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 experi- mental 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 de- scribed 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 in- cident 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 mathe- matical formulation, an example of the adaptation of the ex- perimental 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. is generated by the distribution of distances within the indi- The use of constraints is a part of bridging and, in fact, a very vidual molecules. In addition to Iint(s), there is a steeply falling important part. It was discussed as a separate matter here be- background intensity function, Ib(s), upon which the molecular cause of the close connection between constraints and the scattering function is superimposed. The total intensity of problem of ambiguity. In this article, the characteristics of scattering, It(s), is several diffraction techniques will be described and their ap- plications will be illustrated. In the description, examples of the It(s) = Iint(s) + Ib(S). [1] use of constraints and other aspects of bridging should be easily The molecular scattering intensity Im(s) is obtained from It by recognized. forming (4) Im(S) = [It(s)/Ib(s)] - 1 GASEOUS MOLECULES = Iint(S)/Ib(S) [2] Electron diffraction is the technique of choice for the investi- It may be written gation of gaseous molecules because of the relatively high nEn Co scattering power of atoms for electrons compared to that for I-m(s) = E U P1j(p)(sin sp/sp)dp (i j), [3] x-rays and neutrons. t=1 1=1 0 Diffraction experiments are, in essence, quite simple. The in which n is the number of atoms in the molecule, the coeffi- incident beam is made to intersect a target material, and the cients cij are characteristic of the ith and jth atoms, Pjj(p)dp scattering pattern of the beam formed by the target is recorded. is the probability that the distance between the ith and jth A diffraction pattern for gas molecules is radially symmetric atoms has a value in the interval p and p + dp, sin sp/sp is an and appears to the eye as having diffuse maxima and minima interference function representing the contribution from a pair Downloaded by guest on September 28, 2021 Applied Physical and Mathematical Sciences: Karle Proc. Nati. Acad. Sci. USA 74 (1977) 4709 of atoms separated by the distance p, and s is the scatt rug superimposed on a steeply falling background was enhanced variable given in terms of earlier defined quantities, by the use of a rotating sector introduced by Finbak (9) and P. P. Debye (10). In addition, a device for rotating the photo- s = [4r sin (0/2)]/X. [4] graphic plates while they are being traced on a microdensi- Debye (5) showed that the application of Fourier transform tometer was developed (11) which minimizes the effects of the theory to Eq. 3 would give valuable information if the coeffi- graininess of the emulsion and enhances the quality of the cients cqj were independent of the scattering variable s. The densitometer traces. Fourier transform function is The optimization of the shape of the background function, Ib(s), was effected by the introduction of a concept that has D(r) = (2/ir)1/2 J'sIm(s) sin srds [5] played a valuable role in structure research. In addition to maintaining the smoothness of the function in the course of the in which D(r) is a function of the distance variable r. For cij adjustment-i.e., the background must be featureless compared independent of s, integration of Eq. 5 gives to the oscillations in the molecular intensity function-im- n n portant use was made of a special characteristic of Eq. 6. It is D(r) = (7r/2)1/2 E Ecn P0(r)/r (i # J) [6] noted that the ci1 are positive numbers and the probability i=1 j=1 functions Pi are non-negative. Therefore D(r) is a non-negative The calculation of Eq. 5 with the use of the molecular intensity function. The requirement that D(r) be non-negative affords function, Im(s), leads to a function rD(r) that is shown by Eq. a strong constraint on Ib(s) because it must generate an Im(s) 6 to represent a weighted sum of the probability distributions in Eq. 2 that, when substituted into the integrand of Eq. 5, for all the interatomic distances in a molecule. These probability produces a non-negative function. distributions comprise a representation of the structure. For In practice, a damped integral is computed instead of Eq. simple molecules, the use of chemical insight permits the de- 5, but the same non-negativity criterion applies. The integral duction of the atomic arrangement easily from a knowledge is of the equilibrium interatomic distances. There were several problems that had to be solved before this f(r) = (2/ix)1/2 ,f ax sIM(s) exp(-as2) sin srds. [7] theory could be applied