Scattering Intensity Distribution

The scattering of light, X-rays, or neutrons by atoms in matter provides information on the structure and dynamics of the material being sampled. The scattering experiments depend upon measuring the scattering intensity distribution, generally as a function of scattering angle. Presented here is a brief description of the theory for the elastic, coherent scattering of X-rays and neutrons. Coherence refers to the fact that the amplitudes of the scattered waves are additive and hence can produce interference. Elastic scattering means that there is no energy difference between the incident and scattered waves. When X-rays or neutrons are scattered coherently and elastically by atoms, the scattered waves interfere in a manner related to the spatial distribution of the atoms in the sample. This interference provides the foundation for small-angle scattering and crystallography applications. The basic principles of light scattering are the same as those for X-rays, but the wavelengths are much longer and different formalisms are used. Light scattering is therefore discussed as a separate topic in this series.

X-rays and neutrons can be considered as plane waves with wavelengths, l, and can be represented as

where kt = 2p/l is the amplitude of the incident wave vector which is taken to be in the direction z. A wave scattered by an atom at a position in space designated by the vector r will be a spherical wave of the form

where kf is the scattered wave vector, and A is the scattering amplitude for the atom. X-rays are electromagnetic radiation and are scattered via interactions with the electrons in a sample. Thus, for X-rays, A is proportional to the number of electrons in the atom and is generally designated as f Because X-ray wavelengths are of the same order as the dimensions of the electron clouds of the scattering atoms, there is an angular dependence to X-ray scattering amplitudes. Neutrons are neutral particles and are scattered by atomic nuclei. Neutron scattering amplitudes are generally given in units of length and are designated as b. Because nuclei have dimensions much smaller than the wavelengths of neutrons used in scattering experiments, b values have no dependence on scattering angle. The total coherent scattering from a molecule made up of atoms is the summation of the coherent scattering over all atom pairs (i,j):

whereis the scattering vector equal to the difference between the incident and scattered wave vectors; its amplitude is 4p(sin q)/l, where 2q is the scattering angle (Fig. 1).

Q is also the momentum transfer vector. In scattering experiments, one can describe molecules as continuous density distributions rather than sums of discrete point atom scatterers. Scattering densities are calculated as the sum of the scattering amplitudes of atoms within a finite volume element divided by its volume,The total coherent, elastic scattering, /(Q), from a molecule in a vacuum then can be written as

where r(r) is the scattering density distribution within the molecule, and the integration is over the volume of the molecule. For a randomly oriented molecule in a solvent with scattering density rs, equation (4) becomes

The brackets <H> indicate averaging over all orientations of the particle, and Dr(r) = r(r-rs) is the scattering density difference or "contrast" between the molecule and the solvent (1, 2). Equation (6) is known as the Debye equation 3 and is the basic equation for scattering. The total scattering intensity distribution for a solution of monodisperse molecules will be directly proportional to both the number density (concentration) and the square of the molecular weight of the particles. Analysis of the scattering intensity distribution at small angles (small-angle scattering) yields structural parameters such as the radius of gyration, R molecular weight, M, and the vector length distribution function, P(r), for the scattering particle. Contrast variation techniques involve the deliberate manipulation of contrast in order to increase the information content of the scattering experiment. For polydisperse solutions, or a solution of nonidentical particles, the time- and ensemble-averaged scattering intensity distribution function is measured; hence average structural parameters are determined.

Figure 1. Geometrical representation of the scattering vector Q. k ; and ky are the incident and scattered wave vectors, respectively, of a wave that interacts with an atom at a point r from an arbitrary origin O.

For scattering from samples ordered in one, two, or three dimensions, the scattering from an individual molecule is convoluted with the repeating lattice structure to yield a diffraction pattern that has discrete intensity maxima. These diffraction patterns can give information on the repeat distances in the lattices, as well as provide higher resolution structural information on the ordered molecules. For well-ordered three-dimensional crystals, complex diffraction patterns are obtained that can be indexed according to the crystal lattice indices h, k, l. The unit cell scattering is described mathematically in terms of the square of the structure factor F^, which is the ratio of the radiation scattering by any real sample to a point scatterer at the origin:

where x, y, and z are the coordinates of each atom in the crystallized molecules in one unit cell, and the summation is over all atoms. Diffraction data from three-dimensional crystals of biological polymers (proteins, polynucleotides) can be used to solve the structure of the crystallized polymer at high resolution (crystallography) if one knows both the phases and amplitudes of the structure factors Fhki The structure factor amplitudes are readily calculated as the square root of the measured intensities of the diffraction peaks. These amplitudes are combined with experimentally and/or theoretically deduced model phases in order to calculate (by Fourier transformation) an electron density distribution function that is usually interpreted using the known contiguous sequence of chemical groups in the polymer.