Mean free path approximation

Figure 6 shows the MD predicted in-plane and out-of-plane thermal eonduetivities at 376K (Fig. 6a) and lOOOK (Fig. 6b) as a function of film thickness. It is seen that both the in-plane and out-of-plane thermal conductivities are affeeted by the thiekness of the film. For thiekness smaller than the phonon mean free path (approximately 300 nm and 30 nm at 300K and lOOOK, respeetively), both the in-plane and out-of-plane thermal eonduetivities deerease with deereasing thiekness, an effeet attributed to the scattering of phonons with the boundaries of the thin film. This effeet is more pronounced in the out-of-plane direction, where the dimensions of the thin film make the phonon transport ballistic. At large thicknesses, the thermal conductivities approach the bulk value (shown as dashed lines in Fig. 6). The bulk value is reached at smaller thicknesses at lOOOK due to the smaller phonon mean free path at this temperature.

The subscript av represents an average over all states with energies E. Equation (4.10) is called the Kubo-Greenwood formula. For a long mean free path approximation, Eq. (4.10) reduces to the Drude model. 

Here f denotes the fraction of molecules diffusely scattered at the surface and I is the mean free path. If distance is measured on a scale whose unit is comparable with the dimensions of the flow channel and is some suitable characteristic fluid velocity, such as the center-line velocity, then dv/dx v and f <<1. Provided a significant proportion of incident molecules are scattered diffusely at the wall, so that f is not too small, it then follows from (4.8) that G l, and hence from (4.7) that V v° at the wall. Consequently a good approximation to the correct boundary condition is obtained by setting v = 0 at the wall. ... 

As is to be expected, inherent disorder has an effect on electronic and optical properties of amorphous semiconductors providing for distinct differences between them and the crystalline semiconductors. The inherent disorder provides for localized as well as nonlocalized states within the same band such that a critical energy, can be defined by distinguishing the two types of states (4). At E = E, the mean free path of the electron is on the order of the interatomic distance and the wave function fluctuates randomly such that the quantum number, k, is no longer vaHd. For E < E the wave functions are localized and for E > E they are nonlocalized. For E > E the motion of the electron is diffusive and the extended state mobiHty is approximately 10 cm /sV. For U <, conduction takes place by hopping from one localized site to the next. Hence, at U =, )J. goes through a... 

Vacuum Flow When gas flows under high vacuum conditions or through very small openings, the continuum hypothesis is no longer appropriate if the channel dimension is not very large compared to the mean free path of the gas. When the mean free path is comparable to the channel dimension, flow is dominated by collisions of molecules with the wall, rather than by colhsions between molecules. An approximate expression based on Brown, et al. J. Appl. Phys., 17, 802-813 [1946]) for the mean free path is... 

Two mechanisms which contribute to GMR have been identified, a "non-local" mechanism and a "quantum" mechanism. To understand the first or non-local, mechanism it is necessary to understand that on the scale of the electron mean free path (possibly 10 to 20 nanometers at room temperature) electrical conduction is a non-local phenomenon. Electrons may be accelerated by an electric field in one region and contribute to the current in other regions. To a good approximation they may viewed as contributing to the current until they are scattered. 

Assuming that A << /o and that /o varies appreciably only over distances x L, it is easy to show that A//o —XjL, where A is the mean free path length i.e. /o is a good approximation if the characteristic wavelengths of p, T and u are all much greater than the mean free path. The exact solution / can then be expanded in powers of the factor X/L. This systematic expansion is called the CAia.pma.n-Enskog expansion, and is the subject of the next section. 

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. 

Because molecular velocities increase with rise of temperature T, so also does the diffusivity which, for a gas, is approximately proportional to T raised to the power of 1.5. As the pressure P increases, the molecules become closer together and the mean free path is shorter and consequently the diffusivity is reduced, with D for a gas becoming approximately inversely proportional to the pressure. 

