Structural and magnetic properties of Mn1−tTitAs

J Phys Chem Solrds Vol 46, No 3, pp 275-286,

1985

W22-3697/85 $3 00 + 00 0 1985 Pergamon Press Ltd

PnntedmthelJSA

STRUCTURAL

AND MAGNETIC OF Mn 1_,Ti,As

PROPERTIES

ANDRZEJ ZIFBA Institute of Physics and Nuclear Techmques, Umverslty of Mmmg and Metallurgy, 30-059 Krakow, Poland

and HELMER FJELLVAG and ARNE KJEKSHUS Department of Chemistry, Umverslty of Oslo, Bhndem, Oslo 3, Norway (Recewed 17 November 1983, accepted 12 Aprd 1984) Abstract-The pseudo-bmary TIAs-MnAs system has been mvestlgated by X-ray and neutron dIffractIon as well as magnetic measurements (mcludmg high fields and apphed pressures), The phase diagram IS dommated by the NIAS type structure, which prevads up to a mlsclblhty gap at t c 0 95, m a ferro- or m a paramagnetlc state Two seqarate domams of the MnP type structure, a paramagnetlc state above Tc and a metastable hehmagnetlc state at low temperature, are observed for a small amount of TI The relation to other Mn,_,T,As (T = V, Cr, Fe and Co) phases reveals that TI substituted MnAs behaves as a ‘negative’ pressure system The first-order ferro- to paramagnetlc transItIon m Mn,_,TI,As becomes contmuous at 1 = 0 10, and Tc,falls to zero at t m 0 65 The Tc vs t dependence IS drscussed rn terms of a magnetostnctrve model where the ddutlon Imposed by the non-magnetic TI atoms plays a major role The dllutlon suppresses the range of the first-order transItIon The slgmficance of the amsotropy of the elastic propertles and the exchange mteractlons IS examined 1. INTRODUCTION

gtve a supplement to, and in part a revtston of, the findings of Ido et al [6].

Solid state chemists and phystctsts have been interested m manganese monoarsenide and ternary denvauves thereof for a long time Most of the Interest stems from the pecuhar structural and magnetic transmons which MnAs, Mn,_,T,As (T = tranntton metal) and

2. EXPERIMENTAL

MnAs,_& (X = metalloid) undergo wtth temperature, pressure, composltton (vartatlon of T, t, X or

x) and magnetic field Among the ternary (pseudo-binary) solid solutron MnAs-TAs systems attention has hitherto largely been focused on T = V, Cr, Fe and Co (see Ref [ 1] and references therein), the mam motivation for this choice being that substmmon of these T metals mto MnAs stabthzes the MnP type structure at lower temperatures The chemical substttutron thus stmulates the same effect as an apphed hydrostatic pressure [2]. If, on the other hand, T = Tt, the NrAs type structure is stabthzed, and the substttutton of this element accordingly stmulates the effect of a ficttttous negative pressure. A stmtlar srtuatton arises for MnAs,_$b, [3, 41. The present paper gives an account of a structural and a magnetic study of Mn,_,Tt,As. When the work was started only the sparse mformatron provrded by the somewhat differently armed mvesttgatton of Val’kov et al [5] was at hand, and recently a parallel paper on the SubJect by Ido et al [6] has appeared. These papers consider only the ferro- to paramagnettc transttron whereas the present contnbutron attempts to gtve an account of the overall phase diagram. The expenmental results as well as their interpretation

Samples were made from 99 97% Tr (sponge; Johnson, Matthey & Co, London, England), 99 99% Mn (crushed flakes, Johnson, Matthey & Co ) and 99 9999% As (lumps, Koch-Light Laboratones, London, England). Equthbna are most readily attamed when TtAs and MnAs are used as mtermedrates for the ternary sample preparations MnAs was synthettzed as described m Ref. [7] TrAs (cw-TtAs)was obtained after two heat treatments at 1OOO’C with intermediate crushing. Care was taken to vary the temperature slowly dunng heating (- 150°C d-r) and cooling, and the sample was kept at the maximum temperature for 3 d. Ternary samples were made stmrlarly from the bmary compounds by two or three heat treatments (intermediate crushmgs) at 900 (for t < 0 30 m Mn,_,Tr,As) or 1000°C. The samples were kept at the maximum temperature for 5 d. Expenmental detarls concemmg X-ray and neutron dnfractron and DSC measurements are described m Ref. [S]. The nuclear scattenng lengths (m lo-l2 cm) b,, = -0 37, &, = -0 34 and bAs = 0 64 were taken from Ref. [9], and the magnetic form factor for manganese from Ref. [lo]. A vanety of different magnetometnc techniques was utdlzed an a.c susceptrhhty arrangement (see Ref. [l] for establishment of transition temperatures), two different kmds of Faraday balances, a vtbratmg sample magnetometer wrth a 60 kOe superconductmg

275

276

A ZIFBAet

al

magnet (for determmatlon of saturation moments), a 250 kOe pulsed field magnetometer (for constructlon of magnetic field vs temperature phase diagrams) and a 15 kbar pressure equipment with an a c susceptlb&y sensor (for stlpulatlon of pressure vs temperature phase diagrams) 3. PHASE DIAGRAM Summanzmg phase diagrams for the pseudo-bmary systems TlAs-MnAs and the MnAs nch region of VAs-MnAs are presented m Rg 1 The TlAs-MnAs diagram 1s constructed from the present X-ray and neutron dlffractlon, magnetic susceptlblhty and magnetization measurements For t 5 0 70 the TlAsMnAs diagram IS entirely consistent with the data for Tc gven m Refs [5, 61 The VAs-MnAs sector 1s based on earlier findmgs [ 1, 2, 111, supplemented with unpubhshed data for re-entrant Cune temperatures ( Tc r) by Zach [ 121 and some new values for the MnP,P * NlAs,P type transition temperature T, The Juxtaposltlon m Fig 1 1s made to emphasize the close structural and magnetic connections between the two systems In this way the findings on each side of the borderline (MnAs) mutually lend support to each other The contmuous vanatlon of the phase boundanes through the common MnAs and their shapes also support the pressure analogy m Section 1 Apart from the possible transltlon to a spm glass state for t > 0.60, the composite phase diagram contains, we believe, all existing phase boundanes below 1000 K 4. CRYSTAL STRUCWRE

DATA

The extent of the mlsclblhty gap in the TlAsMnAssystem(Flg 1,092~002It4098fO02

Mn,.,T+

As ‘00

D

D -t

t-

Fig 1 Composite diagram of phase relauons In the pseudobmary TIAs-MnAs and MnAs-VAs systems Data for Mn,-,V,As are taken from Refs [I, 2, 11, 121 Structural state IS mdlcated by type deslgnatlon, magnetic state by F ferro, H helical and P para To denotes the MnP,P 4 NlAs,P type (dlstortlon) transrtlon and Tc,,, rc,c and Tc,r dlstmgulsh the Cune temperatures on heating, coohng and re-entrant condltlons, respectively Results marked 0, 0, Cl, V are obtained from magnetic susceptlblhty, A by X-ray dlffmctlon measurements

;j/cr --__ ---_ _ ,’

t’

/’

I’

a P

F

00

02

04

. . .

