Magnetic correlations in disordered magnetic materials

Journal of Magnetism and Magnetic Materials 14 (1979) 197-206 © North-Holland Pubhshing Company

MAGNETIC CORRELATIONS IN DISORDERED MAGNETIC MATERIALS B.D. RAINFORD * Blackett Laboratory, Imperial College, Prmce Consort Rd., London SW7, UK

The investigation by neutron scattering techniques of the static and dynamic magnetic correlations in disordered magnetic materials is discussed, with reference to recent experiments on CuMn, CrFe and disordered PdaMn alloys.

1. Introduction

2. Static correlations

General interest in disordered magnetic materials has increased greatly in the last five years. The development and application of new techniques in neutron scattering, for example polarisation analysis, small angle scattering and ultra-high energy resolution spectrometers have had a significant impact in this area. It is the purpose of the present paper to outline what can be learnt about the static and dynamic spin correlations in disordered magnetic materials using these methods. Much of the recent work has been reviewed elsewhere [1 ], so I intend to discuss some of our own recent experiments which illustrate most aspects of this field. Reference to related experiments will be made in the appropriate place. Static correlations are discussed in section 2 with particular reference to CuMn alloys, studied by polarisation analysis, and to disordered Pd3Mn. This leads to a discussion of the dynamic correlations in Pd3Mn in section 3. Finally, in section 4 1 discuss the transition from a spin glasslike state to disordered ferromagnetism in CrFe alloys. In what follows, I make an operational definitmn of a spin glass as a material which has a sharp maximum in its susceptibility as a function of temperature, without possessing long-range magnetic order. I will therefore be using the term "spin glass" in contexts where other authors might prefer alternatives like "mictomagnet", "speromagnet" or "cluster glass".

2.1. Relation to neutron cross sections The static magnetic correlations (Sz • S1) between spins at sites R i and Rj may be obtained from total scattering measurements, i.e., from the integral of the neutron cross-section over all energy transfers: dg__~° d E , _ ( T e 2 1 2

× ~ exp iQ. (R, - R i ) ( S i • $fl. 0

(1)

For this expression to be valid, the quasi-static approximation must hold, that is the inelasticity of the scattering must be small compared to the energy of the incident neutron. Total scattering measurements may also be related to the generalised susceptibihty x(Q) if any truly elastic Bragg or diffuse scattering is first subtracted from the total scattering. Thus, x(Q)--N-~BT(g/aB)2 ~il {(S," S,) - (S,)" (S,)} × exp IQ" (R, - R t ) .

(2)

There is often much more information in ×(Q) than in the bulk susceptibility X(0). The basic problem with disordered magnetic materials is to separate the magnetic scattering from the nuclear incoherent and Bragg scattering. In cases where the spins are essentially uncorrelated at high temperature, this may be done using a temperature difference technique [2]. Otherwise, polarlsation analysis techniques must be

* Address after 1st October 1979: Department of Physics, University of Southampton, Southampton SO9 5NH, UK. 197

B D Ramford /Correlattons in disordered magnets

198

used. Examples of each of these methods is given in the following.

2.2 Polarisation analysis techniques." CuMn alloys In CuMn alloys a spin glass phase exists in quenched alloys for Mn concentrations up to 72%. Above this concentration there is long range antiferromagnetism. It was shown recently [3] that there is a maximum in the spin glass temperature Tg near 45% Mn. This was tentatively attributed to a decrease in the Mn atomic moment as overlap of nearest nelghbour d wavefunction increases. Since the early neutron diffraction measurements of Meneghetti and Sidhu [4] there has been an enormous amount of work on the nature of the diffuse scattering which is seen in the vicinity of the (110) reciprocal lattice point in these alloys. We have recently carried out extensive total scattering measurements on polycrystalline CuMn alloys [5] using the polarisation analysis method. This allows an unambiguous separation of nuclear and magnetic diffuse scattering. Some of the data obtained with the D5 diffractometer at ILL, Grenoble are shown in fig. 1, The nuclear and magNUCLEAR I

