STUDY OF SURFACE STRUCTURE BY THE METHOD OF ELECTRON DIFFRACTION1

G. I. Finch and H. Wilman, London

Contents. I. Historical introduction. II. Scattering of fast electrons. III. Experimental technique. IV. Interpretation of electron-diffraction patterns. V. The case of anomalous diffraction. VI. Estimation of crystal size. VII. Influence of crystal size on lattice parameters. VIII. Structure of metallic films and surfaces. IX. Structure of a polished surface. X. Conclusion. XI. Bibliography.

I. Historical Introduction

The first experiments on the scattering of an electron beam by a metallic surface were apparently carried out by Campbell Swinton[^2] in 1899. However, although in the main his apparatus possessed all the essential features necessary for observing directions of preferential scattering, he succeeded in obtaining only diffuse scattering, the reason for which was, of course, the unsuitable choice of object, which in this case was a polycrystalline platinum disk. The results of a whole series of observations on the scattering of electrons by matter, carried out over the following 20 years, all proved to be in agreement with classical or quantum mechanics, and only in 1921 did Davisson and Kunsman[^3] succeed in detecting directions of preferential scattering of an electron beam; this effect, however, these authors attempted to explain on the basis of purely corpuscular mechanics. Although de Broglie[^4] had formulated his theory of the connection between wave systems and moving particles as early as 1924, and although Elsasser[^5] soon afterward put forward the supposition that the interaction between an electron beam and a single crystal might be used to test this theory, it was apparently only accidentally made observations that induced Davisson and Germer to undertake the study of the scattering of slow electrons by the surface of a single crystal of nickel; and only in March 1927 did they publish a preliminary communication[^6] on the results obtained, followed in December of the same year by a detailed communication[^7], which served as a confirmation of de Broglie’s theory.

At the same time Thomson and Reid, who did not know of the experiments of Davisson and Germer, were studying the scattering of fast electrons by thin films; moreover, as early as May 1927 they published a report[^7] on the diffraction of cathode rays by a thin film of celluloid, illustrated by a photographic record of the distribution of scattered electrons. Here again it is interesting to note that, although Thomson had previously been acquainted with de Broglie’s theory,[^8] it was not this theory, but rather certain anomalous results of work on the scattering of electrons in helium according to Diamond’s observations[^9] and Thomson’s own experiments[^10] on the scattering of positive rays in gases, that gave rise to his experiments on electron diffraction. Somewhat later Thomson[^12] published the results of his further experiments on the diffraction of fast electron beams by thin metallic films, which already quantitatively confirmed de Broglie’s equation.

Further progress was achieved by Kikuchi and Nishikawa,[^13] who passed electron beams through thin crystals of mica and observed spotted patterns and phenomena of secondary scattering (“Kikuchi lines”), which in origin proved to be similar to the lines observed by Rutherford and Andrade[^14] in 1914 in the scattering of γ-rays. This was followed by Nishikawa and Kikuchi’s[^15] obtaining of the first patterns in the reflection of fast electrons from the faces of single crystals (calcite, mica, topaz, zinc blende, and natural faces of quartz), and by the experiments of Muto and Yamaguti[^16] on the diffraction of electrons by crystalline powders placed on thin films of an amorphous substance. The next major advance in experimental technique, which may be regarded as the starting point in the study of surface structure by means of electron diffraction, was achieved when Matsukawa and Shinohara,[^17] on the one hand, and Thomson,[^18] on the other, independently discovered that polycrystalline surfaces, when an electron beam falls on them at a very small angle, give coherent diffraction patterns. This fact was correctly interpreted by Thomson as the effect of the passage of electrons through crystalline peaks above the mean level of the surface. Apparently Thomson himself[^18] was the first to point out the possibility of electron diffraction as a means of studying the problem of surface structure.

Despite intensive research on the diffraction of slow electrons, which continued after the work of Davisson and Germer, this technique—unlike the technique of working with fast electrons and chiefly because of the complexity of the observed phenomena—still does not, up to the present time, have great importance for the study of surface structure; therefore, in what follows there will be no further references to the diffraction of slow electrons.

Another step of historical importance is connected with the work of Wierl,[^19] who demonstrated the advantage of electron beams over X-ray…

...by X-rays in investigations of the structure of gas molecules. Reviews of progress in this field of application of electron diffraction, which will not be considered further in this article, have recently been given by Brockway[^20] and by G. Eston[^21].

II. Scattering of Fast Electrons

Taking into account the strong scattering of electrons by matter, one might have expected in advance that the method of electron diffraction would be especially suitable for studying the structures of thin films and surfaces, as distinct from the bulk structures of matter. Such an intensity of scattering is caused chiefly by the interaction of the strong local electrostatic fields surrounding atomic nuclei with the electron beam; owing to their own electric charges, the electrons undergo strong deflection. Therefore crystalline films only a few atoms thick produce strong diffracted beams; moreover, even those electrons which move with a velocity equal to half the speed of light are deflected and lose energy in repeated collisions with atoms so rapidly that, in the reflection method at almost grazing incidence, they can give coherent diffraction effects from surface layers extending into the depth only to a distance of the order of 20 Å.[^22]

The de Broglie equation $\lambda = \frac{h}{mv}$ gives only the wavelength $\lambda$, associated with a material particle of mass $m$ and velocity $v$, where $h$ here denotes Planck’s constant. However, in 1926 Schrödinger[^23] derived, from de Broglie’s law, a wave equation for the motion of a particle, expressed in terms of its total and potential energy and applicable to phenomena of atomic dimensions,

\[ \nabla^2 \psi + \frac{8\pi^2 m}{h^2}(E - V)\psi = 0, \]

where $\psi$ is a function of the spatial coordinates $(x, y, z)$, and $|\psi|^2$ denotes the probability of finding the particle at the given point of space, while $m$, $E$, and $V$ are respectively the mass, the total energy, and the potential energy of this particle. Thus, in the case of a beam of electrons of uniform velocity, elastically scattered by aggregates of atoms, $E$ denotes the constant initial and final energy of the electrons, determined by the accelerating potential, while $V$ is the spatial distribution of the electrostatic potential through which the electrons move; in this case $|\psi|^2$ gives the distribution of the electrons in space, i.e. both in the primary and in the diffracted beams.

At approximately the same time, Heisenberg[^24] constructed a new atomic mechanics based on the principle of using, as a basis for calculation, only observable quantities; the mathematical connection between these quantities was chosen in such a way that the fundamental laws of quantum theory followed as...

of this latter kind, experimental data or various theoretical data^34. By Hartree’s method one calculates \(F\) for atoms with \(Z<40\), while the Thomas–Fermi method gives good results for heavier atoms. Mott and Massey^27 gave, for electrons, a table of values of

\[ I(\theta)=\frac{e^4}{4m^2v^4}(Z-F)^2\operatorname{cosec}^4\frac{\theta}{2} \]

for elements from Li to A, calculated by Hartree’s method, and a table of values of \(\dfrac{10}{Z^{2/3}}\) for heavier atoms, calculated by the Thomas–Fermi method. For most elements the values of \(F\) are given in the form of a table by Bragg^35.

The intensity of inelastic, i.e. incoherent, scattering of electrons has a course entirely different from that in the case of x-rays, since it falls rapidly from its maximum value at \(\theta=0\), whereas inelastic x-ray scattering slowly increases as \(\dfrac{\sin \frac{\theta}{2}}{\lambda}\) increases up to a maximum value proportional to \(Z\). For atoms from H to Ca, the intensity of inelastic scattering was calculated and given in tabular form by Compton and Allison^36. Likewise Heisenberg^37 gave a method for calculating the intensity of x-ray scattering based on the Thomas–Fermi theory, and Morse^38 showed that, with certain modifications, this method can also be applied to the study of inelastic scattering of electrons by an atom. According to this method, the intensity \(I\) at a distance \(R\) from the atom will be

\[ I(\theta)_{\text{electr.}}=\left(\frac{0.116}{R}\right)^2 \frac{1}{Z^{5/3}}\cdot \frac{G(v)}{v^4} \]

and

\[ I(\theta)_{\text{x-ray}}=\left(\frac{2.81\cdot 10^{-5}}{R}\right)^2 \frac{1+\cos^2\theta}{2}\,ZG(v), \]

where \(v=\dfrac{4\pi}{\lambda}\,b\sin\dfrac{\theta}{2}\), with \(b=\dfrac{0.176}{Z^{2/3}}\,\text{\AA}\) (\(R\) is likewise measured in \(\text{\AA}\)). The function \(G(v)\) was calculated by Bewilogua^39 and is given in tabular form in Debye’s review^40. Owing to the arbitrary change of wavelength in inelastic collisions, in order to find the total intensity of inelastic scattering by a given substance, the intensity of scattering by a single atom should simply be multiplied by the number of atoms \(N\).

To solve the question of the intensity of elastic scattering of electrons by a single crystal, we may first consider the corresponding expressions for x-rays. If, as a single crystal, we take a parallelepiped with the numbers of nodes \(M_1\), \(M_2\), \(M_3\) in rows parallel to the axes \(a\), \(b\), \(c\), and assume that the incident ray makes angles \(\alpha_0\), \(\beta_0\), \(\gamma_0\) with these axes, then the intens-

the scattering intensity \(I\) in a direction making angles \(\alpha, \beta, \gamma\) with the axes \(a, b, c\) at a distance \(R\) from the crystal will be

\[ I=\frac{S^{2}}{R^{2}}\frac{\sin^{2} M_{1}h\pi}{\sin^{2} h\pi}\cdot \frac{\sin^{2} M k\pi}{\sin^{2} k\pi}\cdot \frac{\sin^{2} M_{3}l\pi}{\sin^{2} l\pi} =\frac{S^{2}}{R^{2}}\cdot f, \]

where \(f\) is Laue’s interference function, and

\[ h=\frac{a}{\lambda}(\cos \alpha-\cos \alpha_{0}), \qquad k=\frac{b}{\lambda}(\cos \beta-\cos \beta_{0}), \]

\[ l=\frac{c}{\lambda}(\cos \gamma-\cos \gamma_{0}). \]

Further, here \(S^{2}\) denotes the structure factor

\[ S^{2}= \left[\sum_{1}^{n}\Psi_{n}\cos 2\pi\left(\frac{hx}{a}+\frac{ky}{b}+\frac{lz}{c}\right)\right]^{2} + \]

\[ + \left[\sum_{1}^{n}\Psi_{n}\sin 2\pi\left(\frac{hx}{a}+\frac{ky}{b}+\frac{lz}{c}\right)\right]^{2}, \]

where the \(n\)-th atom in the elementary cell has coordinates \([(x,y,z)]\) and scattering amplitude \(\Psi_{n}\). It follows from this relation that the principal maxima are obtained near those positions where \(h\), \(k\), and \(l\) are simultaneously integers.

For the reflection of X-rays by a crystal face, the total intensity of the diffraction maximum arising when the crystal is rocked through a small interval, i.e., the “integral reflection” for the entire diffraction region with interference indices \(hkl\), will be

\[ \rho_{hkl}=\frac{1}{2\mu}\left\{(NS)^{2}\frac{\lambda^{3}}{\sin\theta}\cdot \frac{1+\cos^{2}\theta}{2}\right\} =\frac{Q}{2\mu}, \]

where \(\mu\) is the linear absorption coefficient of X-rays in the crystal, \(N\) is the number of elementary crystal cells in \(1\ \mathrm{cm}^{3}\), \(S^{2}\) is the structure factor, \(\theta\) is the angle of deviation of the ray, and the third factor in parentheses is the polarization factor.

For an X-ray intensity equal to \(I_{0}\) per \(1\ \mathrm{cm}^{2}\), diffracted by a volume \(V\) of crystalline powder, the total amount of radiation \(P\) diffracted along a cone with aperture angle \(2\theta\) will be

\[ \frac{P}{I_{0}}=Q\frac{p\cos\frac{\theta}{2}}{2}V, \]

where \(p\) is the number of equivalent planes giving the diffraction ring. However, electron diffraction, owing to very strong scattering by the nuclei, is more complicated than X-ray diffraction; but, if no anomalies arise due to the shape of the crystals, their size or orientation, then, for comparing the theoretical intensities of different rings on a powder diagram, as a rough approximation there may serve the inten-

intensities calculated from the formulas for X-rays, in which the polarization factor may be neglected. If, however, it is required to know only relative intensities, then it would be quite sufficient to regard the intensity at a given point on a ring as proportional to \(\dfrac{S^{2}p}{\sin^{2}\theta}\) or \(S^{2}pd^{2}_{hkl}\). That such an approximation in the case of a pattern from a disordered powder is often sufficiently good is shown by the fact that Thomson, and also Mark and Wierl, as indicated above, obtained good agreement between the quantity \(F_{e}(\theta)\), calculated from Mott’s formula, and the quantity obtained experimentally from ring intensities. Probably, a considerable deviation may occur in the case of randomly arranged crystals when the pattern contains one or several rings of much greater intensity than the others, since in such cases the effects of “multiple scattering” may strongly change the intensities, and also when the shape or size of the crystals gives rise to anomalous diffraction phenomena or when forbidden (or “extra”) reflections arise. By “multiple scattering” is meant here the effect of scattering of rays that act as new primary rays and are scattered again by the same or neighboring crystals\(^{41}\). The expression \(I=\dfrac{S^{2}}{R^{2}}\) for the intensity of scattering by a single crystal, given by the simple kinematic theory, is applicable only to a camera of infinite length; it is not entirely correct for fast electrons with energy of about 50 kV, owing to their small wavelength (about \(0.05\ \mathring{\mathrm A}\)) when the dimensions of the crystal transverse to the beam are more than \(15\,000\ \mathring{\mathrm A}\) and the camera length is 50 cm.

For a more rigorous calculation of electron scattering by a crystal, one should take into account rather the repeated scattering and the interaction of the diffracted rays within the crystal than proceed from simple scattering and the associated strengthening of the secondary ray; i.e., it is necessary to draw rather the “dynamic” than the above-indicated “kinematic” picture of diffraction. Such a dynamical theory was developed for X-rays by Darwin\(^{42}\) and Ewald\(^{43}\) and expressed in geometrical form by Laue\(^{44}\). An analogous dynamical theory for electron scattering was given by Bethe\(^{45}\), who uses for this purpose the wave equation of Schrödinger and substitutes into it not a constant potential but a periodic function of the coordinates \(V(\mathbf r)\), which can be represented in the form of a triple Fourier series, so that

\[ \frac{8\pi^{2}me}{h^{2}}\,V(\mathbf r)=U(\mathbf r)=\sum_{\mathbf g} v_{\mathbf g}e^{2\pi i(\mathbf g,\mathbf r)}, \]

where \(\mathbf r\) denotes the position vector of the given point, and \(\mathbf g\) is the vector of one of the nodes of Ewald’s reciprocal lattice. In this case the Schrödinger equation describes the entire stationary system of plane

waves in dynamic equilibrium inside the crystal. To solve the resulting equation, Bethe expands \(\psi\) in a Fourier series analogously to the potential and, from the boundary conditions, finds relations between the coefficients and an expression for the incident wave. For simplicity he assumes that, apart from the primary beam, at the given moment only one strong diffracted beam appears. Laue\(^{46}\) showed that the potential at the boundary of the crystal does not fall immediately to zero, as Bethe assumes, but decreases in a more complicated manner. According to Bethe’s theory, as also according to the dynamical theory of X-ray scattering, there exists a definite range of angles of incidence within which total reflection occurs (“Anregungsfehler”).

