On the Molecular Beam

N. N. Semenov.

§ 1. Production of a molecular beam and the critical temperature of condensation of molecules. § 2. Oriented growth of crystals. § 3. Attempts to explain the mechanism of the phenomenon of the critical temperature of condensation. § 4. Direct determination of the velocities and mean free paths of molecules. § 5. Direct determination of the magnetic moments of atoms. Conclusion.

§ 1. Production of a Molecular Beam and the Critical Temperature of Condensation of Molecules

A molecular beam is a stream of molecules flying in one direction in a high vacuum. Such a beam of molecules was first realized by Dunoyer [¹] in 1911 and was investigated in more detail by Wood [²,³] and Knudsen [⁴,⁵] in 1915 and 1916.

Wood’s apparatus is shown in Fig. 1. The entire tube is evacuated by a pump to a good vacuum. At the end \(A\) is placed a metal, heated by an external electric furnace. The evaporating molecules of the metal rush out into the wide part of the tube and spread downward. If the whole apparatus is immersed in a Dewar vessel with liquid air, the following phenomena appear in this form. From above, beginning at the boundary of the liquid air, the inner walls of the glass tube become covered, over time, with a thick layer of the distilling metal. The density of this layer gradually decreases downward, reaching zero at a distance of \(1—2\) cm below the boundary of the liquid air. Farther on, down to the very constriction \(B\) of the tube, the metallic deposit is absent. The upper cone of the constriction \(B\) is again covered with a deposit of metal. The lower cone and the whole bulb \(C\) remain completely clean (no matter how long

Fig. 1.

continued the experiment), except for a small, sharply outlined spot around the point lying directly under the constriction \(B\). The diameter of the spot is determined by the intersection of the dashed straight lines \(aa\) and \(bb\), passing through the point of intersection of the upper tube with the boundary of the liquid air and the edge of the constriction. Such a picture quite obviously proves the following character of the phenomenon. Escaping from \(A\), the vapor strives to distill into the lower parts of the apparatus. But the vapor molecules, striking the walls of the glass cooled by liquid air to a temperature of \(-140^\circ\) C, adhere to the surface after the very first impact, forming a layer of metal. Without explanation it is obvious that the density of the deposit in this case must, decreasing, come to nothing as one moves away from the boundary of the liquid air. Near the constriction \(B\), through the wide tube, evidently, there will fly those molecules which, without colliding with the wall, pass through the entire space from the boundary of the liquid air to the constriction. Some of these molecules condense on the cone of the constriction \(B\), while some pass through the constriction, forming in the bulb \(C\) a sharply bounded beam of vapor molecules moving in one direction1; the boundaries of this beam are determined, evidently, by the cone formed by the straight lines \(aa\) and \(bb\), connecting the points of the section of the tube at the boundary of the liquid air and the points of the section of the narrowest part of the constriction.

Falling on the bottom of the bulb \(C\), the beam condenses, forming a sharply outlined circle. The complete absence of deposit on the remaining parts of the bulb proves our fundamental assumption, that glass cooled to a temperature of \(-140^\circ\) C captures a metal molecule at the very first impact. Moreover, this proves that collisions of the molecules of the beam with one another may be neglected. If the apparatus is filled with a gas, e.g. hydrogen at \(0.01\) mm, then the boundaries of the spot become blurred, and the deposit covers the whole bulb. This, evidently, occurs as a result of collisions of the molecules of the beam with the gas molecules.

If the vessel with liquid air is removed, then the phenomenon proceeds quite differently. At first the whole tube becomes covered with a deposit of metal, as does the whole bulb \(C\), and only after this does the thickness of the layer begin to grow more strongly around the point located directly under the constriction than in the remaining parts of the bulb. One may also carry out the experiment as follows: first cover with a deposit of metal the entire tube above the bulb, while leaving the bulb clean, and apply cotton wool moistened with liquid air to the lower part of the bulb. Then again a sharp spot is obtained around \(O\) of the same size as in the presence of the Dewar vessel, while the rest of the bulb remains clean. These experiments prove: 1) that in the absence

liquid air, when the walls of the glass are at room temperature, the metal molecules only after multiple reflections finally condense on the walls; 2) that the walls of the glass, covered with a layer of metal, at room temperature behave in the same way as clean glass walls at \(-140^\circ\) C, i.e., the metal molecules adhere to the same metallic surface immediately, i.e., after the very first collision, even at room temperature.

If the whole apparatus is lowered into liquid air, as shown in Fig. 1, but the construction of the apparatus is changed in such a way as to introduce into the bulb a glass stem, heated to room temperature by wires sealed into it through which a current passes, and to place the expanded end of the stem opposite the constriction, i.e., in the path of the beam, then the molecules will be reflected from the surface of the stem and enter the upper part of the bulb, where they will immediately condense, since the temperature here is below \(-140^\circ\) C. The whole lower part of the bulb below the plane of the polished surface of the stem should remain completely clean. This is exactly what happens in the experiment.

From the kinetic theory of gases it follows that the direction in which a molecule incident on a surface is reflected may be any of those allowed by the form of the surface1. However, the probability that the molecule will be reflected in such a way that its direction of motion will lie within some solid angle \(d\Omega\) depends on the angle \(\varphi\) between the direction \(d\Omega\) and the normal to the reflecting surface; namely, this probability will be proportional to:

\[ W = k' \cos \varphi\, d\Omega . \]

For a large number of reflected molecules, the number of those which are reflected within a given angle \(d\Omega\) will be proportional to the probability \(W\). Consequently,

\[ dn = k \cos \varphi\, d\Omega . \]

If all the reflected molecules condense after the very first impact on a sphere of radius \(R\), at whose center the reflecting surface is situated (Budd’s experiment), then the density of the deposit on an element of the surface \(dS\) of the sphere, cut out by a cone of solid angle \(d\Omega\), will be

\[ \rho = \frac{k \cos \varphi\, d\Omega}{dS} = \frac{k \cos \varphi\, d\Omega}{R^2 d\Omega} = A \cos \varphi . \]

That is, $\rho$ will be proportional to the cosine of the angle between the perpendicular to the reflecting surface and the vector connecting the center of this surface with the point on the sphere where the density is measured. Of course, this reasoning is valid under the assumption that the reflecting surface is small in comparison with the radius $R$ of the sphere. In Wood’s experiments this, strictly speaking, was not the case. But it could be shown by calculation that, under the conditions that obtained in Wood’s experiment, the influence of the error introduced by the finite dimensions of the surface was nevertheless very small.