06

06

10

t

Fig 2 Room temperature umt cell dlmenslons of Mn,_;Tl,As as a function of composltlon c/2 IS plotted for the TIP type structure of TIAS nch samples Concemmg the broken curve portlons see text Error hmlts do not exceed twice the size of the symbol (I A = IO* pm ) for slowly cooled samples) has been determined from the vanatlon of the hexagonal unit cell dlmenslons with the composltlonal parameter t (Fig 2), the solublhty limits bemg also confirmed by application of the dlsappeanng phase pnnclple on the X-ray powder data The mcomplete mlsclblhty m Mn, _,TI,As 1s apparently not observed by Ido et al [6] However, the present result 1s fully consistent with the fact that TlAs and MnAs take related, but different structure types (TIP and NlAs, respectively [ 131) at room temperature The reported [ 13- 151 high temperature and/or excess TI stablhzed moddication of TlAs (pTlAs) with a NlAs type structure was not observed m this study which was confined to temperatures less than 1OOO’C and metal/non-metal atomic ratios of 1 00 As seen from Fig 1 there 1s a change of the magnetic state (namely F to P) near room temperature for t < -0 40 (Tc > 293 K for 0 00 5 t 5 0 30 and Tc < 293 K for t L 0 40) This 1s not reflected m the vanatlon of the umt cell dlmenslons vs t curves shown m Fig 2, where a slight cusp m V vs t at t = 0 10 1s the only sign of u-regulanty (vlde znfra) The unit-cell dlmenslon vs composition relatlonshlps m Fig 2 are consistent with the room temperature values read off from the lllustratlons of Ido et al [6] The overall temperature vanatlons m the unit cell dlmenslons of MnO g8Tb 02A~ and Mn, 96Tb,,4A~ (Rg 3, as denved from Debye-Scherrer X-ray photographs obtained m a Umcam camera at fixed temperatures) may illustrate the thermal expansion m the Mn nch samples The dlscontmuous, combmed crystallographic and magnetic NIAs,F to MnP,P type transltlon

Structural and magnetic properttes of Mn,_,Tt,As

277

of the Umcam camera, the MnP type regton could not be established by this technique. However, on turning to the focusing condrttons of an EnmfNomus Guuuer Simon camera the slight MnP type drstortron manifests r&elf as hnebroadening (hnesphttmg for MnAs) of charactenstrc reflections The hnebroadenmg (hnesphttmg) gtves a measure of the deviation of (c/&,,,~ from ,/3, the latter providing a sufficient, but not necessary condttton [8, 161 for an MnP type atomic arrangement (The Pnma setting IS used for the MnP type unit cell, and m an NtAs to MnP type transttion aMnp = cN,&, hnp - aNIASand

(4 z 569

,I

q;/, 900

TM)

1100 1300

CM~P

=

J(3)a~,~s;

CMnP ’

&nP

’

&nP

)

Maximum dtstortton IS expected to be found Just above T, (TC,h or T,,= dependmg on heating or cooling condmons) [ 16, 181 Examples of thermal expansion curves as obtamed by the Guuuer Simon technique are gtven m Ftg 4, but the minute orthorhombic MnP type dtstortton of the very nearly NtAs type arrangement could not be represented m these diagrams The vanatrons of the hnebroadenmg (hnesphttmg) with temperature and cornpositron are m accordance wrth the above descnpnon and the relevant sector of Fig 1 The followmg data were e&mated from the Guuuer Simon photographs t, TC,h, T,, (c/b)MnPat TC,h, 0 00, 315 + 5 K, 395 + 10 K, 1 737 + 0.001, 002, 310 k 5 K, 365 & 10 K, 1.735 +O.OOl; 0.04, 310 f 5 K, -, 1732 Z!Z0.001; 0.06, 310 + 5 K, -, J3,0.08, 310 + 5 K, -, J3 (TC,h = 320 zk 3 K was obtamed by DSC for 0 00 5 t SOOOS)

g’,*/,I 200

LOO

600

T(K)

800

lcoo

12co

Fig 3 Thermal expansron of (a) Mno98Tloo2Asand (b) MQ,~~TI~&s as denved from Debye-Sherrer X-ray photographs Error hmlts do not exceed twice the size of the symbol Hysteresis region of the NIAs,F to Mn,P type transItIon IS not shown, see Fig 1 Concernmg the MnP,P Fi NIAs,P type transltlon see text (I 8, = 102pm )

( Tc,h; Tc,c see Fig. 1) 1s here evtdent as an abrupt change m the umt cell dtmensrons The dlscontmumes m a vs T are unequivocally established for 0 00 i t < 0 10 (Figs 3 and 4(a), see also Ref [6]), whereas those m c vs T are less definite due to scarcity of decisive reflectrons m the X-ray photographs. The first-order component of the NtAs,F to MnP,P type transmon gradually vamshes wtth mcreasmg Tt content, and seems to disappear at t = 0.10. With the available expenmental means tt 1s drfficult to settle whether the transttton 1s of first order or of enhanced second order Due to the small &storttons involved m the secondorder MnP,P @ NiAs,P type transition (at T,, see Fig. 1 and Refs [ 16, 17]), and the limited resolution

Thermal expansion curves were also utilized to estimate the magnitude of the magnetostnctrve effect at the NtAs,F to MnP,P or NrAs,P type transttton on the room temperature unit cell dimensions of Mn,_,Tt,As. The broken tarls on the a, c, V vs t relattonshrps m Fig. 2 are denved by extrapolatmg the structural sttuanon m the (high temperature) NrAs,P and/or MnP,P type state to room temperature The compontronal range concerned ts 0.00 $ t d 0.35 which IS the domam of the F mode at room temperature (see Section 5 and Fig. 1) The crystal structure 1s confirmed to be of the NtAs type wrth random (long-range order) dlstnbunon of Mn and Tt atoms over the metal sublattrce. An appreciable and probably fluctuating degree of shortrange order (namely metal paumg etc ) IS absolutely present m the samples, but this aspect could not be pursued by us Structural data at 10 K as obtained by neutron dtffractton are gtven m the caption to Ftg 4.

