Dynamic Structure Function Research Articles

Antisymmetric part of the dynamic structure function of liquid 4He.

At large momentum transfers it is convenient to express the dynamic structure function $S(k, \ensuremath{\omega})$ as the sum of a symmetric part about $\ensuremath{\omega}={k}^{2}$ and an antisymmetric part. The latter is zero in the impulse approximation, and its leading contribution is given by $\frac{{S}_{A}(y)}{{(2k)}^{2}}$, where $y=\frac{(\ensuremath{\omega}\ensuremath{-}{k}^{2})}{2k}$ is the usual scaling variable. We calculate the integrals of ${S}_{A}(y)$, weighted with $y$, ${y}^{3}$, and ${y}^{5}$ in liquid $^{4}\mathrm{He}$ using sum rules as suggested by Sears. Polynomial expansions are used to construct models of ${S}_{A}(y)$ which appear to be in qualitative agreement with the observed antisymmetric part at large values of $k$.

Spin dynamics of EuO in the paramagnetic phase.

The spin dynamics of the semiclassical Heisenberg model on the fcc lattice, with ferromagnetic interactions up to the second-nearest-neighbor shell, is studied in the paramagnetic phase at the temperatures 1.68T/sub c/ and 2.0T/sub c/ using the Monte Carlo--molecular-dynamics technique. The important quantities calculated are the dynamic structure function S(q,..omega..), the spin autocorrelation function , and the static correlation functions. Our results for S(q,..omega..) show the existence of purely diffusive modes in the low-q regime. For q close to the zone boundary, our calculated S(q,..omega..) shows a three-peak structure, signifying damped propagating modes. This result disagrees with the earlier experimental observation of Mook and the theoretical result of Lindgard, where a two-peak structure was obtained near the zone boundary. Our results for S(q,..omega..) in the entire q space are in good qualitative and quantitative agreement with the recent neutron scattering experiments by Boeni and Shirane and also with the predictions of Young and Shastry. Our calculated autocorrelation function shows a diffusive behavior temporally.

Correlated basis theory of nuclear matter response functions

A comprehensive correlated basis theory for calculating the response functions of nuclear matter from realistic nuclear interactions is presented. It is used to calculate the dynamic structure function of nuclear matter in an approximation in which only correlated 1p–1h excited states are retained. The effect of 2p–2h excitations are approximately included via real and imaginary parts of the optical potential. It is argued that at large values of k the longitudinal response function R L( k, ω), observed in electron scattering experiments, can be obtained from the dynamic structure function. The calculated R L ( k, ω) at k = 300 and 400 MeV/ c is in fair agreement with the data on 40Ca, 48Ca and 238U nuclei. At k = 550 MeV/ c, the calculated response is in agreement with experiments on 40Ca and 56Fe at small values of ω, while it is larger than experiment at large ω.

Scaling and deep inelastic neutron scattering from quantum liquids and solids.

A new scaling variable is proposed for investigation of deep inelastic neutron scattering from quantum liquids and solids. The method improves the usual impulse approximation, accounting for a shift of the quasielastic peak with respect to the free recoil energy ${\ensuremath{\Elzxh}}^{2}$${q}^{2}$/2m and for an asymmetric behavior of the wings in the dynamic structure function. The approach is successfully employed to investigate the experimental data in liquid $^{4}\mathrm{He}$ at q=10 A${\r{}}^{\mathrm{\ensuremath{-}}1}$.

Spin wave damping in Heisenberg ferromagnets near the percolation threshold

The decay of zero temperature spin wave excitations near the percolation threshold in bond-diluted d-dimensional (d>1) Heisenberg ferromagnets is investigated using a Green function equation-of-motion method. The diagrammatic perturbation theory is convergent in the hydrodynamic limit (k xi <<1) and the following expression is obtained there for the damping, or decay rate: Gamma c(k, xi ) varies as (p-pc)-vd+(t- beta ) kd+2,(k xi <<1). This form satisfies the requirements of the dynamic scaling principle, by which Gamma c(k, xi )=kzg(k xi ) for general k xi , and z=2+(t- beta )/ nu . An explicit expression, applicable in the hydrodynamic regime and consistent with dynamic scaling, is also obtained for the infinite cluster transverse dynamic structure function. The approach is extended to incorporate the ensemble of finite clusters and to treat the case of non-zero temperature and also the experimentally more relevant situation of site dilution.

Temperature dependence of the structure factor of liquid 4He

Temperature dependence of the liquid structure factor of superfluid 4He is studied using the moments of the dynamic structure function with a realistic interaction potential in the effective mass approximation.