By photometering the cadmium deposit, Wood showed that the cosine law is justified. However, the photometering was associated with large errors. An exact verification of the cosine law belongs to Knudsen, who, working with mercury, determined the amount of condensed mercury in different parts of the little sphere by direct weighing of the mercury drops after the liquid air had been removed and the apparatus had filled with air. In doing so he made use of the following ingenious consideration. If the reflecting surface is some element $dS$ of the inner surface of the little sphere (and not the center, as in Wood), then, as is not difficult to show by a simple geometrical argument, the cosine law distributes the deposit of reflected molecules uniformly over the whole sphere. And it is much easier to establish the uniformity of the deposit than to verify the cosine law under Wood’s conditions. In practice, Knudsen heated from outside, with a metal rod, one element of the little sphere immersed in liquid air—precisely the one on which the beam fell.

Knudsen further clarified the question of the degree of accuracy with which one may assume that the surface of glass at a temperature of $-140^\circ\mathrm{C}$ actually absorbs every metal molecule that strikes it. He carried out experiments with cadmium, mercury, and zinc. He worked with a special, rather complicated apparatus; the surface on which the beam fell was the outer wall of a test tube sealed into the interior of the little sphere. By pouring liquid oxygen or various cooling mixtures into this test tube, it was possible to maintain the reflecting surface at any strictly defined temperature (with the aid of a toluene thermometer).

He found that at $t=-140^\circ\mathrm{C}$, for all metals, on average, no more than two out of 10,000 molecules incident on the surface are reflected. The same value, in Knudsen’s opinion, is the number of molecules reflected from a layer of metal, but already at room temperature. This latter conclusion, however, is rather doubtful. Thus, in one of the experiments with a metallic surface, Knudsen himself obtained the number of reflected molecules as 10 per 100 incident.

In the case of a mercury beam, Knudsen carried out a series of experiments at various temperatures of the reflecting surface, and in the experiment at $t=-77.5^\circ\mathrm{C}$ (with carbon dioxide and ether in the test tube) the surface

glass reflected just as well as at room temperature, i.e. almost completely. Carrying out a series of measurements between \(-140^\circ\text{ C}\) and \(-77.5^\circ\text{ C}\), Knudsen expected to find a gradual increase in the coefficient of reflection from a value close to zero at \(-140^\circ\text{ C}\) to a value close to unity, found for \(-77^\circ\text{ C}\). In fact, however, it turned out otherwise: down to a temperature of \(-130^\circ\text{ C}\) the coefficient of reflection remained \(=1\) (according to the accuracy of Knudsen’s measurement, no more than two out of 10,000 incident molecules stuck). At \(-140^\circ\text{ C}\), on the contrary, the coefficient of reflection became equal to 0; more precisely, out of 10,000 incident molecules no more than 2 were reflected. Knudsen did not investigate the interval between \(-140\) and \(-130^\circ\). He thus did not succeed in finding the transition region. If it exists, then in any case it is narrower than \(10^\circ\text{ C}\). Thus, for the surface of glass, in the sense of the reflection of nonmetallic molecules incident upon it, there exists, as it were, a certain “critical temperature,” above which all molecules are reflected, while below it all stick. For cadmium, zinc, and magnesium Knudsen established only the interval in which the critical temperature lies. The same he did in special experiments for silver and copper. We give his data:

If the surface on which the beam falls is at a temperature higher than the critical one, no trace of the beam appears. On the basis of this phenomenon Wood also determined a number of critical temperatures. His results are as follows:

However small the probability may be that molecules stick to glass at a temperature above the critical one, it evidently has some value; and it is precisely this circumstance that makes understandable the always observed phenomenon of vapor condensation on walls. Because of the small coeffi-

...the coefficient of reflection of metal from metal, a molecule that has accidentally adhered becomes a center of vapor condensation. In the works of Wood and Knudsen there are indications that, at temperatures above the critical one, the metallic deposit, when examined under a microscope, consists of crystallites separated from one another, whereas at temperatures below the critical one the deposit is continuous or almost continuous.

§ 2. Oriented growth of crystals.

Folmer \[6, 7, 8, 9\] carried out a series of experiments to study the condensation of a molecular beam on a glass plate whose temperature was above the critical one. He performed his work mainly with cadmium and partly with zinc and mercury. The glass plate on which the beam falls was placed at various angles to the direction of the beam. During the first few minutes of the experiment no image of the beam was obtained on the plate; because of reflection from the walls of the vessel, the whole plate was covered with a faint film of condensing metal. However, after the walls had become covered with a light layer of cadmium, they (in agreement with Knudsen’s results) ceased to reflect, and on the plate a film of metal began to grow only in the place on which the beam fell.

Fig. 2.

Examining the plate after the experiment, Folmer found that, generally speaking, the matte film of metal, at a certain angle of view, suddenly begins to reflect light like a mirror. This angle was completely independent of the position of the plane of the plate with respect to the direction of the beam, but depended only on the direction of the beam. Microscopic investigation of the deposit showed the cause of such an astonishing phenomenon. It turned out that the film consists entirely of individual small crystals having the form of prisms (cadmium crystallizes in the hexagonal system). The edges of these prisms are directed toward the beam. The upper face of these prisms is perpendicular to the edges and, consequently, perpendicular to the direction of the beam. This face is very regularly formed, and it is this face that reflects the light. It is therefore natural that the angle of reflection observed at first corresponds to a mirror with a plane perpendicular to the beam, and does not depend on the orientation of the plate. The role of this plate is purely passive; it cuts the prisms at an angle to their axis, as shown in Fig. 2.