Morse\(^{47}\) used the following expansion for the potential at the point \((x,y,z)\) in the crystal:

\[ V(x,y,z)=\sum_{-\infty}^{+\infty}{}_{h_1} A_{xh_1} e^{\frac{2\pi i h_1 x}{a_1}} +\sum_{-\infty}^{+\infty}{}_{h_2} A_{yh_2} e^{\frac{2\pi i h_2 y}{a_2}} +\sum_{-\infty}^{+\infty}{}_{h_3} A_{zh_3} e^{\frac{2\pi i h_3 z}{a_3}}, \]

where \(h_1, h_2, h_3\) are the Miller indices of the plane, and \(a_1, a_2, a_3\) are the axial lengths of the coordinates \(x,y,z\) in three mutually perpendicular directions. He showed that finite eigenfunctions, as solutions of the Schrödinger equation, exist only in certain regions of energy levels, and calculated the formula for the distribution of scattering intensity. Morse’s calculations for the case of diffraction of slow electrons by the \((111)\) face of a nickel single crystal proved to be in agreement with the experimental results of Davisson and Germer\(^{48}\), who worked with slow electrons and found that the positions of the diffraction maxima of the lower orders deviate somewhat from those positions which follow from Bragg’s law.

Bethe’s dynamical theory was applied by Shinohara\(^{49}\) to determining the change of intensity within the “Kikuchi lines” obtained in diffraction from thick single crystals; although he adopted somewhat arbitrary boundary conditions, the calculated distribution turned out to be of the same form as that observed experimentally. On the other hand, Hiyasi\(^{50}\) obtained an approximate solution of the same problem for the case of a rhombic crystal, treating it as a phenomenon of secondary scattering and assuming, first, that the primary spherical wave scattered by an atom \(O\) has at the point \(Q\) an amplitude proportional to an expression of the form \(\frac{e^{ikr_Q}}{r_Q}\cos\theta_Q\), where \(\theta_Q\) denotes the angle of deviation from the initial ray, and \(r_Q=OQ\), and, second, that the atom at \(Q\) scatters this wave and sends out a secondary spherical wave with amplitude proportional to \(\frac{e^{ikr_{PQ}}}{r_{PQ}}\), where

\(r_{PQ}=PQ\). The first expression for an inhomogeneously scattered spherical wave was given by Wentzel and Sommerfeld\(^{51}\) for the distribution of \(K\)-photoelectrons emitted in the interaction of polarized X-rays with atoms. The intensity distribution calculated by Hayasi by summing all such secondary scattered waves in a crystal, the dimensions of which were taken to be infinite, proved to be of the same form as that observed experimentally.

The width of the reflections and the theory of the refractive index of crystals for electrons were discussed by Kikuchi\(^{53}\) for the case of cubic crystals on the basis of a method analogous to Bethe’s method. The diffraction spots observed from the cleavage plane of ZnS turned out to be approximately 20 times broader than the theory predicts, but the calculated change in the refractive index of the crystal with diffraction order proved to be similar to what was observed experimentally. Beeching\(^{53}\) compared the width of the (111) reflections of diamond with that given by Bethe’s theory and found that the (333) spot has a width, corresponding to half intensity, equal to \(30'\), whereas the calculated value is only about \(2'\). These and other discrepancies are probably caused by the small elastic penetration of the electron beam in the direction normal to the surface of the crystal, although it is also possible that in this case adsorbed gas played a role.

Recently, Harding\(^{54}\) developed a dynamical picture of electron diffraction based on Darwin’s theory for X-ray scattering and analogous to the method of Kronig and Penney\(^{55}\) for the interaction of an electron beam with a potential barrier, with both gas adsorption and the possible change of the lattice constant near the boundary of the crystal being taken into account in the calculation. Making a quantitative comparison of the conclusions obtained with Beeching’s experimental data, Harding came to the conclusion that the indicated general character of the experimental results can be explained on the basis of the assumption that several planes near the surface have reduced Bragg spacings in comparison with the normal ones, and that the magnitude of this effect indicates that the contraction of the spacings is caused rather by a layer of adsorbed gas than by contraction of the crystal itself near the surface.

III. Technique

The equipment used for studying the structure of thin films and surfaces by means of electron diffraction consists, first, of a chamber in which the directions and intensities of the beams scattered as a result of the interaction of a thin beam of electrons, moving with uniform velocity, with the specimen are recorded, and, second, of auxiliary apparatus required for obtaining the electron beam and maintaining the necessary vacuum.

INVESTIGATION OF SURFACE STRUCTURE BY THE METHOD OF ELECTRON DIFFRACTION

Camera. The source of electrons may be either a hot cathode, operating by thermionic emission, or a cold cathode, where electron emission is obtained as a result of bombardment of the cathode by positive ions arising when electrons pass through gas at low pressure in a discharge chamber. In particular, Finch^56, Kikuchi^57, Shinohara^58, Thomson^59, and Trendelenburg^60 with their collaborators make use chiefly of a cold cathode, while Aminov^61, Germer^62, Kirchner^63, Trilat^64, and Yamaguti^65 prefer a cathode in the form of a heated filament. The relative advantages of both these systems are enumerated in the papers of Finch^66 and Kirchner^67. To illustrate present-day tendencies in the construction of cameras for electron diffraction, it will suffice briefly to describe the most modern type of apparatus now used in our laboratories. The basic requirement for the apparatus is high precision combined with economy and speed in operation; moreover, one has to take into account the design of such parts as can be made in the workshops of an average scientific laboratory. Thus, for example, it is necessary to prefer, instead of conical ground joints, flat metal-to-metal ground joints, likewise working with a grease of low vapor pressure. Good fitting of such surfaces does not require special skill and is easily carried out on a steel plate with the aid of an ordinary carborundum paste; only a matte finish of the surface is required. As a lubricant it is recommended to use white vaseline, free of salts and neutral, thickened to the required consistency by boiling in vacuum together with the addition of about 2% white rubber.

The general design of the camera (Fig. 1) has been described in detail earlier^68, and therefore here one may confine oneself merely to mentioning the new improvements. The anode diaphragm now consists of a drilled nickel or platinum disk attached to a massive tube with a wide opening^69. A small magnet of cobalt steel on a short bent lever, fitted directly beneath the anode, serves to set the electron beam along the axis of the apparatus below the anode. The system of diaphragms is made movable by means of external control, as in the case of ordinary crystal holders. The upper diaphragm, situated directly below the fixed anode diaphragm, consists of a thin brass plate with a whole series of openings of different sizes, each of which, as required, can be set on the beam. This setting is carried out by the simple rotation of the brass plate, arranged in the form of a carousel. Under the first diaphragm there is a sufficiently large simple diaphragm, also mounted on an ordinary holder. Its purpose is to block any scattering caused by the first diaphragm. A third, comparatively coarse diaphragm, mounted on a holder directly beneath the focusing coil, is also sometimes used for eliminating

of random scattering. The diameter of the diaphragms, counting from the anode diaphragm, is approximately equal to 0.03–0.2, 0.5 and 0.5 mm.

The focusing coil rests on three adjusting screws supported by a supporting plate and lies midway between the anode and the screen. The coil is adjusted by means of two long hinges, which makes it possible, simultaneously with shifting the coil, to observe the screen itself.

Fig. 1. Finch electronograph

The specimen holders are, in the main, not different from those described earlier^68. Their design is calculated so that the specimen can have six degrees of freedom. For work with single crystals a somewhat different type of holder is more convenient, allowing independent movements in height and azimuth during the exposure^1). The cassette for one plate is made in such a way that it can be removed from the apparatus in daylight. A fluorescent screen of zinc sulfide on an aluminum plate is riveted to a shutter at the top of the cassette. Other authors have apparatus designed for several photographic plates.

The camera is constructed entirely of nonmagnetic materials—brass and bronze; the wall thickness of the tubes is about 1.5 mm. Cast parts, generally speaking, contain pores, and therefore their inner portions are forged with a spherical hammer. The inner walls of the camera sections are blackened by spraying with colloidal graphite in order to reduce reflection of light from the screen.

The vacuum regulation is essentially the same as in the previously described apparatus^68, except that the glass capillary has been replaced by a copper tube with a small aperture, having

^1) The drawings of the holder and cassette, as well as some details about them, have been omitted in the translation. —Ed.

meter length and flattened with plane-tooth pliers to such an extent that, at a pressure in the gas-supply reservoir of 100 mm Hg, the rate of flow of air into the discharge chamber is sufficient to maintain a potential drop from cathode to anode of about 50 kV under normal working conditions, i.e., at a discharge current of about 0.5 mA. To protect the chamber from the entry of mercury vapor into it, a trap is placed above the high-vacuum part of the pump; it is cooled with water and, in addition, contains a certain quantity of tin leaves. This trap is closed at the top by a ground-glass cover, in which there is a mirror glass attached with the aid of picene for observation. For evacuation a three-stage diffusion mercury pump is used, or, in case of extreme necessity to avoid the risk of contamination by mercury vapor, a Burch diffusion oil pump, which, in order to prevent loss of vacuum in it, can be isolated by means of metal stopcocks with large openings. In some cases it is possible to obtain successive photographs, counting not only the breaking of the vacuum, the changing of plates, and the evacuation, but also the changing of the specimen, in 5–8 min. with a mercury pump and 3–5 min. with an oil (diffusion) pump.

All branch tubes at different points of the tube have a diameter of 5 cm and are designed, on the one hand, so that object—plate distances of 50, 25, and 20 cm can be made at will, and, on the other hand, so that, as needed, the holder for the specimen can be replaced by an observation window or a filament can be introduced into the chamber, etc.

Since the chief source of errors in precise work is the difficulty of accurately measuring the accelerating voltage, one may use a cassette with two shutters, which make it possible to record successively, one after the other, on one and the same photographic plate two halves of diffraction patterns, one of which belongs to a standard substance^70. In this way the effective length of the chamber is determined and the error is eliminated in the same way as when using arc spectra of iron in optics or rock salt and calcite in radiography. In some cases, to eliminate this error, simple superposition of two complete diffraction patterns upon one another on a single photographic plate is used.

Preparation of specimens. The most important data on the technique of preparing specimens have already been given in previous papers^68, so that only a few details can be added here. Experience has shown that, in investigations of the surface of massive single crystals, cleavage planes and natural faces give poor results. Good surfaces are obtained in many cases by sawing the crystal in the given crystallographic direction and subsequent polishing. In some cases, in order to remove the amorphous layer formed on the surface, it is useful to employ cautious heating, which causes strict orientation of the recrystallized substance.

When electrons are transmitted through them, structures giving good cleavage, such as, for example, mica, brucite, etc., can be prepared in the form of comparatively large sheets. The author, together with Wilman[^71], succeeded in obtaining intact sheets of graphite and molybdenite of very small thickness. Crystals of other types can, under suitable conditions, be grown in the form of thin layers from solutions, by electrodeposition, by condensation of vapor on a fine mesh or on an inert substrate such as collodion, and also on an active substrate such as rock salt or other single-crystal surfaces.

IV. Interpretation of Electron-Diffraction Patterns

Transmission of single crystals. Electron-diffraction patterns produced by single crystals are fundamental for understanding the patterns of all other types. If an approximate calculation is made, as is done in the case of X-rays, assuming that the incident plane wave is scattered coherently by each atom of a complex three-dimensional lattice, then summation of the secondary spherical waves for all atoms of the lattice leads to the following expression for the intensity distribution of the scattered rays:

\[ I=\frac{S^2}{R^2}\frac{\sin^2 M_1 h\pi}{\sin^2 h\pi}\frac{\sin^2 M_2 k\pi}{\sin^2 k\pi}\frac{\sin^2 M_3 l\pi}{\sin^2 l\pi} =\frac{S^2}{R^2}\cdot f. \tag{1} \]

The Laue interference function shows that strongly diffracted rays will normally appear only in directions close to those for which the three Laue conditions with integer numbers \(h, k, l\) are simultaneously satisfied:

\[ a(\cos\alpha-\cos\alpha_0)=h\lambda, \tag{2a} \]

\[ b(\cos\beta-\cos\beta_0)=k\lambda, \tag{2b} \]

\[ c(\cos\gamma-\cos\gamma_0)=l\lambda. \tag{2c} \]

These conditions mean that, for reinforcement, the path difference of the waves scattered in the direction under consideration by two neighboring atoms of a linear lattice parallel to the axis \(a\) must be equal to an integral number of wavelengths, and that the same conditions must be satisfied for the axes \(b\) and \(c\). The three factors in the expression \(f\) correspond to the fact that the intensity function for a row of \(M_1\) atoms with spacing \(a\) between them is equal to

\[ I=\frac{\Psi^2}{R^2}\frac{\sin^2 M_1 h\pi}{\sin^2 h\pi} =\frac{\Psi^2}{R^2}\frac{\sin^2 \dfrac{M_1\delta}{2}}{\sin^2 \dfrac{\delta}{2}}, \tag{3} \]

where \(\delta\) is the constant phase difference of waves scattered by two consecutive atoms of the row. Therefore, if each atom of this

if the row is replaced by a row of a linear lattice of type \(b\), then a plane net is obtained, whose scattering intensity will be

\[ I=\frac{\Psi^2}{R^2}\frac{\sin^2 M_1 h\pi}{\sin^2 h\pi}\cdot \frac{\sin^2 M_2 k\pi}{\sin^2 k\pi}. \tag{4} \]

If, however, each atom of this plane net is replaced by a row of a lattice of type \(c\), then we have a simple three-dimensional lattice, to which the function \(f\) with all three factors is applicable. Since any three rows not lying in one plane may define the lattice, we can construct the diffraction pattern for any position of the crystal by finding the directions which simultaneously lie on the reinforcement zones (Laue zones) of any three conveniently arranged rows of atoms of the lattice, as in Fig. 2, whence, by comparison with the diffraction spots obtained experimentally, the latter may be indexed.

Fig. 2. Construction of a diffraction pattern in transmission

All directions satisfying condition (2a) lie along the generators of a family of cones with solid angle \(2\alpha\) about the linear lattice \(a\), and, as a result of the intersection of the screen or photographic plate, give a family of curves. These curves are hyperbolas, ellipses, or circles, depending on the angle of inclination of the linear lattice to the beam, whose direction is assumed normal to the screen, as is also the case in practice. The maximum of zero order \(h=0\) always passes through the undeviated spot, i.e., coincides with the direction of the primary ray. The function (3) has a minimum, as shown in Fig. 3, where \(h\) is equal to half-

Fig. 3. Curve \(\dfrac{h\pi\sin^2 M_1}{\sin^2 h\pi}\) for \(M_1=8\)

positive or negative integer multiple of the quantity \(\frac{1}{M}\), principal maxima at \(h=1, 2, 3\), etc., and secondary maxima at intermediate values of \(h\), close to \(\frac{3}{2}M_1,\ \frac{5}{2}M_1,\ \frac{7}{2}M_1,\ldots\), \(1-\frac{3}{2}M_1,\ 1+\frac{3}{2}M_1,\ 1+\frac{5}{2}M_1\), etc.

