Non-coherent scattering in subordinate lines: III. Generalized redistribution functions

The concept of the redistribution function, which was originally introduced in order to describe a correlation between frequencies and directions of the absorbed and emitted photon in resonance scattering, has been extended to other resonance two-photon processes including resonance Raman scattering, resonance two-photon absorption and emission, and inverse Raman scattering. We have derived, within the frame of the impact approximation, the appropriate form of the generalized redistribution function. Using a suitable formalism, the generalized redistribution function takes the same form for all types of two-photon processes and contains all the redistribution functions, considered previously, as various limiting cases. In analogy to Hummer's original scheme of redistribution functions, we have derived a similar set of generalized redistribution functions, denoted as p_i(i = I, II, III, IV, V), and we have shown that the most general case is described by a linear combination of p_III and p_V, analogously to the previous results. Explicit formulae for the velocity-averaged (laboratory-frame) generalized redistribution functions p_i(i = I, II, III, IV, V) are given and possible numerical methods for their evaluation are briefly indicated.     

Effects of nitrogen-doping configurations with vacancies on conductivity in graphene

We investigate electronic transport in the nitrogen-doped graphene containing different configurations of point defects: singly or doubly substituting N atoms and nitrogen–vacancy complexes. The results are numerically obtained using the quantum-mechanical Kubo–Greenwood formalism. Nitrogen substitutions in graphene lattice are modelled by the scattering potential adopted from the independent self-consistent ab initio calculations. Variety of quantitative and qualitative changes in the conductivity behaviour are revealed for both graphite- and pyridine-type N defects in graphene. For the most common graphite-like configurations in the N-doped graphene, we also consider cases of correlation and ordering of substitutional N atoms. The conductivity is found to be enhanced up to several times for correlated N dopants and tens times for ordered ones as compared to the cases of their random distributions. The presence of vacancies in the complex defects as well as ordering of N dopants suppresses the electron–hole asymmetry of the conductivity in graphene.     

Correlation functions in the multiple Ising model coupled to gravity

The model of p Ising spins coupled to 2d gravity, in the form of a sum over planar φ³ graphs, is studied and in particular the two-point and spin-spin correlation functions are considered. We first solve a toy model in which only a partial summation over spin configurations is performed and, using a modified geodesic distance, various correlation functions are determined. The two-point function has a diverging length scale associated with it. The critical exponents are calculated and it is shown that all the standard scaling relations apply. Next the full model is studied, in which all spin configurations are included. Many of the considerations for the toy model apply for the full model, which also has a diverging geometric correlation length associated with the transition to a branched polymer phase. Using a transfer function we show that the two-point and spin-spin correlation functions decay exponentially with distance. Finally, by assuming various scaling relations, we make a prediction for the critical exponents at the transition between the magnetized and branched polymer phases in the full model.     

Numerical results for the thermal scattering functions

A recent formulation in radiative transfer defined the thermal scattering functions that characterize radiative transfer from a general, plane-parallel, finite medium driven solely by an internal distribution of thermal sources. Exiting diffuse intensities are expressed as space convolutions of the thermal scattering functions with any thermal source distribution. A parametric study is presented to obtain the basic structure of these scattering functions. The independent variables of these azimuthally independent functions are the direction consine μ and source location t, while the parameters are the single scattering albedo ω, total optical depth t₀, and the asymmetry factor g in the Henyey-Greenstein phase function. The basic functional trends are discussed using various parametric plots, and selected tabular results are given to allow numerical checks. The computational method is invariant imbedding. As a particular application, these functions are used in the following companion paper to obtain exiting intensities from inhomogeneous and nonisothermal media.     

Boundary correlation functions of the six and nineteen vertex models with domain wall boundary conditions

Boundary correlation functions of the six and nineteen vertex models on an N×N lattice with domain wall boundary conditions are studied. The general expression of the boundary correlation functions is obtained for the six vertex model by the use of the quantum inverse scattering method. For the nineteen vertex model, the boundary correlation functions are shown to be expressed in terms of those for the six vertex model.     

