PH YSICA L RE VIEW B VO LUME 15, N UMBER 9 1 MA Y 1977

Spin fluctuations in random magnetic-nonmagnetic two-dimensional antiferromagnets.I. Dynamics

R. A. Cowley» and G. ShiraneDepartment of Physics, Brookhaven National Laboratory, ~ Upton, New York 11973

R. J. BirgeneauDepartment of Physics, Massachusetts Institute of Technology, ~ Cambridge, Massachusetts 02139

H. J. GuggenheimBell Laboratories, Murray Hill, New Jersey 07974

(Received 26 October 1976)

The magnetic excitations in the system Rb~Mn, Mg, ,F4 with c = 1, 0.57, and 0.54 have been measured using

neutron-inelastic-scattering techniques. In pure Rb,MnF4 the magnon dispersion relation is accurately

described by simple spin-wave theory with dominant Heisenberg antiferromagnetic nearest-neighbor exchange,

a much weaker antiferromagnetic next-nearest-neighbor interaction, and a small single-ion anisotropy field. In

the specimens with c = 0.57 and 0.54 the scattering shows a broad response but with substructure which may

be associated with the different Ising energies of Mn ions in different environments. The measurements are in

excellent accord with the results of computer simulations but theories based on the coherent-potential

approximation do not give a satisfactory description of the measured spectra.

I. INTRODUCTION

In the past several years a number of studiesof the dynamics of concentrated insulating mag-netic alloys have been reported. ' The explicitsystems stud'ied have been almost exclusivelytransition-metal-fluoride binary alloys with thetransition-metal constituents either both mag-netic or magnetic-nonmagnetic. The reasons forthis ubiquitous choice of these fluoride hosts are(i) the transition-metal ions are all quite similarchemically so that true random alloys can besynthesized with little or no clustering effects,while at the same time, the ions may be quitedissimilar magnetically. (ii) The exchange inter-action is typically isotropic and between nearestneighbors only; thus, the mixed-crystal Hamil-tonian involves only a small number of parametersall of which are known from measurements onthe pure systems and/or very dilute alloys.(iii) The spin-wave energies are in the rangemost easily measured by conventional neutron-scattering techniques. (iv) In favorable caseslarge homogeneous single crystals may be grown,so that the dynamic structure factor' S(Q, &o) maybe measured with high resolution throughout theBrillouin zone. All of these factors combine toproduce a nearly ideal situation in which a directconfrontation between experiment and theory ispossible.

The current status of experiment and theory inthese substitutionally disordered fluorides hasbeen summarized by Cowley. ' For the mixedmagnetic systems, experiments on real crystals,

computer-simulation experiments, "and variousanalytic theories, most notably those based onthe coherent-potential approximation" (CPA),seem to be in general accord. The most glaringdiscrepancy is in the intensities at low frequenciesand small wave vectors in the two-dimensional(2-D) mixed antiferromagnet Rb,Mn, ,Ni, ,F,.'In that case there is much more weight at longwavelengths than predicted by any of the theoriesincluding the computer simulations. Als-Nielsenet al. ' have also shown that in Rb,Mn, ,Ni, ,F„the static critical behavior is identical to that inthe pure systems.

The situation in mixed magnetic-nonmagneticsystems is rather more complex. Whereas inmixed magnetic systems the excitations split intotwo well-defined branches, in the dilute Heisen-berg magnets S(Q, ~) generally becomes muchbroader and ill-defined; thus it is less meaning-ful to talk in terms of spin waves or a "dispersionrelation. " Correspondingly, the CPA theoriesare more difficult for dilute systems as comparedto mixed magnetic systems. Until recently, oneof the principal issues in these random crystalstudies was the existence or nonexistence ofresidual structure in the overall broad response;this substructure represents the remnants of theIsing cluster modes reflecting the different pos-sible environments of the magnetic ion. Very re-cent high-resolution experiments on Mn, Zn, ,F„'however, indicate that there is indeed some barelyresolvable structure in S(Q, &u), in that system atleast, at wave vectors near the magnetic zoneboundary.

15 4292

15 SPIN F LUCTUATIONS IN RANDOM. . .I. . . 4298

The most subtle, and certainly the least under-stood aspects of dilute magnets, relate to percola-tion' "rather than to the dynamics. It has al-ready been shown in a number of experiments in

the transition-metal fluorides" that when themagnetic ion concentration c is reduced below acertain level, the system does not order down tovery low temperatures, that is, there is no long-range order down to at least 1 K. This almostcertainly corresponds to the classical percolationproblem. In this case for nearest-neighbor inter-actions alone and for c&c~, the percolation con-centration, the system breaks up into finiteclusters. As T -0, these clusters will freezeinto some static configuration, but the differentclusters will be statistically uncorrelated. Thereis currently virtually no experimental informationon the temperature dependence of the correlationsin real magnets near the percolation concentra-tion. Similarly, rather little is known about thelong-wavelength dynamics in these percolativeclusters.

Motivated by the above considerations, we haveundertaken an extensive study of the mixed mag-netic-nonmagnetic 2-D antiferromagnetsRb2MncMg l F4 These are iso mo rphous to themuch studied pure antiferromagnets K,NiF„Rb,MnF„etc. Previously, detailed experimentshave been performed on the mixed magnetic sys-tem Rb,Mn, ,Ni, ,F„' and this present work rep-resents an extension of that to the diluted case,as well as an extension to two dimensions ofearlier studies on the three-dimensional (3-D)magnets Mn, Zn, ,F,."'

