Theory of the local field correction in an electron gas

The wave-vector- and frequency-dependent dielectric function ?(k,ω) of an electron gas can be expressed in terms of Lindhard's function and a complex local field correctionG(k,ω) which incorporates all the effects of dynamic exchange and correlation in the system. The general properties ofG(k,ω) are discussed, in particular the static and high-frequency limits. It is shown that for smallk, bothG(k, 0) andG(k, ∞) vary ask ², with different coefficients, but both determined by the average kinetic and potential energies per particle. For largek,G(k, ∞) varies again ask ² and it is argued that the same holds true forG(k, 0), with both coefficients (though different) determined by the average kinetic energy per particle. General formulas for the plasma dispersion relation and damping, involving, respectively, the real and imaginary parts ofG(k,ω), are given. The term in the plasma frequency which is proportional tok ² is given directly in terms of the average kinetic and potential energies per particle, a result true at all temperatures. A calculation of the frequency dependence ofG(k,ω), starting from the exact equation of motion for the particle-hole operator and employing a decoupling approximation introduced previously by Toigo and Woodruff, is presented. Explicit results forG(k,ω) are obtained for smallk and allω. The complete expressions forG(k, 0) andG(k, ∞) in this approximation have been obtained and are plotted.     

Extension to velocity dependent gauge transformations

In previous papers it was shown that in phase space a generator of gauge transformations for a singular Lagrangian \(L(q,\dot q)\) is given byG[ε^α]=ε _k ^α φ _α ^k (q,p,t) where φ _α ^k are first class constraints andG is subjected to a stationarity condition. A non-trivial extension from velocity (or momentum) independent gauge transformations to velocity dependent ones is realized, by replacing gauge functions ε^α(t) with momentum dependent functions ε^α(q,p,t), as long asG satisfies the stationarity condition. Inversely, it is proved in a classical framework that, within velocity dependent gauge transformations, all generators of gauge transformations can be expressed in terms of the linear combinations of φ _α ^k .     

Calculation of the hyperfine structure of the 42 S 1/2, 42 P 1/2, 42 P 3/2 states in the 3d 10 nl configuration of Cu

The hyperfine structure of the 4² S _1/2, 4² P _1/2, 4² P _3/2 states in the 3d ¹⁰ nl configuration of Cu has been evaluated using many-body perturbation theory. Polarisation effects were included in all orders and correlation to third-order. By the use of iteration methods, a large number of higher order diagrams were also included. The correlation effects between the valence electron and the 3d shell were found to be very important. The results forA(² S _1/2) andA(² P _1/2) 5827MHz and 440 MHz, respectively, are in good agreement with the experimental results, whereas the result forA(² P _3/2)=83 MHz is far from the experimental value. No explanation was found for the discrepancy. The quadrupole values were found to be ?206 mb for⁶³Cu and ?185 mb for⁶⁵Cu.     

Exact solutions for a forced Burgers equation with a linear external force

We investigate the solutions of the Burgers equation , where F(x,t) is an external force and Φ(x,t) represents a forcing term. This equation is first analyzed in the absence of the forcing term by taking F(x,t)=k₁(t)−k₂(t)x into account. For this case, the solution obtained extends the usual one present in the Ornstein-Uhlenbeck process and depending on the choice of k₁(t) and k₂(t) it can present a stationary state or an anomalous spreading. Afterwards, the forcing terms Φ(x,t)=Φ₁(t)+Φ₂(t)x and Φ(x,t)=Φ₃x−Φ₄/x³ are incorporated in the previous analysis and exact solutions are obtained for both cases.     

Computation and analysis of the dynamic structure factor S(k, ω) for small wave vectors

Molecular dynamics (MD) calculations have been performed for the Lennard-Jones (12-6) potential function using 2048 particles. Using conventional parameters the results may be compared with those for liquid argon.

