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THE SCATTERING FUNCTION S(q,ω) OF RbMnF3
M. Evans, C. Windsor
To cite this version:
M. Evans, C. Windsor. THE SCATTERING FUNCTION S(q,ω) OF RbMnF3. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-614-C1-616. �10.1051/jphyscol:19711208�. �jpa-00214033�
JOURNAL DE PHYSIQUE Colloque C I, supplbment au no 2-3, Tome 32, Fborier-Mars 1971, page C 1 - 614
THE SCATTERING FUNCTION S(q,o) OF RbMnF,
M. T. EVANS
Cavendish Laboratory, Cambridge, England and
C . G . WINDSOR
A. E. R. E., Harwell, Berks, England
Rksume. - Au moyen de la mtithode du temps de transit nous avons determine S(q, a), la transformke de Fourier de la fonction de corrtlation de spin de RbMnF,, compost antiferromagnktique de structure cubique simple a 77 O K (0,93 TN) et a 97 OK (],I7 TN) dans toute la gamme de frkquence, le vecteur q dckrivant le plan (1 10). On sait que 1'Ha- miltonien de RbMnF3 est bien approche par le modele d'echange &Heisenberg entre premlers volslns avec une energie d'interaction J = 0,28 meV et S = 512.
A I'aide de cet Hamiltonien, nous avons calcult S(q, w) pour les deux temperatures citks et par une methode nume- rique de la dynamique d'un arrangement 10 x 10 x 10 de spins classiques.
Pour les valeurs de S(q, o) mesurks et calculees, on constate une allure fortement structurke avec de nombreux maxima surtout pour les valeurs de q voisines de la couche limite de la phase antiferromagnetique ; on peut etablir la courbe donnant la correspondance de cette structure avec les maximas des ondes de spin observkes ii 0,93 TN.
La dtcomposition d'un triplet observe pour une valeur de q, en pics distincts perpendiculaires et paralleles est ega- lement obtenue a partir des fonctions de corrtlation des spins calculks.
Abstract. - The time-of-flight method has been used to measure S(q, w ) , the Fourier transformed spin correlation function, for the simple cubic antiferromagnet RbMnF3 at temperatures of 77 OK(0.93 TN) and at 97 OK(1.17 TN), over the full range of frequency and for values of q throughout the (110) plane. The Hamiltonian of RbMnF3 is known to be closely approximated by the nearest neighbour Heisenberg exchange model with an interaction J == 0.28 meV and S = 512. Using this Hamiltonian, S(q, w) has been calculated at each of the above temperatures by numericalsolutionof the dynamics of a 10 x 10 x 10 array of classical spins. Both the observed and calculated S(q, to) at 1.17 T X show appre- ciable structure, particularly for values of q near the antiferromagnetic phase zone boundary, which may be correlated with the spin wave peaks observed at 0.93 Tx. The decomposition of a triple-peaked spectrum, observed at one value of q, into distinct parallel and perpendicular peaks is shown to arise from the computed spin correlation functions.
1. Introduction. - The Van Hove scattering func- tion S(q, o), measured as a function of wave vector q and frequency o , provides an almost complete des- cription of the spin dynamics of a magnetic material.
It is the fourier transform over space and time of the pair correlation functions < S,(O).S,(t) >/S(S + 1) which, from a classical viewpoint, have the physical interpretation of the probability that a given spin is parallel to its neighbour a t distance R observed a time t later. The direct measurement of S(q, w) is possible through the inelastic neutron scattering cross-section with a momentum transfer hq and an energy transfer ho. Previously the scattering from RbMnF,, the archetypal cubic Heisenberg antiferro- magnet,- has been measured at very low tempera- tures [I] and at the comparatively high temperature of 3.5 TN [2] and in the critical region 131. The present experiments and calculations at 0.93 TN and 1.17 TN explore regions of reciprocal space which have so far received little detailed attention. The most interesting region appears to be close to the zone boundary of the antiferromagnetic reciprocal lattice structure, where traces of the famous triple-peaked lineshape, observed by the Brookhaven group in the critical region (4) are seen to persist even at 1.17 TN. To substantiate these results, calculations of S(q, o ) for a finite compu- ter model of classical spins has been performed. These calculations show good absolute agreement with the experimental measurements, and enable the interpre- tation of the triple peak structure into spin wave-like and diffusive components to be confirmed, while the origin of these peaks may be seen from the form of the spin correlation functions.
2. The Inelastic Measurements. - Most of the data was obtained using the twin-rotor time-of-flight spectrometer on PLUTO reactor at Harwell. This instrument, with its thirty separate counters is parti- cularly suited to making surveys of S(q, o ) over a wide range of q and o. The inset of figure 1 represents the scattering diagram for a typical setting where the incident neutron wave vector ko makes an angle
ct = 13O to the (001) axis within the (1 10) reciprocal lattice plane. k' represents the scattered neutron wave vector for a given time-of-flight channel of a given counter. The size of the resolution ellipsoid, whose section is indicated at k', is of order 0.1 A in wave vector and 1 meV in energy (FWHH). K = ko - k' is the scattering vector, and q = K - z is the reduced scattering vector relative to the nearest nuclear reci- procal lattice point z. At each temperature some 10 runs were recorded at different crystal angles a and the results normalised together and averaged within intervals of q and o matching the instrumental resolution. An advantage of the time-of-flight method is that, since the counters are static, the background spectrum may be reliably subtracted and the cross- section established on an absolute scale using a vana- dium specimen [2].