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netic cross-sections were derived from the nonspln flip and spin flip scattering, respectively. It can be seen that the (1~0) diffuse peak, near [Q[ = 20 nm - I , has both a magnetic and nuclear component. Analysis of the nuclear scattering shows that atomic shortrange order (st•) predominates below 60% Mn, whilst clustering occurs at higher concentrations. In most of the alloys the width of the magnetic diffuse peak is less than that of the nuclear peak, showing that the spatial extent of the magnetic correlations is greater than that of the atomic short range order. Since highenergy neutrons (116 meV) were used in these experiments, the quasi-static approximation is satisfied. The magnetic scattering was analysed using an expression of the form / e 2,), \2

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(3)

where/3i = c + (1 - c) %, c is the concentration of Mn and a, is the Cowley st• parameter for the ith shell. (So • Si) is the corresponding static spin pair correlation. The first and second neighbour spin correlations were found to be antiferromagnetic and ferromagnetic, respectively. The atomic moment per Mn was also derived from the self term S(S + 1) in eq. (3). This was found to be close to 4/aB for all concentrations. It follows that the decrease of Tg beyond 45% Mn cannot be associated with a decrease in the moment. For alloys with high Mn content it is not possible to estimate the Mn moment from bulk susceptibility measurements, since the susceptibility does not follow a Curie-Weiss law. The temperature dependence of the spin correlations was measured for the 17% Mn alloy. This is shown in fig. 2. Although the spin glass temperature is near 75 K in this alloy it can be seen that strong magnetic correlations persist up to and beyond room temperature. It follows that any attempt to use a temperature difference technique to extract the magnetic correlations would seriously underestimate the magnetic component of the scattering in this case. The temperature dependence of the pair correlations may be used to estimate the exchange interactions in the alloy [2]. Taking the pair Hamiltonian to be

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B D. Rainford / Correlations in disordered magnets

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and ignoring the molecular field, we find for S = 2
=

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(5)

where x = exp(JiJ2kBT ). The solid lines in fig. 2 show a fit of this expression to the measured temperature dependence, yielding Jl = - 1 4 6 K, J2 = 102 K. This approach could be improved by inclusion of the molecular field using one of the standard methods [6]. 2.3. Static correlations in disordered Pd3Mn With appropriate heat treatment PdaMn orders into an antiphase domain structure based on a fourfold repeat of CuaAu type units [7]. Cable et al. [7] showed by neutron diffraction that the ordered compound is an antiferromagnet with a N~el temperature of 170 K. The magnetic unit cell is the same size as the nuclear cell with the moments lying in the (001) plane, perpendicular to the unique axis. The Mn and Pd moments were estimated to be 4/1s and 0.2/~B, respectively. Neutron diffraction measurements [8] on a quenched powder of PdaMn showed nuclear peaks consistent with a disordered face centred cubic lattice, with some indication of short-range atomic order. At low temperatures no magnetic Bragg scattering was observed, but there was substantial scattering arising from magnetic short-range order. Magnetisation measurements [9] were made on a quenched (disordered) single crystal of PdaMn using a vibration

199

sample magnetometer with a field of 10 Oe. These showed a sharp cusp at Tg = 30 K, characterimc of spin glass order. Time-dependent effects were observed below Tg similar to those reported by Guy [10] for AuFe alloys. The spin glass phase here results from a competition between positive and negative exchange interactions arising from the positional disorder of the Mn atoms. In the ordered compound, by contrast, the Mn moments lock into the peaks and troughs of the RKKY interaction, and long-range antiferromagnetism results. Thus, we have in Pd3Mn the possibihty of comparing the static correlations in the spin glass with the known magnetic structure of the ordered antiferromagnetic alloy. In principle the distribution of exchange interactions in the compound could be varied by successive heat treatments designed to produce different degrees of short- or long-range order. We have made a detailed study of the static correlation in a large single crystal of disordered Pd3Mn [9], as a prelude to the investigation of the dynamical correlations, discussed in the following section. Measurements were made with the MKV1 two-circle diffractometer at AERE Harwell over a wide range of temperatures below 10 K and room temperature. Fig. 3 shows longitudinal scans through the (001) reciprocal lattice point. At room temperature a broad diffuse peak is seen centred on (001) corresponding to the short-range atomic correlations in the alloy. As the temperature is lowered, extra intensity appears, resulting in the development of two diffuse peaks centred at (0, 0, 0.85) and (0, 0, 1.15). The temperature dependence of the scattering at (0, 0, 0.85) is shown as the upper curve in fig. 4, where the room temperature scattering has been subtracted as background. It can be seen that the short-range magnetic order persists up to temperatures of the order of TN for the ordered compound. By the spin glass temperature (Tg = 30 K) the short-range order is almost fully developed. The arrows in fig. 3 indicate what would be the positions of the magnetic reciprocal lattice points ((0043-) and (004s-) indexed on the cubic unit cell) of the antlferromagnetic structure of ordered Pd3Mn. It can be seen that the sro peaks are shifted inwards towards (001). This suggests that the exchange interactions in Pd3Mn are such that J(q) has a maximum value J(Q) for Q = 0.85 (2rr/a). The observed magnetic structure of ordered Pd3Mn (cor-