5. MAGNETIC PROPERTY DATA The account of the magnetic properties of Mn,_,Tt,As conveniently starts wtth the NrAs,P type state which prevails over considerable ranges of composttron and temperature (see Ftg 1) Figure 5 shows the high temperature charactenstrcs of the inverse magnenc suscepnbthty (x;‘(T)) for selected

218

A

ZIE+BA

et al

i

(4

"

572 04

E

571t/ 372 .* 371E

370-

*: z m

169 -

a \

a 3.663671, 260

/ ,

,

,

300

320

340

, , 360

,j 380

3671, 260

, ( , , 300

320

, , 340

, , 360

360

3660t,

1

260

,

/

,

300

320

T(K)

T(K)

,

I , , 340

, , ,I

360

360

400

T(K)

Fig 4 Thermal expansron of (a) Mn,98T~02A~, (b) M~I,,~TI,,,&s and (c) MII,,~TI,~& as denved from Gunuer Simon X-ray photographs See also the captron to Frg 3 (Umt cell dlmenslons at 10 K denved from neutron dlffmctlon data (m pm) t, (I, c 0 00, 372 20(5), 566 25(8), 0 04, 372 10(4), 568 35(7), 0 08. 371 96(2), 571 71(4), 0 20, 368 72(3), 580 68(3), 0 50, 366 14(2), 598 88(3))

samples as determined by the Faraday technique The Cune-Weiss law IS fulfilled for 0 00 5 t $ 0 60 (TtAs exhtbtts a vutually temperature independent (Pauh) paramagnetum, see also Ref [ 191) Although the Cune constant C,,, decreases wrth increasing values oft, the effective (para)magnettc moment per Mn atom (CL,@= ,/(SC,,,,,)/( 1 - t)) remams constant (wtthm +O 2pa, see Ftg 6) at 4 45~~ quoted [2] for MnAs The latter value corresponds to 2s = 3 7 -t 0 2 (S IS the spm quantum number denved accordmg to the ‘spin only’ approxtmatton, peff = gJ(S(S + 1)) with g = 2) which m turn 1s somewhat htgher than the equally constant saturatron magnettzanon moment at 4 2 K and 58 kOe steady field (CL= 3 45 + 0 05j~a per Mn atom, 0 00 I t < 0 60). The constancy of p=ffand M1s an important key to the understandmg of the magnetic properties of Mn,_,Tt,As, since this suggests that only the Mn Mn,_,Tl,As

atoms

contnbute

to the magnetic

moment

(I e the

Tt atoms behave as magnettcally featureless diluters) The vanatrons of 8 (extracted from Ftg 5) and Tc (determined from a c suscepttbihty data, Ftg 7) wtth t are also mcluded m Ftg 6 Smce the x&T) curves for Mn,_,Tt,As (Fig. 7) change rapidly m the cnttcal regron around T, the actual choice of the cntenon for the sttpulatron of Tc (namely the mflectron pomt, which was m fact adopted m this work, the half value, the extrapolated kmk pomt, etc ) 1s of minor importance The x&T) curve for t = 0 04 (with expanded temperature scale) shows that the NtAs,F to MnP.P

60 50 40 30 I 20 IO 0 300

LOO

600

500

700

600

T(K)

Fig 5 Reciprocal magnetic susceptlblhty of Mn,_,T1,As in the NlAs,P type state

Frg 6 Effective magnetic moment (P.R, estimated error limit f 0 lla), saturation moment at 4 2 K (p, error f 005&, Weiss constant (8, error f 10 K) and Cune temperature (Tc, error tncreasmg from fl to -15 K for increasing t) as functions of composktion Inset gwes magnetization (M(H)) curves for t = 0 60 and 0 70

%ructural and m~netIc properties of M~,.+TI,As

279

Mri,.tTit AT’

E-

x

T (Kf

Fig 7 Magnetic susceptibdtty xAc vs temperature around Tc for selected samples of Mni_,TIIAAsNote that the temperature scale IS considerably expanded for t = 0 04

type transition for this sample is accompamed with a hysteresis loop of - 1 K width. Thus, the present xAC data cot&m (at least for 0.00 6 t I; 0 04) that thts transttton IS of first order In the TI rich part of the homogeneny range (Frg I), Tc drops to zero for 0.60 < t < 0 70, as evidenced by the magnetl~tlon vs applied field curve for t = 0 70 (see inset to Fig. 6) whtch shows no spontaneous magneti~tlon. This suggests that the magnenc moments on the Mn atoms (which stall take appreciable values of pL,kand 8, cf Fig 6) may disclose spur glass behavtour at low temperatures The observed constancy of the magnetic moment per Mn atom agrees with the findings of Ido et al [6] However, the latter authors display Tc values well above 100 K for samples with t = 0.70 and 0 80, as opposed to the present study which gtves no evtdence for ferromagnetrsm for t 2 0 70 The competttton between the NIAS and MnP type structures m the phase sector between the boundaries Tc and Tn (Fig 1) 1s an mterestmg point about the Mni _(T,As phases The x;‘( 2”)curve for Mn, 98Tfo&s (Fig. 8) shows deviation from the Curre-Weiss law Just m the range between Tc and TD, but the departure ts less conspicuous

for the MnP,P

kOe) IS here taken as the mflecnon pomt on the x(T) curve As seen from the values given on Fig 8 the distortion temperature determmed by the magnetometnc and X-ray dtfhactton means agrees withm the estimated error hmtts. In the domam of the MnP,P type state between Tc and TD it is also possrble to mduce the NiAs type atomic a~ngement by the apph~~on of a sullenly strong magnetic field [ 1, 17, 201 Hence, the H,T phase diagrams of Mn,+TltAs and Mn,_(V,As wtth small t were explored The values of the transttron

fields for the field mduced MnP to NiAs type transformation were determmed from the mflection pomts of magnetmatron vs apphed field [l] The thus deduced H,T phase diagrams for Mno98Ti,,02As and Mnc,99VoolAs are grven m Fig 9 together wtth the diagram for MnAs [20]. The transition field IS found to decrease wtth mcreasmg t for Mn,+Tt,As and to increase with mcreasmg t for Mni_tVgAs. The latter

