Derivation of the Rayleigh Ratio

Here M is the mass of a particle, c is the particle mass concentration (c = pMIN and is the Avogadro number). The quantity (dnldc) presents the refractive index gradient of the suspension and is measured by refractometer of high sensitivity. Combining Equations 5.401 and 5.404, we can derive the following expression for the Rayleigh ratio of a suspension of independent scatterers ... 

Even if we accept that we cannot go to the Hartree-Fock Limit the question of establishing an appropriate and effective criterion for determining the combinatorial coefficients, cr, of equation (2-1) still remains with us. For this purpose, the criterion which has been used is called the Variation Method. This is a fundamental principle of Quantum Theory which we merely adopt here without proof. According to this principle, the e, are chosen in order that a certain quantity (called the Rayleigh Ratio because it is derived chiefly from the work of Lord Rayleigh at the beginning of the present century) shall be stationary with respect to variation of the cr. ... 

The Rayleigh ratio simplifies on collecting all constants, the independently measured refractive index, and its change with concentration into a constant, K. Even simpler is the result when measuring at 6= 90° when 1 + cos 6=. If more species of molecules with molar masses Mj and concentrations c, are present, the mass-average is to be used. It is derived as follows from Sect. 1.2 ... 

The insertion of Eq. (4) into Eq. (3) and the simplification of the expression for ig is indicated at the top of Fig. A.3.5. The Rayleigh ratio of the light which is scattered into the direction d is the main outcome of the present derivation. It is shown at the bottom as Eq. (5). The Rayleigh ratio eliminates the influence of the measuring distance r and the incident light intensity. Both are instrumental calibration parameters. When all of the constants of the to-be-analyzed system are... 

The theory of light scattering in polymolecular polymer solutions enables one to determine the derivative (d(A/io)/dw).ir using the Rayleigh ratio Integrating this... 

Both modes of relaxation contribute to the scattered intensity measured by SLS, hence to the total Rayleigh ratio Rg. The Rayleigh ratio Re,fast for the faster mode of relaxation has been derived according to equation 3 (see also refs [30] and [31]). 

What meaning do these two-point resolution criteria have in describing the deconvolution process, that is, resolution before and after deconvolution Although width criteria may be applied to derive suitable before-after ratios, the Rayleigh criterion raises an interesting question. Because the diffraction pattern is an inherent property of the observing instrument, would it not be best to reserve this criterion to describe optical performance The effective spread function after deconvolution is not sine squared anyway. 

Equation 13 reduces to the Rayeigh equation (3) when the ratio of the gas-phase diffusivities, , is unity. Since gas-phase diffusivity is inversely proportional to the square root of the reduced mass, in the case of fission product-sodium systems where sodium has the smallest molecular weight, the above diffusivity ratio is less than unity. Therefore, the Rayleigh equation, which was derived on the basis of equilibrium vaporization, in fact represents an upper limit for the fractional fission-... 

The excess Rayleigh ratio Rg can be derived from the raw galvanometer readings Ig and Igg when the sample container contains die solution and the solvent, respectively, with the photocell of Fig. 4.11 positioned at an angle 9 in both cases. Hie equation used is... 

A simple alternative to the Rayleigh equation can be derived for an ideal mixture based on the relative volatility. Although the temperature in the still increases during a batch distillation, the relative volatility, which is the ratio of vapor pressures, does not change much and an average value can be used. From Eq. (18.34),... 

An expression for Rayleigh s ratio /f,> wllli duo account of polymolecularily. intra- and intcrmolecular interference, the spiuodal proximity, and appropriate dependence, of the interaction parameter has been derived by Vrij (197 1, 1978) and will be rlisciissetl later. 

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