H + HtO. At larger wavevectors (e~o ~ e~,2,3 <~ E20) all terms between square brackets (but the unity) are small. At large wavevectors (e2, 3 ~ e20) the two last terms in eq. (3.18) become dominant. The general expression for the amplitude qbp12;2 of the confluence of a lowerand a higher-branch magnon into a higher-branch magnon by a fluctuation fl (1 + 2--> 2) is cumbersome and will not be written here. This process becomes operative at temperatures T ~> e20 and in this region gives the same contribution into the damping of the lower-branch magnons as the confluence process inside the lower branch (1 + lf--~ ~ 1).

4. The relaxation frequencies of the lower branch magnons in AFEP In this section we calculate m a g n o n - m a g n o n contributions into the damping of lower-branch magnons in A F E P in the lowest orders of a perturbation theory. The majority of the results deals with the case el0 - e~ < T which is typical for microwave experiments. The simplest among the magnon scattering processes are those of magnon elastic scattering or transformation into the other branch magnon by fluctuations of longitudinal spin components. For weakly diluted antiferromagnets, the longitudinal spin correlation function ~ i p~p2 = ( A S zpl AS~ p2) ( P I , P2 = 1, 2 are the sublattice indices) may be calculated in the mean field approximation by summing up the chain diagrams. For small values of the canting angle O with the use of the diagram technique for diluted magnetic systems (see e.g. ref. [18]), one arrives at the following results for the Fourier transformed correlators: 2[PqP = cB'

~pap2

1 1 - (cB '/3o¢q)2

= cn t

1 --

+ c(1 - c ) B 2

1 + (cB'/3Jq) 2 (1 - (cB '/3cr~q) 2 ) 2

c B t/3cr~q (cn'/3~rSq) 2 + c ( 1 - c ) n 2

'

(4.1) (1

2 c B ' /3~ q - (cn'/3~q)2) 2 '

P l ~: P2 ,

where /3 = 1/T, c is the concentration of magnetic ions, 1 - c = c' ~ 1 is the vacancy concentration. The magnon damping due to fluctuational processes of various types (elastic scattering and interbranch transformation) is given by the expression

D.A. Garanin et al. / Magnon-magnon interactions

n

"Ym;n(k) = O0

f (2,tr)3 dq E

PlP2

~PlP2~)Pl k-q m;n ~)P2 n;m "IT~(Eq -- ~k)

541

(4.2)

where o0 is the unit cell volume. As is seen from eqs. (3.3) and (3.6), the ftuctuational scattering amplitudes (m = n) do not depend on the index p, whereas the amplitudes of the interbranch transformation (m ~ n) by fluctuations of the " u p " ( p = 1) and the down ( p = 2) sublattices have opposite signs. As a consequence, the relaxation frequencies corresponding to these two types of processes have substantially different temperature dependences in the vicinity of the phase transition point T N. In particular, for the damping of lower-branch magnons at small wavevectors with the use of eqs. (3.17) and (4.1) one gets the result

n

r/3

~/';l(k) = 16-~-B2

( 1-

cB' c(_l-- c)B 2 ~ 4(sk) 4 "~- (HHD) 2 sk cB'fl~ o + (1 - cB'flJo) 2/ oJ~ e"-~k' (4.3)

where 77 = ol/3toE/S is the structure dependent factor (7 = 21/331/2 for sc lattice and 7/= 2 for bcc lattice). In eq. (4.3) the term in large brackets taking into account longitudinal correlations is singular as 1/~- (~- = (T N - T)/TN) near the phase transition point (T---~ TN). At larger wavevectors (ek ~> e20) the process of the lower-branch magnon transformation into an upper-branch one becomes allowed by the energy conservation law. With the use of eq. (3.6) for e~ i> ez0 one obtains fl 'Yl;2 =

cnt ,73 ( 4~--B2 1 + cB'[3~o +

c(I~c)Ba

2 \3/2 ~ ek2 (1-e--~z2 °) .

(1 + c B ' f l ~ o ) 2] to--~

(4.4)

ek

One can see that the fluctuational interbranch transformation process taking into account longitudinal correlations does not lead, in contrast to eq. (4.3), to an additional singularity at T N. Nevertheless, in general, this result exceeds the previous one by a factor of the order of (tOE/ek)2 >>1. Now we will proceed to the consideration of three-magnon processes in AFEP. The only process allowed by the energy conservation law is the confluence of two lower-branch magnons into an upper-branch one (1 + 1--->2). The contribution of this process into the damping of the lower-branch magnons is of the form:

"Yll;2(k)=°of ( ~ ) 3 qblv,2c192;lln;ln,+q(nq + 1)~r~(e, + eq- ~,+q).