Our experiments subdivide straightforwardlyinto two distinct categories: (i) those concernedwith the general dynamical response S(Q, ~)across the Brillouin zone at low temperatures,and (ii) studies of the static temperature and con-centration-dependent correlations for sampleswith c-c~ together with some high-resolution mea-surements of the long-wavelength dynamics. Themain advantage of the system, Rb,Mn, Mg, ,F„for part (i) is that for the simple square latticeZ = 4 the near-neighbor coordination is quite low.This, in turn, implies that the "Ising clustermodes" which are a very subtle feature in theMn, Zn, ,F, system should become a gross featureof the dynamics here.""As we have indicatedabove, there is al.most no experimental informa-tion in the literature on the correlations nearpercolation. The main advantage of these 2-D sys-tems is that in two dimensions the deviation frommean-field behavior is as large as it can be inthe percolation problem. " Thus, these experi-ments should provide important information toguide and select between theories of magnetism

around the percolation point. We have, in fact,already published a brief note on certain aspectsof the percolation work. '4

In this paper we discuss our experiments and

theory for the dynamics in Rb,Mn, Mng F4 In asecond paper to follow this, (hereafter labeled II)we give a complete description of the percolationexperiments. The format of this paper is asfollows. In Sec. II we describe the sample char-acterization, crystal and magnetiC structure, and

other basic details. Section III describes mea-surements of the spin-wave dispersion relationsin the pure material Rb,MnF4. These are a nec-essary prerequisite to our detailed studies of thedilute samples. The measurements on the dynam-ics of the dilute crystals are given in Sec. IV. Adetailed comparison with computer simulation and

CPA theories is made in Sec. V. Finally, theconclusions are given in Sec. VI.

II. PRELIMINARY DETAILS

A. Crystal growth

In order to grow large single crystals that arerandom and whose final composition correspondsclosely to that desired, considerable care mustbe taken in the growth technique. The need forespecially large crystals in these dynamic studieshas therefore led us to explore new methods ofcrystal growth for this family of compounds. Wehave indeed developed a new technique which hasbeen quite successful. We describe this verybriefly below. A full report will be given else-where. "

In the past, crystals of Rb, XF, (X = Ni or Mn)were prepared primarily by horizontal zone melt-ing or the Bridgman method. " Because thesecompounds melt incongruently, the general methodof preparation has been the mechanical separationof the 2-D phase from the other phases aftergrowth. While this technique often produced goodmaterial, the single crystal size was almostalways rather less than 1 cm' in volume. Asnoted above, rather larger crystals are requiredfor studies of the sort reported in this paper.Recently, we have found that by using a mixturecontaining an excess of RbF (40 mole/q), the phaseequilibria are shifted sufficiently to produceRb, XF, as the first phase to solidify. Therefore,good single crystals could be grown on an orientedseed crystal. This was done by using the Czochral-ski pull method. A typical crystal preparationwas as follows. A mixture of single crystals ofRbF, MnF„and MgF, (each previously zonerefined) was weighted to produce the stoichiomet-ric equivalent of Rb,Mn, Mg, ,F, plus 40 mole%excess RbF. This mixture was melted in a platin-

4294 COWLEY, SHIRANE, BIRGENEAU, AND GUGGENHEIM 15

um boat in a hydrogen fluoride atmosphere. Thecharge was then transferred to a vitreous carboncrucible in a crystal puller. A quartz envelopewas used to contain a nitrogen atmosphere. Acrystal was pulled on a seed crystal at 1-5 mm/hwith a rotation speed of 20-50 revolutions perminute.

B. Sample characterization

TABLE I. Lattice constants at 5 K.

Rb2MnF4Rb2MBp gvMgp 43F4Rb2Mng 54Mgp 4gF4

4.2153(8)4.1500(4)4.1457 (4)

13.801(15)13.801(15)

For these studies we grew crystals with threenominal compositions: Rb,MnF„Rb, Mn, „Mg, „F„and R02Mnp SpMgp pF4 Several boules of eachwere grown and individual samples with optimalmosaic spread and size were selected for detailedstudy. The crystals were typically 2 cm' in volumeand had a mosaic spread of several minutes. Ingeneral, the crystals have a laminar morphology.Typical Bridgman crystals (for example, theRb,Mn, ,Ni, ,F, described in Ref. 7) actually con-sist of a large number of slightly misalignedplatelets. This platelet stratification was muchless pronounced in these pulled crystals.

We now consider our estimates of the actualversus the nominal concentration, in two samples,one nominally 50% Mn and the second nominally59%. From previous studies of random fluoridesone knows that the final composition may differmarkedly from the composition of the start-ing material. Further, from the work onRb,Mn, ,Ni, „.F, and Mn, Zn, ,F, we know that thecomposition may vary across the boule itself byas much as 10%. Chemical analysis of separatesamples taken from the 50% and 59% boulesyielded actual Mn atomic concentrations of 55%and 60.5%, respectively. As discussed in Ref.14, we know that the 60.5% estimate must beslightly high for the "59%"neutron sample sincethe neutron studies of the spin fluctuations in thenominal 59% sample show that there is no mag-netic long-range order down to 1 K. This inturn necessitates that c(c~= 0.59. The simplestmethod of estimating the concentration is fromthe lattice constant themselves. We show inTable I the measured lattice constants at 5 K forour samples. All three samples were confirmedto have the K,NiF, structure. " At room tempera-ture a(Rb,MnF, ) -a(Rb, MgF, ) = 0.17 A so that themixed samples have lattice constants in the range

expected on the basis of the nominal concentra-tions. From Table I we can estimate accuratelythat the two mixed samples differ by 3% in Mnconcentration (as opposed to 9% nominally, and(5+ 2)% from the chemical analysis on the separatesamples). We should add that this interpolationtechnique is not reliable enough to give accurateabsolute concentrations.

