DIFFRACTION OF MOLECULAR BEAMS FROM CRYSTALS

M. N. Flerova, Leningrad

INTRODUCTION

According to wave mechanics, an atom (or molecule) of mass $m$, moving with velocity $v$, corresponds to a probability wave $\psi$, whose direction of propagation coincides with the direction of motion of the atom. The wavelength is related to the velocity of the atom $v$ by the de Broglie relation:

\[ \lambda = \frac{h}{mv}. \]

The square of the modulus of the wave function $(\psi)^2$ is proportional to the probability of finding the atom at a given point in space. For a free atom moving by inertia, the function $\psi$ is a plane wave:

\[ \psi = Ae^{\frac{2\pi i}{h}(Et - px)}, \]

where

\[ E = \frac{mv^2}{2}, \quad p = mv, \]

$x$ is the coordinate of the moving atom.

The collision process, in which an atom enters the sphere of action of the force field of a crystal lattice or of another atom, is treated by wave mechanics as the scattering of a “probability wave” by a medium with a variable refractive index.

In a rough way, this process should be imagined as follows. A plane wave, incident upon the force field, is scattered by its individual elements. The scattered waves, differing in phase and amplitude, interfere with one another and as a result give the distribution of the probability of scattering of the atom in various directions. Such a rough treatment, in many cases where an exact mathematical analysis cannot be carried out, proves sufficient for clarifying the main features of the phenomenon.

Let us analyze from this point of view the process of scattering of an atomic or molecular beam from the surface of a crystal. The wavelengths

de Broglie wavelengths for atoms or molecules of light substances (H, H₂, He), moving with the mean thermal velocity at room temperature (\(T \simeq 300^\circ \mathrm{K}\)), are of the same order of magnitude as the distances between the lattice points of most inorganic crystals. Owing to this, the surface of a crystal constitutes a natural diffraction grating for atomic waves (we assume that the diffracted atoms do not penetrate into the interior of the crystal and that, therefore, only the surface layer of the crystal takes part in the scattering).

An atom approaching a crystal moves in a spatially periodic field created by atoms situated on the surface of the crystal. The elementary probability waves scattered by individual regions of this periodic field behave in complete analogy with ordinary light waves scattered by the nodes of a plane diffraction grating. Interfering, they produce maxima and minima of intensity in those directions which are required by simple geometrical conditions determining the phase difference of the scattered and incident waves.

All these conclusions of quantum mechanics have been brilliantly confirmed by experimental studies of the diffraction of atomic and molecular beams from single crystals. The chief significance of these experiments lies in the fact that they point to a new path of investigation in that complicated field which is at present represented by the physics of phenomena occurring at the surface of a solid. Diffraction of atoms gives us a direct method for analyzing the force fields near the surface of a crystal and the geometrical structure of the surface layers. In addition, this method makes it possible to follow the various changes undergone by the surface lattice of a crystal in such processes as, for example, the formation of a monomolecular film of gas.

Thanks to the intensive study of the phenomena of reflection and diffraction of atoms in recent years, it has been possible to establish certain very important facts concerning the energy properties of particles adsorbed on the surface of a crystal. The regularities in the anomalies of the scattering coefficient, studied in the recent works of Stern and Frisch, indicate the existence of a connection between the energy levels of atoms that are in metastable binding with the surface of a crystal and the diffraction conditions.

Of great interest for modern physics is also the investigation of elementary acts of coherent scattering of material particles, i.e., the processes of elastic collision of free particles (atoms and molecules) with one another. When these processes have been well studied, we shall be able better to learn the character and origin of the forces acting between neutral particles when they are at a very small distance (of the order of \(10^{-8}\) cm).

Experimental investigations of the diffraction of atoms and molecules require a very refined and complex technique. The principal problem is the production of intense atomic and molecular beams,

containing particles with a definite direction of motion and desired mean velocity, and the measurement of the intensity of these beams. Such beams are usually called “molecular beams.”

1. Production of molecular beams and measurement of their intensity

The method of producing molecular beams reduces in principle to the following.

Let us take a vessel \(A\) (Fig. 1) and divide it by partitions into three separate chambers. We make a small aperture in each of the partitions, then fill the first chamber with the gas or vapor whose molecular beams we wish to obtain, while the second and third chambers are pumped out separately by powerful vacuum pumps. If the pressure in the first chamber is not too high and the aperture \(S_1\) is sufficiently small (the diameter of the aperture must be less than the mean free path of the gas molecules in the first chamber), then the outflow of gas through this aperture will take place in the form of a molecular stream. The gas molecules that have emerged from the aperture \(S_1\), on their subsequent path to the walls of the second chamber, will practically undergo no collisions, since the mean free path in the second chamber is very large. Therefore they will move rectilinearly in directions diverging from the aperture.

Fig. 1.

The molecular stream emitted by the aperture \(S_1\) is entirely analogous to the stream of light rays emitted by a small source, which we may mentally place at the position of the aperture. From the purely geometrical point of view the analogy will be complete, which also permits us to call the stream of molecules emerging from the aperture \(S_1\) “molecular beams.”

The aperture in the second partition selects from the broad cone of diverging molecular beams a narrow cone \(LL\), in which the coordinated motion of molecules in a definite direction is realized. In the third chamber there will exist only this isolated beam of molecules, or molecular ray.

If now, in the path of the beam in the third chamber, a plate is placed on which condensation of the beam molecules can occur with the formation of a visible deposit, then after some time a sharply outlined spot will form on the plate, the dimensions of which will be determined by the geometry of the “optical system,” i.e., by the sizes of the apertures and the distances from the plate to the source of molecular beams (the aperture \(S_1\)) and to the second partition.

The method described was first carried out in practice by Dunoyer (1911), who used it as an elegant demonstration of the correctness of the basic ideas of the kinetic theory of gases.

Subsequently the technique for obtaining molecular beams developed rapidly and, in the hands of Stern and his students, reached a high degree of perfection.

A molecular beam is characterized by the number of molecules emerging per unit time from the aperture of the first chamber, and by the distribution of the molecules with respect to velocities and directions.

The total number of molecules emerging per unit time from the aperture of the first chamber, according to the kinetic theory of gases, is determined by the formula:

\[ N=\frac{n\Delta s\bar{v}}{4}, \tag{1} \]

where \(n\) is the total number of molecules per unit volume, \(\Delta s\) is the area of the aperture, and \(\bar{v}\) is the mean velocity of the molecules at the given temperature. Expressing \(n\) and \(\bar{v}\) in terms of the pressure and temperature of the gas in the first chamber, we obtain for the molecular flux in moles per second:

\[ N_1=\frac{5.83\cdot 10^{-2}}{\sqrt{MT}}\,p\Delta s, \tag{2} \]

where \(M\) is the molecular weight, \(T\) the absolute temperature, and \(p\) the pressure in millimeters of mercury.

The law of distribution of the beam molecules with respect to velocities will be:

\[ n(v)\,dv=Be^{-\frac{v^2}{\alpha^2}}v^3\,dv, \tag{3} \]

where \(\alpha\) is the most probable velocity.

Knowing the distribution of molecules with respect to velocities, one can determine the spectral composition of the molecular beam, i.e. the distribution with respect to the lengths of the phase waves. Let \(\varphi(\lambda)\) be the distribution function with respect to wavelengths \(\lambda\). Then

\[ \varphi(\lambda)=B\cdot e^{-\frac{v^2}{\alpha^2}}v^3\frac{dv}{d\lambda}. \tag{4} \]

From de Broglie’s formula we have:

\[ v=\frac{h}{m\lambda},\qquad \frac{dv}{d\lambda}=-\frac{h}{m}\cdot\frac{1}{\lambda^2}. \]

Substituting these expressions into formula (4), we find:

\[ \varphi(\lambda)=C\lambda^{-5}e^{-\frac{\beta^2}{\lambda^2}}, \tag{5} \]

where \(C=B\left(\frac{h}{m}\right)^4\) and \(\beta=\frac{h}{m\alpha}\) is the wavelength corresponding to molecules of the most probable velocity \(\alpha\).

The wavelength \(\lambda_{\max}\), to which the maximum intensity in the spectrum corresponds, is obtained from the condition \(\dfrac{d\varphi(\lambda)}{d\lambda}=0\).

This condition gives:

\[ \lambda_{\max}=\sqrt{\frac{2}{5}\,\beta}=\frac{19.5\cdot 10^{-8}}{\sqrt{MT}}\ \text{cm}. \tag{6} \]

The distribution curve by wavelengths is shown in Fig. 2.

The distribution of the emitted molecules by direction obeys Knudsen’s cosine law.