(4.5)

H e r e the amplitude ~2;11 is given by eq. (3.12), and the conjugate one reads

542

D.A. Garanin et al. / Magnon-magnon interactions

(]~lI ;2 = 2 Btb2;l l- After integration over the angular variables expression (4.5) is reduced to a quadrature, "Yll;2(k)- I ~ B

sinh

toESkek2

E+ ×

f

(6, + %): +

deq s i n h ( e q / 2 T ) sinh((e,

6q)/2T) '

(4.6)

e_

where in the limit 61o ~ 620 the upper (e+) and the lower ( 6 ) boundaries of the thermal magnon energy eq a r e the following:

2 e+

-

(4.6b)

e2°

2(e k -Y-s k ) "

In the case T ~ 6_ from (4.6) for the magnon damping one gets the result ylt

773 H2 ( ~ E ) 2

;2

-8--~-B

sk

6 exp(--~-){1-(6+t2exp[-~(e2°)2]/ \ 6 / \710 / dJ"

(4.7)

For "moderate" temperatures: e ~ T ~ e+ the damping of the lower-branch magnons is of the form "fiT/3 H2 ( T ) 2 %,;2- 2 ~ e k ~EE "

(4.8)

For high temperatures (T>> e+) one gets

],/3 ( H ) 2 (F.2O)2T ")tll;2-- 8-~--B ~1o \toe / "

(4.9)

Among the four-magnon processes in the temperature range T < e20 the scattering within the lower branch (1 + 1---> 1 + 1) is dominant:

~/11;11

--

2 + 3H2) 2 T 2 r/ 6 (ex0 3 × 29~B 2 to 4E e k A(ek/elO) ,

24f t _t21

(4.10)

1

A(x)=l+~-~

~

~/ x2--- ]- l n t ,

A(1)=1,

A(oo)=4.

l/x At temperatures T ~> e20 the major role is played by the processes of scattering of lower-branch magnons by upper-branch magnons (1 + 2--> 1 + 2). In the

543

D . A . Garanin et al. / Magnon-magnon interactions

region 62o ,~ T ,~ e_ one gets --

"~12;12

q'l"06

~-'O2

T(

6

4

4

(4.11)

620/ \ O.)E/

where

f(x)

= (1 + x2)-2{(1 + 4x2)(V1 + x 2 - x) 2 + ( 1 / 6 x ) [ 3 ( V ~ ' + x 2 + x) 5 - (V~ + x 2 + x) + ~-(Vc]- + x 2 - x)S]},

fl0)=l,

f(~)=16/5.

(4.12)

In the range e_ ~ T ,~ e+ the damping of the lower-branch magnons due to the " c r o s s " process is of the form

3'12;~2

72-~-Bz WE ~E

,

(4.13)

dz - ln(sinh z ) = 1.49. sinh z z

(4.13b)

In ~ _

- C

where

1( )2 C=~+

~(3)

z2

- --

~r

0

A t higher t e m p e r a t u r e s (e+ ~ T "~ O.)E) one gets

T/6 e 2 (T)3 ~/12;12 = 72-~---Bz toE where x =

x2.+ 4[ln(2_~oT) _ C + X2 +-----'~

g(x)]

(4.14)

sk/elo,

g(x) = x~/1 + x

2 1 + 4x 2 ln(V~'+ x 2 + x) 3 + 4X 2

and T o --- e2z0/2el0 . A t t e m p e r a t u r e s T ~> e the process of transformation of two lower-branch m a g n o n s into two super-branch ones (1 + 1--> 2 + 2) becomes operative. In the ranges e ~ T "~ e+ and e+ ~ T ~ toE the corresponding relaxation frequency Y~1;22 is given by eqs. (4.13) and (4.14) but without the term containing ff(3)(3/'rr) 2 (see eq. (4.13b)). T h e a b o v e given expressions (4.10)-(4.13) describe the m a g n o n damping due to f o u r - m a g n o n processes in the long-wavelength region elk ~ ezo p r o b e d

D.A. Garanin et al. / Magnon-magnon interactions

544

by microwave experiments, as was mentioned at the beginning of the present section. In the opposite case elk >>e20 the characteristic temperature T o (as well as characteristic energies e±) falls out of consideration, and for T~> e20 the earlier obtained results [10] for magnon damping in the isotropic case are recovered. In the range e k ~ T ~ toE the summarized contributions of two processes 1 + 2---* 1 + 2 and 1 + 1 ~ 2 + 2 make up

y(k)

-

ln--+A

36~---B2 tOE

,

ek

(4.15)

where A = ( 8 / ~ 2 ) [ ~ ( 2 ) - ( 1 5 / 1 6 ) ~ ( 3 ) ] . - ~ 0 . 4 2 . In the case of high magnon energies T ~ e k ~ tOE, the two four-magnon processes under consideration give y(k) =

1v7/6 e k ( T ) 4 43"-~-~B2 ~

(4.16) "

Eqs. (4.15), (4.16) are essential for the interpretation of neutron scattering experiments in antiferromagnets. At low temperatures ( T ~ e20), the three-magnon and the interbranch four-magnon processes result in exponentially small damping. As the quantity Yl~:H (4.10) contains small numerical factor, the temperature dependent part of the lower-branch magnon damping for not too pure crystals may be governed by the three-magnon fluctuational process inside the lower branch (1 + 1 ~ 1) (see fig. 6): fl

in2 Z |C d p d q ),--

")/ll;1 ~- C ~

m

m

-1

m t~11;lt~l;11np(nq ÷ 1 ) n k

DO d ~

~t~(ek + eq - - e p ) ,

(4.17) where c ' = 1 - c ~ 1 is the vacancy concentration. For e ~ 0 - e k ~ T in the amplitude ~1~;1 (3.18) the second and third term are small; at T ~ T o the denominators of the two last terms are close to e~0. In the temperature range under consideration a simple calculation gives

.2 2[

"Yll;1 -

48rrB

w3

1 +

t

5

-~2o" + ~ \ ~ 2 o

/ J

"

(4.18)

:L

Fig. 6. The self energy part corresponding to the three-magnon fluctuational process.

