Finite-size scaling from the self-consistent theory of localization

Accepting the validity of Vollhardt and Wölfle’s self-consistent theory of localization, we derive the finite-size scaling procedure used for studying the critical behavior in the d-dimensional case and based on the consideration of auxiliary quasi-1D systems. The obtained scaling functions for d = 2 and d = 3 are in good agreement with numerical results: it signifies the absence of substantial contradictions with the Vollhardt and Wölfle theory on the level of raw data. The results ν = 1.3–1.6, usually obtained at d = 3 for the critical exponentν of the correlation length, are explained by the fact that dependence L + L ₀ with L ₀ > 0 (L is the transversal size of the system) is interpreted as L ^1/ν with ν > 1. The modified scaling relations are derived for dimensions d ≥ 4; this demonstrates the incorrectness of the conventional treatment of data for d = 4 and d = 5, but establishes the constructive procedure for such a treatment. The consequences for other finite-size scaling variants are discussed.     

Electrostatic lattice energies of some ABX3 polytypes with close-packed AX3 layers

Electrostatic lattice energies (Madelung and polarization energies) for five ideal ABX₃, structure types with close-packed AX₃ layers (X = F, Cl, Br, I, O, S) are calculated. Stability regions for the ideal structure types are indicated as function of the anion radius and the anion polarizability. The effect on the electrostatic lattice energy due to trigonal deformations of the AX₃ layers in the 2L and 3L stacked structures is discussed for the anions F and Cl.     

A quantitative mean-field theory of the hydrophobic effect of neutral and charged molecules of arbitrary geometry

A self-consistent two-length scale theory of the interaction between a hydrophobic molecule and a water environment is considered. This theory allows the width of the hydrophobic layer to be calculated for molecules of arbitrary geometry by explicitly taking into account the water structure through the correlation function of a pure liquid. This approach is used to calculate the density profile ρ(r) around a molecule of arbitrary geometry and the solvation free energy ΔG(R) related to the transport of the molecule from a vacuum to a liquid. The model parameters are adjusted by comparing the results of numerical Monte Carlo simulations taken from the literature with predictions of the model for molecules of spherical geometry. The free energy of the interaction Δ G(D) between two spheres of radius R separated by distance D is also determined using the developed approach. The model is generalized to electrostatic interactions within the framework of a self-consistent scheme in which water is modeled by a gas of point dipoles. Analysis of the derived equations shows that this theory coincides with the electrostatic theory of a continuous medium with an effective permittivity in the limit of weak electric fields.     

Calculation of electrostatic fields in ionic crystals based upon the Ewald method

Formulas for the electrostatic potential, field strength and field gradient due to monopoles, dipoles and quadrupoles are given, based upon the Ewald summation method. A convenient choice of the convergence parameter in the Ewald method is demonstrated in some actual calculations. A method to calculate self-consistent electrostatic fields due to self induced dipoles is evaluated. Monopole and self-consistent fields are calculated in Pb₂O₃, βMn₂O₃, FeTiO₃, MnF₃ and Pb₃O₄. The influence of the anion parameter on the electrostatic field is investigated for idealized CdCl₂ and TiO₂. Results of Madelung and induced dipole polarization energy for a series of arbitrary compounds are given. The effect of spontaneous polarization in polar crystals is discussed.     

Finite Fermi systems theory and self-consistency relations

The self-consistent theory of the finite Fermi systems is outlined. This approach is based on the same Fermi liquid theory principles as the familiar theory for finite Fermi systems (FFS) by Migdal. We show that the basic Fermi system properties can be evaluated in terms of the quasiparticle Lagrangian L_q which incorporates the energy dependency effects. This Lagrangian is defined so that the corresponding Lagrange equations should coincide with the FFS theory equations of motion of the quasiparticles. The quasiparticle energy E_q defined in the terms of t he quasiparticle Lagrangian L_q according to the usual canonical rules is shown to be equal to the binding energy E_o of the system. For a given Lagrangian L_q the particle densities in nuclei, the nuclear single-particle spectra, the low-lying collective states (LCS) properties, and the amplitude of the interquasiparticle interaction are also evaluated. The suggested approach is compared with the Hartree-Fock theory with effective forces.     