The intensity of the beam traveling in the direction of the normal to the aperture at a distance \(r\) from it will be:

\[ I=\frac{1.86\cdot 10^{-2}}{r^{2}\sqrt{MT}}\,p\Delta s\ \frac{\text{moles}}{\text{sec}\cdot\text{cm}^{2}} . \tag{7} \]

Formulas (1)—(7) are valid, however, only under certain conditions, one of which we have already mentioned, namely: the mean free path of the gas molecules in the first chamber must be greater than the dimensions of the aperture. When the aperture is given the form of a narrow slit, this condition reduces to the requirement that the mean free path be greater than the width of the slit. The second condition is that the thickness of the walls must be much smaller than the dimensions of the aperture. In practice, however, this condition of “ideality” of the aperture is not always fulfilled, and in many cases the wall thickness cannot be neglected; it is then necessary to take into account the so-called \(\chi\)-factor, i.e., the ratio of the intensity of the beam emerging from an ideal aperture to the intensity of the beam passing, under equal conditions, through a real aperture with the same cross section. Obviously, the \(\chi\)-factor is always greater than unity.

Fig. 2. Distribution of beam molecules by wavelengths.

Depending on the kind of gas and the experimental conditions, the first chamber (in the German literature the name “oven” has become attached to it, although this name does not in all cases correspond to the actual design) may be given various forms.

In the case when the source of molecular beams is readily condensing vapors of metals (K, Na, Hg, etc.) or of organic substances, the evaporated substance is placed inside the oven, which thus serves as a reservoir of evaporating molecules. Fig. 3 shows an oven for obtaining atomic

of mercury beams (Zahl and Ellet¹). Since the upper and lower sections of the furnace, separated by a quartz gasket, can be heated separately, the intensity of the atomic beam and the velocities of the emerging atoms can be varied independently of one another. The constriction \(C\) serves to reduce the diffusion of atoms between the lower and upper sections of the furnace.

When working with gases that are difficult to condense (\(\mathrm{H_2}\), He, Ne, Ar), the furnace is given an entirely different form. Thus, in all the work on the study of reflection and diffraction of molecular beams carried out by Stern and his collaborators, the furnace had the form of a narrow long tube (Fig. 4), connected with a reservoir containing the gas under investigation at the proper pressure.

Fig. 3. Furnace for producing atomic beams of mercury.

Fig. 4. Furnace for producing molecular beams of He and \(\mathrm{H_2}\).

The end of the tube could be heated by means of the spiral shown in the figure. The tube could also be cooled by coming into contact with the vessel \(R\), into which liquid air was poured. In this way it was possible to vary the temperature of the molecular flow, i.e., the mean velocity of the molecules of the beam.

To measure the intensity of a molecular beam selected by apertures in the two partitions, receivers of various types may be used. In the overwhelming majority of cases only relative values of the intensity of the molecular beam, expressed in arbitrary units, are important. Owing to this, the problem of constructing receivers—measuring instruments for molecular beams—is considerably simplified.

In those cases where it is required to obtain merely simple qualitative proof of the existence of a molecular beam, one may use

be used as condensation-type receivers, whose principle of operation is as follows. The molecular beam is directed onto a glass or metal plate whose temperature is sufficiently low that each molecule striking the plate is adsorbed on its surface. The adsorbed molecules, accumulating, form a deposit, which may be visible directly, or may become visible after “development,” i.e., special chemical treatment with suitable reagents.

The use of condensation-type receivers for experiments on the diffraction of atoms is very advantageous, since with the aid of these receivers one can obtain at once the entire diffraction pattern, with maxima and minima of intensity, i.e., the entire geometry of the phenomenon.

Condensation receivers have very high sensitivity; nevertheless, they are unsuitable for precise quantitative measurements, since the process of deposit formation is very complex and little studied, and they can give only the roughest idea of the numerical ratios of the intensities of the individual molecular beams.

There are also receivers based on the chemical action of molecular beams. The molecules of the beam falling on the plate enter into a chemical reaction with the material of the plate. The reaction product must give a good color contrast with the surface of the plate not subjected to the action of the molecular beams. Thus, for example, atomic hydrogen falling on a plate of \(\mathrm{MoO_3}\) gives blue \(\mathrm{MoO_2}\) on the pale-yellow background of \(\mathrm{MoO_3}\).

For quantitative measurements of the intensity of molecular beams, in most cases receivers are used whose action is based on the accumulation of the beam molecules in a closed space and on the measurement of the pressure produced by them. The molecular beam enters a slit, which is the only opening in a vessel closed on all sides, and creates in this vessel a certain pressure, which increases until equilibrium is established, i.e., until just as many gas molecules enter the vessel as leave it in an equal interval of time.

The quantity of gas flowing out per 1 sec. from the slit is the smaller, and at the same time the final pressure is the greater, the greater the resistance of the slit to outflow. The equilibrium pressure, and consequently the sensitivity of the method, can be increased many times by giving the receiving slit the form of a channel. This does not affect the incoming beam, since all the molecules of the beam have a definite direction (along the axis of the channel), but it increases the resistance to the exit of uncoordinatedly moving molecules from the vessel. The magnitude of the equilibrium pressure is directly proportional to the intensity of the molecular beam, i.e., to the number of moles falling per second on \(1\ \mathrm{cm^2}\), directly proportional to the \(\chi\)-factor of the receiver slit (the ratio of the resistance to outflow of the real and ideal slit—see

…above) and does not depend on the surface of the slit (if the transverse dimensions of the beam are greater than the surface of the slit) or on the volume of the receiver.

Since there are fluctuations in the pumping speed (up to 10%), we accordingly have pressure fluctuations in the receiver. The influence of these fluctuations can be eliminated by using a compensating instrument for measuring the pressure, which should be as nearly identical as possible to the receiver. This compensating instrument is connected with the observation chamber (the third chamber, see Fig. 1). The intensity of the molecular beam is taken to be the difference between the readings of the receiver and of the compensating instrument. In this way the fluctuations are automatically eliminated.

Since in receivers of the type under consideration what is ultimately measured is the pressure produced by the molecular beam in a certain closed volume, they may be called “manometers.” The difference between them lies only in the methods of measuring the pressure.

In experiments on the diffraction of molecular beams of light gases, the intensity is usually measured with the aid of a Pirani manometer.[^2]

Sometimes, for measuring the intensity of molecular beams, an ionization manometer is used, which is especially convenient for investigations with heavy gases.[^3]

The receivers listed above have the disadvantage that each of them can be used only for a narrow group of substances, except, perhaps, the ionization manometer. Only recently have Stern and Estermann[^4] succeeded in developing a design of an ionization manometer that may be called universal. This manometer responds with sufficient sensitivity to light gases, to metal vapors, and to vapors of organic compounds. In it, for measuring the pressure, use is made of the phenomenon of compensation of the negative space charge, formed near the cathode during thermoelectronic emission, by positive ions of the gas.*

A major advantage of the Stern and Estermann manometer is the rapid establishment of equilibrium pressure. In practice it is established within a few hundredths of a second. Thanks to this, the sensitivity can be increased still further by applying the sharp-resonance method of Tykocinski-Tykociner. This method consists in the fact that, by means of a sector rotating at a definite frequency, the molecular beam is alternately opened and closed, and the resulting periodic pulsations of the current in the manometer are selectively amplified by means of a vacuum-tube amplifier.

The sensitivity of the manometer with respect to measuring the pressure produced by molecular beams of mercury is of the order of \(10^{-11}\) mm Hg. Under optimal conditions, the change of pressure in the manometer by

* This phenomenon was first investigated by Kingdon, who showed that it can be used for very accurate measurement of low pressures.

$10^{-6}$ mm Hg corresponds to a change in the emission current of several milliamperes. This sensitivity, as Stern and Estermann indicate, can be increased still further.

When working with atomic beams of alkali metals, a detector is often used whose action is based on the phenomenon of surface ionization. Langmuir and Kingdon showed that each atom of an alkali metal (K, Rb, Cs), falling on the surface of an incandescent tungsten wire ($T > 1600^\circ\mathrm{K}$), loses an electron and evaporates in the form of a positive ion. This occurs because the ionization potential of these atoms is less than the work function of electrons from tungsten. If the wire is surrounded by a negatively charged cylinder, then the positive ionic current to the cylinder will directly give the number of atoms falling per unit time on the surface of the wire. By making a slit in the surface of the cylinder and placing the entire system in the path of the beam, one can, knowing the diameter of the wire, measure the absolute intensity of the beam, i.e., the number of atoms passing per unit time through $1\ \mathrm{cm}^2$.

This method was developed by Taylor[^5] in Stern’s laboratory. It is distinguished by extremely high sensitivity. If the wire has a diameter of $0.05$ mm and the length of the slit in the cylinder is $4$ mm, then, at a molecular-beam intensity equal to $2\cdot 10^{14}\ \frac{\text{atoms}}{\text{sec}/\mathrm{cm}^2}$, $4\cdot 10^{11}$ atoms fall on the filament per second. If one assumes that each atom then evaporates as a positive ion, the corresponding positive ionic current will be equal to $6\cdot 10^{-8}$ A.

The surface of the tungsten wire must, as Taylor indicates, be very clean. Vapors of grease present in the apparatus must be frozen out with liquid air.

