PHYSICAL REVIEW B VOLUME 50, NUMBER 9 1 SEPTEMBER 1994-I

Enhancement of anisotropy due to fluctuations in quasi-one-dimensionalantiferromagnets

Alexander G. AbanovJames Ehxnch Institute, the University of Chicago, 56/0 S. Ellis Aoenue, Chicago, Illinois 80887

Oleg A. PetrenkoDepartment oj Physics and Astronomy, McMaster Unioersity, 1880 Main Street, West, Hamilton, Ontario, Canada L8S)M1

(Received 28 February 1994)

It is shown that the observed anisotropy of magnetization at high magnetic fields in RbMnBr3, aquasi-one-dimensional antiferromagnet on a distorted stacked triangular lattice, is due to quantumand thermal Suctuations. These Quctuations are taken into account in the framework of linear spin-wave theory in the region of strong magnetic fields. In this region the divergent one-dimensionalintegrals are cut off by magnetic field and the bare easy-plane anisotropy. Logarithmical dependenceon the cutofF leads to the "enhancement" of the anisotropy in magnetization. Comparison betweenmagnetization data and our theory with parameters obtained from neutron scattering experimentshas been done.

I. INTRODUCTION

The effect of fiuctuations on the magnetic properties ofquasi-one-dimensional antiferromagnets has been inten-sively discussed in the last few years. Particular attentionhas been given to the cases where there is a stacked tri-angular lattice and an antiferromagnetic interaction, sothat there is a &ustration in going &om one-dimensionalto three-dimensional (3D) ordering.

This situation is apparent in xnaterials with hexagonalCsNiC13-type crystal structure having the general for-mula ABX3 where A is an alkali metal, B is a biva-lent metal of the 3d group, and X is a halogen. Whilethe spin dynamics of CsNiCls and RbNiClz (8=1, easy-axis anisotropy) has been studied to examine the Hal-dane conjecture, ~'2 the compounds of the same groupCsMnBrs, RbMnBrs (8=5/2, easy-plane anisotropy) canbe considered as appropriate for checking the applicabil-ity of standard spin-wave theory for the case of largehalf-integer spins.

CsMnBr3 has been investigated by various experixnen-tal techniques as an example of a &ustrated antiferro-magnet on a stacked triangular lattice exhibiting a num-ber of unusual magnetic properties. These are the field-induced phase transition &om the triangular phase to thecollinear one, a critical behavior associated with a chiraldegeneracy and an unusual phase diagraxn. In additionto numerous neutron scattering experiments and to elec-tron spin resonance (ESR) measurements, 4 the magneti-zation process has been studied in detail. The resultsare in good agreement with classical spin calculationsexcept for two discrepancies. Namely, (i) the measuredmagnetic torques are significantly smaller than the the-oretical oness and (ii) in large magnetic fields (H )H„where H s the critical field of the transition to thecollinear phase) there is a considerable anisotropy be-tween magnetization when magnetic field is applied along

(M ~~') and perpendicular (M~ ') the c axis of the crys-tal. This is absent in the theory when spins are treatedas classical arrows. s

It was suggested that these peculiarities of anisotropyare due to quantum fiuctuations. In this paper we presentthe data of magnetization experiments on another com-pound RbMnBr3. We show that these data can be quan-titatively described by linear spin-wave theory and dis-cuss how both quantum and thermal fiuctuations af-fect the magnetization of a quasi-one-dimensional non-collinear antiferromagnet.

II. CRYSTAL STRUCTUREAND MAGNETIC PROPERTIES OF RbMnBr3

Powder neutron scattering measurements by Glinkaet al.s showed that RbMnBrs exhibits antiferromag-netic order below T~ = 8.8 6 0.1 K. Single crystalmeasurements '~ show that the magnetic structure ofRbMnBr3 is incommensurate with Mn + xnoments lyingin the basal plane and with antiferromagnetic ordering inthe c direction. The commensurate ordering in CsMnBr3gives scattering peaks of type (s, s, 1), while in the in-commensurate RbMnBrs these peaks are replaced by twotriads of peaks near to (s, s, 1) with threefold symme-

try around ( s, s, 1). Because the incommensurate peakssurround a comxnensurate position in reciprocal space,it is likely that the incomxnensurate structure is locallysixnilar to the 120 triangular structure of CsMnBr3, butwith angles between neighboring spins in the basal planesomewhat larger than 120 .

