R. D. Lowde and G. L. Tindle- On spin waves and spin disorder in face-centred-cubic Mn73Ni27

8/3/2019 R. D. Lowde and G. L. Tindle- On spin waves and spin disorder in face-centred-cubic Mn73Ni27

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doi: 10.1098/rspa.2003.1168, 1213-12414602004Proc. R. Soc. Lond. A

R. D. Lowde and G. L. Tindle27

Ni73On spin waves and spin disorder in face-centred-cubic Mn

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10.1098/rspa.2003.1168

On spin waves and spin disorder in

face-centred-cubic Mn73Ni27

B y R. D . L o w d e a n d G . L. T i n d l eThe UK Atomic Energy Authority, Harwell OX11 0RA, UK

([email protected])

Received 5 December 2002; accepted 8 April 2003; published online 18 February 2004

Data obtained from neutron scattering are presented for the generalized perpendic-ular magnetic susceptibility Im (q, ) at room temperature along the 001-axisof a single crystal of face-centred-cubic Mn73Ni27 whose elastic, magnetic and crys-tallographic characteristics have also been closely examined. Here q stands for thewavevector and for the frequency of an excitation in the crystal. Plotted against qand , the measured function shows a broad ridge that sweeps up from = 4 meVat q = 0 to ca. 140 meV at the zone boundary qmax. We take this to be the track of anexceptionally diffuse spin-wave mode. The broadening of this excitation, however, isso great that beyond energies of ca. 40 meV it would have to be said that the wholeBrillouin zone is involved in the setting-up of an excited state at any specified energy.To investigate that feature, attempts were made to analyse the measured Im as awell-defined spin-wave dispersion law of elementary form, broadened either in q orin by some simply-expressed interaction. A successful outcome would indicate, inthese two extreme cases, that the broadening arose from the excitation being limitedby considerations respectively of space or of time. Our efforts to fit with broaden-ing in ran into problems, making it difficult to argue for a picture of well-definedspin waves interacting strongly with phonons, electrons or other quanta. By con-trast, the entire set of 15 spectral surveys could be fitted to a smooth theoretical

function with a statistical chi-squared per degree of freedom of 1.57 if we took asharp antiferromagnetic spin-wave dispersion law subject to a generalized OrnsteinZernike broadening-in-q function based on a spatial correlation function betweenspin magnitudes varying as er/r, where r is an interatomic distance in A and = 0.59 was the best-fitting exponent. The dispersion law thus obtained was linearbetween 0.15qmax and 0.65qmax, over which range the fit indicated a propagationvelocity of 34 km s1. Up to a rather higher limit of wavevector, was also quiteaccurately linear in q, specifically = (0.25 0.01)q. Correspondingly, linewidthswere accurately linear in q out to about 0.8qmax. Thus the tremendous broadeningof the spin-wave line in the 27% nickel alloy is preponderantly due to the spin cor-relation function suffering severe decoherence as a function of distance in all spatialdirectionsan effect we attribute to the magnetic structural irregularities induced bythe alloying. A spin wave in these materials is therefore a rather localized oscillationconfined to a restricted region of mean dimension just a few angstroms.

Keywords: manganese alloys; spin waves; magnetic disorder

Deceased.

Proc. R. Soc. Lond. A (2004) 460, 12131241

1213

c 2004 The Royal Society

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1214 R. D. Lowde and G. L. Tindle

1. Introduction

The spin waves displayed by face-centred-cubic (fcc) manganese alloys are of interestfor a number of reasons. The magneto-elastic coupling of the manganese spin to adisplacement of the manganese atom from its equilibrium position in the unit cellis very strong, and this produces an interaction between magnons and phonons inthe crystal that softens both, leading as a function of temperature to crystal struc-ture transitions as well as magnetic ones. The MnNi phase diagram in particularshows a complex of cubic, tetragonal and orthorhombic phases extending from puremanganese to the alloy with 22% Ni, beyond which the fcc antiferromagnetic stateprevails out to 74% Ni. From a theoretical point of view this phase diagram can beunderstood, thanks to the efforts of N. A. Cade (1981, unpublished research), M.Sato (1981, unpublished research) and Long & Yeung (1986a, 1987b, c). Fundamen-tally, the effects may be traced to the fact that the antiferromagnetism is frustratedin these materials (Long 1993).

Yet the dynamics of the alloys remain to be clarified. Taking the example of thesimplest structure transition, namely the second-order cubic-to-tetragonal transfor-mation that we have already studied (Lowde et al. 1981a, b; Hesketh et al. 2003;

Long et al. 2004), Long & Yeung (1987b) have argued theoretically that it is drivenby the softening, as a function of temperature, of the 001 magnon at small wave-vector q. One would therefore expect to find both magnons and phonons broadenedin energy, certainly in the 001 set of directions. In our work on this topic so far, wehave concentrated on Mn85Ni9C6; and if, as believed, the carbon in that material isinterstitial the sample approximates to a 10% nickel alloy. We duly found the broadphonons, and they proved to have a highly anomalous temperature dependence (Hes-keth et al. 2003). It would be of particular interest to extend that work by observingthe magnons in Mn85Ni9C6 through the same transition, thus laying the basis for aneventual understanding of the remarkable dynamics of these alloys.

The magnons, however, interact significantly with many systemsnot only withphonons, but with one another, with conduction electrons and with the static spin

disorder that is an inevitable consequence of the alloying. An essential part of theinvestigation must therefore be to separate the total observed magnon interactioninto these components. In the study now reported here we make a modest start onthat programme by establishing the spin-wave dispersion relation at room tempera-ture for one alloy, Mn73Ni27, and confirm that its spectral distribution is indeed verybroad. We explore the possibility of diagnosing the interactions by spectral lineshapeanalysis. The end result is quite striking: at the 27% nickel concentration, alloy disor-der proves to be overwhelmingly the dominant cause of line broadening and thereforeof the diffuse nature of magnetic response in these materials. Our conclusion har-monizes with the long-held idea that disorder is the cause of the prominent elasticdiffuse scattering observed with this substance and with comparable alloys (Moze &Hicks 1981).

This intense interaction of the magnons with spin disorder is indeed of fundamen-tal importance, because it can be argued that the alloy disorder is the very agentstabilizing the high-symmetry magnetic structures that occur. According to Long(1993), spin-deficient nickel impurity sites allow the manganese spins to lower theirenergy below that of a perfectly-aligned structure by developing a disturbed angulararrangement where they have nickel neighbours (Long 1989). To do this they take

Proc. R. Soc. Lond. A (2004)



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On spin waves and spin disorder in fcc Mn73Ni27 1215

on multiple-spin-density-wave structures, culminating, as one increases the nickelfraction and moves to the right on the alloy phase diagram, in a chemically fccstructure having the four spins of the unit cell, on average, in a tetrahedral angulararrangementthat is to say where the magnetic moments at the corner sites of thetetrahedral repeat unit of pattern all point to the centre of the tetrahedron (Long

1990). In a later development, Long & Bayri (1993) arrived at the same conclusionusing a completely different representation based on the supposition of local momentsand analysing the effect of an isolated impurity atom.

