Electron Diffraction by Gas Molecules*

L. O. Brockway, California, USA

IV. INTERPRETATION OF ELECTRON DIFFRACTION PATTERNS

Quantitative determinations of intensity

The most widely used methods for interpreting electron diffraction patterns were developed for the purpose of accurately determining internuclear distances. Since a quantitative estimate of intensities proved sufficient for this purpose, a complete quantitative comparison of the theoretical dependence between intensity and scattering angle with experimental data has still not been made by anyone. Accurate measurement of intensities requires the use of electrometric methods, which is associated with two serious difficulties. First, obtaining a diffraction pattern for a continuous series of scattering angles requires a large number of separate measurements, while maintaining a constant amount of gaseous scattering substance in space, sufficiently small to obtain sharp diffraction effects, appears, as indicated above, to be extremely difficult. Second, an electrical recording device in a vacuum apparatus of ordinary dimensions would be located too close to the object under investigation and would capture all scattered electrons within a small solid angle, without distinguishing them by direction. Photographic film, on the other hand, records the entire diffraction pattern at once and makes it possible to distinguish the change in intensity for directions differing by extremely small angles. Since the angles at which diffraction maxima appear play the principal role in determining molecular dimensions, precise geometrical measurements on the electron diffraction pattern prove more important than quantitative determination of intensities. Therefore the photographic method is the most convenient in practical work.

* Continuation; see Uspekhi fizicheskikh nauk, 17, 175, 1937.

Approximate measurement of the intensity of scattered electrons can be carried out on a photographic plate. Some work has already been done on studying the behavior of various emulsions under the action of electrons[^36]. The reciprocity law, establishing that blackening is a function of the product of the intensity of the incident radiation by the exposure time (i.e., in the case of electrons, of the total electric charge incident on \(1\ \mathrm{cm}^2\)), remains valid for fast electrons within approximately a hundredfold change in the duration of exposure. If one plots along the abscissa axis the logarithms of the charge incident on \(1\ \mathrm{cm}^2\), and along the ordinate axis the corresponding values of the blackening, defined as the decimal logarithm of the opacity*, one obtains a characteristic curve similar to the curves obtained for visible light. Small exposures correspond on the curve to a nonlinear portion with increasing curvature; this is followed by a steeper, approximately linear portion, corresponding to an increase of the charge from tenfold to hundredfold. Further on, the saturation effect already appears. The contrast (or, what is the same thing, the slope of the linear part of the curve) increases, while the inertia (or the segment cut off by the continuation of the linear part) decreases with increasing voltage, so that the maximum blackening is reached sooner at a lower charge density, but at a higher voltage of the accelerating field. This effect, however, has a maximum lying near \(100\ \mathrm{kV}\), and for very high voltages the blackening, referred to unit charge, decreases as the accelerating potential increases. The form of the blackening curve for photographic plates used in certain special cases depends mainly on the speed of the electrons, the type and thickness of the emulsion, and, possibly, also—in the case of very fast electrons—on the nature of the material on which the emulsion is applied.

From these experiments it follows that, at those electron speeds which are ordinarily used in work with gaseous substances, on the blackening diagram for the emulsions studied the portion of the curve from a blackening of 0.2, below which microphotometric measurements are inaccurate, to a blackening of 1.5–2.0, above which the emulsions were not investigated, corresponds to a change in exposure within a range from fiftyfold to hundredfold. It will be shown below that the intensity of scattered electrons changes by more than a thousand times when the angle is varied within limits of interest in many cases. A quantitative measurement of the intensities could be attempted on a series of electronograms taken with different exposure times, although so far no one has succeeded in obtaining, sufficient for microphotometric measurements, blackening in the outer part of an electronogram that is not accompanied by a background darkening the whole picture.

* Opacity is equal to the ratio of the amount of transmitted light to the amount of incident light. (Translator’s note.)

Thus, for example, although maxima of twelve orders can be observed on the electronogram for \(\mathrm{CCl}_4\), microphotometry could be carried out only up to the seventh order.

Theoretical intensity curves for \(\mathrm{CCl}_4\)

The intensity distribution to be compared with the electronogram is given by formula (20). To illustrate this, let us apply this formula to the case of electron diffraction by \(\mathrm{CCl}_4\) molecules. To calculate the intensity it is necessary to base the calculation on a definite model of the molecule. Proceeding from chemical ideas about the tetrahedral direction of the valences of carbon, and taking into account that the dipole moment of the molecule is zero, we test a model in which the carbon atom is located at the center of the tetrahedron, while the chlorine atoms occupy the vertices of the latter. Equation (20) then assumes the following form:

\[ I=K\left\{f_C^2+f_{\mathrm{Cl}}^2\left(4+12\frac{\sin 2.87s}{2.87s}\right)+8f_Cf_{\mathrm{Cl}}\frac{\sin 1.76s}{1.76s}+ \frac{S_C}{s^4}+\frac{4S_{\mathrm{Cl}}}{s^4}\right\} \tag{21} \]

The distance \(\mathrm{Cl—Cl}\) was taken to be \(2.87\ \text{\AA}\), and the distance \(\mathrm{C—Cl}\), \(1.76\ \text{\AA}\). The ratio of the first to the second is determined by the geometry of the model and is equal to 1.633, but the absolute magnitudes of the distances have been chosen arbitrarily in the present equation. Definite magnitudes were chosen in order to obtain \(I\) as a function of \(s\). If the chosen model corresponds to reality, then the absolute dimensions are determined by shifting the curve representing this function along the axis of abscissae until it coincides with the intensity curve constructed on the basis of the electronogram. If the geometry of our model is incorrect, the two curves cannot be made to coincide by a simple shift along the axis of abscissae. If the model is chosen correctly, but the curve leads to distances noticeably different from those adopted at the beginning, then new values can be substituted into formula (20) and a second curve derived. In this way, by successive approximations, one can approach the true values of the interatomic distances.

In order to illustrate the relative significance of the various terms of equation (21), in Fig. 7 coherent atomic scattering, incoherent atomic scattering, and molecular scattering are each shown separately; the total effect, obtained by summing all three intensities, is represented by the fourth curve. The separation was carried out as follows:

\[ I_1=f_C^2+4f_{\mathrm{Cl}} \]

— coherent atomic scattering,

\[ I_2=\frac{S_C}{s^4}+4\frac{S_{\mathrm{Cl}}}{s^4} \]

— incoherent atomic scattering,

\[ I_3=12 f_{\mathrm{Cl}}^2\frac{\sin 2.87s}{2.87s}+8 f_C f_{\mathrm{Cl}}\frac{\sin 1.76s}{1.76s} \]

— molecular scattering

\[ I_4=I_1+I_2+I_3. \]

The scale of ordinates was chosen in such a way that the constant \(K\) in equation (21) is equal to unity. The values of \(F\) entering into the expressions for \(f\) were taken from the work of Pauling and Sherman\(^{18b}\); the values of \(S\) were taken from the tables of Bewilogua\(^{25}\). For values of \(s\) greater than 8, the scale of ordinates is increased one hundredfold. The curves are carried out to the value \(s=17\), which corresponds to the first seven “maxima” observed on the electronogram for \(\mathrm{CCl}_4\). The intensity of both kinds of atomic scattering decreases very rapidly with increasing angle; however, as is evident from the figure, the curve of incoherent scattering falls off still more rapidly. At very small angles the intensity of incoherently scattered electrons constitutes a large part of the total intensity, but already at \(s=1.8\) its magnitude falls for ras-

Fig. 7. Theoretical electron diffraction curves for \(\mathrm{CCl}_4\). For \(s>8\) the ordinates are increased 100 times.

of the substance under consideration below the corresponding value of the coherent atomic curve. For \(s=4\) the incoherent part of the scattering intensity amounts to no more than \(10\%\) of the total value. In other words, incoherent scattering will be intense near the central spot, while in the outer parts of the electron diffraction pattern it plays only an insignificant role.

Of special interest to us is the curve of molecular scattering \(I_3\), since the intensity of the latter is the only component of the total intensity that depends on the structure of the molecule. It is obtained as the result of the addition of strongly damped sinusoidal curves, and already after the first maximum its amplitudes become smaller than the weakening of the background produced by atomic scattering in the corresponding region; thus the curve of the total intensity shows not the actual intensity, but only its oscillations about the atomic background. The general form of this curve is confirmed by microphotometric measurements on the electron diffraction pattern (Fig. 8).

Fig. 8. Microphotometric curves of electron diffraction patterns of \(\mathrm{CCl}_4\), showing—\(A\) (left) the first three and \(B\) (right)—the first five visible maxima

As was mentioned above, this curve has not yet been checked by comparing observed and theoretical intensities for a continuous series of angles. The reason for this is partly the difficulties associated with precise quantitative measurement of intensity; thus, in passing from the first maximum, appearing at \(s \simeq 2.3\), the total intensity decreases from \(4.3\) to \(0.0027\), which corresponds to a ratio of \(1600:1\). Another, more important reason is the circumstance that, for the purposes of investigating molecular structure, precise measurement of intensities is not required. Molecular scattering accounts for such a small part of the total intensity that its oscillations are reflected only relatively weakly in the shape of the summation curve. In other words, the calculated changes of the total intensity are not proportional to the changes in the parameters of the molecular model that cause them. Even with good numerical agreement of the theoretical and experimental intensity curves, one still cannot assert that the structure of the molecule has been determined with the corresponding accuracy. For this it is first necessary to verify to what extent the molecular model can be varied without violating the agreement

between the corresponding theoretical curves and the experimental data.

