Holstein Equation Research Articles

Radiative transfer of excitation: Lévy flights and self-similarity

The Green function of the Biberman–Holstein integral equation, both nonstationary (instantaneous point flash) and stationary (stationary point source) is studied. The point flash produces a self-similar spherical wave of excitation in homogenous gas medium. The transfer of excitation is due to the Lévy flights of resonance line photons. The properties of the wave – its shape, time evolution and propagation – are found for arbitrary line profile. The cases of Doppler, Lorentz and Voigt profiles are considered in some detail. For the Doppler broadened line, the structure of the self-similar wave is given by a rational function. The properties of the Green function of stationary Biberman–Holstein equation are reviewed. Some unexpected parallels with non-stationary case are found. The spherically-symmetric version of the longest flight approximation is discussed and the condition of its validity is formulated.

Atoms and Photons: Kinetic Equations with Delay

In his recent works, the author drew attention to the fact that the extraction of a part of a closed Hamiltonian system turns the original well-known Liouville differential equation into an integro-differential equation with a delayed time argument, which describes the dynamics of the selected subsystem as an open system. It was shown that the integral operator can be represented in the form of a fractional differential operator of distributed order. In this paper, we show how the kinetic theory of the system “atoms+photons” is transformed for the subsystem consisting of excited atoms. We present the derivation of the telegraph equation with delay, derive the Biberman–Holstein equation in the fractional differential form (with the fractional Laplace operator), and discuss boundary effects in the nonlocal transport model. The final section is devoted to laser technologies, which include free-electron lasers and laser cooling of atoms.

Self-Similar Solutions in the Theory of Nonstationary Radiative Transfer in Spectral Lines in Plasmas and Gases

Radiative transfer (RT) in spectral lines in plasmas and gases under complete redistribution of the photon frequency in the emission-absorption act is known as a superdiffusion transport characterized by the irreducibility of the integral (in the space coordinates) equation for the atomic excitation density to a diffusion-type differential equation. The dominant role of distant rare flights (Lévy flights, introduced by Mandelbrot for trajectories generated by the Lévy stable distribution) is well known and is used to construct approximate analytic solutions in the theory of stationary RT (the escape probability method is the best example). In the theory of nonstationary RT, progress based on similar principles has been made recently. This includes approximate self-similar solutions for the Green’s function (i) at an infinite velocity of carriers (no retardation effects) to cover the Biberman–Holstein equation for various spectral line shapes; (ii) for a finite fixed velocity of carriers to cover a wide class of superdiffusion equations dominated by Lévy walks with rests; (iii) verification of the accuracy of above solutions by comparison with direct numerical solutions obtained using distributed computing. The article provides an overview of the above results with an emphasis on the role of distant rare flights in the discovery of nonstationary self-similar solutions.

Transfer and Trapping of Resonance Radiation in a Two-Level System

The population kinetics of a two-level system has been theoretically investigated taking into account the transfer of spontaneous radiation under continuous optical pumping. It is shown that, when the photon mean free path in a resonantly absorbing medium is smaller than the characteristic size of this medium or comparable to it, the spontaneous radiation may become an additional pump source. The contribution of spontaneous photons to the kinetics of level has been considered within two approaches. The first approach consists of solution of the radiation transfer equation within the diffusion approximation. The second approach is related to solution of the Biberman–Holstein equation.

Automodel solutions for Lévy flight-based transport on a uniform background

A wide class of non-stationary superdiffusive transport on a uniform background with a power-law decay at large distances of the step-length probability distribution function (PDF) is shown to possess an approximate automodel solution. The solution for the Green’s function is constructed using the scaling laws for the propagation front (relevant-to-superdiffusion average displacement) and asymptotic solutions far beyond and far in advance of the propagation front. These scaling laws are determined essentially by the long-free-path carriers (Lévy flights). The validity of the suggested automodel solution is proved by its comparison with numerical solutions in the one-dimensional (1D) case of the transport equation with a simple long-tailed PDF with various power-law exponents and in the 3D case of the Biberman–Holstein equation of the resonance radiation transfer for various (Doppler, Lorentz, Voigt and Holtsmark) spectral line shapes.