The appearance of incomxnensurate magnetic struc-ture may be due to the presence of crystal structuredistortions. According to an x-ray study by Von Finkand Seifert as the temperature decreases the crystalstructure of RbMnBr3 changes at 470 K &om CsNiC13-

0163-1829/94/50(9)/6271(6)/$06. 00 50 6271 1994 The American Physical Society

6272 ALEXANDER G. ABANOV AND OLEG A. PETRENKO 50

type structure (space group P6s/mmc) to the so-calledKNiCls-type structure (space group P6scm). Measure-ments of the bire&ingence and recent neutron scatter-ing experiments show that another structural phasetransition takes place at about 220 K. As reported inRef. 14 that below this temperature the crystal has or-thorhombic structure with cell dimensions a=14.680 A,b=12.805 A, c=6.516 A at 12 K.

At low temperatures a field of 3.0 T applied along thea axis produces a first-order phase transition to a com-mensurate structure, ' so that the lattice parameterof the magnetic cell is 8 times that of the nuclear cell. io

In a magnetic field of about H, = 3.9 T and above ap-plied in the basal plane, the magnetic structure becomescollinear, as in the case of CsMnBr3 with H, = 6.4 T.This conclusion can be derived &om a previous studyof the magnetization process and ESR spectrum inRbMnBr3.

The crystal distortions lead primarily to a change inthe antiferromagnet coupling between chains. But for thepurpose of this paper the small distinction between theRbMnBrs magnetic structure and a simple 120' struc-ture is not so important. Moreover, confining ourselvesby the range of large magnetic fields we assume that theeffects of interchain exchanges and particularly the dis-tortion of a simple 120' magnetic structure in the absenceof magnetic fields will give only quantitative correctionsnot changing the qualitative picture. This assumptionappears to be true and we estimate these corrections forlarge magnetic fields [see (19)—(22) below].

III. EXPERIMENTAL PROCEDUREAND RESULTS

A vibration-sample magaetometer, similar to the onedescribed in Ref. 18, was used to measure the magneti-zation in RbMnBr3. A magnetic field up to 6 T was gen-erated by two superconducting coils. Simultaneous mea-surements of two mutually perpendicular components ofthe magnetization of the sample, one of which (Mi s) isparallel and another (Mt, ) is perpendicular to the direc-tion of magaetic field, were done by using three pairs ofmeasuring coils.

The absolute accuracy of the magnetization measure-ments was about 5%. The crystal was oriented with anaccuracy of 1 -2 .

The investigation was performed on single crystals hav-

ing approximately 1.5 x 1.5 x 1.5 mm volume and approx-imately 30 mg mass in the temperature range 1.7—12 K.

The magnetization component M~ „g parallel to thefield as a function of the field H for H ~~c and H J c atT=1.7 K is shown in Fig. l. In fields above the crit-ical field (H, = 3.9 T) the magnetizations Mi

' and

MP+ are not the same, but differ by about 7%. Thisanisotropy of magnetization cannot be explained just bytaking into account the easy-plane anisotropy in classicalcalculations, because it leads to M& „' ( M&H„+' whileexperimentally the situation is opposite. Moreover, cor-rections to the classical magnetization due to easy-plane

e 1000

800

600

4000

200

I I I

I

I I I II

I I I I

II I I I

I

I I I I

T= 1.7K

0:-0 2 3

Field (tesla)

FIG. 1. The magnetization M~ „~ parallel to the directionof the magnetic field as a function of field. Circles (o) de-note M~~ (H) for H

~~c and squares (0) denote Mi (H) for

H 0 c. Solid and dashed lines are the results of calculationwith parameters J = 186 GHz, D = 1.3 GHz, H, = 3.9 T,J' = 0.22 GHz using formulas (21), (22) and theory of classi-cal spins (Ref. 5) with J = 199 GHz, D = 2.2 GHz, respec-tively.

anisotropy have to be of order D/ J 1% which is muchsmaller than the observed anisotropy (5—10%).

The anisotropy in magnetization is shown in Fig. 2where the magnetization compoaent Mq, perpendicularto the magnetic field H as a function of the field is plottedat different angles y between the field and the basal planeand at T=4.2 K. Due to the anisotropy of the magneti-zation at H ) H„a nonzero transverse-magnetizationsignal occurs at &p g 0.

100

50

I I II

I I I

I

I I l

7—42K

0 —50

~ —100I I I I I I I I I I I

Field (tesla)

FIG. 2. The magnetization Mt, perpendicular to the mag-netic Geld as a function of H with the Geld making smallangles p with the basal plane. y = 36' (1), y = 26' (2),

16' (3), y = 0.5' (4), y = —14' (5), &p= —24' (6),

y = —34' (7).