In determining the structure of a molecule, it is desirable to obtain such agreement between theory and experiment as would respond with sufficient sensitivity to changes in the molecular-scattering curve. This condition is fulfilled in the neighborhoods of well-defined peaks on the intensity curve, which correspond exactly or approximately to actual diffraction maxima (as, for example, the first hump of the curve in Fig. 7). The correct theoretical curve must reproduce the shape and position of the maxima of the experimental curve in such a way that, by varying only the dimensions of the molecular model, one can obtain a sufficiently accurate coincidence. This method has so far been applied only by Pauling and Brockway in determining the structure of benzene[^14]. Microphotometry of the electron diffraction pattern showed a well-defined second maximum, reproducing the corresponding maximum on the theoretical microphotometric curve obtained by microphotometric reproduction of the theoretical curve referred to the intensity of the incident electrons. The result corresponds to the correct hexagonal model, in which the distances between neighboring carbon atoms are equal to 1.39 Å. The electron diffraction patterns showed an additional background, more uniform than the incoherent scattering in the corresponding region, owing in origin to multiple reflection, as well as to reflection from gas molecules scattered in the apparatus. However, this method, irrespective of the nature of the substance under investigation, can be applied only to one small segment of the curve of total intensity, for the maxima of the latter, generally speaking, are so broad and flat that even a small inaccuracy in the observed intensity is sharply reflected in the shape of the hump, and the parameters of the satisfactory molecular model fluctuate within excessively wide limits.

The desired accuracy can be achieved by varying, in different ways, another method. In this method the quantities \(s\) are determined for several points corresponding to characteristic features of the theoretical curve and are compared with the corresponding points of the experimental curve. In doing so, one dispenses with quantitative measurement, limiting oneself only to a qualitative estimate of the experimental and theoretical intensities being compared. Such methods require the expenditure of considerably less effort than methods connected with quantitative measurement of intensity, since the presence of an additional background, decreasing more slowly than the background caused by atomic scattering, does not affect the calculations. To verify the correctness of all these methods, a structure obtained by the electron-diffraction route is compared with the results of investigating the same substance by other methods—in particular, by analysis of band spectra and by X-ray methods.

The different variants of this method differ from one another in the manner of choosing the points to be compared on the intensity curves.

The relative merits of these depend on the accuracy of the results and the simplicity of application.

Visual Method

The first method, applied already by Wierl \(^{10}\), is the so-called “visual” method. In visual observation the electronogram reveals a system of sharply bounded ring-shaped regions, evidently corresponding to maxima and minima of photographic blackening. Generally speaking, such maxima and minima of blackening do not exist, but the eye perceives not the total intensity of the light transmitted through the negative being examined, but rather the relative change in the transmitted intensity (referred to the total intensity). The behavior of the eye in viewing electronograms is discussed in detail in the work of Pauling and Brockway \(^{37}\), from which it follows that the limits of sensitivity of the eye to fluctuations of intensity about some background are approximately determined by the ratio of the magnitude of the fluctuation to the intensity of the background; this proposition remains valid over wide limits of variation of the background intensity. These considerations explain the visual impression produced by electronograms and indicate that the eye tends to exaggerate the part of the total intensity \((I_{\mathrm{e}})\) due to molecular scattering, i.e., that the visual impression of an electronogram is determined to a considerable degree by the structure of the molecule.

In quantitative treatment, the positions of the visible maxima and minima on the electronogram are compared with the positions of the maxima and minima on the varied curve of molecular scattering. In accordance with the reasoning mentioned in the preceding paragraph, the theoretical curve (for the case of \(\mathrm{CCl}_4\)) has the form:

\[ \frac{I_{\mathrm{mol.}}}{I_{\mathrm{at.}}} = \frac{I_3}{I_1 + I_2} = \]

\[ = \frac{ f_{\mathrm{Cl}}^{2}\dfrac{12\sin 2.87\,s}{2.87\,s} + f_{\mathrm{C}}f_{\mathrm{Cl}}\dfrac{8\sin 1.76\,s}{1.76\,s} }{ f_{\mathrm{C}}^{2}+4f_{\mathrm{Cl}}^{2}+\dfrac{S_{\mathrm{C}}+4S_{\mathrm{Cl}}}{s^{4}} } = \]

\[ = \frac{ (Z-F)_{\mathrm{Cl}}^{2}\dfrac{12\sin 2.87\,s}{2.87s} + (Z-F)_{\mathrm{C}}(Z-F)_{\mathrm{Cl}}\dfrac{8\sin 1.76\,s}{1.76\,s} }{ (Z-F)_{\mathrm{C}}^{2} + 4(Z-F)_{\mathrm{Cl}}^{2} + S_{\mathrm{C}}+4S_{\mathrm{Cl}} }, \tag{22} \]

the factor \(\dfrac{1}{s^{4}}\), appearing in every term of the numerator and denominator, is omitted here. In practice formula (22) is reduced to a simpler form

\[ I=\sum_i\sum_j Z_i Z_j \frac{\sin s r_{ij}}{s r_{ij}} = Z_{\mathrm{Cl}}^{2}\frac{12\sin 2.87\,s}{2.87\,s} + Z_{\mathrm{C}}Z_{\mathrm{Cl}}\frac{8\sin 1.76\,s}{1.76\,s}, \tag{23} \]

the prime on the double summation sign means that the constant terms corresponding to \(i=j\) (exactly equal to \(Z_i^2\)) have not been taken into account. Equality (23) is obtained directly from (22) (apart from an insignificant constant factor) if the incoherent scattering constitutes a small part of the atomic background and if

\[ \frac{Z-F_i}{Z_i} \]

is constant for all atoms. We emphasize that, with such an approximate derivation of formula (23), nothing is known to us concerning the accuracy of the results obtained when it is applied; the latter must be determined empirically.

Fig. 9. Theoretical electron-diffraction curves for \( \mathrm{CCl}_4 \), illustrating different methods of choosing points for comparison of theory and experiment

The curve calculated for \( \mathrm{CCl}_4 \) by means of formula (23) is shown in Fig. 9 (curve \(A\)). The positions of the maxima and minima on it (not counting the first two) differ by less than \(0.2\%\) from the positions obtained from formula (22). Curve \(A\) is similar to the molecular-scattering curve \(I_s\), Fig. 7, with the excep-

by the fact that, owing to the replacement of \(f_i\) by \(Z_i\), the sharp damping characteristic of the latter is absent. Although it does not show a general decrease in intensity with increasing angle, as is observed on the electron diffraction pattern, curve \(A\), nevertheless, on the whole, correctly conveys the qualitative features of the latter—as, for example, the fact that the fourth and sixth maxima are lower than, respectively, the fifth and seventh.

The theoretical points for comparison are taken from this curve; the experimental points are obtained by measuring the diameters of the visible maxima and minima. The original negative is illuminated in some standard way, and for each ring, along the diameter of the electron diffraction pattern, the distance between the two points of greatest visible blackening is measured. The measurement is perhaps most conveniently carried out with the aid of an instrument in which the film is fastened on a ground-glass plate illuminated from below, while the measurement itself is made with two pointers having mutually independent micrometric motion in mutually perpendicular directions. From the ring diameters the values observed in the experiment are calculated, which are then compared with the theoretical ones. In Table 2, taken from the work of Brockway and Wold, the theoretical and experimental values of \(s\) for the case of \(\mathrm{CCl}_4\) are thus compared. Multiplying the interatomic distance adopted for

TABLE 2

Carbon tetrachloride. The distance to the film is \(12.19\ \mathrm{cm}\),
\(\lambda = 0.0604\text{--}0.0619\ \text{Å}\)

in calculating the formula, for the ratio of the theoretical values of \(s\) to the experimental ones, one obtains a value of the internuclear distance that agrees with the electron diffraction pattern. The numbers in the last column of Table 2 are

\[ \frac{S_{\text{calc.}}}{S_{\text{obs.}}}\cdot 1.76 \]

for each ring. The distance 1.76 was adopted for convenience of calculation, but any other value would have led to the same values of the observed internuclear distances. The small discrepancy between the values obtained from different rings confirms the correctness of the chosen method of comparison. Possible sources of error in the mean value are considered below.

The comparison described above makes it possible to determine directly the dimensions of the molecule, but not its shape. In the general case it is necessary to try several different models of the molecule. For each model, on the basis of various internuclear distances corresponding to variations of the molecular configuration, a special curve is constructed. A qualitative comparison of the curves with the electron diffraction pattern usually makes it possible to choose among the different models. Thus, for example, a maximum appearing on the electron diffraction pattern as more intense or weaker than both neighboring ones must be obtained in the same way on the curve. The same applies to doublets (or to two closely spaced rings), to rings showing a sharp fall of intensity on one side and a gradual fall on the other, to rings with a border, etc. A model whose internuclear distances agree in magnitude with the calculations described in the preceding section must, if it is correct, also reproduce the qualitative features of the electron diffraction pattern. Thus the determination of the molecular configuration is carried out here not only on the basis of comparing individual points, but also with a certain account of the distribution of intensities. This criterion for choosing a model proves very useful and constitutes one of the advantages of the visual method, since the judgment here is based chiefly on effects depending on molecular scattering.

