Structure Determination of Gaseous and Amorphous Substances

Proc. NatI. Acad. Sci. USA Vol. 74, No. 11, pp. 4707-4713, November 1977 Applied Physical and Mathematical Sciences Structure determination of gaseous and amorphous substances by diffraction methods: Philosophical concepts and their implementation (A Review)* (molecules/films/glasses/electron scattering/x-ray scattering) JEROME KARLE Laboratory for the Structure of Matter, Naval Research Laboratory, Washington, D.C. 20375 Contributed by Jerome Karle, August 12,1977 ABSTRACT Structure determination by diffraction methods useful constraints and their implementation are intimately is reviewed in terms of the philosophical and mathematical associated with progress in the field of structure analysis. A main aspects of the techniques used to investigate the atomic ar- purpose of this article is to discuss the analyses of the structures rangements in gaseous and amorphous substances. Both theory and experiment must be adapted to each other's limitations. This of gases and amorphous substances in the context of the philo- is accomplished often only after much time and study have been sophical and mathematical aspects of appropriate constraints devoted to the problems. Some areas of application are favored and the manner in which such constraints find application. by an overdeterminacy in the data, which helps to simplify the Crystalline substances will be discussed in a future article. mathematical aspects of the analysis. Other areas have relatively Another type of ambiguity concerns the possibility of ob- few data compared to the number of unknown parameters. In taining more than one structural result that satisfies the various such cases, special analytical procedures that use, for example, chemical and structural information are indispensible. Appli- criteria for an acceptable physical model. Usually, the more cations cover a broad range of problems and afford fundamental accurate and extensive the diffraction data are, the more readily information to many fields of science. a unique physical answer can be distinguished. However, we may ask, what if the physical answer is not unique for the type Structure analysis by diffraction methods is concerned with the of data obtained, no matter how extensive and accurate the data determination of the geometric arrangements of atoms in are? This occurs in crystal structure analysis, for example, when materials. At times the resolution is not sufficient to permit the no attempt is made to collect data at an x-ray wavelength that location of atoms. In such cases, the positions of aggregates of is sensitive to the absorption edges of some of the constituent atoms are determined. The average thermal motion of the atoms. Under such circumstances it is not possible to distinguish atoms also often can be described. This is achieved by an in- a structure from its enantiomorph (mirror image) when the terpretation of the patterns formed by suitable incident beams enantiomorphs are distinct. However, by use of a technique that are scattered by the substances of interest. The beams developed by Bijvoet (1) which uses additional information commonly used for such purposes are composed of electrons, from x-ray absorption, it is possible to distinguish enantio- x-rays, or neutrons, giving rise to the names of the various dif- morphs. This technique has played an important role in asso- fraction techniques. ciating the correct configuration (the absolute configuration) This article concerns mainly the philosophical aspects of with the handedness of optically active substances. It serves as structure determination-i.e., concepts and problems-rather an example of how an additional constraint-namely, addi- than the numerous applications. The latter can be found in tional experimental information-can facilitate the achieve- other articles referred to throughout the text. ment of a unique result. As far as other cases of ambiguity among physically accept- BACKGROUND able structures are concerned, such events have rarely, if ever, constraints occurred in the structure analysis of crystals with unit cells of Ambiguity and small to moderate dimensions, in which case the number of As a general rule, the information contained in diffraction diffraction data is large relative to the complexity of the patterns is ambiguous-i.e., it is possible to formulate non- structure. As the relative number of data decreases (for exam- physical structural models that satisfy the diffraction data. Some ple, for unit cells of increasing size involving macromolecular examples of non hysical features are interatomic distances of substances), the opportunity for physically acceptable am- unacceptable value and electron density distributions that are biguities to occur increases. For amorphous materials, which negative or have improper shapes. The existence of numerous show no periodicity, the diffraction data can be quite limited nonphysical models imposes a severe burden on procedures for relative to the number of known structural parameters, and the direct analysis of diffraction data. Clearly, it is necessary to possibility