Equilibrium Maxwellian Distribution Research Articles

Nonlocal effects of alpha particles on ICRF heating

A model based on the linearized Vlasov-Maxwell equations, taking into account the nonlocal interactions of particles due to their finite Larmor radii has been developed. Assuming an inhomogeneous 1D slab plasma, Maxwellian equilibrium distribution functions and ky = 0, this model leads to a system of one first-order and two second-order integro-differential equations for Ex and Ey, Ez, respectively. These equations are valid for arbitrary values of k⊥ρσ, where k⊥ is the perpendicular wave number and ρσ the Larmor radius of species sigma . Therefore, the code SEMAL, which solves these equations, is appropriate for studying the effects of alpha particles on heating in the ion cyclotron range of frequencies. These effects are shown to be much less significant for heating at the second harmonic of deuterium than is expected from local models. Also investigated are other heating scenarii of deuterium, as well as the influence of kz, Tα, non-Maxwellian distribution functions and nα/ne. The results indicate under which conditions the power absorption by alpha particles begins to dominate and therefore to degrade the heating efficiency

Diode laser probing of I*(2P1/2) Doppler profiles: Time evolution of a fast, anisotropic velocity distribution in a thermal bath

The relaxation of a nonthermal translational population distribution of fast I*(2P1/2) atoms dilutely dispersed in a gaseous bath at thermal equilibrium is studied by time-resolved Doppler spectroscopy. The fast, anisotropic velocity distribution of I* atoms is produced by pulsed laser photolysis of n-perfluoropropyl iodide (n-C3F7I) at 266 nm. A frequency-narrowed, GaAsInP diode laser is tuned across the iodine (2P1/2,F=3←2P3/2,F=4) transition at 1315 nm to measure the Doppler gain profile of the I* photofragments. The velocity distribution is expressed as a separable product of a radial speed function and an angular function describing the anisotropy. The collision-induced time evolution of both the speed and anisotropy components of the nascent velocity population distribution relaxing to form a 300 K Maxwellian equilibrium distribution is determined. The thermalization dynamics of I* are studied for a heavy bath gas (n-C3F7I) and a light (He) bath gas. In the case of the heavy bath gas the anisotropy is removed by collisions 2.5 times faster than the speed is thermalized, while for the light bath gas the anisotropy and speed relaxation occur on the same time scale. The velocity and angular distributions of the I* photofragment from the 266 nm photolysis of n-C3F7I are also reported.

A nonlocal analysis of electrostatic waves in hot inhomogeneous bounded plasmas

A numerical code, seal, solving the full form of a second-order integrodifferential equation that describes electrostatic waves in a slab plasma is presented. No expansion in the smallness of the ion Larmor radius is made. The plasma may have arbitrary density and temperature profiles and is immersed in a nonuniform magnetic field. Only small magnetic field gradients, Maxwellian equilibrium distribution functions, and ky =0 are assumed. First the integral equation is derived in Fourier space using the linearized Vlasov and Poisson equations, and then it is transformed back into real space, which enables us to treat the case of bounded plasmas. The two boundary conditions specified simulate an antenna at one end of the plasma and wave-reflecting walls. Solutions having wavelengths smaller than the ion Larmor radius have been found. Comparison with experiments where ion Bernstein waves are launched in argon and barium plasmas shows very good agreement with the solution of the code seal. A positive-definite formula for the local power absorption is also derived and computed.

Electromagnetic waves in a relativistic plasma stream: fusion instability

We derive the dispersion formulae for electromagnetic waves including relativistic kinematics and a corresponding Maxwellian equilibrium distribution function without any approximations and having anisotropy in the streaming velocity. For non-relativistic temperatures, waves propagate when the streaming velocity is much smaller than the thermal velocity of the species, with varying thermal modes.

Doppler Broadened Spectral Functions and Time Correlation Functions for a Gas in Non-Equilibrium

Abstract Under non-equilibrium conditions, the velocity distribution function of a gas differs from the Maxwellian equilibrium distribution. The effect of this deviation from equilibrium on the time correlation function and the spectral function associated with a Doppler broadened spectral line is calculated in full generality for stationary non-equilibrium processes. As special cases the heat conduction and the viscous flow in the hydrodynamic regime are discussed.

