Frequency-moment Sum Rules Research Articles

Sum rules and dynamical properties of two-dimensional classical electron liquid

Expressions for the fourth frequency moment sum rule for the longitudinal and transverse current correlation functions are obtained for a classical system of electrons trapped on the surface of liquid helium. The expressions involving static triplet correlation functions are considerably simplified so that the numerical results for the moments can be easily obtained for all momentum transfers. The results for the moments as a function of wavenumber k are presented for 2.23<or= Gamma <or=90.0. The authors make use of these results to analyse the recent molecular dynamics (MD) data on the dynamical structure function S(k, omega ) and the transverse current correlation function Ct(k, omega ) using the Mori formalism. It is found that a relaxation kernel essentially with single decay constant provides an overall good description of MD data over the entire range of Gamma .

Sum rules and electron correlations in two-dimensional electron liquid

The low order frequency moment sum rules of the density response function for quantum 2-dimensional electron liquid are given. These are used to obtain an integral equation for the static structure factor. This integral equation is solved self-consistently for different electron densities. Results obtained for the pair-correlation function are positive for all inter-electron separations even at physical densities.

Dynamical structure factor of an electron liquid. I

Electron correlations of an electron liquid at metallic densities are studied by means of a diagrammatic method, the primary aim being the interpretation of the spectral structure observed for several metals in recent inelastic electron and x-ray scattering experiments. Several sum rules and requirements such as the positivity of the pair distribution function, the compressibility sum rule, the frequency moment sum rules, and the conservation law of the local electron number which have hitherto been regarded as guiding principles in theoretical formulation are first examined and discussed in a critical manner. Detailed examinations are made of the proper polarization function within the Hartree-Fock approximation. The lowest-order perturbation expressions giving two-pair excitations and one-pair---one-plasmon excitations are next examined. It is pointed out that these expressions are indeed divergent in the usual one-pair excitation region; the renormalization of the one-particle Green's function is indispensable there. From the above considerations and further discussions, the quasi-one-pair excitation approximation is presented for the purpose of describing the spectral shape in the dynamical structure factor $S(q,\ensuremath{\omega})$ in the intermediate wave-number and intermediate frequency regions. In the quasi-one-pair excitation approximation the proper polarization function is constructed from a coherent part of the renormalized one-particle Green's function and a static part of the irreducible particle-hole interaction including strong short-range correlations.

Sum Rule and Hydrodynamic Analyses of the Velocity Autocorrelation Function in Strongly Coupled Plasmas

The memory function for the velocity autocorrelation function in a strongly coupled, one-component plasma is analyzed in the short time and long time domains, respectively, with the aid of the frequency-moment sum rules and the hydrodynamic consideration evoking the idea of the generalized Stokes friction. A series of interpolation schemes with successively improved accuracies are then introduced. Numerical investigations of those interpolation schemes clarify the physical origin of the three different types of the velocity autocorrelation function observed in the molecular dynamics simulation at different regimes of the coupling constant.

Dielectric formulation of strongly coupled electron liquid at metallic densities: Longitudinal response

The longitudinal dielectric response function for a strongly coupled plasma is formulated in such a way that it satisfies a set of physical requirements representing strong-coupling effects, namely, the equation of continuity, long-time relaxational behavior, static and dynamic local-field corrections, and the frequency-moment sum rules. Dynamic effects arising from Coulomb collisions are considered in terms of the long-time relaxation rate and the multipair excitation spectrum; explicit expressions for these quantities are obtained as functionals of the static form factor. The theory provides a quantitative description of the long-wavelength excitations, such as the plasmons, the single-pair and multipair excitations, in a way consistent with the known qualitative features.

Correlational properties of two-dimensional electron systems in the surface states on liquid helium

Abstract The static structure factor and the correlation energy of the two-dimensional electron liquid are computed through numerical solution of the self-consistent integral equation which has been obtained in a way analogous to the corresponding three-dimensional problem; the result shows a good agreement with the Monte-Carlo data. The dielectric response function and the equation for the dynamic structure factor are derived in such a way as to satisfy the frequencymoment sum rules exactly up to those terms involving the pair correlation function; they also take account of the relaxation effect in the long-time behavior. Variational calculation of the surface states is re-examined, affecting the theoretical estimate on the lifetime of surface state electrons.

Correlational properties of two-dimensional electron systems in the surface states on liquid helium

The static structure factor and the correlation energy of the two-dimensional electron liquid are computed through numerical solution of the self-consistent integral equation which has been obtained in a way analogous to the corresponding three-dimensional problem; the result shows a good agreement with the Monte-Carlo data. The dielectric response function and the equation for the dynamic structure factor are derived in such a way as to satisfy the frequencymoment sum rules exactly up to those terms involving the pair correlation function; they also take account of the relaxation effect in the long-time behavior. Variational calculation of the surface states is re-examined, affecting the theoretical estimate on the lifetime of surface state electrons.