to the investigation of molecular structure. The coefficients of the interference function in Iint(s), The value of a that is chosen is just large enough to permit the as obtained from experiment, are rapidly decreasing functions replacement of o by Smax without significant error. of s. The experimentally measured molecular intensity function Molecules usually undergo harmonic vibrations to a good required for Eq. 5 terminates at some value of s, Smax, before approximation. Some cases of internal librations about bonds its contribution to the integral becomes negligible. Finally, the can be significant exceptions. In the case of harmonic vibrations, background intensity is not precisely represented by theoretical rD(r) or Eq. 6 can be written functions because of the effects of chemical binding and ex- n perimental error. The relatively weak signal given by Iint(s) is rD(r) = (ir/2)1/2 E Cq(hj/rr)1/2 difficult to measure accurately when superimposed on the i=l j=1 steeply falling Ib(s) of Eq. 1. X exp[-hqj(rjj - r)2] (i # i) [8] Ways were found to overcome these problems. The purpose in mentioning them in a brief review is not only to afford insight in which (2hij)-1/2 = (12, ) 1/2 iS the root mean square ampli- into the manner in which theory and experiment have been tude of vibration between the ith and jth atoms. The molecular combined into a workable system but also to illustrate how the intensity function to be used with Eq. 5 to obtain D(r) defined philosophic implications of the solution to one of the problems by Eq. 8 is given to good approximation by led to the solution of a key problem in another area of structure Im(S) = z exp(-(l2,J)s2/2)sinsrq/srqj (i 6 J). [9] analysis-namely, crystal structure determination. i=1 j=12cii A molecular intensity function having reasonably constant coefficients over most of the experimental range was obtained It has been shown (4, 12) that, if rD(r) is defined by Eqs. 5 and from It(s) or Iint(s) by dividing by the background function (4), 8, rf(r) defined by Eq. 7 has the form as shown in Eq. 2. Calculations show that the so obtained (re cij rf(r) = (r/2)1/2 F L cij(hij/ir)1/2 exp[-hsj(rij - r)2 Eq. 3) are fairly constant, except for the first few values of s, i=1 J=1 if the atomic numbers of the ith and jth atoms do not differ greatly. As the discrepancy in atomic number increases, further /(4ahij + 1)]/(4ahij + 1)1/2 (i $ j). [10] alterations in the handling of the data are required to maintain Examination of Eq. 10 shows that the effect of introducing the essential constancy of the coefficients. This matter has been exp(-as 2) is to broaden the peaks representing interatomic analyzed by Bonham and Ukaji (6) and Kimura and Iijima (7), distances in a known way and that the positions of the maxima and a suitable calculation has been proposed by Bartell et al. are preserved. (8). To handle the region of very small s units where the in- An application of this theory is illustrated by the investigation tensity data may not be measured and the variation of the of the structure of the molecule of dimethyl diselenide by coefficients would be the greatest, there is attached a theoretical D'Antonio et al. (13). The radial distribution function rf(r) is function computed from constant coefficients (4). shown in Fig. 2. It was computed from the experimental mo- As described above, the problem concerning the termination lecular intensity function, Im(s), with the use of Eq. 7. The of the molecular intensity data, Im(s), at some Smax is handled function is seen to be non-negative and its composite peaks are by a damping function (2, 3), exp(-as2), in which a is a suitably decomposed into individual component distances. The de- chosen constant, that is introduced into the integrand of Eq. 5. composition is fairly readily carried out because the shapes and A subsequent theoretical analysis eliminates the effect of the areas for the individual interatomic distances are known. Dis- damping function on the final results (4). tances involving the weakly scattering hydrogen atoms appear The relatively weak signal from the interference scattering in Fig. 2. The direct information afforded by this experiment Downloaded by guest on September 28, 2021 4710 Applied Physical and Mathematical Sciences: Karle Proc. Nati. Acad. Sci. USA 74 (1977)

Se-Se ~Se--HS Se--HLI rf(r) C--Se C-Se Se ...HL2 C-H 11Se..HL3 A/ 'S' - I 0.0A 1.0 2.0 3.0 4.0 5.0 6.0 r FIG. 2. Experimental radial distribution curve, rf(r), for dimethyl diselenide, partially decomposed into the contributions from com- ponent distances. Note the essential non-negativity of this function which is expressible in terms ofthe probability of finding interatomic distances in the molecule. [From D'Antonio et al. (13), by permis- sion.]