Plasmon dispersion and linewidth in aluminium

Plasmon dispersion in Al is estimated using the expression for the dynamic structure function,S pl(K,ω), corresponding to the plasmon excitations in a many-electron system derived earlier. An evaluation of its plasmon linewidth is also presented. It is observed that for Al both the dispersion and linewidth agree fairly well with experiments.

Critical dynamic response of the dilute antiferromagnetic chain

The transverse dynamic response of the dilute classical Heisenberg antiferromagnetic chain is calculated near the percolation threshold as k to 0 and k to pi . In the critical limit the universal dynamic structure function, which is calculated in closed form, exhibits dynamic scaling with exponents z=1, eta T=1. This function displays the crossover from spin wave (k- pi ) xi p>>1 to hydrodynamic (k- pi ) xi p<<1 response. The relevance of the results to the experimental situation is discussed.

Role of two-phonon excitations in the inelastic scattering of neutrons from He II

The suggestion that the multi-excitation component of the dynamic structure function is dominated by the two-phonon mode is analysed on the basis of a model for the two-particle excitations resulting in a qualitative fit to the experimental data. A better agreement is anticipated if a physically relevant potential is employed and higher (three or more) correlations are also considered.

Universal Scaling in the Motion of Random Interfaces

The dynamics of interfaces where the normal component of an interface velocity is proportional to the curvature is studied. The dynamic structure function due to the motion of random interfaces is shown to satisfy a scaling law. The results are compared with Monte Carlo simulations of the kinetics of the order-disorder transition in a quenched system.

The dynamic structure function for electron gas

The dynamic structure function S(q, omega ) of the homogeneous electron gas is calculated in a formalism where both the collective and particle-hole excitations of the electron system are described in terms of interacting bosons. The author presents numerical results only for the electron density corresponding to aluminium. The curve S(q, omega ) as a function of omega for a fixed momentum transfer q does not quite agree with the experimentally observed curve, although a definite shift is found in the peak value towards lower energies plus a tail. Also the plasmon dispersion curve is discussed.

Study of two-phonon excitations in superfluid 4He

We explain the second branch of excitations in superfluid 4 He observed by Cowley and Woods, by accounting for two-phonon contributions to the dynamic structure function, S( k , ω). Our theory gives a good fit with the experimental data in the high energy region for several values of momentum transfer. It is observed that the contribution to S( k , ω) due to two-phonon excitations is of the order of k 2 as against its k 4 dependence reported in earlier theories.

Deep inelastic scattering from quantum fluids

A simple and general argument is given for the deep inelastic behavior of the dynamic structure function of quantum fluids at zero temperature. The resulting high-energy tail is shown to give a reasonable fit to existing neutron data for liquid 4He.

Collective Variable Approach to Many-Boson Systems at High Density

Green's functions of phonons for many-boson systems at high density are obtained from the phonon Hamiltonian given in a previous paper on the basis of the coherent-state repre­ sentation. A dispersion equation derived from these Green's functions gives rise to the excitation spectrum leading to the Feynman energy in the low-momentum limit. It is shown that the dynamic structure function obtained here satisfies the sum rules in the low-momentum limit. As an illustrative example, the one-dimensional Lieb-Liniger model is considered. The relationships to the previous theories, in particular, to the velocity field approach of Yamasaki, Kebukawa and Sunakawa and to our density and phase operator approach are elucidated by the use of a unitary transformation. Discussion will be made on the D-wave portion of the roton-roton interaction in relation to the resonance state of roton pair studied in one of our previous papers. Finally, it is pointed out that the zeroth Fourier transform of the phase operator is related to the superfluid velocity in the two-fluid theory of liquid helium II.

Surface contribution to the dynamic structure function of liquid He

The liquid-He dynamic structure function $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})$ at wave vector $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$ and frequency $\ensuremath{\omega}$ is calculated within the framework of quantum hydrodynamics of an infinite "slab" of a viscousless, compressible fluid bounded by a free, sharp surface of area $A$ and by a wall at depth $\frac{V}{A}$, where $V$ is the volume. A simple technique of representing momentum sums to $O(\frac{A}{V})$ is presented. $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})$ is thereby represented as $S={S}_{\mathrm{bulk}}+{S}_{\mathrm{rip}}+{S}_{\mathrm{phon}}^{\mathrm{coh}}+{S}_{\mathrm{phon}}^{\mathrm{inc}}$, where ${S}_{\mathrm{bulk}}$ is the usual bulk contribution to $S$, and the last three terms are $O(\frac{A}{V})$ corrections. ${S}_{\mathrm{rip}}$ is a coherent ripplon contribution which diverges in the long-wavelength limit; ${S}_{\mathrm{phon}}^{\mathrm{coh}}$ is a $\ensuremath{\delta}$ function at the phonon frequency $\ensuremath{\omega}=qc$, where $c$ is the sound speed; and ${S}_{\mathrm{phon}}^{\mathrm{inc}}$ is a broad incoherent function, which behaves as ${\ensuremath{\omega}}^{\ensuremath{-}3}$ for $\ensuremath{\omega}\ensuremath{\rightarrow}\ensuremath{\infty}$, and which has a square-root integrable divergence at a threshold of $\ensuremath{\omega}={q}_{\ensuremath{\rho}}c$, where ${q}_{\ensuremath{\rho}}$ is the component of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$ parallel to the free surface. Both limiting forms of ${S}_{\mathrm{phon}}^{\mathrm{inc}}$ provide probes of the boundary conditions at the wall. Relevant sum rules are discussed---both formal and explicit verifications of the $f$-sum rule are presented; the static structure function $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ is discussed; finally, a surface tension sum rule analogous to the compressibility sum rule is given.