Evidence confirming this tetrahedral magnetic symmetry in the cubic alloys hasbeen assembled by Long et al. (2004); further experiments by Long & Moze (1990)and by Kawarazaki et al. (1988) reach the same conclusion. It is a symmetry thatreduces the magnetic Brillouin zone (BZ) to a small cube having one-quarter thevolume of the familiar dodecahedral BZ that would be appropriate to a nonmagneticstate.

These considerations show that in any fundamental discussion of manganese alloysone must confront directly the issue of magnetic structural disorder. Yet it is arequirement that brings tremendous difficulties. Every relevant atomic property,including for present purposes the spin magnitude and its direction, must varyirregularly from site to site. As everybody knows, a common way of dealing withthis problem has been to ignore it, emphasizing instead that the band theory ofmetals involves collective electron orbitals and employing, in effect, a statisticallyaveraged uniform structure. Local moments are accounted for by some extension ofthe idea of Wannier orbitals, assumed capable of describing the details. But signifi-cantly, the concept of an array of localized momentsundoubtedly applicable to themanganesenickel alloysallows us to illuminate the problem of alloy disorder withthe concepts of percolation theory. We do not know the critical concentration thatensures a spin disturbance on a manganese atom in MnNi will percolate an infinitedistance; but the idea that the magnetic connectivity is based upon a spine or uponrestricted pathways with branches having dependent clusters gives us a basis fordeveloping intuition about spin-wave broadening in disordered alloys. In the 1970s

and 1980s Cowley and co-workers (Cowley et al. 1975, 1977, 1980) extensively inves-tigated the spin dynamics of certain mixed-ionic salts (of decidedly local-momentcharacter) in the light of these ideas. The appropriateness of percolation theory wasconfirmed, and they also found that even in their magneticnonmagnetic examplessuch as Rb2MncMg1cF4, close to the critical concentration the excitations were inaccordance with linear spin-wave theory. In our own study herein of Mn73Ni27which, as we shall see, is also a magneticnonmagnetic examplewe do not go intothe mathematics of percolation theory, but still find it a useful insight that in aconcentrated alloy one may have a meaningfully well-defined spin wave and it bediffraction-broadened in a calculable way.

Returning to the collective-electron picture: the spin-wave spectrum of a triplespin-density-wave fcc antiferromagnet has been analysed theoretically by Long &

Yeung in a remarkable series of papers (Long & Yeung 1986b, 1987a, b). They showthat the system offers three acoustic spin waves, the q-dependence of the energyin every case having a leading term proportional to q and all with a pronouncedbroadening in energy proportional to q due to the phonon interaction we have dis-cussed and also due to decaying into electronhole pairs (the interaction with theStoner modes). In addition there is a fourth mode that couples to neutrons only




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very weakly. Each spin density wave creates susceptibility in a characteristic set ofone-third of the non-chemical small magnetic BZs; or, alternatively expressed, in anyone magnetic BZ the susceptibility reflects the dynamics of just one particular spindensity wave. In any one such set of BZs there are three Goldstone-mode spin wavesthat couple to neutrons, but their intensity distribution follows rules that allow them

to be observed individually, especially easily in high-symmetry directions. We haveexploited these rules by studying the acoustic spin wave (acoustic except for a smallanisotropy gap) along [001], which in fact stems from the spin density wave polar-ized also along [001]. In this axial direction the cross-section for neutrons takes thesame form as the more familiar expression calculated for an insulating substance.The basic symmetry argument underlying these conclusions is set out in Long et al.(2004), and details of the susceptibility distribution may be found in Long & Yeung(1987b).

Onto this elegant theory we have to graft some kind of an account of the spin-wave energy broadening due to static magnetic disorder. Two factors contribute: thepresence of spin defect sites (at the nickel positions), and the existence of aberrantspin angles at the majority sites. An unusually heavy contribution to the broaden-ing might therefore be expected. The fact that magnons in Mn85Ni9C6 do have anextremely broad energy spectrum is known from our earlier work; but ideally, theforegoing ideas should be investigated by measuring the spin-wave spectra of MnNialloys over a wide range of conditions and analysing them into the contributions fromall different processes.

We have thought it prudent to begin with a study of the spin wave in a compara-tively featureless alloy well to the right of 22% Ni on the metallurgical phase diagram,where the effects of magneto-elasticity must be dramatically less than to the left ofthat point, and where it might be hoped that the consequences of magnetic struc-tural disorder plus the magnonelectron interaction intrinsic to an itinerant-electronantiferromagnet would dominate the scene. Such is the study that we report here.A crystal composition Mn73Ni27 was chosen, whose magnetic structure is cubic withNeel temperature TN = 445 K, for the very good reason that it offers effectively

a null matrix for coherent nuclear scattering of thermal neutrons (the manganeseand nickel amplitudes cancelling out), so that in a fully disordered alloy there isno nuclear contribution to Bragg reflections or to coherent phonon intensity. Therewill still be coherent phonon intensity in the magnetic BZs due to the effective mag-netic scattering length of the manganese ion; but (looking ahead to figure 10) itsdispersion relation is dramatically separated from that of the antiferromagnon. Ourobservations are restricted entirely to room temperature; but comparable studieswhich extend the measurements to elevated temperatures up to 1.6TN have sincebeen undertaken on the same crystal by Moze et al. (2004).

In analysing the data, one essential idea we bring is the contrast between magnoninteractions with quanta, envisaged in the standard way as involving a wave of infi-nite extent enduring for a limited time, and magnon interactions with the static spin

disorder, represented mathematically by imposing on the spin-correlation functiona spherically-symmetric decoherence function. Representing the spin-wave disper-sion in the standard way by means of a surface in (q, )-space, the former set ofprocesses might be expected to induce broadening in the -dimension, the latter toinduce broadening in the q-dimension. The two predictions differ sufficiently thatin a favourable case it was hoped the data would discriminate between the two dif-




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ferent kinds of process that render the magnon energy uncertain. Our result, thatstatic magnetic spin structural disorder is overwhelmingly the main contributor tothe magnon linewidth in Mn73Ni27, might be thought disappointing from one pointof view, in that we might otherwise have hoped to gain information about the inter-actions with quanta. In fact, the data bring out an entirely different featurethe

extremely closely localized nature of a spin-wave oscillation in these alloys due tospatial incoherence.

2. The crystal

Our single crystal, in the form of a cylinder of diameter 14 mm and length 31 mm,was cut with a crystallographic 110 direction along the cylinder axis from a 25 mmdiameter ingot kindly prepared by Dr D. Hukin of the Clarendon Laboratory, Uni-versity of Oxford, by annealing at 1000 K and quenching into water. The fcc unitcell dimension at room temperature was a0 = 3.680 A.