D . A . Garanin et al. / Magnon-magnon interactions

545

At the temperatures T >> e20 one must account also for a process involving fl . . . upper branch magnons (1 + 2--->2). In the hmlt T >> e20 this process gives the same contribution into the relaxation frequency: y12;2~ f' f' y11;1. If T ~> To, then the pole character of the amplitude (3.18) is revealed. This means that the "pure" three-magnon process is switched on. In this region these two types of three-magnon processes may not be considered separately; the rescattering of thermal magnons by fluctuations must be allowed for in all orders of a perturbation theory.

5. The three-magnon processes in AFEP with rescattering by fluctuations In the temperature range T >> g20 for not too pure crystals the damping of thermal magnons in AFEP is determined mainly by the process of interbranch transformation by fluctuations (1 f-~l2) (see eq. (4.4)): fl ¢(.OX ~ V , ; 2 i, 1 - -

fl "~2;l(O.))~'~(tO)~---

~

3 "Y]

2 O)

~

-

,

-

O)E

cB'

c ( 1 - c)B 2 ~------ l+cB,/3~0 + (i+~-~,--~o)2

(5.1)

] ,

e20"CtO~tOE.

This damping must be allowed for in the Green's functions entering the three-magnon confluence loop. Besides this, the process of the type 1 + 2---> 1 (the confluence of a lower-branch and an upper-branch magnon into a lowerbranch one) which is otherwise ruled out by the energy conservation law becomes operative. And, at last, the three-magnon confluence vertices are renormalized by the ladder-type sequence of diagrams. Concrete calculations may be carried out in the framework of the Keldysh diagram technique [19]. In the "turned" version of this technique (G ~2 ~- G R, G 21 ~-- G A, G 2 2 ~ G °, G I1 ~ 0 ) both the Dyson equations and the ladder equations for the vertices are diagonalized with respect to Keldysh indices. The self energy part ~ R allowing for two types of confluence processes is given by the sum of four diagrams represented in fig. 7. The ladder equations for the vertices A2;~l and A1;~2 are given by fig. 8. In figs. 7 and 8 the "dressed" retarded and advanced Green's functions are given by the following expression: •

Cm'

(k, ,o) = 1 " -- ~m~ --

-1

(5.2)

the distributive GF, G o, being expressed through them by the spectral identity

546

D . A . Garanin et al. / M a g n o n - m a g n o n interactions

2

.~

-4- .z 2_±

ha.

Fig: 7. The self energy part ~R of the lower-branch magnons corresponding to the confluence process with rescattering of intermediate magnons by fluctuations. Each arrow (Green's function) is supplied with Keldysh indices (1 or 2) at the ends and the magnon branch indices (1 or 2) in the middle.

G D = (2n~, + 1)(G R - GA),

(5.3)

where n o = ( e x p ( w / T ) - 1 ) -1. It is seen that because of the interbranch transformation of magnons by fluctuations the ladder equations for the vertices Az;~l and A,;12 are intertangled with each other. These integral equations are in fact algebraic ones since the fluctuation transformation amplitude qoP1;2 is constant to a high degree of accuracy (see eq. (3.6)). The other simplification occurs in "bare" three-magnon confluence vertices (see eqs. (3.12) and (3.13)) which for e k ~ ep "~ eq depend only on the energy of the "probe" (incoming) magnon: A~°) ~ q~2.11~ 2;11

,

)1/2

H wE 2B \-~-~k/

~- Akm

a~0 ) 7 1 ; 1 2

--- A1;12 •

(5.4)

Note also that conjugate vertices All;z and A12;I are equal to 2 B A k. Having introduced the designations A1;12 ~---A AR ,

Az:11=__A~ R

~~

~~ ,= "3~ , ~ , ~

(5.5)

+'~L.~/,~.A i .

Fig. 8. The ladder equations for the magnon confluence vertices.

D.A. Garanin et al. / Magnon-magnon interactions

547

one obtains from the diagram equations given in fig. 8 the system of coupled algebraic equations:

A AR ~"

A k d-

7ARAAR

~1 ~'2

A2AR = Ak + 7ARAAR ~2 "'1 '

,

(5.6)

where

z A R = l J o ~ O f ~dp

GA(p, t o , ) G R ( p + k , to,+to ) (5.7)

R = 1 ¢o2 eOo

f

dp

G2A ( p , . ,

, ) G R ( p + k , to , + .,).

The solution of the system (5.6) reads

AAR = Ak

1-

1 + Z AR 7AR7AR

~1 ~2

AR '

A2

1 + Z A~ = Ak

7AR7AR

1-- ~1 ~2

(5.8) "

Now one may write down the expression for the self energy part X R (see fig.

7):

f d,o' f ~-)3 dp

~vR=½i-~--O

0

2BAk[(GA(p, to') GD(p + k, to' + w)

+ G ~ ( p , w ' ) Gz(p+k,a o9' + t0))AAR+(I<=>2)] .

