Interpretation of small-angle scattering functions of dilute solutions and gases. A representation of the structures related to a one-particle scattering function

Small-angle scattering gives a much poorer resolution of the structure than does diffraction by perfect crystals, i.e. the loss of information due to the random orientations of the scattering molecules is far greater than that known from the phase problem. For a quantitative comparison the scalar field functions in physical and reciprocal space are expressed as a series of spherical harmonics Y_lm. From the rotational properties of spherical tensors it is deduced that the orientation of the partial structures described by the sum of the multipole components belonging to the same l has no influence on small angle scattering. There are no interference terms between these partial structures, i.e. the partial small angle scattering functions arising from the partial structures superimpose independently. Structures giving the same small angle scattering can be generated by displacing the coordinate system and rotating the partial structures in an arbitrary manner and sequence. The calculations are greatly facilitated by the properties of the 3-j and 6-j coefficients widely used in nuclear physics. The Hankel transformations of the multipole components are reduced to an algebraic problem by the introduction of Laguerre polynomials.