Why, in the case of a molecular beam, can the crystals grow only with one definite face, corresponding to the base of the hexagonal prisms? Undoubtedly, the embryonic crystals may be oriented arbitrarily with respect to the beam, and, it would seem,

any face could, with the same success, grow toward the beam. Analyzing this question, Volmer comes to the following conclusion: the probability \(\alpha\) of adhesion of a molecule to a face of the crystal corresponding to the base of the prism is close to 1, whereas for the other faces it is much smaller. From the other faces the molecules are more often reflected than adhere. Therefore, of all possible embryonic crystals, only those develop which have a face—the base of the prism—situated perpendicular to the beam.

It cannot be said that such a conclusion was a direct consequence of Volmer’s experiments, but it is the only explanation so far available for the results of the phenomenon under consideration. Observing how the crystallites grow, Volmer noticed that at first they grow much faster in width than in height, while, however, all the time retaining a properly formed face perpendicular to the beam. This growth in width is at first almost 1,000 times greater than the maximum possible theoretical value. Having reached a certain size, the crystal begins to grow ever more slowly in width and, finally, ceases altogether. Correspondingly, the growth in height becomes ever greater, finally reaching a certain definite value corresponding to that calculated theoretically under the assumption that \(\alpha = 1\). To explain this phenomenon Volmer also makes the following supposition. An atom of the metal, arriving at the surface of the crystal, does not at once enter organically into the composition of the crystal, does not at once find its place in the space lattice, but first enters with the surface of the crystal into a certain intermediate bond, of approximately the same character as the bond of gases absorbed by a surface in the phenomenon of gas adsorption. In this state it can move over the surface and, when the crystal is of small dimensions, reaches the boundary of the surface and there already enters definitively into the crystal lattice, causing growth of the crystal in width. When the crystallite is sufficiently large, it does not manage to reach the boundary, since before that it has time to bind organically with the crystal on the plane of the main face, causing growth in height. This, in itself very hypothetical, point of view finds confirmation in the conclusions which Langmuir \([10]\) drew from his experiments on adsorption and condensation. But this will be discussed below.

§ 3. Attempts to Clarify the Mechanism of the Phenomenon of the Critical Temperature of Condensation.

We turn to clarifying the question of what, then, is the mechanism of the remarkable phenomenon discovered by Wood and Knudsen. In the work just cited, Langmuir arrives at the following conception of the mechanism of reflection of molecules from the surface of a solid—

bodies1. In his opinion, between the moment when a molecule strikes a surface and the moment when it leaves it—when, as has been said up to now, it is “reflected” from the surface—there passes some finite time $\tau$. Upon impact the molecule becomes bound to the surface and remains there until, by an accidental thermal jolt of a molecule of the solid body, our molecule is pushed out from the surface. This is what we take to be reflection. The phenomenon is entirely analogous to the behavior, for example, of a metallic vapor over a metal. Here no one, of course, assumes that the vapor molecules are always immediately reflected when they strike the surface of the metal. On the contrary, we consider that in this process they condense, but that in return some other molecule flies out of the metal. Thus a certain mobile equilibrium is established, when the number of condensing molecules is equal to the number of evaporating ones, and this determines the elasticity of the vapor. If we artificially remove the vapor, then the number of condensing molecules will be less than the number of evaporating ones, and the metal will continue to evaporate until it has all flown off.

According to Langmuir, a completely analogous phenomenon also occurs when a metal molecule strikes an inert substance, for example glass. It must only be said that very thin layers of metal evaporate extremely rapidly; this indicates that the bond between the atoms of the metal and the glass is much weaker than the bond of the atoms of the metal with one another. Consequently, the time during which an atom of the metal remains bound to the surface of the glass is much shorter than the time of its bonding with the metal. It remains also to point out that the rate of evaporation, as is known, increases very strongly with increasing temperature.

Thus, let us suppose that we have a metallic beam falling upon the surface of glass having temperature $T$. If the number of metal atoms evaporating per unit time from the surface of the glass at temperature $T$ is less than the number of atoms brought per unit time by the beam, then, obviously, we shall observe no deposit and shall consider that all the molecules are reflected as a whole. Only with sufficiently long exposures may it turn out that two or several metal atoms meet accidentally on the surface of the glass and, owing to their stronger mutual bond, remain longer on the surface. During this time several more atoms may have time to adhere to them, and thus a crystallite will begin to grow. Such, probably, is the origin of the crystals observed by Folmer.

Let us now imagine that, with the very same intensity of the beam, we have lowered the temperature of the glass plate so much that

that the number of evaporating atoms has become slightly smaller than the number of incident ones. Then, at every instant, there will be on the plate a certain number of metal atoms; the atoms arriving from the beam will bind with these metal atoms and, owing to the strength of the metal–metal bond, will not evaporate. The newly arriving atoms will continue to increase this metallic deposit, until the entire plate is completely covered with a very finely crystalline layer of metal, which thereafter will grow without hindrance, since at this temperature the rate of evaporation of metal from metal is negligibly small. Thus, such a temperature, at which the rate of evaporation of the metal from glass will be less than the rate of condensation of the metal from the beam, will possess all the properties of the critical temperature of Wood and Knudsen. However, with this interpretation the value of this temperature will depend on the intensity of the beam, i.e., on the number of molecules brought by the beam to unit surface area, and will be the smaller the lower the intensity. No fact of this kind was observed either by Knudsen or by Wood.

Recently I have carried out experiments \[12\] which apparently confirm Langmuir’s point of view. At the same time Ya. I. Frenkel \[11\] developed Langmuir’s point of view theoretically and arrived at numerical results coinciding with the data obtained by me.

