From Current Literature

X-Ray Scattering at Small Angles in Colloidal Systems*)

In the three papers under review, one and the same problem is considered—the relation of the distribution of the intensity of X-radiation within the range of angles \(0.2\)–\(2.5\) degrees to the size, shape, and size distribution of colloidal particles.

The purpose of all these works is to calculate scattering curves for certain models and to compare them with experiment. Kratky distinguishes two types of systems, for which the character of the theory will be substantially different: a system of closely packed colloidal particles (the distance between particles is smaller than the particle size) and a dilute system. Kratky assumes that in the second case the distance between particles must be greater than the particle size. The American authors believe that this circumstance has no direct bearing on the theory, provided only that scattering of one particle by another may be neglected.

Be that as it may, it is precisely on this latter assumption that all three authors carry out their calculations. The validity of assigning a system to the class of dilute systems in each individual case may be tested experimentally, more precisely by comparing the data obtained in the experiment under consideration with data from other experiments.

For comparison one may use determination of the particle weight by the ultracentrifuge method. In some cases the shape of the particles can be judged from double refraction in flow. One may also, as the American authors do, compare the X-ray small-angle scattering data with data on line broadening for an analogous fine-crystalline material. Finally, direct information on the sizes and shapes of particles is provided by the electron microscope. At the same time, the authors emphasize the indispensability of the small-angle scattering method in a whole series of cases. The action of the electron microscope does not make it possible to study the solutions themselves. Removal of the solvent, as well as the action of fast electrons, may lead to sharp changes, for example, in such particles as protein particles. Moreover, sizes below 50 angstroms are not yet fully accessible to the electron microscope. For protein substances such sizes play a major role. The ultracentrifuge method makes it possible to judge only the particle size and gives no information about its shape or about the distribution of particles by size. The method of broadening of Debye lines is not suitable for colloidal systems. Thus, for scat—

) C. G. Shell and L. C. Roess, X-ray scattering at small angles by finely divided solids, J. Appl. Phys. 18, 295 (1947); L. C. Roess and C. G. Shell, X-ray scattering... II. Ibid 18, 308 (1947); O. Kratky, Die Abhängigkeit der Röntgenkleinwinkelstreuung von Grösse und Form der kolloiden Teilchen, Monatshefte f. Chemie 76*, 325 (1947).

of the method under consideration has its own particular field of application: protein substances, various forms of cellulose, aluminum hydroxide gels, metals in colloidal form, the most varied catalysts, etc.

In the paper by Schull and Ross the following problems are solved: scattering-intensity curves of X-rays are calculated for systems consisting of spherical particles, assuming that the particles are distributed by size: a) uniformly, b) according to Maxwell’s law, c) according to Gauss’s law, and d) according to the rectangular law. In the paper by Ross and Schull the problem is solved for spheroidal particles. By changing the parameters of the spheroid one can obtain results for particles of very different shape, ranging from disk-shaped to needle-shaped. The scattering problem is considered for uniform, Maxwellian, and rectangular distributions. In the calculations it was assumed that: 1) the total intensity is the sum of the intensities scattered by the individual particles (“dilute medium”); 2) the particles are oriented at random; 3) the primary beam is monochromatic, parallel, and of small cross section; 4) there is no refraction; 5) inside the particle the electrons are distributed uniformly; 6) there is no absorption within the particle. For all the cases considered, series of curves are given, so that the results of the calculations can be applied directly to experimental data.

The results of determining particle size from their calculations are compared by the American authors primarily with measurements of the broadening of the Debye lines of the corresponding crystalline material. This was done for hydrous aluminum oxide gel and microcrystalline bœhmite. For the average size of the particle and of the crystal an excellent agreement was obtained; namely, the corresponding figures obtained were 36 angstroms and 38 angstroms. Sometimes a large discrepancy is obtained, but predominantly in the natural direction—the sizes of the particles prove larger than the sizes of the crystals. This is natural because agglomeration of several crystallites into a colloidal particle can always be expected.

The authors also compared their data with the results of gas-adsorption measurements at low temperatures. These measurements make it possible to calculate the specific surface, i.e., the surface per unit mass. It is not difficult to show that the product of this quantity by the density of the particle is inversely proportional to the mean diameter. Experiment confirms the authors’ calculations.

Kratky, in his work, calculates scattering curves for particles of various shapes. The character of the particle-size distribution is not taken into account here, i.e., all particles in the colloidal system are considered identical. Kratky represents particles of various shape as an aggregate of touching spheres. He calculates the scattering intensity of such a system by Debye’s formula for gas molecules, taking as the “atomic factor” an expression calculated for a sphere with uniformly distributed electrons (in this respect the fundamental idea of Kratky’s calculation and that of the American authors are one and the same). A long particle is approximated by a linear row consisting of several spheres (6, 10, 12). A very long particle is regarded as an infinite row of spheres. A double row of spheres should, in the author’s view, represent a ribbon-like particle; these calculations are given for 4, 6, 8, 12, and an infinite number of spheres. Particles composed of spheres are also considered in the form of a semicircle, a ring, a spiral, etc. All calculations are carried through to the end, and numerous tables and curves are presented.

Only one example of the application of the theory is given, namely, the scattering curve of asbestos is calculated and compared with experiment.

Both in Kratky’s paper and in those of the American authors special calculations are made that take into account the influence of the shape of the slit on the scattering pattern; in Kratky this is done for a rectangular slit, and in the American authors’ papers—for rectangular and circular slits.

The experimental technique is discussed in all the papers; in greater detail, in the papers by the American authors. The experiment must be carried out very carefully. It is necessary to use filtered radiation (the American authors use radiation reflected from a crystal; Kratky proposes using a system of filters cutting off the radiation on both sides of the characteristic wavelength). The recording of the X-ray patterns must be carried out in a vacuum chamber. The system of diaphragms is important, as are the means of blocking the primary beam. In connection with the foregoing, it is clear that, even with good X-ray tubes, exposures may reach as long as 30 hours.

A. I. Kitaigorodskii