Isotropic scattering radiation

Unlike the typical laser source, the zero-point blackbody field is spectrally white , providing all colours, CO2, that seek out all co - CO2 = coj resonances available in a given sample. Thus all possible Raman lines can be seen with a single incident source at tOp Such multiplex capability is now found in the Class II spectroscopies where broadband excitation is obtained either by using modeless lasers, or a femtosecond pulse, which on first principles must be spectrally broad [32]. Another distinction between a coherent laser source and the blackbody radiation is that the zero-point field is spatially isotropic. By perfonuing the simple wavevector algebra for SR, we find that the scattered radiation is isotropic as well. This concept of spatial incoherence will be used to explain a certain stimulated Raman scattering event in a subsequent section. 

In Raman spectroscopy the intensity of scattered radiation depends not only on the polarizability and concentration of the analyte molecules, but also on the optical properties of the sample and the adjustment of the instrument. Absolute Raman intensities are not, therefore, inherently a very accurate measure of concentration. These intensities are, of course, useful for quantification under well-defined experimental conditions and for well characterized samples otherwise relative intensities should be used instead. Raman bands of the major component, the solvent, or another component of known concentration can be used as internal standards. For isotropic phases, intensity ratios of Raman bands of the analyte and the reference compound depend linearly on the concentration ratio over a wide concentration range and are, therefore, very well-suited for quantification. Changes of temperature and the refractive index of the sample can, however, influence Raman intensities, and the band positions can be shifted by different solvation at higher concentrations or... 

A quantitative description does become possible, however, if the system under examination satisfies special conditions. These include diffuse, monochromatic illumination, homogeneous pigmentation, isotropic scattering in the coating, no difference in refractive index between vehicle and air, and a coating so thick that the substrate has no effect on the exiting radiation. This is the special case treated by the Kubelka-Munk theory. 

Classical electromagnetic theory shows that the intensity of light radiated by a small isotropic scatterer is... 

This expression defines the polarizability, a, which has the dimensions of a volume and which is a scalar for an isotropic spherical particle. From the energy of the electric field produced by the oscillating dipole, an expression can be derived for the intensity of the scattered radiation ... 

If the polarizable element is anisotropic, the mean flux is isotropic and it is obtained by averaging over all the orientations of the polarizable element in this case, the scattered radiation is partially depolarized It is convenient to express P3 and P4 in terms of the contributions Px and Py... 

Hale and Bohn [252] measured the scattered radiation from a finite sample of reticulated alumina from an incident laser beam at 488 nm. They then matched Monte Carlo predictions of the scattered radiation calculated from various values of extinction coefficient and scattering albedo and chose the values that best matched the experimental data for reticulated alumina samples of 10, 20, 30, and 65 ppi. A scattering albedo of 0.999 and an assumed isotropic scattering phase function reproduced the measured data for all pore sizes. The large reported albedo value indicates that alumina is very highly scattering and that radiative absorption is extremely small for this material. 

The absorbed dose buildup factor B(r) in Equation 8 is the ratio of the actual dose in the dosimeter at a point P, r cm. from a point isotropic 60Co source imbedded in large water container, to the dose that would be measured at the same point if there were no scattered radiation. In this equation I0 is the energy emitted by the source Zs is the scattered radiation flux at P m is the total absorption coefficient of water (0.0632... 

The above expression is a differential equation, since the value j/K (named as the source function) depends on the intensity of the radiation at each point. A solution to this expression can be obtained only by approximation. In the exact solution, the equation requires the division of the radiation field into a large number n of linear differential equations. The detailed solution has been presented by S. Chandrasekhar (2). Such a rigorous solution is practically never used for the calculation of isotropic scattering in the thin layer. 

Equation (3.10) gives the reflectance behavior for isotropic scattering when two oppositely directed radiation fluxes are assumed in the direction of the surface normal. The function ((1 - Roo) /2Roo) is commonly known as the Kubelka-Munk (K-M) function and is usually given the symbol f(Roo), although it is interesting to note that in their original paper [21], Kubelka and Munk did not derive this expression. Kubelka actually published the derivation of this equation [22] 17 years later. 

It should be noted here that Kubelka and Munk define scattering differently than does Mie (or Schuster). Mie defines scattering as radiation traveling in any direction after interaction with a particle. Kubelka and Munk defined scattered radiation as only that component of the radiation that is backward reflected into the hemisphere bounded by the plane of the sample s surface. In effect, the defining of S as equal to 2s makes S an isotropic scattering coefficient, with scatter equal in both the forward and backward directions. 

The distribution of scattered radiation is isotropic so that all regular (specular) reflection is ignored. 

A few points might be made about the assumption of isotropic scatter. If there is specular reflection in the scatter, there may be preferential directions of travel through the sample, and the assumption of diffuse radiation will be violated. Assumption 3 points out that the effect of front-surface reflection is ignored in their treatment. This assumption has often been interpreted as meaning that forward and backward scatter from a particle are assumed to be equal. As stated above, related to Equation (3.24) and Equation (3.25), the assumption of isotropic scatter of this kind is also built into their treatment. 

For the analysis of time-dependent fluctuations by scattering of coherent laser radiation and the analysis by sophisticated time-correlation techniques, cf. [75C1, 76B1]. For small isotropic scatterers undergoing Brownian motion the position autocorrelation function C (r, q) exponentially decreases... 

The theory of Raman scattering simplifies dramatically in the far-from-resonance (FFR) limit, where the exciting laser radiation is far from the lowest allowed excited electronic state of the molecule [15]. In this limit, the interaction of the light with the molecule is approximately the same for both the incident and the scattered radiation. There, the Raman tensor becomes symmetric and this symmetry reduces the number of Raman invariants from three to two, the isotropic and (symmetric) anisotropic invariants. The antisymmetric anisotropic invariant vanishes, by its antisymmetric definition in Eqs. (7) and (9), and we have... 

Fig. 1.33. Scattered radiation generated by the induced dipole moment in an isotropic and an anisotropic molecule.

---