The mean free path

If an electric field F acts on an electron in a state with wave function of the form (7) then k increases according to the equation 

The easiest way to prove this is by invoking the conservation of energy the rate of increase of the energy W is 

For the Boltzmann formulation we introduce, for each point on the Fermi surface, the relaxation time t, defined so that any disturbance 5/ of the equilibrium Fermi function / decays according to the law 

Then in the presence of a field F along the x-axis, when a steady current is flowing, the Fermi surface will be shifted in the direction of the field by an amount 

If l is constant and the Fermi surface is spherical with surface area SF, this reduces to 

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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. 

The average time between collisions is then v and in this time tlie particle will typically travel a distance X, the mean free path, where... 

This is the desired result. It shows that the mean free path is mversely proportional to the density and the collision cross section. This is a physically sensible result, and could have been obtained by dimensional... 

A3.1.2.2 THE MEAN FREE PATH EXPRESSIONS FOR TRANSPORT COEFFICIENTS... 

One of the most usefiil applications of the mean free path concept occurs in the theory of transport processes in systems where there exist gradients of average but local density, local temperature, and/or local velocity. The existence of such gradients causes a transfer of particles, energy or momentum, respectively, from one region of the system to another. 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

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 ... 

We consider the motion of a large particle in a fluid composed of lighter, smaller particles. We also suppose that the mean free path of the particles in the fluid, X, is much smaller than a characteristic size, R, of the large particle. The analysis of the motion of the large particle is based upon a method due to Langevin. Consider the equation of motion of the large particle. We write it in the fonn... 

Such ideal low mean free paths are the basis of FEED, the teclmique that has been used most for detennining surface structures on the atomic scale. This is also the case of photoelectron diffraction (PD) here, the mean free path of the emitted electrons restricts sensitivity to a similar depdi (actually double the depth of FEED, since the incident x-rays in PD are only weakly adenuated on this scale). 

Because a set of binding energies is characteristic for an element, XPS can analyse chemical composition. Almost all photoelectrons used in laboratory XPS have kinetic energies in the range of 0.2 to 1.5 keV, and probe the outer layers of tire sample. The mean free path of electrons in elemental solids depends on the kinetic energy. Optimum surface sensitivity is achieved with electrons at kinetic energies of 50-250 eV, where about 50% of the electrons come from the outennost layer. 

The strong point of AES is that it provides a quick measurement of elements in the surface region of conducting samples. For elements having Auger electrons with energies hr the range of 100-300 eV where the mean free path of the electrons is close to its minimum, AES is considerably more surface sensitive than XPS. 

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...

Since they are just as elementary as the "mean free path" theories. 

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. ... 

The limiting cases of greatest interest correspond to conditions in which the mean free path lengths are large and small, respectively, compared with the pore diameters. Recall from the discussion in Chapter 3 that the effective Knudsen diffusion coefficients are proportional to pore diameter and independent of pressure, while the effective bulk diffusion coefficients are independent of pore diameter and inversely proportional to pressure. 

Knudseci s very careful experiments on a long uniform capillary show that N L/ Pj -p ) passes through a marked minimum when plotted as a function of (P +P2)/2, at a value of the mean pressure such that the capillary diameter and the mean free path length are comparable. At higher values of the mean pressure, N L/(pj " 2 rises linearly, as in the case of a porous medium. 

When the mean free paths are long compared with all pore diameters, condition (i) above determines the form of the flux relations completely in isothermal systems, and no further modelling is required if is regarded... 

To be specific let us have in mind a picture of a porous catalyst pellet as an assembly of powder particles compacted into a rigid structure which is seamed by a system of pores, comprising the spaces between adjacent particles. Such a pore network would be expected to be thoroughly cross-linked on the scale of the powder particles. It is useful to have some quantitative idea of the sizes of various features of the catalyst structur< so let us take the powder particles to be of the order of 50p, in diameter. Then it is unlikely that the macropore effective diameters are much less than 10,000 X, while the mean free path at atmospheric pressure and ambient temperature, even for small molecules such as nitrogen, does not exceed... 

In molecular distillation, the permanent gas pressure is so low (less than 0 001 mm. of mercury) that it has very little influence upon the speed of the distillation. The distillation velocity at such low pressures is determined by the speed at which the vapour from the liquid being distilled can flow through the enclosed space connecting the still and condenser under the driving force of its own saturation pressure. If the distance from the surface of the evaporating liquid to the condenser is less than (or of the order of) the mean free path of a molecule of distillate vapour in the residual gas at the same density and pressure, most of the molecules which leave the surface will not return. The mean free path of air at various pressures is as follows —... 

Mean Free Path. The mean free path of a gas moiecuie I and the mean time between coiiisions T are given by... 

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