50 ENHANCEMENT OF ANISOTROPY DUE TO FLUCTUATIONS IN. . . 6273

I I Ii

I I Ii

I I I

IV. THEORY

404

3

2

In this section we show that the anisotropy of magne-tization and reduced magnetic torques can be describedby linear spin-wave theory using the microscopic Hamil-tonian

R = 2J) S;.S;~~ +2J') S; ~ S+~

20~ W

tg0

0 2 4Field (tesla)

FIG. 3. The field dependence of the magnetization compo-nent Mt, , at difFerent temperatures with the field making asmall angle y with the basal plane. T = 2.4 K (1), T = 4.2 K(2), T = 6.0 K (3), T = 7.1 K (4).

I I II

I I II

I I I

I

I I I

60 —H=4. 5T

N~ W

0~ ~

N

t}g)

pHO~

pAA ~

20 ~

40—

0 I I

0I « I I I I I I

2 4 6Temperature (K)

FIG. 4. The magnetization Mt, (H = 4.5 T, rp = 20') isshown as a function of temperature. The dotted line is a guideto the eye.

According to Fig. 2 the field dependence Mt, (H) ex-hibits a small hysteresis. This hysteresis takes place whenthe projection of the magnetic field on the basal planereaches the value of 3.19 + 0.10 T in increasing fieldsand 2.67 6 0.07 T in decreasing fields. It corresponds tothe first-order phase transition from the incommensuratemagnetic structure to the commensurate one.

In Fig. 3 the field dependence of the magnetizationcomponent Mt, at difFerent temperatures is shown. Atlarge magnetic fields we can see that Mt, increases withtemperature supporting the idea that the anisotropy ofthe magnetization at H & H, is due to fluctuations.

This is even clearer from Fig. 4 where the value ofMt, (H = 4.5 T, y = 20') is shown as a function oftemperature.

+D) (S;)' —h) S;.

The intrachain antiferromagnet exchange constant J isthe largest parameter in the Harniltonian (1) and isabout two orders of magnitude larger than the exchangeconstant between chains J' and easy-plane single-ionanisotropy D, 0 J )) J', D and J, J', D ) 0. h is themagnetic field in the units of energy and it is equal toh = gp~H where H is a magnetic Beld in some standard~~nits, p~ is a Bohr magneton, and g is Lande factorwhich is equal to 2 in the compound under considera-tion. For h = 0 the classical ground state of the Hamil-tonian (1) is the 120' structure with spin vectors form-ing equilateral triangles in the basal plane. The case ofa nonzero magnetic field was considered classically byChubukov. s For h along the z (c) axis the transversecomponents of the spins conserve 120 structure and theclassical magnetization

hMP, II, = (S') = —,

when h ( 8JS neglecting by J'/ J,D/J (here and thereon M means M~ where the opposite is not stated ex-plicitly). The case of the magnetic field perpendicular tothe z axis is more complicated. For a sufficiently strongeasy-plane anisotropy, spins do not leave the planes butthe 120' structure is not conserved anymore and a spinHip of the two magnetic sublattices takes place at therelatively small magnetic Beld h, = v48JJ'S . Ath, ( h ( SJS we have a collinear spin structure and

glmagnetization has to be the same (up to small &, &cor-

rections) as in the case of field along z axis. s

In this section we take into account quantum and ther-mal Quctuations in the &amework of linear spin-wavetheory and show that the anisotropy of magnetizationat h & h, due to these Buctuations is of the right sign(Maj (MaII, ) and is much larger than the one expectedft.om calculation with classical spins.

In what follows we confine ourselves to magnetic fieldsh ) h where we have a collinear spin structure for amagnetic field perpendicular to the c axis of the crystaland we neglect the J' term in the Hamiltonian (1) in thissection. Below we argue that this terxn gives correctionsto the gap in a spin-wave spectrum of order of J'/ J anddoes not change the qualitative picture. Thus we considerthe Hamiltonian of a single magnetic chain instead of(1) keeping in mind that our results will be applicableonly for the h & h = /48J J'Sz region. We can writeHamiltonians for two directions of magnetic field in theform

6274 ALEXANDER G. ABANOV AND OLEG A. PETRENKO 50

'Rh~~, = 2J) S;.S;+& —h) S; + D) (S;), (3)

