Nonlinear Raman scattering of photons by a channeled particle

Channeled particles are characterized by the discrete spectrum of bound transverse motion. The interaction of photons with channeled particles in single crystals can be accompanied by energy transitions between the levels of transverse motion of the channeled particle. The Raman scattering of photons at a quasibound channeled particle leads to the appearance of a combination of frequencies: the incident radiation frequency ω₀ and the frequency Δω_{m, n}, i.e., ω = ω₀ ± Δω_m,n where Δω_m,n = 2Δε_m,nγ²; Δε_{m, n} is the energy of the transition between quantum states (m and n) of the transverse motion of the channeled particle; and γ = E/mc² is the Lorentz factor of the channeled particle. The appearance of a violet satellite (the anti-Stokes component) in the Raman scattering spectrum is analyzed. The three-photon Raman-type transition, which is the process of the simultaneous absorption of two photons with the frequency ω₀ with the emission of a photon with the frequency ωs = 2ω₀ ± 2Δε_m,nγ², is considered. The conditions for resonance observation during the formation of the second harmonic (ω = 2ω₀) are discussed.     

KMS condition in a Schrödinger picture of the dynamics

Given the C¹-algebra $\text{A}$ of observables, the KMS condition is formulated in terms of the time evolution α¹_t of a set $\text{S}$₀ ? $\text{S}$($\text{A}$) of “physical” states subject to certain natural conditions. α¹_t need not be defined by an automorphism group of $\text{A}$. It is shown that, for a KMS state ω, α¹_t induces a 1-automorphism α^ω_t of the von Neumann algebra π_ω($\text{A}$)″ generated by the representation π_ω belonging to ω.     

The determination of hopping rates and carrier concentrations in ionic conductors by a new analysis of ac conductivity

The a.c. conductivity σ(ω) of ionic materials takes the form, σ(ω) = σ(0) + Aωⁿ. The carrier hopping rate, ω_p, is obtained from the new expression σ(0) = A ω_pⁿ and the carrier concentration is estimated from σ(0). The contribution of creation and migration terms to the activation energy for conduction may be determined from the thermal activation of σ(0) and ω_p and the corresponding entropy terms quantified. Data have been analyzed for four widely different ionic materials: single crystal Na β-alumina, polycrystalline Li₄SiO₄, Ag₇l₄AsO₄ glass and Ca(NO₃)₂/KNO₃ glass and melt. For each, the carrier concentration and hopping rates have been obtained.     

Asymptotic properties of the solutions of a differential equation appearing in QED and QCD

Stimulated by the study described in the preceding paper, we establish the asymptotic behaviour of the ratio h′(0)/h(0) for g → ∞, where h(r) is a solution, vanishing at infinity, of the differential equation h″(r) = igω(r)h(r) on the domain 0 ≤ r ≤ ∞ and ω(r) = (1 − √rK₁(√r))/r. Some results are valid for more general ω's.     

Measurement of the surface density of states by field ionization

It is shown that recent measurements of field ion energy distributions from clean tungsten surfaces probe the density of metal states in the vicinity of the surface. We find j(ω) = (2π/kh)Σ_m| ∫ d³rψ_m(r)γ_z|²δ(ω??_m), where j(ω) is the ion current a ω, ψ_mand ?_m are electronic metal eigenfunctions and eigenvalues in the presence of the external electric field used in field ionization and γ(z) is a function which is large near the noble gas atom. An explicit expression for γ(z) is given in the text. It is estimated that tungsten metal states with values of k₆ at least as large as 0.5 Å^?1 make an appreciable contribution to j(ω) where k₆ is the electron momentum parallel to the surface.     

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.     

What is measured in a time-resolved stimulated Raman experiment?

We have performed four-beam time-resolved stimulated Raman measurements on liquid CS₂. To interpret our results we have applied the response formalism for the third-order polarization to such an experiment. It turns out that the measured quantity is proportional to (a convolution of) Abs[χ⁽³⁾(-ω_s, ω_s, ω_l, -ω_l)]. As a consequence in this four-beam e xperiment one picks up electronic contributions in contrast to a two-beam experiment where Im[χ⁽³⁾] is observed.     