After our experiment was completed, the crystal was further examined by Moze& Hicks (1982) at Monash University, who studied the diffuse elastic scattering ofpolarized neutrons. Their investigation confirmed an earlier observation by the sameauthors, using polycrystals, that the moment on the nickel atom, in terms of theBohr magneton B, is (0 0.1)B (Moze & Hicks 1981). The average manganesemoment (when 27% of nickel is present) was shown to be (2 .0 0.1)B, which maybe compared with a rival figure of (1.8 0.1)B obtained by M. W. Stringfellow(1974, unpublished research) with a polycrystal.

Moze & Hicks (1982) also measured the Cowley short-range-order parameters(r) for our crystal, where r is an interatomic distance, out to the tenth atomicshell around a nickel atom. In a later publication (Moze & Hicks 1984a) they plotthese values graphically, and derive an effective chemical pair-interaction energyV(r) between nickel atoms that would produce them. V(r) is seen to be repulsiveat nearest-neighbour positions (specimen coordinate a0(

12 ,

12 , 0)), but attractive for

the second-neighbour shell (specimen coordinate a0(1, 0, 0)), with much diminished

absolute magnitudes at greater distance. Thus the nickel atoms tend to anticluster,instead tending to order at the corner sites of the face-centred cube.

The Monash data reveal that chemically, our crystal is very nearly in the per-fectly disordered state. Beyond the second shell of neighbours to a nickel site it isdifficult to detect any effect significantly above statistical uncertainty; but in theimmediate neighbourhood of a nickel atom, taking a statistical average, it can beshown that in broad terms the state of our crystal has been modified away from per-fect disorder essentially by transferring a mere (6 12 112)% of the nickel atoms in thenearest-neighbour shell of the perfect case to the second-neighbour shell. The mutualrepulsion of the nickel defects is thus apparent; but its effects have successfully beenkept very small.

A more complex situation is encountered on examining the magnetic moments.

By considering the diffuse elastic magnetic scattering, Moze & Hicks (1981, 1982,1984b) showed that a single nickel impurity atom, taken in isolation, subjects thesurrounding manganese moments to a large disturbance out to third neighboursand beyond. The superposition of these defect patterns in a concentrated alloy suchas ours creates in the manganese matrix a spatially irregular fluctuation of the mag-netic moment. However, a careful analysis of the situation by Long & Moze (1990)




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accounted for the diffuse intensity as arising from coherence between the amplitudesscattered by a certain pattern of spin disturbances that is thought to be set uparound each nickel impurity.

Finally, in 1993 Saunders & Salleh (1993) at the University of Bath published anextensive study of the elastic properties of our crystal, or pieces from it, as a function

of elevated temperature and pressure.

3. Preliminaries

(a) Technical considerations. The apparatus

The theoretical studies of fcc manganese (Gillan 1973; Cade & Young 1977; Young1977; Long & Yeung 1987b), although approximate in different ways, concurred insuggesting that the lowest spin-wave branch, as a function of ( q, ), would be foundto rise from a small energy gap to show a linear region of slope ca. 30 km s1; at themagnetic zone boundary the energy would attain 100200 meV. The mode would befound to have an enormous half-width of ca.20%.

Now in practice, to make neutron scattering measurements at these high energies

and at low enough momentum transfer to avoid serious intensity losses arising onaccount of the magnetic form factor, it is essential to use incident energies up toat least 0.5 eV. When we first began this enquiry there were no neutron sourcesoffering enough incident intensity in that energy range. Thus in our earliest work(Haywood et al. 1969, 1971) we were able to follow the magnon energy only up to50 meV, although measurements were made along both 001 and 110 and at 77 Kas well as room temperature. Later (Hennion et al. 1976) we reached 110 meV in aninvestigation using a copper monochromating crystal on the triple-axis spectrometerat the high-flux beam reactor of the Institut Laue-Langevin, Grenoble, France. Thelast-mentioned investigation detailed the great widths of the magnon spectrum lines,and offered a first, but mathematically unsatisfactory, attempt to determine theirtrue shapes by deconvolution from the instrumentally broadened form.

The situation was transformed when, in 1975, the Oak Ridge group and par-ticularly Dr H. A. Mook very helpfully lent their large high-reflecting berylliummonochromating crystal to the Institut Laue-Langevin, where it was installed on thespectrometer IN1 employing the hot source operating at 2000 K within the high-fluxbeam reactor. We were privileged to use this equipment, and took a further datasetextending up to energies of 180 meV. An analysis of the spin wave is now at lastpresented, with apologies for the quarter-century delay.

(b) Precautions

Even with this very-high-quality apparatus, the significant counts at our highestenergies were only of the order ca. 2 counts min1 above a background of the orderof 5. Inevitably, to make accurate surveys of rather featureless, gently undulat-ing spectra at these intensities a range of technical precautions must be observedrigorously. The enclosure and collimation of the neutron beam paths with use ofabsorbing materials are only the most obvious of these. We employed as analyserthe 002-planes of a pyrolytic graphite assembly engineered into the form of a slablying across the neutron beam with the [002] normal in the plane of the slab, so thatthe significant rays all passed through the analyser in a geometry that varied little




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with the neutron energy selected even up to the highest values. We had to considerthe well-known hazards of triple-axis spectrometer experiments that arise when, typ-ically, the crystal under study rotates during the course of a run. During such a run itis possible for a Bragg reflection to flash out and create a gas of secondarily scatteredneutrons in the specimen chamber that must not be allowed to reach the counter.

More importantly the Bragg reflection would alter the degree of extinction of theincident beam in the crystal, thereby creating a dip in the recorded spectrum. Sucha Bragg reflection could be produced by higher harmonics of the incident wavelengthas well as the fundamental one. Most serious of all, it is perfectly possible for a Braggreflection evoked in the sample by any order of the monochromating crystal planes tohave the same take-off angle as that to which the spectrometer is set at the momentthe Bragg reflection appears, so that it intensely irradiates the analyser crystal andproduces spurious counting if there is any disordered or incoherent scattering bythat analyser. Finally, in addition to the hazards from Bragg scattering there is thedanger that some combination of orders of reflection in the monochromating andanalysing crystals will set up an unintended parallel survey of phonon or magnonintensity, possibly in a different BZ, whose counting rate will significantly deform thespectrum the instrument is notionally set to record.

The remedy for these ills is to plan every survey meticulously in advance, so thatthey cannot occur. We did this, much assisted by the fact that the (10 .1) planes of theberyllium monochromator have zero structure factor for third-order intensity, whilefor fourth order the processes feared take place at such large momentum transferthat the square of the magnetic form factor effectively removes the problem by beingso tiny.

There is one more consideration, in that a non-zero intensity will be contributedby the neutrons creating individual electronhole pairsthe so-called Stoner scat-tering. But in the light of theoretical deliberations by Blackman et al. (1987), andalso our own experience with nickel (Lowde & Windsor 1970), we concluded thatthis process will merely create a background without conspicuous features and anorder of magnitude weaker than the spin wave scattering.

