MODE-COUPLING APPROACH TO THE SPIN DYNAMICS OF EUROPIUM COMPOUNDS

(1)

HAL Id: jpa-00228959

https://hal.archives-ouvertes.fr/jpa-00228959

Submitted on 1 Jan 1988

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

MODE-COUPLING APPROACH TO THE SPIN

DYNAMICS OF EUROPIUM COMPOUNDS

A. Cuccoli, S. Lovesey, V. Tognetti

To cite this version:

(2)

JOURNAL DE PHYSIQUE

Colloque C8, Suppl6ment au no 12, Tome 49, dBcembre 1988

MODE-COUPLING APPROACH TO

THE

SPIN DYNAMICS OF EUROPIUM

COMPOUNDS

A. Cuccoli (I), S. W. Lovesey (') and V. Tognetti (I)

(I) Dipartimento di Fisica UniversitaJ, L. F e m i , 2 50125 Firenze, Italy (') Rutherford Appleton Laboratory, Ozfordshire 0 x 1 1 OQX G.B.

Abstract. - The results of a numerical solution of the mode-coupling equations for EuO and EuS taking into account the microscopical properties of the compounds are presented. Good agreement with the available neutron scattering data is found and predictions for further experiments on EuS are given.

The results of high resolution neutron scattering experiments on insulating ferromagnetic Europium compounds EuO and EuS appeared in literature in recent years [I-41. The particular attention devoted to such compounds, is due to the fact that already in the ear- lier neutron scattering comprehensive investigation on E n 0 and EuS [I] it was shown that the magnetic behaviour of these systems is very well described by a simple isotropic Heisenberg exchange interaction. The availability of experimental data at and above the or- dering temperature Tc has encouraged a renewal of

theoretical studies on critical and paramagnetic spin fluctuations in Heisenberg ferromagnets.

The most successful theories appear to be those ones based on Renormalization Group (RG) or on the mode-coupling approximation. RG techniques consti- tute a well established method to study the static and dynamic critical properties of a system; they allow to evaluate critical exponents as well as universal functions which characterize the critical behaviour. Very recently an attempt has been made t o extend RG cal- culations to temperatures above Tc [5], but some

problems persist to make possible a quantitative comparison with experimental data. The mode coupling approximation, in spite of its heuristic character, has revealed very useful in many problems of condensed matter physics [6]. For pure Heisenberg magnets, for example, it gives for the critical dynamical scaling ex-

,-

ponent z the value z = 171, in a agreement with 2

the experiments. Moreover it has been shown very recently [8] that by taking into account the dipolar macroscopic interaction, the mode-coupling theory can be able to interpret the unexpected simple exponen- tial decay obtained in neutron spin-echo experiment on EuO at T = Tc and very small wave vector [4],

which remained until now inexplicable.

However the mode-coupling theory appears also suitable t o investigate the dynamical behaviour of ferromagnetic systems in all the paramagnetic region and throughout the Brillouin Zone. In this paper we present the results of a numerical solution of the mode- coupling equations for EuO and EuS taking into ac-

count the microscopical features of these compounds. This allows to show some distinctive characters which distinguish EuS from EuO that cannot be accounted by previous approximate solutions, which were limited t o simple cubic nearest neighbours interaction [7] or confined to the continuum limit [6].

Europium compounds can be described by a Heisen- berg exchange hamiltonian:

I for an FCC lattice of magnetic ions with S =

-.

The

2 experiments [I] have shown that the interaction is re- stricted to nearest neighbours (J1) and next nearest

neighbours ( J z )

.

For EuO: J1 = 1.22' K , J2 = 0.25' K

and a = 5.12 A; for EuS: J1 = 0.48' K , J2 = -0.24' K and a = 5.95 A, a being the lattice constant.

The final result given by the mode-coupling theory is the following integr~differential equation:

for the relaxation function Fq ( t )

,

whose Fourier Trans- form is related to the measured scattering function S (q, w ) by the relation:

W

(q' W, = 1

-

exp (-v/KBT)

.x

(4) F, ( w )

.