The dynamic structure factor S(k, ω) has been determined both by Fourier inversion of the intermediate scattering function F(k, t) and from the longitudinal current-current correlation function C _∥(k, t). Particular attention was paid to the recurrence time of the system. The results for S(k, ω) by the two methods agree within 5 per cent for the whole region of small k-vectors considered. Double Fourier inversion of the van Hove function G(r, t) led to insufficiently accurate results for these small k-values. In view of the present data, the MD-results of Levesque et al. [1] for S(k, ω) have only a qualitative character. These latter data appear to contain truncation errors due to incomplete Fourier transformations.

Using a hydrodynamic assumption for F(k, t) we were able to extract the transport coefficients, the velocity of sound and the ratio of the specific heats in the limit of large wave lengths or small k. The velocity of sound was obtained by exploiting the MD generated anomalous dispersion curve of sound waves. Anomalous dispersion was found to set in for kσ ～ 0·25. A sound speed of 880 ms^-1 has been determined which is in excellent agreement with experimental values for liquid argon. The total error for the MD value amounts to about 5 per cent. In contrast, the ratio of the specific heats γ and the transport coefficients D _T and Γ (thermal diffusivity and sound attenuation) were determinable only with an accuracy of 15 per cent due to the need for larger extrapolations. Nevertheless, we found D _T, Γ and γ in agreement with experimental values within 5-10 per cent.     

Superconductivity and magnetic ordering in YBa2(Cu1−x Fe x )3O7

Transverse-and zero-field μSR measurements have been made for YBa₂(Cu_1−x Fe _x )₃O₇ withx=0.04, 0.08 and 0.12. The temperature range studied was from approximately 7.5 K to 100 K. The onset of magnetic ordering commences at about 7.5 K forx=0.04, 10 K forx=0.08 and 20 K forx=0.12. The Gaussian depolarization parameter, σ ofG _x (t) = exp(−σ² t ²/2), is depressed by a factor of about 0.6 forx=0.04, but for thex=0.08 sample σ is depressed by a factor of 10 and increasing suppression is seen as the temperature is lowered below 45 K. This decrease in σ is interpreted in terms of decreasing electronic mean free paths.     

The gamma decay of high spin states of39K

First will be exhibited, for the stationary case, a connection between the probability, to find a particle in the region of interaction, and the derivative of the scattering phase shift for the momentum. From the idea, that in stationary scattering a resonance is linked with an appreciable increase of this probability, one obtains new and quantitative criteria for the behavior ofδ _l(k). For instance, the nonresonant behavior can be characterised by the condition 2k dδ _l(k)/dk<1. The maximum of probability for the particle to be in the region of interaction, is considered in accordance with the criterium of maximal change of the phase shift, as a function ofk. This characterises the location of the resonance. Wigner's lower limit fordδ _l (k)/dk allows to formulate a resonance condition directly for the cross section. Furthermore will be discussed some connections with the pole approximations of theS-matrix and the “collision lifetime” of Smith.     

The nuclear magnetization as the origin of transient changes in the magnetic field in pulsed NMR experiments

The very measurement of the NMR signal in a pulsed NMR experiment disturbs the nuclear equilibrium magnetization. A small change in the magnetic field results, which can be of the order of −0.0025 ppm in protonated solvents. It is shown that the resonances in pulsed high-field NMR consequently are non-Lorentzian. The FID is properly described by S(t) = S(0)exp(−iω_0tt)exp(−iΔωf{t}) is the small resonance shift which is experienced immediately following the pulse, and f {t} characterizes the subsequent return toward zero by the response of the lock system and by spin-lattice relaxation. In particular in samples containing large concentrations of protons, shifted and distorted resonance lines can be observed.     

Four-wave mixing and hyper-Raman scattering in CuCl

Non-degenerate four-wave mixing using two non-collinear laser beams with frequencies (wavevectors) ω_p, ω_t (k_p, k_t) respectively is studied in CuCl. Two emission lines at frequencies ω⁽¹⁾=2ω_t-ω_p, and ω⁽²⁾=2ω_p-ω_t are observed. Their excitation spectrum is sharply peaked if the phase-match condition k⁽¹⁾=2k_t-k_p is fulfilled. This is the case, if ω_p coincides with the hyper-Raman lines (R⁺_T, R^-_T) of the laser labelled (t) in a well-defined geometrical configuration.     

