Averaging 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... 

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... 

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

However, because of the Maxwellian distribution function for molecular speeds, not all H2 molecules will be faster than all 02 molecules and some H2 molecules will have velocities near 0 m/s. Some 02 molecules will be moving faster than the average speed of H2 molecules. 

In more refined calculations, 2 is replaced by its average over the Maxwellian velocity distribution and is a slowly varying function of t. 

If one follows the approach of Landau and Teller [11], who in dealing with vibrational relaxation developed an expression by averaging a transition probability based on the relative molecular velocity over the Maxwellian distribution, one can obtain the following expression for the recombination rate constant [6] ... 

A sufficiently rarefied gas, or a mixture of gases, consists of a number of neutral molecules of species 1 and 2 (which may or may not be the same). We may assume a distribution of velocities (measured in the laboratory frame), fi ( ) d3u, that may be modeled by a Maxwellian distribution function, with i = 1 or 2, as long as the duration of the average collision is short compared to the time between collisions. For binary collisions, one usually transforms from laboratory coordinates, Vj, to relative ( >12) and center-of-mass (1>cm) velocities,... 

These latter measurements led only to relative cross-section values. However, by comparison with absolute values of velocity-averaged cross sections, they can be put on an absolute scale. To do this, the absolute values obtained in FA measurements were used because here the velocity distribution is exactly known—a Maxwellian distribution /(t>, T) with the temperature of the buffer gas. Denoting the velocity-dependent relative total ionization cross section, obtained in the beam experiment, by oKl(v) and the absolute total ionization rate constant obtained in the FA experiment by R(T), then a normalization k may be determined by... 

All the experiments described in this chapter were performed on molecules in gas cells, and, unlike in beam experiments, the results obtained are averaged over all possible orientations of the molecules with respect to the relative velocity vector, and over the Maxwellian distribution of the relative velocities of the colliding particles. Nevertheless, the experimental data stimulated interest in the development of theoretical models of elastic... 

We start to follow the relative motion between two particles of mass mp separated by a specified initial distance. This initial distance is chosen to be large compared to the correlation length Xr (the average distance over which the relative velocity of the particle pair is correlated). Therefore, the initial relative motion of the particle pair, which is considered as the initial condition in the calculations, is uncorrelated and its initial relative velocity distribution is Maxwellian. The correlation length Xr is given by (1)... 

The relative Brownian motion between the constituents of doublets consisting of sufficiently small equal-size aerosol particles is described by a one-dimensional Fokker-Planck equation in the particle energy space. A first passage time approach is employed for the calculation of the average lifetime of the doublets. This calculation is based on the assumption that the initial distribution of tire energy of the relative motion of the constituent particles is Maxwellian. The average dissociation time of doublets, in air at 1 atm and 298 K, for a Hamaker constant of 10 12 erg has been calculated for different sizes of the constituent particles. The calculations are found to be consistent with the assumption that the... 

Here ey is the cross section for a collisionally induced transition and v is the thermal velocity of the colliding particle, < av > is the average value of ov for a Maxwellian velocity distribution. Assuming a typical dipole moment of 1 Debye, a = 1CT15 cm"2, v = 5 x 104 cm/s, one obtains the density n % 103/X3. Thus for the detection of an emission line in the centimeter-wave region (X = 1 cm) the density within the cloud is expected to be of the order of 103 cm-3. On the other hand, a detection of a millimeter-wave transition in emission at X = 1 mm requires densities of the order of 10s to 106 particles/cm3. 

Considering the Ik>i rates for even values of k, say k = s 2, and averaging these rates with a Maxwellian distribution for the neutral particle velocity, then, up to normalizing factors, the above mentioned 12S, integrals result ... 

E. Collision Frequency between Maxwellian Molecules. Finally, we can calculate the average number of collisions made by a molecule going through a Maxwellian gas if the molecule does not have a fixed velocity V, but has instead a velocity distribution which is itself Maxwellian. This may be done by multiplying Zc [Eq. (VII.8D.4)] by the Maxwellian distribution function and averaging over all values of Vx ... 

In order to calculate the flow we must know something about the distribution of molecular velocities in the gas. Since the gas is not at equilibrium but only in a steady state, we cannot say that we have an equilibrium distribution. However we can make the approximation of assuming that the velocity distribution is flocally Maxwellian, i.e., that the molecules at any given point distant Z from the fixed plate have the normal distribution of velocities with respect to an average which is not zero but is given by the macroscopic stream velocity at that point. Thus at a point Z from the fixed plate the distribution is to be taken as... 

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