CRITICAL DYNAMICS AND DIPOLAR INTERACTION IN EuO

Recent neutron scattering studies of the critical behaviour of isotropic ferromagnets revealed the apparent contradiction between a good agreement with dynamic scaling at T = T, and drastic deviations from scaling at T > Tc. In EuO the experimental findings are consistent with latest theoretical results taking into account dipolar interactions. The ferromagnetic-paramagnetic phase transition at the Curie point of simple isotropic ferromagnets probably is the most common second order phase transition and a lot of effort has been devoted to its understanding. The study of static and dynamic scaling properties is crucial in this respect. However, it became clear by now that just in this particular case the real critical behaviour can only be observed in a very restricted sense. This is due to what we can call the "dipolar paradox" . Namely, the well known demagnetization effects of magnetostatics (in other words dipolar interactions) inevitably become very substantial as the intrinsic susceptibility X : diverges at the critical wavevector q = 0 on T -+ T,. This simply means that the magnetic fields produced by the critical fluctuations tend to reduce these fluctuations and lead to an apparent susceptibility: where the X: = C/ (K: + q2) Omstein-Zernicke form has been assumed with C being a constant and nl = 5-I the inverse correlation length. m is the demagnetization factor, and the so called "dipolar wavevector" q d = m characterizes the relative strength of the dipolar effects. Equation (1) implies a fundamental breakdown of the scaling hypothesis, since represents a second characteristic length beyond <. Thus earlier efforts to establish scaling behaviour at the ferromagnetic Curie point were rather too optimistic. Experimentally q, can be calculated from the measured macroscopic susceptibility and n1. For EuO, which is considered to be the best model Heisenberg ferromagnet, qd=0.15 A-' (see [l] and Refs. therein) is paradoxically so substantial that there is virtually no room left for the true exchange critical regime. (For comparison in Fe q,=0.045 A-'.) Until recently the dipolar anomaly expressed by equation (1) escaped observation, since in usual critical neutron scattering experiments around the forward scattering direction the demagnetization factor m is identically zero. Namely, for the transverse fluctuations to which neutrons couple, the magnetization direction M l q stays in the quasi-infinite planes of the wave with wavenumber q. In contrast, around Bragg peaks in single crystals neutrons can also couple to the longitudinal fluctuations M 11 q with m = l, and equation (1) has been actually verified in a recent neutron polarization analysis experiment [2]. With finite and constant m, instead of the usual scaling form X : = q2g(nl/q), we can write equation (1) as a "two parameter scaling'' law X, = q2gL (nl/q, nl/qd). The same way, instead of normal dynamic scaling, the behaviour of the relaxation rate I' can be expected to be given as For a given sample this "two parameter scaling" implies nothing experimentally relevant, since there are only two parameters (g and T) anyway. Relevant is if the function f; can be predicted theoretically, as it is the case in the trivial example of equation (1). Experimentally the dynamics of the transverse fiuctuations could only be studied by now, lacking good enough single crystals. Two, rather contradictory features emerged for both Fe and EuO [l, 3-51: (a) at T = Tc there is no appreciable deviation from the exchange scaling prediction r (q ) X q5/' in the accessible and critically relevant q range of qd/8 Tc there are huge deviations from the scaling predictions in the range qd/8 Tc data in EuO, cf. figure l. The new theoretical results also resolved another controversy. At T = T, triple-axis spectrometer scans Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19888703 C8 1538 JOURNAL DE PHYSIQUE Fig. 1. Scaling plot of the temperature dependent relaxation rates of the transverse (m = 0) fluctuations in EuO [6]. rc is the relaxation rate at Tc=69.3 K. Exchange scaling theory would require that all data points fall on the lowest line (RP) of reference [7]. The other lines are results of the latest dipolar theory [8]. The data presented were taken in the q range 0.02 0.15 A-', and /cl 0.64 ( T / T ~ ~ ) ~ . ~ ~ A-'. at q 2 0.15 A-' [5] showed non-Lorentzian inelastic lineshapes in agreement with predictions of critical theories [g], while Neutron Spin Echo (NSE) scans at much smaller g's produced Lorentzian lineshapes 141. This is illustrated in figures 2 and 3 by the most precise data obtained by now [6]. (The q = 0.3 W-' data in figure 3 also show that this wavenumber is already Fig. 2. Small wavenumber lineshapes at T = Tc as directly measured in the time domain by Neutron Spin Echo scans [6]. The lines show the exp (-r (g) t) (Lorentzian) lineshape averaged over the rather broad q resolution of the spectrometer, about 0.006 A-' HWHM. In the inset representative error bars are only indicated once on each curve. Fig. 3. High resolution triple-axis r~eutron spectrometer scans in the w domain [6]. The dashed and continuous lines show the lineshape predicted by critical theories (e.g. [g]) and the Lorentzian one, respectively, both convoluted with the instrumental w resolution function (with 0.015 meV HWHM the corrections are hardly detectable). too large to show undistorted critical behaviour). This lineshape crossover at Tc has now been shown to be also due to the dipolar ineractions [l.O]. In sum, due to the strong dipolar (i.e. demagnetization) effects the critical behaviour in isotropic ferromagnets violates both static and dynimic scaling. Latest theoretical results give gratifyingiy proper account of the observed dynamics in the canonical Heisenberg model system EuO. Note that in Fe the agreement is less good [ll], and other interaction.^, e.g. spin-orbit pseudodipolar effects might also be effective [12]. [l] Mezei, F., J. Magn. Magn. Maier. 45 (1984) 67. [2] Kotzler, J., Mezei, F., Gorlitz, I). and Farago, B., Europhys. Lett. 1 (1986) 675. [3] Mezei, F., Phys. Rev. Lett. 49 (1982) 1096. [4] Mezei, F., Physica B 136 (1986) 417. [5] Boni, P. and Shirane, G., Phys. Rev. B 33 (1986) 3012. [6] Mezei, F., Farago, B., Hayden, S. M. and Stirling, W. G., Physica B, in press. [7] Resibois, P. and Piette, C., Phgs. Rev. Lett. 24 (1970). [8] Frey, E. and Schwabl, F., Phys. Lett. A 123 (1987) 49. [g] e.g. Folk, R. and Iro, H., Phys. Rev. B 32 (1985) 508. [l01 Frey, E., Schwabl, F. and Thoma, S., Phys. Lett. A 129 (1988) 343. [l11 Frey, E.. and-~chwabl, F., 2. Phrys. B 71 (1988) 355. [l21 Aberger, C. and Folk, R., Physitxz B, in press.