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responding to Q = 0.75 (27r/a)) probably arises from the perturbation of the Fermi surface by the formation of superzone boundaries in the antiphase domain structure. In a preliminary analysis of the low-temperature data we have assumed the scattering to be predominantly from the Mn moments, and have fit-

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where gn represents magnetic short-range order parameters between a central Mn atom and a plane of Mn atoms distance ~a away. The quasi-static approximation is well satisfied here, as will be seen below. The interplanar parameters gn are shown in table 1, where they are compared in sign with those deduced from the observed magnetic structure of ordered Pd3Mn. It can be seen that there is a close correlation. Analysis of the temperature dependence of the magnetic short range order will be presented elsewhere [9]. Blech and Averbach [11] have pointed out that information about the average directions of magnetic moments with respect to crystallographic axes may be extracted from analysis of magnetic short-range order in disordered materials. This method has been applied by Nagele et al. in their study of the magnetic correlations in a Mn alumino-silicate glass [2].

B.D. Rainford /Correlattons in dtsordered magnets Table 1 Interplanar magnetic short-range order parameters for disordered Pd3Mn, compared with moment per layer for ordered Pd3Mn Layer number n

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3. Dynamic correlations

3.1. Introduction Much of the recent interest in disordered magnets has been centred around the spin dynamics of spin glass materials and around the question of whether or not there is a true phase transition associated with the spin glass state. The mean field theory of Edwards and Anderson [12] suggested that there is a phase transition, leading to a sharp cusp in the susceptibility. On the other hand, the work of Moore et al. [13], based on computer simulations and backed up by renormalisation group calculations, suggests that there is no true order parameter: rather, that the spin glass state is characterised by extremely long relaxation times. The N6el blocking model [14], derived for the case of superparamagnetic particles in rock magnetism, has been invoked by experimentalists [10] as a mean of introducing the rapid onset of long relaxation times. Wohlfarth [15] has indicated the relationship between a distribution of blockmg temperatures and the measured form of ×(7) in this model. With the N6el model, however, the blocking temperature should depend on the frequency in a susceptibility measurement. For many alloy systems no frequency dependence is seen [16]. Inelastic neutron scattering provides the means of measuring the dynamic response in frequency ranges well above those employed in standard ac susceptibility measurements. The relevant parameter here is the

201

energy resolution of the neutron spectrometer. For a conventional triple axis or time-of-flight spectrometer, the energy resolution might be 1 meV (0.24 × 1012 Hz) which means that relaxation rates of 1012 Hz or greater may be measured, but that rates slower than 1011 Hz would appear indistinguishable from elastic scattering. New techniques have allowed a great improvement in the range of available energy resolutions: thus the backscattering spectrometer IN10 at ILL has a resolution of about 1 ~eV, while for the spin echo spectrometer IN11 the resolution is perhaps one order of magnitude better still. 3.2. Dynamical correlations in disordered Pd ~ l n In many ways disordered PdaMn is an ideal material in which to study the spin dynamics of the spin glass state since the magnetic sro scattering is well separated from the nuclear diffuse scattering (unlike CuMn), yet it is well localised in reciprocal space. This means that there will be good contrast between the magnetic scattering and the nuclear incoherent "background". Similar measurements to those described below have been performed on CuMno.o8 by Murani and Heidemann [17]. Preliminary measurements [9] on the Pluto triple axis spectrometer at AERE Harwell showed the scattering to be almost entirely quasi-elastic, with an energy width less than the resolution of the spectrometer (1 meV). There was no evidence of inelastic magnetic excitations at any scattering vector, for temperatures down to 10 K. Further measurements [9] were made with the IN8 triple axis at ILL using a spectrometer configuration which gave an energy resolution of 0.I 5 meV. The scattering was observed to be independent of temperature above 200 K. The 200 K scattering was therefore taken to be the nuclear incoherent background, and was subtracted from the data measured at lower temperatures. The results for Q = (0, 0, 0.85) are shown in fig. 5(a). As the temperature is lowered the quasi-elastic peak is seen to increase in intensity and narrow in energy until, at the lowest temperature, the width equals the spectrometer resolution. The deconvoluted widths F (FWHM) are shown in fig. 6 compared with a fit to an Arrhenius law. F = Fo exp(-A/kBT),