Dynamic structure function of liquid 3He

The dynamic structure function S(Q, ω) of liquid 3He is evaluated in the impulse approximation, based on a theoretically calculated momentum distribution n( k). The gap of n( k) at the Fermi surface gives rise to discontinuities in the slope of S(Q = const., ω) which may be detectable in neutron- or γ-ray scattering experiments.

Direct variational approach to the calculation of dynamic structure functions

We consider the problem of calculating the dynamic structure function $S(k,\ensuremath{\omega})$ via a properly posed initial-value problem for a linear kinetic equation. A variational principle is introduced and shown to give a direct bounded estimate of $S(k,\ensuremath{\omega})$. The variational trial functions for the problem are the space-time transforms of the phase-space correlation function. The choice of such trial functions are discussed and general expressions for $S(k,\ensuremath{\omega})$ are developed which are applicable to simple gases. Numerical results are presented for the specific case of hard spheres, these results being compared to previous first-principle calculations based upon kinetic equations with and without the inclusion of memory effects. Excellent agreement with previous results, at all ratios of fluctuation wavelength to mean free path, is obtained with considerably less computational complexity. This computational efficiency of the variational approach suggests that it should be a valuable technique, from the standpoint of feasibility, in attempts at first-principle calculations for more complex many-body systems. In this regard, further study and improvement of the variational techniques themselves appear to be warranted.

Application of field-theoretic, collective-coordinate, and correlated-basis-function methods to many-boson systems

The collective-coordinate and correlated-basis-function approaches are systematically applied to the determination of the elementary excitation spectrum $\ensuremath{\omega}(k)$ and the static structure function $S(k)$ for two models of a Bose gas which can be characterized by a single small expansion parameter $g$. Results are obtained to first order in $g$, the leading order in which multiexcitation processes are present, and are compared with those found by the microscopic-dielectric-function approach. It is found that a collective-coordinate calculation of $\ensuremath{\omega}(k)$ by Sunakawa, Yamasaki, and Kebukawa is incomplete to first order in $g$ and that the convolution approximation for the three-particle distribution function in the correlated-basis-function formalism leads to incorrect results for overlap matrix element and for $S(k)$. A form for the overlap matrix element is proposed which leads to correct first-order results. Various properties of $\ensuremath{\omega}(k)$, $S(k)$, and the dynamic structure function are discussed.

Dynamic structure function of a Bose gas atT=0

The dynamic structure function S (k,ω), as a function of wave vector k and frequency ω, is calculated rigorously for a simple Bose gas at T = 0. The dielectric formulation is used to satisfy general symmetry requirements for a Bose system, and the calculation is carried out in the first approximation beyond Bogoliubov taking into account the three-phonon process. An analytic solution in terms of elliptic integrals is obtained for the width of S (k,ω) valid for arbitrary k and ω. Qualitatively, S (k,ω) at finite but small k is found to have a square-root behavior near the threshold frequency for two-phonon production, a peak at the elementary excitation frequency, and a long power-law tail at high frequencies. Illustrative numerical results are presented for the width of S (k,ω), the imaginary part of the phonon spectrum, and S (k,ω) itself. Finally, the impulse approximation and the extraction of the condensate density n0 from S (k,ω) in the large-k limit are discussed.

Sum Rules and High-Frequency Behavior of Dynamic Structure Function of Quantum Fluids

Using a novel diagrammatic perturbation-theory approach, it is found that the dynamic structure function of quantum fluids at high frequencies is independent of quantum statistics, depends only on the interparticle potential, and has a long tail as a function of frequency. In particular, it is found that the fifth and higher frequency moments diverge in charged quantum systems. This is demonstrated to hold in general by an investigation of the fifth-moment sum rule.