Fluctuations and Rayleigh scattering

If a smectic A sample is homeotropically aligned between two glass plates having a separation L, the boundary conditions require that = mnIL, where m is an integer. We shall coniine the discussion to m = 1. When = 0 we have from (5.3.6) 

We know that the intensity of light scattering is proportional to the mean square fluctuation of the director (see 3.9)  

The elastic energy is minimized when q = q, which represents the optimum wavevector. When q = 0, 

Strictly speaking, one should take into account the contributions of k and fcjj, since the layers are assumed to be compressible. Following the procedure outlined in 3.9, the fluctuations may be decomposed into two modes, and choosing the wavevector q in the xz plane, one gets the general expressions 

These modes are highly damped, the relaxation time r being of the order of 10 s. To discuss the damping quantitatively we have to consider the hydrodynamics of smectic A. The most general formulation of the theory is due to Martin, Parodi and Pershan but we shall present the relevant equations in a simplified form using de Gennes s notation. 

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Rayleigh scattered light from dense transparent media with nonuniform density. If these nonuniformities are time-independent, there will be no frequency shift of the scattered light. If, however, time-dependent density fluctuations occur, as e. g. in fluids, due to thermal or mechanical processes, the frequency of the scattered light exhibits a spectrum characteristic of this time dependence. The type of information which can be obtained by determining the spectral line profile and frequency shift is described in an article by Mountain 235). 

Stimulated Rayleigh scattering from localised thermal fluctuations in gases 258) and liquids 259) has been reported with measurements of the line shifts, thresholds and critical absorption coefficients. 

Light scattering from a solution is due both to the scattering from local density fluctuations and to the scattering from the solvent [9,18], This scattering may be described by the Rayleigh scattering ratio [9,18] ... 

In order to be able to use the fluctuation of the intensity around the average value, we need to find a way to represent the fluctuations in a convenient manner. In Section 5.3b in our discussion of Rayleigh scattering applied to solutions, we came across the concept of fluctuations of polarizabilities and concentration of scatterers and the role they play in light scattering experiments. In the present section, what we are interested in is the time dependence of such fluctuations. In general, it is not convenient to deal with detailed records of the fluctuations of a measured quantity as a function of time. Instead, one reduces the details of the fluctuations to what is known as the autocorrelation function C(s,td), as defined below ... 

Here, Instantaneous Value X = Mean Value X + Fluctuation Value X LV denotes laser velocimetry RS, Raman scattering, RayS, Rayleigh scattering and pdf, probability density function. 

As shown in Table IV, for a given altitude in the troposphere there are significant seasonal and latitudinal fluctuations in the total column of ozone, and these lead to similar fluctuations in the ultraviolet flux. The maximum ozone column and minimum transmitted flux occur in late winter and early spring, while the minimum column and maximum flux occur in the fall. This effect is most pronounced at high latitudes, and much less so near the equator [cf. Junge (128)]. The seasonal and latitudinal variation in Rayleigh and aerosol scattering is much less and is not important to this discussion. 

The different symmetry properties considered above (p. 131) for macroscopic susceptibilities apply equally for molecular polarizabilities. The linear polarizability a - w w) is a symmetric second-rank tensor like Therefore, only six of its nine components are independent. It can always be transformed to a main axes system where it has only three independent components, and If the molecule possesses one or more symmetry axes, these coincide with the main axes of the polarizability ellipsoid. Like /J is a third-rank tensor with 27 components. All coefficients of third-rank tensors vanish in centrosymmetric media effects of the molecular polarizability of second order may therefore not be observed in them. Solutions and gases are statistically isotropic and therefore not useful technically. However, local fluctuations in solutions may be used analytically to probe elements of /3 (see p. 163 for hyper-Rayleigh scattering). The number of independent and significant components of /3 is considerably reduced by spatial symmetry. The non-zero components for a few important point groups are shown in (42)-(44). 

For Rayleigh scattering / = 0 at 90°. As R increases, theory shows that X is a periodic function of diameter for monosize particles, and this has been used to measure particle size [78] specifically the size of aerosols in the size range 0.1 to 0.4 pm [79]. It has also been used to determine the sizes of sulfur solutions [80] In this work, transmission and polarization methods yielded results in accord with high order Tyndall spectra (HOTS) for sizes in the range 0.365 to 0.62 pm. In the limited region where (0.45[Pg.537]

Density and composition fluctuations are inherent in glasses and lead to scattering. For example, Rayleigh scattering in fused silica amounts to about 0.7 dB/km at 1 pm. Since scattering scales as 1 /A, it becomes most important at shorter wavelengths. 

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