(5.9)

SO as the "dressed" vertices A AR (i = 1, 2) do not depend on the wavevector p, the integral over p in eq. (5.9) with the use of eq. (5.3) may be expressed through the quantities Z AR already introduced in eq. (5.7):

.,y R

-

2~2BA~ f do)' i¢ . { 2 + Z~ R + Z AR _ 2) -~-(no,,-n,o+.,,)k -I~-~-z~eR •

(5.10)

The integration in eq. (5.7) gives

zAR = ¢i + iAi , where

vz 2 i n ( ( / - 7 z/z_)2 + v2z__~6~ A1'2 = 4x

\(1

-7 Z / Z + ) 2 + p 2 Z 6 ] '

2

~1,2 = VZ2x [arctan(2vxz4/A1,2) A1, 2 = (1 -7- z/z_)(1 -7 z/z+)

+

"rrO(-Al'2)]

+ /)2z6 ,

(5.11)

548

D.A.

G a r a n i n et al. / M a g n o n - m a g n o n

interactions

2

and z = t o ' / T o, x = sk/elo, z~_ = e ± / T o = V ~ + X 2 -I- X , T O = e20/2el0. In eq. (5.11) the numerical factor v is the measure of the rescattering effect:

/, m_

2y(to=To) elo

-

~73 [ 83

2

cB'

2 ]e20 (e20t 3.

c(1-c)B

1 + c B ' f l J o + (1-+~'-7~0)2J w--~ e~0/ (5.12)

In particular, for FeBO 3 (toE ~ 400 K, e20 ~ 20 K, el0 = 1 K) for only concentrational fluctuations v = 102c ' (c' --- 1 - c), so if the concentration of vacancies c' is of the order of 1%, the relaxational broadening of intermediate magnon states caused by concentration fluctuations is of the order of the "probe" magnon energy e k. The resulting expression for the damping of lower-branch magnons may be written in the form ac 773

H 2

f

to

r2

dto' sinh2(w'/2 T) q~(to'),

(5.13)

~ie2)(2 + ~1 "~ e2) -~ 2A, B2-- A ~ 2 - A~el ) - - ~ - - ~ 1 ~ + ~--£-22)~+ ~-~-~2 + ~2~1-~ --2 .

(5.14)

_Fk = - I m ER(k, ek) -- 32-~-B Tskto 2 0

where 2x ~ -- ,rrvz2

((1-

In eq. (5.13) we take into account that actually for "probe" magnon energies e k the inequality e k ~ to', T is fulfilled. In this sense the last expression for the magnon damping due to three-magnon confluence processes is simplified in comparison with the previously obtained one in the pure limit (4.6). A more important difference consists in the replacement of the "window" function q~0(to') = 0(to' - e_) 0(e+ - to') which is actually adopted in eq. (4.6), by the function q~(to') given by eq. (5.14) and allowing for the rescattering effect. The structure of the latter is rather complex and depends crucially on the rescattering parameter v and dimensionless wavevector x = sk/elO. In the case v ~ 1 the function q~ describes the two "windows" of the three-magnon confluence process: a slightly smeared "clean window" q~0and an additional "dirty window" corresponding to higher energies w' of magnons involved. Generally, the maximum of the "dirty window" corresponds to v z 2 = - 2 y ( t o ' ) l e ~ o - 1, i.e. t o ' - - Tolv L/2. It is seen that in the clean limit (v---> 0) the "dirty window" goes beyond the Brillouin zone edge tomax~ toE and falls out of consideration. However, passing to this limit may not be traced quantitatively since the analytical result (5.13), (5.14) is valid only in the longwavelength region (to' < toE)" The concrete form of the "dirty window" of

D.A. Garanin et al. / Magnon-magnon interactions

549

the three-magnon confluence process in A F E P is controlled by the " p r o b e " magnon wavevector. In the limit x = sk/elO ¢ 1, the " a r e a " S = J" q~(z) dz of the "dirty window" is strongly reduced as the function ~o acquires an additional factor x 2: 4x 3 /,Z 2 ~P-- 3"rr l + ( v z 2 ) 2 '

Z~>Z_+.

(5.15)

In the opposite limit (x >> 1) the function q~ is close to unity in a wide region of frequencies: ~ 1 - - -

- -

Iv

(2x) - l ~ v z

l+ln 2

+

2x/,z 2

2~2x,

-27

/,z z

'

z>z+~2x>>l.