Setting ourselves the task of determining the critical temperatures exactly, we, on the advice of Academician A. F. Ioffe, used the following method. A copper plate was taken, \(1\ \mathrm{cm}\) wide and \(10\text{--}15\ \mathrm{cm}\) long. Opposite its middle, at a distance of \(3\text{--}10\ \mathrm{mm}\), a nichrome wire was placed, on which a layer of the metal under study (\(Cd, Zn\)) was deposited electrolytically. In Fig. 3 the middle part of the plate is shown enlarged; in the middle one can see, out of focus, the nichrome wire coated with a layer of metal. These metallic parts were placed inside a test tube immersed in liquid air; from the upper part the air was pumped out by powerful pumps—so that the air pressure in the apparatus did not exceed \(10^{-5}\ \mathrm{mm}\). The lower end of the plate rested on the bottom of the test tube, into which a small quantity of mercury had been poured. When immersed in liquid air the mercury froze and, through thermal conductivity, kept the lower end of the plate at a temperature close to \(-140^\circ\mathrm{C}\).

At the upper end of the plate a small nichrome oven was wound, heated by a current, which maintained the temperature of the upper end at a certain temperature. Usually we choose a temperature of about \(10^\circ\mathrm{C}\). Thus, the lower end of the plate was at a temperature of \(-140^\circ\mathrm{C}\), and the upper at \(10^\circ\mathrm{C}\). Along the plate, owing to thermal conductivity, there occurred a drop in temperature from \(10^\circ\) to \(-140^\circ\). A series of thermoelements soldered to the outer side of the plate showed,

that the temperature drop along the plate is almost exactly uniform. By heating a nichrome wire with a current, we heat the layer of $Cd$ or $Zn$ covering it, and the molecules of the evaporating metal fly off in all directions from the surface of the wire. In doing so they may reach either the surface of the plate or the inner glass walls of the test tube. Since the test tube is immersed in liquid air, the glass surface has a temperature known to be below the critical one; therefore every metal molecule that reaches the glass adheres to it. As for the molecules that reach the surface of the plate, if they reach the upper parts of the plate, where the temperature is above the critical one, they are reflected and, after being reflected, are absorbed by the glass; but if they reach the lower parts, where the temperature is below the critical one, they adhere, forming a metal layer. Knudsen and Wood, only for $Hg$, determined that the temperature interval above which the molecules are reflected from the glass and below which they adhere does not exceed $5^\circ$. For $Cd$ and $Zn$ this interval was determined with much less accuracy. With our method we could clarify the question of the size of the temperature interval in which both reflection and adhesion may occur, by the degree of blurring of the boundary of the deposit.

Fig. 3.

In addition, by covering the copper plate with thin layers of pitch, paraffin, and mica1, we were able to investigate the phenomenon of critical temperature for a number of new substances besides glass. The first experiments were made with pitch, on whose black surface the metal layer is especially clearly visible. The photograph (Fig. 3) shows the result we obtained. The metal layer had a perfectly sharp and distinct boundary. The transition region extended no more than $1^\circ$–$2^\circ$. However, the following remarkable phenomenon immediately caught our eye. The boundary was not a straight line, but an arc whose convexity was directed upward. Such a direction of convexity is opposite to what might have been expected under the assumption that the wire heats the surface of the plate by radiation, this heating being greater in the middle than at the sides.

Changing the relative position of the wire and the plate, we found that the convexity of the arc increases as the filament approaches the plate and becomes asymmetric when the wire is placed—

is displaced not opposite the middle of the plate, but toward the edge, while the upper part of the arc is always opposite the wire.

The only explanation of this phenomenon should be considered to be the assumption that the critical temperature depends on the number of molecules falling on the surface, i.e. on the density of the molecular flux1, which corresponds to the viewpoint of Langmuir and Frenkel, set forth below. The shape of the arc calculated on this assumption from Frenkel’s theory generally agrees well with that obtained from experiment. We tried to verify this result directly by strongly varying the intensity of the flux (by heating the wire) and determining the critical temperature of the upper part of the arc. Here, however, we encountered great difficulties: at very low intensity the deposit either does not appear at all or is extremely indistinctly expressed; at very high intensity the entire plate is completely covered with deposit. The latter is evidently connected with the high probability of encounters of two or several metal molecules in different parts of the plate and with the consequent appearance of crystallization centers (as was the case with Volmer) even at temperatures below the critical one. In general it turned out that appearance in the form shown in the photograph is produced only within very narrow limits of variation of the flux intensity; this probably explains the rather small fluctuations, determined by us from different experiments, in the critical temperature corresponding to the upper boundary of the arc. Besides the intensity of the flux, the exposure time plays some, not yet fully clarified, role in the good reproduction of the phenomenon.

We observed exactly the same phenomenon in the deposition of Cd and Zn on mica and paraffin. Covering half of the plate with paraffin and half with mica, we noticed that the boundary on the mica is considerably higher than on the paraffin, this difference corresponding to a difference of \(10^\circ\mathrm{C}\) in the critical temperatures of deposition of Cd and Zn on mica and paraffin. In conclusion I shall point out that, at the request of Ya. I. Frenkel, we carried out an experiment in which the position of the wire during the experiment was determined as carefully as possible; then, by weighing the wire before and after the experiment, the number of evaporated molecules was determined, and, in addition, the critical temperature \(T_{max}\) at the top of the arc and \(T_{min}\) at the edge was measured. Using our data, Frenkel calculated, by his formulas, the value \(T'_{max} - T_{min}\), which almost exactly coincided with the value measured by us (\(5^\circ\mathrm{C}\)).

§ 4. DIRECT DETERMINATION OF THE VELOCITIES AND FREE PATHS OF MOLECULES

The idea of a direct experimental determination of the mean velocity of molecules is as follows.

