Planck Distribution Research Articles

Lattice-Boltzmann modeling of phonon hydrodynamics

Based on the phonon Boltzmann equation, a lattice-Boltzmann model for phonon hydrodynamics is developed. Both transverse and longitudinal polarized phonons that interact through normal and umklapp processes are considered in the model. The collision term is approximated by the relaxation time model where normal and umklapp processes tend to relax distributions of phonons to their corresponding equilibrium distribution functions-the displaced Planck distribution and the Planck distribution, respectively. A macroscopic phonon thermal wave equation (PTWE), valid for the second-sound mode, is derived through the technique of Chapman-Enskog expansion. Compared to the dual-phase-lag (DPL) -based thermal wave equation, the PTWE has an additional fourth-ordered spatial derivative term. The fundamental difference between the two models is discussed through examining a propagating thermal pulse in a single-phased medium and the transient and steady-state transport phenomena on a two-layered structure subjected to different temperatures at boundaries. Results show that transport phenomena are significantly different between the two models. The behavior exhibited by the DPL model, as thermal wave behavior goes over to diffusive behavior, tau_{T}-->tau_{q} is incompatible with any microscopic phonon propagating mode. Unlike the DPL model, in which tau_{T} only has an effect on the transient phenomena, in the PTWE model tau_{T} shows effects on phenomena at both transient and steady state. With the intrinsic compatibility to the microscopic state, discontinuous quantities, such as a jump of temperature at a boundary or at an interface, can be calculated naturally and straightforwardly with the present lattice-Boltzmann method.

Effect of a Planckian radiation field on population of bound-electrons

The basic atomic physical processes of laser beams interacting with a solid target is checked and the effects of a Planckian radiation field on the population of bound electrons are studied. The results illustrate that the radiation field, if it has a truly Planckian distribution, will influence on the physical processes in plasmas. For instance, it can ensure the distribution of the population on the different ionization stages under the LTE conditions even when the plasmas density is low enough that the collision rates do not dominate over radiative rates.

Semi-analytical formulas for the Hertzsprung-Russell diagram

The absolute visual magnitude as function of the observed color (B-V), also named Hertzsprung-Russell diagram, can be described through five equations; that when calibrated stars are available means eight constants. The developed framework allows to deduce the remaining physical parameters, mass, radius and luminosity. This new technique is applied to the first 10 pc, the first 50 pc, the Hyades and to the determination of the distance of a cluster. The case of the white dwarfs is analyzed assuming the absence of calibrated data: our equation produces a smaller ?2 with respect to the standard color-magnitude calibration when applied to the Villanova Catalog of Spectroscopically Identified White Dwarfs. The theoretical basis of the formulas for the colors and the bolometric correction of the stars is clarified by a Taylor expansion in the temperature of the Planck distribution.

An existence result for a quantum BGK model

A class of kinetic BGK models for Bose–Einstein statistics are considered for the stationary frame in a slab. Existence of bounded measure solutions is proven for Planckian diffuse boundary conditions with a given total inflow. Compactness properties are extracted from the L 1 part of the generalized Planckian distribution function and the boundary behavior.

Non-Planck equilibrium radiation in plasma model of early Universe

Consideration of the adiabatic character of radiation expansion in early Universe leads to the conclusion that equilibrium distribution of the primordial radiation in the presence of charged particles could be different from the Planck distribution in some regions of the spectrum. The equilibrium distribution of electromagnetic radiation (the black body radiation) is generalized for the system containing an extremely dense fully ionized plasma. The conditions of the adiabatic expansion of radiation for the model of the early Universe are found.

Emission spectroscopy of ablation plumes in liquid for analytical purposes

The emission intensity of Ag(I) lines obtained by the laser ablation of an Ag plate immersed in water was examined. Due to an intense self-absorption in the plume, the spectral intensity of the emission approaches the Planck distribution, which is a function of temperature, but not a function of the population of the levels involved in the transition. The temperature obtained by assuming the Planck intensity agrees well with the previous study on the plume temperature in water. The result that the emission suffers from the intense self-absorption is a disadvantage for the use of emission spectra from the laser ablation plume in liquid as a quantitative elemental analysis of solid surfaces in liquid.