The influence of resonance radiation transport on the contraction of a glow discharge in argon

The role of resonance radiation transport in the contraction of a positive column in an argon glow discharge is studied numerically. The theory is based on the self-consistent solution of the ambipolar diffusion equation for electrons, the diffusion equation for metastable atoms and the Biberman–Holstein equation for resonance atoms. To calculate the ionization and excitation rates, the Boltzmann equation is solved in a local approximation taking into account elastic, inelastic and electron–electron collisions. A solution method for a boundary problem is developed which allows one to obtain a hysteresis of the parameters during a continuous transition from a diffuse mode to a contracted mode through an unstable branch. At small currents there is a diffuse discharge where the role of radiation transport is inessential because the radial distributions of electrons and excited atoms are close to the fundamental modes of the corresponding equations. Under these conditions, the traditional approximation of ‘effective lifetime’ is accurate enough. For a contracted discharge, this approximation is not applicable because the higher diffusion and radiation modes play a notable role and a more strict description of radiation transport is required. It is shown that, when radiation transport is taken into account, the width of a filament in a contracted discharge significantly exceeds that obtained in the traditional ‘effective lifetime’ approximation. The critical current, when the discharge abruptly turns into a contracted mode, is shifted towards higher current values. The results obtained in this paper can also relate to a discharge in other inert gases.

The role of visible and resonance radiation in the energy balance of LTE plasma in argon

Energy losses due to radiation in an LTE arc plasma in argon are investigated in the temperature range from 5000 to 15 000 K. Calculations of the radiation transport require us to know the absorption spectra; for that purpose, free–free, bound–free and bound–bound transitions are taken into account. The energy losses due to resonance transitions with large absorption coefficients are analyzed. The transport of visible radiation with small absorption coefficients is calculated by accurate integration over a whole spectrum. The resonance radiation transport in LTE plasma is described by the Biberman–Holstein equation, which is usually used in non-equilibrium plasma. As the temperature grows, radiation processes become the main channel of energy losses. It is shown that, despite the resonance radiation being trapped, the energy losses due to resonance radiation escape can achieve 10% of the total radiation losses.

Glomerular filtration rate in cows estimated by a prediction formula.

To testify the relevance of Jacobsson's equation for estimating bovine glomerular filtration rate (GFR), we prepared an integrated formula based on its equation using clinically healthy dairy (n=99) and beef (n=63) cows, and cows with reduced renal function (n=15). The isotonic, nonionic, contrast medium iodixanol was utilized as a test tracer. The GFR values estimated from the integrated formula were well consistent with those from the standard multisample method in each cow strain, and the Holstein equation prepared by a single blood sample in Holstein dairy cows. The basal reference GFR value in healthy dairy cows was significantly higher than that in healthy beef cows, presumably due to a breed difference or physiological state difference. It is concluded that the validity for the application of Jacobsson's equation to estimate bovine GFR is proven and it can be used in bovine practices.

Technical note: Use of a simplified equation for estimating glomerular filtration rate in beef cattle1

This study was performed to clarify whether a formula (Holstein equation) based on a single blood sample and the isotonic, nonionic, iodine contrast medium iodixanol in Holstein dairy cows can apply to the estimation of glomerular filtration rate (GFR) for beef cattle. To verify the application of iodixanol in beef cattle, instead of the standard tracer inulin, both agents were coadministered as a bolus intravenous injection to identical animals at doses of 10 mg of I/kg of BW and 30 mg/kg. Blood was collected 30, 60, 90, and 120 min after the injection, and the GFR was determined by the conventional multisample strategies. The GFR values from iodixanol were well consistent with those from inulin, and no effects of BW, age, or parity on GFR estimates were noted. However, the GFR in cattle weighing less than 300 kg, aged<1 yr old, largely fluctuated, presumably due to the rapid ruminal growth and dynamic changes in renal function at young adult ages. Using clinically healthy cattle and those with renal failure, the GFR values estimated from the Holstein equation were in good agreement with those by the multisample method using iodixanol (r=0.89, P=0.01). The results indicate that the simplified Holstein equation using iodixanol can be used for estimating the GFR of beef cattle in the same dose regimen as Holstein dairy cows, and provides a practical and ethical alternative.

