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



upon intraparticle interference effects) increases as tlie wavelength of scat- 

 tered radiation decreases. By the use of radiation in the X-ray region it is 

 possible to determine sizes, and, to some degree, shapes of particles as small 

 as the smallest viruses, since at very short wavelengths the intensity of 

 scattering from such particles falls off very rapidly with the angle of scattering 

 6. In fact, the scattering dissymmetry is so great for ordinary X-radiation 

 that values of 6 only extremely close to zero can be measured, hence the term 

 "low-angle scattering". It would be convenient if considerably longer wave- 

 lengths could be used, but since aqueous fluids are quite opaque to these 

 the experimental difficulties of measuring at ^ «i; 0° must be put up with 

 (see review by Edsall, 1953). 



a. Particles of Any Shape. The derivation of the relevant scattering 

 equations follows the same pattern as that for the light-scattering equations, 

 except that individual electrons are taken as the source of the scattered 

 radiation instead of induced oscillating dipoles within the macromolecules. 

 Also interparticle interference effects are neglected m the derivations, as in 

 the case of light scattered by the molecules of a gas. The general form of the 

 scattering envelope for molecules large compared with the wavelength of the 

 X-rays has a maximum centered at ^ = 0° with minima and subsidiary 

 maxima as 6 increases. The central maximum has a shape that is approxi- 

 mately Gaussian for particles with centrosymmetry, and in this angular 

 region the equation for the scattered intensity can be written (Schmidt 

 et al., 1954; Guinier, 1939): 



<t>^-{h, R) = e" 3 (34) 



where </> is the amplitude of scattering, h = r , and R is the radius 



A 



of gyration of the centrosymmetrical particles. This equation is valid what- 

 ever the shape of the particles. Therefore, if log ^ is plotted against (sin ^/2)2 

 the shape of the straight line will yield a value of R. To obtain from a know- 

 ledge of R the actual radius of the particle requires either additional informa- 

 tion or an assumption regarding its shape. For spherical particles R^ == f a^, 

 where a is the particle radius. 



6. Spherical Particles. Low-angle X-ray scattering has been used particu- 

 larly for determining the size of spherical virus particles for which an addi- 

 tional equation (not restricted to small angles of scattering) is applicable: 



"3 (sin ha — ha cos ha 



nh, a) = 



(35) 



(ha) 

 :7T sii 



and minima which can be tabulated as functions of 6 and of a. Thus, by 



4-77 sin ^/2 



where a is the particle radius, and h = r . This equation has maxmia 



A 