INFRARED MEASUREMENTS OF POSSIBLE IR FILTER MATERIALS

A Fourier Transform Infrared Spectrometer (FTS) was used to obtain the transmission spectra of candidate materials for use as infrared (IR) filters in cryogenic receivers. The data cover the range from 50 cm−1 (∼1.5 THz), well below the peak of the 300 K black body spectrum, to 5000 cm−1 (∼150 THz), Z-cut quartz, Goretex, Zitex G and Zitex A, High Density Polyethylene (HDPE), Teflon (PTFE), Fluorogold and Black Polyethylene were measured. The relative effectiveness of each material as a filter is determined by integrating the transmission spectrum multiplied by the Planck distribution to obtain a normalized attenuation for the mid-IR band. Measurements at both room temperature and 8 K are compared.

The effect of modified dispersion relations on the thermodynamics of black-body radiation

In the study of loop quantum gravity and of models based on noncommutative geometry, there has been strong interest in some candidate modifications of the energy-momentum dispersion relation. In this paper we consider the effects of modified dispersion relation (MDR) on the thermodynamics of black-body radiation. We find a generalized Planck distribution and then apply it to find generalized Stefan–Boltzmann and generalized Wien’s law. As a consequence, we study the effects of these modifications on the entropy and specific heat of black-body radiation. We investigate also the relation between our findings and the results of E-infinity theory by considering distribution and temperature of cosmic microwave background radiation.

Straight-edge diffraction of Planck radiation

The irradiance diffraction profile of a straight edge is given as a Taylor series in powers of the distance from the geometrical shadow boundary to any point in the profile for monochromatic radiation. The coefficients of the series, which are obtained as simple analytic expressions, are proportional to the real part of a complex number whose phase cycles through a complete period every eight terms in the series. Integration of this series over a Planck distribution of radiation yields the power series for the Planck profile; this derived series has a finite radius of convergence. The asymptotic series for the Planck profile far from the shadow boundary and beyond the radius of convergence of its power series is obtained by analytic continuation of the power series with the aid of a Barnes type of integral representation.

Uncertainties in source distribution temperature and correlated colour temperature

Uncertainties in the distribution temperature (DT) and correlated colour temperature are estimated for common sources and typical measurement uncertainties of a spectral transfer from a reference lamp. Uncertainty sensitivity coefficients for both parameters in terms of measured values of spectral irradiance are derived using a generalized matrix inverse. The uncertainty values are compared with differences in the source temperature parameters. DT is calculated using the CIE definition; shifts in DT due to the alternative of a direct fit of Planck's distribution, and to including weights in the process, are compared with the estimated uncertainties.

A generalized planck distribution

We introduce a generalization of the standard Planck distribution discussed by Johnson and Kotz (1970, Section 33.6.1). This generalization results in a very flexible family which contains the gamma distribution as a particular case. In this paper we provide a comprehensive treatment of the mathematical properties of the family. We derive expressions for thenth moment, moment generating function, characteristic function, mean deviation about the mean, mean deviation about the median, Renyi entropy, Shannon entropy and the asymptotic distribution of the extreme order statistics. Estimation and simulation issues are also considered.

Comparison between blackbody radiation and continuous X-ray spectra produced by electron deceleration

The continuous X-ray spectral profiles are compared with the Planck's distribution law for blackbody radiation. The energy distributions of continuum X-rays produced by bombardments of 20 and 30 kV electron beams onto iron target are crudely reproduced by the blackbody radiation at about 10 7 K temperature at their absolute intensity scale.