concerns interatomic distances and their root mean square amplitudes of vibration, as represented by the peaks for the individual component distances. Given the distances, it is pos- sible to construct a model of the molecule shown in Fig. 3. Various views of the molecule are presented in Fig. 4 which shows, particularly in Fig. 4b, the characteristic approximately 900 dihedral angle formed by the atoms attached to a pair of (C) Group VIB elements bonded together. In this case the dihedral FIG. 4. Views of dimethyl diselenide. (a) Along one C-Se bond; angle formed by CSeSeC was found to be 87.50 + 4°. (b) along Se-Se bond; (c) along line connecting the two carbon atoms. A calculation of the molecular intensity can be based on the [From D'Antonio et al. (13), by permission.] structural model by use of Eq. 9 and values of the Ciq computed from theory, investigation of molecular structure show that the adjustments of the theory and the treatment of the experimental data have Cj (fi(S)fi(s)I/ E1SI(s) + I(S)12] led to a technique that provides detailed and accurate infor- mation concerning the structures and average internal motions X cos[n(s) - n(s)] [11] of molecules. Applications have concerned problems of mo- lecular configuration, variations in as in which fi/S2 is the atomic scattering factor for electrons for bonding, bond distance a function of atomic environment, motion atom is is the inelastic scat- vibrational including the ith which tabulated (14), Si(s) special studies of motion tering function for the ith atom which is also tabulated and internal torsional and the associated (15), potential barriers hindering rotation, preferred orientation in the are numbers which be written = fj complex may f, If, I conformers, conjugation, aromaticity, and unusual species such exp[ini(s)]. A calculation based on Eqs. 9 and 11 was performed for diselenide and with the as free radicals and molecular fragments. There are several dimethyl compared experimental reviews covering various aspects of these subjects (16-22). molecular intensity curve. The result is shown in Fig. 5, where A fruitful area which has been receiving increased attention the difference curve indicates good agreement. in recent years has been the combination of spectroscopic and The numerous applications of electron diffraction to the electron diffraction data to enhance the accuracy of molecular structure results. Spectroscopic information of interest concerns normal vibrational frequencies and rotational constants. Root mean square amplitudes of vibration can be computed from force models combined with knowledge of normal vibrational frequencies, as shown by Morino and collaborators (23, 24) and discussed extensively by Cyvin (25). Rotational constants are valuable for obtaining accurate structures of simple molecules and for providing additional constraining information in the

0.0[ SIM(S) 0o0

DIFF. 0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 36.0 40.0

S FIG. 5. Experimental and theoretical molecular intensity func- FIG. 3. Molecular parameters for dimethyl diselenide. [From tions for dimethyl diselenide and their difference. [From D'Antonio D'Antonio-et at. (13), by permission.] et al. (13), by permission.]

