Infinite inhomogeneity

The classical method of solving scattering problems, separation of variables, has been applied previously in this book to a homogeneous sphere, a coated sphere (a simple example of an inhomogeneous particle), and an infinite right circular cylinder. It is applicable to particles with boundaries coinciding with coordinate surfaces of coordinate systems in which the wave equation is separable. By this method Asano and Yamamoto (1975) obtained an exact solution to the problem of scattering by an arbitrary spheroid (prolate or oblate) and numerical results have been obtained for spheroids of various shape, orientation, and refractive index (Asano, 1979 Asano and Sato, 1980). 

Finally, dopant inhomogeneities due to statistical dopant density fluctuations are also problematic, but this problem is shared with the analogous bulk materials and is not unique to the nanocrystals. Equation 3 describes the probability of finding N nearest neighbors around a central dopant ion in an infinite crystalline lattice for a given dopant concentration, x, where M is the number of cationic sites in the first shell around the central cation (12 in II-VI and III—V semiconductors) (39)... 

Follow an outside-in strategy. Consider a semi-infinite body with permittivity eout, coated with an inhomogeneous layer of constant thickness D and permittivity e(z), facing a medium em of variable thickness Z. Subscripts for a, b, L, and R will be added later (Fig. L3.14). 

It should be noted that the spectral emission is influenced by the. self-absorption of the emitted radiation by the sample. If the temperature distribution is homogeneous, this effect is already included in the determination of the absorptivity. In inhomogenous samples, the self-absorption may be neglected if the absorptivity is below 5%. In this case, the overall emission can be treated as the sum of the emission of all infinitely thin layers into which the sample can be divided (Pepperhoff and Grasz, 1955). Otherwise, the emission of all inner layers must be corrected by transmission factors before summation. For practical calculations, the sample volume can be divided into different layers, each of which is assumed to be in thermal equilibrium. 

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