Qg~, ——2J) S, . S;+g —h) S;+D) (S, ) . (4)

DSN— 2JS (1+2sin P) — cos

2

+DS sin

Here we slightly changed notations for simplicity. Nowmagnetic 6eld is always applied along the z axis and theeasy plane is the xy plane for the parallel case and theyz plane for the perpendicular one. Let us apply theDyson-Maleev transformation

IS' = S —ataata

S =/2Sat/1—2Sp '

S+ = +2Sa

to (3), (4) with the axes of quantization z' directed along"classical spins" as is shown in Fig. 5. Choosing theangle P to cancel terms linear in a and at we reproducethe classical results for magnetization. The terms of (3),(4) quadratic in a and at will give us the linear spin-wavetheory corrections to these results. %e have

h

2S(4J+ D)'

hsing~ = 8JS'

and both Hamiltonians (3), (4) after Fourier transforma-tion can be written in the form

(dying(k)= 4JS(1 —sin icos k) + DS cos

~2ii(k) = 4JS cos icos k + DS cos

for a magnetic 6eld parallel to c, and

N~ —2JS (1+2sin P)—DS It

2

uq~(k) = 4JS(l —sin Pcosk) + DS,~2i(k) = 4JScos Pcosk —DS,

lj2(~, + g~,'-~,'&ag ——

2 Qtdg —td2

Z/2

we diagonalize (8) and get

'R = p + ) e(k)~

b„bi, + —~—~g(k)2J

(io)

for one along the easy plane. Making the Bogolyubovtransformation

'R = p+ ) (ug(k)at„ag—A:

with p, ~q, and ~2 equal to

~2(k)2

(a&a & + aga I,), (8)

or

e (k) = (ui(k) —ur2(k)

k) 4JS(1—cos k)

x 4JS(1+ cos 2P cos k) + 2DS cos

with the spectrum of the spin waves given by

Y'

e~(k) = [4JS(1 —cos k) + 2DS]x 4JS(l + cos 2P cos k). (12)

Now we can 6nd the magnetization M = —& .'

Y'

Bp 8 ( t 1 5 1 BurqM = ——+) ——e(k)

~(b„4)+ —

~

+-t9h Bh ( " 2) 2 Bh

(i3)or after simple manipulations using ((b&bt) + 2)—coth 27

YI

dk 0M = S sing — ———e(k) coth

2~ Bh 2 2T.(14)

FIG 5. Quantiz. ation axes and coordinate systems used inthe theory.

Here the 6rst term describes the classical part of magneti-zation while the second one comes from the contributionof quantum and thermal Quctuations.

Let us show now that Eq. (14) leads to enhancedanisotropy between Mh, ~~,

and Mg~~. Consider the caseof zero temperature for simplicity. Substituting 1 for

coth+& and taking derivatives using (6), (7), (11), (12)we have

50 ENHANCEMENT OF ANISOTROPY DUE TO FLUCTUATIONS IN. . . 6275

dk 1 (1 —cos k) cosk+ D/4J2~ 2 1+cos2$cosk+ D/2Jcos2$ 1+D/4J

" dk 1 1 —cosk+ D/2JMg~ = S sin P + sin P cos k.2' 2 1+cos2gcosk

dk 1M —Ssing —sing

2m gk&+g2' (16)

with

h2 DII 16J282 + J'

2 = h2(is)

Only one of the two soft modes, corresponding to k = sr,

gives a big contribution to the magnetization. This arisesfrom the mode that corresponds to Buctuations of the an-

gle between spins and magnetic field. Another soft modeat k = 0 corresponds to azimuthal Buctuations of spinsaround the direction of magnetic field and almost doesnot contribute to the magnetization. This mode howevercontributes signi6cantly to the value of the average spinon the site. ~s Expanding the integrand in Eq. (15) in thevicinity of k=z' and neglecting D/J in comparison with1 we get

would diverge for g = 0]. This gap is larger for the caseof magnetic field along the c axis due to bare easy-planeanisotropy in (1).