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v \- J\I / Si~ 0 4 8 12 16 0 4 8 12 16 S r FIG. 6. A regular tetrahedron of oxygen atoms surrounding a FIG. 7. (Upper Left) Experimental interference intensity, si(s), silicon atom (black) located at the center of gravity of the tetrahedron, for 100 A amorphous Si film prepared by vacuum deposition (Stage equidistant from each oxygen atom. The grouping forms a silicate I). (Upper Right) Radial distribution function, rG(r), (Stage I). group found, for example, in silica glass (SiO2). (Lower Left) Experimental interference intensity, si(s), for the same film after ambient oxidation for 2 years (Stage II). (Lower Right) study of complex molecules by electron diffraction. This subject Radial distribution function, rG(r) (Stage II), indicating, by the has been reviewed by Kuchitsu (26) and discussed further by growth of relevant peaks, the formation of silicate tetrahedra. [From Kuchitsu and Cyvin (27). D'Antonio et al. (29), by permission.] AMORPHOUS MATERIALS regions provides the clue that structural ordering may be The word "amorphous" evokes the concept of formlessness, present. For ordered regions, the presence of some values for even at atomic dimensions. However, this is not necessarily true the distances is much more probable than it is for others. in the case of amorphous materials and quite often is not so. Ordered regions may occur as isolated units separated from Substances regarded as amorphous do not possess one-, two-, one another by a disordered matrix. It is also conceivable that or three-dimensional periodicity, but there are many structural they may have an essentially continuous bonding topology. The possibilities in the range between a completely random ar- uniformity of the radial distribution function at large distances rangement of atoms and periodicity. It is often found, for ex- then could be due to long-range distortions of ordered regions. ample, that the basic structural unit in an amorphous material Liquids as well as solids can show ordering. Small groupings of is a regular arrangement of atoms, such as the silicate tetrahe- atoms can form regular geometric arrangements that persist dron in silica glass in which each silicon atom is found essentially in the liquid state. equidistant from four oxygen atoms situated at the corners of The same problems that arose in bringing theory and ex- a regular tetrahedron (Fig. 6). Such groupings of atoms fur- periment together in the application of electron diffraction to thermore can form extended ordered regions consisting of rings the structure determination of gaseous molecules also arise in or chains of regularly placed structural units. This type of or- the diffraction investigation of amorphous materials. They have dering has been termed "structural ordering" (28) and contrasts been treated in much the same way, although there are some with crystallographic ordering which includes the property of additional considerations because of the much greater com- periodicity. plexity of amorphous materials compared to gaseous molecules. Amorphous materials are usually isomorphous in the mac- This is manifested by the large numbers of interatomic distances roscopic sense. This implies that interatomic distances occur that are unresolved in the radial distribution functions and their uniformly over all orientations in space in a typical sample unlimited range. subjected to diffraction analysis. As a result, a radially sym- X-ray and neutron diffraction as well as electron diffraction metric diffraction pattern is obtained similar to that obtained have been applied to amorphous substances. Electron diffrac- for gas molecules (Fig. 1). Again, to the eye, the diffraction tion at an accelerating voltage of about 50 kV can, in fact, only pattern appears to be composed of diffuse maxima and minima. be applied to surfaces or thin films with thicknesses of not more This contrasts with diffraction patterns from polycrystalline than several hundred angstroms. The beam would be com- materials which have sharp maxima. As the number of unit cells pletely absorbed at greater thicknesses. Multiple scattering is in the crystallites of a polycrystalline material becomes small, an additional complication in electron diffraction of solids. For the diffraction patterns become diffuse and resemble patterns thick samples, x-ray and neutron diffraction become the ordinarily attributed to amorphous materials. techniques of choice. Optimal thicknesses for x-rays depend The radial distribution of distances for