(7)

where P0 = 1250 bteV (3.0 × 1011 Hz) represents the

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attempt frequency, while A = 120 K represents the height of the energy barrier which needs to be surmounted by thermal activation. This form of the temperature dependence of the widths would arise in the NCel picture of blocking of superparamagnetic

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clusters, where A would represent an anisotropy energy against reversal of the cluster spins. The attempt frequency F• obtained from this fit is somewhat higher than that normally assumed in the N6el theory (109 Hz). While the observed temperature dependence is well represented by eq. (7), it is obvious that this relationship does not describe all the features of the spin dynamics, viz. (i) the above parameters predict a blocking temperature near 4 K, much lower than the spin glass temperature (30 K). (ii) the integrated intensity under the quasi-elastic scans falls off with temperature much faster than the total scattering (fig. 4). This means that there must be a component of the scattering lying outside the energy window (+0.5 meV) of the spectrometer, and that the proportion of this component increases as a function of temperature. To investigate this further we made measurements using the IN 10 back-scattering spectrometer with an energy resolution (FWHM) of 1.2/~eV [9]. Here the scattering was found to be temperature independent above 70 K and the spectrum measured at 150 K was taken as the incoherent nuclear background. The data in fig. 5(b) result from subtracting the 150 K spectrum from spectra measured at lower temperatures. An obvious quasi-elastic component does not develop until 50 K, this rapidly narrows and grows in intensity as the temperature is lowered further. The spectra below the glass temperature have a width equal to the energy resolution of the spectrometer within experimental error (inset in fig. 6). The integrated lntenmies of the IN10 quasi-elastic peaks are shown in fig. 4. They decrease much more rapidly with temperature than either the total scattering of the integrated intensities of the IN8 quasi-elastic peaks. Note also that the component of the scattering seen within the narrow IN10 energy window (+2 MeV) has a much smaller width, at 40 K for example, than the 60 jueV estimated from the IN8 data at the same temperature. It is apparent then that a single expression such as eq. (7) is inadequate to describe the spin dynamics, and that we should rather think of there being a range of activation energies. The present data then is not inconsistent with the N4el model, as it IS applied to spin glasses, with a distribution of blocking temperatures. In this picture, the essentially elastic scattering observed below 30 K corresponds to the component of the magnetic clus-

B.D. Rainford /Correlattons in disordered magnets

ters with relaxation rates slower than 108 Hz. However we should also consider whether the phase transition model [12] is also compatible with these observations. The susceptibdity, from eq. (2), may be generally written x(Q, T) - N k B T

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203

concentration cf. There has been considerable interest in the magnetic properties of such systems, but before discussing the bulk measurements it is instructive to outline what has been learnt about the spin correlations from small angle scattering of neutrons. CrFe alloys are discussed in the following, but similar studies have been made on AuFe [21] and on the insulating system (Sq_eEue)S [22].