M”oxYoo,A=

I

than for MnAs. The value of TD

F? NtAs,P type transitton

(at 12 6

300

320

340

360

380

T(K)

0

300 1

350

400

T(K)

ml

1’ O

Rg 8 Magnetrzatton and reciprocal magnetic susceptlbthty for MG~~T~,,~As as functions of temperature Increasmg and decreasrng temperature condmons are mdrcated by arrows

Fig 9 H, T phase diagrams for M% 98T~ozAs, Mm %VoolAs and MnAs The diagram for MnAs IS quoted from Ref [20] Sohd data points correspond to the MnP(P) to NlAs (spm aligned state) type transmon, open data points to the reverse WanWon The arrows at the bottom of the dlustfa~on @ve r,, and Y&i,,for MG~~T~~~As and MQ,~~V~O,AS A gtves To determmed by X-ray ddfiictzon

280

A

ZI~BA

observatton is m agreement wtth the earher findmgs [1], and the data for both phases moreover concur with the expectations on the basis of the composite phase dtagram for TiAs-MnAs-VAs (Fig 1) It should be noted that the H,T diagrams for Mn, 98Tic,,,*As and Mn, 99V0o,As (Fig 9) are deduced from data collected under nearly adiabatic condtttons by the pulsed field technique, whereas the MnAs diagram is obtained under the steady field conditton The phase boundanes of MnoggTiOOZAs and Mn,, 99V0olAs are accordingly shifted m temperature due to the magnetocalonc effect [1] For the truly isothernnc condttion the transition field boundanes should start at the temperatures Tc,c and Tc ,, (vrde sup-u) which are indicated by the arrows m Fig 9 The Mn,_,Tt,As samples with t = 0 04 and 0 06 dtd not show stgmotdally shaped magnetization vs applied field curves (1 e the field induced MnP to NtAs type transition was not detected) above T, The deviattons from the Cune-Weiss law for the x;‘(T) curves of these samples are less pronounced than for MQ, ssTb 02A~and correspond to the findings for nearly any ordinary ferromagnet lust above Tc As pointed out m Section 1, the MnP type atomic arrangement also prevails at low temperature m the MnAs-TAs systems wtth T = V, Cr, Fe and Co, connected with the hellmagnetic H, mode [8] For Mn,_,Tt,As the MnP,H, state is metastable at atmosphenc pressure, but tt can be released at low temperature by application of an external pressure which is strong enough to suppress the NtAs,F mode [2] The minimum pressure necessary to induce the MnP,H, state in Mn,,96Tb04As is -4 kbar (Fig 10) as compared with -2 kbar for MnAs The re-entrant Cune temperature Tc,r 1s obtained when the MnP,H, phase transforms mto the NiAs,F phase on heating Figure 10 is included merely to illustrate how the data for Tc,r m Ftg 1 are denved A more detailed account of the pressure induced properties of Mn,_,Tt,As will be gtven elsewhere [21] It should also be noted that the speaficatton of the hellmagnetic mode to H, (although very likely correct) is subject to future expenmental venfication for Mn,_,Ti,As, t f 0 The assumption has recently been confirmed for t = 0 [22]

P(kbar)

Fig 10 P,T phase diagram for Mn, 96Tb&s Abbrewatlons are defined m the caption to hg 1

el

al

28 (“1 FIN I I Neutron powder dd?iict~on diagram of Mn,,92Tb08As at 10 K (A = 187 7 pm) The AI reflections originate from the Dlsplex coolmg system

6. THE FERROMAGNETIC STRUCTURE Neutron diffraction experiments were performed on selected powder samples of Mn,_,Ti,As m order to obtain mformatton about the ordered magnetic state Smce the powder neutron diffractton patterns of Mn,_,Ti,As (Fig 11 shows the diagram for t = 0 08 at 10 K as an example) contam relatively few, mixed nuclear and magnetic reflections, special consideration had to be given to the Rietveld [23] analysis of these data Both the metal and non-metal atoms of the NiAs type atomic arrangement [ 131 are m fixed positions, and the structure is accordingly completely specified when the dtmenstons of CI and c are given The accommodation of an F mode within the NiAs type lattice requires (when limited to powder neutron diffraction) specification of the components of the magnetic moment (pp) parallel and perpendicular to the hexagonal c axts (the onentatton of the moment within the basal plane bemg undetermined) Other vanables for the Rtetveld analysis are two atomic, isotropic thermal vibration parameters (& and Bx). three half-width parameters and one scale factor The different vanables are coupled, and m order to permit comparison between the different compositions examined, tt is therefore essential that the same vanables are used m the final refinements The magnetic form factor of manganese also enters as a pseudo-vanable which affects the denved value of Pi? The form factors for Mn, Mn+, Mn2+ and Mn3+ [lo] were tried m prehmmary calculattons for vanous refinement models and different sets of neutron dtffractton data Generally the form factor for Mn*+ came out with the lowest rehabihty factors (R,, R, and Rp, cf. Ref [23]) and the most reasonable values for kLFas Judged from comparison with p m Fig. 6 The form factor for Mn3+ gave too low and Mn and Mn+ too high ~1~values Although these findings were somewhat unexpected m view of the clear preference gtven to Mn+ (for MnAs) by Haneda et al [24], the Mn2+ form factor was adopted m the following calculations

Structural and magnetrc propertles of Mn,_,Tl,As The relatively many (although certamly not equally important and physically srgmficant) varrables compared wtth only 8-12 observed (namely wtth nonzero mtensity) reflections led to secondary mnnma m the rehabrhty functrons (m thts respect functtons rather than factors) and/or other convergence problems. This made it necessary to put constramts on some of the vanables The thermal vrbratton parameters were consequently fixed at values (Br = Bx = 0 2 X lo4 pm* at 10 K, Br = 0.7 X lo4 pm’, Bx = 0 9 X lo4 pm* at 293 K) esttmated from the Debye temperature for MnAs (310 K [25]) Each set of neutron dtffractron data was then subject to refinements m terms of two essentiahy different models, one where pr was allowed to form any angle (Ywith the hexagonal c axis and one wtth pr constrained to the basal plane (1 e (Y = 90”) Several refinement cycles, wtth quite different input values, were camed out in order to ascertain that the calculatrons were not terminated at a secondary mmtmum of the rehabthty functtons. The rehabthty factors obtamed after this treatment were generally low (R, = 0.02-O 05, R, = 0 02-0.07 and Rp = 0.06-O 11) and only slight, msigmficant (e.g according to the Hamilton [26] test) improvements were obtamed for the model wrth the addmonal degree of freedom (A comparison of observed and calculated integrated mtensmes for the dtffractron dragram depicted m Fig. 11 1s grven m Table 1.) The shtfts of pr out of the basal plane (65” 5 (Ys 85”) exceed (true enough) usually three or more standard devtatrons, but this appears to be an arttfictahty in view of the mentioned msrgmficant tmprovements in the rehabthty factors. This rmphes that magnetrc structure data m Ref. [I] should be corrected to (Y = 90” (for or see Fig 12) The vanatrons of or wrth t at 10 and 293 K are shown in Fig. 12 (The overall ,.@values denved for the most relaxed refinement model are included in Ftg 12 to demonstrate that the two models gtve nearly tdenttcal moments.) The linear k(F vs t relatronshtps show that the magnettc moment per Mn atom 1s constant as IS also illustrated for the 10 K