V. COMPARISONW'ITH EXPERIMENT AND DISCUSSION

To compare our theory with an experiment we needto know the values of the parameters of the Hamiltonian(1). They can be obtained from inelastic neutron scat-tering experiments. As reported in Ref. 10 J = 199 GHzand D = 2.2 GHz. These values were found as the bestto fit formulas for spin-wave dispersion derived withouttaking into account quanta' fiuctuations to the data ofinelastic neutron scattering. Bare values of J and D from

(1) differ from these J, D by quantum corrections. Using(4) and (5) from Ref. 5 we take into account these renor-malizations by quantum Buctuations and we have fromgiven J, D

or taking integrals finally J = 186 GHz, D = 1.3 GHz.

hMh IIg

hMh~, —

1 ( h2 Dl2.S i6J S"J ~

1+ '12z S q 16J'S' y'

(19)

(20)

Now let us estimate the corrections to (19),(20) due to3D (J') effects. These efFects give corrections of order ofJ'/J under logarithm in (19), (20) (Ref. 20) and can becalculated in the &amework of the same linear spin-wavetheory. Calculations which we do not present here showthat 3D Buctuations will efFectively increase the gap of1D system giving instead of (19), (20)

hMhII, ——

hMgg 8J

ln ~, , + —+3—~, (21)

h' J')1+»I, , +2—IE16J S (22)

The formulas (21), (22) are good at sufficiently bigmagnetic 6elds where Buctuations and efFects of three-dimensional ordering are small enough to guarantee theapplicability of linear spin-wave theory for a quasi-one-dimensional system.

Now from (19)—(22) the origin of the enhancement ofanisotropy is clear. Big Buctuations of the angle betweenspins and magnetic field give a signi6cant contributionto magnetization which depends strongly on the corre-sponding gap in the spectrum of spin waves due to lowdimensionality of the system [the integral in Eq. (16)

Using these values of J and D and the expressions (21)and (22) with J' = 0.22 GHz from H, = /48J J'S3.9 T we have calculated the magnetization of RbMnBrsfor both H

~~c and H J c. Formulas (21) and (22) are

applicable for H ) 4 T. For H ( 4 T the efFects of three-dimensional ordering and next-order corrections to thelinear spin-wave theory become significant.

The magnetizations (21), (22) are plotted in Fig. 1 to-gether with the experimental data for M~~~, and M~~, .Also we plotted the magnetizations calculated using for-mulas for classical spins and the values J = 199 GHz,D = 2.2 GHz. We can see that the formulas (21) and(22) give much better absolute values of the magneti-zations than the classical expressions and also describethe anisotropy in magnetization. The experimental de-pendence MHII is slightly nonlinear with the slope in-creasing with H (this is not clearly seen from Fig. 1 butcan be checked by 6tting the experimental data by linearfunctions in difFerent ranges of H) The effect can. be ex-plained naturally by the suppression of Buctuations withincreasing H and it is seen in (21).

The graphs obtained from (21), (22) in Fig. 1 lie be-low experimental points. They will lie even lower if weuse exact integrals instead of approximate formulas (21),(22). This can be improved by considering interactionsbetween magnons which will give the second order in 1/Scorrections to magnetization. The contribution of thesecorrections is of the opposite sign to the first-order cor-rections and it is not very small ( 5%%uo) because the

6276 ALEXANDER G. ABANOV AND OLEG A. PETRENKO 50

Buctuations are quite big in the experimental range ofparameters. For example M = 30% at I= 4.5 T.b MHg

class

VI. CONCLUSION

An effect of enhancement of anisotropy in quasi-one-dimensional spin systems due to quantum and thermalHuctuations was found. It explains the anisotropy inmagnetization seen in experiments. The essence of theeffect is that the Buctuational part of the magnetizationis suKciently big and is determined mostly by the loga-rithm of the gap in the spin-wave spectrum. This gap isanisotropic due to the bare easy-plane anisotropy. Thelogarithmical dependence of the fluctuational part of themagnetization on this gap (essentially the divergence ofone-dimensional fluctuations) leads to the strongly "en-hanced" anisotropy of the magnetization. The same Huc-

tuations explain the slightly nonlinear character of MH ~~,

and the excessive values of magnetization calculated us-

ing formulas for classical spins.

This theory can be applied to the compounds witheasy-plane anisotropy and quasi-one-dimensional mag-netic structure such as RbMnBr3, CsMnBr3, andKNiClq. %'e presented in this paper the comparisonof our theory with the results of magnetization mea-surements for RbMnBr3. One can check that the samescheme applied to magnetization data for CsMnBr3 &OIIl

Ref. 8 and neutron scattering data for exchange constants&om the erst paper in Ref. 3 also gives a reasonableagreement between theory and experiment.