(5.16)

The "wings" of the "dirty window" are represented by the expressions

4x/,Z2"rr

~

[

1+

\~/

j,

vz2~min(1,(2x) -1) , z>>z+_,

(5.17)

and 4x

x2

~ - 3~rvz 2 l + x 2 '

vz2>max(l'2x)"

(5.18)

One can see from (5.16) that with the increasing of x the "clean" (z_ ~< z z+) and the "dirty" ( z - - v -1/2) windows of the three-magnon confluence process tend to merge together. This occurs at z + ~ 2 x - ( 2 v x ) -~/e, i.e. 2X-

/,-1/3

One more manifestation of rescattering processes in our result (5.13), (,3.14) consists in the function ~ having non-zero values at z < z_ (i.e. oJ' < e_): ~-

8X/'Z 4

(5.19)

T~

This fact actually corresponds to taking into account three-magnon fluctuational processes of the types 1 + 1 f-~l 1 and 1 +2f-~l 2 considered above and is important at low temperatures ( T ~ T O=- e~0/2el0 ), where one gets from eq. fl fl fl (5.13) the result 'Y11;1 + Y12;2 ~ 2711;1 (see eq. (4.18) at T~> g20)" At large values of the rescattering parameter v (v ~>min(1, (2x)-3)) the interval between two "windows" of the three-magnon confluence process in A F E P disappears. In the large /" limit the function ~ is simplified to

550

D.A. Garanin et al. / Magnon-rnagnon interactions 4

8x uz ~o ~ - ~r 1 + 4 ( 1 + x 2 ) v 2 z 8 '

v> 1

(5.20)

One can see that here the characteristic frequency to' ~ To/1J 1/4, and the " a r e a " of the "window" S ~ v-1/4 are diminishing slower with the increasing of v than the corresponding parameters of the "dirty window" in the case v ~ 1. The behavior of (5.20) at small values of the " p r o b e " magnon wavevector (x ~ 1) is trivial; an additional "hydrodynamic" factor of the type x2~ (sk) 2 does not appear (compare with eq. (5.15)). T h e temperature and wavevector dependence of the lower-branch magnon damping (5.13) is explained by the properties of the function ~ considered above. In the case v ~ 1 and T ~ T Oexpression (5.13) becomes the sum of two terms: the three-magnon damping in the pure limit (4.7) and the doubled last term of formula (4.18). In the temperature range T o ~ T ~ To/u 1/2 (only the " c l e a n " window of the three-magnon confluence process is populated by thermal magnons) one returns to the result (4.9). If the inequality T O To/lpl/2"~ T ~ f o E is satisfied, then both "windows" are populated and the magnon damping F exceeds its clean-limit value (4.9) acquiring an additional factor of the order of v-1/2 > 1. In particular, at small wavevectors (x ~ 1) in this temperature range with the use of (5.15) one obtains:

773 T(H)2(E20)211 ~- 1 I'-- 8VB ,El(l~ ,fOE/ k ~

( s k )2] ,~lo/ a'

v~l

(5.21)

(compare with eq. (4.9)). At low values of the rescattering parameter v the last term of eq. (5.21) may be large. At x - 1 this term reaches its maximum and then diminishes as (vx) -1/2. If the value of v is so small that the integration in eq. (5.13) is cut by the Brillouin zone edge, then this diminishing of F at x ~> 1 is even more pronounced: F ~ / , ' - l / 2 x - 1 In the high-temperature limit ( T ~< foE) the magnon damping F as a function of v reaches its maximum at 1.,-1/2 ~----foE/To =_ 2foEel0/e202 (the maximum of the "dirty window" coincides with the edge of the Brillouin zone). So, the most pronounced rise of the lower-branch magnon damping in A F E P due to the rescattering effect at small values of the rescattering parameter (v < 1) may be observed in materials with high values of the exchange energy toE and relatively low values of the quantities el0 , e2o and T O-= ez0/2e10 2 (e.g. in a - F e 2 0 3 in the temperature range slightly above the Morin point TM). If, on the contrary, the rescattering p a r a m e t e r is large (v > 1), the resulting lower branch magnon damping F is lower than in the clean limit and diminishes with the increasing of the probe magnon wavevector. With the use of (5.20) one gets

D.A. Garanin et al. / Magnon-magnon interactions

F=

1 3 __ 16'rrB ~/2(1 + X,'~)

T (H)2(~Eo)

2(~K0) 5/4

551

Toll 11/4 ~ T .

/.pl/4

(5.22) T h e w a v e v e c t o r dependence of the lower-branch magnon damping in A F E P in the high-temperature region for different values of the rescattering p a r a m e t e r v calculated numerically according to (5.13), (5.14) and (5.11) is represented in fig. 9. T h e r e are several different types of t e m p e r a t u r e dependence of the lowerbranch m a g n o n damping in A F E P depending on the relation between the p a r a m e t e r s e~0, T O and toE and on the values of the dimensionless p r o b e m a g n o n wavevector x = sk/elO and the impurity concentration c ' ~ 1 - c. If T ~ toE and x - 1 then, generally, in the t e m p e r a t u r e range T o ~< T "~ T 1, the quantity F ( T ) increases faster than Foc T (see eq. (4.9)) due to the stepwise population of the "dirty window" in eq. (5.13). The characteristic t e m p e r a t u r e T 1 is analogous to that introduced in ref. [15] and is determined from the condition that the "dirty window" begins to be thermally populated: 2yn(T~, to = T 1) ~ el0 (see eq. (5.1)). Since El0 ,~ toE, this transcendental equation gives T 1 considerably lower than toe (for el0 = 1 K and toE = 1200 K ( a - F e 2 0 3 ) in the case of purely thermal (c' = 0) fluctuations one gets T 1 = toE/3.7). In the region T ~> T 1 the m a g n o n damping has a plateau or slightly diminishes due to the decreasing of the "dirty window a r e a " ( S ~ v l / 2 ( T ) , see eq. (5.12)) with t e m p e r a t u r e . A t higher temperatures, T ~> T 2, where T 2 corresponds to v -- 1

8

dsmping, arb. units

2

i 4 _

]

_

7

2 3 4 X Fig. 9. The wavevector (x =- sk/elo ) dependences of the lower-branch magnon damping in AFEP due to the three-magnon confluence process with rescattering by fluctuations in the temperature region To, Toy -~lz ~ T .~ c% for different values of the rescattering parameter v: (1) v = 0.001; (2) v =0.003; (3) v=O.O1; (4) v =0.03; (5) v=O.05; (6) v=O.1; (7) v =0.3; (8) v = l .