Table 1 Observed and calculated neutron dlffractron data for MnoplTbosAs (10 K) Space group P6Jmmc; Mn/Tl m 2a and As in 2c, c(F = 3 25(3)pa perpendicular to c axis R,=O020,R,=O025andR,,=O078 hkt

I

I

In

Illl

100

14022

6672

20694

20524

002

7974

3102

11076

11114

101

13892

0

13892

14457

102

70

5314

5384

5699

110

773

968

1741

1636

103

5323

0

5323

5198

C&C

obs

200

4018

503

4521

4409

112

16365

1119

17484

17429

201

4875

0

4875

5196

281

Mn14Ti,As 40-

oo-, 000

010

020

t

, 030

, a40

,050

Fig 12 Ferromagnetic moment (c(~) at 10 and 293 K as function of t m Mn,_;Tl,As Open symbols refer to a refinement model where p(Fcould take any angle a wth the c axls, sohd symbols to a model with Q constramed to 90’ The bars correspond to calculated standard devlatlons

data on the top of Fig. 12 The value & = 3 6~Lgper Mn atom at 10 K IS slightly higher than P = 3.45 f 0 05~~ found by the magnetometnc method, but attention should not be paid to this dtscrepancy m view of the uncertamty imposed by the choice of the magnetic form factor. The gp vs t lines for 293 and 10 K in Fig 12 extrapolate to pr = ORBfor t = 0.45 and 0 95, respectively Hence, the ,.& data are fully consistent with the other results (notably m Figs 1 and 6) obtained m this study. With the magnetrc moments for the F phase of Mn,_,Tr,As confined to the basal plane there is a notable drstmctton to the F phases of the crystallographically isostructural compounds MnSb and MnBt [27, 281 These compounds show changes in the dtrectton of the magnetic moments from a perpendrcular to a parallel onentatron with respect to c as the temperature mcreases. As a further point of interest tt can also be mentroned that Take1 et af [27] report that the preparation procedure has a pronounced effect on the parameters specttjmg the magnetic structure as well as on Tc. It may be worthwhtle to check this posstbthty also for Mn,-,Tt,As, but since this has to be done systematically the question IS left open m the present paper 7. PHENOMENOLOGY FERRO-PARA

OF THE

TRANSITION

In mterpretmg the ferro to paramagnenc transrtton m MnAs only two essentrally different models have been introduced, one phenomenologtcal model and one qualitative, electromc band structure based model [29, 301. The phenomenologtcal approach was mtroduced by Bean and Rodbell ([29] hereafter BR), and this model has been successful m accountmg for many of the properties accompanymg the phase transitron m question Although ObJections have been rarsed against the isotroptc BR model [2-4, 29-311 its stmphctty encourages an attempt to extend the model to a case hke MnI_zTt,As. Some of the objet-

A ZlipA et al

282

trons to the basrc concepts of the BR model are drscussed at the end of this sectton The key feature of the BR model 1s the very strong magnetostnctrve couphng between the magnettzatron and the lattice According to BR the Cune temperature IS gtven by T, = T,(l + @w)

(1)

where o = (I’ - V&I’, denotes the relatrve change m lattrce volume, T0 the Cune temperature for an lncompre~ible lattice and /3 ts a model constant (see Ref f29f) Equation (1) contams the essential physrcs of the BR model For the rsotroprc BR model (using the mean field approximation (MFA)) the Gibbs free energy per unit volume takes the form (for spin quantum number 4)

G = -NkBTo( 1 - t)=(I + /3whr2/2 + 02/(2K)

I- (P + P*)w - N!caTfl - Q&,2(a)

(3)

The equation of state follows by solvmg the equrhbnum condmons dG/& = 0 and aG/& = 0 The sequence of calculatrons as well as the answers wrll be as presented by BR when m all formulae To IS replaced by ( 1 - #To, T by ( 1 - t)T and P by P + P* The temperature dependence of the relattve magnetrzatton 1s gven (m the inverted form) by -fIcr) = (t - t)Z’&[ 1 - (P + P*)K@ i- (I - Q2na2/3]/tanh-‘u

and the accompanied

(4)

change m the volume by

w = NkBToKp(l - t)=a=/2- (P + P*)K

(5)

G = -NkBTca2/2 + w2/(2K) The latter expressron describes the magnetostnctron (as usual) as proporttonal to the square of the magnetrzatron The effective parameter n = ~~)~k~T~~*( I Imphcn m eqn (2) 1s the assump~on of loeahzed - t)’ of the dtlutron modified BR model determines moments The first term describes the exchange the order of the ferro-para transition The transltlon energy where N denotes the number of moments, LT IS of the first order when the relative magnetization and kB the Boltzmann constant Tc 1sdefined by eqn (1) The second term n> l-(P+P*)K@ (6) gtves the elastic energy, where K 1sthe compressrbrhty, the thud deals with the couphng to the pressure P, The effect of the drlutlon 1sto favour the converston and the last refers to the spm entropy (T 1s the of the transmon from the first order to the second absolute temperature and L?,,~relates to spin quantum order The concentration and pressure dependence number i) Terms descnbmg the lattice entropy (conof the paramagnetrc 6 temperature 1s.grven by nected with thermal expansion) and the effect of an external magnetic field are here ignored for srmphaty @(t, P) = (1 - t)T,[l - (P + P*)K/?] (7) (These terms are conadered m Refs [29, 311) f PO - Nk,T&(a)

(2)