In measurements on an electron diffraction pattern there is usually a subjective error in determining the position of maximum visible blackening if the intensity of the transmitted light on the two sides of it is sharply different. This effect was first investigated by St. John and Uhler \(^{39}\), who measured the intensities of asymmetric spectral lines with a recording microphotometer and a filar micrometer and found that in subjective measurement the maximum is always shifted toward the side of greater contrast. In measuring a doublet, its visible splitting is always greater than the true one. The St. John effect is also observed in measurements on an electron diffraction pattern. Internuclear distances obtained from asymmetric rings of the type described above as an example of distinctive qualitative features of an electron diffraction pattern

may differ by several percent from the values obtained by measuring sharp, well-defined rings. The interatomic distance comes out larger than the actual one if the edge (or ridge on the curve) is located along the outer edge of the ring (in the direction of increasing scattering angle), and smaller than the actual one if it is located along the inner edge. In one case (\(\mathrm{SF}_6^{40}\)), as was found from measurements of the outer sharp rings, both rings of a close doublet proved, on measurement, to be shifted in opposite directions by approximately equal distances. The presence of such asymmetric features on the electronogram, while useful for choosing the correct molecular model, cannot, however, serve for the determination of interatomic distances; and since it is impossible to predict the magnitude of the correction, the distances obtained from such rings should not be taken into account in the final calculation of the size of the molecule.

An error is also usually observed when measuring the first two or three rings, as can be seen from Table 2. The reason for this undoubtedly lies in the rapid increase in the intensity of the scattered electrons in the vicinity of the central spot. Here the background rises much more steeply than in the outer part of the electronogram, and the St. John effect shifts the visible maxima of blackening toward larger \(s\). That this effect is not due solely to subjective error was shown by Cosslett \(^{41}\), who observed the same phenomenon on microphotograms. His method for compensating the steeply falling background is described below in this chapter. Cosslett attributes this phenomenon to saturation effects in the photographic emulsion; however, a certain subjective error is also observed on underexposed negatives, whose central part, apparently, is sufficiently bright to fall on the linear portion of the characteristic curve. To some extent this effect is probably caused by contrasts in the very dense central part of the image. Cosslett also suggests using positives for measurements, which, however, is hardly applicable in the general case because of the difficulty of copying the light outer part of the electronogram. Errors in the vicinity of the central spot are not of serious importance in determining the structure, owing to the presence of a sufficient number of outer rings accessible to measurement.

Since, in the investigation of a whole series of chemical compounds, it is necessary to construct a very large number of curves, it is desirable that the calculations be made as simplified as possible. In this respect the visual method has a clear advantage over all the others, since it does not require the use of \(F\) factors. Usually the formulas used are

\[ I = \sum_i A_i \frac{\sin a_i x}{a_i x}, \tag{24} \]

where \(A_i\) and \(a_i\) are constant quantities. For convenience

DIFFRACTION OF ELECTRONS BY GAS MOLECULES

means of interpreting the electronogram. Sherman^42 computed tables of values of \(\frac{\sin x}{x}\). Later there appeared tables of values of \(\frac{\sin ax}{ax}\), arranged on long strips of paper^43; in calculation one selects strips with the corresponding values of \(a\) and lays them side by side, after which multiplication by the constants \(A_i\) and summation are carried out with the aid of a calculating machine. A formula of type (24), containing ten or more terms, can be calculated up to \(s=20\) at intervals equal to 0.2 in the course of 3 or 4 hours.

The accuracy of the visual method was checked by comparing the results obtained with its aid for the diatomic molecules of chlorine, bromine, and iodine chloride (\(\mathrm{Cl}_2\), \(\mathrm{Br}_2\), \(\mathrm{JCl}\)) with the results of an analysis of the rotational absorption spectra of the same compounds^37. The error in the electronographic determination of the internuclear distances for these molecules is, respectively, \(+1.1\), \(+0.4\), and \(-0.6\%\). Possible sources of error are considered below.

Among the special advantages of the visual method should be included the simplicity of its application, the broad quantitative use of the intensity ratio over a considerable range of variation of the scattering angle, and the larger number, than in other methods, of maxima and minima used in the interpretation. The first of these advantages is very important in practice in the investigation of a large number of different substances. The second advantage makes it possible to choose the model correctly. The third increases the number of independent determinations of the molecular dimensions and, because the outer rings are more sensitive to variations of the molecular model than the inner ones, facilitates the choice of the latter. A disadvantage of the visual method is the need to take into account the possibility of subjective errors in measurement.

Bauer^138 proposed an analytical method for comparing visual measurements with the theoretical intensity curve corresponding to equation (23). The results of a visual measurement of the rings are compared with the corresponding maxima and minima of the curve of equation (23) by means of the following analytical expression for the position of maxima and minima:

\[ \sum_{ij} Z_i Z_j \left( \cos s_k r_{ij} - \frac{\sin s_k r_{ij}}{s_k r_{ij}} \right) = 0, \tag{24a} \]

where \(s_k\) is the observed value of \(s\) for the \(k\)-th ring. If the chosen model is correct, then each of the \(k\)-expressions formed according to (24a) will be equal to zero; in the general case these equations give a series of residuals. Bauer showed how corrections must be introduced into the values of \(r_{ij}\), assumed in advance, in order to obtain values of the interatomic distances satisfying equation (24a).

This method may prove useful if a large number of sharp, well-defined rings is observed on the electronogram. In this case its application will require less expenditure of effort than interpretation by the methods described above. On the other hand, the analytical method

does not permit the use of qualitative features of the electronogram (asymmetric rings, relative intensities, etc.) in choosing a molecular model. In view of the possible Debye–Scherrer effect in the measurement of the error, some rings have to be discarded altogether. These limitations are sufficiently serious to cast doubt on the usefulness of the analytical method for obtaining more perfect electronograms.

Microphotometric Method

Another basic method of interpreting electronograms consists in comparing points of the complete theoretical intensity curve with points of the microphotometric curve. The theoretical curve is calculated for the model under test and is drawn, like curve \(C\) in Fig. 9. After this, a smooth background curve without oscillations is drawn so that it touches the intensity curve from below. Up to the present time three ways of choosing points for comparison have been proposed. Maxwell, Hendricks, and Mosley \(^{44}\) choose the points of intersection with the intensity curve of vertical lines that bisect the areas enclosed between the latter and the smooth background curve. Cosslett \(^{41}\) and others chose, in each such interval, the points whose relative height above the background curve was maximal. Brockway and Pauling \(^{37}\) used the points of tangency themselves. The first of these methods is probably less subject than the others to inaccuracies caused by possible variations in the way the background curve is drawn; however, the possible inaccuracy is small in all three cases. The tangent curve can also be drawn so that it touches the intensity curve at its maxima, and the same methods can be applied. In this case the first two methods will correspond to measuring minima in the visual method. Degard and Vander Grinten \(^{45}\) use, instead of the smooth background curve, the curve of atomic scattering (including both coherent and incoherent scattering, i.e. \(I_1\) and \(I_2\)). The points of intersection of the latter with the intensity curve are used by the authors for comparison with analogous points of the microphotometric curve.

Table 3 gives a comparison of different methods for choosing the points to be compared in the case of \(\mathrm{CCl}_4\); the first column contains “visual” values, while the next two columns are calculated by means of the first two of the methods described. With the exception of the first two maxima, where the visual values are several percent greater than the others, the discrepancies are very small. It has been noted that this comparison indicates the lower accuracy of results obtained by the visual method. This would be true only if the theoretical points chosen by the different methods were compared with the same experimental points. In practice, each determination of theoretical intensity is compared with the corresponding experimental determination, and the accuracy of the final result from the different methods is subjected to empirical

... check by comparison with the molecular structure obtained from entirely different experiments. The error in the results obtained from the first two rings cannot be explained by a discrepancy between the visual and theoretical values of \(s\), since this error has the opposite sign.

TABLE 3

Comparison of different methods of selecting corresponding points on the curves

The selection of experimental points on the microphotometric curve when using the methods described above is carried out in the same way as in the visual method (Table 2); the comparison of the observed values of \(s\) with the theoretical ones is also identical in both cases. The possibility of certain discrepancies between the microphotometric curves and the curves representing the experimental electron intensities is not excluded; however, the influence of these discrepancies on the values of \(s\) corresponding to the points selected by the above-described methods is hardly likely to be appreciable. Maxwell, Hendricks, and Mosley \(^{44}\) tested this effect on electron diffraction photographs of 4,4′-diiododiphenyl ether and phosphorus (\(P_4\)). They calibrated the photographic emulsion by exposing it to the electron beam for various lengths of time. It was found that the points lying on the vertical lines that bisect the areas enclosed between the smooth background curve and the experimental-intensity curve agree well with the points obtained from the microphotometric curve. In this case no corrections were made for the experimental background caused by multiple scattering, scattering from gas molecules distributed in the apparatus, etc., but these corrections play a role only in a quantitative comparison of intensities.