(3)

Not with standing formal differences in the various derivations, behind equation (2) is the following fun- damental approximation: correlation and relaxation functions among more than two spin operators have been decoupled. Equation (2) is only one of the forms in which this equation is obtained, however all such forms are equivalent if we use for the static susceptibility that one given by the spherical model:

(3)

C8

-

1572 JOURNAL DE PHYSIQUE

This susceptibility turns out t o be compatible with the dynamical equation (2).

The temperature dependent parameter X appearing in equation (4) can be evaluated by using the sum rule:

(I/N)

C

T X ~ =

s

(S

+

1) /3. k

The comparison with the experimental data [I], of the calculated X , = 0 and the correlations length KT' which appears in the low q approximation of X,, X , z

A shows good agreement.

":

-

q2

The method followed to obtain the solution of equation (2) has been described in detail elsewhere [9].

An extensive comparison of the results of our calculation with the available experimental and spin dynamics simulation data

[lo]

has been made [9], and a good agreement has been generally found both with constant q and constant energy spectra. For constant energy spectra at T = T, the agreement is not very

satisfactory at low energy transfer, where the experimental data are very well reproduced by RG, which otherwise fails at high energy. At constant wave vectors our line shapes reproduce very well the experimental ones a t various temperatures and wave vectors. A typical result is shown in figure 1.

I

F(w) ( rneVF1 EuO T=1.68Tc q=1.06 A-'

Fig. 1. - Comparison of our numerical results with spin dynamics calculation

[lo]

and neutron scattering data [2] at T = 1.68 T, and q = 1.06

A-'

for EuO. Full line: our numerical results; dashed line: three poles approximation using the parameter fitted in 121; filled quad: neutron data; open triangle: spin dynamics simulation.

The most interesting feature is the strong depen- dence of the line shape on the direction of the wave vectors in EuS. Such behaviour, which is a direct conse- quence of the competing exchange interactions present in EuS, is here obtained for the first time in the frame- work of the mode-coupling theory. In figure 2, a comparison is shown of the line shape a t

T

= T, for q = qm, along the three principal directions for EuO and EuS, qm, being the wave vector corresponding to the zone boundary in the (1, 1, 1) direction. In EuO the line shape is the same along all the directions and only

Fig. 2.

-

Mode-coupling theory results for the line shape at T = T, for EuO (la) and EuS (lb) for three different directions of the wave vector. The value of q corresponds in both cases to the zone boundary along: the (1, 1, 1) direction. Full line: q along (1, 1, 1); dashed line: q along (1, 1, 0); dot-dashed line: q along (1, 0, 0).

small differences, less than 10 %

,

appear in the inten- sity and the line width; on the contray in EuS we see that a t the zone boundary we have a drastic change in the line shape, with the propagation along (1, 1, 1) which conserves a diffusive character whereas that one along (1, 0, 0) approach a damped spin wave like behaviour. Some experimental data [I11 seems to confirm these results, but new experiments are called t o test also the quantitative accuracy of our predictions.

[I] Passell, L., Dietrich, 0. W. and Als-Nielsen, J.,

Phys. Rev. B 11 (1976) pp. 4897,4908 and 4923. [2] Boni, P. and Shirane, G., Phys. R'ev. B 33 (1986)

3012.

[3] Bohn, H. G., Kollmar, A. and Zinn, W., Phys.

Rev. B 11 (1984) 6504.

[4] Mezei, F., Physica 136B (1986) 417. [5] Iro, H., 2. Phys. B 68 (1987) 485.

Lovesey, S. W., Condensed Matt,er Physic: Dy- namic Correlations, Frontiers in Physics, Ben- jamin Cummings, vol. 61 (1986).

Hubbard, J., J. Phys.

C

4 (1971) 53.

Rey, E. and Schwabl, F., Convnunication to MECO (Wroclaw) April 1988.

Cuccoli, A., Lovesey, S. W. and Tognetti, V., Preprint 1988.