2. Experimental studies of the diffraction of molecular beams from crystals

To obtain a sharp diffraction pattern it is necessary first of all that the wavelength associated with the incident atom or molecule be of the same order of magnitude as the parameter of the surface lattice of the crystal. At very small wavelengths the entire diffraction pattern contracts around the direction of the specularly reflected beam, and it is practically impossible to separate the individual diffraction maxima. In addition, at small wavelengths the influence of irregularities present on the surface of the crystal, and the associated decrease in the intensity of the diffraction maxima, must be manifested more strongly. The second, no less important, condition for obtaining a diffraction pattern requires that the molecules or atoms of the gas under investigation not be able to condense on the surface of the crystal.

Both of these conditions are well satisfied for molecular beams of light gases—He, H$_2$, and H. For them, as follows

from Stern and his co-workers, coherent scattering does in fact exist. The de Broglie wavelengths for helium and hydrogen atoms at room temperature are of the same order as the lattice parameters of commonly used crystals (for NaCl the lattice parameter is \(3.03\ \text{Å}\)). Indeed, from formula (6) it follows for the most probable wavelength at \(T = 300^\circ\mathrm{K}\):

\[ \lambda_{\mathrm{He}}=\frac{19.5\cdot 10^{-8}}{\sqrt{4\cdot 300}}\simeq 0.57\cdot 10^{-8}\ \text{cm}, \]

\[ \lambda_{\mathrm{H}_2}=\frac{19.5\cdot 10^{-8}}{\sqrt{2\cdot 300}}\simeq 0.82\cdot 10^{-8}\ \text{cm}. \]

Owing to these properties of helium and hydrogen, the principal and most complete investigations of the diffraction of molecular beams have been carried out precisely with these gases.

The distinctness of the diffraction pattern depends not only on the nature of the molecular beams, but also on the properties of the crystalline surface. For reproducibility of the results, careful cleaning of the crystal surface from adsorbed gases is necessary. Stern and his co-workers showed that the crystals most suitable for studying the diffraction of molecular beams are LiF and NaCl, especially the former. Their numerous investigations are based almost entirely on the use, as the scatterer, of a natural cleavage plane of LiF, degassed by prolonged heating.

Fig. 5. \(K\)—crystal, \(R\)—receiver, \(O\)—capillary supplying the gas.

The experimental arrangement is in principle the same for all work in this field (Fig. 5). The differences, determined by the specific conditions of the experiment, are reduced chiefly to the choice of special types of manometers and the construction of mechanical devices for rotating and moving the receiver and the crystal in vacuum.

The interpretation of the principal results of the experimental investigation of the diffraction of atomic and molecular beams must be based on the guiding principles of quantum mechanics. However, one cannot go very far in this direction, since the complexity of the process of interaction of a free atom (or molecule) with the surface of a crystal does not permit an exact mathematical calculation of the scattering probability for any particular case. For a complete solution of the problem it is necessary to know the potential energy of the atom in the field of the surface lattice of the crystal. At present we know only that the potential energy of an atom in the field of a crystal possesses periodicity of the same type as the periodicity of the surface lattice of the crystal, and with the same

periods, and that it practically goes to zero already at a very small distance from the surface of the crystal. Since the slow atoms or molecules of the beam, possessing thermal velocities, cannot penetrate into the interior of the crystal, move freely within it, and then emerge without losing their velocity, volume interference for molecular rays must be regarded as excluded.

Although this information concerning the form of the potential function is quite insufficient for one to be able, using it, to calculate the absolute magnitudes of the probability of scattering of an atom in different directions, it nevertheless makes it possible to draw certain conclusions concerning the geometry of the phenomenon, namely, to establish the conditions determining the positions of the diffraction maxima.

Considering the surface of a crystal as a system of scattering centers located at the nodes of a lattice, one may, by analogy with optics, obtain the diffraction conditions by a purely geometrical method, determining the phase differences of the individual rays. In the case when the surface lattice of the crystal is a rectangular net with parameters \(d_1\) and \(d_2\), the diffraction conditions will have the form:

\[ d_1(\cos\alpha-\cos\alpha_0)=m\lambda, \]

\[ d_2(\cos\beta-\cos\beta_0)=n\lambda, \]

where \(\alpha_0\), \(\beta_0\), \(\alpha\), and \(\beta\) are the angles made by the directions of the incident and scattered atoms with the \(x\) and \(y\) axes, which are situated along the principal axes of the lattice; \(\lambda\) is the wavelength associated with the atom; and \(m\) and \(n\) are any integers.

These equations are completely identical with the equations determining the positions of diffraction maxima for the reflection of light rays from a plane grating (see, for example, Sommerfeld, Atomic Structure and Spectra, p. 136).

The first experiments on the reflection of atoms from a crystal were carried out by Ellett and Olson,\(^6\) who investigated the reflection of mercury, cadmium, sodium, and hydrogen from the cleavage surface of NaCl. The atomic beam fell upon an NaCl plate, was scattered by it, and struck a surface cooled by liquid air, forming a deposit. The angular distribution of the scattered atoms could be judged from the spatial distribution of the density of the deposit. Ellett and Olson found that mercury and cadmium undergo specular reflection from a clean NaCl surface. The reflected atoms produced on the surface cooled by liquid air a deposit with sharply outlined edges at the angle of specular reflection, while the remaining part of the surface remained clean. The angular divergence of the reflected beam was of the same order as the angular divergence of the primary beam. Periodic variations in the density of the deposit, which would have indicated the existence of diffraction, were not observed, probably owing to the low resolving power of the crystal and also, possibly, owing to insufficiently clean experimental conditions.

Diffraction of Molecular Beams from Crystals

If the crystal had not previously been degassed by heating in vacuum, then specular reflection did not occur. The deposit was obtained over the entire hemisphere of the receiving surface facing the crystal, and was denser in the direction normal to the crystal, as should be expected for a process of adsorption followed by evaporation. Sometimes the maximum density of the deposit was displaced from the direction of the normal toward the direction of specular reflection, probably because some of the atoms nevertheless underwent regular reflection.

Attempts to detect specular reflection of sodium and atomic hydrogen from NaCl were unsuccessful.

The experiments of Ellett and Olson are the first step toward experimental proof of the wave nature of molecular beams and toward the investigation of the reflection and diffraction phenomena of atoms and molecules conditioned by it. While showing that specular reflection of atoms from the surface of a crystal exists, these experiments do not yet, however, give indications of the existence of diffraction phenomena for atomic beams. These phenomena were first discovered and studied in detail in the works of Stern and his collaborators, along with the reflection of molecular beams from crystals. We now turn to an account of these works, begun by Stern in 1927, even before the publication of the first experiments on electron diffraction.

In the first work (Stern and Knauer)\(^{7}\), the reflection of molecular beams of H\(_2\) and He from well-polished metallic surfaces and from cleavage surfaces of various crystals (chiefly NaCl) was investigated. Here we first encounter indications (although still rather unclear ones) of the existence of diffraction of molecular beams from the surface of a crystal.

Experiments on the reflection of He and H\(_2\) from optically polished surfaces were carried out with the aid of a mirror whose inclination relative to the direction of the molecular beam could be changed by means of a screw. It was found that for well-ground surfaces (having irregularities of the order of \(10^{-5}\)—\(10^{-6}\) cm) specular reflection occurs at very small grazing angles of the order of \(10^{-3}\). This result is in agreement with de Broglie’s theory. If a molecular beam falls on a specular surface, then regular reflection will occur only for such a value of the grazing angle at which the projection of the height of the surface irregularities onto the direction of the incident molecular beam is at least several times smaller than the de Broglie wavelength corresponding to the molecules of the beam. This is clear from consideration of the roughly schematic Fig. 6. Let \(d\) be the height of a protrusion on the surface of the mirror on which the molecular beam falls. The path difference of rays 1 and 2, equal to the path difference of rays 2 and 3, will be:

\[ BC + BC' = 2AC \sin \alpha = 2d \sin \alpha . \]

For regular reflection to exist, it is necessary that

\(2d \sin \alpha < \dfrac{\lambda}{2}\), i.e. \(d \sin \alpha < \dfrac{\lambda}{4}\). Since for hydrogen and helium \(\lambda = 10^{-8}\) cm, while the irregularities on the surface of the mirror are \(10^{-5}\)—\(10^{-6}\) cm, the glancing angle at which specular reflection becomes appreciable should be of the order of \(10^{-3}\), as indeed was the case.

The intensity of reflection of the molecular beam was measured with a Pirani manometer; in order to determine it quantitatively,

Fig. 6.

from the value of the intensity obtained with the mirror there was subtracted the value of the intensity measured in the absence of the mirror (background). The presence of this background is explained partly by the presence of scattered molecules of the beam, and partly by the fact that the front slit itself, owing to the high pressure in the furnace space, served as a source of molecular rays. Mirrors of glass, steel, and speculum metal were used, the latter giving the largest coefficient of reflection.* The intensity of the reflected ray decreased sharply with increasing glancing angle (see Table 1). Increasing the wavelength (achieved by lowering the temperature of the molecular beam) caused an increase in the reflection coefficient, which made it possible to work at large glancing angles. All these results are in complete agreement with the idea of the wave nature of molecular rays.

