Spin Fluctuations in Paramagnetic Nickel

The temperature dependence of the cross section for inelastic magnetic scattering of neutrons in constant-q scan has been studied using results of renormalization group theory. The temperature behaviour of the peak in the-cross section and its width are in good agreement with the results of computation of the cross section for paramagnetic Ni using acurae energy bands. The spin fluctuations above Curie temperature in 3d ferromagnetic metals are still unresolved problem. Experimentally those fluctuations have been widely studied by means of inelastic neutron scattering (usually Fe and Ni are used in such experiments) [I, 21. The cross section for scattering of neutron in the constant-energy scan has well defined peak in paramagnetic phase, which dispersion law is similar to dispersion of spin waves below Tc. It has been interpretated as the propagating spin excitations above Tc. The persistance of spin waves into paramagetic phase in ferromagnetic metals predicts Moriya's theory [3] and they naturally appear in paramagnetic phase in the local band theories [4]. However, relatively large half-width of this peak, the absence of any peak in cross section in a constant-q scan (at least for small momentum transfer q,q 5 0.3 A-' for Ni) and possibility to derive it using dynamic scaling arguments [5] sugest that this peak should be ascribed to the conservation law of magnetic moment rather than to spin wave like excitations. The experimental results in constant-q scan for large momentum transfer q(q >_ 0.3 A-' for Ni) are controversial. There are mesuraments which indicate existence of the peak in cross section above T, for relatively large q vector (q 2 0.3 A-' for Ni at T = 650 K) [2] but in other measurements [I] no peak has been found. In view of those controversies the computer calculation of cross section in constant-q scan has been espetially desired. It gives possibilities to study system in controversial range of parameters. In our previous papers 161 we reported results of computation of neutron scattering cross section, in constant-q scan for Ni in paramagnetic phase using accurate energy bands. The calculations were based on computation of the dynamical susceptibility x (q, w) which is straightforward related to the scatering function S (q, w) that is proportional to the neutron cross section. = 2x (q) .w (1 e ~ ' ~ ~ ) ' F (q, a), (1) where F (q, w) is the spectral weight function. The dynamical susceptibility x (q, w) of ferromagnetic metals can be written as Here Xo (q, w) is susceptibility of free electrons and Ief (q, w) describes correlations between spin fluctuations. In RPA Ief (q, w) --+ I where I is Coulomb integral. To evaluate the temperature dependence of Ief the dynamical susceptibility X (q, w ) was calculated in the approximation in which the vertex part of X (q, w) , I? (q, w) was approximated by two full dressed spintriplet scattering amplitude. That approximation is justified for temperatures not too far from T,. It has been found that Ief T~~~ M and consequently the Curie-Weiss law is satysfied by x (0,O). With that temperature dependence of Ief the scattering function S (q, w) was computed for several q and wide range of temperatures. At T = 1.03 Tc (for Ni Tc = 631 K), for large momentum transfer q >, 0.31 A-', there is peak in the cross section for non zero energy transfer w. With increasing q this peak shifts to higher energy, it becomes better defined and over a substantial range of q its width is proportional to q2.'. Its position and width is well described by critical dynamics but its detail structure depends on the energy bands structure of Ni. As temperature increases from 1.03 Tc to 1.5 Tc the peak shifts to higher energy and its width increases. However, temperature dependence of the width is stronger than temperature dependence of the peak position. The same behaviour of the positioln of the cross section maximum and its width is observed in constant-energy scan. This peak can not be interpretated in terms of propagating spin excitations due to its relatively large width compare to its position. Morover, that peak is not accompanied by any peak in Im x (q,w). In this note we explaine the temperature dependence of the cross section for magnetic scattering of neutrons in constant-q scan applying the results of renormalizaArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988813 C8 54 JOURNAL DE PHYSIQUE tion group theory. According to the dynamic scaling hypothesis the spectral function F (q, w) has the form 4 (2, qc) for sufficient small arguments, where charwc acteristic frequency scales as wc (q, 5) = A ~ ~ ' ~ R (qt) . The correlation length E = to (T / Tc I)-" (for Ni t o = 0,62 A, u = 0,7) is decreasing function of temperature. In the region where the dynamic scaling works w << kT the detail balance factor -!!