The background obtained as a result of inelastic scattering is not always taken into account when calculating theoretical

curves. This background is fairly intense in the regions of the electronogram corresponding to small values of \(s\), where it constitutes a considerable part of the scattered intensity; however, it falls off very rapidly, so that for points situated farther along the intensity curve its influence may be neglected. Thus this background should either be taken into account or not, depending on the region in which the measurements are made.

By virtue of the same considerations that were set forth in the discussion of the visual method, testing different molecular models is also necessary here. It should be noted, however, that variations of the molecular model are noticeably reflected in the complete theoretical curve only when they are accompanied by substantial changes in the corresponding values of \(s\) on the molecular-scattering curve (\(I_3\) in Fig. 7). Changes in the relative heights of the humps on the molecular-scattering curve appear as considerably weaker oscillations of the general curve, so that methods of interpretation involving the use of complete theoretical curves are somewhat less sensitive to small changes in the molecular configuration than the visual method.

The accuracy of the methods described may be judged from the value of the interatomic distance \(2.65\ \text{Å}\), obtained for the iodine molecule \(^{46}\) (band spectra give \(2.66\ \text{Å}\)), and from the general agreement of their results with the results of the more thoroughly tested visual method. The advantage of the microphotometric method over the visual one is the impossibility of subjective measurement errors. The number of points considered here is somewhat smaller than in the visual method, since the light outer part of the electronogram is not recorded by the microphotometer; meanwhile this part of the curve is the most sensitive to small changes in the molecular configuration. A serious drawback of the microphotometric method, when it is necessary to investigate a large number of substances, is the laboriousness of the computations it requires in comparison with the substantially simpler theory and computations in the visual method.

“Compensation” methods

Other proposed methods are based on various ways of compensating for the extremely rapid falloff of the intensity of the diffraction pattern with distance from the central spot. The most convenient of these was proposed by Maxwell, Hendricks, and Mosley \(^{44}\), who multiplied the curves of the theoretical and experimental intensities (the latter obtained by densitometric calibration methods) by \(s^2\). Such a curve for \(\mathrm{CCl}_4\) is shown in Fig. 9 (curve \(B\)). Multiplication by \(s^2\) transforms the oscillations of the curve into true maxima and minima, and the resulting curve resembles the curve used in the visual

method (Fig. 9, curve \(A\)). Curve \(B\) reproduces the gradual fall of the background intensity observed on electron diffraction patterns, but somewhat exaggerates the height of the inner maxima in comparison with the outer ones. For comparison of the corresponding theoretical and experimental curves, the \(s^2 \cdot I\) points corresponding to the positions of maxima and minima are taken. These values for \(\mathrm{CCl}_4\) are given in the last column of Table 3. The results of this method do not differ substantially from the results obtained by direct microphotometry.

Kossel\({}^{41}\) used a somewhat different method in studying the deviation of the intensity of the first two rings of the \(\mathrm{CCl}_4\) electron diffraction pattern from the background. He prepared a compensating plate whose density at any given segment represented the complement to the intensity of the uniform background of the electron diffraction pattern in the corresponding region. To do this he microphotometered the original electron diffraction pattern, drew on the microphotogram the curve of the uniform background, made a vessel whose shape corresponded to this curve, filled it with an absorbing liquid, and exposed a photographic plate under the wedge thus obtained. The negative and the compensating plate were put together, and a new plate was exposed beneath them. The compensated electron diffraction pattern thus obtained was a positive image of the deviations of the intensity curve from the uniform background. The photometric curve of this plate showed clearly expressed minima at the places corresponding to the maxima of the electron diffraction pattern. These points were compared with the theoretical points corresponding to the maximum elevation of the theoretical curve of the electron diffraction pattern above its uniform background. The interatomic distances to which this method leads agree excellently with the values obtained from the outer rings of the electron diffraction pattern by the usual microphotometric method. Kossel’s method is applicable only to that part of the electron diffraction pattern in which the intensity falls most steeply, but it is precisely in this region that the correction is most necessary. Kossel attributes the error obtained in the ordinary measurement of these rings to the saturation effect of the photographic emulsion, caused by the very large scattering intensity near the central spot. In another compensation method, developed by Trendelenburg\({}^{47}\), copies from the electron diffraction pattern are printed with the aid of a rotating sector, graduated in such a way as to compensate the background. In this procedure, however, the relative intensity of the rings is altered. This method has been applied to electron diffraction patterns of powders, but has never yet been used in gas electron diffraction.

A contrast microphotometer, constructed by Sirs\({}^{48}\), has also been used in interpreting electron diffraction patterns. In this instrument the negative under investigation moves beneath the beam incident upon it from the photometric slit, and at the same time performs oscillations whose direction coincides with the direction of the steady motion. The light that has passed through the negative falls on a photoelement, the current of which—

therefore, after passing through the amplifier, is recorded by the current-measuring device. This instrument records rather the density gradient than the density itself, and therefore proves sensitive to changes in the curvature of the intensity curve. The density-gradient curve obtained in this way can be compared with the curve representing the first derivative of the theoretical intensity, or else used to locate characteristic points on the intensity curve (in particular, points of inflection). This method was applied to electronograms of benzene and carbon tetrachloride with satisfactory results.

Method of Radial Distribution

All the methods described above for deciphering electronograms are based on testing various, a priori accepted models of the structure of the molecule. The radial-distribution method, described by Pauling and Brockway \(^{49}\), has the important advantage that it makes it possible to carry out the investigation without making any assumptions and without possessing any preliminary information concerning the structure. Directly from the electronogram measurement data, a function is obtained for the distribution of scattering power. This latter represents the product of the scattering powers in volume elements situated at a distance \(r\) from one another, as a function of \(r\).

Since electrons are scattered chiefly by atomic nuclei, the maxima of this function represent internuclear distances in the molecule, equal to the corresponding values of \(r\). Thus the radial-distribution method leads directly to the interatomic distances and, consequently, to the structure of the molecule.

The theoretical intensity, according to formula (17), is equal to:

\[ I(s)=K\sum_i\sum_j f_i f_j \frac{\sin s r_{ij}}{s r_{ij}} \tag{25} \]

In deriving this formula it was assumed that the scattering centers are molecules consisting of separate atoms, and that the summation extends over all atoms. The factors \(f\) contain integrals that arise because of the continuous distribution of scattering power within the atoms. A more general expression is obtained if it is assumed that the scattering power is continuously distributed throughout the whole molecule.

Instead of (25) we obtain:

\[ I(s)=K' \int_{0}^{\infty}\frac{r^{2}D(r)}{s^{4}}\frac{\sin sr}{sr}\,dr. \tag{26} \]

Summation is replaced by integration over \(dr\), and instead of the prod-

of the atomic factors there appeared the function \(\dfrac{r^2 D(r)}{s^4}\), in which \(r^2D(r)\) represents the product of the scattering powers in volume elements located at a distance \(r\) from one another. Rewriting (26) in the form

\[ s^5 I(s)=K' \int_0^\infty rD(r)\sin sr\,dr, \]

we are convinced that the integral admits inversion:

\[ rD(r)=K'' \int_0^\infty s^5 I(s)\sin sr\,ds, \]

or

\[ D(r)=K'' \int_0^\infty s^6 I(s)\frac{\sin sr}{sr}\,ds. \tag{27} \]

The calculation of this integral is possible if quantitative determinations of the experimental intensities are available. It is more useful, however, to replace the integral by the sum

\[ D(r)=\sum_k I_k \frac{\sin s_k r}{s_k r}, \tag{28} \]

each term of which corresponds to a definite ring on the electronogram. Here \(s_k\) denotes the value of \(s\) for the \(k\)-th ring (determined in the same way as in the visual method), and \(I_k\) represents the visually determined intensity of the \(k\)-th ring. \(D(r)\) is calculated with the aid of tables for \(\dfrac{\sin ax}{ax}\), which were discussed above.

The applicability of formula (28) is discussed in detail in the original paper. It was found that variations in the visually determined intensities affect the positions of the maxima of the function \(D(r)\) only to a small extent, so that the interatomic distances are derived chiefly from the diameters of the rings of the electronogram. The method described was tested in investigations of chlorine, bromine, and more complex molecules, in which the ratio of interatomic distances is determined by the geometry of the molecule, such as benzene, carbon disulfide, etc. Generally speaking, the highest maximum of the function \(D(r)\) makes it possible to calculate the corresponding interatomic distance with an accuracy of up to one percent; the other maxima give less accurate values. The radial distribution function for \(\mathrm{CCl}_4\), determined from ten observed maxima, is reproduced in Fig. 10. The sharp maximum at \(2.86\ \text{Å}\) corresponds to the distance between chlorine atoms, the value of which agrees with that obtained by the usual methods.