After some study we realized that the concen-trations of the explicit samples studies could beestimated most accurately from our own quasi-elastic neutron-scattering measurements. Inparticular the pair-connectivity correlation lengthas measured in Ref. 14 depends sensitively on theMn" concentration. As discussed extensively inPaper II and briefly in Ref. 14, this techniqueyields c = 0.54+ 0.01 and c = 0.57 ~ 0.01 for the twosamples. The former value, c =0.54, is satis-factorily close to the chemical analysis result ofc = 0.55 + 0.02 while the difference value of 3%%d

agrees exactly with out lattice-constant estimate.Also, as discussed in Sec. V of this paper, ifone regards c as an adjustable parameter in fittingcomputer simulation theory to the measured zoneboundary S(Q, &u), then one obtains independentestimates of c=0.54~0.03 and c=0.57+0.03 in thetwo samples. Clearly, neither of these estimatesis theory independent, but the close internalagreement, together with the reasonable consis-tency with the chemical analysis, leads us tobelieve that they are quite accurate. Henceforth,we refer to the separate samples studied as "con-centrated, 5y% and 54%."

C. Crystal and magnetic structure; randomness

The crystal and magnetic structures of Rb,MnF„together with the (010) and (001) magnetic zonesof the reciprocal lattice, are shown in Fig. 1. Inall cases only those nuclear Bragg peaks appro-priate to the K,NiF, structure are observed, there-by confirming that the mixed crystals do indeedretain the pure crystal structure. The magneticordering in Rb,MnF, has been extensively dis-cussed by Birgeneau et al."; the spin waves andsublattice magnetization have been discussed byde Wijn et al." Above T~=38.4 K, Rb,MnF, ex-hibits pronounced 2-D magnetic short-range-ordereffects in which near neighbors are antiferro-magnetically aligned. This short-range ordermanifests itself as lines of diffuse magnetic scat-tering in the (010) zone extending in the t directionalong [h0l], centered about h=1, 3, . . . . At T„= 38.4 K the system undergoes a rather unusualphase transition to 3-D magnetic order. In thesame crystal, Birgeneau et al."observe two dis-tinct 3-D ordering patterns. In the first, whichis illustrated in Fig. 1, the central spin on the

SPIN F LUCTUATIONS IN RANDOM. . .I. . . 4295

(b) (c) III. MAGNETIC INTERACTIONS IN R12MnF4J.

MO (03I ) 302 (033) 3IO' 320

CI'' n

&iI

II

(~'

l I I I I

~ MH'o F~ Rb+

n

)IOO (Oll)

I

I

0 (b)lI

I02 (OI3)

~002

2IO

IIO

I I

I

"OIO

b%

~220

I 20

+20

FIG. 1. (a) Crystal and magnetic structures ofR12MnF4. Inverting the central spin exchanges the 5and b magnetic axes. Here we show only the K2NiF4-

type magnetic structure; the Ca~Mn04-type structuremay be generated by inverting the spina of the nnn layer.(b) (010) and (100) magnetic zones of the reciprocal lat-tice; again we show only K2NiF4-type Bragg positions.Nuclear Bragg peaks are indicated by double circles.(c) (001) zone of the reciprocal lattice: K2NiF4-type mag-netic scattering is observed only at the filled-circlepositions.

near-neighbor (nn) plane may point either up ordown, while the spins along the e axis of next-nearest-neighbor (nnn) planes are parallel. Thisgives rise to the magnetic superlattice peaksshown in Figs. 1(b) and 1(c). This magnetic struc-ture is commonly denoted as the K,NiF4 structure.However, a second structure also occurs in whichthe spins of nnn planes along the e axis are anti-parallel (the so-called Ca,MnO, structure"). Thisgives rise to superlattice scattering at (h, h, ',I)—with h+ k, l odd. These two structures otherwisehave identical magnetic properties, thereby show-ing that the magnetism is determined completelyby the 2-D in-plane interactions. In the dilutedcrystals since c & e~ = 0.59 no magnetic Braggscattering is anticipated. However, there shouldbe lines of diffuse magnetic scattering along [hOI],reflecting the development of magnetic correla-tions in the percolation clusters, in exact corres-pondence to the critical scattering in the concen-trated system. This diffuse scattering is the sub-ject of Paper II.'4

Finally, we must consider the degree of random-ness in the mixed crystals. As we noted in theintroduction, Mn" and Mg" are quite similarchemically so no pronounced clustering is antici-pated. A variety of neutron scans were made inexact analogy with those described by Als-Nielsenet al. ' in their crystallographic characterizationof Rb,Mn, ,Ni0 .„F4. As in that case, and in thecase of Mn, Zn, ,F„'we find a null result; thatis, we see no features in the diffuse backgroundscattering indicative of positional short-range-order effects. We conclude therefore that theMn'+ and Mg'+ are indeed randomly distributed.

800

NEUTRON GROUPS IN Rb&Mn F+ AT 4.2K1 ]

( I.0,0.25,0)—E

100 ~~

400PO

(AI-

OV

0 )

5.5

(1.0,400—

I ] ]

6.0 6.5 7.0

0 I I I

5.5 6.0 6.5T

I I

7.0 5.5 6.0ENERGY (meV)

E

—IOO (g)

6.5

FIG. 2. Typical neutron groups for spin waves at 4.2Kin Rb2MnF4. The widths are determined by the resolu-tion of the spectrometer.

Since the interpretation of our measurementson the system Rb,Mn, Mg, ,F, requires a knowledgeof the exchange interactions between the Mn"ions, we have measured the spin-wave excitationsin a crystal of pure Rb,MnF4. The experimentswere performed at the Brookhaven high flux BeamReactor. The crystal at 4.5 K was oriented with ac axis vertical on a triple-axis crystal spectrom-eter; measurements were thus in the (001) zone[Fig. 1(c)]. The measurements were performedwith pyrolytic graphite crystals as both mono-chromator and analyzer and a constant incidentneutron energy of 14.8 meV. A pyrolytic graphitefilter was used to reduce the higher-energy con-taminants in the incident neutron beam. The colli-mation was typically 10 min throughout.