TABLE 1

Reflection of H\(_2\) from speculum metal

Glancing angle	Reflection coefficient in %
\(1 \cdot 10^{-3}\)	5
\(1.5 \cdot 10^{-3}\)	3
\(2 \cdot 10^{-3}\)	\(1 \tfrac{1}{2}\)
\(2.25 \cdot 10^{-3}\)	\(\tfrac{3}{4}\)

An attempt was also made to obtain the diffraction of He and H\(_2\) from an optical ruled grating, which ended in failure. The difficulties consisted in the following: diffraction maxima lying close to the reflected beam could not be observed because of the diffuseness of the latter, while diffraction maxima lying far from it

* With the given design of the apparatus (see Fig. 7), it was possible to investigate only diffraction spectra lying in the plane of incidence.

…from the reflected beam, could not be measured because of their low intensity.

To investigate the diffraction and reflection of molecular beams from crystals, Stern and Knauer used the apparatus shown in Fig. 7. Here \(O\) is the slit of the furnace, to which gas is supplied through the tube \(a\). The gas can be cooled near the slit to the temperature of liquid air or heated to \(500—600^\circ\), depending on the requirements of the experiment. \(Ab\) is the reflecting slit; \(Kr\) is the crystal, mounted on a table that can be rotated from outside by means of \(S_2\). The reflecting surface is parallel to the axis of rotation. The crystal can be heated or cooled as desired. \(Af\) is the receiving slit. The receiver can be rotated by means of the ground joint \(S_1\) about the same axis as the crystal. The receiver is connected to a Pirani manometer by the tube \(R_1\). In front of the receiving slit there is an electromagnetic valve \(Kl\), which blocks or opens access of the beam to the receiver. The tube \(R_2\), passing through the ground joint \(S_1\), leads to a compensation manometer. The adjustment of the slits and of the crystal was carried out first optically and then with the aid of the molecular beam itself.

The pressure in the furnace space was \(2 \cdot 10^{-4}—4 \cdot 10^{-4}\) mm Hg, and the pressure in the irradiation space was \(1 \cdot 10^{-5}\) mm Hg.

The experiments were carried out with crystals of rock salt, calcite, and galena, and with the gases \(\mathrm{H_2}\), He, Ne, Ar, \(\mathrm{CO_2}\); good results were obtained only for the reflection of He and \(\mathrm{H_2}\) from NaCl. The main investigations were performed with helium. It was possible to obtain sharp specular reflection, the intensity of the reflected beam amounting to 8% of the intensity of the primary beam.

Fig. 7.

It was found that the reflection coefficient depends to a large extent on the state of the crystal surface. Before the measurements the crystals were thoroughly degassed by heating in vacuum, as a result of which the reflection coefficient increased noticeably. The reflection coefficient also depends on the absolute temperature of the crystal—the lower the latter, the higher it is. Moreover, if the reflecting surface of the crystal is rotated in its own plane, while keeping the angle of incidence unchanged, the reflection coefficient changes. Two cases of orientation of the NaCl crystal surface were investigated with respect to…

with respect to the molecular beam: “direct” orientation (Fig. 8) and “oblique” orientation (Fig. 9). In the first case the plane of incidence of the beam intersects the surface of the crystal along one of the principal crystallographic axes. In the second case the plane of incidence intersects the surface of the crystal along the axis (1,1,0). The “oblique” orientation is obtained from the “direct” one by rotating the crystal by 45° about an axis perpendicular to the reflecting surface. The reflection coefficient is considerably higher for the oblique orientation, and therefore the investigations were carried out chiefly for this case. As the measurements showed, the reflection coefficient is also a function of the angle of incidence and of the wavelength (beam temperature). The angle of incidence was varied from 5 to 45°. It was found that the reflection coefficient, even at large angles (∼30°), is 2–3%, whereas for polished surfaces (glass, mirror metal) it is practically zero already at angles of incidence of 10–15°. We do not give the quantitative data obtained by Stern and Knauer, in view of their insufficient accuracy.

Fig. 8. “Direct” position of the crystal.

In the same work the diffraction of He from NaCl was studied. At all angles of incidence investigated, diffraction maxima were obtained.* However, these maxima were very indistinct, especially in the case of the direct orientation of the crystal, which made it impossible to index them accurately. Their intensity increases with increasing angle of incidence.

Despite the importance of the results obtained by Stern and Knauer, these results are still insufficiently convincing as a quantitative test of the de Broglie relation and still do not make it possible to understand the geometry of the phenomenon. These results were further developed in the brilliant work of Estermann and Stern,⁸ in which the reflection and scattering of molecular beams of hydrogen and helium from LiF were studied in detail.

Fig. 9. “Oblique” position of the crystal.

* By the reflection coefficient in the present case is meant the ratio of the intensity of the specularly reflected beam to the intensity of the incident beam.

The design of the apparatus used in the experiments of Stern and Estermann is somewhat different from the apparatus used in the above-described experiments of Knauer and Stern. The new construction of the apparatus makes it possible to study scattered beams lying outside the plane of incidence. In addition, in the new apparatus one can investigate the change in the diffraction pattern when the orientation of the crystal lattice is changed relative to the plane of incidence.

The crystal is fastened on a special table, which is arranged in such a way that adjustment can be carried out in vacuum by rotating the crystal about two axes perpendicular to the axis of the holder. The rotation is effected by means of the screws \(S_1\) and \(S_2\). The crystal is positioned so that the axes of rotation coincide with the axes of the lattice surfaces of like-named ions. Owing to this, the most varied displacements of the crystal surfaces are possible. The receiver can be rotated about an axis coinciding with the axis of the crystal holder.

The height of the receiving slit was made small (\(\sim 1.5\) mm) in order to separate the reflected beam from the diffraction maxima located close to it. The receiver does not perform any other motions except rotation about the axis of the holder.

Fig. 10.

The very first experiments of Estermann and Stern with the diffraction of helium and hydrogen from NaCl showed that, for the diffraction of molecular beams, essentially only the lattice of ions of one sign plays a role. The axes of this lattice, as is seen from Fig. 10, are parallel to the diagonals of the face of the cube that represents the unit cell of the NaCl crystal. The parameters of this lattice are \(\sqrt{2}\) times larger than the parameter of the surface lattice formed by ions of both signs. The very same result is obtained in the investigation of the diffraction of He and \(H_2\) from LiF; in this case also, not the entire surface lattice formed by Li and F ions takes part in the diffraction, but only the lattice composed of identical ions.

Before presenting further experimental results, it is necessary to clarify the pattern of the spatial arrangement of the diffracted beams and the method by which it can be obtained experimentally. Let us place the origin of coordinates at the point of intersection of the molecular beam with the surface of the crystal, and let us arrange the \(x\) and \(y\) axes along the principal axes of the square lattice under consideration. Let the direction of the incident beam be determined by the angles \(\alpha_0\), \(\beta_0\), and \(\gamma_0\), which it makes with the axes \(x\), \(y\), and \(z\), and the direc-

…direction of the diffracted beam—respectively the angles \(\alpha,\ \beta\), and \(\gamma\). The diffraction conditions will have the form:

\[ \begin{aligned} \cos \alpha - \cos \alpha_0 &= \frac{n_1 \lambda}{d},\\ \cos \beta - \cos \beta_0 &= \frac{n_2 \lambda}{d}, \end{aligned} \tag{8} \]

where \(n_1\) and \(n_2\) are integers determining the order of the diffraction maximum, \(\lambda\) is the wavelength, and \(d\) is the grating parameter, i.e., the distance between two neighboring ions of the same sign.

Each diffracted ray represents the line of intersection of two cones described about the axes \(x\) and \(y\), with a common vertex and opening angles \(2\alpha\) and \(2\beta\) (see Fig. 11), where \(\alpha\) and \(\beta\) are angles satisfying equations (8).

Fig. 11.

Let us consider first of all the case in which the plane of incidence coincides with the plane \(xz\)—the zero position (“oblique” orientation).

In this case the second of equations (8) takes the form:

\[ \cos \beta = \frac{n_2 \lambda}{d}, \tag{9} \]

Putting now \(n_1 = n_2 = 0\), we obtain the specularly reflected ray, for which

\[ \alpha = \alpha_0,\qquad \beta = \beta_0 = 0, \]

Fig. 12.

This ray lies at the intersection of cone \(A\), with aperture \(2\alpha_0\), and the plane \(xz\) (see Fig. 12).

The most intense, and therefore best studied, maxima correspond to the orders \((0,+1)\) and \((0,-1)\). The rays corresponding to both these maxima lie symmetrically with respect to the specularly reflected ray on the same cone \(A\).

