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Submitted on 1 Jan 1988
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DYNAMICAL CROSSOVER IN THE NEUTRON
SCATTERING OF ISOTROPIC FERROMAGNETS
C. Aberger, R. Folk
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, Supplement au no 12, Tome 49, dkembre 1988
DYNAMICAL CROSSOVER IN THE NEUTRON SCATTERING OF ISOTROPIC
FERROMAGNETS
C. Aberger and R. Folk
Institut fiir Theoretische Physik, UniversitcTt Linz, A-4040 Linz, Awtria
Abstract. - We study the shape crossover of the spin spin correlation function within mode coupling theory. At Tc the dipolar interaction is taken into account and the crossover in the transversal and longitudinal correlation function is studied. We also mention briefly the case T
2
Tc for pure exchange interaction.Although in general the dipolar interaction is be- the transversal correlation length
<
and the overall lieved to be unimportant with respect to the exchange time scale A we haveinteraction in ferromagnets like Fe, Ni, EuO or EuS, d
-f
"
(2, y, a ) = because of its smallness, it becomes the dominant in- duteraction in the critical region determining the critical =
-
6'
du'k" (x, Y, u-
u')f a (z, y, u')
.
(2)properties [I]. In statics the dipolar interaction leads to a crossover from isotropic to anisotropic spin fluc- tuations. The longitudinal spin fluctuations (parallel to the wavevector q) are suppressed in the limit q + 0 and T
+
Tc; this has been observed recently in EuOand EuS [2]. At T, the static crossover takes place
at the value of wavevector q = qd = $I2 given by the ratio of the strength of dipolar interaction to ex- change interaction. At the same value also a crossover in dynamics from model J ( z = 2.5) to model A ( z = 2)
(for the notation see [3]) was expected. However in tfie transversal w-width of conventional neutron scattering no crossover was observed [4, 51. On the other hand at larger values of wave vector a non Lorentzian (non ex- ponential) shape of the transversal correlation function has been observed [4] in agreement with renormaliza- tion group theory [6-91, whereas at smaller wave vec- tors an Lorentzian (exponential) has been measured in the spin echo experiments [5]. A part of the puz- zle was solved within mode coupling theory (with the approximation of a Lorentzian for the shape) in
[lo],
where it was shown that the dynamical crossover in the transversal w-width sets in at wavevectors an or- der of magnitude smaller then in statics. We consider here the shape crossover by solving the mode coupling equations [lo] without the Markovian approximation. It turns out that the crossover in the shape of the transversal correlation function sets in at a wavevec- tor value different (larger) from the w-width and in agreement with the experiments [ll].We consider here the Hamiltonian containing ex- change and dipolar interaction only
&j,"
Pseudodipolar terms and terms of cubic symmetric are neglected, The mode coupling equations for the longitudinal and transversal spin relaxation function
F a
(q,E ,
g, t ) (a = L, T) have been derived in [lo] (for the generalisation including pseudodipolar interaction see [12]). WithThe kernel k" (x, y,u) reads (for the vertices v&
and the susceptibilities X" and detailed notation see [lo, 111).
x
f
( P ~ . ~ u , ZIP-,YIP-)
(i) The transversal or longitudinal scattering cross section at T, is given by the relaxation function in theform
S" (9, g, w) = const. (q2
+
g & ~ ) - lF a
(q, g, w).
(4)Thus the shape and the halfwidth of the scatter- ing intensity are directly given by the numerical so- lution of equation (2). At small scaling arguments y = g1/2/q the results of the pure exchange interac- tion case 113-151 is reproduced. Both the transversal and the longitudinal scattering intensities are given by the same non Lorentzian shape with an energy line with
:
?
I
q2'5. Increasing y the transversal correla- tion function shows a crossover to a Lorentzian shape a t y--
1. The transversal energy width crosses over to dipolar behaviour,:
?
I
-
q2, at scaling arguments roughly one order smaller. This has already been found in the Markovian approximation[lo].
The shape is most directly measured in the spin echo experiments [5] and the comparison for EuO with our numerical r e sult showed very good agreement [Ill.The longitudinal scattering intensity at Tc on the other hand does not show such a crossover t o a Lorentzian shape but remains a non Lorentzian [ll]. The longitudinal energy line width however crosses over to a constant roughly a t qd.
The position of the maximum of the scattering in- tensity at constant energy transfer is a sensitive quan-
C8 - 1568 JOURNAL DE PHYSIQUE
tity with respect to the shape of the correlation func- tion [4, 9, 161. In the case of pure exchange interac- tion, because of the conservation property of the spin, the scattering intensity at q = 0 is always zero and there is always a well defined peak at nonzero waveveo tor. The dipolar interaction destroys the conservation law for the spin. Therefore contrary to the pure ex- change interaction case the scattering intensity of both the transversal and longitudinal scattering is finite at
q = 0.
sT
(0, W) increass like w-2 for small energies, be- cause of the crossover t o the Lorentzian shape a t small q and the q2 behaviour of the line width.sL
(0, W) (see Fig. 1) however is finite for w = 0, because the line width is finite for q = 0. For larger values of wavevec- tor we recover the results of pure exchange model with a decrease of the intensity like q-4'5 in both scatteringintensities. As a result we find that for not too small energies the peak position in the transversal scattering intensity is only weakly affected by the dipolar inter- action. A more pronounced effect is seen in the peak
position of the longitudinal scattering intensity (see Fig. 2). Deviations from the isotropic result already
Fig. 1. - Typical constant energy scans showing the longi- tudinal scattering intensity as function of wavevector q for different energies. The value of A is that for EuO.
Fig. 2. - Reduced peak position qo ( w / ~ y ( 0 ) ) - ~ . ~ =
Q (w) w-0.4 as function of the scaled energy w = w / ( ~ y (0) g1.25) ( y (0) - 7.5) for the transversal (dashed curve) and longitudinal (solid curve) intensity.
appear at wavevectors or order g1I2 that means ener- gies of the order of g5I4.
(ii) Let us turn now to the pure exchange interaction case [13], where we study the shape cirossover from the critical to the hydrodynamic region This crossover modifies the energy width compared to the Markovian case studied by Resibois and Piette [17]. Our result is in reasonable agreement with recent RNG calcula, tions [8]. Other quantities of interest are the scaling functions for the peak position and width of the con- stant energy scans and the energy width of the con- stant wavevector scans. The differences between the mode coupling result and the recent result of RNG theory [8] lies within the scatter of .the experimental data [13].
Note added i n proof:
The shape crossover at T, was also considered in a recent paper by Frey, Schwabl and Thoma (Phys. Lett. A 129 (1988) 343).
Acknowledgment
Work supported by the Fonds zur Forderung der wissenschaftlichen Forschung.
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[ll] Aberger, C. and Folk, R., Phys. Rev. B 38 (1988) 7207.
[12] Aberger, C. and Folk, R., ICNS Grenoble (1988)
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