Turning now to the Mn,_,Tt,As case, the effect of drlutton has to be examined Firstly tt IS assumed that the dtlutton does not affect the exchange mteractrons between the locahzed (Mn) moments, consequently the factor (1 + fiw) still holds The dtlutron merely decreases the number of moments by a factor (1 - t), t being the share of the non-magnettc dtlutmg atoms The entropy term must then be multrphed by (1 - t) and the exchange term by (1 - t)’ (Accordmg to the MFA (1 - t)= appears in the exchange term because both the magnetrzatron and the molecular field must be scaled by the factor (1 - t) ) When the couphng to the lattice 1s neglected such an MFA type treatment of the ddutron would produce a lmear decrease of Tc reachmg zero for t = 1, I e the occurrence of a percolatton hmrt would not be reprboduced by the model From the crystal structure data (Section 4) n IS natural to assume that substrtutron of TrAs mto MnAs generates a ‘negatrve’ pressure The pressure term in eqn (2) will then be modtfied to compnse the external pressure Pas well as the mternal pressure P*, generated by the presence of the ddutmg atoms It 1s assumed that P* 1s propomonal to the drlutron, 1 e P* = tt Thus, the expresston for G converts mto

When the tranwtron ts of second order Tc = 8 For the first-order case, eqn (7) grves the stab&y limit on cooling, I e Tc,= = 8 The stab&y hmrt on heatmg, TQ,, and the equrhbnum transmon temperature, Tc,~, can be obtained numencally from eqns (3) and (4) For the present model calculatrons the values of the parameters To, K, @, and N and w, were taken from BR [29], and E = -50 kbar was adopted for Mn,_ZTr,As (The assessment of c was made by a comparison of Rg 10 wrth the corresponding data for MnAs m Ref [2] This shows that t = 0 04 (namely 4% Ti) shafts the phase borders of the pressure vs temperature diagram by about -2 kbar compared to the MnAs case, and hence f = P*lt gives t = -2/O 04 = -50 kbar ) The model parameters are likely to vary somewhat wrth the composmonal parameter t (vrde mficz) and, hence, the calculated, partial phase diagram m Fig 13 should be regarded as semi-quantitative The dashed hnes m Frg 13 gave the calculated phase borders for a srtuatron where the drlutron of the metal sublattrce 1s neglected, whrle the pressure contnbutron IS mamtamed It IS seen that one effect

Structural

and magnetic

of the dllutlon IS to strongly suppress the range of the first-order transition. The calculated tncntlcal point occurs at t M 0.50 for the un&luted situation and at t = 0.17 when dilution is taken mto account The latter, calculated location of the tncntlcal point 1s in reasonable agreement wth the value t = 0 10 suggested by the expenmental data for Mn,_,Tl,As. For the analogous ‘negative’ pressure phase MnAs,_$b, Bamer [3] reports a tncntlcal point at x = 0.58. In this case the ‘negative’ pressure effect (P* = cx) 1s caused by the substitution m the non-metal sublattice However, although the poatlon of the tncntlcal point comes out reasonably well, the observed magnetic phase diagram for MnAs._$b, [3] differs appreciably from that described by the dashed lines in Fig 13. This discrepancy could, on the other hand, be explained as a consequence of the choice of t (where e g t = -10 kbar seems to be appropnate for MnAs,_$b, [4]) and a compoatlonal vanatlon of the model parameters over the large range of x The concentration dependence of Tc and 8 is, perhaps accidentally, well reproduced by these calculations for moderate t values (cf Figs 6 and 13). The approximate constancy of Tc up to t = 0 3 appears to anse from the competltlon between the effects of the ‘negative’ pressure and the dilution The dilution effect should also be operative m the ‘posltlve’ pressure phase Mn,_,V,As In this case the effects of dilution and ‘positive’ pressure should be additive and consequently Tc,, and Tc,hfor Mn,_,V,As decrease rapidly (Fig l), and about two times faster than expected from comparison wth MnAs under pressure. These findings explain the change of slope of Tc vs t at the conJunction (t = 0 00) in Fig 1 as 600 I

I I

u

I

F

I

/ ‘.I

Rg 13 Calculated Tc vs t relatlonslup accordmg to the chlutlon mod&d Bean and Rodbell model (see text) Dashed lmes correspond to a sltuatlon wlthout ddutlon TP and TPDIL denote tncntlcal pomt calculated wlthout and with ddutlon, respectively, Tceq eqmhbnum Cune temperature and PERC LIM gives the expected concentration range for the percolation hmlt For the other symbols see the caption to Rg 1

propertles

of Mn,_;Tl,As

283

0

100

200

300

T(K) Fig

I4 Calculated

(I -

t)‘d

vs temperature

relatlonshlp

for Id 04 opposed to the rather good contmuatlon which 1s observed for Tc,, and TD The mean field treatment for diluted systems overestimates Tc even for small t t When t becomes high enough Tc falls to zero at the percolation hmlt (m the absence of competmg mteractlons). No calculation for the nearest nelghbour site percolation limit of the NlAs type structure has apparently so far been performed. It 1s reasonable to suppose that this percolation hmlt lies somewhere between the values for the simple cubic (1 - t = 0 31) and bee (1 - t = 0.24) lattices [33] This compares rather well with the observed range of the F mode of Mn,_,Tl,As (Fig 1) Figure 14 shows the temperature dependence of (1 - t)*$, which according to eqn (5) should be proportional to the magnetostnctlve effect below Tc This may be compared qualitatively with the thermal expansion data for the a axis gven m Ref [6] and Fig. 4 A quantitative comparison reveals that the magnetostnctlve effect decreases more rapidly urlth t than prescribed by the influence of dilution alone The expenmental data of Ido et al [6] for the pressure dependence of Tc also show that -(dTc/ dp)IT, decreases rather rapidly wth increasing t. The immediate inference of these findings 1s that the combined factor Kb decreases urlth the substitution (see eqns (5) and (7)) It 1s not experimentally established whether this 1s a result of a change m @ or concerns K since the elastic constants of Mn,_,Tl,As for t > 0 are unknown (This 1s discussed further m Section 8.) In any case, a decrease of K/3 with increasing t should produce an ad&tional decrease m q and consequently shift the calculated tncntlcal point towards a lower value oft. The conclusion 1s accordmgly that the dilution modified BR model 1s capable of simulating essential t Usmg the Helsenberg model on the fee lattice Rushbrooke et a/ [32] found -(dT,-/dt)/T, = I 36 and I I5 for spm quantum numbers l/2 and co, respectively, compared to I accordmg to the MFA