Let us project the line of intersection of this cone with a sphere of unit radius and with center at the origin onto the plane \(yz\) [parallel projection] (Fig. 13). The line of intersection will be represented by the circle \(M\), the reflected ray by the straight line \(OP\), di-

DIFFRACTION OF MOLECULAR BEAMS FROM CRYSTALS

the diffracted beams \((0,+1)\) and \((0,-1)\)—by the straight lines \(OP_1\) and \(OP_2\).

With this arrangement of the apparatus, the receiver can rotate only about the \(z\)-axis. Therefore, if the receiver, brought into the plane of incidence, measures the specularly reflected beam, it will not be able to receive the diffracted beams \(OP_1\) and \(OP_2\), since these beams form with the surface of the crystal angles smaller than the angle \(\alpha_0\).

The line connecting the origin of coordinates with the slit of the receiver, when the latter is rotated about the \(z\)-axis, will trace on our diagram (Fig. 13) the horizontal straight line \(SS\). If this line passes through the point \(P\), then it does not touch the points \(P_1\) and \(P_2\). Since the slit of the receiver has a finite length, we must replace the line \(SS\) by the strip shown in our figure by hatching. In Stern and Estermann’s first experiments the slit was of such a length that the strip covered the points \(P_1\) and \(P_2\). In subsequent, more careful experiments the dimensions of the slit were reduced, and in order to investigate the diffracted beams the crystal was turned through a small angle

Fig. 13.                Fig. 14.

\((1—3^\circ)\) about the \(y\)-axis. Fig. 14 shows the parallel projection for the rotated crystal. The coordinate system is assumed to be rigidly connected with the crystal and to rotate together with it. In this case the circle \(M\) becomes larger in view of the increase of the angle \(\alpha_0\) (by the angle of rotation about the \(y\)-axis). The circle described by the receiver is then projected no longer as a straight line, but as a narrow ellipse, which, with a suitable choice of the angle of rotation, intersects it at the points \(P_1\) and \(P_2\).

Thus, by changing the angle of rotation of the crystal about the \(y\)-axis, one can arrange that the receiver, as it rotates about the \(z\)-axis, can measure the intensity of the diffracted beams.

The preceding considerations apply to the case in which the incident beam contains molecules of one and the same wavelength. In Stern and Estermann’s experiments the diffraction of nonmonochromatized beams is investigated. The distribution of molecules with respect to wavelength is determined by formula (5). As a result, each diffraction maximum represents the result of the superposition of diffraction maxima corresponding to the individual monochromatic components of the beam.

In diagrams 13 and 14, which depict (in schematic form) the picture of the diffraction field, the nonmonochromaticity of the beam must be expressed in the replacement of the points \(P_1\) and \(P_2\) by the segments \(l_1\) and \(l_2\) on the circle \(M\) (Fig. 15). The distribution of the intensity of the diffracted rays along each of these segments corresponds to the distribution of the molecules of the beam by wavelengths. In order to study the angular distribution of the intensity of the diffracted rays in an undistorted form, it is evidently necessary, for each given angle of rotation of the receiver about the \(z\)-axis, to find the best rotation of the crystal (about the \(y\)-axis), at which the maximum intensity of the diffracted rays corresponding to the given portion of the spectrum enters the receiver.* This is precisely how Estermann and Stern proceeded in their work.

Fig. 15.

The first experiments were carried out with diffraction of helium atoms from the cleavage surface of NaCl. The results are shown in curves I and II of Fig. 16. Along the abscissa axis is plotted the angle of rotation of the receiver about the \(z\)-axis; along the ordinate axis, the intensity of the diffracted rays, expressed (as in all subsequent curves) in conventional units—in divisions of the galvanometer scale. Both curves were taken with oblique orientation of the crystal and an angle of incidence equal to \(11.5^\circ\). The temperature of the beam is indicated on each curve.

Fig. 16. Diffraction of He from NaCl.

The central maximum corresponds to the specularly reflected rays, and the lateral maxima to the diffracted rays of orders \((0,+1)\) and \((0,-1)\). The arrows mark the positions of the diffraction maxima \((0,+1)\), \((0,-1)\), computed from equation (9), corresponding to the optimal wavelength \(\lambda_m\) [see formula (6)].

* Thus, different points of one and the same diffraction curve are recorded at different angles of incidence. This, however, does not practically distort the results, since in the investigated range of angles of incidence \((\alpha_0 \leq 20^\circ)\) the intensity of the reflected and scattered rays is almost independent of the angle of incidence.

The agreement between the calculated maxima and those found experimentally is satisfactory, since it lies entirely within the limits of observational error.

Still more distinct results were obtained in the study of the diffraction of He and H₂ from LiF. The advantage of this crystal in comparison with NaCl is manifested, first, in the greater sharpness and intensity of the diffraction pattern and, second, in the considerably smaller magnitude of the diffuse background. The results of experiments at an angle of incidence of \(11 \tfrac{1}{2}^{\circ}\) are shown in curves III, IV, V, VI (Figs. 17, 18, and 19).

These curves were taken with improved alignment of the crystal and receiver. The accuracy in determining the angle of rotation is, for them, \(0.5—1^{\circ}\). For all these curves, with the exception of curve VI

Fig. 17. Diffraction of He from LiF, angle of incidence \(11 \tfrac{1}{2}^{\circ}\).

Fig. 18. Diffraction of He from LiF at \(580^{\circ}\) K, angle of incidence \(11 \tfrac{1}{2}^{\circ}\).

(diffraction of H₂ at a beam temperature of \(290^{\circ}\) K), the experimentally found positions of the maxima coincide with those calculated for \(\lambda=\lambda_m\) with an accuracy of up to \(0.5^{\circ}\).

The discrepancy between experiment and calculation for the diffraction of H₂ at \(290^{\circ}\) K is explained by the fact that, because of imperfections in the geometry of the apparatus, measurement of the intensity of rays scattered at large angles to the plane of incidence becomes inaccurate.

The results of the most accurate measurements of the positions of the diffraction maxima are given in Table 2.

In these experiments of Stern and Estermann we thus find complete confirmation of de Broglie’s formula \(\lambda=\dfrac{h}{mv}\) for molecular-

beams. They also show that diffraction of molecular beams occurs from the surface lattice of the crystal, while volume interference is excluded.

If the scattering coefficient does not depend on wavelength and the dispersion within the measured range of angles is constant, then the distribution of intensity in each diffraction maximum must reproduce the distribution of intensity in the molecular beam with respect to wavelengths. This is indeed justified if the necessary corrections are introduced into the experimental curves, determined by the geometry of the apparatus and by allowance for the unequal dispersion for different parts of the spectrum. The latter factor is easy to estimate.

Fig. 19. Diffraction of H₂ from LiF, angle of incidence \(11 \frac{1}{2}^{\circ}\).

The dispersion for the entire series of experiments, the results of which are shown in curves I to VI, is determined by the expression [from formula (9)]:

\[ -\frac{d\beta}{d\lambda}=\frac{1}{d\sin\beta}, \]

where \(\beta\) is the angle of the scattered beam with the \(y\)-axis.

The angle \(\beta\) for this series of experiments, in which maxima of order \((0,\pm 1)\) were studied, is equal to \(90^{\circ}-\varphi\), where \(\varphi\) is the diffraction angle, i.e., the angle made by the scattered beam with the \(xz\) plane. Therefore the dispersion is equal to:

\[ \frac{1}{d\cos\varphi}\sim \frac{1}{d}\left(1+\frac{1}{2}\varphi^{2}\right). \]

TABLE 2

Gas	Angle of incidence in °	Beam temperature in °K	Position of maximum \((0,1)\) in °, found experimentally	Position of maximum \((0,1)\) in °, calculated
He	11.5	290	12	11.75
He	11.5	580	8.75	8.5
H₂	11.5	580	12	11.75
He	18.5	180	14.5	15.5
He	18.5	290	11.5	12
H₂	18.5	590	9	8.75
H₂	18.5	290	17	17
H₂	18.5	580	11	12

DIFFRACTION OF MOLECULAR BEAMS FROM CRYSTALS

When \(\varphi\) is changed from \(5\) to \(20^\circ\), it changes by \(5\%\). Thus the correction for the nonuniformity of the dispersion at different scattering angles \(\varphi\) is small. Consequently, the diffraction curve representing the change in intensity of the scattered beam with the angle \(\varphi\) must have the same form as the curve in Fig. 2, which expresses the law of distribution of molecules in the beam by wavelengths. The experiments of Stern and Estermann confirm this conclusion and at the same time prove that the scattering coefficient is independent (to a first approximation) of the wavelength.

In addition to spectra of order \((0, \pm 1)\), Stern and Estermann also investigated diffraction maxima lying in the plane of incidence. Indistinct indications of the existence of these maxima had already been obtained in the work of Stern and Knauer.

Stern and Estermann carried out investigations of spectra lying in the plane of incidence with the “straight” orientation of the crystal (see Fig. 15). With this orientation, spectra of order \((-1,-1)\) and \((+1,+1)\) lie in the plane of incidence.

These experiments were performed with the apparatus of the first construction (Fig. 7), in which a number of changes had been made that considerably improved the adjustment of the crystal and the receiver and increased the accuracy of the angle measurements.