The permittivity of a monatomic gas with spatial dispersion

The longitudinal, ε^l(ω, k), and transverse, ε^tr(ω, k), permittivities of a monatomic gas were calculated. The frequency ranges in which the permittivity ε(ω) and permeability μ(ω) of a gas without spatial dispersion have a physical meaning were determined. The limiting magnetic susceptibility χ(ω) at ω=0 and the static magnetic susceptibility were found. The question of whether an electromagnetic wave with antiparallel group and phase velocities can propagate through a monatomic gas is discussed.     

Two-dimensional wigner crystal under magnetic field

Dispersion relations of phonons are obtained in the harmonic approximation for the two-dimensional Wigner crystal under the magnetic field. For small wave vector, the eigen-energies of two modes are ω₊ = ω_c = eH/mc and ω_? → ω_lω_t/ω_c, where ω_l and ω_t are frequencies of longitudinal and transverse phonons in the absence of the field.     

Bi-solitons for a class of non-linear equations with quadratic non-linearity. I

We consider the class of non-integrable, non-linear equations,

$\text{L}{}_{\mathrm{q}}\text{K=K}{}^2\text{, L}{}_{\mathrm{q}}\text{=? +}\text{∑}\text{1?i+j?q}\text{aij?}{}^{\mathrm{i}}{}_{\mathrm{x}}{}^{\mathrm{i}}\text{?}{}^{\mathrm{j}}{}_{\mathrm{t}}{}^{\mathrm{j}}\text{, ?≠0,}$

in 1+1 dimensions. We seek rational solutions K(ω₁,ω₂), which we call bi-solitons, with exponential type variables ω_i = exp(γ_ix + ρ_it). In this paper, we restrict to q = 2 and 3, and investigate the general q case in the following paper. We find that these bi-solitons exist when the operator L_q (with ± ?) can be factorized as the product of smaller order differential operators. Besides the trivial factorized bi-solitons, we show that there exist non-trivial ones whenever K may be written as Σ^lmaxx ω^l₂F_l(Z = ω₁ + ω₂). In order to understand the origin of the factorization property, to any polynomial K = Σω^l₂F_l(Z) we associate a linear transformation such that L_qK has only the power ω^l₂ of K². For q = 2 and 3, we find that there exist particular polynomials of this type restraining L_q to be a product of smallr order operators. For the full non-linear equations we verify that all the bi-solitons can be obtained from these particular polynomials.     

An approach to radially symmetrical solutions of the sine-Gordon equation

The solution φ(r, t) of the radially symmetric sine-Gordon equation is considered in three and two spatial dimensions for initial curves, analogous to a 2π-kink, in the expanding and in the shrinking phase, for R(t)j? R(0). It is shown that the parameterization φ(r, t) = 4 arcian exp[γ(r?R(0)] + x(r, t), where R(t) describes the exact propagation of the maximum of φ,(r, t), is suitable. Using an appoximate differential equation, recently given for the propagation of the solitary ring wave, a rough analytic approximation for the correction function x(r = R(t), t) is found and tested numerically. A relationship between the fluctuations in x(r = R(t), t) and those in $\text{R}\text{?}\text{(t), t)}\text{and}\text{R(t)}$ explains why the solitary wave is almost stable. From x(r = R(t), t) and the supposition x(1, t) ≈ x(∞, t) ≈ 0 an assymetry in φ_r(r, t) with respect to r = R(t) is predicted. It also exhibits fluctuations corresponding to those in x(r = R(t), t). The condition for validity of this approximation apparently is also a limit for the stability of the solitary ring wave.     

