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Collective excitations in paramagnets: Confrontation of theoretical and experimental results for gadolinium
Journal of Physics: Condensed Matter
Abstract
Recent high-resolution data for the spin response function S(k, w) of gadolinium at T>or=1.5 Tc, obtained by neutron scattering, is compared with results derived from the coupled-mode theory of a Heisenberg exchange interaction on an HCP lattice. A well-defined collective mode is obtained in a calculation which uses the long-range interaction derived from an analysis of the low temperature spin-wave dispersions. However, the peak position is at a much lower energy than the corresponding structure in the data, even though the estimated Tc agrees very well with the experimental value. Good overall agreement between theoretical and experimental results is found by using a nearest neighbour exchange deduced from Tc and the measured second-frequency moment.
A high-resolution inelastic neutron scattering study of the liquid and plastic crystal phases of oxygen is presented. The separation of the magnetic response from the total cross sections is achieved by means of a computer molecular dynamics simulation which enables an approximate estimation of the magnetic spectral weight functions. The magnetic dynamics of the orientationally disordered plastic crystal is shown to exhibit a characteristic behaviour corresponding to an exchange-coupled paramagnet. The magnetic response of the liquid phase retains the most salient characteristics of the rotationally disordered solid phase.
A coupled-mode theory for the spin response function is used to calculate time-dependent properties of the classical spin chain at infinite temperature. The behaviour of the spin autocorrelation function is consistent with computer simulation data, in that the exponent of the time tail decreases on moving from intermediate to asymptotic time regions. Results for the spin response function and memory function in the two time regions are obtained by a combination of asymptotic analysis and numerical studies. For a large wavevector, near the zone boundary, the spin response function displays a very pronounced oscillatory time dependence. However, there is next to no structure in the corresponding spectral function, in accord with simulation data. The asymptotic analysis shows that, in the infinite-temperature limit, coupled-mode theory does not comply with usual hydrodynamic assumptions, a view that has also emerged from analysis of some of the computer simulation data.
A short perspective of dynamic critical scattering is given, together with an orientation to modern theories of static and dynamic critical phenomena. The materials discussed are magnetic salts that are believed to spontaneously order through correlations induced by a Heisenberg exchange interaction between atomic spin moments.
Results are provided for the spin-spin response function of a three-dimensional, antiferromagnetically coupled Heisenberg magnet covering a range of temperatures from the critical temperature to deep in the paramagnetic phase. The wave vectors considered span the Brillouin zone. Damping rates are given at the zone centre and the antiferromagnetic-ordering wave vector, w, together with copious numerical results for the full response function. The calculations are based on the non-linear, integral-differential equations obtained from so-called coupled-mode theory. In a confrontation between experimental and theoretical findings for RbMnF3 nearly all aspects have a positive outcome. The main exception is found at T, for wave vectors close to w. Here, the measured response comprises three distinct components, reasonably ascribed to diffusive and oscillatory collective processes. In the corresponding predictions, the diffusive component is conspicuously missing. A less pronounced discrepancy if found at the antiferromagnetic zone boundary where, once again, there is more structure in the observed spectrum than in the calculated one.
At normal emission, the bulklike band in gadolinium overlayers on W(110) shifts toward {ital E}{sub {ital F}} from 1.64 eV at 310 K to 1.43 eV at 230 K in photoemission. This is accompanied by an increase of the feature width from 0.8 eV at 320 K to 1.1 eV at 230 K. The surface-state binding energy remains unchanged, though the width increases significantly with increasing temperature. The different effects of temperature upon the photoemission results obtained for the bulklike and surface states are interpreted as magnetic-order-induced band movement and spin mixing, respectively, and are discussed in the light of a quasiparticle theory. The band dispersion along {bar {Sigma}} is also temperature dependent. We postulate that the exchange splitting of the conduction band is not uniform at different positions in {bold k} space.