The results of measurements of these spectra for angles of incidence from \(10\) to \(70^\circ\) are presented in curves \(VIII—XVII\) (Fig. 20). Along the abscissa axis is plotted the angle formed by the diffracted beam with the direction of the specularly reflected beam. The spectrum \((+1,+1)\) corresponds to positive values of these angles, and the spectrum \((-1,-1)\) to negative ones.

At small angles of incidence the measured positions of the maxima coincide with the calculated ones (marked by arrows) to an accuracy of \(2—3^\circ\). At large angles of incidence and for shorter waves (temperature \(290^\circ\text{K}\)) the experimental maxima diverge strongly from the calculated ones. This may be attributed to the influence of irregularities on the crystal surface, which, as we saw earlier, is more pronounced at large angles of incidence and small wavelengths.

In addition to studying the diffraction of molecular beams of He and \(\mathrm{H}_2\), Stern and Estermann carried out a series of experiments to investigate the reflection of these beams from LiF and NaCl.

The dependence of the reflection coefficient on the angle of incidence and the orientation of the crystal was studied.

The dependence on the angle of incidence is shown in Fig. 21. Just as in reflection from an artificially polished surface, the reflection coefficient of molecular beams from a crystal falls sharply with increasing angle of incidence, beginning at \(10—20^\circ\).

It is natural to suppose that the reason for the decrease in the reflection coefficient at large angles of incidence is the influence of microscopic irregularities on the crystal surface. From the curves in Fig. 21 it follows that these irregularities are of the order of \(1\ \text{Å}\), i.e. by

are, in order of magnitude, the same as the amplitude of the thermal vibrations of the ions forming the crystal lattice.

In favor of identifying these irregularities with the thermal vibrations of the lattice is the increase in the reflection coefficient when the temperature of the crystal is lowered.

The results obtained in studying the dependence of the reflection coefficient on the orientation of the crystal (with respect to the plane of incidence of the beam) will be set out below.

The natural next step in the study of the diffraction of molecular and atomic beams was the experiments of Estermann, Frisch and Stern\(^9\) with monochromatic molecular beams. The significance of these experiments is obvious: only by working with monochromatic beams can one obtain the phenomenon of diffraction in pure form; using an ordinary nonmonochromatized beam of molecules, in diffraction we must obtain a total effect in which many essential details of the phenomenon cannot appear.

Fig. 20. Diffraction spectra in the plane of incidence.

Fig. 21. Reflection coefficient of He from LiF.

Monochromatization of molecular beams can be carried out by two methods:

The first method, in principle, reduces to the following: an ordinary nonmonochromatized molecular beam, falling on the surface of a crystal, gives a diffraction spectrum, from which, by means of a slit, a beam of definite direction and, consequently, of definite wavelength is selected.

The second method is based on the use of rotating toothed wheels, by means of which (by analogy with Fizeau’s experiments on determining the velocity of light) a purely mechanical method selects from the molecular-

of the beam there is selected a group of molecules of a definite velocity. Let us first dwell on monochromatization by diffraction from a crystal.

This method may have several different variants. The difference between them lies in the ways in which the mutual displacement of the separate parts of the apparatus (crystals, slits, and receiver), necessary for studying the spatial diffraction pattern, is carried out. The simplest variant, leading to the smallest number of displacements in vacuum, was used in the work of Estermann, Frisch, and Stern. The arrangement employed by them is shown in Fig. 22.

Here \(A\) is the crystal serving as the monochromator, \(B\) is the diaphragm selecting the monochromatic beam, \(C\) is the crystal with which the diffraction investigations are carried out, and \(D\) is the receiver measuring the intensity of the beams scattered by the second crystal. The \(x\)-axis (Fig. 22) is the principal axis of the surface lattice of the like ions. The diaphragm \(B\) is arranged with respect to crystal \(A\) in such a way that, for an oblique orientation of the crystal, it transmits the specularly reflected beam. If now the crystal \(A\) is rotated about the \(x\)-axis, then through this diaphragm there will successively pass the whole diffraction spectrum lying on the cone \(\alpha=\alpha_0\) (\(\alpha_0\) is the angle formed by the incident beam with the \(x\)-axis).

Fig. 22.

To determine the wavelength of the beams transmitted by the diaphragm for some definite rotation of the crystal, let us first note that for the scattered beams*) entering the diaphragm, \(\beta+\beta_0=180^\circ\) (retaining the previous notation). Therefore from equation (9) it follows:

\[ 2\cos\beta_0=\frac{n_2\lambda}{d}, \]

whence

\[ \lambda=\frac{2d\cos\beta_0}{n_2}. \]

The angle \(\beta_0\) is related to the angle of rotation of the crystal \(\varphi\) (\(\varphi\) is reckoned from the position of the crystal when the plane of incidence passes through the \(x\)-axis) by the relation:

\[ \cos\beta_0=|\sin\varphi|\,\sin\alpha_0, \]

therefore:

\[ \lambda=\frac{2d\,|\sin\varphi|\,\sin\alpha_0}{n_2}. \]

For the most intense spectra of order \((0,\pm 1)\) we have

\[ \lambda = 2d\,|\sin \varphi|\,\sin \alpha_0 . \tag{10} \]

The monochromatized beam, after passing through diaphragm \(B\), falls on the surface of crystal \(C\), which can likewise rotate about the \(x\)-axis. The receiver is positioned with respect to the second crystal in the same way as diaphragm \(B\) is with respect to the first. When crystal \(C\) is rotated, receiver \(D\) records all diffraction maxima of order

Fig. 23.

Fig. 24.

Fig. 25.

\((0,\pm n)\), corresponding to the given wavelength \(\lambda\), selected by the first crystal and the slit.

Curve \(I\) (Fig. 23) represents the results of measurements of diffraction from the first crystal when the second is removed (it was taken with the aid of a compensation receiver). Along the abscissa axis, as in all the curves, the angle of rotation of the crystal \(\varphi\) is plotted. The central maximum corresponds to the specularly reflected beam, while the maxima located on both sides of it correspond to diffraction of order \((0,\pm 1)\).

The diffraction curves obtained when rotating the second crystal about the \(x\)-axis are shown in Fig. 25. Each curve gives one half of the diffraction pattern. Along the abscissa axis is plotted the angle of rotation of the second crystal. On the left, everywhere, is indicated the angle of rotation of the first crystal at which the curve was taken. The angle \(\varphi\), corresponding to the diffraction maximum on the diffraction curve, must obviously be equal to the angle of rotation of the first crystal. This is indeed the case, as the curves show. The difference between these curves and curve \(I\) (Fig. 23) is that the diffraction maxima are considerably sharper and narrower with a monochromatized beam. On curves \(II\) and \(III\) the appearance of a second maximum is noticeable, which should be attributed to the order \((0,2)\). At large angles of rotation (curves \(VI\) and \(VII\)) an intermediate maximum appears, which apparently corresponds to a beam with a wavelength 2 times smaller than the wavelength corresponding to the principal maximum. This beam can also pass through the diaphragm if it is scattered by the first crystal in the second order.

It is noteworthy that on diffraction curves obtained with monochromatized beams the maxima of order \((0,1)\) have a higher intensity than the specularly reflected beam.

Mechanical monochromatization is carried out by means of the device shown in Fig. 24 (schematic). Here \(A\) is a pair of disks rotating on a common axis \(S\). Each of the disks has on its rim a large number of notches (the same for both disks). The notches of one disk are located exactly opposite the notches of the other. The direction of the molecular beam is indicated by the line \(MM\).

When the disks are at rest, all molecules that have passed through a notch in the first disk will also pass through the opposite notch of the second disk. When the disks rotate, molecules possessing small velocities, after passing through a notch of the first disk, will in their further motion strike a tooth of the second. The greater the number of revolutions of the rotating system, the smaller the number of molecules that will be able to pass through notches located opposite one another. However, part of the molecules may pass through the next notch of the second disk. The greater the rotational speed of the disks, the greater the velocities of the molecules passing through the disk system by such a path. The velocity of these molecules will obviously be equal to \(\frac{l}{t}\), where \(l\) is the distance between the disks, and \(t\) is the time in which the disk turns by one tooth. If the number of notches on the rim of the disk is equal to \(z\) and the number of revolutions per second is equal to \(\nu\), then

\[ t=\frac{1}{\nu z} \]

and

\[ v=lz\nu . \tag{11} \]

The last formula is valid only in the case when the width of the slit can be neglected in comparison with the distance between neighboring slits. Owing to the fact that the slit has a finite width, a group of molecules is selected whose velocities lie within a certain interval determined by the geometry of the system.

Besides the monochromatic beam of molecules selected in this way, we also have a beam of faster molecules which have passed through slits situated opposite one another. At small rotational velocities the intensity of this beam may be considerably greater than the intensity of the beam of molecules selected by the method indicated above.