3. The Scattering Function. - The measurements were transformed using the following expression for the cross-section
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711208
THE SCATTERlNG FUNCTION S(q, w) OF RbMnF, C 1 - 615
FIG. I . - The scattering function S(q, o ) / h S / S + 1) given in units of (meV)-1 (atom)-' for RbMnF3 at 1.17 and 0.93 TK. The solid lines show the classical model calculations. The shaded portion of the inset shows the position of q in the 110 section of the reciprocal lattice zone. The experimental points include a small
contribution from the nuclear scattering cross-section.
with (r, y)2 = 0.291 bn and with the form factor approximated by F(K) = exp(- 0.08 K2). The factor 213 arises in the present case of a cubic ferromagnet or antiferromagnet from a spherical average over the spin orientation factor occuring in the general expression for the cross-section. In figure 1 we show, by the filled circles and errors, values of S(q, w ) obtained by averaging the time-of-flight data over the range of q near the magnetic zone boundary indicated by the shaded rectangle in the inset. The 0.93 TN results show a clear triple-peak structure. Cooling the sample to very low temperatures identifies the two outer peaks as arising from spin waves and the central peak as being largely magnetic in origin.
At 1.17 TN structure is still seen in the results at this value of q, and also at several other values, mostly close to the magnetic zone boundary. However over much of the zone the energy spectra decrease mono- tonically from the elastic position without structure.
The solid lines in figure 1 are absolute calculations of S(q, o), made without any adjustable parameters, using the classical computer model described in the following section.
4. The Classical Computer Model. - Within the classical approximation, the spin correlation functions may be computed directly by solving numerically the
equation of motion of a large finite array of spins initially in thermal equilibrium [5], [6]. I n the present case a 10 x 10 x 10 simple cubic array with anti- ferromagnetic nearest neighbour Heisenberg interac- tions and periodic boundary conditions was used.
Small portions of the equilibrium arrays at 0.93 TN and 1.17 TN are shown in figure 2. The 0.93 TN array
FIG. 2. - The computed self correlation functions
at 1.17 and 0.93 TN. The insets show portions of the spin arrays at each temperature.
has long range order with the sublattice mag;1etization direction rotated to lie along Oz. Although the mean probability of a spin lying along this axis
it is possible to see local domains where the short range order is nearly perfect, but where the spin direc- tions lie well away from the z axis. Turning to the 1.17 TN array, the extensive short-range correlations present at this temperature are immediately seen.
In fact the nearest neighbour static correlation is some 60 % of its value at 0.93 TN, although the next-nearest neighbour correlation has fallen to 20 % of its previous value.
Having established the initial directions it is straight forward to calculate the pair correlation functions
< S,(O).S,(t) > if, in the classical approximation, the expectation value is replaced by a simple average over equivalent types of neighbour. Correlations for the thirteen inequivalent neighbours R out to (222) were calculated for times up to 4 h/2 JS and the results
C 1 - 616 M. T. EVANS A N D C . G . WINDSOR
at each temperature further averaged over four inequi- valent starting configurations. Figure 4 shows exam- ples of the correlations at 0.93 TN and 1.17 TN. The heavy lines show the self correlation R = (OOO), whose transforms over time are important in determining the form of S(q, o ) at large q . The curves at both temperatures resemble each other, with the 1 .I7 TN curve showing the more strongly damped oscilliatory motion. The origin of these oscillations and their effect on the S(q, o ) spectra may be studied by consi- dering the correlations along and perpendicular to the sublattice magnetization direction. In a cubic anti- ferromagnet this direction is not strictly a constant
of the motion, however if a z axis is defined for the initial spin array we may evaluate the parallel and perpendicular spin correlations < SE(0) S i ( t ) > and
3 < St(0) S i ( t ) + Sg(0) Si(t) >. The faint lines of figures 1 and 2 show the contributions from the two components at 0.93 T N . It is seen that the perpendicular part gives rise to the outer spin wave peaks, while the parallel part gives rise to an appreciable central peak. In the example shown this gives rise to a triple- peak lineshape, but at many values of q the width of the central peak compared with the separation between the spin wave peaks is such that only a double-peak lineshape is seen.
References
[I] WINDSOR (C. G.) and STEVENSON (R. W. H.), Proc. [4] NATHANS (R.), MENZINGER (F.) and PICKART (S. J.), Phys. Soc., 1966, 87, 501. J. Appl. Phys. 1968, 39, 1237.
t2] WrmsOR (C. G.)* BRIGGs (G. A . ) a n d K ~ l G I A N ( M ~ ) ~ [5] WINDSOR (C. G.), PYOC. Phys. Soc., 1967, 91, 353.
J. Phys. C., 1968, 1, 940.
[3] LAU (H. Y.), CORLISS (L. M.), DELAPALME (A.), HAS- [61 WATSON (R. E.), BLUME (M.) and VINEYARD (G. H.),
TINGS (J. M.), NATHANS (R.) and TUCC~ARONE Phys. Rev., 1969, 181, 811.
(A,), Phys. Rev. Letters, 1969, 23, 1225.