It is necessary, however, to consider only the regions of the principal maxima. In all other directions the intensity of the scattered radiation is practically equal to zero. If the middle of a maximum corresponds to an \(h\) equal to an integer, then the extreme rays limiting the maximum correspond to values of \(h\) greater by \(\frac{1}{M}\) and smaller by \(\frac{1}{M}\) than this integer value. It is easy to see that, for the small angles of deviation with which one has to deal in experiments on the diffraction of fast electrons, if the linear grating is close to a position parallel to the plane of the screen, the regions of reinforcement form on the screen approximately straight bands, perpendicular to the projection of the linear gratings on the screen, with distance between them \(\frac{\lambda L}{a\cos\theta}\) and width \(\frac{2\lambda L}{M_1 a\cos\theta}\), where \(L\) is the length of the camera, and \(\theta\) is the angle between the linear grating and the normal to the ray lying in the plane of incidence. When the linear grating is parallel to the primary ray, the regions of reinforcement form concentric circular rings around the undeviated spot, and from

\[ \cos\alpha = 1-\frac{h\lambda}{a} \tag{5} \]

the radii of these circles are also determined:

\[ r=L\left\{\left(1-\frac{h\lambda}{a}\right)^{-2}-1\right\}^{\frac{1}{2}} \]

or, in view of the smallness of the wavelength, and hence also of the small scattering angles, with sufficient accuracy,

\[ r=L\left(\frac{2h\lambda}{a}\right)^{\frac{1}{2}}. \]

In this case the zero-order region broadens the undeviated spot to a circle with radius approximately equal to

\[ L\left(\frac{2\lambda}{M_1a}\right)^{\frac{1}{2}},\quad \text{i.e.}\quad L\left\{\frac{2\lambda(M_1-1)}{M_1T}\right\}^{\frac{1}{2}}, \tag{5a} \]

where \(T=(M_1-1)a\) denotes the length of the linear grating, i.e. the thickness of the crystal in the direction of the ray. The width of the zone of the \(h\)-th order will be approximately equal to the \(\frac{1}{(M_1h)^{\frac{1}{2}}}\)-th part of the width of the zero zone. As we deflect the linear grating from this position, the center or, more precisely, the centers of the circular bands will shift to the side, and the zones will stretch out into more or less

elongated ellipses. The zero-order zone becomes an ellipse passing through the undeviated spot, and the zones with negative values of \(h\) appear inside the zero-order zone. With further inclination of the linear grating the ellipses open into hyperbolas, which near the undeviated spot take the form of equidistant rectilinear bands. At small inclinations of the linear grating to the beam, the zones of reinforcement are still approximately circles whose centers almost coincide with the point of intersection of the linear grating with the screen, and their radii are determined sufficiently accurately from the relation

\[ R^2 = R_0^2 + L^2 \frac{2h\lambda}{a}, \tag{6} \]

where \(R_0\) is the radius of the zero-order maximum.

It can be shown that the three Laue conditions \((2a,b,c)\) are equivalent to Bragg’s law \(n\lambda = 2d\sin\theta\), according to which the primary beam for the direction of a true maximum with Laue numbers \(h\,k\,l\) may be regarded as a beam reflected at the glancing angle \(\theta\) by a group of parallel atomic planes with spacing \(d\) between them, whose Miller indices will be \(h\,k\,l\) or \(\frac{h}{n},\frac{k}{n},\frac{l}{n}\), the reflection in the latter case being of the \(n\)-th order. Indeed, the Laue conditions \((2a,b,c)\) can be expressed by the following three vector equations:

\[ \begin{aligned} \mathbf{a}(\mathbf{s}-\mathbf{s}_0) &= h\lambda, && \tag{7a}\\ \mathbf{b}(\mathbf{s}-\mathbf{s}_0) &= k\lambda, && \tag{7b}\\ \mathbf{c}(\mathbf{s}-\mathbf{s}_0) &= l\lambda, && \tag{7c} \end{aligned} \]

where \(\mathbf{s}_0\) and \(\mathbf{s}\) denote unit vectors parallel respectively to the incident and diffracted beams. But the axial vectors of the reciprocal lattice \(\mathbf{a}^*, \mathbf{b}^*, \mathbf{c}^*\) have the property that one may write

\[ \left. \begin{aligned} \mathbf{a}\cdot\mathbf{a}^* &= \mathbf{b}\cdot\mathbf{b}^* = \mathbf{c}\cdot\mathbf{c}^* = 1,\\ \mathbf{a}\cdot\mathbf{b}^* &= \mathbf{a}\cdot\mathbf{c}^* = \mathbf{b}\cdot\mathbf{c}^* = \mathbf{b}\cdot\mathbf{a}^* = \mathbf{c}\cdot\mathbf{a}^* = \mathbf{c}\cdot\mathbf{b}^* = 0, \end{aligned} \right\} \tag{8} \]

so that \(\mathbf{a}^*=\dfrac{[\mathbf{bc}]}{\mathbf{a}[\mathbf{bc}]}\), and \(\mathbf{b}^*\) and \(\mathbf{c}^*\) are obtained from this by cyclic permutation; i.e., \(\mathbf{a}^*, \mathbf{b}^*\), and \(\mathbf{c}^*\) have magnitudes \(\dfrac{bc\sin\alpha}{V}\), \(\dfrac{ca\sin\beta}{V}\), \(\dfrac{ab\sin\gamma}{V}\), respectively, and lie along directions perpendicular (in the positive direction) to the planes formed by \(bc\), \(ca\), and \(ab\). The angle \(\alpha^*\) between the axes \(\mathbf{b}^*\) and \(\mathbf{c}^*\) of the reciprocal lattice is determined from the relation

\[ \cos \alpha^* = (\cos\beta\cos\gamma - \cos\alpha)\frac{1}{\sin\beta\sin\gamma}, \tag{9} \]

and the other two angles \(\beta^*\) and \(\gamma^*\) are obtained from this by cyclic permutation. It is easy to show that the radius vector \(\mathbf{h}\), drawn from

begun at any point with coordinates \([[hkl]]\) in the reciprocal lattice and given by the expression

\[ \mathbf{h}=h\mathbf{a}^{*}+k\mathbf{b}^{*}+l\mathbf{c}^{*}, \tag{10} \]

is parallel to the normal to the atomic planes with Miller indices \(\left(\frac{h}{n},\frac{k}{n},\frac{l}{n}\right)\) in the crystal lattice, and its length is equal to \(\frac{1}{d/n}\), where \(d\) is the distance between the planes, and \(n\) is the greatest common divisor of the numbers \(hkl\). Further, any vector \(\mathbf{r}\) can be represented by the expression

\[ \mathbf{r}=(\mathbf{r}\cdot\mathbf{a})\mathbf{a}^{*}+(\mathbf{r}\cdot\mathbf{b})\mathbf{b}^{*}+(\mathbf{r}\cdot\mathbf{c})\mathbf{c}^{*}. \tag{11} \]

Hence, and from equations \((7a,b,c)\), we have

\[ (\mathbf{s}-\mathbf{s}_{0}) =\{(\mathbf{s}-\mathbf{s}_{0})\cdot\mathbf{a}\}\mathbf{a}^{*} +\{(\mathbf{s}-\mathbf{s}_{0})\cdot\mathbf{b}\}\mathbf{b}^{*} +\{(\mathbf{s}-\mathbf{s}_{0})\cdot\mathbf{c}\}\mathbf{c}^{*} = h\lambda\mathbf{a}^{*}+k\lambda\mathbf{b}^{*}+l\lambda\mathbf{c}^{*} \]

or simply

\[ \mathbf{s}-\mathbf{s}_{0}=\lambda\mathbf{h}. \tag{12} \]

Fig. 4. Construction of the direction of the diffracted ray

This equation can be represented in the form of the vector triangle in Fig. 4, whence it follows that for principal maxima the directions of the incident and diffracted rays make equal angles \(\theta\) with the atomic plane, and

\[ n\lambda=2d\sin\theta. \tag{13} \]

The conditions for the principal diffracted rays may also be represented in another way. From Fig. 4 it is easy to see that the Laue conditions \((2a,b,c)\) are satisfied simultaneously for all such diffractions with Laue numbers \(hkl\) for which the corresponding points \([[hkl]]\) of the reciprocal lattice lie on a sphere with center \(A\) and radius \(AO\), where the vector \(AO\) is parallel to \(\mathbf{s}_{0}\) and has magnitude \(\frac{1}{\lambda}\). The directions of the diffracted rays are then obtained by joining the point \(A\) to all points \([[hkl]]\) on this sphere. Thus, with the aid of the reciprocal lattice, the diffraction pattern from a crystal can be constructed. This construction may also be used to determine the intensities of the diffracted rays, in accordance with equation (1), if we regard this equation as a triple function of the coordinates \([[hkl]]\) of points in reciprocal-lattice space. Here the intensity of diffraction in any given direction is determined by the value of this function associated with the end of the parallel radius vector of the sphere of reflection. In the first

To the article by G. I. Finch and H. Wilman.

Fig. 5. Graphite

Fig. 6. Thin mica

Fig. 7. Comparatively thick mica

Fig. 11. Reflection from a natural face of diamond,
\[ \varphi = 60^\circ;\quad L = 22\ \text{cm};\quad E = 50\ \text{kV}. \]

In a first approximation we may regard this as an equivalent admission that the points of the reciprocal lattice, i.e. the points with integral indices $hkl$, are replaced by parallelepipedal regions with sides

$$ \frac{2}{M_1},\quad \frac{2}{M_2},\quad \frac{2}{M_3}, $$

parallel respectively to $a^*$, $b^*$, $c^*$, the areas of whose intersection with the sphere of reflection give the magnitude, shape, and intensities of the diffractions arising for any position of the crystal relative to the primary beam. The shape of these regions of intensity around the points of the reciprocal lattice, as was shown by Laue[^72], is connected more with the external shape of the crystal than with its size. A detailed investigation of this question is important for understanding the nature of the origin of certain phenomena of anomalous diffraction described below. In the special case of a thin crystal the points of the reciprocal lattice are drawn out into a system of straight-line segments (rods), normal to the plane of the plate, and have approximately the length $\dfrac{2}{T}$, where $T$ is the thickness of the plate-like crystal, equal to $(M_1 - 1)a$. In the case of a true two-dimensional lattice with axes $a$ and $b$, the reciprocal lattice is a system of infinite straight lines passing through the points in the $ab$ plane of the reciprocal lattice.

The “sphere of reflection,” otherwise called the Ewald sphere, in the case of diffraction of fast electrons has a much larger radius in comparison with the distances between points in the reciprocal lattice and, in a first approximation, may be taken as a plane, especially when considering the position of diffraction spots close to the undeviated beam.

The pattern with diffraction spots from a single crystal, directly surrounding the undeviated spot, is practically equivalent to a section of the reciprocal lattice by a plane (normal to the beam) magnified in the ratio $\lambda L : 1$. This circumstance was used by Bürgers[^73] to construct a simple model by means of which the geometrical features of electron-diffraction effects can be demonstrated optically.

Beginning with Kikuchi’s first experiments with mica, a whole series of other experiments has shown that patterns with spots from single crystals are, generally speaking, in quantitative agreement with the considerations presented above. Thus, for example, the thinnest lamellae of single crystals, having only a few atoms in thickness ($\ll 10^{-6}\ \text{cm}$), give sharp electronograms (Figs. 5 and 6), very similar to those patterns which ought to be obtained from two-dimensional atomic layers. Usually, however, it is possible to detect signs of increased spot intensities in the Laue zones, which becomes especially noticeable in the case of thicker crystals (Fig. 7). With a further increase in thickness these zones become ever narrower, and the number of spots situated in them decreases. With some orientation of a thick crystal these spots disappear altogether. At a certain thickness of the crystals, when the third

the Laue condition for rows of atoms along the direction of the beam has not yet become sharp, effects of secondary scattering arise in the form of bands or pairs of white and black parallel lines, the so-called “Kikuchi lines” (Fig. 8). With a further increase in the thickness of the crystal, a general blackening of the photographic plate occurs owing to strong incoherent scattering. By observing diffraction from mica flakes of known thickness, Kirchner[^74] and Derbyshire[^75] showed that the circular Laue zones remain still comparatively broad even at a thickness of the order of \(10^{-4}\) cm, when the coherent pattern is drowned in the general background. They also found that the blurring of the zero maximum obtained experimentally is greater than that given by the simple kinematic theory of scattering.

Fig. 9. Explanation of the origin of Kikuchi lines

Black and white lines in diffraction patterns, as was shown by Kikuchi[^76], can most simply be regarded as the result of multiple scattering, owing to which the electron beam, without losing its monochromaticity, becomes divergent and is reflected from those planes in the crystal which are at the Bragg angle. Such scattering is shown schematically in Fig. 9. The intensity of the diffuse scattering inside the crystal decreases with increasing deviation of the beam from its initial direction. Therefore the intensity of the \(I_1\)-ray, which makes the Bragg angle \(\theta\) with the reflecting plane and lies on the same side of \(CP\) as the primary ray, will be greater than the intensity of the \(I_2\)-ray, which makes the same angle with the plane but lies on the opposite side. These two rays should have met the screen at the points \(B\) and \(A\), respectively, but after reflection by the group of parallel planes \(CP\) they will in fact meet it at \(A\) and \(B\). Therefore at the point \(B\) there will be a resultant decrease in intensity proportional to \((I_1 - I_2)\), and at the point \(A\) an equal increase in intensity. All such selectively reflected rays will lie on two cones with half-apex angle \((90^\circ - \theta)\) about the normal to the reflecting plane, and the result of their intersection with the screen will be two practically straight branches of a very obtuse hyperbola, symmetrically situated on opposite sides of the line of intersection of the reflecting plane with the screen. In the direction toward the line situated on the same side as the undeviated spot \(O\), there will be a subtraction from the total intensity, while in the direction toward the other line, lying on the side opposite from \(O\), there will be an increase in the scattering intensity; the distance between these lines will be equal to \(\dfrac{n\lambda L}{d}\). The general pattern of such pairs of lines for any orientation of the crystal with respect to the primary beam can serve

may serve as a characteristic of the symmetry and structure of the crystal, since the median of a given pair of lines together with the position of the crystal determines the direction of the corresponding atomic plane, whereas the distance between the paired lines determines the interplanar spacing, and the intensity of each line relative to the neighboring background is a measure reflecting the strength of the group of planes, i.e. it depends on the structure factor.

Indexed patterns from mica in transmission were published by Kikuchi\(^{76}\), Shinohara and Matsukawa\(^{77}\), and Kirchner\(^{78}\); from graphite—by Finch and Wilman\(^{71}\), the intensities found being approximately of the same order as could be expected from calculation of the normal structure factor. The general method applicable to the complete indexing of patterns with Kikuchi lines was developed by Finch, Quarrell, and Wilman\(^{79}\).

Fig. 10. Data for indexing Kikuchi lines

Let the position of the crystal be such (Fig. 10) that the primary beam, directed normally to the screen, forms a glancing angle \(\psi\) with the plane \((u v w)—OX_1X_2,\;Y_1Y_2\). Suppose, further, that the direction \(OY_1\) in the plane \((u v w)\), having azimuth \(\varphi\) with respect to a convenient initial direction \(OY_2\) in the plane \((u v w)\), lies near the plane of incidence of the beam on \((u v w)\). Let \((UVW)\) and \((\lambda\mu\nu)\) be the direction cosines of the normals to \((u v w)\) and to any plane \((h k l)\) with three orthogonal axes \(OX_3,\;OY_3,\;OZ_3\) in a right-handed coordinate system conveniently chosen with respect to the crystal axes, and let the direction \(OY_2\), with zero azimuth in the plane \((u v w)\), also lie in the plane normal to \((u v w)\) and containing the axis \(OY_3\). If we take as the origin of coordinates in the diffraction pattern the point \(O_1\), at which the \(\varphi\)-azimuthal direction in the plane \((u v w)\) intersects the screen, with coordinate axes \(O_1X\), lying in the plane \((u v w)\) (positive direction to the right, as in Fig. 10), and \(O_1Z\), perpendicular to \(O_1X\), then the equation for the pair of Kikuchi lines of the \(n\)-th order, arising from the atomic plane

with Miller indices \((h\,k\,l)\) and distance \(d\), with sufficient accuracy, can be written as follows:

\[ C_2 z=-A_2 x+\left(B_2 l \pm \frac{nhL}{2d}\right), \tag{14} \]

where

\[ \begin{aligned} A_2&=A_3\cos\varphi-B_3\sin\varphi,\\ B_2&=A_3\sin\varphi+B_3\cos\varphi,\\ C_2&=C_3 \end{aligned} \tag{14a} \]

and

\[ \left. \begin{aligned} A_3&=(\lambda W-\nu U)\frac{1}{(1-V^2)^{1/2}},\\[4pt] B_3&=\{-\lambda UV+\mu(1-V^2)-\nu VW\}\frac{1}{(1-V^2)^{1/2}},\\[4pt] C_3&=\lambda U+\mu V+\nu W, \end{aligned} \right\} \tag{14b} \]

where, for planes almost normal to the screen, in the first approximation \(\dfrac{nhL}{d}\) denotes the distance between paired lines. We may note that

\[ \sum \lambda^2=\sum U^2=\sum A_3^2=\sum A_2^2=1. \tag{15} \]

If the position of the crystal and the lattice constants are known, then equation (14) can be solved, and the Kikuchi-line indices can be derived from their inclination and from the intercepts cut off on the \(z\)-axis, and conversely. The angle \(\varphi\) is taken here as the angle through which the crystal must be rotated in the positive direction, i.e. in the direction of rotation of a right-handed screw, about the positive direction of the normal to the plane \((u\,v\,w)\), in order to bring the crystal from the position with the axis \(OY_2\) in the plane of incidence of the ray onto the plane \((u\,v\,w)\) into the position under consideration.