Let us imagine a small closed vessel \(G\) (see Fig. 4), filled with gas. Let the temperature of the vessel be \(T\). The vessel \(G\) is placed inside another large vessel \(V\), from which the air has first been pumped out. A small circular opening \(L\) has been drilled in the vessel \(G\). Then, from this opening, gas molecules will fly out in all directions with velocities corresponding to the temperature \(T\). The circular diaphragm \(B\) selects from this stream of molecules a narrow beam, which falls on the plate \(P\) near the point \(a\) and condenses there, in the form of a sharp circular spot with its center at the point “\(a\).” Such a phenomenon, as we know, will occur if the temperature of \(P\) is below the critical temperature of deposition of the molecules under consideration on the given surface.

Fig. 4.

If we now set the whole apparatus into rotational motion about an axis coinciding, for example, with the perpendicular to the plane of the drawing, erected at the point \(L\), then the spot on the plate \(P\) will be displaced in the direction opposite to the rotation. Indeed, during the time while the molecule flies the distance \(l\) from \(L\) to \(P\), i.e.

\[ \tau=\frac{l}{v}, \]

the point “\(a\)” will move through the distance \(s=2\pi \nu \tau\), and the molecule will meet the plate \(P\) at a point separated from \(a\) by the distance \(s\). Here by \(v\) we denote the velocity of the molecule, and by \(\nu\) the number of complete revolutions of our apparatus per second. According to Maxwell’s law of distribution of velocities, the velocities of gas molecules at any temperature \(T\) are very diverse. However, most molecules possess velocities grouped very closely about the so-called root-mean-square velocity, which is entirely determined by the temperature of the gas and can easily be calculated. Therefore, in what follows we shall assume that all molecules possess the same velocity \(v\).

In that case the spot produced by the molecules depositing on \(P\), while retaining its shape, will be displaced during the rotation of the apparatus so that the center of the displaced spot will be separated from “\(a\)” by the distance

\[ s=\frac{2\pi \nu l^{2}}{v}. \]

By measuring this distance \(s\) and knowing the number of revolutions per second \(\nu\) at which such a displacement has occurred, it is easy to find \(v\).

In essence, the experiment actually carried out by Stern \([13,14]\) (1920) differed only in that, instead of the emitting

molecules from a small circular aperture, a linear source of molecules was used in the form of a thin platinum wire, on which a layer of silver had been deposited (\(L\) in Fig. 5). When the wire was sufficiently heated by an electric current, the silver evaporated, and in this way the emission of molecules by a thin linear source was effected.

As is known, the velocity of molecules is determined only by the temperature and molecular weight; it does not depend on the pressure or on the state of aggregation.

In addition, in Stern’s experiment, instead of one diaphragm two were used, likewise not circular but slit-shaped; Stern’s apparatus is shown schematically in Fig. 5.

It should be noted, however, that such a replacement of the circular source by a linear one causes an essential difference in the calculation. Indeed, since the molecules fly not only in directions perpendicular to the wire, but also in all other directions, then, depending on the angle that the direction of their velocity makes with the wire, they will take different times to fly from the source to the plate \(P\); the corresponding value \(l\) for them will obviously be different. Instead of a sharp strip of deposited silver when the apparatus is at rest, rotation will produce a blurred picture; the calculation will also be greatly complicated. However, if the length of the wire emitting the silver molecules is considerably less than the distance \(l\) to the plate, then the influence of the indicated cause will be small. In Stern’s experiments, where \(l = 6\ \mathrm{cm}\), and the length of the emitter was \(1.5\ \mathrm{cm}\), the magnitude \(s\) could vary for different directions of molecular velocities only within \(3\%\), which was considerably less than the other experimental errors, reaching \(10\text{–}15\%\).

Fig. 5.

Very great difficulties had to be overcome by Stern in order to attain, inside the measuring vessel, a vacuum of \(0.0001\ \mathrm{mm}\) while individual parts of the apparatus were rotating at speeds up to 2700 revolutions per minute.

It has already been indicated above that the critical temperature of adhesion of silver both to metallic and to dielectric surfaces is apparently above room temperature. Therefore, with the apparatus at rest, Stern observed on the plate, under the action of the emitter, the appearance of a sharp and thin (width \(0.3\ \mathrm{mm}\))

strips of deposited silver. When the instrument was rotated, the strip was likewise sufficiently sharp and of almost the same width, but it was displaced, in comparison with the first experiment, by 0.4–0.6 mm. Since this is a very small quantity, Stern, for greater accuracy in determining the magnitude, carried out the experiment in the following sequence. First he obtained a deposit while rotating the instrument in one direction, then in the other, with the same number of revolutions. In this case the distance between the strips was doubled (Fig. 6). This distance between them was: at 1500 revolutions per minute—0.8 mm, at 2400 revolutions—1.12 mm, at 2700 revolutions—1.26 mm. The corresponding values of \(s\) are obtained by division by 2. Hence for \(v\) the following values are obtained: \(v_1 = 640 \dfrac{m}{\text{sec.}}\), \(v_2 = 643 \dfrac{m}{\text{sec.}}\), \(v_3 = 675 \dfrac{m}{\text{sec.}}\), constant within 5%. All these experiments were carried out at one and the same temperature of the wire, \(1200^\circ C\). This temperature was determined with an optical pyrometer.

Fig. 6.

The root-mean-square speed under the conditions of effusion of molecules into a void is

\[ = \sqrt{\frac{4RT}{M}}, \]

where \(R\) is the gas constant, \(M\) is the molecular weight, \(T\) is the absolute temperature. If it is assumed that silver vapor, like other more volatile metals, is monatomic, then \(M\) is equal to the atomic weight of silver.

Calculating \(v\) for \(T = 1200 + 273 = 1473\), we obtain \(672 \dfrac{m}{\text{sec.}}\), i.e., a quantity in close agreement with the experimental data. It should, however, be noted that, in my opinion, the speed determined by Stern must differ by 20% from the root-mean-square speed. According to the calculation I have made, the speed determined by Stern is simply the arithmetic mean speed of the usual equilibrium Maxwell distribution of velocities. It goes without saying, however, that these experiments of Stern are connected with very large errors both in measuring the distance between the strips, which is only three times the width of a strip, and in determining the temperature of the filament, which, by Stern’s own admission, is not uniform along the whole wire because the silver melts and gathers into droplets. Therefore one cannot expect precise quantitative results. But the obtaining of the correct order of magnitude and the sharpness of the effect allow one to hope that in the future this method will provide physicists with a powerful tool. With the aid of this method it will be possible to investi-

to investigate the masses of different molecules, i.e. their composition at high temperatures and low pressures. Since different molecules give different deflections, one can register all the kinds of molecules present.