A basic difficulty of the 'local mean-field' theory in a plasma

An exact formulation of the framework of the so-called 'local mean-field' theory is given in a plasma with an arbitrary number of components and it is proved that an approximate theory with a frequency-independent 'local-field' correction factor Gij(q) has a basic difficulty. The difficulty is as follows. When Gij(q) is chosen so as to satisfy the third frequency moment sum rule, the theory has the unphysical result: (a) 1/ epsilon (q,0) does not vanish as q to 0 in an electrically neutral system without a uniform, neutralising background charge and (b) chi s=0 in a magnetically neutral system, epsilon (q, omega ) and chi s being the generalised dielectric function and the spin susceptibility, respectively. An approximation procedure for removing this difficulty is discussed.

Dynamical exchange effects on the plasmon dispersion in aluminum

AbstractIn the dielectric function, obtained from the dynamical exchange decoupling method, exchange effects are described by a frequency‐dependent local field correction G(q, ω). From the behaviour of G(q, ω) in the static limit ω = 0, it follows that the compressibility sum rule is satisfied, and that the pair correlation function g(r) equals 1/2 at the origin. Also, in the static limit, G(q, 0) shows a sharp peak near q = 2kF. The dynamical behaviour of G(q, ω) at high frequencies fulfills the condition imposed by the third frequency moment sum rule, and it also implies that g(0) = 1/2. The same g(0) thus derives from both the static and the dynamical behaviour of G(q, ω), in contrast to earlier theories. The influence of the dynamical exchange effects on the plasmon dispersion is examined. It is found that, compared to RPA, the plasmon is almost unaffected by exchange outside the continuum, but that in the particle‐hole excitation region exchange effects drastically lower the peak position of the structure factor. This tendency has been experimentally observed in Al, where the RPA peak positions at finite wave vector are found at too large frequencies, but where the dynamical exchange decoupling method yields relatively good agreement with the experimental data.

Extraction of pair-interactions from neutron inelastic scattering data

A self-consistent scheme is discussed for calculating the pair potential of simple classical liquids from a knowledge of the zeroth and fourth frequency-moment sum rules of the coherent scattering function $S(q, \ensuremath{\omega})$. The procedure is tested by using molecular-dynamics data on a liquid-argon-like system interacting through a Lennard-Jones (6-12) potential. A detailed investigation of the propagation of error in the potential due to uncertainties in the measurement of the data has been made which should serve as a guideline to the experimentalist.

Theory of strongly correlated charged-particle systems: Plasma turbulence and high-density electron liquids

A unified theory is developed for describing dynamic and static properties of strongly correlated charged-particle systems, such as a high-density electron liquid and a strongly turbulent plasma, for which the correlation (or fluctuation) energy is comparable in magnitude to the kinetic energy. Based on the microscopic Klimontovich formalism, a systematic renormalization of the single-particle propagators is carried out and vertex corrections arising from strong correlations are taken account of. The resulting theory is examined, and found to be satisfactory, in the light of a number of rigorous criteria, such as explicit inclusion of statistical modification effects in the single-particle orbits, dynamic modification of effective particle interactions brought about by strong correlations, frequency-moment sum rules of the dielectric response function, and reproduction of exact results known in the static properties of the electron liquid; the theory is thus valid for all wave numbers and frequencies. With the aid of a velocity-average approximation, we show that the result can be expressed in a simplified form which still satisfies those criteria. Its relation with the polarization potential model in the theory of condensed matter is thereby noted; explicit expressions for the scalar and vector polarization potentials, the wave-number-dependent effective mass, and the collisional contribution to the response function are obtained.

Conductivity sum rules for the relativistic two component plasma

Abstract Using the Kubo sum rule theorem, external conductivity frequency moment sum rules are derived for the relativistic two-component plasma in a constant external magnetic field.

Viscosities of liquid rubidium

Estimates of the longitudinal and shear viscosities of liquid rubidium have been made on the basis of the memory-function approach coupled with the frequency-moment sum rules. The basic ingredients used were the potential of Price et al. (1970) and the molecular dynamics data of Rahman (1974) for the pair correlation function consistent with this potential. The theoretical results for the viscosities have been found to be in good agreement with the available experimental result. An estimate of the longitudinal viscosity from mean-field theory has also been found to be of the right order of magnitude.

Spatial correlations and sum rules for a Fermi liquid

The effect of strong short-range spatial correlations in a Fermi liquid is discussed in connection with various sum rules. It is found that all mean-field theories with a frequency-independent localfield factor will necessarily violate either the susceptibility sum rule or the third frequency moment sum rule, and the physics behind this is discussed. The necessary modifications in the available theories are briefly discussed and the situation one is facing is demonstrated on the simpler system of a harmonic lattice.

A sum rule for the transverse current correlation function in an electron liquid

An exact expression for the first frequency moment sum rule of the transverse current correlation function in an electron liquid is obtained and discussed.

Structure Factor of the Electron Liquid from the Moment Sum Rules

The structure factor of the electron liquid at metallic densities is calculated from a nonlinear integral equation, derived earlier by Kuglar, using the low-order frequency-moment sum rules of the spectral function of the density response function. The results for the structure factor and the pair-correlation function are compared with those of Vashista and Singwi.