We accordingly believe that the data put forward herein represent pure first-ordercoherent magnetic inelastic scattering from Mn73Nl27, with effectively zero distortionto the statistical accuracy claimed.

(c) Extracting from the data

In conventional notation, the differential cross-section per atom for inelastic scat-tering at temperature T with creation of an [001] antiferromagnon having energy and wavevector q, involving incident and scattered wavevectors k and k thattransmit a momentum K = (k k), is related to the interesting perpendicularsusceptibility Im (q, ) = Im xx(q, ) = Im yy(q, ) of the crystal by

2

E=

4

e

2

mc2k

k |f(K)

|2(n|| + 1)(1 +

K2

z

) Im (K, ) (3.1)

(see the treatise by Lovesey (1987)). Here n|| is the Bose oscillator-occupation num-ber, and the other symbols have their conventional meanings. By appeal to what isin effect a tight binding approximation, our definition of susceptibility factors outthe square of the magnetic form factor f(K), which consequently appears explicitly




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in (3.1); as a result the susceptibility (K, ) is periodic in the chemical unit cell,so that K is equivalent to q + G, where G is a reciprocal lattice vector taking Kback to the first BZ. In practical calculations we used the well-known form factor ofMn2+, and although this cannot be exactly correct it is close enough in the contextof the present experiment.

Observed counting rates are proportional to this cross-section times the incidentbeam intensity and the efficiency of the analysercounter system, but distorted inparticular by two effects. Firstly, beam attenuation in the sample depended on inci-dent wavelength, where, for example, at 1 A the (1/e) length for attenuation was14 mm, equal to the diameter of the cylinder. Correction factors for this effect weretaken from tables compiled by Rouse et al. (1970). Secondly, the finite resolutionof the instrument acted to smear out features of the recorded spectrum and thus,among other things, to displace the magnon peaks significantly. The use of veryshort incident wavelengths down to ca. 12 A put the instrument to maximum disad-vantage in this connection, and for the higher-energy runs, conditions were such thatan infinitely sharp spectrum line would have been broadened by at least 0.1qmax.To deal with this, before every run we employed the program Txres to determinewhich configuration of the spectrometer would give the smallest broadenings, andafter every run the observed curve was subjected to a procedure, described below,that effectively deconvoluted the true spectrum from the instrumental resolutionfunction.

(d) Modelling the susceptibility

A well-known theorem relates the susceptibility to the correlation functionS(q, ) between the components of spin perpendicular to the momentum transferin the circumstances of this experiment: thus

S(K, ) =1

2

r

dt ei(Krt)S(0, 0)S(r, t)T

=4h

(n + 1) Im (K, ). (3.2)

The sum runs over all atoms. To investigate the causes of spin-wave broadening, ourstrategy will be to guess a spatiotemporal correlation function and derive the relatedsusceptibility via (3.2).

4. The experiment

(a) The data

We have studied the generalized susceptibility only as manifested at propagation

vectorsq

along the direction of the tetragonal symmetry axis, i.e. parallel to a unitcube sidespecifically, say, along [001]. The first nuclear reflection encountered along[001] is 002, at wavevector q = 2 with q in units of 2/a0. Along [001] in the smallmagnetic BZ qmax is at 0.5, with absolute magnitude 0.854 A.

The cylindrical axis of the crystal, which may be called [110], was held vertical sothat the [001] direction lay in the equatorial plane of the spectrometer. We made a




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(arbitrarys

cales)

SW

OZOZ

The best-

( )

w

wavevector q (rlu)

scatteredintensity

,OR

arbitrarysc

ales

Figure 1. Spin-wave peaks in the susceptibility of Mn73Ni27, as seen in neutron scattering spec-tra taken by varying the wavevector q along [001] at constant energy. With the BZ boundaryat 0.5 reciprocal lattice units (rlu) it is evident that the distributions are exceptionally broad.Short vertical lines representing the experimental points show raw counting rate data and their

ranges of standard deviation, corrected for background intensity only. A bold curve through thedata in each case shows the best fit obtained by varying the parameters of a broadening func-tion that describes the brief lifetime or the short coherent range of the spin wave in real space,taking into account the instrumental resolution. A more lightly drawn curve below it resultsfrom deconvoluting the true scattering spectrum from the instrumental resolution function andderiving from it the imaginary perpendicular susceptibility (plotted here to an arbitrary scale).The left-side spectrum, at an energy approximately a quarter of the zone boundary energy,shows a striking feature of the susceptibilitynamely, its strong continuity across q = 0 withno pronounced tendency to fall to zero or to low values at the centre of symmetry. At 50 meVthe bold curve shows the best fit given by our most successful decoherence function, a general-ized OrnsteinZernike (GOZ) function describing the supposed collapse of the spin correlationfunction S(0)S(r)T with increasing separation r between two spins in the crystal. Thecurves for other imagined correlation functions differ measurably, as illustrated. At 72.5 meV,two rival best fits are shown, one using the GOZ function as above and representing the idea

that the spin wave is impeded in space by the magnetic disorder but not cut short in time, andthe other representing the idea that an infinite spin wave is created, but quickly extinguishedby interaction with other particles. The latter assumption gives the curve with the deeper Mshape. Depending on which input theory is held to give the most convincing account of the datathere will be differences of interpretation.

series of z-direction scans, using neutron energy loss, in the 110 BZ from 1, 1,0.8to 1, 1, 0.8 and in the 112 BZ from 1, 1, 1.5 to 1, 1, 2.5; in all, 12 surveys varying qat constant energy with values up to 125 meV. The surveys at 110 had a(1 +

K2z) factor very nearly constant and 1. Three constant-q surveys were alsomade for q = 0, 0.65qmax and qmax, varying up to 180 meV.

Representative spectra are shown in figures 1 and 2, where the raw data are dis-

played from constant-energy scans for

= 35, 50, 72.5 and 110 meV, corrected forbackground count only. The curve for 110 meV is unusual, in that a combination ofeffects has produced an asymmetry so marked as almost to suggest a third peak onthe left-hand side. However, our analysis, particularly in regard to the instrumentalresolution function, allows the shape to be explained (Tindle 1984). Figure 3 showsa constant-q scan, and figure 4a, b the q = 0 scan.




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wavevector, q (rlu)

scatteredintensity,OR

Im

0.5 0.51.0 1.00

Figure 2. As with figure 1, a spectrum obtained by varying wavevector q along [001], here atenergy 110 meV. A bold curve shows our best fit to the data points on the assumption that awell-defined spin wave is broadened by having to propagate through disorder; a lighter curvegives the underlying susceptibility Im (to an arbitrary scale). The interpretation allows therather curious shape to be explained, and locates the wavevector q of the fundamental spinwave to better than 5%.

scatteredintensity

100 200h (meV)

Figure 3. A survey of the high energy scattering at 1 , 1, 1 12

, i.e. withwavevector q kept constant at qmax, the zone boundary position.