552

D . A . Garanin et al. / Magnon-magnon interactions

(i.e. 2 " y f l ( T 2 , to = T 0 ) ~ - E l O ) the "dirty" and the "clean" windows merge together, and the resulting magnon damping gets smaller than its clean-limit value (4.9). If the probe magnon wavevector is small (k--)0), then the function q~ given by eq. (5.14) has no "dirty window" (see (5.15)), and the characteristic temperature T~ falls out of consideration (this is not the case if the vertex corrections are omitted; here there is a fictitious "dirty window" even at k =0). So, the magnon damping F(T) begins to deviate down from its clean-limit value (4.9) and has a maximum or a plateau at the temperature T2 introduced above. In fig. 10 we represent some examples of the temperature dependence of lower-branch magnon damping in AFEP due to the threemagnon confluence process with rescattering by inherent thermal and also (fig. 10a) by concentrational fluctuations of magnetization. For substances in which T0 - toe (e.g. in FeBO 3, toE = 4 0 0 K , and for el0 = 1 K one gets To = 200 K) there is no place for the "dirty window" in the Brillouin zone; the "clean window" becomes smeared by the rescattering effects already at temperatures T ~ T0. Here the magnon damping F(T) is considerably lower than the lowest-order result (4.9) in the whole temperature range.

magnetization

SW d a m p i n g , arb, u n i t 9

a

msgnetization

SW d a m p i n g , arb. u n i t s

o

0,8

0.8

0,6

06 2

04

04

0.2

0.2

0

,,, I ,,,,11,,~ O0

o,2

I ,,,,,~1,,

0.4

I i1,,,,,,,

0.6

I ,,,,,,,,

o.e

1.o

02

0,4

06

0.8

T/T.,

"

T/TN Fig. 10. The examples (a)-(d) of temperature dependence of the lower-branch magnon relaxation frequency in a model high-temperature A F E P due to the three-magnon confluence process with rescattering by fluctuations; S = 3, toE = 1200K, e20 = 14.1 K; (a) e ' l O : 0 . 5 K , k = 0 ; (b) /z'lO: 0.5 K, x =--sk/elo = 2; (c) e~o = 2 K, k = 0; (d) el0 = 2 K, x = 2. In (a)-(d): (0) the lowest-order result (4.6); (1) the rescattering effect is accounted for only in the GF's of magnons involved, the vertex renormalization neglected, the vacancy concentration c ' = 0; (2) the rigorous result (5.13), c ' = 0 . In (a): (3) c ' = 0.001; (4) c ' = 0 . 0 0 3 ; (5) c ' = 0 . 0 1 .

553

D . A . Garanin et al. / Magnon-magnon interactions magnetization

SW damping, arb. units

magnetization

SW damping, arb. unit£

C 0.8

0.6 1

0.4

0,4

0.2

0.2

0

0

0.2

0.4

0.6

0.8

T/T

" 0

0.2

0.4

06

08

T/]

Fig. 10 (cont.).

6. Discussion

So, we have calculated the amplitudes of the magnon-magnon interactions and the relaxation frequencies of the lower-branch magnons in a two-sublattice easy-plane antiferromagnet. As distinct from the other papers cited in the introduction, the results have been obtained for arbitrary values of the magnon wavevector k including the case s k ~ el0 typical for the parallel-pumping experiments. In view of the comparison with existing experimental data, the most important formulas are those corresponding to the three-magnon confluence process, including the renormalization effects due to the rescattering of intermediate magnons by thermal and concentrational fluctuations. For small values of rescattering parameter u (see eq. (5.12)) and at low temperatures T < To, there are two additive contributions into the lowerbranch magnon due to the pure three-magnon process (1 + 1--.2) and the processes with rescattering by concentrational fluctuations ( 1 + 1 f-~l1 and 1 + 2J-~ 2). The first one (see eq. (4.7)) in the actual range of magnon wavenumbers sk ~>sko = elo( T/To) simplifies to ~7

Tll;2(k) = 8~rB sk

exp -

(6.1)

and lastly diminishes with the lowering of the temperature. The second group of processes gives a contribution proportional to vacancy concentration c' ~ 1 but has a power temperature dependence (Y ~ T6) instead of the exponential one:

D . A . Garanin et al. / Magnon-magnon interactions

554

f'

t,k. ~

3"11:1~, ) + 3'

fl12;2(k) _-

c ' '1"/6

336"trB

O2T 2 (re,fiT) 4

to3

,,-~-2o/ (1 + F(T/e2o))

(6.2)

