Maxwellian

The rate coefficient for elastic scattering between two species with non-isothennal Maxwellian distributions is then... 

The amount of energy lost in unit time, the energy-loss frequency, is Vgg = kpN (t). The energy-loss rate coefficient for two-temperature Maxwellian distributions is... 

The velocity distribution of the electrons in a plasma is generally a complicated function whose exact shape is detennined by many factors. It is often assumed for reasons of convenience in calculations tliat such velocity distributions are Maxwellian and tliat tlie electrons are in tliennodynamical equilibrium. The Maxwell distribution is given by... 

A similar algorithm has been used to sample the equilibrium distribution [p,(r )] in the conformational optimization of a tetrapeptide[5] and atomic clusters at low temperature.[6] It was found that when g > 1 the search of conformational space was greatly enhanced over standard Metropolis Monte Carlo methods. In this form, the velocity distribution can be thought to be Maxwellian. 

Doppler broadening arises from the random thermal agitation of the active systems, each of which, in its own test frame, sees the appHed light field at a different frequency. When averaged over a Maxwellian velocity distribution, ie, assuming noninteracting species in thermal equilibrium, this yields a line width (fwhm) in cm C... 

Unlike the initial coordinates, which can be obtained experimentally, the only relevant information available about atomic velocities is the system s temperature T, which determines the velocity distribution. In the absence of a better guideline, initial velocities (v , Vy, Vy) are usually randomly assigned from the standard Maxwellian velocity distribution at a temperature T,... 

If particle i has been selected to undergo a collision, obtain its new velocity from a Maxwellian velocity distribution, defined in Eq. (23), corresponding to the desired temperature Tg. All other particles are unaffected by this collision. 

Fig. 3.31. Distributions (i)/(Ee) dEe of electron energy (E ) for a low-pressure HF-plasma (suffix pi, Maxwellian with temperature = 80000 K) and an electron beam (suffix eb, simplified to Gaussian shape with 40 eV half-width) (ii) rTx (Ej) ofthe Ej dependent electron impact ionization cross-section for X=Ti...

Maxwell found that he could represent the distribution of velocities statistically by a function, known as the Maxwellian distribution. The collisions of the molecules with their container gives rise to the pressure of the gas. By considering the average force exerted by the molecular collisions on the wall, Boltzmann was able to show that the average kinetic energy of the molecules was... 

The Burnett Expansion.—The Chapman-Enskog solution of the Boltzmann equation can be most easily developed through an expansion procedure due to Burnett.15 For the distribution function of a system that is close to equilibrium, we may use as a zeroth approximation a local equilibrium distribution function given by the maxwellian form ... 

It was shown, in Eqs. (1-73), (1-74), (1-75), that a = 1, afy r> => 0, a = 0. As the zeroth approximation we shall assume that A mid /a are zero (their effects are negligibly small) if Eqs. (1-86) and (1-87) are multiplied by /a and A, respectively, we obtain the condition that og0 and -oSi are zero higher order equations would show that all the coefficients are zero. Thus, the coefficients are proportional to some power of /a (or A). The zero-order approximation to the distribution function is just the local maxwellian distribution... 

Consider electrons of mass m and velocity v, and atoms of mass M and velocity V we have mjM 1. The distribution function for the electrons will be denoted by /(v,<) (we assume no space dependence) that for the atoms, F( V), assumed Maxwellian as usual, in the collision integral, unprimed quantities refer to values before collision, while primed quantities are the values after collision. In general, we would have three Boltzmann equations (one each for the electrons, ions, and neutrals), each containing three collision terms (one for self-collisions, and one each for collisions with the other two species). We are interested only in the equation for the electron distribution function by the assumption of slight ionization, we neglect the electron-electron... 

For large thermal energies of the electrons (kT predominates in the denominator), the distribution is maxwellian for large electric field strengths, we obtain the Margenau distribution29 (assuming, for simplicity, constant mean-free path) ... 

It is calculated in the S-matrix formalism and averaged over impact distances b and velocities v with Maxwellian distribution f(v)... 

Despite the fact that relaxation of rotational energy in nitrogen has already been experimentally studied for nearly 30 years, a reliable value of the cross-section is still not well established. Experiments on absorption of ultrasonic sound give different values in the interval 7.7-12.2 A2 [242], As we have seen already, data obtained in supersonic jets are smaller by a factor two but should be rather carefully compared with bulk data as the velocity distribution in a jet differs from the Maxwellian one. In the contrast, the NMR estimation of a3 = 30 A2 in [81] brought the authors to the conclusion that o E = 40 A in the frame of classical /-diffusion. As the latter is purely nonadiabatic it is natural that the authors of [237] obtained a somewhat lower value by taking into account adiabaticity of collisions by non-zero parameter b in the fitting law. 

Maxwellian distribution 129 infinite-order sudden (IOS) approximation 155-6 semi-classical calculation 136-7 Sack s model rotational relaxation 19 strong collision model 219 scattering see isotropic scattering spectra ... 

A Maxwellian material can be described by the following constitutive equation ... 

An important specific feature of the present experiment is worth noting. The X-ray photons have energies that are several orders of magnitude larger than those of optical photons. The pump and probe processes thus evolve on different time scales and can be treated separately. It is convenient to start with the X-ray probing processes, and treat them by Maxwellian electrodynamics. The pumping processes are studied next using statistical mechanics of nonlinear optical processes. The electron number density n(r,t), supposed to be known in the first step, is actually calculated in this second step. 

Figure 13a-e shows the EEDFs of a CH4/H2 plasma as a function of pressure. The shape of the EEDF at 50 mTorr corresponding to Figure 13a is different from that of an Ar plasma at the same pressure. There is a hump at " 6 eV. The hump gradually disappears with decreasing pressure. The shape of the EEDF at 30 mTorr in Figure 13c is almost a straight line, which means a Maxwellian distribution. The shape of the EEDF at 20 mTorr in Figure 13d deviates from a straight line and comes close to a Druyveysten distribution. It still keeps a Druyveysten distribution at 10 mTorr in Figure 13e. Accordingly, the transition from a Maxwellian distribution to a Druyveysten distribution occurs at 20 mTorr in a CH4/H2 plasma.

Electron energy distribution function The distribution function of electrons in a plasma. That of a low-pressure radiofrequency plasma generally consists of two Maxwellian distributions, that is, fast and slow electrons. 

For low-pressure plasmas containing mainly inert gases the electrons can be characterized by a Maxwellian electron energy distribution function (EEDF). How-... 

II. 51 Pa for H2, and 0.55 Pa for Si2H6. With these partial pressures, and the data on electron collisions as given in Table II, first the EEDF can be calculated, followed by the calculation of the electron transport coefficients and electron impact rates. From a comparison of a Maxwellian EEDF and the two-term Boltzmann... 

EEDF as calculated here, it follows that the Maxwellian EEDF underestimates the EEDF at lower energies and overestimates it at higher energies [189]. 

In the intermediate region II, assuming a Maxwellian EEDF, the electron retarding current is given by... 

The flow of gas molecules striking the hole area is simply the product of the hole area A and the intensity of the Maxwellian stream given by Eq. (43). However, since the gas samples are characterized by different pressures, Pi and l, as well as by different temperatures, T and T2, the flow of gas sample 1 is not equal to the flow of gas sample 2, and there exists a net flow through the hole,... 

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