Further, by using a microphotometer to investigate the distribution of the thickness of the silver layer along the width of the strip, it would be possible to verify experimentally Maxwell’s law of the distribution of velocities. The discovery of deviations from Maxwell’s law could shed light on the very difficult and controversial question of whether, and in what way, quantum theory should be applied to the rectilinear motion of atoms and molecules.

Almost simultaneously with Stern, Born \[15\] showed that, with the aid of a molecular beam, one can directly find the mean free path of a molecule in a gas. This quantity, too, had never before been measured by a direct method.

At the end \(S\) (see Fig. 7) of the quartz tube \(Q\) a piece of silver is placed. When \(S\) is heated by an electric furnace, the silver begins to evaporate. In this way \(S\) is a source of silver atoms. At first the experiment is carried out in a good vacuum. The diaphragm \(R\) selects from this stream of atoms a beam which, on traveling farther, passes in part through the diaphragms \(P_1, P_2, P_3, P_4\) (all the \(P\)’s are made of brass). The number of atoms \(n_{20}\) that have passed through the diaphragm \(P_2\) will be smaller than the number of atoms \(n_{10}\) that have passed through the diaphragm \(P_1\), owing to the circumstance that the beam of atoms is divergent and part of the atoms that have passed through \(P_1\) strike the metallic edges of the plate \(P_2\) (near the aperture) and condense there. Thus, by the conditions of the experiment, \(n_{10} > n_{20} > n_{30} > n_{40}\) (the subscript \(0\) refers to measurements in a vacuum). We now pass to the case where the vessel \(C\) is filled with gas (in Born’s experiments, air at pressures of about \(10^{-9}\) mm).

Fig. 7.

From the kinetic theory of gases it is known that, out of the number \(n_0\) of atoms which could reach a given surface in a complete vacuum, the number of those which will reach it without collision at a gas pressure \(p\) will be

\[ n_1 = n_0 e^{-\frac{x_1}{\lambda}}, \tag{a} \]

where \(x\) is the distance from the source of atoms to the given surface, and \(\lambda\) is the so-called mean free path of an atom in the surrounding gas. According to the kinetic theory,

\[ \frac{1}{\lambda}=\pi N\sigma^{2}\sqrt{1+\frac{m'T'}{mT}}, \tag{b} \]

where \(N\) is the number of gas atoms per unit volume, \(\sigma\) the sum of the radii of the atom and the molecule of the gas filling the vessel, and \(m, m', T, T'\) are the masses and temperatures of the gas molecules and of the silver atoms. Since \(N\) is proportional to the pressure of the gas, \(p\lambda\) must be a constant quantity. The correctness of the relation \(p\lambda=\mathrm{const}\) is confirmed by a number of facts, such as: the magnitude of internal friction, the thermal conductivity of the gas, etc. From these same quantities the numerical value of \(\lambda\) is computed by the formulas of kinetic theory. However, for cases in which there is the free path of one kind of molecule in a medium of others, as occurs under the conditions of Born’s experiment, we do not have data at our disposal for its calculation.

Fig. 8.

Thus, according to formula (a), the number of silver atoms passing through the diaphragm \(P_1\) in the presence of air at pressure \(p\) will be

\[ n_1=n_{10}e^{-\frac{x_1}{\lambda}}, \]

where \(n_{10}\) is the number of atoms passing through \(P_1\) in vacuum, and \(x_1\) is the distance from \(O\) to \(P_1\). Similarly,

\[ n_2=n_{20}e^{-\frac{x_2}{\lambda}}, \]

and so on. Knowing the numbers \(n_1, n_2, n_{10}\), and \(n_{20}\), one can calculate \(\lambda\). Indeed,

\[ \frac{n_1}{n_2}=\frac{n_{10}}{n_{20}}e^{-\frac{x_1+x_2}{\lambda}} \quad \text{or} \quad \ln\frac{n_1 n_{20}}{n_2 n_{10}}=\frac{x_2-x_1}{\lambda}, \]

whence we determine it. Born finds the quantities \(n_1, n_2, n_{10}\), and \(n_{20}\), etc., in the following way. On each of the plates \(P_1, P_2, P_3, P_4\) he pasted one glass plate in the form of a quadrant, the center of which was placed on the axis of the apparatus. The form and arrangement of the plates are visible in Fig. 8. The quadrant on \(P_2\) was shifted by \(90^\circ\) relative to the quadrant on \(P_1\). The quadrant on \(P_3\) was shifted by \(90^\circ\) relative to the quadrant on \(P_2\), and the same for \(P_4\). Thus, if one looks from \(S\), the projections of the quadrants form a full circle, entirely covering the surface of the diaphragms. Obviously, on each of the quadrants there falls \(1/4\) of the silver atoms passing through the given diaphragm in the absence of the quadrant. All atoms striking a quadrant condense, and from the thickness of the layer it was possible to determine numbers proportional to \(n_{10}, n_1, n_{20}\), and \(n_2\), etc. By eye...

it was seen how the blackening, almost the same in all quadrants in the case of a vacuum, changes abruptly at a gas pressure of \(10^{-4}\) mm; in this case the quadrant at \(b_1\), an almost completely black quadrant, at \(b_4\) reveals an almost complete absence of deposit. The thickness of the layer was investigated with a microphotometer. The values found for \(\lambda\) at different pressures satisfied the relation \(p\lambda=\mathrm{const}\). Substituting the value of \(\lambda\) into formula (b), Born obtained for \(\sigma\) (the sum of the radii of the silver atom and air) the value \(2.6\cdot 10^{-8}\). This quantity is not known from other data, but its order of magnitude is correct. The main difficulty of the method is the determination of the thickness of the precipitate.