As discussed in Sec. II, the magnetic orderingin Rb,MnF4 has been described by Birgeneau et aL."Above 38.4 K there are 2-D magnetic short-rangeordering effects in which the near neighbors areantiferromagnetically aligned. At 38.4 K the sys-tem undergoes a transition to a 3-D magneticallyordered structure which may take one of twoforms, dependent upon the relative orientation ofthe magnetic planes along the e axis. In our speci-men, magnetic Bragg reflections were observedat 100 and 010 reciprocal-lattice points showingthat at least part of the crystal ordered with theK,NiF, structure which is illustrated in Fig. 1.

The frequencies of the spin waves were deter-mined for propagation directions [&00], [&&0],and [&0.50] (t= aqj2v) within the magnetic unitceQ shown in Fig. 1. Some typical scans areshown in Fig. 2 where the widths of the groupsare consistent with the expected instrumental

4296 CO%LE Y, SHIRANE, BIRGKN EAU, AND GUGGENHEIM 15

E

C9

Llj 4z4J

MAGNON DISPERSION RELATION IN Rb~MnF4 AT 4.5K

I I I I,

'I I I I I I I I

(0.5,(,O)

E-,/4S~, =( [l+ ~+ Py, (q)]' —y, (q)'}'",where

a ggeH„/4J, S, P =J,/2Z, ,

(2)

where all the sites on one magnetic sublattice havei even and the others have i odd, and the summa-tions are over nearest-neighbor and next-nearest-neighbor pairs.

The energies of the spin waves of wave vectorq are then obtained from linear spin-wave theoryas

j I I I I I I I I I I I

0 C 0.50 ( 0.5 Q

REDUCED WAVE VECTOR (=(P)FIG. 3. Magnon dispersion relation at 4.2 K in

Rb2MnF4. The solid line is the best-fit theoretical dis-persian wave using the parameters listed in Table II in-cluding J& and 42 and (model II).

resolution. The frequencies of the spin waveswere determined by fitting, in a least-~uaressense, Gaussians to the neutron groups, and theresulting frequencies are shown in Fig. 3. In allcases the relative errors are smaller than thesize of the points on the dispersion curve. Atthe magnetic zone center the measured energy,0.62+ 0.02 meV, is in excellent accord with theresult obtained by antiferromagnetic resonancetechniques, ~ 0.62 + 0.005 meV.

Spin-wave theory for Rb,MnF, has been exten-sively discussed by de Wijn et a/. " These authorshave deduced the nearest-neighbor-exchange con-stant from measurements of the sublattice magneti-zation. We have assumed a model which includesnearest-neighbor exchange J,and next-nearest-neighbor exchange J,within the magnetic planes; theanisotropy originating from the magnetic dipolarinteractions is incorporated as an anisotropy field.The Hamiltonian is then given by

x=z, Q5, 5, +z, Q x, 5,Qn) (nnn&

y, (q) = cos(-,'q, a)cos (-,'q„a),

y, (g) = cos(q, a)+ cos(q„a) —2.A least-squares fit to the experimental results

shown in Fig. 3 was performed neglecting thenext-neighbor-exchange interactions. The resultsare shown in Table II where they are comparedwith the values deduced by de Wijn et al. ' Theresults are in remarkably good agreement withone another.

More distant interactions might play an impor-tant role in determining the behavior of the mag-netic correlations near the percolation point; asnoted previously, the critical percolation concen-tration on a square lattice is 0.59 for nearest-neighbor interactions, "and 0.41 for next-nearest-neighbor interactions. Consequently, one of theprincipal objectives of our study of Rb,MnF, wasto determine the magnitude of the (nnn) exchangeinteractions. This is most readily observed bycomparing the energies of the spin waves forwave vectors at the [&00] and [g)0] zone boundaries.The difference between these two frequencies isgiven within the linear spin-wave approximationas

E(~, 0, 0) -E(E', ~, 0) = 2J2S.

Both of these frequencies were determined at avariety of different wave-vector transfers, twoof which are illustrated in Fig. 2. The resultingestimate for P from the difference between the

TABLE II. Parameters of the spin-wave dispersion relation in R12MnF4.

Parameters

J& (meV)

Goodnessof fitX2

Model I

0.6544+ 0.0014

0.0048 + 0.0010

0.016

Model II

0.6734+ 0.0028

0.0048 + 0.0007

0.0088 + 0.0013

0.005

de Wijn pt al.

0.657 + 0.008

0.0047

SPIN FLUCTUATIONS IN RANDOM. . .I. . . 4297

values is P = 0.010~ 0.003. An alternative estimateof P was obtained by a least-squares fit to all themeasurements shown in Fig. 3. The results arelisted in Table II and give P = 0.0088+ 0.0013.

One possible origin of the next-nearest-neighborinteraction in Rb,MnF, is the magnetic dipolarforces. These interactions are discussed in detailby de Wijn et al."who calculate the lattice sumsfor this structure. Their calculations show that

the dipolar forces split the degeneracy of the zone-boundary modes at the [g(0] zone boundary; the

splitting is 0.08 meV and the mean energy is re-duced by 0.006 meV. The energy of the excita-tions at the [F00] zone boundary is reduced by

0.009 meV. We conclude that dipolar forces giverise to a difference between the energies of the

two zone boundaries which is the opposite signand an order of magnitude smaller than we ob-serve. Consequently, the measurements show

there to be a next-nearest-neighbor antiferro-magnetic exchange interaction J,= 0.012 a 0.002meV. The nearest-neighbor-exchange interactionis 0.673 ~ 0.028 meV which when corrected forthe effects of spin-wave interactions" gives an

exchange constant of 0.653 + 0.027 meV.

IV. DYNAMICS OF EXCITATIONS IN Rb2Mn, Mg&, F4

The experiments were carried out with the sameexperimental configuration as described above ex-cept that coarser collimation was employed. Thecollimation before the monochromator was 20min, and after the monochromator, and both be-fore and after the analyzer, was 40 min. Theanalyzer angle was held constant throughout thescans so that the outgoing neutron energy was14.8 meV and a pyrolytic-graphite filter was in-serted between the specimen and the analyzer toremove those neutrons which would be reflectedby the higher-order reflections in the analyzer.In this configuration, the energy resolution of thespectrometer was typically 0.6 meV full width athalf -maximum.