of having a number of physically acceptable struc- introduce some constraints on the analytical procedures in order tures becomes a serious problem. to limit or eliminate the ambiguities contained in the data. Several decades ago, when diffraction analyses were much less sophisticated than they are at present, it was common Bridging practice to determine structures by the trial and error fitting As is the case for many fields that are dependent upon advances of physical models to the diffraction data. The use of physical in the application of mathematical theory for their progress, models avoided the problem of nonphysical answers and, to a a considerable amount of "bridging" is required between the certain extent, masked their existence. It was only with the development of procedures for the direct extraction of struc- * By invitation. From time to time, reviews on scientific and techno- tures from diffraction data that the virtues of imposing physical logical matters of broad interest are published in the PROCEED- constraints on the analyses became apparent. The discovery of INGS. 4707 Downloaded by guest on September 28, 2021 4708 Applied Physical and Mathematical Sciences: Karle Proc. Natl. Acad. Sci. USA 74 (1977) theory as first obtained and its practical application to experimental data. This generally involves not only the development of useful algorithms but also the treatment of experimental data so as to be suitable for analysis. Bridging perhaps may be described as a mutual process in which theory is adapted to the limitations of experimental data and experimental data are adapted to the limitations of theory. For example, Fourier in- tegral transforms are useful in the analysis of diffraction data. However, they require data over an infinite range of scattering variable, [sin(O)/2)]/X, in which 0 is the angle between the incident and scattered beams and X is the wavelength. Clearly, infinity is not achievable for finite X, but the problems in practice do not arise from this fact. They arise because, even for experimentally useful wavelengths, the scattered intensities at the larger values for the scattering angles become difficult to measure. Nevertheless, such data are important for the Fourier transform theorems. The intensity measurements must be terminated at some value of the scattering variable and the omission of data beyond this value gives rise to "termination errors" that can be serious. There are several ways of handling this problem. In one, the diffracted intensity is multiplied by a damping function, introduced by Degard (2) and Schomaker (3) for gas electron in such a diffraction, way that the intensity FIG. 1. Diffraction photograph from the vapor of CC14. The radial data that are omitted no longer make a significant contribution symmetry is a consequence of the uniform occupation of all orienta- to the Fourier transform. This permits the theory to be used as tions in space by the molecules. if the data were infinite in extent as required by the mathematical formulation, an example of the adaptation of the experimental data to the theory. On the other hand, a question (Fig. 1). When measured, the maxima and minima are found immediately arises concerning the consequences on the results to be superimposed on a steeply falling background. The radial of altering the intensity function by damping it. To answer this, symmetry arises from the fact that the molecules occur in all an adaptation or modification of the applicable theory had to orientations in space. Structural information is obtained from be made. In the case of gas electron diffraction, this was an analysis of the diffraction pattern. achieved (4) by combining the theory of scattering with a There are a number of conditions that need to be attained convolution formula involving Fourier integrals. It was thus and controlled in the experiments, and this leads to some degree possible to use limited data and existing theory in a way that of complexity in the various types of diffraction apparatus. In enhanced the accuracy of the final result, a good example of the investigation of the structure of gases by electron diffraction, bridging. some key requirements are high vacuum, a fine electron beam At times the problems of developing a practical combination (less than 0.1 mm in cross section), control of incident voltage of experiment and theory may take a long time to sort out. to better than 1 part in 104, and cold traps for condensing the Under such circumstances, the mathematical theory generally jet of sample material. The experiments are generally carried remains in limbo. In the field of structure analysis, for example, out at 40-60 kV and usually require seconds or less for an ade- theory per se usually contains no significant lessons for the quate photographic recording. mathematician and, until experiment and new theory can be When a beam of fast electrons impinges on a jet of molecules, joined in a useful fashion, there is little of significance for the a portion of the scattering, the interference scattering, Iint(s), structure analyst.

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