5. A formalism

(a) Procedure

It is already clear that if our surveys were plotted on a (q, )-plane a function wouldbe created that presents a broad swathe of elevated intensity tracing out a pathfrom low to high that could be thought of as a heavily smeared-out spin-wavedispersion law. (It is jumping ahead, but a glance forward to figure 13 shows theconcept that we can already see roughly.) Our method of analysing the data hasbeen to interpret them throughout in the simplest possible light: taking a lead from




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h (meV)

scatteredintensity,OR

Im

(a)

(b)

2 3 4 5 6 7 8 9 10

Figure 4. A survey varying the energy at wavevector q = 0 over the range = 010 meV.Starting from zero, an energy gap is observed before the intensity mounts up. In (a) a successfulfitting curve is seen, derived on the assumption that the spin correlation in real space fallsaway in all directions according to a generalized OrnsteinZernike formula. The more lightlydrawn curve below shows the underlying susceptibility. Corresponding curves in (b) illustratethe best results obtained on an alternative view that the spin wave is broadened not by spatialdecoherence but by interaction with other excitations that reduce its lifetime. In principle thismodel should predict no intensity at q = 0, except at the gap energy, and what is observed hasto be attributed to the effects of imperfect instrumental resolution. But the impossibly bad fitshows that this interpretation cannot be sustained.

the Brookhaven work on mixed-ion salts we assume that the swathe is derived froma perfectly well-defined spin-wave dispersion curve as given by the elementary theoryof an insulating antiferromagnetwhich would constitute a sharp ridge across thediagrambut somehow broadened by plausible functions, again of elementary form.As a final simplification we represent the dispersion law as essentially a straight line,but with parabolic parts joined on at the low- and high-energy ends. This is, ofcourse, an extremely drastic reduction of the existing complexities; but it is justified

by the earlier analysis using an entirely different approach that we presented at theGatlinburg conference of 1976 (Hennion et al. 1976), which shows convincingly thatsuch a picture is satisfactory to the accuracy we expect to attain. Theoretically, too,the approximation proposed is not unreasonable, as we explain in 7.

In relation to each spectral survey the method of analysis is to parametrize allthe mathematical features of the foregoing, thus creating a general algebraical form




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wavevectorwavevector

energy

(a) (b)

Figure 5. Schematic of a sharply defined spin-wave dispersion relation (q) in the form of

a bold curve, broadened by various processes to an extent indicated by hatching. Two specialcases are illustrated, where the processes operate exclusively to create broadening along (a) theq-dimension and (b) the -dimension. Each representation has a weakness, in that (a) cannotbe used to account for any intensity found above max, and (b) is incapable of explainingintensity that might be found along q = 0.

for a distribution of spin-wave scattering power in the four-dimensional reciprocalspace (q, ); then to convolute this with the known four-dimensional instrumentalresolution function in that space; and then to best fit the result to the data. Inthis way we get: a deconvoluted susceptibility distributionshown in figure 1, forexample, by the lower solid line; a (q, ) for the spin wave, and its standard deviation;a curve through the raw spectral data giving our best interpretation of what theyindicate; and a statistical chi-squared factor for quality of fit that serves as a factor

of merit for the fitting parameter set.

(b) Studies of the spin-wave line broadening

In the matter of broadening functions we have again sought the greatest simplicity.Following the spirit of our remarks in 1 we made two analyses, one with broadeningof the spin wave in only (figure 5b), to represent the effects of magnon interactionswith quanta, and one with broadening in q only (figure 5a) to represent the effectsof spatial decoherence. The cost of this simplicity is that each approach has a weakfeature: the q-broadening set cannot cope with diffuse intensity above max, and the-broadening set has no mechanism for attributing susceptibility to the line q = 0.

(c) Change of notation

Two sets of (wavevector/energy) symbols will be required. In all of what followswe shall use dashed symbols q and for the underlying well-defined spin wave andthe space in which it operates, and plain symbols q and for the diffuse scatteringpower. Thus a specific magnon may be located on the dispersion law at q, , but




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A

B

TS

Oh'

h'= f(q')

'

qx

qz

(q,)

'

'

q's

Figure 6. The spin-wave dispersion relation (q) (assumed isotropic) represented schemati-cally by means of a bowl in a space of energy and the z- and x-dimensions of the wavevectorq. The spectrometer can be set to pick up scattering intensity from any desired point, forexample B, subject to finite resolution as represented by an ellipsoid around B giving the locusof the half height of the resolution function. An experimental spectrum is obtained by trackingthe point of observation B through the bowl, possibly along a radial direction as for a q scan.

surveys through that point varying the wavevector will give broad spectra that arefunctions ofq, and surveys through the point varying energy will be functions of .

(d) A formalism

Figure 6 reproduces a familiar illustration of a spin-wave dispersion surface inenergy-momentum space, where, however, to get a comprehensible diagram onedimension of momentum is suppressed. The principal axes are labelled , qx and q

z,

so that the dispersion law of a sharply defined spin wave is = f(q) and appearson the diagram as a kind of bowl. At a height there is a continuum of correspond-ing propagation vectors q, such as OT and OS, lying in a plane as shown, and in thediagram we have simplified the situation by having all their lengths equal, so that thelocus of spin-wave points for energy is the circle A. (In a real case there could bea variation in |q| ofca. 25%). The spectrometer can be set to measure the scatteringpower of the solid at any desired point, not necessarily on the bowl. In the experi-ment this point is tracked along some trajectory that passes through the dispersionsurfacecommonly along a radial line ofq, or at fixed q in the direction of varying .

In an ideally simple case there is a flash of intensity at the exact setting (q

, f(q

)) fora spin wave, to which we may conventionally attribute a scattering power S(q, ).However, in practice an instrument has imperfect resolution, so that at any notionalsetting (q, ) it actually accepts intensity from a significant region of (q, )-spacearound the designated point according to some function R(q q, ), usuallytaken as a Gaussian function and represented on the illustration by a hatched ellip-




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soid around the notional point (q, ). Then the measured scattering intensity will beproportional to the product ofR and the scattering power, integrated appropriately.

It is onto this situation that complications due to spin-wave broadening have tobe superposed. We introduce them through the idea that the scattering power S atany one point (q, ) is itself dispersed over the (q, )-space via two broadening

functions g and h:

S(q, ) =

0

N()g(q, q)h(, ) dq. (5.1)

Integration is along the curve (q). Normalization is ensured if we say

g(q, q) dq = 1,

h(, ) d = 1. (5.2)

As mentioned above, we took only two very simple models: the -spread model,where the broadening of the spin wave is considered to be entirely within the functionh (figure 5b), and a q-spread model, where it is all attributable to g (figure 5a). To

be explicit:

(i) Broadening in only

g(q, q) = (q q); (5.3)S(q, ) = N()h(, ). (5.4)

We employed the Lorentzian

h(, ) =

2

1

( )2 + 142, (5.5)

and Gaussian

h(, ) =

exp

( )

2. (5.6)

Here is a full width at half height and = 2

ln2.Also the LorentzianGaussian convolution:

h(, ) = Re

G

w

( )

G+

i

R

. (5.7)

In the convolution formula, R is the Gaussian to Lorentzian width ratio, and w(z)

is the complex error function. G is a number that can be determined from

1

R

G

= 1 1.12

G

2+ 0.12

G

3. (5.8)

In fitting the -spread model we used these formulae for S, varying and R.




