Distribution function Rayleigh

Rayleigh found the v2 dependence, Jeans later supplied the rest). Their distribution function g(v) increases as v2, with no provision for a fall-off to zero as the frequency and the energy go to infinity ("ultraviolet catastrophe"). 

Forpolydisperse aerosols, the simple expression (5.36) for the autocorrelation function must be averaged over the particle size distribution function. In the Rayleigh scattering range... 

Fig. 10.41. Scaled grain size distribution function for various times in aimealing history of sample of Fe (adapted from Pande and Marsh (1992)). In addition to the experimental results, the figure also shows the lognormal and Rayleigh distributions thought to be relevant to describing the distribution...

As a second example, it is instructive to derive the Kramers stationary flux function which serves as a basis for practical application in the Rayleigh quotient variational method (34,35). In principle there are an infinity of stationary flux functions, as any function in phase space which is constant along a classical trajectory will be stationary. Kramers imposed in addition the boundary condition that the flux is associated with particles that were initiated in the infinite past in the reactant region. Following Pechukas (69), one defines (68) the characteristic function of phase points in phase space Xr, which is unity on all phase space points of a trajectory which was initiated in the infinite past at reactants and is zero otherwise. By definition x, is stationary. The distribution function associated with the characteristic function Xr projected onto the physical phase space is then... 

In summary, the diflBculties in determining aggregate form factors, particle form factors, phase shifts, and distribution functions combine to make the Rayleigh-Debye approach too complicated for practical application. On the other hand, the coalesced-sphere approach using the Jobst approximations to the Mie scattering eflBciencies allows rapid correlation of turbidity with particle size distributions. Consequently, a coalesced-sphere approach was adopted for experimentation in the E. coli-PEI system. 

In such a case this radial distribution function for particle centres may be obtained by Fourier inversion of this equation for the Rayleigh ratio. 

This means that the scattering is elastic, often referred to as Rayleigh scattering. (2) Each ray entering the system is scattered only once. This assumption is essential for obtaining the required relation between the intensity of the scattered beam and the pair distribution function. If multiple scattering occurs, then such a relationship would involve higher-order molecular distribution functions. 

Consider a dilute one-component solution of particles with a molecular weight distribution expressed by the distribution function u/(M), which is defined such that the weight frartion of particles of molecular weight between M and M + dAl is given by u/(M)dM. Then, for such a system, the excess Rayleigh ratio is given from eqn [26] as... 

Measurements of extinction by small particles are easier to interpret and to compare with theory if the particles are segregated somehow into a population with sufficiently small sizes. The reason for this will become clear, we hope, from inspection of Fig. 12.12, where normalized cross sections using Mie theory and bulk optical constants of MgO, Si02, and SiC are shown as functions of radius the normahzation factor is the cross section in the Rayleigh limit. It is the maximum infrared cross section, the position of which can shift appreciably with radius, that is shown. The most important conclusion to be drawn from these curves is that the mass attenuation coefficient (cross section per unit particle mass) is independent of size below a radius that depends on the material (between about 0.5 and 1.0 fim for the materials considered here). This provides a strong incentive for deahng only with small particles provided that the total particle mass is accurately measured, comparison between theory and experiment can be made without worrying about size distributions or arbitrary normalization. 

---