(see eq. (4.18)) at T ~> ~ez0; the function F(x) accounts for switching on the process involving upper-branch magnons (1 + 2---~2); F(0) = 0, F(oo) = 1, In the temperature range To ~< T ,~ T~ (T~ is the hydrodynamic temperature introduced at the end of the previous section) the magnon damping is provided 2 2 by the pure three-magnon process: 3'll;2OcT H / e l o (see eq. (4.9)), while rescattering corrections of the type (6.2) are small. In the region T ~ T 1 the hydrodynamic situation is realized, and the lower-branch magnon damping has a maximum or a plateau (see fig. 10). In this region for sk ~ elo the result may exceed its pure-limit value (see the second term in eq. (5.21)). However, as far as the characteristic temperature T1 is comparable with the exchange energy tOE, the value of rescattering parameter v is no longer small due to switching on the thermal fluctuations. If the rescattering parameter u is large (v ~> 1), which is favored by high enough vacancy concentration, temperature fluctuation level and high values of T o (see eq. (5.12)), there is another type of hydrodynamic regime characterized by the less pronounced dependence of magnon damping on the rescattering parameter v ( F ~ T / u 1/4 for T>> To/t: 1/4 (see eq. (5.22)). The comparison of theoretical results obtained with the available experimental data is not so easy, because there are many different regimes depending on material parameters and experimental conditions, the situation being complicated by competition between various magnon relaxation mechanisms. The picture of relaxation processes is more acceptable for interpretation for lowtemperature antiferromagnets, such as MnCo 3 [20] and CsMnF 3 [21, 22]. In particular, in parallel pumping experiments on MnCo 3 (T N ~32.5 K, toE 43 K, H D = 0.59 K, ezo = 6 K) at the pumping frequency fp = 36 GHz in the helium temperature range (1.2 K < T < 2.1 K) and H ~<4 kOe, the dependence of the lower-branch magnon damping of the type 3' oc H2t 74 was observed. The latter may be interpreted as the sum of two expressions (6.1) and (6.2). The temperature dependences of these two contributions are close in the given short temperature interval. As for the magnetic field dependence of the damping, the three-magnon process with rescattering by fluctuations gives strictly 3' oc H 2, whereas the "pure" confluence process gives the quadratic dependence only at small values of the magnetic field (at higher values of H the dependence 3"(H 2) deviates downwards from the straight line and at e~0 = ½e20 the value of 3' turns to zero). The experimental magnetic field dependence of 3"(H) [20] shows that both processes give comparable contributions into the damping under the experimental conditions. The question of relative importance of the two processes under consideration

D.A. Garanin et al. / Magnon-magnon interactions

555

may be clarified by investigation of the frequency dependence of magnon damping (the expression (6.1) rises sharply with e k through e , while eq. (6.2) is independent of ek). The experiments on CsMnF 3 [22] ( T N ~ 5 3 K, toE --= H E ~ 40.3 K, H D ~ 0, H A ~ 0.333 K, e2o ~- ~ ~- 5.52 K, e102~ H 2 + a / T , 2 is the contribution of the nuclear subsystem, a --~6.8 kOe2/ the last term in el0 K) carried out at H a = 2.8 k O e : in the frequency range 22 G H z < f p < 44 G H z show nearly linear increase of the magnon damping with frequency (the damping at fo = 44 G H z being twice as large as at 22 GHz). The latter indicates that both processes are manifested here, though for complete agreement between the theory and the experimental data one must choose the defect concentration c' = 0.05 (see the temperature and the wavevector dependences of the magnon damping in CsMnF 3 represented in figs. 11 and 12). It must be mentioned that if the defect is not a vacancy but a paramagnetic impurity with the easy axis non-collinear with the equilibrium directions of matrix spins, then the amplitude of the combined processes acquires an additional factor of the order of ~/ex/-~pe~ ( ~ is the exchange interaction between host and impurity spins) [23]. This may cause a substantial increase of the improper contribution into the temperature dependent part of the damping. One more possible source of the rescattering of magnons taking part in the three-magnon confluence processes is the rescattering on the boundaries of the crystal, which for small crystals with rough boundaries may be dominant. The expression (6.1) which plays the key role in the comparison with the experiment is valid in the case e± ~ toE where it gives a non-zero result for all values of the magnon wavevector k. On the other hand, in antiferromagnets Av, MHz

0.I 1.5

I

i

2

2,5

T,K Fig. 11. The temperature dependence of the lower-branch magnon damping (Av---2F) in CsMnF3; fp = 30.4 GHz, H 2= 2.8 kOe2; (1) c'= 0; (2) c'= 0.05; (*) ref. [221.