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y

x

z

A

s u

q

O

Q T

ZB

S

Figure 7. A three-dimensional generalization of the construction in figure 6, concentrating onthe situation at a given energy and showing all three dimensions of the wavevector q. Thespin wave q values, again assumed isotropic, for the energy now describe the sphere A.The spectrometer is considered to be set for the energy and a point q, typically not onthe sphere. In the presence of line broadening due to spatial decoherence, the point q now hasscattering power referred to it by all points such as S on the surface of the sphere, but attenuated

according to the distance u from q.

(ii) Broadening in q only

Here

h(, ) = ( ); (5.9)

S(q, ) =dq

dN()g(q, q), (5.10)

and we have to suggest forms for g. The situation we consider is that where thestatic disorder in the magnetic structure causes the spin correlation in real space todecohere increasingly with increasing separation r between the spins. We may say,quite phenomenologically, that the correlation between spin deviations making up

the spin wave is proportional to

cos(q r t) F(r), (5.11)where F(r) is an isotropic function of distance r that falls away from a peak valueat r = 0. The scattering power is proportional to the Fourier transform of (5.11),namely

1

2

0

dr

dt ei(qrt)(ei(qrt) + ei(q

rt))F(|r|), (5.12)

leading to two values, of which we take the positive- one 12F(q q)( ).To proceed to an expression for g, we must first three-dimensionalize the treatment

of wavevectors q in figure 6. Retaining the approximation that the spin-wave energy

is isotropic, the circular locus A of q

spin waves with energy

becomes thesphere A centred at O in figure 7. Figure 7 shows in perspective a three-dimensionalconstruction, with points S and T on the spherical surface, and the line from O tothe BZ boundary being the [001] axis surveyed in this experiment. The spectrometermakes a constant-energy survey like those of figures 1 and 2 by tracking along thisline while set for energy , and with an uncomplicated specimen discovers intensity




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at the point T, i.e. on the sphere. However, in the presence of line-broadening dueto spatial decoherence, the spectrometer adjusted for a point q not on the spherestill picks up intensity, because all points on the sphere such as S now have theirscattering power dispersed across the diagram. The illustration shows how some ofthe intensity appropriate to S reaches the point q after diminution by a factor F(u),

where u = q s. The total quantity received is the expression (5.10) but with theintegral now over the sphere.On defining the sine and cosine Fourier transforms of F:

FS(k) =

0

F(r)sin(kr) dr; FC(k) =

0

F(r)cos(kr) dr; (5.13)

it is not difficult to determine that the integral of 12F(q s) over the sphere is

g(q, q) =FC(q q) FC(q + q)

qFS(q). (5.14)

The expression (5.14) has been normalized, so that

g dq = 1. It is the formula we

have used in analysing the data for spatial decoherence.

(e) Normalization

To interrelate the different model distributions appropriate to different spin-waveenergies , we have again fallen back on the most elementary theoretical pictureand further normalized the areas of the spectra, to 1/ in the q-spread case and1/ in the -spread case. With use of the best-fitting parameters, the result is amodel prediction for the susceptibility over the whole range of (q, ) surveyed. Andbased on this, an interpolation formula may be constructed covering the gamut of of interest here.

6. Results and analysis

As the trial curves in figure 1 show, different assumptions about the origin of theline broadening lead to significantly differing predictions for the spectra. The dif-ferences, however, are quite subtle; and to discriminate between competing theorieswe shall have to examine the counting statistics closely. In practice this can onlymean inspecting values of the standard chi-squared function, which for any givendistribution studied we shall express per degree of freedom. To avoid confusion withthe susceptibility we refer to the statistical chi-squared per degree of freedom asa cspdf. By way of a reminder: the significance of cspdf depends on the numberof degrees of freedom in the system. Taking as an example the fit of 10 spin-wavedispersion points to a straight line, a cspdf of 0.92 would imply that the observedpoints have just the degree of scatter that one would typically expect if the theory

were correct. A cspdf of two would imply that if the best-fitting line were absolutelycorrect, then the probability that another run would give points more deviant fromit than the present ones is only 0.042. Conversely, if the data are viewed as the givenquantity, a best-fitting cspdf of two could be interpreted as making it unlikely thatthe theory is right. A cspdf of three would mean a probability not 0.042 but a nine-teenth of that. On any reasonable analysis these figures indicate that a cspdf of two




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1

1000

0

50

(meV)h

Figure 8. Best-fitting values obtained when the shapes of spin-wave spectra at constant energy are attributed to the spin correlation in real space S(0)S(r)T falling away as er/r.

is a very bad mark for a trial theory, and three is a calamity. Turning next to a casewith 28 degrees of freedom, which occurs with several of our constant- spectra, the

corresponding figures are: cspdf 0.98, probability 0.5; cspdf 2, probability 0.0013;cspdf 3, probability 107.In describing the fits we have tried, it is convenient to begin with the q-spread

model. We shall find it is successful; and by contrast the inadequacy of the -spreadmodel is the more quickly demonstrated.

(a) Broadening in q only

(i) The parameters and

For F(r) we first experimented with exp{(12r)2} and er, where and would be new parameters to be determined by the fitting. Finding that the latter

gave a consistently better fit, we investigated a slightly more complicated range ofcorrelation functions based upon it, namely rer, where is another arbitraryparameter, allowed to range from 1 to 1. The value = 1 here of course givesthe familiar OrnsteinZernicke function, known to give a good account of the spincorrelations in paramagnetism. With every constant-energy survey a non-zero madea distinct improvement. The preferred values are shown in figure 8 for nine surveysfrom 25 to 125 meV; their evident tendency to cluster around a particular magnitudereinforces the implication that introducing this new parameter has in some way takenus a step nearer the truth. Taking the example of the curve for 50 meV in figure 1b, thebest fit with = 1 offered a cspdf of 1.83; but with = 0.59 it was 0.83. We adopteda value = 0.6; and with this, cspdf values for the nine surveys averaged 1.11.

The best-fitting , displayed in figure 9, is quite closely proportional to q, beingin fact (q) = (0.25

0.01)q with a cspdf of 0.95.

Results from a similar exercise with exp{(12r)2}er were unsatisfactory. Infact, the worst-fitting set of functions in all the foregoing studies was the Gaussianset, whose best spectra gave cspdf values averaging 1.50. If rather loosely we takethis cspdf as a score typically representing the quality of fit of a Gaussian F, tobe compared with our best score of 1.11, we could say the probability that another




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0

0.10

0.05

0.1 0.2 0.3 0.4

(rlu)

q' (rlu)

Figure 9. Best-fitting values obtained when the shapes of spin-wave spectra at constant energy are attributed to the spin correlation in real space S(0)S(r)T falling away as er/r.

experimental run would give worse statistics is 33% with the generalized OrnsteinZernicke function (against an ideal of 50%), but only 4.3% with the Gaussian F(r).