(8) 4.2. Small angle neutron scattering results

where q-il = (St--~" ($1) and the bars denote a configurational average. In the phase transition model for spin glasses [12,18], the peak in the susceptibility occurs when q# becomes non-zero. This corresponds to the presence of a truly elastic diffuse component in the neutron cross-section. It is tempting then to regard elastic scattering, observed in the IN10 measurements below 30 K, as a measure of the spatial Fourier transform of the spin glass order parameter q-0" However, we consider this approach to be naive since (i) all neutron spectrometers have a finite energy resolution, and it is only possible to put a lower limit on a possible relaxation rate, (ii) recent theories of spin glasses [19,20] suggest that low-energy diffusive modes are important. It would be extremely difficult to decide what proportion of the scattering is strictly elastic in this situation. This means, therefore, that in spin glasses or superparamagnetic materials where slow relaxation rates are expected it is not possible in practice to derive a susceptiblhty x(Q) from the neutron cross-sections. Neutron scattering measurements then give important information about the dynamics of spin glass systems, namely by showing the absence of highenergy excitations, and the dominance of the lowfrequency diffusive modes. However, we are unable to cover the important range of relaxation rates between 107 Hz and, say, 10 - I Hz, which would allow us to determine whether the phase transition model is valid for real spin glass systems.

4. Onset of ferromagnetism in C r - F e alloys 4.1. Introduction In alloy systems such as Aul_cFec and Cr l_cFec, ferromagnetism is observed above a certain critical

Extensive measurements of small angle scattering from CrFe alloys have been made using the D11 and D17 small angle diffractometers at ILL. These have covered Fe concentrations from 15 to 25%, temperatures between 2 and 300 K, magnetic fields up to 5 kOe and a range of scattering vectors from 0.003 to 0.1 A -1 [23]. Some of this data is shown in fig. 7 where the intensity at several scattering vectors is plotted as a function of temperature. The peak in the scattering at finite temperatures is identified with the critical scattering occurring at the Curie temperatures Tc of the ferromagnetic alloys. It can be seen that the divergence at Tc becomes weaker as the concentration IS decreased towards cf so that at 19.9% Fe it appears only as a shoulder. By combining the values of Tc

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204

B D Ram ford /Correlattons in dtsordered magnets

from the critical scattering with those from magnetisation measurements we deduce cf to be close to 19% Fe. The other striking feature of the data is the rise in the scattering at low temperatures, due, as will be seen below, to a rapid increase in the correlation range. Many features of the scattering are consistent with a percolation picture for the onset o f ferromagnetism" that is, cf is identified with the percolation concentratlon Cp at which an infite cluster of iron atoms, coupled by nearest nelghbour exchange interactions, first appears. For concentrations below cf only finite clusters of ferromagnetically coupled spins can exist, but the average cluster size diverges as c approaches cf. For c ~ cf the infinite cluster coexists with a distribution o f finite clusters. In this model the magnetlSatlon for c ~ cf is proportional to the fraction of Fe atoms in the infinite cluster while the susceptibility IS related to the mean square cluster size. It has been suggested [24] that the critical concentration cf IS a multlcrltical point, since the susceptibility can dwerge either along tile hne of second order transition Tc (c/> ef) or along the percolation hne, as a function of (c - cf) for T = 0. The behaviour of the spin correlations in the ViCinity o f Cf has received considerable theoretical attention [ 2 4 - 2 6 ] . Several authors [24,26] have suggested that since the clusters in the vicinity o f the percolation concentration are "stringy" or ramified in form, the growth o f correlations will be similar to that m a linear chain, namely the correlation range will tend to diverge towards T = 0. But for c < cf, the correlation range cannot exceed the dimensions of the finite cluster and so will reach a finite maximum at low temperatures. For c > Cr there will be two components to the scattering one from the infinite cluster which will contribute only to the Bragg scattering at T = 0, but will give rise to the critical scattering near Tc; the second component will be due to the finite clusters and the dangling bonds attached to the backbone o f the infinite cluster. The response of this second c o m p o n e n t will be similar to that of the finite clusters for e < of, and will dominate the scattering near cf where the number o f atoms in the infinite cluster is small. This is presumably why the divergence near Tc becomes very weak near cf in fig. 7. The inverse correlation ranges K derived from the