A multi-channel scattering theory for some time dependent Hamiltonians,charge transfer problem

Scattering theory for time dependent HamiltonianH(t)=?(1/2) Δ+ΣV _j (x?q _j (t)) is discussed. The existence, asymptotic orthogonality and the asymptotic completeness of the multi-channel wave operators are obtained under the conditions that the potentials are short range: |V _j (x)|≦C _j (1+|x|)^?2?ε, roughly spoken; and the trajectoriesq _j (t) are straight lines at remote past and far future, and |q _j (t)?q _k (t)| → ∞ ast → ± ∞ (j≠k).     

On asymptotic behaviour of four spin correlation functions in a linear chain of spins

Second and fourth moments of the four spin frequency correlation functions S^Z_i(t) S^Z_i+1 (t) S^Z_j+1 〉_k,ω and 〈S^x_i(t) S^z_i+1 (t) S^x_jS^z_j+1 〉_kω have been calculated in the high temperature limit for an arbitrary spin with an isotropic Heisenberg Hamiltonian.     

Spontaneous emission of a two-level system and the influence of the rotating-wave approximation on the final state. I

By using a modified Robertson projection technique an exact equation of motion for the expectation value of the population inversion operatorS ^z of a single two-level atom in the case of spontaneous emission is derived. Afterwards, by making the Markov approximation, it is shown that the ground state expectation value〈S ^z 〉 _t =? 1/2 fort→∞ will be reached only if the rotating-wave approximation or the Born approximation is made additionally.     

Spin autocorrelation of Ni in K2Mn0.975Ni0.025F4 studied by nuclear spin-lattice relaxation

From the nuclear spin-lattice relaxation of the out-of-layer ¹⁹F nuclei in magnetic fields perpendicular to the c-axis the low-frequency component of the autocorrelation function 〈S^z(t)S^z(O)〉 of Ni in ordered K₂Mn_0.975Ni_0.025F₄ is found to be substantially reduced relative to the Mn host. The experimental rates vs temperature are in accord with those for relaxation involving two spin excitations calculated with local Green's functions.     

Scaling function for order-parameter correlations in expansion to order 1/n: I. Short-range interactions

The order-parameter correlation function $\text{G}\text{?}\text{(}\text{q}\text{, ξ}{}_1\text{)}$ is calculated in the critical region of momentum space $\text{q}$ in terms of a second-moment correlation length ξ₁ by means of perturbation expansion to order 1/n, for an n-vector system with short-range interactions, in zero field above T_c, for 2 < d < 4. The scaling function of the $\text{q}$ dependence is obtained in closed form with a precisely identified cutoff-dependent factor which is the amplitude of the correlation-length dependence of the susceptibility. Both the exponents and the coefficients of the expansion for fixed q as t = (T?T_c)/T_c → 0 are given explicitly and the former are shown to be in accordance with the operator product expansion. The coefficients of order 1/n in the terms associated with a t^k(1?α) dependence of the energy density, for integer k ≥ 1, are expected to be explicitly cutoff-dependent and this is verified by the detailed calculations for k = 1. The behaviour for fixed t and q → 0 is shown to be markedly different from the Ornstein-Zernike approximation. Detailed comparison is provided with the scaling function of the t dependence of the correlations appearing in parallel work.     

Spherical harmonics and Cartesian tensor scalar product expansion in the Fokker-Planck equation

On the basis of the expansion of the distribution functionf(v, r,t) in a sum of spherical harmonics, which is equivalent to a Cartesian tensor scalar product expansion of the distribution function, i.e.,f(v, r, t)=f ₀(v,r,t)+v. f ₁(v,r,t)+vvf ₂(v,r,t)+vvvf ₃(v,r,t)+ wheref _k (k=2, 3) arek-th order irreducible tensors, the Rosenbluth potential functions and the Fokker-Planck collision term are expanded in a similar sum. Collisions termsJ _Fk (k=0, 1, 2) and the equations forf _k (k=0, 1, 2) for the case of the Coulomb interactions are also determined.Technická 2, Praha 6, Czechoslovakia.The autor wishes to express his thanks to Prof. J. Kracík, DrSc. for valuable advice and suggestion.     