The smaller maximum corresponds to the C—Cl distance, equal from here to 1.74 Å, which is approximately 0.01 Å less than the actual value. Further on, in the region of about 3.5 Å, the curve has no meaning; if, in calculating the integral of formula (26), it had not been replaced by a sum, the curve would show in this region an asymptotic approach to zero.

Direct interpretation of electron diffraction patterns by the radial-distribution method is extremely useful, but often does not lead to a complete determination of the structure, since only a small number of interatomic distances can be calculated accurately with its aid. Moreover, if in a molecule there are interatomic distances that differ only slightly from one another, they may prove unresolved on the radial-distribution curve. On the other hand, the application of this method at once restricts the limits within which the values of the molecular parameters under consideration are contained. For the investigation of molecules containing rotating groups of atoms, and of molecules for which the choice of model requires a quantitative estimate of intensities (ClO₂, SO₂, etc.), this method also offers undeniable advantages.

Fig. 10. Radial-distribution function for CCl₄

Determination of molecular structure

Before turning to the discussion of the reliability of the results, we may briefly dwell on the general methods of approach to structural problems and on those types of structure which can be determined by the electron-diffraction method. After the electron diffraction pattern has been obtained, it is first of all recommended to calculate the radial-distribution function by means of visual measurements of \(s\) \(\left(s=\frac{4\pi}{\lambda}\sin\frac{\vartheta}{2}\right)\) and of the same visual estimate of intensity.

After this, theoretical curves are calculated in order to choose among models compatible with the results of the radial-distribution method. This can be done most accurately by comparing experimental and theoretical intensities over the entire range of variation of the angular variable \(s\) (as described above for benzene); however, the general form of the intensity curves and the necessity of a large number of calculations limit the application of this method. It is therefore necessary to choose among various methods of comparing individual points of the theoretical—

tical and experimental scattering curves; there are two such methods: namely, visual and microphotometric. The determination of an interatomic distance on the basis of the measurement of a single ring is almost equally accurate by both methods. The values obtained from the inner rings are in both cases too small (only the compensation method proposed by Cosslett gives correct values). Visual measurement of some rings leads to subjective errors (the St. John effect), but the latter is easy to determine; the number of measurements that must be rejected in determining interatomic distances is always only a small fraction of the total number. In the case where the configuration of the molecule is known from other sources, or where variations of the model require changes in the distances between the heaviest atoms of the molecule, the accuracy of both methods proves to be approximately the same. The visual method permits comparison of a larger number of points, since the eye distinguishes more rings than the microphotometer; moreover, the outer part of the electron diffraction pattern is more sensitive to changes in the molecular configuration than the inner part. Generally speaking, in determining configurational parameters, preference should be given to the visual method.

Simplicity of application is also one of the advantages of the visual method. The laborious calculations of theoretical intensities required by the microphotometric method, which are not rewarded by an increase in the accuracy of the results, limit the number of models subjected to verification in the study of a whole series of compounds.

Some investigators have raised the question of the extent to which various methods of deciphering electron diffraction patterns are “justified.” If a complete quantitative comparison of the theoretical intensity curve with the experimental data is necessary, then none of the methods described above can be considered justified. If what is required is the determination of the dimensions and configuration of a molecule by comparing the values of \(s\) for a small number of characteristic points on the theoretical curves and on the electron diffraction pattern, then any method is justified that gives a well-founded comparison of theory and experiment (i.e., a comparison of theoretical “visual” curves with visual measurements and of complete theoretical curves with the curves of experimental intensity), especially one which, while giving accurate results, is at the same time sensitive to variations of the molecular model. In any case it is very important for the investigator to choose definite methods of deciphering and to test them himself experimentally, by studying compounds of known structure.

The stochastic nature of the methods for deciphering electron diffraction patterns (with the exception of the radial-distribution method) is not always taken into account sufficiently. The best agreement between an electron diffraction pattern and an intensity curve for some definite model does not yet give an unambiguous determination of the structure. The existence of other configurations compatible with the electron diffraction pattern—

program, remains possible until each of them is eliminated by considering the corresponding intensity curve. It is always desirable to restrict the uncertainty in the configurational parameters by testing a whole series of models of the molecule, rejecting them as a clear disagreement with the electron diffraction pattern is found; without this the configuration of the molecule cannot be regarded as unambiguously determined. It is permissible, and sometimes simply necessary, to use parameter values obtained by other methods; but parameters adopted in this way are no longer determined from the electron diffraction pattern. In the interpretation one is limited to showing the compatibility of these parameters with the experimental results. At the same time the investigator must not be satisfied with a rough agreement between theory and experiment; variation of the parameters may lead to a better agreement and, consequently, to a more accurate determination of their values. A report on the results of the investigation will be incomplete if it is not accompanied by a list of the various tested models of the molecule, showing exactly which interatomic distances and angles between bonds were adopted beforehand and which were determined.

As an illustration of the use of previously adopted parameters one may cite the example of the investigation of the benzene molecule (restricting oneself, for convenience, to consideration of only the six carbon atoms). If one assumes that all six carbon atoms are in any possible positions, the number of variables will be too large. The chemical ideas according to which the six carbon atoms of the benzene molecule form a ring and the bonds between the carbon atoms are completely equivalent may be accepted without hesitation. Thus only two parameters remain to be determined: one determining the configuration (i.e., the question whether the benzene ring has a planar or a spatially zigzag form), and the other establishing the dimensions of the molecule, or, what is the same thing, the bond distance. In the course of the investigation, to which a reference has already been given above[^14], the configurational parameter was determined by testing three different models; in each of these models the carbon atoms were arranged three at a time in two parallel planes, the distance between which was respectively 0.2, 0.1, and 0.0 Å. After it had been shown that the last of these three models was the most satisfactory, the size of the molecule was determined by comparing the points on the experimental and theoretical curves.

Some types of structures do not lend themselves to complete determination. Examples of one such type of structure are the molecules ClO$_2$[^50] and SO$_2$[^10,^51]. The central atom of a triatomic molecule here has a considerably greater scattering power than the outer atoms. Therefore the diffraction pattern is almost completely determined by the distances Cl—O (or S—O), and the change of the term containing the distance O—O, accompanying a change of the angle between bonds from 60 to 180°, is very insignificant.

affect the theoretical intensity curve. Even with the aid of the radial-distribution method it has been possible to obtain only an approximate determination of the angle between the bonds; this uncertainty, however, does not affect the accuracy of the determination of the Cl—O distance. In general, if molecules are composed partly of light and partly of heavy atoms (i.e., if the atomic numbers of some atoms in the molecule are at least twice as large as the atomic numbers of others), the distance between the lighter atoms cannot be determined accurately. This limitation makes it impossible, for example, to determine the positions of hydrogen atoms. Although, in calculating scattering curves, hydrogen atoms cannot be entirely neglected, nevertheless even relatively considerable changes in their positions affect the form of the curves only slightly. If the number of atoms in a molecule is large, a complete determination of the structure by electronographic methods is generally impossible, because the probability of the existence of different systems of parameters leading to the same theoretical distribution of intensities becomes too great. The determination of more than three parameters (including the dimensions of the molecule) is likewise impossible; however, by combining the electronographic method with data from other methods of investigation, it has been possible to elucidate the structure of many complex molecules.

Sources of errors

The magnitude of the possible error in results obtained by the electron-diffraction method depends on a number of factors; in favorable cases it may be no more than one percent. The scale of distances is established from the length of the edge of the elementary cube of the gold lattice, 4.070 Å (Neiberger[^29] gives, for 1936, the value 4.0700 ± 0.0004 Å on Siegbahn’s scale). The possible deviation of the de Broglie wavelength from the mean value obtained from the rings of a single electron diffraction pattern, as was found in calibration from photographs of gold in our laboratory, is on the average about 0.3%; with good control of the voltage and calibration at regular intervals of time, the uncertainty of the wavelength does not exceed 0.5%. Errors in measuring the scattering angle \(\theta\) have only relative significance, since the correctness of the points selected for measurement is determined by empirical checking against the resulting interatomic distances. Repeated measurements of the diameter of sharp rings by the visual method with the aid of a comparator can be made with a maximum error of 0.5%. The results of measurements of poorly outlined rings, with fluctuations up to 2%, are usually discarded as unreliable. The determination of the distance from the object to the film, which lies between 100 and 300 mm, can easily be carried out with an accuracy not worse than 0.5%. In general, the probable error of the quantity \(\frac{\sin \frac{\theta}{2}}{\lambda}\) for the selected ...

in the electronogram of the points can safely be estimated at approximately 1%. The theoretical values of \(s\) for any particular molecular model can be determined with the desired accuracy by calculating the corresponding intensity function over sufficiently small intervals of the argument, in order to fix the position of the curve with sufficient precision. In comparing theory and experiment, the agreement of the values of interatomic distances obtained from various rings (with due allowance for the known causes of oscillation) is a measure of the reliability of the result. The deviation of the values of the interatomic distance from the mean value varies on the average from \(3/4\%\) for \(\mathrm{CCl}_4\) to 5% or more for ethylene and other molecules with low scattering power. The probable error is less than this average deviation and, for most of the substances studied, lies between 1 and 2%; this magnitude is also confirmed by comparison with the results of studies of band spectra mentioned above.