small angle data are shown in fig. 8. These show many of the features described above. Firstly for c < cf, K is seen to decrease as the temperature is lowered (corresponding to an increase in the correlation range) reaching a f n i t e value at low temperatures. This value is smaller for concentrations closer to cf. This behaviour is consistent with the maximum correlation range being determined by the mean cluster size, which of course increases as cf is approached. The temperature dependent part of K appears to scale as T 1.2. K also displays a shallow minimum at low temperatures. This is probably an anisotropy effect, due to dipolar interactions, similar to that seen in Z n l _ c MncF2 [27]. For c > cf there is a minimum in K at Tc, then at lower temperatures a rapid decrease to a smaller value. The apparently finite value o f K at Tc ts perplexing, but this could arise either from (i) smearing o f the transition due to microscopic concentration gradients in the alloys, or (ii) from the swamping o f the small divergent component at Tc, arising from the infinite cluster, by the much larger scattering from the finite clusters. The rapid reduction in K below Tc is obviously related to the growth of long range correlations in the finite clusters as described above, with the range tending to diverge towards T = 0 The Inverse correlation range for c > cf, above the Curie temperature, appears to be

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B.D. Ramford / Correlations m dtsordered magnets well described by

Ka = O.5(c - cf)l/2(~TTc C)l]2 ,

(9)

that is, with mean field exponents, though this expression can only obtain away from Tc, since • is not observed to vanish at Tc. For alloys w~th c < cf and for alloys with c > cf above their Curie temperatures the form of the scattermg is close to Lorentzian over the whole range of Q studied. But for c > cf and T < Tc, the scattering falls off faster than a Lorentzlan at larger values of Q. This IS shown in fig. 9, where reciprocal intensity is plotted against Q2 (this plot would yield straight lines for a Lorentzlan line shape). We suggest that these departures from Lorentzlan line shapes arise when the correlation range becomes comparable with the dimensions of the finite clusters; the scattering will then be sensitive to the conformation of the clusters. We have attempted to model this behaviour by approximating the form of the ramified clusters to a Gaussian coiled polymer [28].

4 3. Magnetic properties near cf Low-field magnetisation experiments have shown some interesting features for alloys near the critical

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205

concentration [29]. Alloys with c < cf show spin glass behaviour with susceptibility peaks and timedependent effects. The susceptibility peaks are rather more rounded here than for the dilute alloys or for systems where the exchange is predominantly antiferromagnetic. We suggest that this behaviour of the susceptibility is due to superparamagnetlc blocking of the finite ferromagnetically coupled clusters. It is interesting to note that the effect of an applied field on the neutron small angle scattering is consistent with the clusters behaving like superparamagnetlc entities [23]. Any analysis of the susceptibility along these lines, however, should take account of the temperature dependence of the spin correlations (in N~el's model the ferromagnetic correlations are assumed to be established at a temperature far above the blocking temperature). Low-field susceptibility measurements on ferromagnetic alloys just above cf also show maxima as a function of temperature. The maxima are well below the measured Curie temperatures. Data we have obtained on CrFe alloys with e = 17.5% and c = 19.5% (cf = 19.0%) are practically indistinguishable. This is perhaps not too surprising, since, as we discussed above, the magnetic properties in the vicinity of cf are dominated by the finite clusters. For increasing Fe concentration the shape of the susceptibility maximum approaches that seen in pure Fe (Hopkinson peak). It is apparent therefore that the maxima observed in the susceptibility of these alloys is related to the development of anisotropy effects (N6el blocking for alloys with c < cf, ferromagnetic hysteresis and remanence for c > cf). In ramified clusters an important source of anisotropy will be the magnetostatic energy associated with the one-dimensional branches (shape anisotropy of the cluster). When this shape anisotropy becomes important the ferromagnetic alignment within the cluster will be subdivided into microdomains, to mmimise the magnet•static energy. There is some evidence from neutron diffraction [31 ] that this occurs in (Srl_cEuc)S. The view presented here of a gradual development of the susceptibility in alloys near cf is in contrast to the sharp transitions &scussed by the Leiden group [29]. It seems likely that these sharp transitions result from the large demagnetising fields within these highly susceptible materials.

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B.D. Rain ford / Correlations in disordered magnets

Acknowledgements I have benefited greatly from advice and encouragement from S.K. Burke and J.R. Davis. I am grateful to J. Spalek, A.P. Young and E.M. Gray for useful &scusslons, and to P.D. Lilley and W.G. Stirling for permission to quote unpublished results. This work was supported by the Neutron Beam Research Comrmttee o f the Science Research Council.

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