284

A

ZIFBA et a/

features of the magnetic phase hagram for Mn, _,Ti,As However, two serious pomts of cntinsm raised agamst the BR model must be mentioned (1) The BR model utilizes a localized picture whereas several independent facts pomt towards itmerancy for MnAs [34] Accordmg to Goodenough et al [35] the short mteratomic metal-metal distances along the c axis reflect the metallic character of MnAs. This feature, as well as the non-mtegral value of the magnetic moment, is mamtamed for Mn,_;Ti,As Edwards and Bartel [4] have advanced a simphfied model based on itmerancy for the second order ferro- to paramagnetic transition m MnAs,_,_Sb, Although the latter model has an attractive basis and is capable of explammg some experimental facts it is not yet developed far enough to be compatible with the BR treatment (u) It is well recognized that the magnetostnctive couphng IS stronger w:thm the basal plane of the NiAs type structure than along the hexagonal c axis Hence, the assumption of isotropy for the exchange mteractions and elastic constants represents only an approximation A generahzed amsotropic BR model (for arbitrary crystal symmetry, including exchange mteractions which depend on all SIXcomponents of stram, with thermal expansion, etc ) was introduced by Menyuk et al [2] (see the appendix of Ref [2]) Recently Ido et al [6] used a particular variant of the amsotropic BR model to discuss the anisotropic magnetostnction and its relation to dTc/dP m Mn,_;Ti,As (One is m the fortunate situation that all relevant components of the elastic modulus tensor for MnAs are expenmentally determined by Dorller and Bamer [36] ) The relation between the arusotroptc and isotropic versions of the BR model IS discussed m Section 8 8. ON THE ANISOTROPIC BEAN AND RODBELL MODEL The present treatment of the anisotropic BR model is based on the same assumptions (and uses the same notations) as Ido et al [6] The symmetry of the hexagonal NiAs type latuce suggests that the magnetic mteractions (which m turn rule the Cune temperature) depend differently on the strain parameters along the c axis and within the basal plane Tc = TO(1 + oc’e,, + p’eZZ)

(8)

where e, = Aafa and e,, = AC/Care strain components and (Y’and /3’ are model constants For the umaxial symmetry of the hexagonal lattice e, = e,, Shear strams and correspondmg terms do not enter in the elastic energy expression smce Tc according to eqn (8) depends on normal strains only The MFA Gibbs free energy then reads G(a, e,, e,,) = -2X( 1 + a’e, + @‘e,,)a*

where A = ($)(J/(J + l))NkaTO, C,, is a component of the elastic modulus tensor, C’,, = C,, + Cl2 and C’;, = &/2 (Note that the treatment is now made for a general spm quantum number J) From the eqmhbnum condition aG/‘e, = 0 and aGjae,, = 0 one obtains the equations descnbmg the magnetostnctive effects e,.. = AXa2 - EP ezz = B&T* - FP

(10)

where A, B, E and F are combmations of the elastic constants and CY’and @’ A = (&C;, - fi’C,,)/D, B - cu’C13)/D, E = (c;, - C,,/2)fD, F = = WC;, (C’,,/2 - C&D and D = C’,,C;, - C:, Insertmg eqn (10) mto eqn (9) one obtains an expression for G(o) G(o) = -2X[ 1 - (cu’E + p’F)P]a2 - X*(a’A + /3’B)a4 + Peconst

- TS

(11)

which subject to the conversion K@ -

CX’Ef /3’F

2K/?* ---t &A + /3’S

(12)

becomes identical to the results obtamed for the isotropic BR model This findmg is not unexpected since, according to the MFA the couplmg to the secondary order parameter (1 e m this case the strain) renonnahzes the expansion coefficients for the pnmary order parameter (a), but does not introduce new terms [37] As long as one considers the magnetic charactenstics resulting from eqn (1 l), e g u(T), dTc/dP, x(T), etc , the predictions of the isotropic and anisotropic vanants of the BR model are quantitatively identical provided K and #I are model parameters which are denved from expenmental data The magnetostnctive effect described by eqn (10) is on the contrary anisotropic and differs accordmgly from the predictions of the isotroptc BR model The experimentally determined compressibility (K) of MnAs by Grazhdankma and Burkhanov [38] and Dorfler and Bamer [36] are, however, m good agreement with K established mduectly by DeBlois and Rodbell [31] from magnetic data From eqn (12) it follows that the thus mdirectly determmed K IS, m fact, a function of the elastic moduh and the model constants 01’and p’ Assuming (Y’>> @‘,which seems to be a good approximation for MnAs, one obtams (using data for C,, from DortIer and Barner) K mode,= 2C;JD

= 4 3 X 10m3kbar-’

whereas the true compressibility, umaxial symmetry), equals

-(AV/V)/P

K true = 2E + F = 4 5 X 10e3 kbar-’

(13) (for

(14)

2 +

(C;,e,

+

Gel,

+ +

2C13e~A

P(2e,

+ e,,) - TS

(9)

It IS interesting to note that these two values are practically equal for MnAs

Structural and magnetic propertles of Mn,_,Ti,As

Ido et al [6] uttltzed data for MnSb which has a compressibility in the basal plane, -(Aa/a)/P, 4-5 times smaller than MnAs, to calculate (Y’and 8’ for Mn,_,Ti,As. The denved (Y’is consequently too high. Moreover, the decrease of the coeffictents (11’and @’ with increasing Ti content as reported by Ido et al, was based on the assumption of constant compressibility. We rather believe that the opposite occurs, namely that (Y’and /3’ vary only slightly whereas the lattice compressrbthty is likely to decrease from the excepttonally large value for MnAs to a more ordmary size