an amorphous ma- upon atomic composition, but they may be as large as tenths terial (Fig. 7) differs somewhat from that for a sample of gas- of a millimeter. For neutron diffraction the sample may be an eous molecules (Fig. 2). There is an upper value for distances order of magnitude larger yet. Intensities from x-ray and in a molecule, but there is no upper value for an amorphous neutron diffraction experiments and some electron diffraction material except for the dimensions of the sample which are experiments as well (30-32; M. Fink, P. G. Moore, and D. C. usually very large compared to the distance range showing Gregory, unpublished data) are collected by use of scanning and structural order. Beyond the range of structural order, the radial electronic recording techniques rather than photographic distribution function becomes a positive constant related to the methods. Allowing more time to measure the intensity in terms macroscopic density of the material. It represents the uniform of counts or total charge at the larger scattering angles can re- probability of finding distances of any length greater than the place the use of the mechanical sector which is employed in dimension of the ordered regions. The nonuniformity of the photographic recording procedures. radial distribution function for the distances within the ordered Some alternative procedures have been developed to address Downloaded by guest on September 28, 2021 4712 Applied Physical and Mathematical Sciences: Karle Proc. Natl. Acad. Sci. USA 74 (1977) the problems of data reduction. Warren and Mavel (33) have In the application of the Fourier transform, Eq. 14, the same developed a fluorescence excitation method for removing matters must be addressed as in the case of gaseous molecules. Compton scattering from diffracted x-ray intensities. The As noted previously, the coefficients of the interference func- various apparatus for collecting scattered electrons, referred tions comprising the reduced intensity, i(s), should be essentially to above, have energy filters that can remove inelastically constant and i(s) should be appropriately treated to take into scattered electrons. It is not necessary to do this, however, and account its contributions beyond smax. These matters are con- data reduction procedures have been developed (29, 34) that sidered in the several data reduction procedures cited (29, 34, remove the background scattering without resorting to exper- 35). In these procedures, constraints are based on the charac- imental means. Another feature of these procedures (29, 34) teristics of p(r). At large values of r, p(r) approaches a constant is the removal of termination errors by a special treatment of value representing the uniform distribution of interatomic the shortest distances in the structure, the only ones that con- distances, whereas at small values of r, where distances do not tribute significantly to the interference scattering beyond the exist, p(r) should be zero. Other constraints are based on the range of experimental data collection. Alternative treatments requirements that the value obtained for Po from the structure of the physical constraints in data reduction procedures for analysis be consistent with measurements of the bulk density amorphous materials have been presented by Kaplow et al. (35) and that areas beneath the peaks in the distribution function and Warren (36). be consistent with atomic scattering factors and coordination A brief outline of the treatment of electron diffraction data numbers. affords the opportunity to become familiar with the physical An example of an application of electron diffraction to a thin quantities of interest. The total intensity, I(s), scattered from amorphous film is given by an investigation of a partially oxi- an amorphous material is composed of the coherent interatomic dized film of silicon about 100 A thick by D'Antonio et al. (29). interference scattering, IC(s), which contains the structural The oxidation of the film was environmental. In the first stage, information, the coherent plus incoherent atomic scattering, the film was about 10% oxidized; after 2 years, it was about 47% IA(s), multiple scattering, IM(s), and extraneous scattering, oxidized. The evaluation of the amount of oxidation and its IE(S), characterization was of interest because of the effect on the conduction properties of the silicon film. The reduced intensity I(S) = IC(S) + IA(S) + IM(S) + IE(S) [12] functions and the corresponding radial distribution functions A reduced interference intensity, i(s), is obtained from the total are shown in Fig. 7 for both stages of oxidation. The buildup intensity, I(s), by dividing by a sharpening function, I.