Spin-wave excitations may be measured usingneutron scattering by means of the transversepart of the scattering cross section. If Sru is theneutron energy loss and Q the neutron wave-vectortransfer, the transverse part of the cross sectionfor unpolarized neutrons is given by'

d 0'

kf'(Q)-' 1+=*, S'(Q ~),d~ u meC 0 Q

(3)

where f(Q) is the form factor of the magneticions, and k, and k' are the wave vectors of theincident and scattered neutrons, respectively,and the spin direction has been taken to be the z

axis. The Van Hove scattering function is given

by

S'(Q, (o) =— dt e'2m

x pe'o 'tarn Rn~(S+(0)S„(t)+S (0)S'„(t)),

(4)where the summation is over all the magneticsites at R or R„, and the spin variables have

been written in the Heisenberg representation.The crystals were oriented with [010] axis ver-

tical [Fig. 1(b)] in a variable-temperature cryo-stat, and the spectrometer controlled in the con-stant-Q mode of operation with variable incident

energy. Ideally, the intensity observed in eachscan is then proportional to S'(Q, ~); the 1/k, fac-tor being cancelled by the 1/k, efficiency of themonitor counter. In practice, however, the mono-chromator reflects neutrons with wave vectorsproportional to 2ko 3ko and 4ko as well as thedesired neutrons with wave vector k, . These con-taminant neutrons are also counted, with steadilydecreasing efficiency, by the monitor detector,and it is necessary to correct the observed spec-tra for the effect of these neutrons on the monitor.This correction was estimated as follows": (i) bydetermining the number of higher-order neutronsusing the absorption of different thicknesses ofboron-loaded glass, (ii) by measuring the intensityof phonons in copper and comparing with the theo-retical cross-section, and (iii) by measuring themagnons in K,MnF4, "and comparing the intensitieswith those expected for antiferromagnetic spinwaves. The results of all these methods were inaccord with one another to an accuracy of 5% forincident neutron energies between 10 and 40 meV.The correction factor relevant for this experimentis shown in Fig. 4 arbitrarily normalized to unityfor an energy transfer of 6 meV.

Also shown in Fig. 4 are the observed distribu-tions for the zone-boundary wave vector ofRb,Mn, „Mg, 4,F4 at three different temperatures,1.1, 5.0, and 10.0 K. The background under theobserved distributions is shown by the dotted linesand is believed to arise largely from the roombackground and to be independent of energy trans-fer. These results show that there is considerablemagnetic scattering, but that it is spread between0 and 7.5 meV. Within this frequency interval,however, the intensity is not constant but showspeaks at 3.15+0.15, 4.85+0.15, and 6.40+0.15meV. The frequencies of these peaks may be com-pared with the Ising frequencies rSJ of the Mnions with the number of magnetic neighborsr=1, 2, 3, and 4; namely, using the results ofTable I: 1.63, 3.27, 4.90, and 6.55 meV. The

4298 COW LEY, SHIRANE, BIRGEN EAU, AND G UGGEN HEIM

o) 2Z0I. I

4J0

I.O

NEUTRON GROUPS IN Rb~ Mn g4Mg 46 F4

I' I

200

I 00 200

EOCU

I-(A 0

O

IOO

I 00

I & I i I ~ I

2 4 6 8ENERGY TRANSFER (meV)

2 054 g046 4FIG. 4. Zone-boundary S'(Q, ~) in Rb Mn0 M F

as a function of temperature. The upper part of thefigure gives the A, /2 correction factor C as discussed inthe text. The solid lines are guides to the eye.

agreement between the positions of the peaks andthese Ising frequencies suggests that the structurearises as a result of the characteristic frequenciesof Mn ions surrounded by different numbers ofneighbors. These results are qualitatively similarto those observed previously' in Mn»Znp „F,andMno 68Znp 32F2 but in the present case the struc-ture is very much more pronounced and easierto measure experimentally. A large part of thisis due to the fact that there are only five differentpossible configurations around each magnetic ionin this case, whereas there are nine in the rutilestructure. Figure 4 also shows that there is aconsiderable temperature dependence to the struc-ture. The peaks are less pronounced at 5.9 thanat 1.1 K, and have almost disappeared at 10 K.This temperature dependence is correlated withth e temperature dependence of the correlationlength described in Paper II.

Measurements were made at six different wavevectors along the [F00] direction in the magneticBrillouin zone in R12Mn, „Mg, „F,crystal. Back-ground was subtracted from the results and thecorrection applied for the effect of higher-orderneutrons on the monitor. The results are shownin Fig. 5. As the wave vector decreases towards

NEUTRON SCATTERING FROM Rb Mn IVlg F AT 4,0 K2 "0.54 g0.4t 4,0 ASCOMPARED N!TH COMPUTER SIMULATIONS

II I

II

I I I II

'I

Q-(06 0 29)

0L

4p

2

F) I

Z.'0I-I6

l2

84

IO

20

IO

.8,0,2.9)

Q = (1.0,0, 2.9)

I

~ aS0'4 6 8 2 4 6 8

ENERGY TRANSFER(meV)

FIG. 5. Dynamical response S (Q, ~) as a function of0 54 0 46 4. e so id lines arewave vector in Hb2Mn Mg F . Th 1'd

the Alben- Thorpe computer calculations as d d 'iscusse in

0&2

the zone center, the intensity of the scatteringincreases, and its weight moves to smaller ener-gies. Both of these results are to be expectedfrom a broadened antiferromagnetic spin-waveexcitation. The fine structure, which is observedmost clearly at the zone boundary, persists atall wave vectors, but is very much less pronouncedat the zone center.

