GENERAL CONSIDERATIONS

The object of an X-ray scattering experiment, whether in the small-angle region or in the usual domain of investigation, is the determination of the variation of the intensity scattered by a sample as a function of the scattering direction, this direction in general being defined by two para-meters. In the important particular case in which the scattering is circularly symmetric about an axis coincident with the incident beam, only one parameter, the scattering angle, is involved, and the object of the experiment is simply the determination of the relative value of J(h), with h = (47T sin B)/} •.

Experiments can also furnish a second quantity, less frequently employed, which is the absolute value of the scattering coefficient, a.

This is defined by means of the relation, I = I 0a dm d.Q, where I is the power scattered by the particle of mass dm in the solid angle d.Q, and I ₀ is the intensity of the incident beam striking the sample. The sample is assumed to be small enough to be non-absorbing.

3.1.1. OPERATIONAL PRINCIPLES

The method employed to realize the objectives discussed above is not different in principle from that used in all experiments in X-ray crystal-lography. Special difficulties are encountered, however, in investigating the scattering at very small angles.

1. Geometrical Definition of the Incident Beam. Following the notation of Fig. 22, let MM' be the portion of the sample which is irradiated. Each point of the sample will receive a beam of rays whose divergence depends on the constitution of the incident beam. Rays will converge at the point of observation P which have been scattered by each of the points of the sample through angles varying in an interval 2d() about a mean value, 20. The interval df) is practically independent of B, so that good definition of the scattering angle in relative value is more difficult to obtain, the closer a scattering angle of zero is approached.

Furthermore, there will always be an angular region inaccessible to experiment; this is the region between N and N' of Fig. 22, in which the scattered radiation received at any point is completely overshadowed by the much greater intensity of the direct beam at this point. Thus, to

83

84 SMALL-ANGLE SCATTERING OF X-RAYS

investigate scattering at small angles, it is necessary to reduce both the cross section and the divergence of the primary beam, the restriction being greater, the smaller the limiting angle of observation that is desired. As a result, the beams employed are much less intense than those used in ordinary techniques, so that a determination of the best geometrical conditions is essential.

x-~ra~y~~~~~~~~;::T'~~;;:;:;~~~~;;;;~~~~~R

source

N'

Observation D' plane Fig. 22. Slit system for a small-angle aoattering apparatus.

2. Parasitic Scattering. The measurement of the intensity received at the point of observation is a correct measure of the intensity scattered by the sample only if there is no parasitic scattering. The term parasitic scattering refers to the radiation received at the point of observation when the sample is withdrawn from the beam. If a Geiger counter or ionization chamber is employed as a detector, the parasitic scattering can easily be subtracted from the observed scattering to give the corrected value, but this procedure is acceptable only if the correction is small. If

photographi~ detection is employed, it is very difficult to make the correction by the above procedure. The reduction of the parasitic scattering is thus· the second important requirement, and here again the suppression is more difficult, the smaller the angles at which scattering is to be observed.

Thus we can say that the quality of a small-angle scattering apparatus is characterized by the power of the beam for a given fineness of dimensions and by the angle beyond which all parasitic scattering is eliminated.

EXPERIMENTAL EQUIPMENT 85 A third critical property of such an apparatus is the spectral purity of the primary radiation. We shall consider successively equipment without and with monochromatization, showing the different domains of application of each.

3.1.2. INFLUENCE OF THE MONOCHROMATIZATION OF THE PRIMARY RADIATION

Use may be made of either filtered or crystal monochromated radiation, depending on the nature of the sample to be studied.

1. The total radiation from the anode, with the usual filtering to remove the K{3, may be used in a study of low-angle crystalline diffraction effects that are analogous to the usual high-angle phenomena, differing only in that the effective lattice spacings are very large. These patterns contain lines, spots, or rings at well-defined angles, and the corresponding inten-sities are considerably larger than those at intermediate angles on the patterns. Thus, as with ordinary diffraction patterns, the diffraction effects due to the characteristic radiation emerge from the continuous background of diffraction and scattering caused by the continuous spectrum. Often the primary objective of such a study is to determine the position of the lines or spots, and in these circumstances even a rather strong parasitic scattering may be tolerated.

2. The opposite case is the study of continuous scattering of the type that has been described in the first part of this book. This continuous scattering is often extremely weak and is superposed on the scattering of various other origins, such as the inactive parts of the sample (the solvent, for example, when the scattering of particles in solution is studied).

Given the actual state of the theory, it is essential in this type of problem to have a precise evaluation of the function J(h).

It is easy to see that in certain cases the influence of the continuous spectrum may be considerable, since, in addition to the intensity J(h) due to the principal radiation of wavelength A.0 , one will also observe a scattering of the form

f ^{I ( ~ h)}

where j(A.) is the distribution function of the energy in the continuous spectrum. The effect of all the continuous spectrum can thus be large with respect to that of the characteristic radiation. Experiments by several authors have proved that an investigation cannot be made free from all objection without the use of monochromatized radiation.

When Geiger-counter detection is used, the elimination of the continuous spectrum by the double filter method of Ross (Kirkpatrick (1939)) is often

86 SMALL-ANGLE SCATTERING OF X-RAYS

sufficient, but this method is not easily applied with photographic techniques. The most practical and the most general method of pro-ducing monochromatic radiation involves the use of a cryBtal mono-chromator. Since the use of a monochromator profoundly modifies the geometry of the equipment, we shall study separately the system with collimation, designed for st~dies of crystalline diffraction, and the system wit:ii monochromatization, especially adapted to the study of the continuous scattering.