Boltzmann correction

The signal intensity remains nearly constant after the Boltzmann correction is introduced. 

Raman spectroscopy has also been applied to the identification of different explosives. Most explosives, both nitro-containing (i.e. 2,4,6-trinitrotoluene (TNT)) and non-nitro-containing (i.e. TATP), produce high quality, low fluorescence Raman spectra. Some plastic explosives (i.e. Semtex) have some fluorescence originating from the binder materials but this can be overcome by use of anti-Stokes bands and Boltzmann correction of the data. 

The prefactor Cm = 1 /M is the Boltzmann correction which takes into account the trivial multi-counting of microstates generated by permuting identical particles - provided the particles can be exchanged at all. For a single molecular chain, where monomers are bonded and caimot change their positions within the chain, no Boltzmann correction is needed. 

Examining transition state theory, one notes that the assumptions of Maxwell-Boltzmann statistics are not completely correct because some of the molecules reaching the activation energy will react, lose excess vibrational energy, and not be able to go back to reactants. Also, some molecules that have reacted may go back to reactants again. 

In this expression, cos 0 is the average value of cos 0 the weighting factor used to evaluate the average is given by the Boltzmann factor exp(-V /RT), where R is the gas constant in the units of and T is in degrees Kelvin. Note that the correction factor introduced by these considerations reduces to unity if... 

The Wilson parameters A,, NRTL parameters G,, and UNIQUAC parameters X all inherit a Boltzmann-type T dependence from the origins of the expressions for G, but it is only approximate. Computations of properties sensitive to this dependence (e.g., heats of mixing and liquid/hquid solubihty) are in general only qualitatively correct. 

Moreover, since the mean free path is of the order of 100 times the molecular diameter, i.e., the range of force for a collision, collisions involving three or more particles are sufficiently rare to be neglected. This binary collision assumption (as well as the molecular chaos assumption) becomes better as the number density of the gas is decreased. Since these assumptions are increasingly valid as the particles spend a larger percentage of time out of the influence of another particle, one may expect that ideal gas behavior may be closely related to the consequences of the Boltzmann equation. This will be seen to be correct in the results of the approximation schemes used to solve the equation. 

The right-hand side of this equation includes components with and without exponential Boltzmann factor but their sum equals the total flow of particles from the j th rotational level to the rest of the levels. After this correction both necessary demands, Eq. (4.65) and Eq. (4.66), are satisfied. This result is of great advantage since calculation of the impact operator with the rather simple semiclassical formula, Eq. (5.1), does not lead after correction to any principal difficulties. The set of equations (5.26) and (5.27) determine the operator r(0) consistently but not uniquely. Other recipes may be used as well (see Chapter 7 and [195]). 

The expressions in Eq. 1 and Eq. 6 are two different definitions of entropy. The first was established by considerations of the behavior of bulk matter and the second by statistical analysis of molecular behavior. To verify that the two definitions are essentially the same we need to show that the entropy changes predicted by Eq. 6 are the same as those deduced from Eq. 1. To do so, we will show that the Boltzmann formula predicts the correct form of the volume dependence of the entropy of an ideal gas (Eq. 3a). More detailed calculations show that the two definitions are consistent with each other in every respect. In the process of developing these ideas, we shall also deepen our understanding of what we mean by disorder. ... 

The discrepancy may also be caused by the approximations in the calculation of the EEDF. This EEDF is obtained by solving the two-term Boltzmann equation, assuming full relaxation during one RF period. When the RF frequency becomes comparable to the energy loss frequencies of the electrons, it is not correct to use the time-independent Boltzmann equation to calculate the EEDF [253]. The saturation of the growth rate in the model is not caused by the fact that the RF frequency approaches the momentum transfer frequency Ume [254]. That would lead to less effective power dissipation by the electrons at higher RF frequencies and thus to a smaller deposition rate at high frequencies than at lower frequencies. 

In a fluid model the correct calculation of the source terms of electron impact collisions (e.g. ionization) is important. These source terms depend on the EEDF. In the 2D model described here, the source terms as well as the electron transport coefficients are related to the average electron energy and the composition of the gas by first calculating the EEDF for a number of values of the electric field (by solving the Boltzmann equation in the two-term approximation) and constructing a lookup table. 

Tlj = V Mijcrji/eij, where mu is the mass. In addition, Boltzmann s constant was set equal to unity. The pair potential was cut and shifted at Rcut = 2.5. No tail correction was used. The shift to make the potential zero at the cutoff is necessary for consistency between the Monte Carlo and the molecular dynamics aspects of the computations. 

In summary, Eq. (86) is a general expression for the number of particles in a given quantum state. If t = 1, this result is appropriate to Fenni-rDirac statistics, or to Bose-Einstein statistics, respectively. However, if i is equated torero, the result corresponds to the Maxwell -Boltzmann distribution. In many cases the last is a good approximation to quantum systems, which is furthermore, a correct description of classical ones - those in which the energy levels fotm a continuum. From these results the partition functions can be calculated, leading to expressions for the various thermodynamic functions for a given system. In many cases these values, as obtained from spectroscopic observations, are more accurate than those obtained by direct thermodynamic measurements. 

At this point it might be helpful to summarize what has been done so far in terms of effective potentials. To obtain the QFH correction, we started with an exact path integral expression and obtained the effective potential by making a first-order cumulant expansion of the Boltzmann factor and analytically performing all of the Gaussian kinetic energy integrals. Once the first-order cumulant approximation is made, the rest of the derivation is exact up to (11.26). A second-order expansion of the potential then leads to the QFH approximation. 

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