The energies at which the peaks and shouldersoccur are shown in Fig. 6. There is little dis-persion to the energies of the structure at thehighest energies, but there is considerably moredispersion to the other bands. In Fig. 6, thesefrequencies are compared with the Ising energiescalculated with the exchange constant appropriatefor Rb2Mn, ,4Mgp y6F4 as discussed in Sec. VA.The observed bands lie between the Ising energiesand tend to lie just above an Ising energy at thezone center and just below the energy at the zoneboundary.

Less-detailed measurements were made onthe other sample Rb,Mnp 57Mgp Q3F„because theconcentrations of the two samples are very simi-lar, and no marked qualitative change in the spin-wave scattering with concentration was anticipated.The results for the zone-boundary response,after subtraction of the background and multi-plication by the correction factor (Fig. 4), areshown in Fig. 7. Comparing this result with Fig.5, indeed shows that there is little qualitativechange in the scattering with concentration. Asexpected, in the more concentrated sample theweight of the scattering is moved somewhat tolarger energies, but the energies of the peaks

15 SPIN F LUCTUATIONS IN RANDOM. . .I. ~ . 4299

FREQUENCIES OF THE STRUCTURE IN S(Q,cu)

FOR Rb2 Mn0. 54Mg 0.46 F4

I I

4lKLLI

LLI

2 —II li

00

I

O, I

I

0.2I

0.3I

0.4I

0.5

3.05+0.20, 4.95+0.15, and 6.50+0.15 are thesame as those found in the c = 0.54 sample.

V. COMPARISON WITH THEORY

A. Computer simulations

The first computer simulations of disorderedantiferromagnets were performed by Harris andHolcomb' for the Ma, Zn, ,F, system. Since thensuch techniques have been used to study theK,Mn, Ni, ,F4 system, and more recently theRb2MncMgl F4 system. "Kirkpatrick and co-workers4 calculate an expression for the neutronscattering by direct inversion of a large matrix.The procedure adopted by Thorpe and Alben' ismore allied to molecular dynamics and involvesintegration of an equation of motion. Thorpe andAlben have kindly made their computer programavailable to us, and for this reason, we give adetailed comparison with results obtained usingtheir techniques. From Ref. 4 we judge that close-ly similar, if not identical results, would be ob-tained using the Kirkpatrick method.

WAVE VECTOR ( C,O, O )

FIG. 6. Energies of the Ising peaks in Rb2Mno &Mgo 46F4as a function of wave vector at 4.0 K.

The computer simulation is begun by setting upan antiferromagnet arrangement of spins on asquare lattice. Some of these spins are then re-moved using a random-number generator so as togive the desired concentration of magnetic and

nonmagnetic sites. The excitations are then cal-culated by setting up a spin wave in the structureand numerically integrating the equation of motionof the spin wave which is deduced by using thenormal approximations of linear spin-wave theory.The resulting time evolution is Fourier trans-formed to obtain the correlation function Eq. (4).As noted above, the calculation was performedwith the aid of computer program supplied byAlben. The details are the same as these describedby Thorpe and Alben' except that the damping func-tion used in the Fourier transforms was exp(-At2)instead of exp(-At). The former choice gives aGaussian peak after Fourier transformation whichmore accurately simulates the experimentalresolution function than the Lorentzian form usedearlier. The parameter ~ was chosen so that thewidth of the Gaussian was the energy resolutionof the experiment.

The calculations were performed with a squarelattice containing 80x80 sites and the time develop-ment followed for 91 time steps. The weak next-nearest-neighbor interaction (Sec. III) was neglec-ted in the calculations, and the anisotropy fieldwas chosen to have the same value as in Rb,MnF, .Initially the calculations were performed with thenearest-neighbor-exchange constant having thesame value as in Rb,MnF, . The peaks in the struc-ture were then all at significantly lower energiesthan those observed. In the final calculationsthe nearest-neighbor-exchange interaction wastaken to be 0.72 meV and the results of the calcula-tions are shown in Figs. 5 and 7. The overallintensity normalization of the theory to the ex-perimental curves was determined by makingthe areas under the curves between 1.0 and 7.0meV equal at a single q vector, in this case, thezone boundary. The calculated spectra at the otherwave vectors shown in Fig. 5 were multiplied bythe change in both the form factor and the geo-metric factor (1+g/Q') with wave-vector trans-fer. The inclusion of the geometric factor isappropriate, as will be shown in Paper II becausethe majority of the spins are aligned along the c[001] axis at low temperature. That is, the clus-ters simulate finite-size Neel antiferromagnetsat the lowest temperatures; hence the inelasticscattering should be overwhelmingly transversein character.

The agreement between the observations andthe calculations shown in Figs. 5 and 7 is unusuallygratifying. The calculations accurately reproduce

4300 COWLEY, SHIRANE, BIRGENEAU, AND GUGGENHEIM

I

400—

NEUTRON SCATTERING FROM RbpMApg~Mgp43F4

COMPARED WITH COMPUTER SIMULATIONS

I I I I 1 I

Q = {0.5, 0, 2.9)T= l.22 K

~ 300'2

0

~ 200V)

UJ

Z'

I 00—

/ X

II ))

0—I ) I

4ENERGY (meV)

FIG. 7. Zone boundary S'(Q, u) in Rb2Mno»Mgo 43F4at 1.22 K. The solid and dashed lines are the Alben-Thorpe computer calculations for & =0.57 and 0.50 re-

spectivelyy.

both the frequency dependence and intensity of theneutron scattering for all the measured wave vec-tors. This success shows that linear spin-wavetheory is as appropriate for describing disorderedsystems as for pure systems.

In the calculations reported by Thorpe andAlben' for c = 0.5 and reproduced by the dottedline in Fig. 7, there is a sharp peak with an energyof approximately 1.7 meV, which they associateat least in part with the existence of isolatedtriads of magnetic ions. It is of interest that thispeak is largely suppressed in both the experimentand the calculations when the concentration is in-creased to c=0.54 or 0.57.