(s), ac- of the peaks associated with the oxidation is apparent. An eording to analysis of the peaks implies that the oxidation involves the formation of silicate tetrahedra similar to silica glass as a sepa- i(S) = IC(S)/IS(S) = I(S) - ITH(S) - B(s) [13] rate phase, rather than the diffusion of oxygen throughout the in which I,(s) is the sum of the absolute values of the squares silicon phase. of the elastic atomic scattering factors (14) for the unit of It is apparent that detailed and useful structural information composition, IAH(S) is the theoretical total atomic intensity (14, is obtainable from this type of analysis. However, the major part 15), and B(s) is an empirical background function that is de- of such information is derived from the first few peaks in the termined in the course of the data reduction by applying radial distribution function. The interpretation of the entire physical constraints on the nature of the radial distribution of radial distribution function in terms of the bonding topology distances. This radial distribution function is obtained by ap- over extended ranges of the structure is generally a very diffi- propriate Fourier inversion of the reduced intensity, i(s). If cult problem and one in which only modest progress has been for the of In test IA(S), IM(s), and IE(S) were known accurately, it would evi- made even simplest materials. order to the dently not be necessary to obtain B(s) empirically. IM(S) often suitability of a proposed structural model, a theoretical radial does not play a significant role in x-ray and neutron diffraction distribution function or a theoretical intensity function is experiments. For electron diffraction, it is important to verify computed from the model and compared to the experimental that the coherent is a featureless functions. The realization of a suitable model is based on such multiple scattering smooth, a comparison. function compared to IC(s). An estimate of its character can The arrangements of atoms in amorphous materials appear be obtained by extensive calculations (37, 38) in the realm of to be regular over extended ranges rather than random. The the dynamic theory of electron scattering first investigated by elucidation of this structural order and the long-ranging Bethe (39). bonding topology from one-dimensional scattering data, i(s), The Fourier sine transform of the reduced intensity can be is a current challenge. In addition to obtaining a good fit to the written experimental data, problems of uniqueness also must be con- sidered. 4wr2[p(r) - p0] = r si(s) sin srds [14] Diffraction experiments have been performed on a wide range of amorphous substances, metals and alloys, semicon- in which 4irr2p(r) is proportional to the probability of finding ductors, superconductors, glasses, and liquids. Review articles pairs of atoms separated by distance intervals (r,r + dr) in the have been written on investigations of thin films by Dove (46), sample, and po is a parameter which is definable in terms of the The subtraction of from amounts to the on amorphous metallic alloys by Gokularathnam (47) and bulk density. p0 p(r) Cargill (48), on glass structure by Wright (49), and on amor- subtraction of the uniform distribution of distances that occurs and Leonard at high values of r, so that p(r) -po is equal to zero at large r. phous catalysts by Ratnasamy (50). Details of this theory may be obtained from the early work of Debye (40) and Zernicke and Prins (41) and the texts by Guinier (42) and Warren (36). Theoretical investigations of neutron CONCLUDING REMARKS scattering relevant to structure determination have been made Structure determination by diffraction methods has progressed by Placzek (43) and Van Hove (44) and have been developed greatly over the past several decades in terms of the complexity to include molecular fluids by Blum and Narten (45). of the substances that can be investigated, the facility with Downloaded by guest on September 28, 2021 Applied Physical and Mathematical Sciences: Karle Proc. Natl. Acad. Sci. USA 74 (1977) 4713

