Mean free path, distribution function

Where, /(k) is the sum over N back-scattering atoms i, where fi is the scattering amplitude term characteristic of the atom, cT is the Debye-Waller factor associated with the vibration of the atoms, r is the distance from the absorbing atom, X is the mean free path of the photoelectron, and is the phase shift of the spherical wave as it scatters from the back-scattering atoms. By talcing the Fourier transform of the amplitude of the fine structure (that is, X( )> real-space radial distribution function of the back-scattering atoms around the absorbing atom is produced. 

This is an indication of the collective nature of the effect. Although collisions between hard spheres are instantaneous the model itself is not binary. Very careful analysis of the free-path distribution has been undertaken in an excellent old work [74], It showed quite definite although small deviations from Poissonian statistics not only in solids, but also in a liquid hard-sphere system. The mean free-path X is used as a scaling length to make a dimensionless free-path distribution, Xp, as a function of a free-path length r/X. In the zero-density limit this is an ideal exponential function (Ap)o- In a one-dimensional system this is an exact result, i.e., Xp/(Xp)0 = 1 at any density. In two dimensions the dense-fluid scaled free-path distributions agree quite well with each other, but not so well with the zero-density scaled distribution, which is represented by a horizontal line (Fig. 1.21(a)). The maximum deviation is about... 

Carlos and Latif both fluidised glass particles in dimethyl phthalate. Data on the movement of the tracer particle, in the form of spatial co-ordinates as a function of time, were used as direct input to a computer programmed to calculate vertical, radial, tangential and radial velocities of the particle as a function of location. When plotted as a histogram, the total velocity distribution was found to be of the same form as that predicted by the kinetic theory for the molecules in a gas. A typical result is shown in Figure 6.11(41 Effective diffusion or mixing coefficients for the particles were then calculated from the product of the mean velocity and mean free path of the particles, using the simple kinetic theory. 

A straightforward Fourier transform of the EXAFS signal does not yield the true radial distribution function. First, the phase shift causes each coordination shell to peak at the incorrect distance. Second, due to the element specific back-scattering amplitude, the intensity may not be correct. Third, coordination numbers of distant shells will be too low mainly because of the term 1/r2 in the amplitude (10.10) and also because of the small inelastic mean free path of the photoelectron. The appropriate corrections can be made, however, when phase shift and amplitude functions are derived from reference samples or from theoretical calculations. Figure 10.17 illustrates the effect of phase and amplitude correction on the EXAFS of a Rh foil [38]. Note that unless the sample is that of a... 

From equation 5, it is apparent that each shell of scatterers will contribute a different frequency of oscillation to the overall EXAFS spectrum. A common method used to visualize these contributions is to calculate the Fourier transform (FT) of the EXAFS spectrum. The FT is a pseudoradial-distribution function of electron density around the absorber. Because of the phase shift [< ( )], all of the peaks in the FT are shifted, typically by ca. —0.4 A, from their true distances. The back-scattering amplitude, Debye-Waller factor, and mean free-path terms make it impossible to correlate the FT amplitude directly with coordination number. Finally, the limited k range of the data gives rise to so-called truncation ripples, which are spurious peaks appearing on the wings of the true peaks. For these reasons, FTs are never used for quantitative analysis of EXAFS spectra. They are useful, however, for visualizing the major components of an EXAFS spectrum. 

The mean free path already calculated gives the average over a great many collisions. We would like to find out how the individual paths between collisions vary. To do so, let us first calculate the probability that a molecule will go a distance x after its last collision without making a collision. If f(x) represents this distribution function, then it is seen that it must satisfy the conditions/(O) = 1 and/(oo) = 0. 

The forces of attraction and repulsion between molecules must be considered for a more accurate and rigorous representation of the gas flow. Chapman and Enskog proposed a well-known theory in which they use a distribution function, the Boltzmann equation, instead of the mean free path. Using this approach, for a pair of non-polar molecules, an intermolecular potential, V (r), is given in the potential function proposed by the Lennard-Jones potential ... 

We are now going to use this distribution function, together with some elementary notions from mechanics and probability theory, to calculate some properties of a dilute gas in equilibrium. We will calculate the pressure that the gas exerts on the walls of the container as well as the rate of effusion of particles from a very small hole in the wall of the container. As a last example, we will calculate the mean free path of a molecule between collisions with other molecules in the gas. 

The average velocity of a gas molecule is determined by the molecular weight and the absolute temperature of the gas. Air molecules, like many other molecules at room temperature, travel with velocities of about 500 m s"1 but there is a distribution of molecular velocities. This distribution of velocities is explained by assuming that the particles do not travel unimpeded but experience many collisions. The constant occurrence of such collisions produces the wide distribution of velocities. The quantitative treatment was carried out by Maxwell in 1859, and somewhat later by Boltzmann. The phenomenon of collisions leads to the concept of a free path, that is the distance traversed by a molecule between two successive collisions with other molecules of that gas. For a large number of molecules, this concept must be modified to a mean free path which is the average distance travelled by all molecules between collisions. For molecules of air at 25°C, the mean free path X at 1 mbar is 0.00625 cm. It is convenient therefore to use the following relation as a scaling function ... 

The heterogeneous stabilization of metals, produeed by non-thermal plasma dissociation of hahdes, can be achieved when the mean free path of halogen atoms, A.Ha, related to recombination (7-75) is shorter than the mean free path of metal atoms, A,Me, related to disproportioning (7-76). Assuming the distribution function of halides MeHat over their oxidation degrees /(k) in exponential form (7-85), the ratio of the mean free paths of halogen and metal atoms can be expressed as... 

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