Polar propagating optical phonon states and their dispersive spectra in wurtzite nitride superlattices: Quantum size effect

The polar optical phonon states of propagating (PR) modes in wurtzite GaN/Al_xGa_1−xN superlattices (SLs) are investigated within the dielectric continuum model framework. It is proved that the PR phonon modes only appear in wurtzite GaN/Al_xGa_1−xN SL with a small x$x$ Al mole ratio concentration (such as x<0.34$x<0.34$). The analytical phonon states of PR modes and their dispersive equation in the wurtzite GaN/Al_xGa_1−xN SL structures are obtained. Numerical calculations on the dispersive spectra of PR modes and their electrostatic potential properties as well as the quantum size effect are performed for a wurtzite GaN/Al_0.15Ga_0.85N SL. Results reveal that dispersive curves of PR modes in SLs form frequency bands. As the well width of GaN well-layer increases, the frequency bands of PR modes become steeper and narrower. The discussion of electrostatic potentials shows that the wavelength of PR phonon modes with a phase qL=π$qL=\pi$ (0) is 2L$2L$ (L$L$). With the increase of the SL well width, the wave-node number of the PR phonon modes in the barrier regions increases. The present theoretical scheme and numerical results are quite useful for analyzing the dispersive spectra of PR phonon modes and their polaronic effect in wurtzite GaN/AlGaN SL structures.     

Development of the self-consistent approximation and its application to the problem of magnetoelastic resonance in an inhomogeneous medium

Our previously proposed approximation involving both the first and second terms of the expansion of the vertex function is generalized to the system of two interacting wavefields of different physical nature. A system of self-consistent equations for the matrix Green’s function and matrix vertex function is derived. On the basis of this matrix generalization of the new self-consistent approximation, a theory of magnetoelastic resonance is developed for a ferromagnetic model, where the magnetoelastic coupling parameter ε(x) is inhomogeneous. Equations for magnetoelastic resonance are analyzed for one-dimensional inhomogeneities of the coupling parameter. The diagonal and off-diagonal elements of the matrix Green’s function of the system of coupled spin and elastic waves are calculated with the change in the ratio between the average value ε and rms fluctuation Δε of the coupling parameter between waves from the homogeneous case (ε ≠ 0, Δε = 0) to the extremely randomized case (ε = 0, Δε ≠ 0) at various correlation wavenumbers of inhomogeneities k _c. For the limiting case of infinite correlation radius (k _c = 0), in addition to approximate expressions, exact analytical expressions corresponding to the summation of all diagrams of elements of the matrix Green’s function are obtained. The results calculated for an arbitrary k _c value in the new self-consistent approximation are compared to the results obtained in the standard self-consistent approximation, where only the first term of the expansion of the vertex function is taken into account. It is shown that the new approximation corrects disadvantages of the Green’s functions calculated in the standard approximation such as the dome shape of resonances and bends on the sides of resonance peaks. The appearance of a fine structure of the spectrum in the form of a narrow resonance on the Green’s function of spin waves and a narrow antiresonance on the Green’s function of elastic waves, which was previously predicted in the standard self-consistent approximation, is confirmed. With an increase in the parameter k _c, the Green’s functions calculated in the standard and new approximations approach each other and almost coincide with each other at k _c/k ≥ 0.5. At the same time, the results of this work indicate that the new self-consistent approximation has a certain advantage for studying the problems of stochastic radiophysics in media with long-wavelength inhomogeneities (small k _c values), because it describes both the shape and width of peaks much better than the standard approximation.     

Self consistent theory of fluctuations near D-dimensional superconducting phase transitions

Nonlinear fluctuation contributions in the vicinity of the transition temperature for D-dimensional superconductors (D = 0…3) are treated in self-consistent Hartree approximation to the Ginzburg-Landau free energy. The results concern specific heat, correlation length, and the effect of a magnetic field on the specific heat (for D = 2).     

Ordering behaviors of the F.C.C. and B.C.C. phase alloys in the InMg,LiMg and AlZn systems

Expressions of the band-structure energy and the electrostatic energy characterized by the usual long-range and short-range order parameters are given to a binary alloy of simple metals with the f.c.c.-type or b.c.c.-type structure.The ordering energy and the local ordering energy are calculated for the InMg, LiMg and AlZn systems. The numerical results explain successfully the facts that the InMg system has the Ll₀-type and Ll₂-type ordered phases, each of which exists over a wide range of concentration and that the LiMg system has a local order with a negative short-range order parameter, while the AlZn system has that with the positive one. A lattice distortion in the Ll₀-type ordered structure of InMg is also discussed.     

A theory of quasi-channeling for electrons and positions,quasi-channeling radiation in systems of macroscopic extent,and experimental implications for a short wavelength laser

The theory of electron and positron channeling and radiation production in crystals is extended to macroscopic-scale systems. The new theory examines systems that exhibit periodicity along the axis of beam propagation, specifically, periodic electrostatic fields: the dimensions of the field periodicity (l_z) are much greater than typical interatomic lattice spacing (l_L), e.g., l_z≫l_L. The feasibility of a laser based on this predicted new phenomenon is examined. Differences between the properties of the proposed quasi-channeling radiation and radiation produced by wigglers and free electron lasers (FELs) are discussed. System lengths a factor of 10³ shorter than FELs are expected, with favorable scaling for below 500 eV laser photon output energies.     