The preceding discussion concerns the determination of molecular dimensions. When the configuration is uncertain, the magnitude of the error can be established only by finding the limits of the smallest change in the configurational parameters that produces a definite discrepancy between the corresponding intensity curve and the electronogram. The limits of uncertainty depend entirely on the nature of the substance under investigation, as was already indicated above when the choice of a molecular model was discussed.

Another possible source of error is contamination in the sample of the substance under investigation. If the nature and amount of the contamination are known, its influence can be checked by calculating the intensity curves both for the principal substance and for the impurity, and taking the mean between the two curves in proportion to the molecular concentrations of the two substances. If the scattering power of the impurity is no greater than that of the substance under investigation, concentrations on the order of up to 1% have no appreciable influence on the result.

Results of Electronographic and X-ray Studies of \(\mathrm{CCl}_4\)

The X-ray and electronographic methods are the only widely applicable methods for determining interatomic distances. A comparison of the merits of the two methods leads to the conclusion that each of them has its own field of application. For the investigation of crystalline structures, X-rays are more applicable because of their greater penetrating power; on the other hand, electron diffraction proves more useful in studying the surface structure of solids. For determining the arrangement of atoms in gaseous molecules, electron diffraction proves more applicable, since electrons are scattered predominantly by atomic nuclei, whereas X-rays are scattered by atomic electrons, which

which, especially in light atoms, are more or less concentrated along the bonds. A detailed discussion of the merits of both methods is given by Bewilogua^52.

We give below a brief history of the study of CCl\(_4\) by both methods, guided by the consideration that this compound, widely used as a standard substance in experiment, plays an important role for electron diffraction; it is all the more interesting that the results obtained by X-ray and electron-diffraction methods at first did not agree with one another. The CCl\(_4\) molecule has always been regarded as a tetrahedron, the center of which is occupied by the carbon atom, while the four chlorine atoms are located at the corners. The numbers given below are the C—Cl distance; the Cl—Cl distance is obtained from them by multiplication by

\[ \left( \frac{8}{3} \right)^{\frac{1}{2}} \quad ( = 1.633). \]

In 1929 Debye, Bewilogua, and Ehrhardt^53 published the first X-ray study and gave the value \(2.0\ \text{Å}\). Somewhat later, using a more complete theory of the scattering of X-rays by gas molecules, the value \(1.9\ \text{Å}\)^54 was obtained and, finally, in 1930—\(1.83 \pm 0.02\ \text{Å}\)^55. This value was confirmed in 1933 by van der Grinten^27. The first electron-diffraction determination of the C—Cl distance was carried out by Wierl^56 in 1930 and gave the value \(1.81\ \text{Å}\). In subsequent years various investigators obtained by the electron-diffraction method the following values of the C—Cl distance: 1931—Wierl^112 \(1.83\ \text{Å}\); 1932—Hengstenberg and Bru^94 \(1.83\ \text{Å}\); 1933—Brockway and Pauling^77, \(1.81\ \text{Å}\); Braune and Knoke^65, \(1.78\ \text{Å}\); Dornte^91, \(1.82\ \text{Å}\); Hendricks, Maxwell, Mosley, and Jefferson^93, \(1.83\ \text{Å}\); 1934—Pauling and Brockway^108, \(1.76\ \text{Å}\); Coslett and de Laszlo^84, \(1.75\ \text{Å}\); Coslett^83, \(1.74\ \text{Å}\); 1935—Detard and van der Grinten^87,88 \(1.75\ \text{Å}\); Pauling and Brockway^106, \(1.755\ \text{Å}\). The agreement of the results of the early electron-diffraction studies with the X-ray results is explained by the fact that in them the interpretation of the electron-diffraction patterns was carried out on the basis of X-ray data. With the improvement of experimental technique and with more careful processing of the electron-diffraction patterns, a discrepancy of \(4\%\) between the X-ray and electron-diffraction values became evident. A more careful combined X-ray and electron-diffraction study, carried out by Degard, Pierard, and van der Grinten^87, led to agreement, and the X-ray value of the C—Cl distance in the CCl\(_4\) molecule is at present considered to be \(1.75 \pm 0.02\ \text{Å}\).

Of special interest are two X-ray investigations carried out by the method of absolute intensity measurement. In 1933 Lu\(^{119}\) studied various gases, including CCl\(_4\), making measurements with an ionization chamber and an electrometer; he confirmed the value given above in 1933 (1.83 Å) with a maximum error in the intensity measurement of 5%. Quite recently van der Grinten and Brasser\(^{120}\) measured the first two diffraction maxima for CCl\(_4\) vapor with a Geiger counter. They obtained for the C—Cl distance the value 1.73 Å. The most probable value of the C—Cl distance in the molecule of gaseous carbon tetrachloride is \(1.755 \pm 0.005\) Å\(^{106}\). The care with which this compound was studied permits one to assign to the probable error a smaller value than usual.

V. REVIEW OF RESULTS

The electron-diffraction method has been used in studying the molecular structure of a whole series of compounds. The results of electron-diffraction investigations have been applied to the solution of various problems of structural chemistry. Several examples are given below by way of illustration.

The first investigation of this kind was carried out by Wierl and had as its purpose the elucidation of the difference between the cis- and trans-forms of 1,2-dichloroethylene (C\(_2\)H\(_2\)Cl\(_2\)). According to chemical data, the two carbon atoms of the molecule of either of these two compounds are joined to one another by a double bond, and each of them is also bonded to one chlorine atom and one hydrogen atom; moreover, all the atoms of the molecule are, most probably, situated in one plane. Such a configuration can be realized in two ways, placing both chlorine atoms on one and the same side (cis-form) or on different sides (trans-form) of the line connecting the carbon atoms. The distance between the chlorine atoms will be entirely different in the two cases. The latest electron-diffraction studies show that in one of these compounds the Cl—Cl distance is 3.22 Å, and in the other 4.27 Å; consequently, the first of them will be levorotatory.

One of the subjects of intensive electron-diffraction study has been the characteristic values of the angles between bonds that one atom forms with others. Wierl’s investigations, carried out on a series of hydrocarbons, confirmed the usually accepted tetrahedral angle (109°28′) between the simple bonds of carbon. Possible deviations from this value upon asymmetric replacement of one of the hydrogen atoms by an atom of another element were also investigated; as examples one may cite methylene chloride (CH\(_2\)Cl\(_2\)) and chloroform (CHCl\(_3\))\(^{110}\), in which the angle Cl—C—Cl is only 3° greater than the same angle in CCl\(_4\).

TABLE 4

Continuation of Table 4

Continuation of Table 4

Continuation of Table 4

Continuation of Table 4

Continuation of Table 4

Trivalent molecules containing elements of the fifth group (N, P, As) have a pyramidal configuration, and the angles between bonds lie within the range \(100—110^\circ\) (see \(N(CH_3)_3\), \(PF_3\), \(PCl_3\), \(AsF_3\), \(AsCl_3\) in Table 4); the interesting cases \(P_4\) and \(As_4\) are exceptions,\(^{103}\) where these angles are equal to \(60^\circ\). The angle between oxygen bonds has been studied in the compounds \(F_2O\), \(Cl_2O\), \((CH_3)_2O\), in 1,4-dioxane, paraldehyde, \(CH_3ONH_2\), and in 4,4′-diiododiphenyl ether; it has been found that its magnitude varies from 100 to \(118^\circ\), depending on the size of the atoms or groups that are bonded to the oxygen atom.

The distance of bonds that one atom forms with others is even more important than the angle between bonds. The bond distance is characteristic not only of the element forming this bond, but also of the type of the bond itself. The data of structural analysis have made it possible to compile a table of atomic radii for each of the bond types\(^{57}\) in such a way that the sum of the radii of any two elements is equal to the distance between the atoms of these elements in a bond of the given type. Thus, the bond distance obtained experimentally also determines the type of the latter (the bond may be simple, double, or triple covalent, resonance-covalent, ionic, mixed—ionic and covalent, etc.).

As an illustration of the application of electron-diffraction analysis data to the determination of bond types, let us give the example of chloroethylene.\(^{71}\) According to the commonly used distribution of bonds, in the molecule of any of the chloroethylenes the carbon atoms are joined to each other by a double shared bond and by simple shared bonds to the hydrogen and chlorine atoms. The laws according to which the distribution of bonds in a molecule takes place also allow other arrangements, in which a double bond joins a chlorine atom with one of the carbon atoms. The three possible arrangements of the bonds are shown in Fig. 11.

Fig. 11. Arrangement of shared bonds in 1,1-dichloroethylene.