The axes \(OX_3Y_3Z_3\) may conveniently be chosen in such a way that the \(b\)-axis of the crystal is directed along \(OY_3\), while the \(a\)-axis lies in the plane \(OX_3Y_3\) in the positive \(X_3\) direction, which gives the following values for the direction cosines \((\lambda\,\mu\,\nu)\) of the normal to any atomic plane \((h\,k\,l)\) for a triclinic crystal with respect to the axes \(OX_3Y_3Z_3\):

\[ \lambda=\left(\frac{h}{a}+\frac{k}{b}\cos\gamma\right)\frac{d}{\sin\gamma}, \]

\[ \mu=\frac{kd}{b} \]

and

\[ \nu=\{abl\sin^2\gamma+bch(\cos\alpha\cos\gamma-\cos\beta) +ack(\cos\beta\cos\gamma-\cos\alpha)\}\frac{d}{V\sin\gamma}. \]

where

\[ d=\frac{V}{\left\{\sum a^{2}b^{2}l^{2}\sin^{2}\gamma+\sum 2abckl(\cos\beta\cos\gamma-\cos\alpha)\right\}^{\frac12}} \]

and

\[ V=abc\left(1-\cos^{2}\alpha-\cos^{2}\beta-\cos^{2}\gamma+2\cos\alpha\cos\beta\cos\gamma\right)^{\frac12}, \]

i.e., \(d\) is the interplanar distance, and \(V\) is the volume of the elementary cell. In particular, for a cubic crystal we have

\[ \lambda=\frac{hd}{a},\qquad \mu=\frac{kd}{a}\quad \text{and}\quad \nu=\frac{ld}{a}, \]

where \(d=\dfrac{a}{\sqrt{\sum h^{2}}}\). Hence in equation (14) we shall have \(d=\dfrac{a}{\sqrt{\sum h^{2}}}\); furthermore, \(A_{2}\), \(B_{2}\), and \(C_{2}\) are from equation (14a), and

\[ \left. \begin{aligned} A_{3}&=\frac{hw-lu}{\sqrt{\sum h^{2}}\cdot\sqrt{u^{2}+w^{2}}},\\ B_{3}&=\frac{-huw+k(u^{2}+w^{2})-lvw}{\sqrt{\sum h^{2}}\cdot\sqrt{\sum u^{2}}\cdot\sqrt{u^{2}+w^{2}}},\\ C_{3}&=\frac{hu+kv+lw}{\sqrt{\sum h^{2}}\sqrt{\sum u^{2}}}. \end{aligned} \right\} \tag{16} \]

Analogous line patterns are also obtained in the case of X-rays. They were obtained, for example, by Linnik\(^{80}\) from quartz, using for this purpose a divergent X-ray beam, and by Kossel and co-workers\(^{81}\) from copper. The question of the intensity of the lines in the latter case was discussed by Laue\(^{82}\).

The representation given above of the origin of Kikuchi lines is only a convenient approximation, which gives us a simple picture when indexing them. In reality, however, it turns out that the sharpness of the lines decreases as they move apart from one another; moreover, when it is small, as is the case for planes with large Bragg spacings, pairs of lines take the form of a band with diffuse edges, although the higher orders of diffraction from the same planes become successively sharper. The intensity of such a band decreases from a certain high value to zero and then assumes a negative value with respect to the background. When the band passes symmetrically through the undeviated spot, it practically disappears in the vicinity of this region, but at known distances its intensity may prove considerable. In addition to Shinohara and Khashi\(^{50}\), mentioned above, other authors\(^{79,83}\) have also studied Kikuchi lines.

One of the most important features of patterns with Kikuchi lines is that the lines corresponding to certain groups of planes, each of which is defined as a certain row of the reciprocal lattice, form distinct black or white parabolic or circular envelopes. A discussion of these envelopes is found in the papers of Finch and co-workers, Shinohara, and Lauer \(^{83}\). Well-defined circles were first observed by Emslie \(^{84}\), who treats them as an effect of one-dimensional diffraction from the principal atomic rows of the lattice when these are oriented almost parallel to the beam. The same view is also held by Tillman \(^{85}\). However, Finch and co-workers \(^{79}\) showed that such circles and parabolas can be explained as envelopes of Kikuchi lines, and that under certain conditions such apparent curves can often be resolved into actual polygons, as in Fig. 11. Shinohara \(^{83}\) also arrived at an analogous conclusion; moreover, he was able to show that such envelopes arise around the points of intersection of lattice rows with the screen and have exactly the same positions as the Laue zones of analogous lattice rows in the diffraction of beams initially parallel to these rows.

Reflection from single crystals. The mechanism described above for the appearance of spots and lines when single crystals are transmitted through is also applicable to reflection from them, but here there are two additional factors that influence the form of the pattern, namely: the limited penetration of electrons into the crystal and their refraction on entering the region of the potential of the crystal lattice and on leaving it at comparatively small glancing angles.

For indexing the spot pattern, it is simplest to consider the points of intersection of the enhancement zones (Laue zones) from two of the most important line gratings, approximately perpendicular to the beam, with one of them parallel to the surface of the crystal; the third line grating must be approximately parallel to the direction of the primary beam, as in Fig. 12. Since the width of the beam, equal to about \(0.1\) mm, covers several hundred thousand atoms, the effective length of the line gratings of the first type will considerably exceed \(200\ \text{\AA}\), so that the enhancement lines will, generally speaking, be sufficiently sharp. Conversely, the lines produced by a line grating of the second type, normal to the beam, will always be greatly broadened, since the small angle in glancing incidence, together with the limited penetrability of the beam, permits coherent scattering only down to a depth of the order of \(20\ \text{\AA}\). The third line grating, parallel to the beam, likewise cannot give a sharp diffraction effect because the path length of the beam for coherent scattering will be only of the order of \(150\ \text{\AA}\); therefore at least the innermost circular Laue zone can never be perfectly sharp. Fig. 12 schematically illustrates these conditions, and Fig. 13, obtained

from a silicon carbide crystal, shows the pattern of spots of this type quite well.

The character of the zones and spots may vary depending on whether the surface is ideal or contains submicroscopic peaks. In connection with this, different patterns from ionic and metallic single crystals can be obtained when the surface is treated by grinding and etching, a question that has been investigated by many authors^86.

In addition to patterns from a stationary crystal, it is often desirable to obtain, in reflection, patterns from a rotating crystal analogous to rotation X-ray photographs, on which it would be possible simultaneously to measure the position, shape, and intensity distribution of a considerable number of diffraction spots, without the complication of the pattern that is caused by the Kikuchi effect, or without them. In this case the crystal has to be rotated during exposure of the plate either in azimuth at a constant angle of incidence of the primary beam, or with a change of the angle of incidence at a constant azimuth. The first method gives, generally speaking, a large number of reflections in the region of the undeviated spot, which is very convenient for determining the constants of crystal lattices and for investigating the structure factor, and also for studying the smoothness of a crystalline surface. An example is Fig. 14, which testifies to a considerable width of the horizontal zones of enhancement and, on the contrary, reveals sharp vertical zones. The second method, developed by Yamaguti^87, has been used many times for studying the effect of refraction on the positions of spots, especially those lying in the plane of incidence, i.e. on the “equatorial layer line” of the pattern; however, work by this method is associated with difficulties because of the spreading of the diffracted beams in the direction normal to the axis of rotation, and hence along the layer lines (see below).

Fig. 12. Construction of the diffraction pattern in reflection

Patterns with Kikuchi lines, obtained in reflection, are in form identical with analogous patterns obtained

...formed upon transmission. In both cases, the character of the pattern from a lattice of a given type (for example, cubic) depends on the scattering power of the atoms, on the atomic arrangement, and on other factors.

Practically all the Kikuchi-line patterns published up to the present time were obtained from natural faces and cleavage planes of ionic inorganic single crystals. However, we have succeeded in finding that sharp patterns of a similar character are also obtained from crystals with a layered structure, for example, from graphite^71 and cadmium iodide, from complex ionic crystals such as potassium chromate, and, finally, from organic crystals, for example, from cane sugar, ammonium oxalate, and benzoic acid. In none of these cases were effects of the appearance of diffuse zones observed, which we so often observed in diffraction by many thin organic crystals. Recently Storks and Germer^88 obtained, upon reflection, bands from lard, stearic acid, and methyl stearate deposited from solutions in benzene, but these bands, apparently, are bounded by rows of spots. Distinct Kikuchi lines are also obtained from metal crystals, as was observed, for example, by Fordham in our laboratory on an etched single crystal of copper. The difficulty of obtaining Kikuchi lines from metals depends to a considerable extent on the difficulty of obtaining good specimens. It is easy to see that, with comparatively weak distortion of the crystal, the lines must become blurred and disappear in the general background.

Another factor exerting a characteristic influence on the reflection pattern from a single crystal—namely, the refraction of electrons at the surface of the crystal—apparently causes, first of all, a displacement of the diffraction pattern toward the shadow side; moreover, the magnitude of such a displacement of any point in the pattern from its position corresponding to the absence of refraction rapidly increases with the distance of the point from the boundary of the shadow. Indeed, if an electron, having initially acquired acceleration due to a potential \(E\), moves in an electrostatic field \(V\), then the Schrödinger equation can be written as

\[ \Delta^2 \psi + \frac{8\pi^2 me}{h^2}(E+V)\psi = 0, \]

and its solution

\[ \lambda = \frac{h}{\{2me(E+V)\}^{\frac{1}{2}}} \tag{17} \]

will be equivalent to de Broglie’s law. If we assume that the field outside the crystal is equal to zero, while inside it has the mean value \(V_0\), then the wavelength outside the crystal will be

\[ \lambda = \frac{h}{(2meE)^{\frac{1}{2}}}, \tag{18} \]

To the article by G. I. Fink and H. Wilman.

Fig. 8. Mica of thickness about \(10^3\) Å

Fig. 13. Silicon carbide, face \((001)\), azimuth \((120)\)

Fig. 14. Rotation pattern in reflection from the \((111)\) face of diamond at a constant angle of incidence

Fig. 16. Polished \((110)\) face of diamond, azimuth \((100)\). See the disappearance of the Kikuchi line near the boundary of the shadow

Fig. 17. Ammonium chloride

Fig. 20. Ionic calcium, orientation \((001)\), film inclined to the beam

Fig. 21. Ionic calcium, orientation \((001)\), normal incidence of the beam

and inside the crystal

\[ \lambda'=\frac{h}{\{2me(E-V_0)\}^{1/2}}, \tag{19} \]

whence the refractive index of the crystal, as was shown by Bethe \(^{89}\),

\[ \mu=\frac{\lambda}{\lambda'}=\left(1+\frac{V_0}{E}\right)^{1/2}. \tag{20} \]

The values of the inner potentials hitherto obtained experimentally by means of the diffraction of fast electrons are all positive and are of the order of \(10\ \mathrm{V}\), the smallest value \((4\ \mathrm{V})\) having been obtained for pyrite, and the largest \((21\ \mathrm{V})\) for diamond. The experimental determination of the potential of crystals is of interest not only in connection with the effects arising in electron diffraction, but also as a basis for calculating the work function \(\chi\) in the emission of electrons from a crystalline surface according to Sommerfeld’s equation.

Fig. 15. Refraction of an electron beam

The position of the diffraction maxima is determined from a combination of (20) with Laue’s conditions \((2a,b,c)\) or with Bragg’s law (13). In the latter case, if diffraction in the plane of incidence from planes parallel to the surface is considered, we must replace the external glancing angle \(\theta\) by \(\theta'\) inside the crystal (Fig. 15), where, by Snell’s law, we have

\[ \frac{\cos\theta}{\cos\theta'}=\mu=\frac{\lambda}{\lambda'}, \tag{21} \]

so that

\[ 2d\sin\theta'=n\lambda'=\frac{n\lambda}{\mu}. \tag{22} \]

But since

\[ \lambda=\frac{h}{mv}\simeq\left(\frac{150}{E}\right)^{1/2}, \tag{23} \]

then, if the relativistic change in the mass \(m\) is neglected,

\[ \sin\theta'=\frac{n}{2d\mu}\left(\frac{150}{E}\right)^{1/2}, \tag{24} \]

whence

\[ \sin^2 \theta=\frac{n^2\lambda^2}{4d^2}-(\mu^2-1) \tag{25} \]

or

\[ \sin^2 \theta=\frac{150n^2}{4Ed^2}-(\mu^2-1). \tag{26} \]

Further, by (20),

\[ V_0=\frac{150n^2}{4d^2}-E^2\sin^2\theta, \tag{27} \]

whence, for known \(E\), \(\theta\), \(d\), and \(n\), \(V_0\) can be calculated. If the relativistic correction to \(m\) is included in the calculation, then

\[ \lambda=h\left(1-\frac{v^2}{c^2}\right)\frac{1}{m_0v}= \]

\[ =\frac{\left(\dfrac{150}{E}\right)^{\frac12}} {\left(1+\dfrac{eE}{600m_0c^2}\right)^{\frac12}} = \frac{\left(\dfrac{150}{E}\right)^{\frac12}} {\left(1+E\cdot10^{-6}\right)^{\frac12}}. \tag{28} \]

whence

\[ \sin^2\theta=\frac{\dfrac{150}{n^2}} {4d^2E(1-E\cdot10^{-6})}-(\mu^2-1). \tag{29} \]

Tillman \(^{8}\) gives a somewhat different expression

\[ E_r\sin^2\theta=\frac{150n^2}{4d^2}-\frac{V_0E_r(1-\sin^2\theta)}{E}, \]

where

\[ E_r=E(1+E\cdot10^{-6}). \]

If \(\theta\) is small, then approximately

\[ E_r\sin^2\theta=\frac{150}{4d^2}n^2-V_0', \]

where

\[ V_0'=V_0\frac{E_r}{E}. \]

Calculating \(E\sin\theta\) as a function of \(n^2\), we obtain a straight line whose intersection with the \(E_r\sin\theta\) axis gives \(V_0'\), and whose slope gives the distance \(d\) between the planes.

According to (20), \(\mu\) depends on the electron velocity, but the ratio of the angle \(\theta\) to the angle \(\Theta\), corresponding to the absence of a refractive index, is given by the expression

\[ \sin^2\theta=\sin^2\Theta-(\mu^2-1)=\sin^2\Theta-\frac{V_0}{E}. \tag{30} \]

Consequently,

\[ \frac{\sin^2\Theta-\sin^2\theta}{\sin^2\Theta} = \frac{V_0}{E}\left(\frac{2d}{n\lambda}\right)^2 \cong \frac{V_0\cdot 4d^2}{150\,n^2}. \tag{31} \]

Thus the ratio \(\sin\theta:\sin\Theta\) is practically independent of the electron energy above approximately 100 V, so that fast electrons are just as suitable for the experimental determination of \(V_0\) as slow ones, since the relative displacement of the diffraction is almost the same.