Theoretical Treatment of Nonstationary Scattering by Phonons and Polaritons Part I. Derivation of the Basic Equations

The equations of motion of a two-level system and a harmonic oscillator coupled to a dissipative system are discussed; the dissipative system is assumed to consist of a large number of radiation oscillators. Special equations for the determination of the correlation functions of the fluctuation forces are derived under the condition of large time values, for which the atomic system has “forgotten” its initial state. The expectation values of the correlation functions are connected with the damping constant and the population operator of the excited state of the atomic system is in thermal equilibrium. Taking into account the influence of the coherent radiation field on the atomic system, the basic equations for the treatment of the nonstationary Raman scattering by polaritons are derived; the temporal range of validity is discussed. Using a time-dependent “variable” Fourier transformation, the nonstationary time- and spacedependent spectral densities are related to the correlation functions of the fields; here the Wiener-Khintchine theorem is applied in a nonstationary form. The limiting cases of the stationary scattering process as well as the usually introduced correlators of the slowly varying amplitudes are discussed.     

Adding and invariant imbedding equations in matrix notation for all the scattering functions

Previous work by the author introduced a radiative formulation, containing a delta interior illumination, that allowed scattering solutions driven by internal sources to be handled in complete analogy to those for the standard problem (external delta illumination scattering through a medium). This analogy was made explicit by defining the three levels of scattering functions, S_s-level, $\text{S}$_s- and S_F-level, and $\text{S}$_F-level, that characterize scattering through, into and out of, and within a finite medium, respectively. For an inhomogeneous medium the invariant imbedding method was employed to solve for these functions. This paper continues the work by showing that: (1) Adding equations can be derived for all the scattering functions using one superposition formula. (2) Adding and invariant imbedding computational methods are closely related and should be used in combination for efficient calculations. (3) A new set of functions can be defined that represent scattering out of a medium driven by thermal sources. (4) All scattering functions can be converted to represent a planetary problem by one adding step. References are given for numerical results using this formulation.     

Electron scattering from some even fp- and N50-shell nuclei with implanting the influence of short-range correlation functions

The influence of two body short range correlations on elastic electron scattering charge form factors, charge densities as well as root mean square charge radii of some fp-shell nuclei (for example, ⁴⁸Ca, ⁵⁰Cr, ⁵⁴Fe, ⁵⁸Ni, ⁷⁰Ge and ⁷⁴Se) and some N50-shell nuclei (for example, ⁸⁸Sr and ⁹⁰Zr) is analyzed using the one- and two-body terms in the cluster expansion together with the single particle harmonic oscillator wave functions. The Jastrow-type correlation function is utilized to embed the effect of short range correlations into elastic charge form factors F(q) and charge densities ρ(r). Both F(q) and ρ(r) depend upon the harmonic oscillator parameter b and the correlation parameter β (which initiates from the Jastrow correlation function). Here, the parameters b and β are determined via the fitting to the measured charge form factors. The embedding of short range correlations imitates the measured charge form factors at the high momentum transfers (q ≥ 2 fm⁻¹). It is noticed that the implanting of short range correlations is required for obtaining a remarkable alteration in the computed elastic charge form factors which in turn leads to explain the data of electron scattering astonishingly throughout the entire range of considered momentum transfers.     

Some aspects of microreversibility for open systems

It is verified that microreversibility conditions for higher order correlation functions for a classical or quantum markovian system are satisfied if they hold for second order correlation functions. The phase space version of the conditions is given. It is found that in this formulation the distribution function and Green's function for a given ordering mix with those for inverse ordering. We display explicitly the terms arising due to non-com-mutativity of operators for the Weyl ordering. Finally the microreversibility conditions are used to calculate the stationary solution of the master equation describingn photon absorption and emission.     

Form factors and correlation functions of an interacting spinless fermion model

Introducing the fermionic R-operator and solutions of the inverse scattering problem for local fermion operators, we derive a multiple integral representation for zero-temperature correlation functions of a one-dimensional interacting spinless fermion model. Correlation functions particularly considered are the one-particle Green's function and the density–density correlation function both for any interaction strength and for arbitrary particle densities. In particular for the free fermion model, our formulae reproduce the known exact results. Form factors of local fermion operators are also calculated for a finite system.