If arrangement I in Fig. 11 were the only one corresponding to the normal state of the molecule, then the \(C—C\) distance would have to be approximately \(1.38\ \text{Å}\), as in all compounds of the ethylene type in general, and the \(C—Cl\) distance would be \(1.76\ \text{Å}\), as in chlorine-substituted methanes. The structure of the molecule cannot, however, correspond to arrangement II or III alone either (since their potential energy is somewhat higher than that of I); however, if we assume that in the normal state of the molecules the arrangement

If \(II\) and \(III\) are partially present in the substance, then the observed values of the bond distances must differ from those given above for arrangement \(I\). Since arrangements \(II\) and \(III\) do not differ physically from one another, both C—Cl bonds are obtained of equal length; this length, assuming that \(I\), \(II\), and \(III\) are present in equal amounts, is \(1.64\ \text{Å}\). In reality, interpretation of electron-diffraction patterns of 1,1-dichloroethylene gives a C—C bond distance equal to \(1.38\ \text{Å}\) and a C—Cl distance equal to \(1.69 \pm 0.02\ \text{Å}\). This result shows that model \(I\) does not correspond to the actual structure of the molecule; moreover, the empirical relation\({}^{58}\) connecting the interatomic distance with the character of a double bond indicates that each C—Cl bond is 14% double and 86% single in character. This can be explained by assuming that the double bond is constantly oscillating among the three possible positions, and that the time during which it is in each of these positions is determined by the percentage ratio given above. A detailed analysis of this reasoning and its application to other resonance molecules studied by electron diffraction is given in the work of Pauling, Brockway, and Beach\({}^{58}\).

Various investigators have used electron-diffraction data, sometimes in combination with results obtained by other experimental methods, in solving many other structural problems. References to the corresponding works are given in Table 4.

Table 4 lists all substances whose vapors have been studied by the electron-diffraction method. The table is divided into three parts according to the type of molecules investigated. The first part lists inorganic compounds (not containing carbon), the second part—molecules of aliphatic compounds (i.e., all carbon compounds with the exception of benzene derivatives), and finally the third part—aromatic compounds (benzene derivatives).

The table of inorganic compounds is subdivided into groups corresponding to those groups of the periodic system to which the central atoms of the molecules of the compounds investigated belong. The different columns of the table give: the chemical formula of each compound, a symbol denoting the type of structure, a description of the configuration of the molecule, the interatomic distances and the angles between bonds calculated from the experiment, and references to all works concerning the given substance. It was impossible to list in the table the often mutually inconsistent results of all investigations carried out on each compound. The structural data given by us represent the most probable values selected for each substance on the basis of all investigations concerning it. The symbols of the type of structure given in the second column are a number denoting the number of atoms in the column, together with the designation

of the symmetry class \(^{50}\), characterizing molecules according to Schoenflies. For a given number of atoms, the symmetry of a molecule determines its configuration quite unambiguously. Exceptions are only one or two compounds, for which the corresponding additional data are given in the table. The description of the configuration is placed in the table in order to facilitate interpretation of the symbol characterizing the type of structure; the words “tetrahedral” and “octahedral” mean a regular tetrahedral or octahedral configuration. Bond distances are given from the central atoms of the molecule to the outer ones; the remaining distances can be calculated from these on the basis of the known configuration. Wherever one more parameter is required for this purpose, the angle between the bonds of the central atom of the molecule is given. The molecule \(\mathrm{B}_3\mathrm{N}_3\mathrm{H}_6\) has no central atom, and here the regular hexagon formed by the atoms is fully determined by the distance \(\mathrm{B—N}\). The molecules \(\mathrm{P}_4\) and \(\mathrm{As}_4\) also have no central atoms, and the distances given in the table correspond to the edges of a regular tetrahedron.

The values of bond distances given in the table are the most probable magnitudes, determined by the electronographic method, with corrections for the probable error. Corrections enclosed in parentheses have been added by us either because no correction at all is given in the original investigation, or because this correction seemed to us too small. In those cases where the distance itself is enclosed in parentheses, its true value apparently lies outside the limits indicated in the original work. Such an estimate is always based on comparison with the results of investigations of other compounds of the same type, and it is very probable that a more careful repeated investigation would lead to corrected values. In cases where such a repeated investigation has been carried out, only the corrected results are given. If the value of a distance (or angle) is given without a correction, this means that it has been adopted a priori (usually on the basis of the results of investigations of similar compounds) in order to reduce the number of parameters in the analysis. Several compounds are included in the table without structural data at all, since the conclusions of the authors of the original investigations are not experimentally substantiated.

In the table of aliphatic compounds the subdivision is made according to the number of carbon atoms in the molecule, the \(\mathrm{C}_2\) group being divided, in turn, into subgroups of derivatives of ethane, ethylene, and acetylene. Special symbols for designating the type of structure are not given here, but a description of the configurations is supplied rather often. The distance \(\mathrm{C—H}\) was taken in different cases to be different, within the limits from \(1.04\) to \(1.10\ \text{Å}\). An exact determination of this distance and of the positions of the hydrogen atoms is impossible either because of the presence of more strongly scattering atoms in the molecule, or else, in the opposite case, because the molecule is too light and does not give a sharp electron diffraction pattern; the \(\mathrm{C—H}\) distance, accordingly, is not given for any compound.

...ment. The molecules of ethylene derivatives are assumed to be planar. The parameters are given in the same system as for inorganic compounds.

The table of aromatic compounds contains all benzene derivatives for which structural data have ever been published. The benzene ring is assumed to be planar, and its size is usually adopted for the various derivatives as indicated in the table. It should be noted that in derivatives with heavy substituent atoms, the values obtained by investigators for the C—X distance are in direct dependence on the dimensions of the benzene ring adopted in the interpretation. The appended bibliography includes all works devoted to the diffraction of fast electrons by gas molecules.

LITERATURE

1. P. Lenard, Ber. Berl. Akad. 1893.
2. C. J. Davisson and C. H. Kunsman, Phys. Rev., 22, 242, 1923.
3. C. J. Davisson and L. H. Germer, Phys. Rev., 30, 705, 1927.
4. L. de Broglie, Thesis, Paris, 1924; Phil. Mag., 47, 446, 1924.
5. G. P. Thomson, Proc. Roy. Soc., London A 117, 600; 119, 651, 1928.
6. S. Kikuchi, Jap. J. Physics 5, 83, 1928.
7. a. N. F. Mott, Proc. Camb. Soc., 25, 304, 1929; b. Proc. Roy. Soc. London, A 127, 658, 1930.
8. G. P. Thomson, Proc. Roy. Soc. London, A 125, 352, 1929.
9. H. Mark and R. Wierl, Z. Physik, 60, 741, 1930.
10. R. Wierl, Ann. Physik 8, 521, 1931.
11. A survey of early work on the diffraction of slow electrons by gas molecules is given in the work of J. Hengstenberg and K. Wolf, Hand- und Jahrbuch d. chem. Physik, Vol. 6, Part IA, SS. 129—138 (Akad. Verlagsges., Leipzig, 1935). The literature on later works is as follows: F. L. Arnot, Proc. Roy. Soc., London, A 144, 360, 1934; F. L. Arnot and J. C. McLauchlan, Proc. Roy. Soc. London, A 146, 662, 1934; E. C. Childs and A. H. Woodcock, Proc. Roy. Soc., London, A 146, 199, 1934; H. S. W. Massey and E. C. Bullard, Proc. Cambridge Phil. Soc., 29, 511, 1933; C. B. O. Mohr and F. H. Nicoll, Proc. Roy. Soc. London, A 144, 596, 1934; A 142, 320, 647, 1933; S. N. Van Voorhis, Phys. Rev., 46, 480, 1934; G. M. Webb, Phys. Rev., 47, 384, 1935.
12. H. S. W. Massey, Proc. Roy. Soc., London, A 129, 616, 1930; H. S. W. Massey and C. B. O. Mohr, Proc. Roy. Soc., London, A 135, 258, 1932.
13. For the conference on solid-state diffraction see Trans. Farad. Soc., 31, 1043, 1935.
14. L. Pauling and L. O. Brockway, J. Chem. Phys., 2, 867, 1934.
15. N. F. Mott and H. S. W. Massey, Theory of Atomic Collisions (Oxford Press, 1933).
16. H. Bethe, Ann. d. Physik, 5, 324, 1930.
17. On the improper integral introduced by substitution of the first term of (11) into (9) or (10), see reference b.
18. a. R. W. James and G. W. Brindley, Z. Krist., 78, 370, 1931; b. L. Pauling and J. Sherman, Z. Krist., 81, 28, 1932.
19. J. Macdougall, Proc. Roy. Soc. London, A 136, 549, 1932.
20. P. Debye, Ann. d. Physik, 46, 809, 1915.
21. P. Ehrenfest, Amsterdam. Acad., 23, 1132, 1915.