At high rotational velocities, i.e., when \(v\), calculated from (11), is greater than the optimum velocity corresponding to the maximum on the Maxwell curve, the intensity of this beam of the fastest molecules is small, and its effect may be neglected in the first approximation.

On the other hand, at a high rotational velocity it is possible for rays to pass which contain molecules with velocities equal to \(\dfrac{v}{2}\), \(\dfrac{v}{3}\), etc. Indeed, if a molecule with velocity \(v\) traverses the path between the disks in the time during which they rotate by one tooth, then a molecule with velocity \(\dfrac{v}{n}\) (\(n = 2,3\)) will traverse the same path in the time of rotation by \(n\) teeth and will enter the \(n+1\)-st slit.

The intensity of rays containing these molecules, at large values of \(v\) (the falling part of the Maxwell curve), will be considerably higher than the intensity of the main ray (containing molecules with velocity \(v\)).

Thus it turns out that mechanical monochromatization carried out with the aid of rotating disks is not, in essence, true monochromatization, since it gives not one monochromatic ray but an entire spectrum with wavelengths that are multiples of one definite wavelength, and, in addition, a portion of a continuous spectrum containing very small wavelengths. This, obviously, complicates the diffraction picture. Upon the diffraction spectra of the second, third, etc. orders there will be superposed first-order spectra corresponding to the “additional” rays with wavelengths 2, 3, … times greater than the wavelength of the main ray. In pure form only the diffraction maxima of the first order \((0,\pm 1)\), \((1,\pm 1)\), etc., are obtained.

In order to subject the de Broglie formula to as careful a test as possible, Stern, Estermann, and Frisch carried out a particularly precise measurement of the positions of the diffraction maxima near the optimum (i.e., corresponding to the maximum of the distribution curve by wavelengths) wavelength. In this case the diffraction maxima are obtained especially sharp. The results of the measurements are shown in Fig. 26. The central maximum corresponds to the specularly reflected ray, and the lateral ones—to the diffraction maxima.

orders \((0,+1)\) and \((0,-1)\). The angles along the axis are measured from an arbitrarily chosen zero.

Comparing the wavelength of the molecular beam, found from the distance between the maxima on this curve by formula (10), with the wavelength calculated by the formula:

\[ \lambda=\frac{h}{mv}=\frac{h}{mz l \nu}, \]

Stern, Estermann, and Frisch showed that the two values agree to within \(1\%\).

Fig. 26. Diffraction of He from LiF.

This direct verification of the fundamental formula of wave mechanics—the de Broglie formula—is the most important result of the experiments with mechanical monochromatization of molecular beams.

The investigations of Stern and his collaborators showed that the phenomena of reflection and diffraction of molecular beams from the surface of a crystal have a very complex character. The complexity of these phenomena appears especially sharply in the peculiar anomalies, first discovered by Stern and Estermann in studying the dependence of the reflection coefficient on the orientation of the crystal. The anomaly in reflection consists in the fact that the intensity of the reflected beam (at a given angle of incidence) changes with the orientation of the crystal not smoothly, but by passing through a series of minima and maxima.

Fig. 27. Reflection of He from LiF; direct beam \(370\) cm.

Figure 27 shows a typical curve of the dependence of the reflection coefficient on the angle of rotation of the crystal about an axis perpendicular to its reflecting surface. The angles of rotation are measured from the “skew” position of the crystal. The curve was taken at an angle of incidence of \(11.5^\circ\) and a beam temperature of \(290^\circ\)K.

The original assumption of Stern and Estermann that such a form of the reflection curve is explained by the influence of diffraction spectra entering the receiver together with the reflected beam subsequently proved to be incorrect. It turned out that the complexity of the form of the curve has definite regularities. To clarify these regularities, Stern and Frisch \(^{10}\) recently carried out a special study of anomalies in the diffraction and reflection of molecular beams.

The anomalies in the reflection of molecular He beams from LiF were investigated especially thoroughly. The crystal and the detector could rotate about an axis perpendicular to the molecular beam, whereby the angle of incidence could be varied. In addition, the crystal could also rotate in its own plane, so that the axes of the lattice surfaces could assume various positions relative to the plane of incidence of the beam.

The results of measurements of the reflection coefficient as a function of the orientation of the crystal at various angles of incidence are presented in the curves of Figs. 28 and 29. In these curves \(\eta\) denotes the angle of rotation of the crystal in its own plane relative to the “oblique” position. Examination of the curves shows that, at large angles of incidence, against the background of a smooth increase of the reflection coefficient with the angle \(\eta\), two minima appear: a diffuse minimum at small \(\eta\) and a sharp minimum at angles \(\eta\) of about \(25\text{--}30^\circ\).

Fig. 28. Reflection of He from LiF at various angles of incidence.

These minima appear still more sharply on curves taken at small angles of incidence (Fig. 29). Here the minimum lying at small \(\eta\) becomes especially sharp. The sharpness of this minimum at very small angles of incidence is such that a change of the angle \(\eta\) by \(1^\circ\) in this region of angles leads to a change in the reflection coefficient by a factor of 5–6.

The dependence of the anomalies found on the velocity of the molecules (the temperature of the beam) is characterized by the curves shown in Fig. 30. Raising the temperature of the beam does not change the shape of the curve and leaves the minima and maxima in their places. There occurs only an overall decrease in the intensity of reflection and, in addition, some blurring of the minima and maxima.

The position of the minimum lying in the region of large values of \(\eta\) is determined by a simple geometrical rule, which may be formulated as follows: for any angle of incidence of the molecular beam and any velocity of the molecules composing this beam, the minimum of the reflection coefficient corresponds to such an orientation of the crystal in which the reflected ray lies in the “forbidden plane” \(M\), making some definite angle \(\psi\) with the plane \(xz\). For LiF and an atomic He beam this angle is equal to \(49^\circ\). The angle \(\psi\) is related to the angle \(\eta\) by the relation:

\[ \frac{\sin \eta}{\tg \xi} = \tg \psi . \]

The indicated rule is illustrated by Table 3. At small angles \(\xi\), deviations from it are observed.

For the second minimum, lying in the region of small \(\eta\), such a simple rule is apparently inapplicable.

TABLE 3

\(\xi\)	\(\eta\)	\(\varphi\)
25	32.5	48.9
22	27.5	48.6
20	24.5	48.6
18	22	49.0
14	17	49.0
10	12	48.8
7	9.7	54
4.3	8	62

Experiments with the reflection of a molecular beam of \(\mathrm{H}_2\) from LiF and of an atomic beam of He from NaF have shown that in these cases as well there are sharply expressed anomalies of the type described above.

Stern and Frisch also carried out investigations of anomalies in reflection with a monochromatized He beam. These experiments were performed by means of an apparatus with two crystals (see p. 637). In this case the minimum of the reflection coefficient lying closer to the zero position is found even at large angles of incidence, whereas with a non-monochromatized beam it is almost imperceptible at \(\xi > 20^\circ\).

As Frisch and Stern showed, sharply expressed anomalies also exist in the diffraction curves. In Fig. 31 is shown the diffraction curve of He from LiF, taken with an “oblique” orientation of the crystal. The central maximum corresponds to the specularly reflected beam, the side ones to the diffraction spectra \((0,+1)\) and \((0,-1)\).

Fig. 29.

Fig. 30. Solid curve — beam temperature \(600^\circ\mathrm{K}\). Dashed curve — room beam temperature.

The dashed curve shows the theoretically expected distribution of intensity. When working with high resolution, however, two minima of scattering intensity are revealed—a weakly expressed one at \(\varphi \simeq 20^\circ\) (\(\varphi\) is the angle of rotation of the receiver) and a sharper one at \(\varphi \simeq 26^\circ\). When the orientation of the crystal is changed (rotation of the crystal about an axis perpendicular to its surface), the minima shift asymmetrically with respect to the density of incidence.

A satisfactory explanation of the anomalies in the reflection and scattering of molecular beams is at present lacking, although, as Frisch and Stern point out, “we are dealing here with a very characteristic phenomenon, which evidently must obey simple regularities.”

For the theoretical interpretation of these anomalies, a law of a very general character, established by Frisch,11 is of great importance. Before giving the formulation of this law, let us note that anomalies in the distribution of the intensity of reflected and diffracted beams are observed only in those cases when the direction of these beams makes a small angle with one of the principal axes of the surface lattice of like ions (the diagonal of the face of the cube). Denoting this axis by \(x\), the surface-lattice axis perpendicular to it by \(y\), and the normal to the crystal surface by \(z\), we can express the law established by Frisch as follows:

Fig. 31.

An anomalous decrease in the intensity of a reflected or scattered molecular beam occurs if and only if the molecules of the incident, reflected, or scattered beam have quite definite values of the momentum components \(p_y\) and \(p_z\). The value of the momentum component \(p_x\) plays no role.