Evolution of the Field‐Emission Properties of Individual Multiwalled Carbon Nanotubes Submitted to Temperature and Field Treatments

AbstractField‐emission (FE) electron and ion microscopies (FEM and FIM),I(V) characteristics, and FE energy spectroscopy (FEES) measurements have been made on individual multiwalled carbon nanotubes (MWNTs) at various stages of their evolution under various thermal and field treatments. Before the highest temperature treatments, the FE is produced by nanometer‐scale structures that have specific and known characteristics such as individual peaks in the total energy distributions (TEDs), highly curved Fowler–Nordheim (FN) plots, and strong local heating due to the Nottingham effect. After cleaning at up to 1600 K, the TEDs approached the well‐known form for FN tunneling from a free electron gas. As previously reported, they revealed and quantified strong heating at high FE currents induced by Joule heating along the MWNT. The TEDs gave a quantitative simultaneous measure of the temperature at the end of the MWNT,TA(which reached up to 2000 K), and its electrical resistanceR, thus allowing the determination ofR(TA).R(TA) decreased by ca. 70 % asTAincreased from 300 to 2000 K, indicating a nonmetallic behavior of the MWNT. Changes in the MWNT length due to the induced high temperatures above 1600 K could be followed using theI(V) characteristics. The high temperatures were accompanied by light emission due to incandescence for which the optical spectra was found to follow the Planck distribution. With the simultaneous measurements ofTAandR, the 1D heating problem could be reliably simulated to estimate the dependence of both electrical and thermal conductivities on temperature. This heating results in an excellent current stability by continuous desorption, and defines the maximum nanotube currents for high‐current applications

On the Cauchy problem for a quantum kinetic equation linked to the Compton effect

The Cauchy problem is studied for an homogeneous quantum kinetic equation describing the Compton effect. Since the collision kernel commonly used in physics is highly singular, numerical simulations are performed for related collision kernels to get a preliminary insight into the behavior of the solutions. Some of the numerical results are then given a theoretical explanation. Global existence of a solution to the Cauchy problem is proven when the L 1 initial data are a.e. smaller than the Planck distribution function, and non-existence of solutions to the Cauchy problem is proven when the L 1 initial data are a.e. bigger than the Planck distribution function.

Simulation of the effects of X-ray emission from accretion disks on the interstellar materials

The thickness of target for X-ray sources with approximate Planck's spectral distribution from laser-produced plasmas is optimized using a one-dimensional radiation hydrodynamics code MULTI. The effect of X-ray radiation field with Planck's distribution on the processes of excitation and ionization in a plasma is stadied using a non-LTE ionized material package NIMP. These results are very important in the future experimental design to study the influence of X-ray emission from accretion disks on interstellar matter.

Emission from an active photonic crystal

We investigated theoretically the emission from an active photonic crystal. Redistribution of photon density of states by a photonic lattice was found to greatly influence the thermal emission spectrum, resulting in substantial deviation from the Planck distribution. The calculation predicts that the photonic lattice intensity may exceed that of a blackbody source within certain spectral regions. However, the excess emission may be lost in practice because of nonradiative losses and photonic lattice inhomogeneities.

Study on two-dimensional transfer of radiative heating wave

A two-dimensional (2D) multigroup radiation transfer hydrodynamics code LARED-R-1 is used to simulate a supersonic wave experiment performed earlier by the Livermore group. The main result is that, contrary to the conclusion of Back et al. (2000a), the average-atom opacity model is sufficient to explain the obtained experimental results, provided that an adequate description of the radiation transport was used. The simulation results from LARED-R-1 show the spectrum of radiation in foam with radius and length of several optical depths is not in Planckian distribution and the angular intensity distribution is anisotropic.

Spectral Formation in X‐Ray Pulsars: Bulk Comptonization in the Accretion Shock

Accretion-powered X-ray pulsars are among the most luminous X-ray sources in the Galaxy. However, despite decades of theoretical and observational work since their discovery, no satisfactory model for the formation of the observed X-ray spectra has emerged. In particular, the previously available theories are unable to reproduce the power-law variation observed at high energies in many sources. In this paper we present the first self-consistent calculation of the spectrum emerging from a pulsar accretion column that includes an explicit treatment of the energization occurring in the shock. Using a rigorous eigenfunction expansion method based on the exact dynamical solution for the velocity profile in the column, we obtain a closed-form expression for the Green's function describing the upscattering of radiation injected into the column from a monochromatic source located at the top of the thermal mound, near the base of the flow. The Green's function is convolved with a Planck distribution to calculate the radiation spectrum resulting from the reprocessing of blackbody photons emitted by the thermal mound. We demonstrate that the energization of the photons in the shock naturally produces an X-ray spectrum with a power-law shape at high energies and a blackbody shape at low energies, in agreement with many observations of accreting X-ray pulsars.