(ii) The spin wave

Individual spin wave q, points appear in figure 10. They strikingly confirm thepresumption that the dispersion law has a long straight part, which in fact extendsfrom 17 to

23 of qmax. Each point derives from an analysis presupposing the existence

of the long straight part; but there is no mathematical requirement for the data to fallin with that, and the fact that they do is a confirmation of the idea. A propagationvelocity of 34 1.4 km s1 is indicated, with a cspdf of 1.98 almost all contributedby the top two points.

To obtain the at q = 0, a survey was made along the lower part of the -axis(figure 4a). The modified-exponential assumption with = 0.6 again provided acredible explanation; omitting the lowest point (which is dangerously close to theBragg setting, and also to the dispersion curve for magnon absorption) the curveshown returns a cspdf of 1.1. The spin-wave gap energy is found to be 4.0

0.1 meV.

It remained to fix the high point on the dispersion curve at qmax, which we sawcannot be done by the q-broadening method. For this one point we made use ofthe -spread method, described below, applied to the experimental measurements offigure 3. At the highest energies in figure 3 it was difficult, at the time, to monitorthe incident neutron intensity exactly; but we considered that the data in that figure




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q' (rlu)

50

150

0 0.5

100

(meV)

h'

Figure 10. The dispersion curve of acoustic spin waves along the tetragonal axis in disorderedMn73Ni27 at room temperature. More exactly, the energies

of the spin wave that under-lies the broadened susceptibility distribution observed, as deduced on the assumption that thebroadening of the spectral line occurs entirely in the wavevector dimension rather than in thatof energy, and shown as a function of wavevector q extending from zero at a magnetic reciprocallattice point for (say) the z-oriented spin density wave to the zone boundary at that point plus00 1

2. Broken lines in the lower part of the figure show leading segments of the dispersion curves

for phonons in the alloy. (These also give neutron scattering in the magnetic BZ in virtue of themagnetic scattering lengths and phases associated with the manganese atoms.)

justified our concluding that the underlying spin wave at qmax had energy 140 5 meV.

(iii) The susceptibility, the decoherence parameter

The interpolation formula we have now obtained is mapped in figure 11. It gives agood impression of the spin-wave ridge, despite the tremendous variation in inten-sity from one end to the other. The contours show very clearly the asymmetry of thedistribution, whereby there is a kind of filling of the zone centre, in this sense: thatsections at constant energy have a peak, but to the left of that peak decline by onlyca. 40% towards q = 0. Thus one cannot define a width at half height, because thefunction on the left does not descend to half height.

The amplitude of at the peak of the ridge is plotted logarithmically against q

infigure 12, where it is seen to approach the prediction (q)2 of elementary theoryonly asymptotically with increase ofq to about 0.7qmax. On turning back towards thecentre of the zone the index falls further, to reach 3 at 0.16qmax. Nevertheless, if oneday an approximation were required it might be worth noting that the broken line ofslope 2 in figure 12 and its extension to the left represent a decline proportional to




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50

150

0

100

(meV)

h

0.5

wavevector q(rlu)

Figure 11. Logarithmically spaced contours, in steps of

2, displaying the imaginary part ofthe perpendicular susceptibility of Mn73Ni27 at room temperature, as measured for wavevectorsalong 001 running from a magnetic reciprocal lattice point to the zone boundary. The scale ofmagnitudes is arbitrary.

(q)2 that is within 9% of the truth over the whole straight part of the dispersionlaw in figure 10. (It should be realized that the apparent line of maxima indicated bythe contours of figure 11 is not exactly the same as the spin-wave line in figure 10.)

The function in figure 11 is shown to advantage in figure 13, for which, however,the susceptibility has been multiplied by to reduce the disparity in magnitudebetween the highest and lowest values and to offer a good view of the spin wave.The asymmetry that leads to being higher on the low-q side than on the high-qside is again very apparent.

Between 15 and 110 meV, that is to say along the straight-line part of the disper-sion law in figure 10, the distribution of figures 11 and 13 fits all 13 of our measuredspectra simultaneously with a cspdf of 1.57.

(iv) Line width, line shape

A spectral linewidth is normally of interest in that its inverse is directly relatedto the lifetime or the coherence length of the excitation. For example, if er wereapplicable as the F(r) in our model calculations of 5 there would be a linewidth




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100

10

1

0.07 0.1 0.2 0.3 0.4 ZB

q'(rlu)

Im

(arbitraryscale)

peak

slope 2

slope 3

Figure 12. A loglog plot showing the amplitude of the crest of the ridge in the generalizedsusceptibility of figure 11 as a function of propagation vector q. The form of the dependence onq approaches (q)2 asymptotically.

0

q

ZB120me

V

0

h

Figure 13. To an arbitrary vertical scale: the imaginary part of the perpendicular susceptibilityof Mn73Ni27 as measured along 001 at room temperature, after multiplying by to make moreclearly appreciable the distribution at the higher energies.




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0 0.1 0.40.30.2

0.20

0.10

0.05

0.15

q'(rlu)

(rlu)

w

Figure 14. Spectral half-widths-in-q at half height of the peaks observed in constant-energysurveys of the susceptibility, shown as a function of the magnon wavevector q. The magneticzone boundary is at 0.5 rlu. , the external half-widths w, defined as in figure 1 and measuredfrom computer line drawings; , the half-width-in-q at half height of the best-fitting broadeningfunction F(q) (see equation (5.12) et seq.) when that function is assumed to be Gaussian.

strictly proportional to at small q values and very nearly so even at larger q valueswhere the effects illustrated in figures 6 and 7 come into play. Then an observed

linearity of in q

would mean that the coherence length of the excitation was (q)1, in accord with simple ideas.The actual situation is not quite so straightforward. Firstly, with Mn73Ni27 it is

not immediately obvious how to define profitably the width in q of an extremelybroad and unsymmetrical ridge in . But we may profit from a more general con-sideration. With strictly proportional to q in F(r) = er/r, simple inspectionof equations (5.13) and (5.14) shows that g(q, q) as a function of qand thus, by(5.10), the scattered spectrum as a function of qis subject to a kind of scalingwhereby its shape is invariant, the spectrum merely being scaled in amplitude andbreadth as the q of the spin wave is altered. In particular, over a region in whichthe dispersion law is linear in q, any measure of the width of the spectrum, howeverarbitrarily defined, will also be proportional to q. (With an energy gap at q = 0

these statements will be true asymptotically.)To investigate this rule in the absence of a width at half height we could perhapscontemplate the width at 1/

2 height, which according to figure 11 is indeed reas-

suringly proportional to q. One could define an external width to half height as win figure 1; the measurements in figure 14 show that it too is proportional to q, afterallowing for a small additive constant.