556

D . A . Garanin et al. / Magnon-magnon interactions

A~, MHz 1

o.8

0.,

0 20

25

i 30

i 35

i 40 45 fp, GHz

Fig. 12. T h e f r e q u e n c y d e p e n d e n c e of the l o w e r - b r a n c h m a g n o n d a m p i n g in CsMnF3: T = 2.0 K, H 2 = 2 . 8 k O e 2 ; (1) c' = 0 ; (2) c' = 0.05; ( * ) ref. [22].

with T O> tOE, the three-magnon confluence process becomes prohibited for small k. It would be interesting to trace the threshold switching o n / o f f of this process with varying the magnon energy e k in such materials. The most favorable situation for the realization of hydrodynamic regimes of lower-branch magnon damping takes place, in principle, in high-temperature antiferromagnets, such as FeBO 3 and c~-Fe20 3. But, unfortunately, in these materials the values of the Dzjaloshinsky field H D are rather high (H D 27 kOe in o~-Fe20 3 and H D ~ 110 kOe in FeBO3), so the parallel pumping and A F M R experiments may be carried out only at sufficiently low magnetic fields (G0 =/xv/-H-HD and tOp = 2 e S 1 ~<50 GHz). Under these conditions the contribution of the three-magnon process is suppressed, and the processes of other origins are usually dominant. In particular, at high temperatures in A F E P FeBO 3 ( T u = 348K) the spin wave relaxation is governed by the impurityinduced three-magnon and magnon-magnon interactions [23] as well as by the relaxation absorption of magnons by paramagnetic impurities [24] which results in the characteristic peaks in the temperature dependence of magnon damping [25, 26]. The contributions of four-magnon processes into the lower-branch magnon damping in easy-plane antiferromagnets at low wavevectors, which are also extensively treated in the present paper, have not been revealed experimentally up to now. These contributions should be searched for in high-purity crystals in the cases where the three-magnon process is prohibited: at ~-- 2 To e2o/2eu) ~ tOmax~ tOE (NiCO3) or at high magnetic fields (e~o ~> ½E20 ) .

D . A . Garanin et al. / Magnon-magnon interactions

557

Acknowledgement T h e a u t h o r s t h a n k Prof. D . N . Z u b a r e v for the discussion of the results of the paper.

References [1] V.G. Vaks, A.I. Larkin and S.A. Pikin, Zh. Eksp. Teor. Fiz. 53 (1967) 281, 1089. [2] Yu.A. Izyumov, F.A. Kassan-Ogly and Yu.N. Sckryabin, Field Theory Methods in the Theory of Ferromagnetism (Nauka, Moscow, 1974). [3] E.M. Pikalev, M.A. Savchenko and J. Solyom, Zh. Eksp. Teor. Fiz. 55 (1969) 1404. [4] D.A. Garanin and V.S. Lutovinov, Solid State Commun. 44 (1982) 1359. [5] D.A. Garanin and V.S. Lutovinov, Physica A 126 (1984) 416. [6] D.A. Garanin and V.S. Lutovinov, Zh. Eksp. Teor. Fiz. 85 (1983) 2060. [7] D.A. Garanin and V.S. Lutovinov, Teor. Mat. Fiz. 60 (1984) 133. [8] A.S. Borovik-Romanov and L.A. Prosorova, Zh. Eksp. Teor. Fiz. 46 (1964) 1151; Pis'ma Zh. Eksp. Teor. Fiz. 10 (1969) 316. [9] V.I. Ozhogin, Zh. Eksp. Teor. Fiz. 48 (1965) 1307; 58 (1970) 2079. [10] A.B. Harris, D. Kumar, B.I. Halperin and P.C. Hohenberg, J. Appl. Phys. 41 (1970) 1361; Phys. Rev. B 3 (1971) 961. [11] V.G. Baryakhtar, V.L. Sobolev and A.G. Kvirikadze, Zh. Eksp. Teor. Fiz 65 (1973) 790. [12] V.G. Baryakhtar and V.L. Sobolev, Fiz. Tverd. Tela 15 (1973) 2651. [13] V.I. Ozhogin, Zh. Eksp. Teor. Fiz. 46 (1964) 531. [14] V.L. Sobolev, Thesis Donetsk (1973). [15] V.S. Lutovinov and S.P. Semin, Fiz. Tverd. Tela 20 (1978) 1681. [16] A.I. Akhiezer, Zh. Eksp. Teor. Fiz. 8 (1938) 1318. [17] V.S. Lutovinov, O.A. Olkhov and S.P. Semin, Zh. Eksp. Teor. Fiz. 72 (1977) 2275. V.S. Lutovinov, Thesis Moscow (1977). [18] Yu.A. Izyumov and M.V. Medvedev, The Theory of Magnetically Ordered Crystals with Impurities (Nauka, Moscow, 1970). [19] L.V. Keldysh, Zh. Eksp. Teor. Fiz. 47 (1964) 1515. [20] B.Ya. Kotuzhansky and L.A. Prosorova, Zh. Eksp. Teor. Fiz. 65 (1973) 2470. [21] V.V. Kveder, B.Ya. Kotuzhansky and L.A. Prosorova, Zh. Eksp. Teor. Fiz. 63 (1972) 2205. [22] A.V. Andrienko and L.V. Poddjakov, Zh. Eksp. Teor. Fiz. 93 (1987) 1848. [23] V.S. Lutovinov, Acta Phys. Pol. A 76 (1989) 45. [24] V.S. Lutovinov, Thesis Moscow State University (1985). [25] R.C. le Crow, R. Wolfe and J.W. Nielsen, Appl. Phys. Lett. 14 (1969) 352. [26] B.Ya. Kotuzhansky and L.A. Prosorova, Zh. Eksp. Teor. Fiz. 81 (1981) 1913.