On the dynamic conductivity for an electron-impurity system

The correlation function formula for the dynamic conductivity of a system of non-interacting electrons in the field of impurities is analyzed in terms of proper connected diagrams. By selecting those diagrams appropriate in the region of weak coupling and low impurity concentration, a set of coupled equations for the energy broadening γ (ω, ε, n_s) and the energy shift Δ(ω, ε, n_s) is derived, where both γ and Δ depend on the frequency ω of a probing field, the energy ε of the electron, and the concentration, n_g, of impurities. With the assumption of a finite range potential, these equations are solved. It is found that γ (ω, n_s) is smaller than that extrapolated value which the conventional expression γ₀ for the low-concentration collision frequency would predict, in the entire region studied, that the difference γ₀-γ becomes appreciable when the ratio of the average time between scatterings, τ_c, to the average duration of a scattering, τ_d, is 100 or less, that γ (ω, n_s) decreases monotonically from its static value γ (0, n_s), and becomes vanishingly small in the region ω≈1/τ_d, and that in the static limit (ω=0), γ=γ₀[1?(2/π) (γ₀τ_d)+…], that the energy shift Δ is positive, and increases from 0 and reach a peak of magnitude γ₀ as ω is raised from 0. By using the γ and Δ obtained, the dynamic conductivity σ(ω, n_s) for degenerated electrons is calculated. The deviation, σ-σ₀, from the conventional expression σ₀=(?i) (n_ee²/M) [ω-iΓ₀]^?1, (n_e]=number density of electrons), for 0°K, is appreciable when the ratio τ_c/τ_d is 100 or less. The field-term correction, which arises from the modification of the scattering due to the probing field, is found to be negligible in the entire region studied.     

Non-adiabatic transitions for random processes in a two-level system

Non-adiabatic transitions in two-level systems are investigated theoretically for a random time dependence of ?ω, the energy difference, between the levels. We assumed that ω = ω(x) and the coordinate x = x(t) is a random function of time. Diffusion and Poisson processes (both homogeneous and with a source) for x(t) were assumed. The cases of linear crossing terms (ω = γx) and non-linear terms (ω = ω _e exp (- αx) + ω₀) were considered. Values of the non-adiabatic transition probability per unit time were obtained by perturbation theory for ω₁τ _c ? 1 where τ _c is the correlation time and ω₁ is the off-diagonal matrix element.     

Bi-solitons for a class of non-linear equations with quadratic non-linearity. II

We generalize to any order q, the methods developed in a companion paper for q = 2,3 for finding bi-solitons, solutions of the class of non-integrable non-linear equations L_qK = K²; L_q = ? + Σ_i+j≤qa_ij?_xⁱ?_lⁱ, ? ≠ 0 in 1 + 1 dimensions. We call bi-solitons K(ω₁,ω₂) of the exponential type variables ω_i = exp(γ_ix + ρ_it), i = 1,2 and deal only with the so-called “non trivial” solutions which may be written as a finite sum K = Σ^lmax₀ω¹₂F_i(Z)_, F₁ rational function of Z = ω₁Z = ω₁ + ω₁. To any such polynomial K, we associate a linear transformation such that L_qK has only the power ω¹₂ of K² and we find that there are particular polynomialswhere the above restriction provide a factorization of the linear operator L_q in the product of smaller order differential operators. After this linear phase, we show in a second step that these forms yield solutions for the full non linear equation which can be derived in an intrinsic manner. Examples in the monomial and binomial cases are given.     

Kotani theory for one dimensional stochastic Jacobi matrices

We consider families of operators,H _ω, on ?₂ given by (H _ω u)(n)=u(n+1)+u(n?1)+V _ω(n)u(n), whereV _ω is a stationary bounded ergodic sequence. We prove analogs of Kotani's results, including that for a.e. ω,σ_ac(H _ω) is the essential closure of the set ofE where γ(E) the Lyaponov index, vanishes and the result that ifV _ω is non-deterministic, then σ_ac is empty.     

Estimation of the influence of ionic dielectricity on the dynamic superconducting order parameter

We consider the influence of an ω-dependent ionic dielectric constant ?(ω) on the properties of a superconductor. Assuming that the pairing interaction is proportional to ?² we have solved the Eliashberg equations for this case, both for imaginary and real frequencies. The interaction potential depends on a coupling constant λ and on a longitudinal phonon frequency Ω. The dielectric constant is assumed to be independent of wavevector q, and to depend on frequency through the expression: ?(ω) = (ω² - ω²_long)/(ω² - ω²_trans), where ω_long, ω_trans are the frequencies of optical phonons of the dielectric. We find that along the imaginary frequency axis (but not for real frequencies) the weighted phonon propagator can be modeled by an appropriate choice of a cutoff frequency and an effective coupling constant. The influence of ?(ω) on T_c, the gap δ(ω), and the renormalization function Z(ω) are studied and it is found that these quantities increase significantly with the dielectric constant.     

