Mean free path calculation

The values for mean free path calculated by using Eq. 3.12 represent approximations because measurements of typical molecular diameters are not very accurate. 

However, as pointed out by Silinsh and Capek [20], the argument does not resist further analysis because, at least for temperatures above 100 K, the mean free path calculated from Equation (2.2.5) falls below the distance between molecules in the crystal, which is not physically consistent with diffusion limited transport so the exact nature of charge transport in these crystals is still unresolved for the time being. 

Table 110. Mean free paths calculated using various formulae for D and assuming v = 2 5 x 10 ... 

Figure 10. Electron inelastic mean free paths calculated for InP and GaAs from the model of Tanuma el al. (64. (Reproduced from Ref. 60 by permi.ssirin...

We are now going to use this distribution fiinction, together with some elementary notions from mechanics and probability theory, to calculate some properties of a dilute gas in equilibrium. We will calculate tire pressure that the gas exerts on the walls of the container as well as the rate of eflfiision 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. 

Next we consider the computation of the loss tenn, p - As in the calculation of the mean free path, we need... 

We now compute r by noting again the steps involved in calculating the mean free path, but applying them now to the derivation of an expression for r -... 

If we wish to know the number of (VpV)-collisions that actually take place in this small time interval, we need to know exactly where each particle is located and then follow the motion of all the particles from time tto time t+ bt. In fact, this is what is done in computer simulated molecular dynamics. We wish to avoid this exact specification of the particle trajectories, and instead carry out a plausible argument for the computation of r To do this, Boltzmann made the following assumption, called the Stosszahlansatz, which we encountered already in the calculation of the mean free path ... 

A more accurate calculation will account for differences in the energy dependent mean free paths of the elements and for the transmission characteristics of the electron analyser (see [7]). 

Figure C2.17.13. A model calculation of the optical absorjDtion of gold nanocrystals. The fonnalism outlined in the text is used to calculate the absorjDtion cross section of bulk gold (solid curve) and of gold nanoparticles of 3 mn (long dashes), 2 mn (short dashes) and 1 mn (dots) radius. The bulk dielectric properties are obtained from a cubic spline fit to the data of [237]. The small blue shift and substantial broadening which result from the mean free path limitation are...

When the size of a particle approaches the same order of magnitude as the mean free path of the gas molecules, the setthng velocity is greater than predicted by Stokes law because of molecular shp. The slip-flow correc tion is appreciable for particles smaller than 1 [Lm and is allowed for by the Cunningham correc tion for Stokes law (Lapple, op. cit. Licht, op. cit.). The Cunningham correction is apphed in calculations of the aerodynamic diameters of particles that are in the appropriate size range. 

Values of the total cross-section a a for A1 Ka, radiation, relative to the carbon Is level, have been calculated by Scofield [2.7], and of the asymmetry parameter /Ja by Redman et al. [2.8]. Seah and Dench [2.3] have compiled many measurements of the inelastic mean free path, and for elements the best-fit relationship they found was ... 

Using cm as unit surface and seconds as unit time, n is the number of molecules falling on 1 cm /sec. The number n thus denotes the number of molecules striking each cm of the surface every second, and this number can be calculated using Maxwell s and the Boyle-Gay Lussac equations. The number n is directly related to the speed of the molecules within the system. It is important to realize that the velocity of the molecules is not dependent on the pressure of the gas, but the mean free path is inversely proportional to the pressure. Thus ... 

The same k p scheme has been extended to the study of transport properties of CNTs. The conductivity calculated in the Boltzmann transport theory has shown a large positive magnetoresistance [18], This positive magnetoresistance has been confirmed by full quantum mechanical calculations in the case that the mean free path is much larger than the circumference length [19]. When the mean free path is short, the transport is reduced to that in a 2D graphite, which has also interesting characteristic features [20]. 

When bounding walls exist, the particles confined within them not only collide with each other, but also collide with the walls. With the decrease of wall spacing, the frequency of particle-particle collisions will decrease, while the particle-wall collision frequency will increase. This can be demonstrated by calculation of collisions of particles in two parallel plates with the DSMC method. In Fig. 5 the result of such a simulation is shown. In the simulation [18], 2,000 representative nitrogen gas molecules with 50 cells were employed. Other parameters used here were viscosity /r= 1.656 X 10 Pa-s, molecular mass m =4.65 X 10 kg, and the ambient temperature 7 ref=273 K. Instead of the hard-sphere (HS) model, the variable hard-sphere (VHS) model was adopted in the simulation, which gives a better prediction of the viscosity-temperature dependence than the HS model. For the VHS model, the mean free path becomes ... 

In the kinetic theory, the gas molecules are represented by hard spheres colliding elastically with each other and with the container walls. Details of this theory are given, for example, in ref. [1], An important parameter that can be calculated by this model is A, the mean free path of a molecule between collisions. The mean free path A of molecules is ... 

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