A study of collisional quenching and radiation-trapping kinetics for Rb(5p) in the presence of methane and ethane using time-resolved fluorescence

An experimental study using time-resolved fluorescence techniques together with theoretical simulations has been conducted and used to determine the quenching cross-sections of rubidium–methane and rubidium–ethane. Radiation trapping was significant under many of the experimental conditions (temperatures 40–130 °C and pressures 50–700 Torr) and a detailed analysis of the interplay between radiation trapping and quenching kinetics was carried out. Modifications of the Holstein equation for radiation trapping were implemented to account for the quasi-2 level behaviour of the Rb atom for high buffer gas pressures, the absolute frequency-dependent absorption cross-section for Rb with variable buffer gas pressures which accounts for the hyperfine splitting of 87Rb and modification of the trapping factors so that radiation trapping and quenching by an additive quenching gas could be treated simultaneously. Experimental results supported by theoretical simulations bound the quenching cross-sections (σ) of methane and ethane at 40 °C to be σ ⩽ 0.019 Å2 and σ ⩽ 0.033 Å2, respectively. These values are nearly two orders of magnitude smaller than previously reported.

Simulation of photons from plasmas for the applications to display devices

Numerical modeling of the photon transport of the ultraviolet (UV) and the visible lights are presented for plasma based display devices. The transport of UV lights which undergo resonance trapping by ground state atoms is solved by using the Holstein equation. After the UV lights are transformed to visible lights at the phosphor surfaces, the visible lights experience complicated traces inside the cell and finally are emitted toward the viewing window after having some power loss within the cell. A three-dimensional ray trace of the visible lights is calculated with a radiosity model. These simulations for the photons strengthen plasma discharge modeling for the application to display devices.

A radiation transport coupled particle-in-cell simulation. II. Simulation results in a one-dimensional planar model

The radiation-transport coupled particle-in-cell model is applied to one-dimensional planar model to investigate the radiation trapping phenomena in relatively high- and low-pressure Ar glow discharges. The radiation intensity spectra and the total radiation flux are calculated from the radiative excited-state profile. The simulation results for Doppler and resonance collision broadenings are compared with the eigenmode description of the Holstein equation, with good agreement. Observed are the effects of radiation trapping of photons, diffusion, collisional quenching, and step ionization from excited states on the discharge properties.

Solution of the Holstein equation of radiation trapping by the geometrical quantization technique. III. Partial frequency redistribution with Doppler broadening

We introduce an analytical method to investigate radiation trapping problems with Doppler frequency redistribution. The problem is formulated within the framework of the Holstein-Biberman-Payne equation. We interpret the basic integro-differential trapping equation as a generalized wave equation for a four-dimensional (4D) classical system (an associated quasiparticle). We then construct its analytical solution by a semiclassical approach, called the geometrical quantization technique (GQT). Within the GQT, it is shown that the spatial and frequency variables can be separated and that the frequency part of the excited atom distribution function obeys a stationary Schr\"odinger equation for a perturbed oscillator. We demonstrate that there is a noticeable deviation of the actual spectral emission profile from the Doppler line in the region of small opacities. The problem of calculating the spatial mode structure and the effective radiation trapping factors is reduced to the evaluation of wave functions and quantized energy values of the quasiparticle confined in the vapor cell. We formulate the quantization rules and derive the phase factors, which allow us to obtain analytically the complete spectrum of the trapping factors in 1D geometries (layer, cylinder, sphere) and other (2D and 3D) geometries when the separation of space variables is possible. Finally, we outline a possible extension of our method to treat radiation trapping effects for more general experimental situations including, for instance, a system of cold atoms.

Solution of the Holstein equation of radiation trapping by the geometric quantization technique. II. Two- and three-dimensional geometries

We solve the Holstein equation describing radiation trapping in an atomic vapor by the geometric quantization technique. The treatment is based on studying the integral trapping equation as a wave equation for an associated quasiparticle with a complicated form of its dispersion law. The latter is determined by the spectral properties of the vapor medium. Extending our previous work, which dealt with one-dimensional (1D) geometries, we consider here more realistic two- and three-dimensional (2D and 3D) vapor cell geometries. For this, an explicit representation of the phase factors is derived when the quasiparticle is reflected under arbitrary angles from the surface of the cell confining the vapor. We give closed-form equations to obtain analytically the complete 3D spectrum of escape factors in finite cylinder, sphere, parallelepiped, and finite prismlike geometries for all practically occurring line shapes and opacities. In addition, the relevant semiclassical variational method is applied to develop a perturbation theory for the evaluation of escape factors in more complicated geometries that are sometimes adopted in experiments, including infinite (or finite) elliptical cylinders, prolate and oblate ellipsoids. Comparisons with numerical calculations show that results are accurate within $5%$ for the ground modes and even better for higher-order modes.