With the knowledge of g, we can estimate the inverse mean free path of a phonon with frequency co. As done originally within the TLS model, the quantum dynamics of the two lowest energies of each tunneling center are described by the Hamiltonian //tls = gcTz/2 + Aa /2. This expression, together with Eqs. (15) and (17), is a complete (approximate) Hamiltonian of... 

For the parameters used to obtain the results in Fig. 3, X 0.6 so the mean free path is comparable to the cell length. If X -C 1, the correspondence between the analytical expression for D in Eq. (43) and the simulation results breaks down. Figure 4a plots the deviation of the simulated values of D from Do as a function of X. For small X values there is a strong discrepancy, which may be attributed to correlations that are not accounted for in Do, which assumes that collisions are uncorrelated in the time x. For very small mean free paths, there is a high probability that two or more particles will occupy the same collision volume at different time steps, an effect that is not accounted for in the geometric series approximation that leads to Do. The origins of such corrections have been studied [19-22]. 

X, the thickness of the boundary layer is approximately equal to the mean free path, , of gas molecules in air. 

Using the Ashley approximation, the inverse inelastic mean free path is given, in congruity with the stopping power (see Sect. 2.5.2), as follows ... 

This is a harmonic mean because it is really the mean free path that is relevant. At high temperatures (> 105 K), the main sources of opacity and approximate formulae for them (the first two originally due to H. Kramers) are ... 

Cross sections for the elements have been calculated and tabulated by Scofield [22], Mean free path values are often approximated by a calculating them from a general formula [23], but recently new data have become available which take... 

The above equations do not allow for kinetics associated with the mean free path discontinuity at the particle surface. A correction for this would increase the charging rate. As a first approximation, this can be allowed for by replacing Pp in Eqs. (75) and (77) with the product >p[l + (2A,/Z)p)]. This is based on the reasoning that ions migrating to within a mean free path of the particle surface will be deposited and that the ion concentration will drop effectively to zero within a mean free path of the surface. 

Recent times have seen much discussion of the choice of hydrodynamic boundary conditions that can be employed in a description of the solid-liquid interface. For some time, the no-slip approximation was deemed acceptable and has constituted something of a dogma in many fields concerned with fluid mechanics. This assumption is based on observations made at a macroscopic level, where the mean free path of the hquid being considered is much smaller... 

When considering boundary conditions, a useful dimensionless hydrodynamic number is the Knudsen number, Kn = X/L, the ratio of the mean free path length to the characteristic dimension of the flow. In the case of a small Knudsen number, continuum mechanics will apply, and the no-slip boundary condition assumption is valid. In this formulation of classical fluid dynamics, the fluid velocity vanishes at the wall, so fluid particles directly adjacent to the wall are stationary, with respect to the wall. This also ensures that there is a continuity of stress across the boundary (i.e., the stress at the lower surface—the wall—is equal to the stress in the surface-adjacent liquid). Although this is an approximation, it is valid in many cases, and greatly simplifies the solution of the equations of motion. Additionally, it eliminates the need to include an extra parameter, which must be determined on a theoretical or experimental basis. 

However, in the case of large Kn, the no-slip approximation cannot be applied. This implies that the mean free path of the liquid is on the same length scale as the dimension of the system itself. In such a case, stress and displacement are discontinuous at the interface, so an additional parameter is required to characterize the boundary condition. A simple technique to model this is the one-dimensional slip length, which is the extrapolation length into the wall required to recover the no-slip condition, as shown in Fig. 1. If we consider... 

The gas molecules fly about and among each other, at every possible velocity, and bombard both the vessel walls and collide (elastically) with each other. This motion of the gas molecules is described numerically with the assistance of the kinetic theory of gases. A molecule s average number of collisions over a given period of time, the so-called collision index z, and the mean path distance which each gas molecuie covers between two collisions with other molecules, the so-called mean free path length X, are described as shown below as a function of the mean molecule velocity c the molecule diameter 2r and the particle number density molecules n - as a very good approximation ... 

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