From (30), taking into account the smallness of the angles \(\theta\) and \(\Theta\), we can derive the relation used by Jenkins\(^{90}\),

\[ z^2=z_0^2-4V_0\frac{L^2}{E}, \tag{32} \]

where \(z\) denotes the distance of the spot in the plane of incidence, measured from the undeviated spot, and \(z_0\) is the distance that would occur in the absence of refraction. The order of magnitude of the displacement of the spot is shown in Table 1, calculated from the values \(V_0=10\ \mathrm{V}\), \(E=45\ \mathrm{kV}\), and \(L=28\ \mathrm{cm}\).

TABLE 1

\(z_0\) in cm	\(z\) in cm
7	6.950
6	5.941
5	4.930
4	3.913
3	2.882
2	1.818
1	0.5506

An analogous equation also holds for the displacement of Kikuchi lines, although in this case we must consider only one refraction, undergone by the electrons after leaving the crystal. Indeed, if the surface of the crystal is approximately normal to the screen and if we take the axes \(OX, OZ\) in the pattern, where \(OX\) coincides with the edge of the shadow, and \(OZ\) is normal to it and lies approximately in the plane of incidence of the primary beam, then the ordinate \(z_0\) of any point in the theoretical pattern of Kikuchi lines is decreased to \(z\) owing to refraction in accordance with

\[ z_0^2-\left(1+\frac{V_0}{E}\right)z^2+\frac{V_0(x^2+L^2)}{E}, \]

i.e.

\[ z^2=z_0^2-V_0\frac{L^2}{E}, \tag{33} \]

when \(x \ll L\) and \(V_0 \ll E\). The coordinate \(x\) remains unchanged. If the equation of the Kikuchi lines is \(z=mx+C\) for \(V_0=0\), then for a finite value of \(V_0\), instead of this equation we shall have the hyperbola

\[ z_0-m^2x^2-2mCx+\frac{V_0L^2}{E}+C^2=0, \tag{34} \]

which rapidly approaches the asymptote \(z=mx+C\). The influence of the inner potential \(V_0=10\ \mathrm{V}\) at \(E=45\ \mathrm{kV}\) and \(L=28\ \mathrm{cm}\) is shown in Table 2.

TABLE 2

\(z_0\) in cm	\(z\) in cm
7	6.988
6	5.985
5	4.983
4	3.979
3	2.971
2	1.956
1	0.909
0.5	0.028
0.30	0.000

It is obvious that the refraction effect for Kikuchi lines is small if they are located far from the boundary of the shadow, and is quite insignificant in transmission patterns, where the angles formed by the external faces of the crystal and the beam are large.

In patterns with spots obtained from good crystal faces, the displacement of the spots predicted by equation (25) was in fact observed, and, in accordance with (27), only those spots appeared for which the order

\[ n>\left(\frac{V_0}{E\cdot \frac{2d}{\lambda}}\right)^{\frac12}. \]

On the other hand, in the case of rough crystal faces, where the electron beam is diffracted as a result of passing through crystalline peaks, true transmission patterns were obtained, i.e. there was no displacement of the spots. However, quite often patterns were also obtained in which there was a continuous series of broadenings of the spots toward the shadow from the undisplaced position. This effect is probably caused by an intermediate roughness of the crystal surface, and also by the fact that the potential at the boundary of the crystal decreases to zero gradually rather than discontinuously\(^ {91}\).

The refraction effect is usually not very noticeable in patterns with Kikuchi lines, although it has also been used to estimate the inner potentials of crystals, chiefly in the case of lines arising from groups of atomic planes parallel to the surface. However, in the case of diamond, for which \(V_0=21\ \mathrm{V}\), the curvature of the lines near the boundary of the shadow is quite noticeable, as, for example, in Fig. 16.

The first reliable experimental determinations of refractive indices were made by Yamaguti\(^ {87}\), who caused beams to be reflected from cylindrically bent sheets of mica and molyb-

denite, thus obtaining something like a rotation photograph, the spots of which corresponded to different orders of diffraction from the cleavage plane. Since then it has been accepted by many investigators that an equation of type (25) is applicable to diffractions arising simultaneously from a stationary crystal, and in this way the inner potentials of many crystals were estimated; but another method, proposed by Yamaguti ^87, in which the crystal is rotated with a change in the angle of incidence at constant azimuth, which gives an enormous number of maxima on a single pattern, proved more successful and found application in many determinations of the inner potential. In measurements on patterns of this type from molybdenite, Yamaguti ^92 was the first to observe small apparent values of \(V_0\), calculated from lower orders of diffraction, and found that they asymptotically approach a limiting value at high orders; this effect, apparently, may be due to the exponential decrease of the mean lattice potential near the surface. This effect was subsequently studied by a whole series of authors ^93.

Passage through polycrystalline films. In considering diffraction patterns, a polycrystalline specimen may be regarded as equivalent to a single crystal rotating within the same limits of orientation as exist in the specimen. If there are many crystals with a random distribution in the path of the primary beam, then a pattern is obtained consisting of rings (Fig. 17) and analogous to a Debye–Scherrer powder radiograph. It was shown by Thomson ^94 and by Finch and co-workers ^79 that, owing to the low coherent penetrating power even of fast electrons, patterns comparatively free from background can be obtained only from specimens consisting of very thin crystals; therefore, if a pattern of the type of Fig. 11 were subjected to rotation about the primary beam, the rings arising from the merging of spots would for the most part be obscured by a dense veil. Consequently, in contrast to a powder radiograph, of which it is the prototype, an electron diffraction pattern usually arises from crystals each of which gives a pseudo-two-dimensional pattern with spots, as in Fig. 5, the character of which depends on the orientation of the crystal. However, every diffraction spot reaches the maximum of its intensity only at such a position of the atomic plane as corresponds to reflection of the primary beam at the Bragg angle. Therefore Bragg’s law gives the angular radii (equal to \(2\theta\)) of the diffraction rings on the pattern exactly as in the case of X-rays, and since these angles in the case of fast electrons are small, we have, with sufficient approximation,

\[ d_{hkl} = \frac{\lambda L}{R_{hkl}}, \tag{36} \]

where \(L\) is the camera length, and \(R_{hkl}\) is the radius of the diffraction ring corresponding to the plane \(hkl\). The indexing of diffraction

maxima and the calculation of lattice constants is carried out in the same way as in the treatment of X-ray photographs of powders, and with the same restrictions as those imposed by the lowering of the symmetry of the crystals.

When Debye–Scherrer patterns from powders are used in connection with the determination of the crystalline structure of a substance, measurements of intensities are of enormous importance; but the interpretation of these intensities on electronograms is complicated not only by the pseudo-two-dimensional nature of the patterns and by the effects of multiple scattering or dynamical interaction, but also by the circumstance that crystals in thin films usually have at least one more or less well expressed preferred direction of orientation. The most frequently encountered type of orientation corresponds to the case in which all crystals tend to have one degree of orientation, i.e. tend to arrange atomic planes of some type parallel to one another, with a random distribution of all the other possible orientations. This type of orientation is determined either by establishing the indices of those atomic planes which prove to be parallel, or by establishing the indices of the linear lattice normal to this plane. Such a linear lattice may be called the axis of orientation or the axis of texture. In this case the specimen is equivalent to a single crystal which is thin in the direction of the orientation axis and which rotates about this axis in the path of the beam. Thus any given diffraction maximum \(hkl\) may be regarded as the result of reflection of the primary beam from the corresponding group of atomic planes when they assume the position in which they form the Bragg angle \(\theta\) with the beam. It is evident that the diffraction maximum \(hkl\) must lie precisely where the ring of the Debye–Scherrer pattern would have passed, and moreover at a point usually determined with the aid of the azimuthal angle \(\delta\) (Fig. 18), measured from the projection of the orientation axis onto the photographic plate,

\[ \cos \delta=\frac{+\cos \beta-\cos \alpha \sin \theta}{\sin \alpha \cos \theta}, \tag{37} \]

where \(\alpha\) is the angle of inclination of the orientation axis to the beam, \(\beta\) is the angle between the normal to the plane \((hkl)\) and the linear lattice \([uvw]\), or the normal to the equivalent plane determining the orientation axis, and \(\theta\) is the Bragg angle. For fast electrons the angle \(\theta\) is small, so that we have

\[ \cos \delta=\pm \frac{\cos \beta}{\sin \alpha}, \tag{38} \]

where this approximation is equivalent to admitting that the diffraction \(hkl\) arises when the plane \((hkl)\) is parallel to the primary beam. Thus each atomic plane gives, generally speaking, four spots or arcs lying in the position of the ring \(hkl\), and if

The symmetry of the crystal is such that the same interplanar spacing is characteristic of several types of planes; thus, for each of them, such a series of arcs arises.

A simpler graphical method for determining the positions of diffraction maxima consists in making use of the fact that each maximum \(hkl\) must lie at the point of intersection of the ring \(hkl\) with the line of the layer of the \(l\)-th order, or the “zone of reinforcement,” caused by a linear lattice parallel to the axis of orientation or “rotation” \([uvw]\), where

\[ l = uh + vk + wl. \tag{39} \]

Fig. 18. Formation of spots upon orientation

In the general case, in electron diffraction the layer lines are practically straight, with the distance between them

\[ \frac{\lambda L}{c' \cos \theta}, \]

when the axis \(c' = [uvw]\) forms an angle \(\theta\) with the normal to the beam lying in the plane of incidence, where \(c'\) is the distance between lattice points along \([uvw]\); the width of the layer lines is approximately

\[ \frac{2\lambda L}{M_0 c' \cos \theta}, \]

if the crystals have an average number of lattice nodes in the direction of the row \([uvw]\) equal to \(M_0\). Diffraction maxima, whose position is determined in the manner indicated above, also fall into other groups with a simple relation of indices, forming places analogous to “row lines” on rotation X-ray photographs. Thus, the three Laue conditions for diffraction can be represented by the conditions that the maxima must lie at the points of intersection of the above-mentioned layer lines, caused by the lattice row \(c'\), with the positions of diffraction maxima corresponding to two other axes \(a'\) and \(b'\), conveniently chosen as two principal lattice rows \([u_1v_1w_1]\) and \([u_2v_2w_2]\) in the plane normal to the orientation axis \([uvw]\). The lines \(h'k'\) of the two-dimensional reciprocal lattice, caused by some one crystal, form a series of straight lines perpendicular to the plane \(a'b'\), i.e. parallel to \([uvw]\), and the directions of reinforcement are determined by the intersection of these lines with the reflection sphere. Hence the positions of the reinforcement directions for all crystals in the specimen can be obtained by

Fig. 19. Explanation of ellipses in two-dimensional diffraction

rotation of this system of reciprocal-lattice lines about an axis passing through the origin and parallel to the orientation axis, so that each line \(h'k'\) describes a cylinder whose intersection with the sphere of reflection, which is an approximately plane surface normal to the beam, is in practice an ellipse, as in Fig. 19. Thus it is easy to see that the loci \(h'k'\) on the photographic plate will be sufficiently close to a family of ellipses, whose major axes are parallel to the projection of the orientation axis on the screen, while the minor axes are the radii, normal to them, of the diffraction rings with indices \(h'k'0\) (taken with respect to \(a'b'c'\)), corresponding to planes parallel to the orientation axis \([uvw]\), the ratio of the major axis to the minor axis for each ellipse being

\[ \frac{1}{\sin \theta}. \]

By virtue of the third Laue condition for the axis \(c'\), i.e. \([uvw]\), the diffraction maxima \(h'k'l'\) appear only when these elliptical loci intersect the layer lines \([uvw]\), and all diffraction maxima with the same indices \(h'k'\) lie on one and the same elliptical locus. The relations between the indices \(h'k'l'\) and the indices \(hkl\), corresponding to the axes usually used, may be expressed as follows:

\[ \begin{aligned} h' &= u_1h+v_1k+w_1l,\\ k' &= u_2h+v_2k+w_2l,\\ l' &= u_3h+v_3k+w_3l. \end{aligned} \tag{40} \]

An example of layer lines and elliptical loci is provided by the electron diffraction pattern of cadmium iodide (Fig. 20), which, during condensation, is oriented with the face \((001)\) parallel to the collodion substrate. Here the layer lines correspond to \(l=0,1,2,3,\ldots\), while the elliptical loci pass respectively through all the diffraction maxima \(10l,\ 11l,\ 20l,\ 12l,\ 30l\), etc., which can appear at the given inclination \(\theta\) of the specimen\(^{78}\). Clearly expressed patterns with layer lines were also obtained from organic substances; moreover, Mott and Trillat\(^{95}\) showed that, in the case of hydrocarbons with long chains, the distance between layer lines corresponds not to a true translation of the crystal lattice, but to the alternating \(C—C\) distances along the chains.

A pattern of this type, analogous to a rotation X-ray photograph, is useful for resolving diffraction maxima into groups with related indices, which makes it possible to study interplanar spacings and intensities more conveniently than in the case of ordinary Debye–Scherrer patterns. In addition to the fact that the layer lines on a flat photographic plate normal to the beam are almost parallel, there is another important feature of electron diffraction patterns which is not encountered in X-ray photographs, namely the appearance of strong continuous rings (the limiting case for ellipses at \(\theta=90^\circ\)), when the orientation axis is parallel to the electron beam; in this case all rings with indices \(h'k'0\),

To the article by G. I. Finch and H. Wilman.

Fig. 22. Platinum obtained by cathodic sputtering

Fig. 23. Nickel, orientation \((110)\)

Fig. 25. Passage through a single crystal of silver grown on rock salt

Fig. 26. Bent crystal of molybdenite; the axis of curvature is normal to the beam

Fig. 27. Mica powder

Fig. 28. Combined pattern from graphite (left) and gold (right)

the corresponding planes, parallel to the beam, have high intensity, while all the others are weak or absent altogether. This constitutes another example, encountered in electron diffraction, of a pseudo-two-dimensional type of pattern, combined with the usual greater or lesser imperfection of orientation; here the formation of rings can readily be understood as equivalent to the rotation, about the undeflected spot, of the pattern of the type indicated in Fig. 5. Such an anomalous distribution of intensities has sometimes been the cause of erroneous judgments about structures, as was pointed out by Finch and Quarrell^96. Orientation can be detected by tilting the specimen relative to the beam (Figs. 20 and 21), as a result of which arcs appear in the pattern.

Transmission patterns have often been used to study growth and orientation in polycrystalline specimens with one degree of orientation by a number of authors^79,95,97. It turns out that most substances, both inorganic and organic, when condensed from vapors, crystallized from solutions, electrodeposited, cathodically sputtered, or deposited from colloidal solutions, or in some other way onto various solid or liquid surfaces, very often exhibit preferred orientations of the type mentioned above.

Restriction of the second degree of orientational freedom in a polycrystalline specimen is equivalent to a tendency toward the formation of a mosaic crystal; moreover, if this orientation is clearly expressed, the specimen gives a spot pattern which is practically indistinguishable from that for a single crystal. More often, however, from the presence of diffraction belonging to zone axes other than those parallel to the beam^98, or from the smearing of spots into arcs^96, such patterns make it possible to establish that the specimens are in fact polycrystalline. The best-known examples of this type of orientation are the patterns with arcs given by forged metallic films of Au, Ag, Al, Cu, Pd, Pt, W, etc. Other interesting cases are represented by patterns obtained upon electrodeposition of various metals on etched foils; here it turns out that the orientation of the deposited metal adapts itself to the orientation of the crystals of the substrate^99, and also by patterns from metallic films on the cleavage plane of rock salt, where polycrystallinity or one degree of orientation is found in the case of a cold substrate, and a tendency toward the formation of a thick single crystal, giving Kikuchi lines, with a corresponding increase in the temperature of this substrate.