§ 5. Direct determination of the magnetic moments of atoms.

Passing a molecular beam through a magnetic field, very inhomogeneous—that is, with a large gradient of the force \(H\) in directions perpendicular to the motion of the atoms—and observing the deflection of the beam from rectilinear motion, one can find the magnitude of the magnetic moment of the atom.

It is known that the phenomenon of paramagnetism can be explained by supposing that each atom or molecule of a paramagnetic substance is a small magnet. These magnets tend, under the action of an external magnetic field \(H\), to turn in the direction of \(H\). Thermal motion hinders such orientation. Ultimately, under the action of \(H\), a certain equilibrium position of the directions of the magnetic moments of the atoms is established, with the greater part of them turned toward \(H\), which causes the magnetization of the body.

At \(H=0\) the magnets are equally often directed both one way and the other; therefore a body in the absence of a magnetic field, generally speaking, does not exhibit magnetic properties. The greater \(H\) is, the more the orientation of the magnets in the direction of the field predominates; the higher the temperature \(T\) of the substance, the stronger are the impacts of the molecules disturbing the orientation, and the smaller is the magnetization \(J\) at the same \(H\). Proceeding from the hypothesis of molecular magnets, Langevin showed theoretically that \(J\) is proportional to \(\dfrac{H}{T}\), which agrees with experiment for all paramagnetic substances. Only for iron, nickel, and cobalt at large \(H\) are deviations observed, which is connected with the quite exceptionally large magnetic susceptibility of these substances. The magnitude of the coefficient of proportionality between \(J\) and \(\dfrac{H}{T}\), according to Langevin’s formula, is very simply related to the magnetic moment \(\mu\) of the molecule. Knowing the experimental value of \(J\), one can calculate \(\mu\), which for different substances turned out to be different, having the order \(10^{-21}\) CGSM.

Another method of determining the magnetic moment is suitable only for iron, cobalt, and nickel. The point is that for these bodies the quantity \(J\), at first increasing with increasing \(H\), then begins to increase more and more slowly and, finally, ceases to increase altogether, remaining constant upon further increase of \(H\). This phenomenon, known as magnetic saturation, from the point of view of the theory of elementary magnets must be explained as follows: at such large \(H\) all molecular magnets are turned completely in the direction of \(H\) and thus exhibit the maximum possible magnetization. If \(n\) is the number of molecules in \(\mathrm{cm}^3\), and \(\mu\) their magnetic moment, then \(J_{\max}=n\mu\), whence \(\mu\) can be found. Since the advent of the electronic theory of atomic structure it at once became clear why atoms may possess a magnetic moment. Indeed, moving in a closed orbit around the nucleus, an atomic electron, in the sense of magnetic action, is entirely equivalent to a closed current whose circuit coincides with the orbit, and the current strength is \(i=\dfrac{e}{\tau}\), where \(e\) is the charge of the electron, and \(\tau\) is the period of revolution in the orbit. It is well known that, in the sense of magnetic actions, a closed current is completely equivalent to a magnet whose magnetic moment is \(\mu=is\), where \(s\) is the area of the current circuit; thus \(\mu=\dfrac{es}{\tau}\), where \(s\) is the area of the orbit.

When there are many electrons in an atom, each of the orbits \(N_k\) has its own magnetic moment \(\mu_k\). The total resultant moment of the atom is \(\mu=\sum \mu_k\), this sum being geometrical. Apparently, the electrons in the atom are arranged so that the magnetic moments of them compensate one another, so that the total moment of the atom is 0 and the body is not magnetic. However, when the number of electrons is odd, such compensation usually cannot occur, and as a result the atom possesses a magnetic moment \(=\dfrac{es}{\tau}\), produced by the electron farthest from the nucleus. About 10 years ago Barnett in America and Einstein in Germany, almost simultaneously, proved by direct experimental means that it is precisely the rotation of electrons that creates the magnetic moment of atoms.

It must be noted, however, that a more careful theoretical analysis of the question from this point of view revealed such difficulties in the theoretical respect and such discrepancies with the experimental material on magnetization that the domain of the theory of magnetism proved to be one of the most confused and unfortunate domains of physics. The very formula of Langevin, from which the value of \(\mu\) was determined, was called into question. Therefore it was especially important and necessary to find a method for the direct measurement of the magnetic moments of atoms. The idea of applying a molecular beam for this purpose was first expressed by the author of this article in 1920. The calculation of the experiment was made by him jointly with P. L. Kapitza and sent in December 1920 to Ger—

...for publication in scientific collections issued in Berlin by the Scientific-Technical Department of the Supreme Council of the National Economy [16]. Unfortunately, this article was printed only in 1922, after a more detailed analogous calculation, made by Stern, had appeared [17].

Let us suppose that we have a magnetic field produced by the prismatic pole pieces of an electromagnet of the form shown in Fig. 9, where the cross-section of the pole pieces is given by the plane \(xy\), perpendicular to the edges of the prism. Let the circle \(O\) represent the cross-section of a molecular beam flying inside a cylinder perpendicular to \(O\), i.e. parallel to the edges of the prisms. How will the magnetic field act upon the motion of a freely flying molecule?

Fig. 9.

It follows from Larmor’s theorem that, if the magnetic moment of the molecule is produced as a result of motion along the orbits of intra-atomic electrons, then, in the absence of collisions with other molecules, the magnetic field will act in a very peculiar way upon the direction of the magnetic moment of the molecule. This action is shown in Fig. 10. If \(H\) is the direction of the magnetic field, and \(\mu\) the direction of the magnetic moment when the molecule enters the magnetic field, then, in its further motion, the electronic orbit begins to precess about the direction \(H\), just as a top precesses about a vertical direction; as a result of this precession the vector representing the magnitude and direction of \(\mu\) will rotate about the axis \(H\), describing a circular cone, as shown in the drawing. The component of \(\mu\) along the axis \(H\) will remain the same all the time, while the components along the two axes perpendicular to \(H\) will change, taking on now positive, now negative values, and, on the average over a large number of revolutions, will therefore be equal to 0.