Observation of radiation trapping in a dense Cs magneto-optical trap

We present an experimental and theoretical investigation of the resonant radiation trapping phenomenon in a magneto-optical trap (MOT) for cesium atoms. We have resonantly excited the 6 2S 1/2( F=4)−6 2P 3/2( F′=5) transition with a probe laser and measured directly its effective radiative lifetime. We compare our experimental data with a model based on an approximate solution of the Holstein equation and with a Monte Carlo simulation of the process. The inadequacy of the standard theory for thermal atomic samples in reproducing the data is pointed out.

Solution of the Holstein equation of radiation trapping in one-dimensional geometries by the geometric quantization technique

We solve the Holstein equation of radiation trapping in an atomic vapor cell by the geometric quantization technique (GQT). In the GQT, the rate equation for the excited-state density is transformed into an ``equivalent'' Schr\odinger equation for an associated quasiparticle. The problem of finding the complete set of radiation escape factors is thus reduced to searching quantized energy values for the quasiparticle locked into the vapor cell. We combine already known solutions for the trapping factors at high opacities with new results for the phase jump at the vapor cell boundary to arrive, within the framework of the GQT technique, at solutions that are valid at all opacities. The phase factors are independent of geometry, and we derive an explicit representation for them by the Wiener-Hopf technique in the most simple geometry, a half-space. Our approach enables an analytical computation of the trapping factors in all practically occurring line shapes (including Voigt lines and hyperfine-split lines), all opacities, and all modes in one-dimensional (1D), 2D, and 3D geometries allowing for variable separation. We present results obtained in 1D geometries at all opacities that show discrepancies within 5% for the lowest-order trapping factor and even less (at the level of approximately 0.1%) for higher-order modes, in agreement with the predictions of the GQT theory that has been developed.

THEORETICAL TREATMENT OF RADIATION TRAPPING IN A MULTIPLE LINES COUPLED SYSTEM

The purpose of this paper is to propose a theoretical treatment of radiation trapping in a multiple lines coupled system in which coupling includes collisional energy transfer and cross absorption between the excited states. First, to describe the coupled system, the Holstein equation of radiation trapping is extended to an integral differential master equation group in which the vector and matrix form is adopted. Then by use of operator manipulations, the asymptotic solution in the form of a series expansion in terms of operators is presented. Finally, the asymptotic solution is reduced, leading to a result which can not only give the system a complete description, but also has far more advantages than the simple cut off solution. A numeral illustration of a two excited states system is also presented.

Linearized quantum transport equations: ac conductance of a quantum wire with an electron-phonon interaction.

A linear response formalism is developed for evaluation of transport coefficients in quantum many-particle systems. The method is based on a systematic linearization of nonequilibrium Kadanoff-Baym transport equations in an integral form in ac electric fields. Simple and consistent approximations can be performed in the resulting set of transport equations, which replace vertex equations in the Kubo formalism. The method is illustrated by the example of an ac and dc conductivity and conductance for a quasi-one-dimensional electron system with an electron--phonon interaction for three-dimensional acoustical phonons. After approximations standard in metals, the solved transport equations coincide with the Holstein equation. \textcopyright{} 1996 The American Physical Society.

Spatial effects of radiation trapping in an optically thick atomic vapor excited by a laser beam

We describe the influence of radiation trapping on the spatial distribution of the excited-state population in sodium vapor, interacting with a linearly polarized Gaussian laser beam. Counterintuitively the excitation volume experiences a strong spatial shrinking at high optical depth. The interpretation of this nonlocal effect is based on the numerical solution of the stationary Holstein equation, which takes into account the full frequency dependence of the reabsorption probability within the process of diffusion of radiation. The calculations performed are in good qualitative agreement with the spatial effects observed experimentally. \textcopyright{} 1996 The American Physical Society.

Radiation trapping in a saturated atomic vapor in cylindrical geometries

We treat radiation trapping in a saturated atomic vapor in a cylindrical vapor cell. When saturation (i.e. stimulated emission) sets in, the Holstein equation of radiation trapping becomes nonlinear, and the absorption coefficient can become inhomogeneous. We first present a new method for the efficient computation of the reabsorption rate with an inhomogeneous distribution of absorbers in a cylinder. We then use this method as the basis of an algorithm for the computation of trapping in a saturated vapor.