The transverse-spin-wave dispersion relation and neutron-scattering cross section for strongly anisotropic magnets is formulated in terms of effective renormalized exchange and anisotropy parameters. The spin-wave dispersion for the cone structure is considered in detail, including a diagonalization of off-diagonal wave-vector-dependent terms and renormalization effects caused by the crystal field. The derived expressions require no a priori knowledge about the crystal-field and magnetoelastic parameters. This is required for the application of the standard-basis-operator theory. A comprehensive analysis of the spin-wave data on Gd, Tb, Dy, and Er yields the interatomic exchange parameters and the effective anisotropy constants. The exchange parameters obey the de Gennes scaling and decrease as the cubed inverse distance, as expected for the isotropic Rudermann-Kittel interaction. Therefore, there is not much room for an unresolved two-ion anisotropy. The cone spin-wave data for Er can be accounted for on the basis of an isotropic exchange interaction and crystal-field parameters with magnitudes in quantitative agreement with those derived from static measurements on dilute Erc-Y1-c alloys. No convincing evidence for a giant non-symmetry-breaking two-ion anisotropy has been found in Tb, Dy, and Er. A qualitative detection of a small symmetry-breaking two-ion anisotropy has been made in Tb and possibly also Dy and Er. The calculated and measured neutron-scattering intensities for Er are in good agreement, and the renormalization parameters calculated from known crystal-field parameters compare closely with those obtained from the fit to spin waves.
The inadequacy of the standard coupled-mode theory of one dimensional paramagnets is resolved by renormalisation of the exchange coupling and time scale, in such a way that the refined theory is consistent with some static spin cumulants. With the proposed refinements the coupled-mode theory gives the exact spin dynamics in the limit of vanishing temperatures. For low temperatures the theory is studied numerically to achieve comparisons with computer simulation data that vindicate the renormalisation scheme.
Spectra of paramagnetic spin fluctuations in insulating and metallic ferromagnets, obtained by inelastic neutron scattering, are interpreted in terms of appropriate Heisenberg models. A mode coupling approximation is used which takes full account of the lattice structure and the extended range of the exchange interaction, the form of which is obtained from the interpretation of spin wave spectra measured in the ordered state. The favourable outcome of confronting theoretical spectra with experimental data for EuO, Pd2MnSn and Fe leads the authors to conclude that the mode coupling theory provides a more than adequate description of paramagnetic spin fluctuations. The dependence of the spectra on an increasing neutron wave vector is a change from a Lorentzian shape to a distinctly squared-up shape, but there is no strong evidence in any of the three systems of a damped collective mode contribution even for zone boundary wavevectors.
Oxford University Press Vols 1, 2 (1988)
The magnetic excitation spectra of Gd were measured by neutron inelastic scattering over the entire Brillouin zone in the <110> direction at temperatures from 250 to 850 K (/ital T//sub /ital c//=293 K). The data were fitted to a damped-harmonic-oscillator form for the spectral weight function and were placed on an absolute cross-section basis. Wave-vector-dependent susceptibilities were obtained by integration over energy. They clearly show the presence of static spin correlations even at /ital T/=850 K. At /ital T/greater than or equal to/ital T//sub /ital c// we observe a crossover from spin-diffusive motion at small /ital q/ to spin-wave behavior at large /ital q/. The wave vector /ital q//sub /ital c// at which this crossover occurs is determined by the ratio of the damping parameter to the second moment, <..omega../sup 2/>, of the frequency distribution. Surprisingly, <..omega../sup 2/> remains independent of temperature above /ital T//sub /ital c//, and the temperature dependence of /ital q//sub /ital c// is therefore determined by a gradual change in the damping. This results in a very weak /ital q//sub /ital c//-versus-/ital T/ dependence, which is not consistent with a magnetic short-range-order interpretation for the existence of spin waves above /ital T//sub /ital c//.
The frequency- and wave-vector-dependent spin response function is calculated for a Heisenberg ferromagnet using the mode-coupling theory. Results are presented for the critical and paramagnetic regions. The work is motivated largely by recent, high-resolution neutron spectroscopy studies of EuO and EuS. In view of this, the Heisenberg model used incorporates nearest- and next-nearest-exchange interactions between spins on a face-centered-cubic lattice. The coupled-mode theory is shown to be successful in describing the measurements. Additional features are predicted as possible subjects of future experimental work. We present a review of various derivations of mode-coupling equations and demonstrate the equivalence of some prescriptions. The question of incorporating an external magnetic field in the theory is addressed, and we conclude that readily attainable fields have a minimal effect on the spin dynamics.
An ultrafast, vectorized spin-dynamics method is used to study the
time-dependent properties of classical XY and Heisenberg spin chains at
infinite temperature. The decay of the energy-energy and spin-spin
correlation functions is oscillatory for short times and at long times
is consistent with classical diffusion, although the approach to the
asymptotic behavior is extremely slow. We have also calculated
S(q,ω) and find clear indication of spin-wave peaks in both
models.
References Cable J W 1991 private communication Cable J W andNicklow R M
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