Radiation transfer in an anisotropically scattering plane-parallel medium with space-dependent albedo ω(x)

A method of analysis is presented for solving radiation-transfer problems involving space-dependent albedo ω(x) for an absorbing, emitting and anisotropically scattering plane-parallel medium with reflecting boundaries. The albedo is represented in terms of Legendre polynomials in the form ω(x) = Σ^R_r=0D_rP_r(x/L), where x is the optical variable, L is the half optical-thickness of the slab, P_r(x/L) are the Legendre polynomials and D_r are known expansion coefficients. The effects of spatial variation of albedo on the reflectivity and transmissivity of a medium having a slab geometry are examined for the cases of both forward and backward anisotropic scattering over a wide range of system variables. The effects of ω(x) on the angular distribution of radiation are also shown for some representative cases.     

The effect of the spatial variation of the order parameter on the tunneling density of states of superconducting alloys

In the first part of the paper we derive expressions of the Ginzburg-Landau (GL) type for the local tunneling density of states of superconducting alloys. These expressions are quite generally applicable at high excitation energies. One can see immediately that the density of states,N(r, ω), at any positionr and high energiesω is always larger than the local BCS density of states if the space dependence of the order parameter is governed by the GL-equation. This effect is largest for long mean free pathsl. In the second part of the paper we calculate the spatial average of the density of states,¯N, at all energiesω for a lattice of vortex lines in a magnetic field slightly below the upper critical field. The resulting curve of [¯N? N(0)]/N(0) versus co shows no gap and has a zero at about the gap value in zero field. Its value at ω=0 depends onl like ln(ξ₀/l) for l?ξ₀ [N(0) denotes the normal density of states, and ξ₀ is the BCS coherence length].     

Lower bounds on the slope parameter

A rigorous lower bound on the slope parameter γ(s, t) = d ln A(s, t)/dt is derived for 0 < t < t₀ where A(s, t) is the absorptive part of the elastic scattering amplitude and t₀ is related to the right extreme of the Lehmann-Martin ellipse. When A(s, t) has high-energy behavior like s^α(t)ln^η(t)s…, this lower bound on α(s, t) is used to obtain lower bounds on α′(t) for 0 < t < t₀, which saturate for ‘parabolic trajectories’. We also obtain a lower bound on γ(s, t) for t < 0 which can be used to find the nearforward region in which γ(s, t) cannot vanish.     

Long-wavelength excitations in a Bose gas at zero temperature

The long-wavelength excitations in a simple model of a dilute Bose gas at zero temperature are investigated from a purely microscopic viewpoint. The role of the interaction and the effects of the condensate are emphasized in a dielectric formulation, in which the response functions are expressed in terms of regular functions that do not involve an isolated single-interaction line nor an isolated single-particle line. Local number conservation is incorporated into the formulation by the generalized Ward identities, which are used to express the regular functions involving the density in terms of regular functions involving the longitudinal current. A perturbation expansion is then developed for the regular functions, producing to a given order in the perturbation expansion an elementary excitation spectrum without a gap and simultaneously response functions that obey local number conservation and related sum rules.Explicit results to the first order beyond the Bogoliubov approximation in a simple one-parameter model are obtained for the elementary excitation spectrum ω_k, the dynamic structure function $\text{S}$(k, ω), the associated structure function $\text{S}$_m(k), and the one-particle spectral function $\text{A}$(k, ω), as functions of the wavevector k and frequency ω. These results display the sharing of the gapless spectrum ω_k by the various response functions and are used to confirm that the sum rules of interest are satisfied. It is shown that ω_k and some of the $\text{S}$_m(k) are not analytic functions of k in the long wavelength limit. The dynamic structure function $\text{S}$(k, ω) can be conveniently separated into three parts: a one-phonon term which exhausts the f sum rule, a backflow term, and a background term. The backflow contribution to the static structure function $\text{S}$₀(k) leads to the breakdown of the one-phonon Feynman relation at order k³. Both $\text{S}$(k, ω) and $\text{A}$(k, ω) display broad backgrounds because of two-phonon excitations. Simple arguments are given to indicate that some of the qualitative features found for various physical quantities in the first-order model calculation might also be found in superfluid helium.