l%ysica 13613 (1986) 417-419 North-Holland, Amsterdam

CRITICAL DYNAMICS IN EuO AT THE FERROMAGNETIC CURIE POINT

F. MEZEI Hahn-Meitner Institut and Technische Universitdt Berlin, Postfach 390128, D-IO00 Berlin 39, Germany

The relaxation rate of critical fluctuations at the ferromagnetic Curie point in EuO has been studied as a function of the wavenumber q using Neutron Spin Echo (NSE) spectroscopy. Taking advantage of the high energy resolution capability of NSE, the critically relevant q ~ 0 limit has been established for the first time. The results shed new light on some aspects of the critical dynamics and happen to bein substantialcontrast to expectations: (a) At T = T~theF = Aq~/21awwasfoundtobevalidwithanA value in reasonable qualitative agreement with mode coupling theory. (b) Dipolar effects are absent in the critical dynamics as probed by q = 0 neutron scattering. (c) The temperature dependence at T > T c is consistent with dynamical scaling, but the scaling function is distinctly different from the one found in Fe. (d) The inelastic lineshape is consistent with a Lorentzian, and inconsistent with the Hubbard-Wegner shape.

EuO is generally considered as the best model system for isotropic Heisenberg ferromagnets. Its critical behaviour at the Curie point has been extensively studied previously by both elastic and inelastic neutron scattering [1]. With the advent of high resolution inelastic scattering methods it has now become possible to extend these studies to critical fluctuations of substantially longer wavelengths. This is of paramount importance in particular in the following respects: (a) theories of critical phenomena are relevant to the long wavelength limit ( q ~ 0); (b) the dipolar interac- tion is expected to manifest itself close to Tc at small q's; (c) subtle details of the inelastic lineshape can only be looked at in the absence of substantial resolution corrections. The present paper is a complete account of the Paramagnetic Neutron Spin Echo (PNSE) results having been obtained in two experiments done 'at Institut Laue-Langevin within the past 3 years with the same isotopic 153EUO powder sample as used in ref. 1. A partial account of the T = T~ results has been published earlier [2]. A discussion of the significance of the results will be published elsewhere.

The experiments have been performed on INl l NSE spectrometer [3] using A = 6.6 ,~ incoming neutron beam with 18% FWHM velocity spread and 97% degree of polarization. Scattering angles between 0.8 ° and 4 ° have been investigated. The beam was collimated by using 28 x 28 mm dia-

phragms as both the exit window of the polarizer and the entrance window of the analyser, both at 3.l m distance from the 11.5 x 24mm size sam- ple. In the data reduction process the exact beam geometry has been taken into account by a num- erical simulation method. The instrumental ener- gy broadening was 0.05/zeV, i.e. much smaller than any of the linewidths measured.

T~ = 69.3 K has been identified as the tempera- ture of the peak of the critical scattering intensity observed at two scattering angles (q = 0.013 and 0.024 A-l ) . The rounding of the peak, about 0.07 and 0.14 K, respectively, was about the amount expected for the susceptibility at finite q's. Thus the relative accuracy of the T c determination (including temperature inhomogeneities) was estimated to be of _+0.02 K. At T c the depolariz- ation of the neutron beam traversing the sample was 8.5%, changing at a rate of 30% per K. Monitoring the beam depolarization lends itself for a precise checking of the temperature stability and reproducibility with an accuracy of ---0.01 K.

The scattering results at T/> T c have been interpreted in terms of the double Lorentzian scattering function

1 F q S(q, o J ) ~ ~ ~ 2 (1) q +Kz F q + w

which has been found, similarly to other authors, e.g. [1,4], as an adequate approximation

0378-4363/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

418 F. Mezei / Critical dynamics in EuO

(to < kT). Thus the Fisher exponent ~ was taken to be zero (instead of the theoretical 0.04). The inelastic lineshape has been checked at T = T~ and q = 0.024 ,~ i with a high statistical accuracy, and it was found to be compatible with the Lorentzian form postulated in eq. (1), as shown in fig. 1. Note that in the time variable domain of an NSE experiment the Lorentzian corresponds to exp(-Fqt) , which form will be slightly modified for t>~ l/Fq due to the finite collimation (i.e. integration over a q range). The continuous line in the figure represents the best Lorentzian fit taking the finite q resolution effects into account, while the dashed and dotted lines give similar fits for the Hubbard-Wegner shape [5, 6] and for the phenomenologically modified Lorentzian with a = 0.05 as introduced by the BNL group in order to explain their q/> 0.2 A data [4], respectively. (Note, that the modified Lorentzian with a = 0.1 practically coincides with the Hubbard-Wegner lineshape.) Remembering that the lineshape gets modified further on approaching the Brillouin zone boundary at q = 1.06 ,~ 1 [4, 7], we have to conclude that the to lineshape is not the same at all q's at T c and that the Lorentzian provides the best approximation in the q--* 0 limit representing

i . E u O t~t, T - T

~k ".x *-1 .,., q : 0 .024 A

"..,

C)-

LO

0 I I I

0 1 t [nsec]

Fig. 1. Measured S(q, t) lineshape at T = T~. The continuous line represents the best Lorentzian fit (Fq = 0.79 ~eV), while the dashed and dotted curves show best fits to the Hubba rd - Wegner [5, 6] and modified Lorentzian (c~ = 0.05) lineshapes [4], respectively.

the " t rue" critical behaviour. Data in fig. 1 allow for the first time a direct distinction to be made between these lineshapes.