which the analyses can be performed, and, in many sI the 2 . Morino, Y., Kuchitsu, K. & Shimanouchi, T. (1952) J. Chem. accuracy with which the final results can be obtaine "This Phys. 20, 726-733. development has been facilitated by the introduction of con- 24. Morino, Y., Kuchitsu, K., Takahashi, A. & Maeda, K. (1953) J. straints into the analyses. As the field of structure determination Chem. Phys. 21, 1927-1933. has progressed, fundamental information has become available 25. Cyvin, S. J. (1968) Molecular Vibrations and Mean Square Amplitudes (Elsevier, Amsterdam). to the many associated fields of science, often with dramatic 26. Kuchitsu, K. (1972) in MTP International Review of Science, results. A report has been prepared (51) containing, among Molecular Structure and Properties, ed. Allen, G. (Medical and other topics, a description of the range of application of the Technical Publ. Co., Oxford), pp. 203-240. various aspects of structure determination, their present status 27. Kuchitsu, K. & Cyvin, S. J. (1972) in Molecular Structures and and future potential. Vibrations, ed. Cyvin, S. J. (Elsevier, Amsterdam), pp. 183- 211. 1. Bijvoet, J. M. (1954) Nature 173, 888-891. 28. Karle, J. & Konnert, J. (1974) Trans. Am. Crystallogr. Assoc. 10, 2. Degard, C. (1937) Bull. Soc. R. Sci. Liege 6, 383-396. 29-43. 3. Schomaker, V. (1939) Meeting of the American Chemical So- 29. D'Antonio, P., Moore, P., Konnert, J. H. & Karle, J. (1977) Trans. ciety, Baltimore, MD. Am. Crystallogr. Assoc. 13,43-66. 4. Karle, I. L. & Karle, J. (1949) J. Chem. Phys. 17, 1052-1058. 30. Grigson, C. W. B. (1968) in Advances in Electronics and Electron 5. Debye, P. (1941) J. Chem. Phys. 9,55-60. Physics, Supplement 4, eds. Marton, L. & El-Kareh, A. B. (Ac- 6. Bonham, R. A. & Ukaji, T. (1962) J. Chem. Phys. 36,72-75. ademic Press, New York), pp. 187-290. 7. Kimura, M. & Iijima, T. (1965) J. Chem. Phys. 43, 2157- 31. Dove, D. B. & Denbigh, P. N. (1966) Rev. Sct. Instrum. 37, 2158. 1687-1689. 8. Bartell, L. S., Brockway, L. 0. & Schwendeman, R. H. (1955) J. 32. Graczyk, J. F. & Moss, S. C. (1969) Rev. Sct. Instrum. 40, 424- Chem. Phys. 23, 1854-1859. 433. 9. Finbak, C. (1937) Avh. Nor. Vldensk. Akad. Oslo, Mat. Natur- 33. Warren, B. E. & Mavel, G. (1965) Rev. Sci. Instrum. 36, 196- vidensk. Kl., No. 13. 197. 10. Debye, P. P. (1939) Phys. Z. 40, 66 and 404-406. 34. Konnert, J. H. & Karle, J. (1973) Acta Crystallogr. Sect. A 29, 11. Karle, I. L., Hoober, D. & Karle, J. (1947) J. Chem. Phys. 15, 702-710. 765. 35. Kaplow, R., Strong, S. L. & Averbach, B. L. (1965) Phys. Rev. A 12. Karle, J. & Karle, I. L. (1948) Am. Mineral. 33,767. 138,1336-1345. 13. D'Antonio, P., George, C., Lowrey, A. H. & Karle, J. (1971) J. 36. Warren, B. E. (1969) X-ray Diffraction (Addison-Wesley, Chem. Phys. 55, 1071-1075. Reading, MA). 14. Cox, H. L., Jr. & Bonham, R. A. (1967) J. Chem. Phys. 47, 37. Cowley, J. M. & Moodie, A. F. (1957) Acta Crystallogr. 10, 2599-2608. 609-619. 15. Tavard, C., Nicolas, D. & Rouault, M. (1967) J. Chim. Phys. 64, 38. Goodman, P. & Moodie, A. F. (1974) Acta Crystallogr. Sect. A 540-554. 30,280-290. 16. Bartell, L. S. (1971) in Physical Methods of Chemistry, eds. 39. Bethe, H. (1928) Ann. d. Physik 87,55-129. Weissenberger, A. & Rossiter, B. W. (Wiley-Interscience, New 40. Debye, P. (1915) Ann. d. Physik 46,809-823. York), pp. 125-158. 41. Zernicke, F. & Prins, J. A. (1927) Z. Phys. 41, 184-194. 17. Bastiansen, O., Seip, H. M. & Boggs, J. E. (1971) in Perspectives 42. Guinier, A. (1963) X-ray Diffraction (W.H. Freeman, San in Structural Chemistry, eds. Dunitz, J. D. & Ibers, J. A. Francisco, CA). (Wiley-Interscience, New York), pp. 60-165. 43. Placzek, G. (1952) Phys. Rev. 86,377-388. 18. Bauer, S. H. (1970) in Physical Chemistry, An Advanced Trea- 44. Van Hove, L. (1954) Phys. Rev. 95,249-262. tise, ed. Henderson, D. (Academic Press, New York), pp. 741- 45. Blum, L. & Narten, A. H. (1976) J. Chem. Phys. 64, 2804- 805. 2805. 19. Haaland, A., Vilkov, L., Khaikin, L. S., Yokozeki, A. & Bauer, S. 46. Dove, D. B. (1973) Phys. Thin Films 7, 1-41. H. (1973) Top. Curr. Chem. 53,1-119. 47. Gokularathnam, C. V. (1974) J. Mat. Sci. 9,673-682. 20. Hilderbrandt, R. L. & Bonham, R. A. (1971) Annu. Rev. Phys. 48. Cargill, G. S. III (1975) Solid State Phys. 30, 227-320. Chem. 22, 279-312. 49. Wright, A. C. (1974) Adv. Struct. Res. Diffr. Methods 5, 21. Karle, J. (1973) in Determination of Organic Structures by Physical Methods, eds. Nachod, F. C. & Zuckerman, J. J. (Aca- 50. Ratnasamy, P. & Leonard, A. J. (1972) Catal. Rev. 6,293-322. demic Press, New York), pp. 1-74. 51. National Academy of Sciences (1976) Status and Future Po- 22. Karle, J. (1978) in Lecture Notes in Chemistry (Springer-Verlag, tential of Crystallography, Report of a Conference (National Berlin), in press. Academy of Sciences, Washington, DC). Downloaded by guest on September 28, 2021