Reflection from polycrystalline surfaces. Patterns obtained from polycrystalline surfaces when the beam is incident at a very small angle, as was shown already by Thomson in 1930, arise through the penetration of the electron beam through crystalline peaks on the surface, through which it can pass without appreciable loss of energy. This is confirmed by

by the fact that in the position of the rings no appreciable displacement due to refraction is usually observed. However, if the crystals have comparatively smooth surfaces of small curvature, then the ray may pass through a part of the surface inclined to it only at a small angle, so that, owing to the small depth of penetration, the diffraction maxima must be strongly blurred, while refraction must cause a displacement and further broadening of the rings near the shadow toward smaller angles. Such a pattern from randomly arranged large crystals should approximately correspond to rotation about an undeviated ray (as in Fig. 14). In practice we have not encountered such patterns, although according to Kirchner^100 and Germer^101 it is precisely this effect that explains the blurring of reflection patterns from certain thin layers of condensed metals and zinc sulfide. An illustration of this effect may be provided by the pattern from oriented crystals of condensed platinum (Fig. 22), where the arcs are displaced toward the shadow from their normal position.

The natural, undisturbed orientation obtained in thin layers of many crystals when they are deposited on inert substrates is best studied by the reflection method, using as substrate glass, fused quartz, polished metals, spinel, and other crystals that give amorphous polished layers. The orientation obtained is almost always such that the crystals are arranged with planes of one type parallel to the surface of the substrate, while otherwise retaining a random arrangement.^1

The indexing of patterns with layer lines from specimens with a single degree of orientation is similar to the indexing of the transmission patterns described above, except that in the former case one has to deal with only one half of the pattern. If the orientation is such that the plane \((uvw)\) is always parallel to the surface of the substrate, then the layer lines are parallel to the boundary of the shadow and the diffraction orders from \(uvw\) lie in the plane of incidence; therefore the orientation indices are in general determined very easily. An example of layer lines is provided by Fig. 23 from a nickel surface, which at first gave a pattern with continuous rings corresponding to a random distribution of crystals, and which began to show orientation after the nickel had acted as an electrode in a two-normal solution of \(\mathrm{H_2SO_4}\) at an alternating voltage of \(12\ \mathrm{V}\) between it and a carbon electrode. The layer lines \(l' = 0, 1, 2, 3,\ldots\), where \(l' = h + k\), here correspond to the interplanar distance

\[ \frac{\lambda L}{T_{[110]}}=\frac{\lambda L}{a\sqrt{2}}=\frac{2\sqrt{2}\lambda L}{4a}=\frac{R_{220}}{4}, \]

^1 H. A. Shishakov and V. I. Kasatochkin (ZhETF, 8, No. 10, 1938) succeeded in observing on unilaterally polished brass a case where parallel bands are arranged not at some plane, but at the axis \([011]\), i.e., the plane \([011]\) lies normal to the substrate and normal to the direction of grinding. — Ed.

where \(R_{220}\) denotes the radius of the 220 ring. A diffraction spot or arc is formed at the point of intersection of the given layer line \(l'\) with the position of the Debye–Scherrer ring, if the indices of one or more reflections \(hkl\) producing the ring satisfy the relation \(h + k = l'\) (Fig. 24).

In addition to revealing the general tendency of crystals toward orientation in the case of specimens obtained by condensation, cathodic sputtering, electrodeposition, evaporation of a solution, or chemical reaction on a massive substrate, the reflection method shows that strong orientation also often occurs when surfaces solidify from the liquid state, arising at the moment during polishing[^102]. Of particular interest may be the following observations: a) several preferred orientations may exist in one and the same specimen[^99],[^103]; b) heating of the substrate often helps to improve the orientation[^103],[^104],[^105]; c) the direction of the orientation axis in condensed layers depends on the direction of the incident vapor stream[^105]; d) along with orientation, effects of pseudomorphism may occur[^105]; e) oxide films on the surface are very strongly oriented[^96],[^106]; f) hydrocarbon molecules with long chains stand upright on the surface on which they have been spread or deposited from solution or from vapor, and therefore give horizontal layer lines with superposed spots, the interplanar spacing giving the layer lines corresponding to the alternating C—C distance along the chains, while the distribution of spots makes it possible to determine the distance between adjacent hydrocarbon chains[^107].

Fig. 24. Face-centered cubic lattice; \((110)\) parallel to the surface

V. Anomalous diffraction phenomena

Advances in the experimental technique of electron diffraction over the last five years have led to the discovery of a number of cases of anomalous diffraction, where the observed patterns cannot be connected

with the known structure under consideration. However, the majority of these anomalies have nevertheless been explained and classified into groups according to the nature of their origin.

One of the first anomalous effects observed is the presence of certain superfluous rings, the so-called “extra” rings, in patterns from polycrystalline materials, which formerly were regarded as diffraction with fractional indices, analogous to what was observed in the case of slow electrons, although Thomson \(^{108}\) had already indicated that some of these reflections may be caused by Kikuchi lines. However, many of the reflections in early patterns of this type, obtained with fast electrons, proved possible to ascribe also to known kinds of contamination. Thus, for example, the innermost ring, sometimes observed in transmission patterns from zinc-oxide films, is now ascribed to contamination, possibly zinc silicate, in which the surface oxide film on the molten metal is usually rich \(^{109}\).

Patterns with sharp spots or rings, which various authors \(^{110}\) formerly ascribed to the crystalline structure of cellulose and its derivatives, were later found \(^{111}\) to belong to contamination by hydrocarbons and fats, to which the “extra” rings in patterns from thin metallic films can also often be attributed. This system of rings is very characteristic and changes only very little with the chain length of hydrocarbons or their derivatives, such as, for example, acids, alcohols, and ethers. In this way the pattern from “fats” is always easily recognized, and can also always be eliminated by trying to wash the specimen with a suitable solvent. Another group of “extra” rings appearing in patterns from metals \(^{112}\), and especially from those sensitive to heating, is now attributed \(^{112}\) to amalgams, which, as it turns out, have the most varied and complex structure.

Another group of “extra” rings, associated with a characteristic system of bands, as Finch, Quarrell, and Wilman \(^{110}\) showed, appears chiefly in patterns from face-centered cubic metals and is connected with the normal rings characteristic of the metal. Quarrell \(^{113}\) ascribes these rings to a hexagonal close-packed modification of the metal, arising in the first stages of deposition, both electrolytic and from vapor, after which, together with the growth of the film thickness, there occurs a gradual change through intermediate rhombohedral structures into the normal cubic form, and, as a result of the transition from one structure to another, the system of bands appears. Incidentally, Quarrell succeeded in obtaining, by electrodeposition, a thin film of silver which exhibited a pattern characteristic of a hexagonal close-packed metal with lattice parameters consistent with the atomic diameter of silver. In the case of gold, nickel, and cobalt, such patterns were obtained only together with the pattern from the normal cubic form. In connection

With this we may note that, analogously to what was described earlier, by means of X-rays a hexagonal form of cobalt was found on thin nickel films obtained by cathodic sputtering—its hexagonal modification with close packing.

A very important class of “extra” diffraction is diffraction of the “forbidden” type, which, according to kinematic theory, should normally be extinguished owing to restrictions associated with the structure factor, but which nevertheless appears because of the strong dynamical interaction of electron waves within the crystal. The formation of these “forbidden” reflections can be pictured on the basis of the assumption that each ray corresponding to a spot in the pattern from a single crystal, as in Fig. 5, acts as a new primary ray and again undergoes diffraction in the crystal. If the first diffracted ray has Laue indices \(hkl\), then the point \([[hkl]]\) of the reciprocal lattice will lie on the Ewald sphere of reflection; the latter will also be a sphere of reflection for the secondary diffraction of this ray \(hkl\). Therefore, if the crystal is not exceptionally small, the newly diffracted ray will give rise only to those further diffracted rays which correspond to points lying on this sphere, i.e., in this process diffracted rays may appear only with integral indices, although some of them may, by their position, correspond to diffracted rays forbidden by virtue of the structure factor. If the diffracted ray \(hkl\), i.e. the ray “reflected” from the plane \((hkl)\), is again reflected by the plane \((h_1 k_1 l_1)\), then the resulting ray will, evidently, have the indices \(h + h_1,\ k + k_1,\ l + l_1\). Such an effect was observed by Rater \(^{114}\) in reflection from a pyrite crystal. Besides producing “forbidden” reflections, multiple scattering tends to smooth out the intensity of the spots in the pattern, and the width of the curved Laue zones becomes greater than that calculated on the assumption of simple scattering by a crystal of known thickness, as was shown experimentally by Kirchner \(^{110}\) and Derbischaire \(^{115}\). Distinct “forbidden” reflections were also obtained by Finch and Wilman \(^{116}\) in point photographs from graphite, molybdenite, and mica.

Of great interest are transmission patterns obtained by Lassen and Brück \(^{117}\) from silver films grown on cleavage planes of heated crystals of rock salt. These patterns contain reflections to which integral indices cannot be assigned, although they evidently do belong to the normal structure of silver. By means of reflection patterns these authors found that, by its orientation, silver adapts itself to the orientation of rock salt, despite the difference between the constants of their lattices (4.08 and 5.63 Å) and the different arrangement of atoms in the planes of contact. We have recently succeeded in observing an analogous orientation and a similar type of pattern not only on heated rock salt, but also on cleavage planes of bromide

potassium (Fig. 25), although in this latter case the differences in lattice dimensions were even greater than between rock salt and silver. The bands lying along the two principal rows of spots and passing through the diffraction spots 200 give a noticeable region of broad bands in the case of a polycrystalline specimen, as we had previously been able to observe for various electrodeposited metals, especially in patterns from silver films electrodeposited on cadmium. We had already long ago noticed that, when the crystals are large and only a limited number of them lie in the path of the beam, the bands reveal a cross-shaped linear structure; moreover, strong spots are obtained where the lines intersect with the normal positions of the rings. This circumstance, as well as the appearance of analogous, though thinner, bands in patterns from cobalt electrodeposited on thin copper foil, suggests that the formation of strong bands in these cases, at least in part, is due to diffraction bands associated with the anomalous-type pattern obtained by Lassen and analyzed theoretically by Laue¹¹⁸.

Kirchner and Lassen¹¹⁹ showed that anomalous diffraction spots appear in patterns obtained by reflection from layers of silver deposited on cleavage planes of rock salt and having a thickness of about 300 Å, whereas in the case of films about 1000 Å thick a pattern with spots of the normal type is obtained together with diffuse Kikuchi lines. The positions of the spots in the patterns proved to be in agreement with the idea that the silver crystals are bounded by octahedral planes, as a result of which a two-dimensional diffraction pattern arises, expressed in the reciprocal lattice by lattice lines parallel to the diagonals of the body-centered cubic cell of the reciprocal lattice. Patterns analogous to Lassen’s pattern from silver were also observed by Brück¹¹⁷ when transmitting through Ag, Al, Au, Cu, Co, Ni, Pd, Cr, and Fe condensed on heated cleavage planes of rock salt.

Laue¹¹⁸ investigated the form of the “regions of intensity” around reciprocal-lattice points when the diffraction intensity in any direction with indices \(hkl\) is represented as a function of the coordinates of the point \([[hkl]]\) in the reciprocal lattice. He found that these points are surrounded by identical “regions of intensity,” in which the intensity function is greater than zero, and that the shapes of these regions are almost completely determined by the external shape of the crystal and are centrosymmetric with respect to the lattice points; moreover, increasing the crystal by \(p\) times in the direction normal to the plane \((hkl)\) decreases the size of the region of intensity in the reciprocal lattice by \(1/p\) times in the \([hkl]\) direction of the reciprocal lattice. Consequently, if the crystal is a flat disk, then the region of intensity expands in the direction perpendicular to the disk. According to Laue, Kirchner and Lassen’s assumption about the octahedral shape of the silver crystals in their films

and their construction of spots by means of the reciprocal lattice are correct; moreover, there is no need to assume that octahedral planes produce a two-dimensional effect. The shape of the intensity regions in the reciprocal lattice for an octahedral crystal was calculated by Laue and Riewe[^118]. The patterns from nickel and cobalt obtained by Cochrane[^120], which show a splitting of the spots into “satellites,” were likewise reduced by Laue to the development of the boundary plane (111).

In the particular case when the crystal has the form of a very thin layer of large area, the diffraction pattern approaches that from a purely two-dimensional lattice. The latter, for any orientation, must give diffraction rays \(hk\) with continuously varying positions as the angle of inclination of the ray changes, as follows from the intersection of the Ewald sphere with continuous lines of the reciprocal lattice. Naturally, the positions of the diffraction rays from such a lattice must correspond to a completely unlimited series of values of the index \(l\). And indeed, Finch and Wilman[^116] succeeded in obtaining patterns of this type from graphite, molybdenite, mica, and cadmium iodide, all of which give continuous diffraction bands when the crystal plate is bent or rotated during exposure (Fig. 26). Such two-dimensional bands were also observed by us in patterns from very thin organic crystals.

Crystals that are easily cleaved into the thinnest layers, such as, for example, mica, if taken in the form of a powder, give a pattern (Fig. 27) consisting of circular bands blurred, owing to the indicated two-dimensional effect, in the direction of increasing angles and having sharp inner boundaries corresponding to the reflections \(hk0\)[^121]. Therefore a considerable part of the crystalline lamellae in such specimens must be extremely thin. Analogous bands were obtained by Burgers[^122] from strongly etched thin layers of an iron–nickel alloy, by Kirchner[^110] from cadmium iodide, and Finch and Wilman[^116] showed that such patterns from powders of pseudo-two-dimensional crystals can be obtained when their thickness is less than 50 Å.

According to the calculations of Laue[^118], the intensity of the waves scattered by a powder consisting of randomly oriented two-dimensional lattices can be expressed as follows:

\[ I_{hk}=\frac{C}{\sin \frac{\chi}{2}\left(\sin^2 \frac{\chi}{2}-\sin^2 \frac{\chi_0}{2}\right)^{\frac{1}{2}}}, \tag{1} \]

where \(C\) is a constant, \(\chi\) is the angle of deviation from the direction of the primary beam, and \(\chi_0\) is the angle determined by the equation

\[ 2\sin \frac{\chi_0}{2}=\lambda\left|ha^{*}+kb^{*}\right|, \tag{2} \]

where \(a^*\) and \(b^*\) are the axes of the reciprocal lattice, obtained from the axes \(a\) and \(b\) of the two-dimensional lattice with any other arbitrary, but for convenience usually perpendicular, axis. For small scattering angles, instead of (1) we have

\[ I_{hk}=\frac{C'}{R^2(R^2-R_0^2)^{\frac{1}{2}}}, \tag{3} \]

where \(I_{hk}\) is the intensity of the diffraction band at the end of radius \(R\), and \(R_0\) is the radius of the ring \(hk0\), corresponding to the three-dimensional lattice. And indeed, the radial distribution of intensities in a pattern with rings from hexagonal two-dimensional lattices proves here to be similar to the intensity distribution observed by Shteingel’ \(^{121}\), with the sharp edges of the bands being in accord with the positions of the \(hk0\) reflections in the case of a normal three-dimensional lattice. Trendelenburg and collaborators \(^{123}\) found that in patterns from powders of crystals with a single well-developed cleavage direction, such as graphite, kaolin, etc., diffraction from this plane is usually weak or altogether absent, although in the case of calcite the corresponding diffraction has nearly the expected intensity. It is obvious that the flakes of layered crystals have such large dimensions in directions parallel to the cleavage plane that coherent diffraction is hardly possible, except for diffraction in the interaction with these faces when they are situated almost parallel to the beam; in that case, however, even the lowest orders tend to disappear as a result of refraction.

