General principles of light scattering
المؤلف:
Peter Atkins، Julio de Paula
المصدر:
ATKINS PHYSICAL CHEMISTRY
الجزء والصفحة:
ص657-658
2025-12-16
42
General principles of light scattering
When the oscillating electric field of electromagnetic radiation interacts with the electrons in a particle, an oscillating dipole moment develops with a magnitude proportional to the polarizability of the particle and the strength of the field (Section 18.1). Elastic light scattering is observed as the oscillating dipoles in the particle radiate at the same frequency as the frequency of the exciting electromagnetic radiation. The term elastic refers to the fact that the incident and scattered photons have the same frequency and hence the same energy. If the medium is perfectly homogeneous, as in a perfect crystal, the scattered waves interfere destructively in all directions except the direction of propagation of the exciting radiation. If the medium is inhomogeneous, as in an imperfect crystal or a solution of macromolecules, radiation is scattered into other directions as well. Scattering of light by particles with diameters much smaller than the wavelength of the incident radiation is called Rayleigh scattering (Fig. 19.3). This type of scattering has several characteristic features.
1 The intensity of scattered light is proportional to λ−4, so shorter wavelength radiation is scattered more intensely than longer wavelengths.
2 The intensity of scattered light is proportional to the molar mass of the particle.
3 The intensity of scattered light depends on the scattering angle θ (Fig. 19.3). In practice, data are collected at several angles to the incident laser beam (Example 19.3).
4 For very dilute solutions excited by plane-polarized light, the Rayleigh ratio, R θ, a measure of the intensity of scattered light at a given scattering angle θ, is defined as
where I is the intensity of scattered light, I0 is the intensity of incident light, r is the distance between the sample and the detector, φ is the angle between the plane of polarization of the incident beam and the plane defined by the incident and scattered beams (see the inset in Fig. 19.3). For a solution of a polymer of mass concentration cP, the Rayleigh ratio may be written as
Here nr,0 is the refractive index of the pure solvent (see Comment 18.6 and Appendix 3), (dn/dcP) is the change in refractive index of the solution with concentration of polymer, V is the volume of the sample, and NA is Avogadro’s constant. The para meter Pθ is the structure factor, which takes into account the fact that scattering may occur from different sites of the same molecule and interference between scattered rays becomes important when the wavelength of the incident radiation is comparable to the size of the scattering particles. When the molecule is much smaller than the wavelength of incident radiation, Pθ ≈ 1. However, when the size of the molecule is about one-tenth the wavelength of the incident radiation, we show in Further in for mation 19.1 that
where Rg is the radius of gyration of the macromolecule, the radius of a thin hollow spherical shell of the same mass and moment of inertia as the molecule (Fig. 19.4 and Section 19.8). Table 19.1 lists some experimental values of Rg.
Fig. 19.3 Rayleigh scattering from a sample of point-like particles. The intensity of scattered light depends on the angle θ between the incident and scattered beams. The inset shows the angle φ between the plane of polarization of the incident beam and the plane defined by the incident and scattered beams. In a typical experimental arrangement, φ = 90°.
Fig. 19.4 (a) A spherical molecule and (b) the hollow spherical shell that has the same rotational characteristics. The radius of the hollow shell is the radius of gyration of the molecule.
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