9. CONCLUSIONS The description of the magnetism m the 3d metals should, in pnnciple, be based on their electronic band structures. The current theory describes quote well the ground state of the 3d metals and their compounds (mcludmg crystallographically ordered alloys), but the properties of the crystallographtcally random alloy phases and phenomena depending on the temperature are far from explained A localized model IS nevertheless often capable of explammg many properties of magnetically ordered alloys This also applies to Mn,_,Ti,As where the followmg features would be consistent with a localized prcture: (1) The magnetic moment counted per Mn atom remains constant up to (at least) t = 0 60 and it thus seems as if the Ti atoms act merely as diluters which carry no magnetic moment The available mformation on Mn,_,TIAs with T = V, Cr, Fe and Co [I] gve no clues since the stability ranges of the NiAs,F phase are very narrow (The apparent decrease of the magnetic moment at the verge of the stabthty range of these phases may JUSt reflect the situation in two-phase regions ) (u) The long range magnetic order m Mn,_,Ti,As disappears at a t value closely correspondmg to the nearest neighbour site percolation limit, and the Mn moments could effectively interact via short-range Heisenberg type interactions. (m) The successful extension of the Bean and Rodbell [29] model for the dtscontmuous ferro- to paramagnetic phase transition m Mni_,Ti,As (by mcludmg the dilution effect from the non-magnetic Ti atoms and adding the assumption that Ti substitution produces a ‘negative’ pressure on the lattice). The dilution suppresses the range of the first-order transition substantially, m accordance with the expenmental findings Despite the accomphshments of the locahzed approach this can only be an approxtmation because Mn,_,Ti,As is certamly metalhc. The full temperature vs composmon phase dtagram for Mnr_,Tt,As is uncovered m Fig. 1 It seems that the phase transition at Tc converts to the secondorder kmd at t = 0.10 (1.e. the phase diagram has a tncrincal point at this composition). The extent of the MnP,P domam (lust above the first-order Tc

285

boundary) shnnks to zero for a t value between 0.04 and 0 06 (i.e. defining a cntmal end point). The present limited observatrons on the topology of the phase boundanes concur with the theory of phase transittons m that the first- and second-order Tc curves are tangentially joined at the tncritical point, as well as that the distortion temperature curve TD intercepts the first-order Tc boundary at an angle (at the cntical end point). However, the present observations are scattered and relatively far m composrtton from the points concerned. Moreover, the possibihty that the tncntical point and the cntical end point coincide cannot be ruled out, because uncertamty stems from some degree of smeanng of all observed transitions m Mnr_,Tt,As. In the phase diagram of Banter [3] for MnAsr_$b, the tncntical point and the critical end point are well separated at x = 0 50 and -0 10, respectively. At lower temperatures and low Tr content the domain of the metastable, MnP,H, mode (the type being not venfied for t # 0) extends at least up to t = 0.04 For large Ti contents, 0 70 < t -c -0 90, spur glass properties probably occur at low temperatures, but this aspect has not been explored m this study A narrow mrsctbthty gap around t e 0.95 appears because of the difference m crystal structure between MnAs and TiAs. authors are grateful to the Norwe8ian Research Council for Science and the Humamties

Acknowledgements-The

for financml support The authors also msh to thank S Foner (M I T , Cambndge), Z Obuszko (A G H , Krakbw), R DUEIJ and R Zach (P K, Krakbw), G Kozlowslu and M Hodorowslu (M L S P M IN T , Wroclaw) for theu generous help m obtammg various expenmental results REFERENCES

6 7 8 9 10 11 12 13 14 15 16

Selte K ,KJekshus A, Andresen A F and Zleba A, J Phys Chem Sohds 38, 7 19 (1977) Menyuk N , Kafahs J A, Dw&t K and Goodenough J B , Phys Rev 177,942 (1969) Bier K , Phys Stat Sol A5,405 (1971) Edwards L R and Bartel L C , Phys Rev B 5, 1064 (1972) Val’kov V I, Zavadsku E A and Todns B M , FIZ Tver Tela 19, 235 (1977) (Sov Phys Soled State 19, 134 (1977)) Ido H , Suzuki T and Iguchl I, J Mugn Mugn Mat 31-34, 159 (1983) Selte K ,KJekshus A. and Andresen A F , Acta Chem Stand A28,61 (1974) FJellvig H and KJekshus A, Acta Chem Stand A38, 1(1984) Bacon G E , In Yelon W B , Ed, Neutron Dtffractton Newsletfer, Columbia (1977) Watson R E and Freeman A J , Acta Crystallogr 14, 27 (1961) Selte K , KJekshus A, Valde G and Andresen A F , Acta Chem Stand A30, 8 (1976) Zach R., Pnvate commumcatlon KJekshus A and Pearson W B , Progr Solid State Chem 1, 83 (1964) Boller H and Nowotny H , Monatsh Chem 96, 565 (1965) Berger R , Acta Chem Stand A31, 5 14 ( 1977) Se.lte K and KJekshus A, Acta Chem Scand 27, 3195 (1973)

Zleba A, Selte K , KJekshus A and Andresen A F , Chem Stand A%!, 173 (1978) Selte K . Klekshus A and Andrew A F +Acta Chem Aia

&and

i7, &07

(-1973)

Sobczak R , Boiler H and Blttncr I-F, Munatsh

Chem

99,222Y (1968)

Ztqba A, Shapua 3’ and Foner S , Ph.v,sL&t A%, 243 (1982) &$a A, Zach R , Fs&?& H and Qekshus A. To be pibhshed Aildxesen A F J Ffeffvsig H axLdLebeclI B > J &@a@ iwffg8z _lwt 43, iSS If984f R%tveld H M , J A$ ~~~~~~~~~~ I& 6.5(1368) Haneda S , Kazama N , Yamaguchr Y and Watanah H , _I Phys Sue Jpn 42, t201 (1977f, 42, 1212 (1977) Grenvoid F , Smldaf S and Westrum E F , Acta Chem &and 24, 285 (1970) t1amdton.W C , Acta Crystakgr I& 502 (1965) Taker W J . Cox D E and Shmme G 1Phvs ” Rev 129. 2008 (1963) Andersen A F , Halg W , Fischer x1 and St011E , Arta Chem Stand 21, 1543 (1967)

29 Bean C P and Rodbell D S , l’hys Rev 126, 104 (1962) 30 Goodenou& J B and Kafalas J A, Phys Rev 157, 389 (1967) 31 DeBlois R W and Rodbell D S , Phys Rev 130, 1347 (1963) 32 Rushbrooke CI 5, Muse R A and Stephenson R I I 2 Phys C S&d State Phys 5,33?1 (1972) 33 Z&ante Y K S and Kxkpatnc~ S j Adv Whys 50, 325 34

l3:Rldiey D H and N&man W A a Frnc Inr Cmf h4a,pettsm, Nottmgham, p 542 (1964) 36 Dotier M and Barner K , Phys Star Sol Al?, 141 35 G0adertoug.b J

(1973) 37 Bruce A D and Cowley R A, Structural Phase Dan* nttons Mcthuen, London (198 1) 38 Grazhdankma N P and Burkhanov A M , Zh l%.y~ Teor Az fi0, 1519 (1966) (Sov Phys JETP 23, 1013 (1966) 1