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1000 50

(meV)h'

100

50

hwhh(meV)

Figure 15. Half-widths in at half height of peaks in the energy spectra of Im

at constantq, plotted against the corresponding , and derived on the basis that the broadening occursin the direction.

An interesting measure of the width of the spin wave is , when e(r/2)2

isthe form used for F(r). The Fourier transform is a constant times eq

2/2, so thatfor small broadenings would be the wavevector magnitude for 1/e height. In thepresent experiment the observed peak is not Gaussian in shape, the Gaussian F(r)does not give a very convincing fit, and the significance of is not quite so direct. Butthe Gaussian fit does have one great advantage, namely that it allows the computerto specify a width, even though it is the width of a Gaussian that does not quitefit the peak in the susceptibility. There is a slight misfit; but if for any reasonshould be found proportional to q, the scaling phenomenon ensures that the degree

of misfit is invariant over the interesting range of that variable and thus that theproportionality to q is strictly significant. Our measurements of , seen in figure 15,very nearly conform to this case in that is strikingly linear up to the topmostreadings, and departs from proportionality only in virtue of a tiny additive constant0.014qmax.

(b) Broadening in only

We made a study with a GaussianLorentzian convolution function, as in (5.7), inwhich the ratio of the half height widths of the contributing Gaussian and Lorentziancomponents may be called R. Making use of the earlier work we set up a dispersionlaw that was parabolic between a gap energy of 4 meV and 15 meV and which then

joined with continuous slope onto a straight line, all to be fitted to the data up to110 meV. The linewidth of Im at half height, measured in the direction, wasgiven the form pq{1 [8/( + 4)]2}, in which p is an adjustable parameter. (Thesuitability of this expression may be appreciated by looking at figure 15.)

It might at first be thought that spectral surveys along q at constant hardlyconstitute an ideal basis for fitting trial shapes in the direction. But it must




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be realized that the specification of in the previous paragraph is complete; anygiven set of parameters fixes the entire function in (q, )-space, just as the fittingparameters of the previous section produced that of figure 11. Thus a procedure ofbest-fitting the model to any survey will produce an interesting set of results.

A specimen fit, together with its underlying Im , is shown in figure 1c. At this

constant energy of 72.5 meV the data, viewed casually, might be thought as welldescribed by the -spread method as they are using q-spread, the cspdf being 1.67as against 1.47 for q-spread. The fitting lines exemplify a general tendency: thatthe curves produced by both theories of broadening fit the data bars more or lesspresentably, but in the -spread case exhibiting a more pronounced peak-and-troughshape, and with the peak at smaller q to help the model to account for the substantialscattering intensity observed in that region. However, a telling weakness in the for-spread as printed out by the computer is that it fails to drop nearly as far towardszero at q = 0 as it must. The aberration is an instance of the difficulties we foresawon looking at figure 5 for the -spread theory at small q.

In general, free fitting of the -spread model to individual spectra was reasonablysuccessful. The cspdfs of the constant- surveys, analysed individually, averaged 1.27.Moreover the half-widths at half height in the direction are seen in figure 15 tobehave well, confirming their necessary linear dependence on q and validating themodel formula used. But as a case for favouring the -broadening model for theseexcitations, the argument falls to pieces when the data are collated all together.Firstly, the optimum R values chosen by the computer for the individual spectrafluctuate violently and unsystematically from one instance to another over a rangefrom 0 to 2.9; when an average figure of 1.5 is adopted for R and the complete setof surveys fitted collectively with one set of parameters the cspdf rises to 2.99aquite unacceptable result. Secondly, the dispersion law emerging from the foregoingprocedures is much worse defined than with q broadening, in that the fit of 10 pointsfrom 15 to 110 meV to a straight line returns a propagation velocity of 383.3 km s1with a quite hopeless cspdf of 7.1. Even if a convinced proponent of the -spreadtheory argued in some way that the most deviant point should be ignored he would

get this cspdf down only to 3.4.Finally, the experimental survey at q = 0 along the -axis up to 10 meV is decisive

against the -spread model. The best free fit is shown in figure 4b. The computerprogram has evidently understood that there is non-zero intensity only at (q, ) (0, 4); it presents as its best offer a symmetrical line about a gap energy (incorrectlyoptimized at 4.6 meV rather than 4.0), and the result is a disaster.

Within the framework of this investigation we therefore have to consider the -broadening picture a failure. Of course, there might be simple modifications we couldmake to the model susceptibility, paralleling our introduction of in the q-spreadanalysis, that would result in an improvement; but we did not manage to find anysuch modifications.

7. Discussion

According to the analysis presented here, the low-lying magnetic excitation mode inMn73Ni27 can be characterized as, fundamentally, a well-defined spin wave of unex-pectedly simple form. But the spectrum line of this excitation is heavily broadenedin virtue of certain identifiable factorsfactors that we have been able to account




26/30


for phenomenologically. Among the causes of broadening, the disorder of the anti-ferromagnetic structure brought about by having 27% of the sites almost at randomwith zero spin is overwhelmingly preponderant. Actually, a substantial contributionto the broadening from interaction of the spin wave at least with the Stoner modesmust be present; but such effects are swamped by those of the alloy disorder and we

have not been able to isolate them.The dispersion law found has an uncomplicated shape rather similar to thatof a simple-cubic insulating antiferromagnet with interactions between nearest-neighbours only, for which the energy in the [001] direction is proportional to

c2 d2 cos2 qa0 in such a way that where anisotropy is negligible the energy variesas q to leading order. In part this is because we imposed such a form on our resultsby demanding that the fitted dispersion curve be a straight line with parabolic piecesat the two ends. But of course that procedure is satisfactory only because the datasupport it. Using our preferred q-spread theory of the line shape the dispersioncurve may be said to start at q = 0 with an energy gap of 4.0 meV and to rise to140 5 meV at qmax, the section between 15 and 110 meV being straight with slope225 9 meV A, corresponding to a wave velocity of 34 1.4 km s1.

The nearest we can get to a theory that might illuminate these measurements isthe 1987 study by Long & Yeung (1987b). These authors considered (among otherthings) the dynamics of a triple spin-density wave in a one-band model metal whoseparameters were adjusted as far as possible to reproduce the density of states andthe magnetization of an appropriate manganese alloy. The Hubbard Hamiltonian forthis system was treated in RPA, and the problem rendered tractable by means ofa group-theoretical tour de force. Two representative model cases were selected forcomputation, and results for the spin waves appear in Long & Yeungs figure 11(1987b, p. 1212). One must bear in mind that the wavevector q in their formalism(hereafter denoted qL&Y) is scaled to be twice the more conventional q used by us;and on that understanding the part of their calculation to be compared with ourexperiment is the left half of each graph in figure 11, i.e. the part with 0 < qL&Y

Documents

R. D. Lowde and G. L. Tindle- On spin waves and spin disorder in face-centred-cubic Mn73Ni27