This law is clearly illustrated by the curves of Fig. 32, where the values of the momentum components \(p_y\) and \(p_z\) of scattered and reflected molecules are plotted, calculated from the position of the anomalous intensity minima on the diffraction curves and the monochromatized reflection curves of He from LiF (by a reflection curve we mean a curve showing the dependence of the reflection coefficient on the orientation of the crystal). The numbers standing near each point indicate the value of the component \(p_x\). The fact that, instead of a star-shaped field of points, this diagram gives two lines along which the points are correctly arranged confirms the first part of the law established by Frisch—the dependence of the position of anomalous minima on the diffraction and reflection curves on \(p_y\) and \(p_z\).

and the scatter of the values of \(p_x\) speaks in favor of the validity of the second part of this regularity, the independence of the anomalies from \(p_x\).

From all that has been set forth it follows that if a molecule, flying toward the surface of a crystal or leaving it as a result of diffraction, has suitable values of the momentum components, then the probability for it to fall out of the diffracted beam (to become trapped on the surface of the crystal) is considerably greater than for molecules whose momentum components do not fall on one of the lines shown in Fig. 32.

It is most natural to picture this “trapping” of molecules as short-lived adsorption followed by escape in an arbitrary direction. Since this adsorption is selective in character, one should think that here we are dealing with a peculiar quantization of the states of the adsorbed atom in the field of the crystal lattice. Indeed, in this case it is easy to imagine that, at certain values of the momentum, a free atom passes into a quantized state and for some time proves to be bound in its motion, as a result of which the phase relation necessary for diffraction is destroyed. The mathematical development of this hypothesis has not yet led to interesting results.

Fig. 32.

Stern’s investigations and those of his co-workers established, in its main features, the geometrical picture of the diffraction of light atoms. In the technique of execution and in the results achieved, these investigations stand at a much higher level than all the other experimental studies of the diffraction of atoms and molecules.

In connection with Stern’s work one should also mention Johnson’s experiments\(^{12}\) on the diffraction of atomic hydrogen by LiF, interesting chiefly for their method. The distribution of the scattered atoms was recorded by a plate coated with \(\mathrm{MoO_3}\). In the places struck by hydrogen, reduction and blackening of \(\mathrm{MoO_3}\) occurred; the resulting bands were photographed. Without giving any quantitative results, this method has the advantage that the entire diffraction pattern is obtained on the photographic plate. The photographs obtained in this way, as well as Stern’s experiments, indicate diffraction from a flat

lattice. Diffraction spectra of the orders \((0,\pm 1)\) and \((\pm 1,0)\) were recorded.

Studies of the diffraction and reflection of heavy atoms (Cd, As, Zn, Hg, alkali metals), carried out mainly in America, have so far not led to definite results. Only for Ne and Ar was Zabel\(^3\) able to observe diffraction from NaCl. However, even in this case the diffraction maxima are very weakly expressed. The crystals with which the investigations were performed were obtained artificially. The diffraction curves were taken with the aid of an ionization detector, with an oblique orientation of the crystal. \(\Phi = 90^\circ - \beta\), where \(\beta\) is the angle formed by the incident beam with the \(y\)-axis (see above). The lattice constant, calculated from the position of the maximum on the diffraction curve for neon, proved to be equal to \(2.05\) Å,

Fig. 33.

which approximately corresponds to the distance between neighboring rows of Na and Cl ions, while the lattice constant corresponding to the maximum of the argon curve proved to be equal to \(3\) Å. The latter result is probably explained by the fact that the diffraction maximum of the argon curve is shifted toward small \(\Phi\) owing to the rapid decrease in the intensity of the diffusely reflected beam, on which the diffraction maximum is superposed.

Zabel also investigated the influence of treatment of the crystal surface on the reflection and diffraction of He from NaCl. In Fig. 33 the diffraction curve 1 was taken with a crystal that was cleaved in dry hydrogen; curve 2—with a crystal that was cleaved in dry air; curve 3—in humid air. From these curves it is seen that the intensity of reflection and diffraction for the humid surface is considerably less than for the dry one, and that the diffraction maximum under the influence of water vapor is shifted toward smaller angles (the latter corresponds to an increase in the lattice parameter). From Zabel’s experimental data it follows that, when the crystal is not subjected to the action of moisture, diffraction occurs from a plane lattice formed by ions of both signs, which for the chosen direction of the axes (see above) corresponds to the parameter \(1.99\) Å. On the surface of a humid crystal, in Zabel’s opinion, there exist two lattices whose action is manifested in diffraction—one with parameter \(1.99\) Å, the other with double parameter \(3.98\) Å. For non-monochromatic beams the action of both these lattices cannot be separated experimentally. It is manifested

only in an apparent increase of the parameter, which for crystals exposed to moisture lies between 1.99 and 3.98 Å. The question of the reliability of Zabel’s results must, however, be regarded as unresolved, since they clearly contradict the much more detailed and precise experiments of Stern and his collaborators on the diffraction of He from NaCl. In all his experiments Stern obtained diffraction of He only from the lattice of ions of the same name, which corresponds to the lattice parameter 3.98 Å.

Of the other works on the scattering of heavy atoms, we shall dwell only on the work of Zall and Ellet,¹³ who investigated the scattering of an atomic beam of mercury by crystals of NaCl, KCl, KBr, and KJ. In the apparatus which they used, the crystal and the receiver could rotate about a common axis perpendicular to the plane of incidence. The intensity of the beams scattered in various directions was measured with an ionization manometer.

Fig. 34. Crystal temperature 350° C, temperature of the incident molecular beam 170° K. The arrows indicate: 1—the direction of the incident beam, 2—the direction of specular reflection, 3—the direction of the normal to the crystal surface.

The distribution of mercury atoms scattered from NaCl, KCl, KBr, and KJ was studied as a function of the angle of incidence and of the temperature of the crystal and of the incident beam.

First of all, the reflection of mercury from rock salt was investigated as a function of the angle of incidence of the beam. The scattering curves obtained in this case are shown in Fig. 34. As we see, the direction corresponding to the maximum intensity on the scattering curves is deflected from the direction of specular reflection toward the normal to the crystal surface. When a correction is made for diffuse scattering (see below), this deflection, although it becomes smaller, still remains. It increases as the angle of incidence decreases.

The effect of changing the temperature of the crystal and the temperature of the beam on the scattering of mercury from NaCl is shown in Fig. 35. The true scattering curves \((A_2, B_2)\) are obtained from the experimental curves \((A_1, B_1)\) after introducing a correction for diffuse scattering, which is represented by the curves \(A_3, B_3\).

From Fig. 35 it is seen that, with a decrease in the temperature of the crystal and with an increase in the temperature of the beam, the position of the intensity maximum on the scattering curves shifts toward specular reflection. In addition, with a lowering of the crystal temperature there occurs a narrowing of the directed beam, accompanied by an increase in the relative magnitude of the diffuse scattering.

Similar curves were obtained for the reflection of mercury from KBr and KCl. The KJ crystal scattered the mercury beam quite diffusely, following the cosine law.

The changes in the scattering curves as a function of the orientation of the crystal were very small and lay within the limits of experimental error.

Fig. 35. a — curve \(A_{1,2,3}\): crystal \(350^\circ\)C, beam \(170^\circ\)C; \(B_{1,2,3}\): crystal \(50^\circ\)C, beam \(170^\circ\)C. b — curves \(A_{1,2,3}\): crystal \(50^\circ\)C, beam \(500^\circ\)C; \(B_{1,2,3}\): crystal \(50^\circ\)C, beam \(170^\circ\)C.

The results of the experiments of Ellett and Zabel do not yet have a sufficiently well-founded explanation. In any case, one can hardly agree with the authors, who suppose that the obtained distribution of the intensity of the scattered atomic Hg rays over angles is the result of volume interference. On the other hand, the form of the scattering curves also cannot be explained by diffraction from a plane grating, since the de Broglie wavelength for Hg is very small, and the diffraction maxima should therefore lie so close to the reflected beam that they could broaden it only very slightly, and not in the way that actually occurs. Moreover, if the scattering curves were due to diffraction from a plane grating, they would have to depend on the orientation of the crystal, which in fact was not observed.

References

1. Ellett and Zahl, Phys. Rev., 38, 977, 1931.
2. Stern and Knauer, Z. Physik, 53, 766, 1929.
3. Zabel, Phys. Rev., 42, 218, 1932.
4. Estermann and Stern, Z. Physik, 85, No. 3–4, 1933.
5. Taylor, Z. Physik, 57, 242, 1929.
6. Ellett and Olson, Phys. Rev., 31, 643, 1928.
7. Stern and Knauer, Z. Physik, 53, 779, 1929.
8. Estermann and Stern, Z. Physik, 61, 95, 1930.
9. Estermann, Frisch and Stern, Z. Physik, 73, 348, 1931.
10. Frisch and Stern, Z. Physik, 84, 430, 1933.
11. Frisch, Z. Physik, 84, 443, 1933.
12. Johnson, Phys. Rev., 35, 1929, 1930, 37, 847, 1931.
13. Ellett and Zahl, Phys. Rev., 38, 977, 1931.

Reviews

Fraser, Molecular rays.
Frisch and Stern, Handbuch d. Physik, XXII, II.