Landau damping of surface waves at a plasma— vacuum interface

The exact surface-wave dispersion relation is expressed in terms of elementary functions for a plasma characterized by a ‘resonance’ velocity distribution function. An approximate form of the relation is derived for the case when the thermal velocity spread is much less than c. The pure surface wave obtained by dropping the term responsible for Landau damping is compared with that predicted on the basis of a fluid model of the plasma. The effect of Landau damping is then investigated, both by analytic approximations and by computation. Two branches of the solution to the dispersion relation are found; and it is shown that the surface wave suffers increasingly severe damping as the frequency grows beyond 1/ √ 2 times the plasma frequency. It is argued that qualitatively similar damping would be present were the plasma to have a Maxwellian equilibrium distribution function.

Linearized Variational Analysis of Single-Species, One-Dimensional Vlasov Plasmas

Analysis of linearized Vlasov plasmas by means of Hamilton's variational principle is illustrated by application to the initial-value problem for a neutralized, single-species plasma in one-spatial dimension with periodic boundary conditions. For any degree of approximation and for arbitrary equilibrium velocity distributions, the problem is reduced to the solution of a system of ordinary differential equations in time with constant coefficients and time-dependent driving terms; an exact particular solution has been found. The equations are discussed, including the relation of the equations to Eulerian treatments of the problem, and the use of the equations to obtain numerical results. Numerical examples have been worked out for a Maxwellian equilibrium distribution function, equilibrium distribution functions that are sums of two Maxwellian distributions, and for nonanalytic equilibrium distribution functions that are represented by cubic spline functions. Comparisons are made between numerical results from the variational method and precise solutions of the exact plasma dispersion relation.

The diffusion cooling of neutrons in a finite moderator

When a pulse of fast neutrons is thermalized in a finite moderating medium, the asymptotic spectrum at long times is slower than an equilibrium Maxwellian distribution, because of preferential leakage of higher-energy neutrons. This phenomenon of “diffusion cooling” can be observed by examining the variation of the asymptotic neutron lifetime with the size of the moderating medium. In this paper, the phenomenon is investigated within the context of energy-dependent diffusion theory. A variational expression for the decay constant is formulated and evaluated for trial solutions of Maxwellian form for the neutron spectrum. For small deviations from thermal equilibrium, an expression of the form proposed by von Dardel is obtained. The present analysis relates the diffusion-cooling term in this expression to the energy dependence of the transport mean free path, and to the mean square energy transfer between thermal neutrons and moderator. An analysis of the available experimental results for beryllium, graphite, and water is carried out and yields information on the effects of chemical binding on the energy transfer between neutrons and moderator. The results for graphite are consistent with the model of Krumhansl and Brooks for the specific heat. The results for beryllium, however, are not consistent with a Debye model which fits the specific heat, and it is suggested that the diffusion cooling is actually smaller than the available experimental value. The analysis for water is less sound theoretically, but it indicates a considerable hindering of the molecular rotations in the liquid.

The Electron Velocity Distribution in Gaseous Nebulae and Stellar Envelopes.

view Abstract Citations (41) References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Electron Velocity Distribution in Gaseous Nebulae and Stellar Envelopes. Bohm, David ; Aller, Lawrence H. Abstract A general method is given for solving Boltzmann's equation for the steady-state velocity distribution of electron-ion gases, such as those present in gaseous nebulae or stellar atmospheres. In a typical plane- tary nebula it is found that the velocity distribution is very close to Maxwellian. Because the average lifetime of an electron in the assembly is about 10 years and because its energy is reshuflied nearly every second by electrostatic encounters, the deviations from the equilibrium Maxwellian distribution must be very small. The non-Maxwellian distributions obtained by Hagihara are merely first approximations to an expansion of a Maxwellian distribution in terms of the wrong temperature. Section I gives a general summary of the results and the physical picture behind them. Section II lists the formulae for the probabilities of all processes involved; in Section III Boltzmann's equation is set down, and Hagihara's work is discussed. Section TV gives a general method for solving Boltzmann's equation when electrostatic interaction is the dominating term Publication: The Astrophysical Journal Pub Date: January 1947 DOI: 10.1086/144890 Bibcode: 1947ApJ...105..131B full text sources ADS |