Van de Hulst approximation

Sl in (1966), Volkov and Baranov (1968) provide reviews of light scattering on anisotropic particles. 

In the general case, light scattering in matter is caused not only by density fluctuations but also by anisotropy fluctuations and fluctuations of the optical axis orientation of anisotropic areas. Such fluctuations arise as a result of mutual orientation of anisotropic molecules, or of their aggregates, or owing to the internal stresses in solid matter. In this case, the polarizability of scattering elements is represented by a polarizability tensor and two correlation functions (density correlation auid orientation correlation) are introduced (Goldstein and Michalik, 1955 van Aartsen, 1972). 

Van de Hulst s approximation is applicable to quite big particles and, what is more, to nonspherical and heterogeneous particles (Tvorogov, 1965 Khlebtsov, 1980) and to absorbing particles (van de Hulst, 1957 I vorogov, 1965 Bryant and Latimer, 1969 Sukhachyova and Tvorogov, 1973). 

Hy some estimates (Shifrin, 1951 van de Hulst, 1957 Heller, 1963 Kerker and Farone, 1963 Moore et al., 1968 Kerker, 1969), the results of the soft pajticle approximation are qualitatively true within, at le.a,st, 0.8 m 1.5 if Q 1 when m 1.15 they are valid quantitatively with a slight error. Hence, Rayleigh-Debye and van de Hulst s approximations are very fruitful in studying the heterogeneous structures of polymer and biological origin. 

---

The premi.ses and main results of the van de Hulst approximation are given and used (as well as the rigorous Mie theory) in the turbidity spectrum method applied for the characterization of ill-defined disperse systems. 

The modification to the van de Hulst approximation was to use a real refractive index spectrum of the PMMA instead of using a constant value such as 1.4. To illustrate this concept here, we have simulated a resonant Mie scattering efficiency curve of a theoretical spherical particle of Matrigel. The n spectrum used is Figure 8.4, with a value of 1.3 added to it to act as an average real refractive index, this value is considered to be typical of biomedical samples. Using a particle radius of 4 pm, theoretical Q curves can be computed and are shown in Figure 8.5. 

Box and McKellar (1978) derived the sum rule (4.81) under the assumption of a constant refractive index and within the framework of the anomalous diffraction approximation of van de Hulst (1957, Chap. 11). 

Considering an incident plane wave on a sphere of no net surface charge, the scattering and extinction efficiency factor for the field far away from the particle can be approximated by [Van de Hulst, 1957]... 

LIGHT SCATTERING BY SMALL PARTICLES. H.C. van de Hulst. Comprehensive treatment including full range of useful approximation methods for researchers in chemistry, meteorology and astronomy. 44 illustrations. 470pp. 514 x 8)4. 64228-3 Pa. 9.95... 

The value of Q t can be estimated by using Van de Hulst s (1957) approximation for transparent spheres... 

Figure 5.2 Extinction curves calculated from the theory of Mie for m = 1.5 and nt — 1.33 (van de Hulst, 19.57). The eurves show a sequence of maxima and minima of diminishing amplitude, typical of nonabsorbing spheres with 1 < m < 2. Indeed, by taking the ab.scissa oi the curve for m = I. .5 to be 2x m — I), all extinction curves for the range I < m < 2 are reduced to approximately the same curve.

For very large particles, i.e. a 1 or Dp > 4-A., the laws of geometrical optics (also called the Fraunhofer regime) are applicable (van de Hulst 1981). The light scattering intensity varies approximately with the square of the particle diameter. 

In van de Hulst s approximation, Equations 90 and 96 are followed by (Shchyogolev and Klenin, 1971b Klenin et aJ., 1977a)... 

The direct and reverse problems of the turbidity spectrum method for systems of aniso-diametric particles were solved by simulation in Rayleigh-Debye s (2 < Up < 4) and van de Hulst s ( p < 2) approximations (Khlebtsov and Shchyogolev, 1977ab Khlebtsov et al, 1977, 1978abc Shchyogolev et al., 1977 Khlebtsov, 1980 Shchyogolev, 1983). 

Figure 2.23. Dependence of n for spherical particles (1) and Up for elongated ellipsoids of revolution (averaging over all the orientations) with the axes ratio p = 2 (2), 4 (5), 6 (4), and 10 (5) on the equivalent parameter p (see caption to Figure 2.21). Van de Hulst s approximation (Khlebtsov and Shchyogolev, 1977b)...

Since anomalous diffraction is only relevant for forward scattering, it is not meaningful to discuss other cross sections than the extinction one. A first order approximation for real refractive indices was provided by van de Hulst (1981, p. 176) ... 

---