Band structure of platinum from angle resolved photoemission experiments

Angle-resolved and angle-dispersed ultraviolet photoelectron spectroscopy (ARUPS) has been used to determine the experimental band structure of platinum at various points along the ΓX, ΓKX and ΓL directions in the Brillouin zone. Various methods have been employed to obtain the E(k) points, among them absolute methods which determine the energy and the momentum independently without any assumption about the final state dispersion. The experimental points are compared with a self-consistent relativistic band structure calculation for energies below and above the Fermi energy. Good agreement between theory and experiment is found.     

Conductance of finite systems and scaling in localization theory

The conductance of finite systems plays a central role in the scaling theory of localization (Abrahams et al., Phys. Rev. Lett. 42, 673 (1979)). Usually it is defined by the Landauer-type formulas, which remain open the following questions: (a) exclusion of the contact resistance in the many-channel case; (b) correspondence of the Landauer conductance with internal properties of the system; (c) relation with the diffusion coefficient D(??, q) of an infinite system. The answers to these questions are obtained below in the framework of two approaches: (1) self-consistent theory of localization by Vollhardt and Wölfle, and (2) quantum mechanical analysis based on the shell model. Both approaches lead to the same definition for the conductance of a finite system, closely related to the Thouless definition. In the framework of the self-consistent theory, the relations of finite-size scaling are derived and the Gell-Mann-Low functions ??(g) for space dimensions d = 1, 2, 3 are calculated. In contrast to the previous attempt by Vollhardt and Wölfle (1982), the metallic and localized phase are considered from the same standpoint, and the conductance of a finite system has no singularity at the critical point. In the 2D case, the expansion of ??(g) in 1/g coincides with results of the ??-model approach on the two-loop level and depends on the renormalization scheme in higher loops; the use of dimensional regularization for transition to dimension d = 2 + ?? looks incompatible with the physical essence of the problem. The results are compared with numerical and physical experiments. A situation in higher dimensions and the conditions for observation of the localization law ??(??) ?? ?i?? for conductivity are discussed.     

Excitation and nonlinear saturation of stimulated backscatter in a bounded dissipative plasma

Nonlinear spatial theory of the backscatter decay interaction predicts that the reflectivity, under conditions of absolute instability and zero noise, cannot exceed 1 ? L₀²/4L_a², where L₀ is the basic gain length and L_a the absorption length of the excited electrostatic wave.     

Self-consistent approaches in the microscopic theory of the nucleus: Static moments of odd-odd nuclei

Self-consistent mean-field theory and the method of the energy density functional, which are two modern self-consistent approaches in the microscopic theory of the nucleus that possess the highest predictive power for describing unstable nuclei, are briefly discussed. Themost recent results of calculations performed within these approaches are presented. The mean energies of E1 excitations in the range of 0–30 MeV are calculated for 15 stable and unstable tin isotopes (A = 100–176) on the basis of the self-consistent version of the generalized theory of finite Fermi systems by employing SLy4 Skyrme forces. A parameter-dependent expression that takes into account the existence of a pygmy dipole resonance is obtained for this quantity. The density-functional method is used within the self-consistent theory of finite Fermi systems on the basis of the Fayans-Tolokonnikov-Trykov-Zawischa functional in order to calculate the ground-state static quadrupole and magnetic moments of odd and odd-odd stable and unstable spherical near-magic nuclei. Good agreement with available experimental data is attained. The respective features are predicted for unstable nuclei.     

Band structures of Ni3Pt and NiPt3

The results of self-consistent, spin-polarized LMTO band structure calculations are shown for the compounds Ni₃Pt and NiPt₃, ofL12 (Cu₃Au) structure. Lattice constants are reported together with bulk moduli, and the electronic structure is studied in relation to magnetism in both cubic compounds. Covalent magnetism is shown to act against the magnetization in Ni₃Pt.     

On the relation between XPS and AES relaxation energies in metals

XPS (R₁) and AES cross-relaxation (R_c) energies are calculated by means of a Pseudopotential linear-response electron-gas method using norm-conserving Pseudopotentials, including intraatomic non-linearity contributions. The results are compared with excited atom model calculations and with self-consistent local density results from the literature. It is pointed out that deviations from the linear response rule R_c = 2R₁ are largely due to a shrinking of the core in the presence of the spectator hole. The remainder is due to non-linear response of the electron-gas.     

Quadrupole moments of spherical semi-magic nuclei within the self-consistent Theory of Finite Fermi Systems

The quadrupole moments of odd neighbors of semi-magic lead and tin isotopes and N = 50 , N = 82 isotones are calculated within the self-consistent Theory of Finite Fermi Systems based on the Energy Density Functional by Fayans et al. Two sets of published functionals are used to estimate systematic errors of the present self-consistent approach. They differ by the spin-orbit and effective tensor force parameters. The functional DF3-a leads to quadrupole moments in reasonable agreement with the experimental ones for most, but not all, nuclei considered.