Fig. 2 shows the measured Fq values at T = Tc, together with previous results of ref. 1. Contrary to the conclusion in ref. 1 the new results tend to t h e I~q = Aq 5/2 law as q--~0 (dashed line), as predicted by dynamical scaling for the exchange model, and the coefficient A = 8.7 meVA 5/2 is close to the prediction of the Riedel theory (i.e., 7 .1meV/~ 5/2) [8], similarly to Fe and Ni [2]. Recent reinvestigation of the q > 0.12 ,~- ~ range at Brookhaven gave about 2 times bigger Fvalues than those of ref. 1 included in fig. 2 and thus the 8.7q 5/2 law was found also valid at larger q's, too [41.

The most surprising aspect of this result is that below q < qa = 0.145 A 1 the static susceptibility Xq is bigger than 1/47r and therefore strong dipolar effects are to be expected. In particular, a crossover has been predicted from the q5/2 ex-

• 2 • change behavlour to the F_ = Bq dloolar one, q L

1 / 2 2 • where B = Aqd = 3.3 meV ~ (dotted hne) [91. This dipolar cross-over is thus definitively ruled out by this experiment [10].

1000

L00

100

40

L~, L.0

o o

o

o o

o

i / o 8 7 q 2 5 / o

/ /

J

, , / • /

. " i ~

, / / • . ° " EuO / s

10 "" T=Tc .."" #"" o D telrich et. ul.

"" {, present work i I 0.4 ~" ,~/

k ,)/ /

001 004 01 q [,~-1] 0.4

Fig. 2. Relaxation rate vs. wavenumber at T = T c. The dashed line is a fit to the new results only, and appears to be the low q limit. The previous data (open) circles are believed to be less accurate [4]. The dotted line gives a theoretical prediction taking dipolar effects into account [8].

F. Mezei / Critical dynamics in EuO 419

The temperature dependence of the relaxation rates Fq at T > T~ has also been investigated, in particular in view of the scaling hypothesis.

Fq(T) : AqSJ2f(K1/q), (2)

where the Tdependence on the right-hand side only enters through that of K 1 . Fig. 3 shows experimental values of f obtained at various temperatures between T c and T c + 2 K, which do not contradict the existence of anf(x) function (e.g. the dotted line is one possibility). The data rule out the universality of f(x) scaling function, since the theoretical prediction (continuous line) [11] has been con- firmed with considerable precision in Fe [12] (with certain limitations though [2]). Various obser- vations by the BNL group [1, 13] also indicated the nonuniversality of f(x).

It is of particular interest to compare the Fq values obtained from this study with the tempera- ture dependence of F 0 = Fq_ o, as observed in the high frequency magnetic relaxation work of K6tz- ler et al. [14]. These authors find that F 0 follows the F 0 = BK ~ law predicted by Finger [9] for the dipolar regime with B = 4.3 meV ~2, close to the

theoretical value Aq~/2 given above. This observ- ation is in strong apparent contradiction with the present neutron results. For example, at T= T c + 0.99K (K 1 =0 .34A -1) we obtained at q = 0.017 and 0.024 A 1 respectively, about 10 and 5 times smaller Fq values than the ~ observed in ref. 14. We have to remember, however, that the two methods do not measure the same thing. At small scattering angles neutrons only couple to the components of the magnetization perpendicu- lar (transverse) to the wave vector q. In magnetiz- ation measurements the role of q is taken over by the sample shape, leading to a complicated mix- ing of transverse and longitudinal effects. In any case, the absence of dipolar thermal broadening in neutron results at T > T c is just consistent with the above-discussed absence of the dipolar cross- over at T= To.

Acknowledgement

The author is indebted to the BNL group for the loan of their 153Eu0 sample and for valuable discussions.

I I

I I

X v

C) L . . I I i I

0 I 2 × : x l / q

Fig. 3. Scaling plot of the temperature dependence of the relaxation rates Fq. For more clarity, the open circles from the left to the right give the change of Fq at q = 0.024 A-1 for temperatures T c + 0.07, 0.021, 0.49 and 0.99 K, respectively. The continuous line stands for the Resibois-Piette prediction [11], while the dotted line is a tentative guide to the eye only.

References

[1] L. Passell, O.W. Dietrich and J. Als-Nielsen, Phys. Rev. B l l (1976) 4897, 4908 and 4923.

[2] F. Mezei, J. Magn. Mat. 45 (1984) 67. [3] See, Neutron Spin Echo, F. Mezei, ed. (Springer,

Heidelberg, 1980). [4] P. B6ni and G. Shirane, J. Appl. Phys. 57 (1985) 3012,

and references therein. [5] F. Wegner, Z. ffir Physik 216 (1968) 433. [6] J. Hubbard, J. Phys. C:,Solid St. Phys. 4 (1971) 53. [7] H.A. Mook, Phys. Rev. Len. 46 (1981) 508. [8] E.K. Riedel, J. Appl. Phys. 42 (1971) 1383. [9] W. Finger, Phys. Lett. 60A (1977) 165.

[10] This is even more so for the Fq ~ q dipolar prediction by S.V. Maleev, Sov. Phys. JETP 39 (1974) 889.

[11] E Resibois and C. Piette, Phys. Rev. Lett. 24 (1970) 5•4. [12] F. Mezei, Phys. Rev. Lett. 49 (1982) 1096. [13] G. Shirane, private communication. [14] J. K6tzler, W. Scheithe, R. Blickhan and E. Kaldis, Solid

State Commun. 26 (1978) 641.