ACKNOWLEDGMENTS

The authors of this paper thank L.A. Prozorova, V.L.Pokrovskii, P. Wiegmann, M. Zhitomirsky, M.F. Collins,and especially A. Chubukov for fruitful discussions. Weare indebted to Yu.M. Tsipenyuk and M.F. Collins forsending us copies of their work prior to publication. Oneof us (O.P.) is grateful to A.N. Bazhan for using his in-stallation. A.A. was supported in part by Grant No.NSF-DMR-MRL 8819860.

F.D.M. Haldane, Phys. Rev. Lett. 50, 1153 (1983).R.M. Morra, W.J.L. Buyers, R.L. Armstrong, and K. Hi-

rskawa, Phys. Rev. B 38, 543 (1988);W.J.L. Buyers, R.M.Morra, R.L. Armstrong, M.J. Hogan, P. Gerlach, and K.Hirakawa, Phys. Rev. Lett. 56, 371 (1986); T. Ohyama andH. Shibs, J.Phys. Soc. Jpn. 62, 3277 (1993),and referencestherein.B.D. Gaulin, M.F. Collins, and W.J.L. Buyers, J. Appl.Phys. 61, 3409 (1987); B.D. Gaulin, T.E. Mason, M.F.Collins, and J.Z. Larese, Phys. Rev. Lett. B2, 1380 (1989);T.E. Mason, B.D. Gaulin, and M.F. Collins, Phys. Rev. B$9, 586 (1989); T.E. Mason, Y.S. Yang, M.F. Collins, B.D.Gaulin, K.N. Clausen, and A. Harrison, J. Magn. Magn.Mater. 104-107, 197 (1992).I.A. Zaliznyak, L.A. Prozorova, and S.V. Petrov, Zh. Eksp.Teor. Fiz. 97, 359 (1990) [Sov. Phys. JETP 70, 203 (1990)].A.V. Chubukov, J. Phys. C 21, 441 (1988).B.Ya. Kotyuzhanskii and D.V. Nikiforov, J. Phys. Condens.Matter 3, 385 (1991).T. Goto, T. Inami, and Y. Ajiro, J. Phys. Soc. Jpn. 59,2328 (1990).S.I. Abarzhi, A.N. Bazhan, L.A. Prozorova, and I.A. Zal-

iznysk, J. Phys. Condens. Matter 4, 3307 (1992).C.J. Glinka, V.J. Minkiewicz, D.E. Cox, and C.P. Khattak,in Magnetism and Magnetic Materials, AIP Conf. Proc.No. 10, edited by C.D. Graham and J.J. Rhyne (AIP, New

York, 1973), p. 659.L. Heller, M.F. Collins, Y.S. Yang, and B. Collier, Phys.

Rev. B 49, 1104 (1994).T. Kato, T. Ishii, Y. Ajiro, T. Asano, and S. Kawano, J.Phys. Soc. Jpn. B2, 3384 (1993).H. Von Fink and H.-J. Seifert, Acta Crystallogr. B 38, 912(1982).T. Kato, K. Yio, T. Hoshino, T. Mitsui, and H. Tanaka,J. Phys. Soc. Jpn. Bl, 275 (1992).J.B. Forsyth, R.M. Ibberson, Yu.M. Tsipenyuk, and S.V.Petrov, Phase Transit. (to be published).S. Kawano, Y. Ajiro, and T. Inami, J. Magn. Magn. Mater.104-107, 791 (1992).A.N. Bazhan, I.A. Zaliznyak, D.V. Nikiforov, O.A. Pe-trenko, S.V. Petrov, and L.A. Prozorova, Zh. Eksp. Teor.Fiz. 103, 691 (1993) [Sov. Phys. JETP 7B, 342 (1993)].I.M. Vitebskii, O.A. Petrenko, S.V. Petrov, and L.A. Pro-zorova, Zh. Eksp. Teor. Fiz. 103, 326 (1993) [Sov. Phys.JETP 7B, 178 (1993)].A.N. Bazhan, A.S. Borovik-Romanov) and N.M. Kreines,Prib. Tekh. Eksp. 1, 412 (1973) [Instrum. Exp. Tech.(USSR) 16, 261 (1973)1.It is seen from this argument that the attempts of I.A. Za-

liznyak, Solid State Commun. 84, 573 (1992) to relate thesuppression of magnetization with the suppression of av-

erage spin by fluctuations are qualitatively wrong becausethese efFects arise from the difFerent spin-wave modes.

gtExpressions like ln J were obtained in D. Welz, J. Phys.Condens. Matter 5, 3643 (1993) for spin reduction but can-not be applied to our case (see Ref. 19).