272 H. K. SCHACHMAN AND R. C. WILLIAMS 



tliird method, involving the examination of crystal structures by X-ray 

 diffraction, utilizes scattered radiation that has come from ordered arrays of 

 particles and in which there is a high degree of coherence, resulting in pre- 

 dominant interference effects, A fourth method, electron microscopy, 

 utilizes electrons that have been scattered, but the optical arrangements are 

 such that these electrons can form a real image of the scattering particles. 

 a. Small-Particle {Rayleigh) Scattering in a Gas. If a single particle, sus- 

 pended in a medium with an index of refraction different from its own, is 

 illuminated with electromagnetic radiation, an oscillating dipole is estab- 

 lished whose strength is a function of the amplitude of the incident wave 

 motion and of the spherical polarizability of the particle. (We shall see later 

 that the polarizability can be related to the relative indices of refraction of 

 the particle and of the medium, but the polarizabilities of molecules are also 

 a function of their volumes. The larger the molecule the more easily is there 

 a separation of charges upon it, hence the greater is its polarizability.) This 

 oscillating dipole acts as a source of radiation, effectively "emitting" light 

 without change of frequency. If the incident radiation is impolarized the 

 expression for the intensity of the observed scattered light is: 



H = ^^^ (1 + cos2 6), where (28) 



ig = intensity of light scattered at the angle d 



lo^^ intensity of incident light (proportional to the square of its ampli- 

 tude) 



a = spherical polarizability of the particle 



r, = distance from particle to point of observation 



A = wavelength of the incident light in the medium 



6 = angle between the forward direction of the incident light and the 

 line between particle and observer 

 'ir. 



The ratio 



2n 



/. 



is usually written as Rq, Rayleigh's ratio. 



The scattered light is plane-polarized at ^ = 90°, with the degree of 

 polarization decreasing symmetrically as 0° and 180° are approached, at 

 which angles it vanishes. The intensity of scattering is a minimum atd= 90° 

 and has a value twice this minimum at 0° and 180°. The envelope of the 

 intensity of the scattered radiation, in any plane containing the incident 

 beam, has a shape somewhat between that of a dumbell and an ellipse, with 

 the major axis lying along the incident beam. 



A collection of randomly arrayed particles in a medium whose own 

 scattering is negligible will scatter light proportionately to the number of 

 scattering centers per miit voliune, allowing a simple addition to be made 

 of the scattering contributions of each particle. To write an expression for 