Spectral Formation in X-Ray Pulsar Accretion Columns

We present the first self-consistent model for the dynamics and the radiative transfer occurring in bright X-ray pulsar accretion columns, with a special focus on the role of the shock in energizing the emerging X-rays. The pressure inside the accretion column of a luminous X-ray pulsar is dominated by the photons, and consequently the equations describing the coupled radiative-dynamical structure must be solved simultaneously. Spectral formation in these sources is therefore a complex, nonlinear phenomenon. We obtain the analytical solution for the Green's function describing the upscattering of monochromatic radiation injected into the column from the thermal mound located near the base of the flow. The Green's function is convolved with a Planck distribution to model the X-ray spectrum resulting from the reprocessing of blackbody photons produced in the thermal mound. These photons diffuse through the infalling gas and eventually escape out the walls of the column, forming the observed X-ray spectrum. We show that the resulting column-integrated, phase-averaged spectrum has a power-law shape at high energies and a blackbody shape at low energies, in agreement with the observational data for many X-ray pulsars.

Nature of the Optical and Ultraviolet Variability of the Nucleus of NGC 4151: A New Concept

We have analyzed the optical (U BV) and ultraviolet (λ1000–2700 A) observations of the nuclear variability of the Seyfert galaxy NGC 4151 in the period 1987–2001 (the second cycle of activity). The fast (tens of days) and slow (∼10 years) components of the nuclear variability, F and S, respectively, are shown to be completely different, but thermal in nature. We associate the S component with the formation and evolution of an accretion disk and the F component (flares) with instabilities in the accretion disk and their propagation over the disk in the form of a shock wave. The S component is present not only in the optical, but also in the ultraviolet range, with its amplitude being comparable over the entire range λ1000–5500 A under study. The amplitude of the average flare (the F component) doubles as the wavelength decreases from 5500 to 1000 A, while the rise time of the brightness to its maximum Δt (the variability time scale) decreases from Open image in new window to 6d ± 2d. The brightness decline (flare decay) time decreases by a factor of 16. The extinction in the ultraviolet is shown to have been grossly underestimated: beginning from the first IUE data, only the extinction in our Galaxy, Open image in new window , has been taken into account. A proper allowance for the total extinction, i.e., for the extinction in the nucleus of NGC 4151 as well Open image in new window leads to a large increase in the luminosity of the variable source in the nucleus of NGC 4151: L = (6–8) × 1046 erg s−1. The spectral energy distribution for the variable source (λ950–5500 A) agrees well with two Planck distributions: Te = 65 000 (λmax = 450 A) and 8000 K. The radiation with Te = 8000 K is the reprocessing of the bulk of the ultraviolet radiation by the accretion disk with a lag of 0.5–0.6 days in the V band. The lag in the U-B variability of the slow component revealed the existence of an extended broad line region (EBLR) at an effective distance of 1.5 lt-years, as confirmed by spectroscopic data obtained at the Crimean Astrophysical Observatory. This yields the following mass of the central object in NGC 4151: Mc = (1–3) × 109M⊙. The luminosity of the variable source then accounts for 50–60% of LEdd rather than 1–2%, as has been thought previously. In general, the pattern of ultraviolet and optical variability in NGC 4151 agrees excellently with the theory of disk accretion instability for a supermassive black hole suggested by N. Shakura and R. Sunyaev 30 years ago: the energy release is at a maximumin the ultraviolet (in the case under consideration, at λ450 A), the luminosity is ∼1047 erg s−1 for Mc ∼ 109M⊙ (several tens of percent of LEdd), and the variability time scale ranges from several days to many years.