The phonon dispersion relations of '6Ar have been measured in the [100], [110], and [111]symmetry directions at 10 K using a triple-axis neutron spectrometer. A comparison of the measured dispersion curves and the best available theoretical calculations based on the Bobetic-Barker (88) potential shows small systematic discrepancies for the [100]L, [110]T2, [110]L, and [111]Lbranches. Otherwise the agreement between experiment and theory is remarkably good. The data are well described by several harmonic Born-von Kkrmin models the most general of which {a three-neighbor general-force model with nine parameters) has been used to deduce values of zero-sound elastic constants. At 10 K (lattice constant a = 5.313 + 0.002 A), these are c» --{424 + 5)X10', c» --(239 + 5)x10', and c44 = {225 + 1)X10 dyncm ' which imply a deviation from the Cauchy relation with 8 = -0.06 + 0.02. From the energies of phonons of small wave vector measured just below the melting point at 82 K (a = 5.463+ 0.002 A}, the zero-sound elastic constants were determined to be c» --(248 +6))&10, c» --(1S3+5)X10', and c44 = (124+4)X10 dyncm ' (S = -0.19+0. 04).In both the 10and 82-K measurements, particular care has been taken to remove the systematic errors introduced by finite instrumental resolution. The measured elastic constants have been compared with several calculations and with results obtained from ultrasonic and Brillouin scattering experiments.Calculations of elastic constants heed on the 88 potential with the Axilrod-Teller three-body interactions are in good agreement with our data at both temperatures. The energies of some typical phonons have been measured at temperatures of 10, 35, S5, and 75 K. In the only case in which these measurements can be compared with an existing calculation {the [100]T mode at g = 0.4), our results show a slightly larger sMt of the phonon energy than that computed by Klein e t a l. from the 8$ potential. Goldman et al. ' and Niklasson'6 have both used the Lennard-Jones (LJ) potentials to compute phonon energies. More realistic computations based on the Bobetic-Barker (BB) potential and including the Axilrod-Teller three-body interactions have been carried out by Barker et al. " and have yielded reasonable agreement with the available measurements. Unfortunately, however, the existing data are not of sufficient accuracy to provide a sensitive test of the BB potential. The temperature dependence of phonon energies provides a further useful test of theoretical models of interatomic forces. In 1971, Batchelder et al. ' measured such temperature dependences for pho-10 3647
We have studied the dynamics of the classical Heisenberg chain at finite and infinite temperatures by computer simulation of an array of 4000 spins. The evaluation of the spin-spin and energy-energy wave-vector-and time4ependent correlation functions out to time 10J made it possible to determine the spin and energy diffusion constants. At infinite temperature we find Ds = (1.33 + 0.10)JSa' and D~= (3 + 1)JSa . Values of Ds have also been determined, with less precision, for ferromagnetic and antiferromagnetic interactions at temperatures k s T+~= 1.0 and 0.5. Our results are compared with various theoretical estimates. In addition, we calculate the time-dependent spin and energy self-correlation functions. I~INTRODUCTIONThe hydrodynamic formalism has often been a successful starting point for the description of the response of many-body systems to low-frequencylong-wavelength disturbances. ' Originally developed for classical liquids, it has been applied in recent years to the characterization of various transport phenomena in solids. The essential feature of these theories is the identification of a set of densities, e. g. , mass density, energy density, momentum density, whose spatial integrals are constants of the motion. Differential equations, the hydrodynamic equations, are developed to describe the temporal evolution of these densities. The normal-mode solutions to these equations are typically characterized by velocities of propagation and damping constants. The velocities are expressed in terms of parameters characterizing the thermodynamic state of the system. The damping constants, on the other hand, are combinations of phenomenological parameters.The success of the hydrodynamic description has spurred attempts to derive the equations using a microscopic formalism. ' In such a program one obtains explicit expressions for the damping constants in terms of the microscopic properties of the system. In addition, information is gained about the range of validity of the hydrodynamic theory.In the case of magnetic insulators the appropriateness of a hydrodynamic description of the transport of excitation in the magnetic lattice was apparently first pointed out by Bloembergen. ' The essence of his argument is that in systems where the dominant spin-spin interaction is of the Heisenberg form Z -JtrSt~Sr, «,y) 2 the total spin, g& S&, is a constant of the motion. Assuming no long-range order, the fluctuations in the total spin, S(tl, t)=g&e s'&S&(t), in the hydrodynamic region should be governed by the diffusion equation = -Dsq~S(tl, f), (l. l) where D& is the spin-diffusion constant. In addition, the energy density E(tl, t) obeys a similar equation, in which Ds is the energy (or thermal) diffusion constant. It is important to note that the particularly simple form for the hydrodynamic equations, (1.1) and (1.2&, is a direct consequence of the assumption that there is no long-range order in the system. When this is not the case there is a coupling between the spin and energy densities as well as propagating m...
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