Three-dimensional speed distribution

The properties of the three-dimensional speed distribution can be readily derived as well. This speed distribution turns out to be ... 

FIGURE 7.5 Schematic illustration of a velocity space. Each possible velocity corresponds to a single point (vx, vy,vz) in three dimensions. The total number of possible velocities (the volume) in a shell near some speed 5 is proportional to s2. Thus the three-dimensional speed distribution has an extra factor of s2. 

Use the three-dimensional speed distribution to show that. S most probable = y/2kT / m and that (s) = VSkT/Tzm. 

We went from the one-dimensional velocity distribution to the three-dimensional speed distribution by adding in an extra v2 factor to account for the added degeneracy in velocity space. The one-dimensional diffusion equation 7.36 has a form which is mathematically very similar to the one-dimensional velocity distribution... 

Chapter 7 covers the kinetic theory of gases. Diffusion and the one-dimensional velocity distribution were moved to Chapter 4 the ideal gas law is used throughout the book. This chapter covers more complex material. I have placed this material later in this edition, because any reasonable derivation of PV = nRT or the three-dimensional speed distribution really requires the students to understand a good deal of freshman physics. There is also significant coverage of dimensional analysis determining the correct functional form for the diffusion constant, for example. 

Often it is easier to model the experiment with an assumed KERD. But before discussing this, let us consider the TOP distribution, or the equivalent Doppler profile, of a nondissociating gas at room temperature. The three-dimensional speed distribution at a given temperature, T, is given by... 

Due to the simple product form of the Maxwell-Boltzmann distribution, the derivations given above are easily generalized to the expression for the relative velocity in three dimensions. Since the integrand in Eq. (2.18) (besides the Maxwell-Boltzmann distribution) depends only on the relative speed, we can simplify the expression in Eq. (2.18) further by integrating over the orientation of the relative velocity. This is done by introducing polar coordinates for the relative velocity. The full three-dimensional probability distribution for the relative speed is... 

Three-dimensional velocity distribution function for molecules of species A Speed distribution function for molecules of species A... 

More quantitative information can be obtained from the images when the full three-dimensional speed and angular distributions are reconstructed using mathematical transformations of the crushed two-dimensional images, or alternatively by using forward convolution simulation techniques. If the initial three-dimensional distribution has cylindrical symmetry, a unique transformation - the inverse Abel transform - can be used to reconstruct the initial three-dimensional velocity distribution. As the photolysis laser vector defines automatically an axis of cylindrical symmetry, the inverse Abel transformation can usually be used, as long as the plane of the position-sensitive detector is placed parallel to the laser polarization vector. 

Runnels and Eyman [41] report a tribological analysis of CMP in which a fluid-flow-induced stress distribution across the entire wafer surface is examined. Fundamentally, the model seeks to determine if hydroplaning of the wafer occurs by consideration of the fluid film between wafer and pad, in this case on a wafer scale. The thickness of the (slurry) fluid film is a key parameter, and depends on wafer curvature, slurry viscosity, and rotation speed. The traditional Preston equation R = KPV, where R is removal rate, P is pressure, and V is relative velocity, is modified to R = k ar, where a and T are the magnitudes of normal and shear stress, respectively. Fluid mechanic calculations are undertaken to determine contributions to these stresses based on how the slurry flows macroscopically, and how pressure is distributed across the entire wafer. Navier-Stokes equations for incompressible Newtonian flow (constant viscosity) are solved on a three-dimensional mesh ... 

Therefore the three-dimensional Maxwell-Boltzmann distribution of molecular speeds is... 

In Section 5.2, we will derive the three-dimensional Maxwell-Boltzmann distribution n(v)dv of molecular speeds between v and v + dv in the gas phase ... 

Since this method captures film thickness information from various areas covered by the CCD camera, the thickness distribution of an area can be measured at extremely high speed. For example. Fig. 6-a shows a three-dimensional plot of the thickness distribution around the Fig. 5-a test pattern taken with the CMP Semi-lnSitu Monitor. Please compare this with Fig. 6-b which shows a three-dimensional plot of an 8-inch wafer film measurement taken with another simultaneous film thickness measurement system with different optical specifications. 

To obtain Eqs (7.18) and (7.21) we need to assume that f vanishes as x oo faster than x. Physically tliis must be so because a particle that starts at x = 0 cannot reach beyond some finite distance at any finite time if only because its speed cannot exceed the speed of light. Of course, the diffusion equation does not know the restrictions imposed by the Einstein relativity theory (similarly, the MaxweU-Boltzmann distribution assigns finite probabilities to find particles with speeds that exceed the speed of light). The real mathematical reason why/ has to vanish faster than x is that in the equivalent three-dimensional formulation/(r) has to vanish faster than r as r oo in order to be normalizable. 

The model is three-dimensional. In order to comprehensively analyze the distribution of flow field, 260 sampling points for 4 levels in goaf are sampled in a short time, and then the gas concentration of samples was analyzed and tested by GC-4000 A gas chromatograph. The sampling points are also used as pressure measurement points. The static pressure of points are absorbed by the YJB-1500-type compensator micro-manometer, and the wind speed and static pressure of intake and return airway were measured by electronic wind instrument and U-type differential pressure meter. 

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