Finch and Wilman \(^{116}\) noted that in patterns from polycrystalline graphite specimens the band and ring system beyond ring \(100\) can be resolved into a group of four clearly visible rings, of which the innermost and the third correspond to normal diffraction from \(100\) and \(101\). It turned out that these four rings are characteristic of patterns from all graphite powders, irrespective of their origin, and that impurities therefore have nothing to do with it. Since the radii of these rings and the distribution of arcs in the case of oriented specimens closely correspond to diffraction from planes with fractional indices \(10\frac{1}{2}\) and \(10\frac{4}{3}\), it was necessary to carry out an investigation of patterns obtained by reflection and by transmission from graphite single crystals; complete agreement was obtained with structural investigations of graphite by means of X-rays. The resolution was aided by the discovery of continuous diffraction bands in the complex pattern with spots, obtained from thin curved flakes, as well as of intermediate spots whose radial distances from the undeviated spot correspond to fractional indices or to integral indices forbidden by the structural factor. Finch and Wilman showed that this phenomenon is limited to extremely

thin lamellae (less than 50 Å), and they attributed this effect to the limitation of the lattice in the direction of the \(c\) axis, as a result of which, in groups of parallel net planes, the number of atoms contained becomes unequal, so that the effective periodicity, or repetition period, of these planes becomes a multiple of the corresponding value in an infinite lattice; hence the diffraction “fractional orders” arise. Similar patterns with forbidden reflections were also obtained from molybdenite lamellae. Recently Thomson\(^{124}\) proposed another explanation of the observed phenomenon, whose cause may be considered to be auxiliary maxima of the interference function; moreover, as applied to the case of graphite and molybdenite, it gives satisfactory agreement with experiment both as regards the positions and as regards the intensities of the observed anomalous diffraction phenomena.

VI. Estimation of the size of crystals

The sharpness of electron diffraction patterns usually does not depend on the thickness of the crystal. Indeed, for a linear lattice of length \(T\), parallel to the beam, the half-width \(B\) is defined as the angular distance between two points on the diffraction maximum with intensities equal to half the intensity between them, and for very small values of \(T\) is given by the equation

\[ B=\left(\frac{2.8\lambda}{\pi T}\right)^{\frac{1}{2}} =0.944\left(\frac{\lambda}{T}\right)^{\frac{1}{2}} . \tag{1} \]

Since \(T\) is limited by the depth of coherent penetration of the electrons, approximately 150 Å, \(B\) has a minimum of \(1.7\cdot10^2\) radians (\(\lambda=0.05\) Å), which corresponds to a width of 8.5 mm for a camera length of 50 cm. Such resolving power is insignificant in comparison with the resolving power of even a much shorter lattice situated normally to the beam. Consequently, in considering the influence of crystal size on the broadening of electron diffraction patterns from powders, we may confine ourselves to considering atomic rows with a very steep inclination to the beam.

For a linear lattice of length \(T\), normal to the beam, the intensity \(I\) at a point situated at an angular distance \(\varepsilon\) from the maximum intensity \(I_{\max}\), for small diffraction angles, will be approximately

\[ \frac{I}{I_{\max}}=\frac{\sin^2\varphi}{\varphi^2}, \]

where \(\varphi=\pi T\varepsilon \frac{1}{\lambda}\). If \(\frac{I}{I_{\max}}=\frac{1}{2}\), then \(\varphi=1.40\) radians, and \(\varepsilon=0.445\frac{\lambda}{T}\). Hence for the half-width \(B=2\varepsilon\) we obtain

\[ B=0.89\,\frac{\lambda}{T}\ \text{radians}. \tag{2} \]

This equation is in fact also valid for a number of linear lattices2 and therefore can be used to calculate, from the half-width of the diffraction maxima, the \(hkl\) length of the crystal in the direction normal to the planes \((hkl)\), provided only that the section of the crystal normal to the electron beam is a rectangle with one side in the plane \((hkl)\), or, in the general case, a parallelogram one side of which is normal to the planes \((hkl)\).

By an analogous method one may also obtain an approximate solution in the case of a diffraction ring corresponding to a disordered arrangement of two-dimensional crystals. Assuming that each two-dimensional lattice is a parallelogram with sides \(T_a\) and \(T_b\) and with angle \(\alpha\) between them, and taking as the lattice translations the lengths \(a\) and \(b\) along the sides of the parallelogram, we have approximately

\[ \frac{T_a \cos^2 \omega_a + T_b \cos^2 \omega_b}{\cos^2(\omega_a-\psi_a)} = 1.59\,\frac{\lambda^2}{B^2}, \tag{3} \]

where

\[ \tg \omega_a = T_a\left(\frac{1}{T_b \sin \alpha} - \frac{1}{T_a \sin \alpha}\right), \]

and

\[ \tg \psi_a = \left(\frac{ak}{bh}-\cos\alpha\right)\frac{1}{\sin\alpha}, \]

and an analogous expression for \(\omega_b\).

For a rectangular two-dimensional lattice we hence have

\[ \frac{bh}{T_a}+\frac{ak}{T_b} = \frac{1.12B}{\lambda}\left(a^2k^2+b^2h^2\right)^{\frac{1}{2}}, \tag{4} \]

which equation is equivalent to (2) for \(k=0\).

For a hexagonal two-dimensional lattice, when \(a=b\) and \(T_a=T_b=T\),

\[ T=\frac{1.54\lambda}{B}\, \frac{h+k}{\left(h^2+hk+k^2\right)^{\frac{1}{2}}}. \tag{5} \]

If \(n\) diffraction maxima of equal intensity and with half-widths \(B_1, B_2,\ldots,B_n\) coincide, then

\[ B= \frac{n^{\frac{1}{2}}} {\left(\frac{1}{B_1^2}+\frac{1}{B_2^2}+\ldots+\frac{1}{B_n^2}\right)^{\frac{1}{2}}}. \tag{6} \]

Scherrer’s calculation of the resolving power for a crystal of cubic system and shape gives

\[ B=2\left(\ln \frac{2}{\pi}\right)^{\frac12}\cdot \frac{\lambda}{T}\cdot \frac{1}{\cos 2\theta} = \frac{0.94\lambda}{T\cos 2\theta}. \tag{7} \]

The general solution, requiring the integration of interference functions that replace the points of the reciprocal lattice for finite crystals, was derived by Laue for parallelepipedal crystals. If the crystal axes are taken parallel to the edges of the parallelepiped, then

\[ B=\frac{3.6\pi}{\cos 2\theta}\,\frac{1}{2k} \left\{\sum \left(\frac{\mathbf{a}^{*}\mathbf{G}}{M_1}\right)^2\right\}^{\frac12}, \tag{8} \]

where

\[ \mathbf{G}=\frac{\sum h\mathbf{a}^{*}}{\left|\sum h\mathbf{a}^{*}\right|} \quad \text{and} \quad k=\frac{2\pi}{\lambda}, \]

and \(M_1\) is the number of lattice points along the axis \(a\). For an orthorhombic lattice we have

\[ B=\frac{0.9\lambda}{\cos 2\theta} \left\{ \frac{\sum \left(\dfrac{h}{M_1a}\right)^2} {\sum \left(\dfrac{h}{a}\right)^2} \right\}^{\frac12}, \tag{9} \]

which, in the case of a rectangular two-dimensional lattice, gives

\[ \frac{h^2}{T_a a^2}+\frac{k^2}{T_b b^2} = \frac{1.23 E^2}{\lambda^2} \left(\frac{h^2}{a^2}+\frac{k^2}{b^2}\right). \tag{10} \]

Brill \(^{126}\) applied Laue’s method to the case of electron diffraction and thereby obtained the equation

\[ B=\frac{0.525}{\omega}\cdot \lambda \cdot \left\{\sum \left(\frac{\mathbf{a}^{*}\mathbf{G}}{M_1}\right)^2\right\}^{\frac12}. \]

Since, however, the constant \(\omega\) must be taken equal to 0.55, as for X-rays, Brill’s equation offers no advantages over Laue’s equation.

Experience shows that even for infinitely large crystals diffraction has a finite angular width \(b\), i.e., it is an apparatus constant. Therefore, if the observed half-width-

the width of the diffraction maximum from the given crystal is equal to \(B'\), then it is usually assumed that

\[ B = B' - b. \tag{11} \]

However, Thomson, Stuart, and Murison\(^{127}\) proposed the relation

\[ B^2 = (B')^2 - b^2, \tag{12} \]

which is theoretically to be preferred if the edges of the diffraction maximum are not sharp.

Besides the limited resolving power, other factors may also affect the broadening of the ring. It is therefore necessary to take into account, for example, one-sided broadening proportional to the diameter of the ring. According to Ewald and Schöbau\(^{128}\), the width of the ring decreases with increasing accelerating voltage, although Kirchner was unable to confirm this. In addition, it is possible that the polychromatism of the beam may also cause broadening of the ring proportional to the diameter of the ring.

In Table 3 below are given some values of \(T\), calculated from equation (2), from the half-width of the diffraction ring for a linear grating, assuming a camera length of \(50\ \mathrm{cm}\) and a wavelength of \(0.05\ \text{\AA}\). Such a table is quite suitable for many purposes. It is useful, however, to remember that the ratio of the ring diameter to its total width is equal to the number of planes in each crystal participating in the diffraction. The chief experimental difficulty lies in measuring the apparatus constant \(b\), since it may vary depending on what part of the electron beam falls because of thick parts of the specimen or passes through the gaps between crystals without diffraction.

TABLE 3

Half-width in mm	\(T\) in \(\text{\AA}\)
0.05	440
0.1	220
0.15	150
0.2	110
0.3	75
0.4	55
0.5	44
0.7	31
1.0	22
2.0	11
5.0	4

The significance of the width of rings in the case of reflection patterns deserves special consideration. We shall not consider here the case of etched single crystals, and shall refer only to the appropriate literature\(^{129}\). As for polycrystalline specimens, we note that the width of the ring gives the length of that part of the surface protrusions which have a thickness of less than \(150\ \text{\AA}\). Therefore, in the case of conical protrusions, the resolving power depends on the angle of the cone and cannot give any indication of the true

the grain size. Thus, for example, large elongated crystals may give broad reflections if the surface peaks are blunt. In addition, when the cone angle approaches \(180^\circ\), refraction effects may occur which cause a broadening of the inner rings, as is seen, for example, in Fig. 22. Despite the considerable width of such reflections, the true size of the crystals may be large.

It must therefore be admitted that, in the case of reflection patterns, when determining particle sizes one can obtain only their minimum value, and that even in the case of transmission specimens the results are associated with various errors and can be regarded only as very approximate, and only for sufficiently thin crystals so that they can transmit electrons. It should also be remembered that, in the case of clear patterns, free of background and showing no orientation, the sizes of the diffracting crystals cannot be greater than \(150\ \text{Å}\) in the direction of the beam, and that the presence of very sharp rings indicates only that the crystals are plate-like or needle-like in shape. Consequently, the indicated methods may be used to obtain approximate values of crystal sizes; absolute values, however, must be spoken of only with extreme caution, especially in the case of crystals with dimensions greater than \(100\ \text{Å}\).

VII. Influence of Crystal Size on Lattice Dimensions

In their method for measuring lattice constants by the successive recording of two different patterns on one photographic plate, Finch and Quarrell \(^{130}\) proposed using zinc oxide as the standard substance. However, determining from it the lattice constant of gold, Finch and Wilman \(^{131}\) found a serious discrepancy between electronographic and radiographic values, while Coslett \(^{132}\) observed that these differences are not constant, but depend on the age of the specimen. A number of other authors \(^{133}\) have reported the existence of such anomalies in the case of other substances as well. Finch \(^{134}\) believes that such anomalous lattice dimensions may arise from the size of the crystals, in accordance with Lennard-Jones’s theory \(^{135}\), which predicts contraction or expansion of the lattice in the case of very small crystals of ionic or homopolar compounds as the crystal size decreases.

In connection with this, Finch and Fordham \(^{136}\) report determining, by the method of electron diffraction, the lattice constants of small crystals of gold and of the lithium, sodium, and potassium halide salts relative to the C—C distance in a hexagonal graphite layer (Fig. 28). The lattice constants of graphite relative to \(a_{\mathrm{Au}} = 4.070\ \text{Å}\) proved to be

\[ a = 2.458\ \text{Å};\quad c = 6.701\ \text{Å},\quad \text{and}\quad \frac{c}{a} = 2.726^{137}. \]

Proceeding from the nature of the crystalline form and the interatomic bonding in hexagonal layers, the distance C—C may be taken as independent of the size of the crystal, which is also confirmed by modern X-ray determinations of the lattice constant of graphite, \(a = 2.456_{8}\ \text{Å}; c = 6.695_{6}\ \text{Å}; \dfrac{c}{a} = 2.726\), which are in agreement with the data cited above. It follows from this that the electron-diffraction lattice constant of gold is likewise in agreement with its X-ray value. Thus gold, as well as graphite, may be used as a standard substance in electron diffraction. However, in the case of halide salts of the alkali metals, considerable differences were sometimes found between electron-diffraction and X-ray data (Table 4).

TABLE 4

Substance	\(a\) by electron diffraction	Excess relative to X-ray values
LiF	4.024	(0.004)
LiCl	5.132	—0.01
LiBr	5.491	(0.002)
LiJ	6.014	0.014
NaF	4.437	0.018
NaCl	5.664	0.036
NaBr	5.978	(0.004)
NJ	6.464	(0.002)
KF	5.352	—
KCl	6.314	0.037
KBr	6.625	0.039
KJ	7.072	0.020
Au	4.067	(—0.003)
ZnO	\(a = 3.258\)	0.015
ZnO	\(c = 5.239\)	0.044

The difference in parentheses must be regarded as insignificant. Thus, if one does not count the one possible exception of LiCl, in all these cases the lattice constants obtained by electron and X-ray methods do not agree with one another; moreover, contrary to the prediction of Lennard-Jones, the difference indicates an expansion of the lattice in the case of small crystals. Such expansion is observed for six halide salts of the alkali metals and in some cases reaches a value of \(0.6\%\), which has not yet been explained theoretically.

VIII. Structure of Metallic Films and Surfaces

The method of preparing metallic films may have a profound influence both on the size and on the arrangement of the crystals. In forged or rolled films the crystals, generally speaking, are more or less strongly oriented; moreover, in the case of cubic face-centered metals the plane \((1\ 0\ 0)\) is arranged parallel to the surface, and, in addition, there exists a second direction of orientation, the purpose of which consists in creating resistance to the rupture of individual crystals. In particular, rolled tungsten \(^{139}\) is oriented with the planes \((1\ 0\ 0)\) parallel...

relative to the surface, while the $(110)$ planes are oriented along the direction of stretching.

To explain the unusually broad X-ray reflections in the case of cold-worked metals, lattice distortion is sometimes considered the cause; but in the case of forged sheets no such effect has been observed by means of electron diffraction. In this respect, the constancy of the lattice dimensions of gold in thin films obtained both by electrodeposition and by condensation or forging deserves attention$^{136}$.

In the case of metal films obtained by condensation from vapor, the nature and temperature of the substrate, as a rule, exert the strongest influence both on the size of the crystals and on their orientation. On a cold inactive substrate, such as cellulose, polished fused quartz, or glass, the size of the crystals is, generally speaking, small, and the orientation is weakly expressed or altogether absent. But if during condensation the substrate maintains a suitable temperature, then the most varied cases of orientation may occur, undoubtedly owing to the increased mobility of the condensing atoms and of the newly formed crystals on the surface. The direction of the orientation axis in these condensed films, generally speaking, indicates the location of the vapor source, provided, of course, that evaporation was carried out in a good vacuum$^{140}$. If a suitably heated substrate is active, such as, for example, the cleavage plane of a rock-salt crystal, then metal films are obtained which give typical two-dimensional patterns$^{117}$. It seems unlikely that such films are true single crystals, since metal films in their orientation conform to the structure of rock salt, and since the two have very different lattice dimensions; it appears more probable that the film is mosaic. It is true that Kirchner and Lassen$^{119}$ succeeded in obtaining Kikuchi lines upon reflection from silver films prepared in this way; this fact rather indicates the accuracy of the alignment of the mosaic crystals.