In evaluating the success of the computer simu-lations one should note the following. There are,overall, three parameters: the concentration, thenearest-neighbor exchange and the intensity scal-ing factor. As discussed in Sec. II, for the 54%sample, chemical analysis on a separate samplefrom the same boule yields c =0.55~0.02 whilethe percolation experiments discussed in Ref. 14and Paper II suggest c = 0.54 a 0.01. Variation ofc in the Alben-Thorpe theory by +2% producesonly a slight change in the theoretical spectra,although the optimal visual fit is indeed obtainedfor c = 0.54. Thus the concentration is not abona-fide adjustable parameter although if one ofthe above estimates were grossly in error onemight question some of the underlying assump-tions. Choosing c = 0.54 for the first sample thennecessitates c = 0.57 for the second; this value

agrees exactly with the percolation experiment onthe second sample. As is evident in Fig. 7, thecomputer simulations for S'(g, ~) (c = 0.57) at thezone boundary are also in very good agreementwith the experiment.

The exchange interaction J, determines thehorizontal scale factor for the theory. Using annn-only spin-wave theory, one finds J, = 0.65 meVfor Rb,MnF, (a=4.214 A) and J, =0.75 meV forK,MnF, (a=4.16 A). Since a=4.15 A for the ran-dom sample, one might have expected J, to beclose to the K,MnF4 value. In fact, the optimalJ1 in the computer simulation is 0.72 meV, inter-mediate between the above two values. In theabsence of an independent estimate of the truemicroscopic exchange, we can only comment thatthe value of J, deduced from the linear spin-wavetheory is most reasonable. Finally, since we havenot measured the scattering in absolute units onecan only evaluate the success of the theory inpredicting relative intensities. As noted above,with the intensity scale factor fixed by the zoneboundary spectra, excellent agreement is ob-tained across the magnetic Brillouin zone. Wenote that much worse agreement' is found in themixed magnetic system Rb,Mn, ,Ni, ,„F,.

B. Coherent-potential approximation

The coherent-potential approximation is theanalytic theory which is most frequently used todescribe the excitations in disordered systems.There are two approaches which have been em-ployed in its application to dilute antiferromagnets.The first approach by Buyers et al,.' was based onthe success of the Ising modeL in providing aqualitative understanding of the experimentalresults especially for wave vectors close to thezone boundary. ' The Ising part of the magneticinteractions gives the characteristic excitationenergy of each magnetic ion in its particularenvironment. The scattering of the excitationsthen arises because these energies are differentfor different environments and the effects of thisscattering are calculated using the single-siteCPA approximation" developed for electrons andphonons in alloys. The difficulty with this approachis in the treatment of the transverse parts of themagnetic interactions, and various different ap-proximations' have been used to incorporate theireffects at least approximately.

The second apprpach was developed by Harriset al."to overcome the difficulty in treating thetransverse interactions. They noticed that indilute ferromagnets and antiferromagnets" thechange in the Hamiltonian due to the presence ofthe nonmagnetic ions could be written as a sum over

l5 SPIN F LUCTUATIONS IN RANDOM. . . I. . . 430l

NEUTRON SCATTERING FROM Rb&Mn 54Mg+&F4 AT 4.0K AS COMPARED

WITH COHERENT POTENTIAI APPROXIMATION CALCULATIONS

I 'I

'I

~I

~ I

Q-(06 0 29)

IA 02

40

2I-u)

0—l6

l2

84

0

0

IO

6 8 2

ENERGY TRANSFER (meV)

Q=(0.8,0, 2.9)

,0,2.9)—

iI

FIG. 8. Comparison of CPA theory for 8'(Q, ~) withdata for Rb2Mno &4Mgo 46F4. The solid and dashed curvesrepresent CPA calculations using the hard-core andempty-core approximations, respectively.

the nonmagnetic sites. They were then able to in-clude both the Ising and transverse parts of theexchange interactions in solving for the scatteringof each nonmagnetic ion surrounded by its neigh-bors. Unfortunately, in this treatment all themagnetic ions are treated equivalently, and sothere is no distinction made between magnetic ionssurrounded by different numbers of nonmagneticions. As a result S'(Q, cu) has a single peak foreach Q, and the model fails to account for the finestructure shown in Figs. 4-7. Detailed calcula-tions for the dilute (2-D) square lattice have notbeen made with this theory. Calculations have,however, been made" with the theory of Buyerset aL

Calculations were performed for a concentrationc = 0.5 with nearest-neighbor-exchange interac-tions. Considerable difficulties were experiencedwith the convergence of the iterative procedureto obtain the coherent potential, and the resultswere obtained only after many (-40) iterations andusing a reduction factor of 0.4 to reduce the cal-culated change in the coherent potential after eachstep. Even then, the potential had not completelyconverged for energies close to %.0 meV.

These calculations employed the same approxi-mation to treat the transverse parts of the ex-change interactions at Buyers et al. ', namely,that the frequency dependence of the off-diagonalpart of the self-energy matrix was the same asthat computed for the diagonal part. Because ofthe approximations inherent in the coherent po-

tential approximation, it is necessary to ensurethat the magnetic excitations do not propagateonto the nonmagnetic sites. One way of achievingthis is to place a large repulsive potential at eachnonmagnetic site" (hard core), and the other isto omit the nonmagnetic ions' (empty core). In

Fig. 8, the results of both of these calculations,convoluted with the experimental resolution, arecompared with the measured spectra. The calcu-lations are only in qualitative accord with themeasurements and give fine structure which isconsiderably more pronounced than that observedexperimentally. This is clearly a much moresevere test of the coherent potential approximationthan earlier measurements for which the CPAtheories appeared to be more successful as re-viewed in Ref. 1. The reason for this may be tiielower coordination number which makes the ap-proximations inherent in the CPA more suspect,and which also makes measurements of the de-tailed spectral shape of S'(Q, &v) easier. We hopethese measurements will lead to efforts to im-prove the CPA theories for these systems.