1. H. S. W. Massey and C. B. O. Mohr, Proc. Roy. Soc. London, A 135, 258, 1932.
2. P. M. Morse, Physik. Z., 33, 443, 1932.
3. W. Heisenberg, Physik. Z., 32, 737, 1931.
4. L. Bewilogua, Physik. Z., 32, 740, 1931.
5. R. W. James, Physik. Z., 33, 737, 1932.
6. W. van der Grinten, Physik. Z., 34, 609, 1933.
7. C. Degard, J. Pierard and W. van der Grinten, Nature, 136, 142, 1935.
8. M. C. Neuberger, Z. Krist., 93, 1, 1936.
9. D. H. Sloan, R. L. Thornton and F. A. Jenkins, R. S. I., 6, 80, 1935.
10. K. C. D. Hickman, J. Frank. Inst., 221, 215, 1936.
11. H. de Laszlo, Proc. Roy. Soc., London, A 146, 662, 1934.
12. V. E. Cosslett, Trans. Farad. Soc., 30, 981, 1934.
13. H. Seemann, Ergeb. techn. Roentgenkunde, 3, 95, 1933; Z. Physik 92, 270, 1934.
14. W. Grether, Ann. d. Physik, 26, 1, 1936.
15. a) Fast electrons, A. Beckera, E. Kipphan, Ann. d. Physik, 10, 15, 1931; b) Electrons of medium velocities, M. Nacken, Physik. Z., 31, 296, 1930; W. Seitz and G. Harig, Physik. Z., 30, 758, 1929; c) Slow electrons, V. Weidner, Ann. d. Physik, 12, 239, 1932.
16. L. Pauling and L. O. Brockway, J. Chem. Phys., 2, 867, 1934.
17. L. O. Brockway and F. T. Wall, J. Am. Chem. Soc., 56, 2373, 1934.
18. C. E. St. John and L. W. Ware, Astrophys. J., 44, 35, 1916.
19. L. O. Brockway and L. Pauling, Proc. Nat. Acad. Sci., 19, 68, 1933.
20. V. E. Cosslett, Trans. Farad. Soc., 30, 981, 1934.
21. J. Sherman, Z. Krist., 85, 405, 1933.
22. These tables have not been published, but they may be obtained from Dr. Paul Cross, at Stanford University. They give values of

\[ \frac{\sin ax}{ax} \]

for \(0.8 \leq a \leq 4.00\) at intervals of 0.01 for \(0 \leq x \leq 40\), and for \(4.02 \leq a \leq 8.00\) at intervals of 0.02 for \(0 \leq x \leq 20\). The intervals in \(x\) are 0.2.
44. L. R. Maxwell, S. B. Hendricks and V. M. Mosley, J. Chem. Phys., 3, 699, 1935.
45. C. Degard and W. van der Grinten, Bull. Soc. Roy. Sci. (Liege) No. 4—5, 1, 1935.
46. S. B. Hendricks, L. R. Maxwell, V. L. Mosley and M. E. Jefferson, J. Chem. Phys., 1, 549, 1933.
47. Trendelenburg, Naturwiss., 21, 173, 1933; Trendelenburg and Franz, Veröff. Siemens Konzern, 13, 48, 1934.
48. F. W. Sears, J. Opt. Soc. Am., 25, 162, 1935.
49. L. Pauling and L. O. Brockway, J. Am. Chem. Soc., 57, 2684, 1935.
50. L. O. Brockway, Proc. Nat. Acad. Sci., 19, 303, 868, 1933.
51. P. Cross and L. O. Brockway, J. Chem. Phys., 3, 821, 1935.
52. L. Bewilogua, Physik. Z., 33, 688, 1932.
53. P. Debye, L. Bewilogua and F. Ehrhardt, Physik. Z., 30, 84, 1929.
54. P. Debye, Physik. Z., 30, 524, 1929.
55. P. Debye, Physik. Z., 31, 419, 1930; L. Bewilogua, Physik. Z., 32, 265, 1931.
56. R. Wierl, Leipziger Vorträge 1930, p. 13.
57. See, for example, O. Pauling and M. L. Huggins, Z. Krist., 87, 205, 1934.
58. L. Pauling, L. O. Brockway and J. Y. Beach, J. Am. Chem. Soc., 57, 2705, 1935.

1. For stereographic projections of crystallographic symmetry classes, see P. Ewald, Handb. d. Physik, 2 Aufl., V. XXIII, T. 2 (Julius Springer, Berlin, 1933), pp. 226—233.
2. H. Boersch, Sitzberich. Acad. Wiss. Wien, 144, 1, 1935.
3. H. Boersch, Monatsh., 65, 311, 1935.
4. H. Boersch, Naturwiss., 22, 172, 1934.
5. H. Braune u. S. Knoke, Z. Physik. Chem., B 23, 163, 1933.
6. H. Braune u. S. Knoke, Naturwiss., 21, 349, 1933.
7. H. Braune u. S. Knoke, Z. Physik. Chem., B 21, 297, 1933.
8. L. O. Brockway, J. Am. Chem. Soc., 57, 958, 1935.
9. L. O. Brockway, Proc. Nat. Acad. Sci., 19, 868, 1933.
10. L. O. Brockway, Proc. Nat. Acad. Sci., 19, 303, 1933.
11. L. O. Brockway, in press.
12. L. O. Brockway a. J. Y. Beach, in press.
13. L. O. Brockway, J. Y. Beach a. L. Pauling, J. Am. Chem. Soc., 57, 2693, 1935.
14. L. O. Brockway a. P. C. Cross, J. Chem. Phys., 3, 828, 1935.
15. L. O. Brockway a. H. O. Jenkins, in press.
16. L. O. Brockway a. H. Levy, in press.
17. L. O. Brockway a. K. Palmer, in press.
18. L. O. Brockway a. L. Pauling, Proc. Nat. Acad. Sci., 19, 860, 1933.
19. L. O. Brockway a. L. Pauling, Proc. Nat. Acad. Sci., 19, 68, 1933.
20. L. O. Brockway a. F. T. Wall, J. Am. Chem. Soc., 56, 2373, 1934.
21. Luis Bru, Rev. Acad. Cienc. Madrid, 30, 586, 1933.
22. Luis Bru, Anales soc. espan. fis. quim., 31, 115, 1933.
23. Luis Bru, Anales soc. espan. fis. quim., 30, 483, 486, 1932.
24. D. C. Carpenter a. L. O. Brockway, J. Am. Chem. Soc., in press.
25. V. E. Cosslett, Trans. Farad. Soc., 30, 981, 1934.
26. V. E. Cosslett a. H. G. de Laszlo, Nature, 134, 63, 1934.
27. P. Cross a. L. O. Brockway, J. Chem. Phys., 3, 821, 1935.
28. C. Degard, C. R., 201, 951, 1935.
29. C. Degard, J. Pierard a. W. van der Grinten, Nature, 136, 142, 1935.
30. C. Degard a. W. van der Grinten, Bull. Soc. Roy. Sci. Liège, Nos 4—5, 1, 1935.
31. R. W. Dornte, J. Am. Chem. Soc., 55, 4126, 1933.
32. R. W. Dornte, J. Chem. Phys., 1, 630, 1933.
33. R. W. Dornte, J. Chem. Phys., 1, 566, 1933.
34. W. Grether, Ann. d. Physik, 26, 1, 1936.
35. S. B. Hendricks, L. R. Maxwell, V. L. Mosley a. M. E. Jefferson, J. Chem. Phys., 1, 549, 1933.
36. J. Hengstenberg a. L. Bru, Anales soc. espan. fis. quim., 30, 341, 1932.
37. P. L. F. Jones, Trans. Farad. Soc., 31, 1036, 1935.
38. H. de Laszlo, Nature, 135, 474, 1935.
39. H. de Laszlo, Proc. Roy. Soc., London, A 146, 672, 690, 1934.
40. H. de Laszlo, Nature, 131, 803, 1933.
41. H. Mark u. R. Wierl, Z. Elektrochem., 36, 675, 1930.
42. H. Mark u. R. Wierl, Z. Physik, 60, 741, 1930.
43. H. Mark u. R. Wierl, Naturwiss., 18, 778, 1930.
44. H. Mark u. R. Wierl, Naturwiss., 18, 205, 1930.
45. L. R. Maxwell, S. B. Hendricks a. V. M. Mosley, J. Chem. Phys., 3, 699, 1935.
46. L. R. Maxwell, V. M. Mosley a. Lola S. Deming, J. Chem. Phys. 2, 331, 1934.
47. L. Pauling a. L. O. Brockway, in press.
48. L. Pauling a. L. O. Brockway, J. Am. Chem. Soc., 57, 2684, 1935.

1. L. Pauling and L. O. Brockway, Proc. Nat. Acad. Sci., 20, 336, 1934.

2. L. Pauling and L. O. Brockway, J. Chem. Phys., 2, 867, 1934.

3. A. Stock and R. Wierl, Z. anorg. u. allgem. Chem., 203, 228, 1931.

4. L. E. Sutton and L. O. Brockway, J. Am. Chem. Soc., 57, 473, 1935.

5. R. Wierl, Ann. d. Physik, 13, 453, 1932.

6. R. Wierl, Ann. d. Physik, 8, 521, 1931.

7. R. Wierl, Physik. Z., 31, 1028, 1930.

8. R. Wierl, Verh. D. Phys. Ges., 11, 29, 1930.

9. R. Wierl, Leipziger Vorträge, p. 13, 1930.

10. R. Wierl, Physik. Z., 31, 366, 1930.

11. P. G. Ackermann and J. E. Mayer, J. Chem. Phys., 4, 377, 1936.

12. S. H. Bauer, J. Chem. Phys., 4, July 1936.

13. S. S. Lu, Chinese J. Phys., 1, 51, 1933.

14. W. van der Grinten and H. Brasseur, Nature, 137, 657, 1936.