A metal condensed from vapor in very thin layers can not only adapt itself to the size and orientation of the crystals of the substrate, but can also form pseudomorphic crystals. For example, aluminum condensed on platinum crystallizes in the tetragonal system with dimensions characteristic of platinum, but the $c$ axis remains the same as in the case of ordinary aluminum$^{141}$. An analogous phenomenon was also observed in the case of nickel electrodeposited on a single crystal of copper$^{120}$.

Metal films obtained by cathodic sputtering, like films condensed from vapors, also always prove to be crystalline. If the substrate is specially cooled, then the crystals, generally speaking, are arranged in a disordered manner. In the case of platinum it turns out that its films obtained under such cooling possess high catalytic activity, whereas films obtained at high tem-

temperature of the substrate, are inert. Films obtained by cathodic sputtering from gold and platinum wires twisted together consist of a mixture of crystals of both these metals, and no signs of alloys are detected^142. This indicates that, in such film formation, the atoms of a given metal tend to be more strongly attracted by the nuclei of the same metal. The gas in the discharge space exerts a profound influence on the nature of the films formed during cathodic sputtering. Finch and co-workers^103,142 found that platinum, palladium, gold, silver, copper, and nickel, sputtered in purified argon, all exhibit a normal structure, with or without orientation, depending on the temperature of the receiving surface; moreover, in no case was it possible to observe any signs of compound formation. However, when a platinum film was subsequently heated in the diffraction camera by radiation from an incandescent platinum filament, the diffraction pattern at first became pale, but then, if the temperature of the filament was sufficiently high, it again became visible. The resulting film was sometimes so thick that it showed interference colors, even if the original film had been very thin. Therefore the secondary appearance of the diffraction pattern may be attributed to platinum condensed as a result of evaporation of the filament. It appears that the deterioration of the diffraction pattern occurred because crystalline peaks were deposited on the surface, probably as a result of the release of absorbed gas. This is confirmed by the fact that such deposited films, after being removed from the substrate with the aid of weak hydrochloric acid, gave sharp crystalline patterns when photographed in transmission.

Trillat and Hirsch^144 showed that the orientation of crystals in a gold leaf increases strongly upon heating in vacuum between 350 and 500°; a hexagonal modification appears, which disappears again at 600°. On the other hand, Preston and Bircumshaw^145 obtained no increase in the size of crystals upon heating gold in vacuum or in inert gases, and Finch and Fordham^142 confirmed this for palladium and platinum. But gold films containing traces of mercury always gave the pattern characteristic of gold amalgam when heated above 550°. Eilmer, Finch, and Fordham^146 found that films of gold, silver, copper, and platinum that had been subjected to the action of mercury have structures that vary greatly with the degree of amalgamation. Heat treatment invariably leads to one and the same crystalline amalgam. Similar patterns from amalgams were obtained by Preston and Bircumshaw^145.

Thomson^147 showed that silver electrodeposited on an etched single crystal of copper adapts itself strongly in its orientation to this crystal. Further, Finch and Sun^148 found that the composition of the bath, as well as the temperature and current density, are the principal factors determining the nature of the orientation in the case of metals electrodeposited on an inert substrate, such as a polished metal surface, but that a crystalline sub-

the substrate in many cases exerts a profound influence both on the size of the crystals and on their orientation in the initial stages of deposition, although subsequently this effect weakens as the thickness of the deposit increases, until finally the orientation and size of the crystals become typical for the given bath composition and electrolyte concentration. For good adhesion, the ability of the deposited crystals to adapt to the size and orientation of the substrate crystals, at least in the initial stages of deposition, is a sine qua non. Finch and Sun also found that iron electrodeposited on gold exhibits exceptionally good adhesion, and they ascribe this to the features of the orientation of the body-centered structure possessed by iron on the face-centered cubic metal, gold. They found that although the substrate crystals and the deposit crystals generally continue to possess one common orientation, showing contact between one set of cube faces, the other cube faces, normal to the surface of the specimen, are not parallel, but are inclined to one another by 45°, so that the planes \((110)\) and \((100)\) of the substrate and deposit, respectively, are parallel to one another. This influence of the substrate on the adaptation of the deposit to it is apparently connected with the tendency toward the greatest possible stress-free arrangement at the substrate–deposit boundary. Judging from the transmission diffraction pattern of such a complex film, where the iron ring \(123\) is obtained free from the gold ring, one must conclude that the cube faces of the iron crystals closely coincide with the faces of the gold crystals when the atoms at the vertices of the iron cubes bisect the edges of the gold cubes. Evidently, when one common direction of orientation, namely \((100)\), has already been given, no other arrangement can allow the two structures to adapt to one another more closely without distortion. It may therefore be considered that in this case the arrangement of the substrate and the deposit is the most free from stresses.

Finch and Sun \(^{148}\) were unable to observe any signs of alloy formation during electrodeposition, even in the first few atomic layers. On the other hand, they found that chemical displacement in solution of one metal by another always led to the formation of alloys. Thus, for example, platinum displaced by copper or silver gives alloy films whose structure, although it remains face-centered cubic, has lattice dimensions intermediate between those of the two pure metals.

In their recent microphotographic and electron-diffraction work, Finch and Williams \(^{149}\) pointed out the unreliability of microscopic data concerning the structure of such deposits. Thus, for example, according to microscopic data it appears that nickel deposited on large copper crystals likewise consists of large crystals growing in columns on the copper substrate, whereas the electron-diffraction method showed that in reality nickel is formed in the form of small crystals enclosed in a mesh

from the boundaries of the pseudomonocrystal, their orientation depending only on the conditions of deposition.

IX. Structure of a Polished Surface

One of the fields in which electronography is applied with great success is the study of the nature and of the structural changes that occur during the mechanical treatment of surfaces. On the basis of microphotographic data, Beilby[^150] came to the conclusion that a polished surface is produced as a result of a true flow of the substance, owing to which the surface, in the course of time, becomes covered with an amorphous layer, and not as a result of the breaking off and smoothing of surface projections. The first electronographic study of a polished surface was undertaken by Thomson[^151]. However, it was only French[^152] who succeeded in showing that, when an electron beam is reflected from a polished surface, a pattern is obtained consisting of blurred rings and analogous to the pattern from liquid metals, a fact which had to be regarded as confirmation of the assumption that this surface is amorphous. Objections to such an explanation were raised by Kirchner[^153], who found that thin films of metals, prepared in such a way that the height of the crystalline peaks on their surface was greatly reduced, give diffuse patterns indistinguishable from the patterns of a polished layer, although examination of such films in transmission shows that they are in fact crystalline. Kirchner therefore believes that the diffuse patterns obtained on polishing are due to the low resolving power of the surface crystals, in which the projecting peaks on the surface have been erased by the polishing process, and only a small depth of the layer is accessible to the electron beam. The difficulty of resolving this question lies in the fact that, because of the smoothness of the surface and the resulting blurring of the rings, it is impossible to establish the influence of the inner potential and the refraction of the beam; similarly uncertain results were obtained in experiments on measuring the thickness of the polished layer by means of its gradual etching[^154].

At present the following data may be regarded as firmly established: a) amorphous surfaces of substances such as glass, silica, and mercury, b) polished metallic surfaces, and c) certain crystalline metallic surfaces—all give the aforementioned diffuse patterns. The last fact contradicts Thomson’s explanation, while the first fact contradicts Kirchner’s explanation. It is therefore necessary to consider that the nature of the polished surface of metals still remains unsolved.

It is impossible, however, not to mention here those new facts which shed light on this question. Finch and co-workers[^155]

showed that the polished surface of metals exhibits a property characteristic of a liquid: to dissolve to saturation the crystals of another metal at room temperature, which is not observed in the case of crystalline surfaces. This fact must be regarded as new evidence that the polished surface of metals is amorphous, i.e., glass-like. Another proof that the material becomes amorphous during polishing was given by Bowden and Ridler ^156, who showed that the mean temperature of the polished layer during its formation rapidly rises to the melting point of the metal and does not depend on a further increase of pressure during the polishing action. If this result is considered in conjunction with the surface flow discovered by Beilby, especially with the reappearance after etching of those scratches that disappeared during polishing, then it indicates that the flow taking place during polishing must be associated rather with annealing of the material, i.e., with its transition into the amorphous state, than with the transfer of crystalline fragments.

Finch, Quarrell, and Wilman ^79 showed that the hand-polished surface of steel, after light rubbing over it with fine emery paper, gives a pattern of $\alpha$-iron with sharp rings, and that, in order to obtain the same effect in the case of the surface of the same steel in the cylinder of an internal-combustion engine, strongly polished as a result of prolonged operation of the machine, prolonged grinding is required. This fact indicates that the thickness of the Beilby layer increases with increasing force and duration of polishing, and is in contradiction with Krämer’s point of view, according to which, in order to obtain a sharp pattern from crystals, light grinding would have sufficed in both cases.

In the light of these facts and arguments one may conclude that the metallic Beilby layer is amorphous and is formed by the surface flow of an amorphous, viscous, or liquid material produced under the action of polishing. In addition, the presence on electron diffraction patterns of diffuse rings from the surfaces of mechanically worked, i.e., polished in various ways and forged, metals ^157 must also be regarded as evidence of the amorphous nature of this layer.

It is now well known that natural faces and cleavage planes of single crystals give, on reflection, characteristic patterns with Kikuchi lines, although such surfaces are, generally speaking, comparatively rough, only rarely being even microscopically smooth. When Germer ^158 found that exceptionally smooth and well-developed basal planes of silicon carbide single crystals give only diffuse rings, he attributed this pattern to the extreme smoothness of the carbide surface and further expressed the view that an adsorbed gas layer, too, possibly plays a known role on such a surface. However, Finch and Wilman ^159 succeeded in giving an entirely different explanation of this result. They,

just as Germer did, invariably obtained the indicated diffraction pattern from an undamaged silicon carbide crystal; but it turned out further that, after slight grinding of such a surface with a fine powder (wet crocus or alumina), softer than carbide but harder than silica, and also after etching (with hydrochloric acid or a solution of caustic soda), capable of acting on silica but not on the carbide, the crystals, whose surface had not changed at all in outward appearance, gave a remarkably clear and bright pattern with Kikuchi lines, characteristic of reflection from the surface of single crystals (Fig. 13). Further, gradual oxidation of the surface of a crystal of such carbide led to the formation of a complex pattern containing a diffuse ring and the pattern of a single crystal, in which the Kikuchi lines became weaker because of the presence of a background from general scattering, which is always present in a pattern with a diffuse ring. With further oxidation only the diffuse rings remained. These results show that the latter pattern, characteristic also of untreated silicon carbide, arises from some amorphous film, probably a film of silica, formed during oxidation of the crystals while they cool after growth.

It should be recalled that Beilby, in many of his experiments, observed phenomena of surface flow also in nonmetals, such as, for example, calcite, and extended his conclusion about the amorphous nature of the polished layer to nonmetals as well. And indeed, the phenomenon of surface flow in the case of calcite can easily be demonstrated experimentally. In this connection the fact observed by Räter[^157] and Hopkins[^160] is quite surprising: the polished layer on the cleavage plane of calcite proves, in structure, to be completely similar to the single crystal lying beneath it. Since the fact that flow exists on calcite indicates that its polishing is not simply a process of abrasion, the question now remains open whether this crystalline surface structure is produced by recrystallization of the amorphous material formed on the surface, with the same orientation as in the crystal itself, or whether this correct arrangement arises through displacement on the surface and the falling into correct positions of crystalline fragments or layers. However, Finch[^161] finds that although the polished layer is crystalline on the cleavage plane of calcite, it gradually becomes less and less crystalline as the polished surface is inclined more and more to the cleavage plane, until finally, on a surface not coinciding with any cleavage planes, this polished layer becomes completely amorphous. Thus these results show that surface flow is characteristic of amorphous material, which, after polishing ceases, recrystallizes if the polished surface coincides with the cleavage plane, but tends to preserve its amorphous state on surfaces steeply inclined to any such plane.

Further experiments showed that the ability of the Beilby layer to recrystallize depends on the forces exerted on it by the surface of the crystal. Thus, for example, the indicated amorphous layer on calcite recrystallizes upon heating; moreover, from the change in the diffraction patterns one must conclude that this recrystallization proceeds from the inner parts of the Beilby layer toward the outer ones.

Recently Finch and Wilman^162 investigated a large number of polished nonmetallic surfaces, and the results obtained proved possible to classify according to the feature of spontaneous recrystallization of the polished layer after polishing has ceased, or the preservation of the amorphous state. In particular, it was found that whereas polished surfaces cut in entirely arbitrary directions on many types of single crystals [quartz, diamond (Fig. 16), natural and synthetic sapphires, almandine, topaz, chrysoberyl, epidote, olivine, andalusite, etc.] give electron diffraction patterns with spots and lines and irrefutably testify to the crystalline structure of the polished layer on such surfaces, other crystals, such as white beryl, zircon, tourmaline, and cassiterite, give patterns with diffuse rings. From other polished crystals (brown beryl, moonstone, orthoclase, and cordierite) complex patterns were obtained, in which diffraction from the single crystal appeared weak in comparison with diffraction from the amorphous layer. Spinels, both synthetic and natural, depending on the degree of polishing, gave either only diffuse rings, or, together with them, also spots weakly distinguishable against the general background. In some cases, as, for example, with calcite, the structure of the polished layer is determined by the nature of the face under consideration. Thus, for example, one face of blue kyanite gave a pattern with diffuse rings, through which monocrystalline spots could be only weakly distinguished, whereas other faces of the same specimen gave a clear single-crystal pattern. No surface flow could be observed in the case of diamond, although in all other cases it undoubtedly occurred. Therefore diamond must be assigned to a separate class, where polishing consists in the simple leveling of peaks protruding on the surface, which are simply abraded from the surface. In all other cases, evidently, the formation of the polished layer occurs in accordance with Beilby’s views and is the result of surface flow. The fact that this layer, however, often proves to be crystalline can quite possibly be attributed to the easy and spontaneous rearrangement of the molecules brought into a disordered state during the polishing process, and to the more or less strict orientation of the forming crystals in accordance with the structure of the substrate, whereas the amorphous nature of the polished surface in other cases is evidence of the difficulty for the liquid layer to crystallize; this difficulty in some cases is evidently determined by the nature of the material, and in others by the crystallographic directions of the polished face.

*

A very important result of these experiments is that, for the study of electron diffraction by single crystals, it is no longer necessary to be confined to the often imperfect natural faces or cleavage planes. For this purpose, by simple grinding and polishing, and, in the case of the formation of an amorphous layer, also by careful etching, a suitable face of any type can be made. Good patterns of single crystals are also obtained from the curved convex surface of a short-focus quartz lens and from screw cleavage of quartz, spinel, corundum, and other crystals^163.

Finch^163 has recently been concerned with the application of these results and with the question of wear of the internal parts of an internal-combustion engine.

X. Conclusion

For lack of space it is impossible to set forth in greater detail the question of the application of fast electrons to the study of surface structures. It can only be indicated that in the 10 years that have elapsed since Thomson’s first experiment in 1927, this new experimental technique has been successfully applied in the most varied fields of surface phenomena, for example, to the study of thermionic and photoelectric emission, the electrical conductivity of thin films, surface catalysis, electrodeposition, crystal growth, the colloidal state, corrosion, lubrication, etc. Its successes are due to the fact that it was the first to make possible the direct examination of the structure and properties of surfaces, previously inaccessible for this purpose.

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1. A somewhat abridged translation of the article by Finch and Wilman, “The Study of Surface Structure by Electron Diffraction,” from Ergebn. d. Exakt. Naturwis., 16, 353, 1937. Translated by N. A. Shinakova. ↩

2. 125. ↩