Effects of finite size and shape on scattering and thermodynamic quantities at 2D Ising critical points

Using recent results by Cardy based on the conformal invariance of critical correlation functions we calculate universal results for scattering functions S(k), susceptibilities, correlation lengths and specific heat correction terms for finite Ising systems in two dimensions with circular and rectangular shapes and free boundary conditions. Our results specify the effect of shape on these quantities at the critical point. In particular, the half-width and lineshape of the scattering function is found to be strongly influenced by geometry. For a circle, S(k) follows the infinite system behavior 1/k^2−η, η = 0.25 only for very large k. For a substantial range of intermediate k values it is well represented by 1/k^2−η_app, with an “apparent” exponent η_app. We also discuss the probable influence of end, edge and domain wall effects on the correlation lengths, susceptibilities and specific heat correction terms. The application of our results to experimental systems and other theoretical models is considered.     

Inhomogeneous broadening of PAC spectra with V zz and η joint probability distribution functions

The perturbed angular correlation (PAC) spectrum, G ₂(t), is broadened by the presence of randomly distributed defects in crystals due to a distribution of electric field gradients (EFGs) experienced by probe nuclei. Heuristic approaches to fitting spectra that exhibit such inhomogeneous broadening (ihb) consider only the distribution of EFG magnitudes V _zz , but the physical effect actually depends on the joint probability distribution function (pdf) of V _zz and EFG asymmetry parameter η. The difficulty in determining the joint pdf leads us to more appropriate representations of the EFG coordinates, and to express the joint pdf as the product of two approximately independent pdfs describing each coordinate separately. We have pursued this case in detail using as an initial illustration of the method a simple point defect model with nuclear spin I?=?5/2 in several cubic lattices, where G ₂(t) is primarily induced by a defect trapped in the first neighbor shell of a probe and broadening is due to defects distributed at random outside the first neighbor shell. Effects such as lattice relaxation are ignored in this simple test of the method. The simplicity of our model is suitable for gaining insight into ihb with more than V _zz alone. We simulate ihb in this simple case by averaging the net EFGs of 20,000 random defect arrangements, resulting in a broadened average G ₂(t). The 20,000 random cases provide a distribution of EFG components which are first transformed to Czjzek coordinates and then further into the full Czjzek half plane by conformal mapping. The topology of this transformed space yields an approximately separable joint pdf for the EFG components. We then fit the nearly independent pdfs and reconstruct G ₂(t) as a function of defect concentration. We report results for distributions of defects on simple cubic, face-centered cubic, and body-centered cubic lattices. The method explored here for analyzing ihb is applicable to more realistic cases.     

Role of metastable atoms and molecules of helium in excitation transfer to hydrogen atoms

The afterglow of a discharge in helium with a small admixture of hydrogen is studied spectroscopically (p=40 Torr, [e]≤10¹¹ cm^?3). The time-resolved measurements of intensities of the first four lines of the Balmer series are performed. The concentrations of metastable helium atoms and molecules are evaluated from the relative intensity of the absorption lines. The ratios of excitation transfer rates from atoms He(2 ³ S ₁) k ₁(n) and molecules of helium He₂(a 2sσ ³Σ _u ⁺ ) k ₂(n) to atomic hydrogen H*(n) are measured to be k ₁(n=3)/k ₂(n=3)=0.04±0.02 and k ₁(n=4)/k ₂(n=4)=0.01±0.02. The ratios of excitation rate constants k ₂(n) corresponding to different states H(n) are measured to be k ₁(n=4)/k ₂(n=3)=0.023±0.01; k ₁(n=5)/k ₂(n=3)≤0.013; and k ₁(n=6)/k ₂(n=3)≤0.007.