Vl. CONCLUSIONS

In this paper we have presented measurementsof the spin-wave excitations in pure Rb,MnF, andthe disordered materials Rb,Mn, Mgy F4 withc= 0.54+0.01 and 0.57+0.01. The results for thepure material show that the nearest-neighbor-exchange constant deduced from macroscopic mea-surements" is very accurately correct. Thereis also a weak antiferromagnetic next-nearest-neighbor-exchange constant.

In the disordered materials, the spin-waveexcitations are damped to the extent that thescattering for each wave vector is spread overthe whole of the spin-wave band. The weight,however, tends to lower frequencies near thezone center. The scattering does show consider-ably more marked structure than was observed'in the analogous 3-D materials. This structurearises from the different characteristic excita-tion energies of magnetic ions with different num-bers of magnetic ions as neighbors. One of themost interesting aspects of our results is theexcellent agreement obtained with computer simu-lations based on linear spin-wave theory. ' Thissuccess shows that linear spin-wave theory doesprovide an appropriate description of the excita-tions in disordered as well as ordered magneticmaterials. Analytic theories based on the coherentpotential approximation are less successful atproviding understanding of our results. Clearlyour experiments suggest that further refinementsof the CPA would be worthwhile. Finally, our

4302 COWLEY, SHIRANE, BIRGENEAU, AND GUGGENHEIM 15

measurements show that the spin-wave spectraof the disordered materials is temperature depen-dent even below 10 K. %e do not know of anytheories of the temperature dependence of mag-netic excitations in disordered systems.

ACKNOWLEDGMENTS

The computer program used to calculate thecomputer simulations was provided by Dr. R. Albenfor whose assistance and encouragement we aregrateful.

*Permanent address: Dept. of Physics, University ofEdinburgh, Scotland.

~Work at Brookhaven performed under the auspices ofthe U.S. Energy Research and Development Agency.

~Work at MIT supported by the NSF MRL Grant No.DMR12-03027-ADB.

'For a review, see R. A. Cowley, AIP Conf. Proc. 29,243 (1976).

W. Marshall and S. W. Lovesey, Theory of ThermalNeutron Scatk~ing (Oxford U.P., New York, 1971).M. F. Thorpe and R. Alben, J. Phys. C 8, L275 (1975);9, 2555 (1976).

S. Kirkpatrick and A. B. Harris, Phys. Rev. B 12, 4980(1975); W. K. Holcomb and A. B.Harris, AIP Conf. Proc.24, 102 (1975); S. Kirkpatrick, ibid. 29, 141 (1976).

W. J. L. Buyers, D. E. Pepper, and R. J. Elliott, J.6

Phys. C 5, 2611 (1972).G. J. Coombs and R. A. Cowley, J. Phys. C 8, 1889(1975).

7J. Als-Nielsen, R. J. Birgeneau, H. J. Guggenheim,and G. Shirane, Phys. Rev. B 12, 4963 (1975); J. Phys.C 9, L121, (1976).

G. J. Coombs, R. A. Cowley, W. J. L. Buyers, E. C.Svensson, T. M. Holden, and D. A. Jones, J. Phys.C 9, 2167 (1976).

O. W. Dietrich, G. Meyer, R. A. Cowley, and G. Shi-rane, Phys. Rev. Lett. 35, 1735 (1975); E. C. Svensson,W. J. L. Buyers, T. M. Holden, and D. A. Jones, AIPConf. Proc. 29, 248 (1976).For reviews, see V. K. S. Shante and S. Kirkpatrick,Adv. Phys. 20, 325 (1971); J. W. Essam, in PhaseTransitions and Critical Phenomena, edited by C. Domb

and M. S. Green (Academic, New York, 1972), Vol. II,p. 197.D. Stauffer, Z ~ Phys. 1322, 161 (1975); H. E. Stanley,R. J. Birgeneau, P. J. Reynolds, and J. F. Nicoll,J. Phys. C 9, L553 (1976); T. C. Lubensky, Phys.Rev. B 15, 311 (1977).D. J. Breed, K. Gilijamse, J. W. E. Sterkenberg, andA. R. Miedema, J. Appl. Phys. 41, 1267 (1970);Physica (Utr. ) 68, 303 (1973).See, for example, S. Kirkpatrick, Phys. Rev. Lett. 36,69 (1976), and references therein.R. J. Birgeneau, R. A. Cowley, G. Shirane, and H. J.Guggenheim, Phys. Rev. Lett. 37, 940 (1976); (unpub-lished).

~5H. J. Guggenheim (unpublished).~6D. Balz and K. Pleith, Z. Elektrochem. 59, 545 (1955).

R. J. Birgeneau, H. J. Guggenheim, and G. Shirane,Phys. Rev. B 1, 2211 (1970).H. W. de Wijn, L. R. Walker, and R. E. Walstedt, Phys.Rev. B 8, 285 (1973).D. E. Cox, G. Shirane, R. J. Birgeneau, and J. B.MacChesney, Phys. Rev. 188, 930 (1969).H. W. de Wijn, L. R. Walker, S. Geschwind, and EI. J.Guggenheim, Phys. Rev. B 8, 299 (1973)~

H. Patterson, G. Shirane, and R. A. Cowley (unpub-lished).R. J. Birgeneau, H. J. Guggenheim, and G. Shirane,Phys. Rev. B 8, 304 (1973)~

23P. Soven, Phys. Rev. 156, 809 (1967).A. B. Harris, P. L. Leath, B. G. Nickel, and R. J.Elliott, J. Phys. C 7, 1693 (1974).W. K. Holcomb, J. Phys. C 7, 4299 (1974).