(1 e-"/") e kT 1 and consequenty we may expect the same temperature dependence of the peak position and the halfwidth because the peak position and the half-width scale in the same manner. Morover, in the region of qE which is considered the scaling function R (x) increases as a result of the deep minimum in R (x) [7]. Therefore the peak position and the width should decrease width temperature. Those discrepancies with results of out computations can be removed in the frame of renormalization group theory. In an experiment and in our computation w / IcT veries in the range 0.2-0.9 and thus in that case the detail balance factor is important [9, 101. We have assumed that in that range of w / kT the spectral function fYlfils scaling law F (q, w) 4 (0, q<) but we wc remaind in scattering function (1) the exact form of detail balance factor. The shape function 4 (s, x) for arbitrary q, w and T Tc, calculated in the first order in E = 6 d has the form [ll] $(s,x) = 2 Rex x (-is + f (x) n (x, isf (x) (1 + x ~ ) ) ) , (2) with n (x, iw) = [(I + F) '/* 0.46 and f (x) = [l 0.51 arctan [0.46 (1 + xe2) x and the scaling function (x) is well approximated by The functions (2) and (3) satisfy asymptotic scaling law and they are in good egreement with experiment [6, 121, despite the fact that they were calculaed far from real dimension. We carried out calculation of cross section (1) in constant-q scan with the shape function (2) in temperatures range of 1.03 Tc 1.5 Tc and in q range of 0.31 0.57 The temperature dependence of the peak position and the half-width at g = 0,46 A-' is shown in figure 1. Fig. 1. The reduced peak position w (T) / wo (wo = w (T = 650 K)) (Lines B) and its half-width Aw / Awo (Awo = Aw (T = 650 K)) (lines A) in constant-q scan for Ni. The 'dashed lines are results of computations using the actual energy bands, the solid curves show the renormalization group theory results. The shape function 4 (s, x) does not have peak for finite s for any (x and E ) , however the cross section (1) does. It originates from competition of detail balance factor and the shape function. Assymmetry in cross section for w -+ -w, whichis introduced by detail balance factor, is less than 10 % in the range of s and z which are considered. It is remarcable the weak temperature dependence of the peak position, relatively stronger temperature dependence of the half-with and they monotonically increase with temperature. Those effects are due t o temperature dependence not only the with but the shape function also. 4 (s, x) changes from non-Lorientzian in critical region i.e. x 4 ca to a Lorentzian in the hydrodynamic region x 4 0. For the peak position the change with temperature in width is compensated by the change of shape. The similar temperature behaviour of peak position and its width is observed in experiment in constant-energy scan [9]. [I] Boni, P. and Shirane, G. , J. Appl. Phys. 57 (1985) 3006. [2] Mook, H. A. and Lynn, J. W., Phys. Rev. Lett. 57 (1986) 150. [3] Moriya, T., J. Phys. Soc. Jpn 40 (1976) 933. [4] Korenman, V., Murray, J. L. and Prange, R., Phys. Rev. B 16 (1977) pp. 4032, 4048, 4058. [5] Folk, R. and Iro, H., Phys. Rev. B 34 (1986) 6571. [6] Rusek, P. and Callaway, J., Phys. Rev. B 36 (1987) 4070; J. Appl. Phys. 64 (1988) 1024. [7] Ramakrishnan, T. V., Phys. Rev. B 10 (1974) 4014. [8] Resibois, R. and Piette, C., Phys. Rev. Lett. 24 (1970) 514. [9] Lynn, J. W., Phys. Rev. Lett. 52 (1984) 775. [lo] Callaway, J., Phys. Lett. A 104 (1984) 487; A 112 (1985) 337. [ll] Iro, H., 2. Phys. B 86 (1987) 485. [12] Shirane, G., Boni, P. and Martinez, J. L., Phys. Rev. B 36 (1987) 881.