Fig. 10.

Molecules whose magnetic moment \(\mu'\) made an obtuse angle with \(H\) will behave in exactly the same way, but their component \(\mu'\) along the axis \(H\) will, obviously, be directed opposite to \(H\). Since, on entering the magnetic field, the \(\mu\)’s were directed equally often both in the direction of \(H\) and in the opposite direction, the field will not act upon freely flying molecules as it does upon molecules in solid, liquid, and gaseous media, i.e. it will not orient them predominantly in the direction of \(H\). Thus, in studying the forces acting on molecules in a magnetic field, we may consider that half of them are directed along \(H\), and half—

in the opposite direction, but the magnetic moment of each molecule is $\mu_1=\mu\cos\vartheta$, where $\vartheta$ is the angle between $\mu$ and $H$. It is well known that a magnet placed in a uniform field $H$, so that $\mu_0$ coincides with or is opposite to the direction of $H$, experiences no force causing it to move. If, however, it is placed in a nonuniform field, and the magnitude of this change of field in the direction of $H$ is determined by the gradient $\dfrac{\partial H}{\partial x}$, then on the magnet as a whole there will act a force

$\mu_0=\dfrac{\partial H}{\partial x}$

and directed toward the increase of the field, if $\mu_0$ coincides with the direction of $H$, and conversely, if it does not coincide. With the arrangement of the pole pieces shown in Fig. 9, the field is very nonuniform at the point $O$ and has a large gradient $\dfrac{\partial H}{\partial x}$. The direction of $H$ at the point $O$ coincides with the direction $x$. Thus, the molecules of the beam will experience forces $=\mu_0\dfrac{\partial H}{\partial x}$, directed partly toward positive $x$, partly toward negative $x$. If the length of the prisms is $l$, then the deflection of a molecule under the action of the field from its path in the absence of the field, upon leaving the field, will be equal to

$s=\dfrac{\mu_0}{m}\cdot\dfrac{\partial H}{\partial x}\cdot\dfrac{t^2}{2}$,

where $m$ is the mass of the molecule, and $t$ is the time of flight in the field $=\dfrac{l}{v}$, where $v$ is the velocity of the molecule, i.e.

$s=\dfrac{\mu_0 l^2}{2mv^2}\dfrac{\partial H}{\partial x}$.

If the temperature of the substance emitting the beam is $T$, then

$2mv^2=6kT$,

where $k=\dfrac{R}{N}$ ($R$ is the gas constant, and $N$ is Avogadro’s number). Finally,

$s=\dfrac{\mu_0 l^2}{6kT}\dfrac{\partial H}{\partial x}$.

Since $\mu_0=\mu\cos\vartheta$, and since $\vartheta$ can vary from $0$ to $\dfrac{\pi}{2}$ and from $\dfrac{\pi}{2}$ to $\pi$, $S$ will take all values from $0$ to

$s_0=\dfrac{\mu l^2}{6kT}\dfrac{\partial H}{\partial x}$.

As a result, if without the magnetic field the molecular beam gave a trace in the form of a small circle, under the action of the field the small circle will spread into a strip whose length will be

$=2s_0=\dfrac{\mu l^2}{3kT}\dfrac{\partial H}{\partial x}$.

By measuring $\dfrac{\partial H}{\partial x}$ and $2s_0$, one can find $\mu$.

Such, approximately, was the calculation carried out by P. L. Kapitza and myself in 1920.

Stern, at the end of 1921, carried out the same calculation, but also took into account the results of applying quantum theory to this case. According to this theory, the motion of an electron inside an atom is subject to the condition that its angular momentum be equal to

$M=n\dfrac{h}{2\pi}$,

where $n$ is an integer, and $h$ is Planck’s constant. The magnetic

the moment \(\mu\) is connected with the quantity \(M\). First, since \(M\) and \(\mu\) are perpendicular to the plane of the orbit, their directions coincide; and, second, in magnitude

\[ \mu = M \frac{e}{2m},^{1)} \]

where \(m\) is the mass of the electron, and \(e\) its charge. Hence

\[ \mu = n \frac{h}{4\pi}\frac{e}{m}, \]

where \(e\), \(m\), and \(h\) are universal constants, and \(n\) is an integer.

Further, the precessional motion of the orbit about the direction of the magnetic field must likewise be restricted by a quantum condition, as was shown by Sommerfeld. We cannot enter here into a more detailed discussion of this question, and shall indicate only the results. It turned out that, if \(n=1\), then there can be only two directions of \(\mu\) in the magnetic field: one along the direction of the field \(H\), the other opposite to the field. If \(n>1\), then there may be several such directions. Thus, in the case \(n=1\), \(\mu\) is directed either along \(H\) or against it. This will evidently change the results of the preceding calculation in the sense that \(\cos \vartheta\) will be equal only to \(\pm 1\). Then, instead of a band, we shall obtain two spots, similar to the primary one and separated from one another by the same amount

\[ 2s_0 = \frac{\mu l^2}{3kT}\frac{dH}{dx}. \]

Stern and Gerlach \([18, 19, 20]\) were the first to carry out experiments for the direct determination of \(\mu\) on the basis of the calculations indicated above. The experiment was performed with a silver beam\(^{2)}\) and pole pieces similar to those shown in Fig. 9. In the absence of a field, a trace of the beam about \(0.6\) mm wide was obtained. In the presence of a field, a splitting of the beam into two was observed, the two being separated from one another by a distance of about \(0.2\) mm. In this way the result of the quantum theory was directly confirmed. Having measured \(\dfrac{dH}{dx}\), it was possible, from the formula

\[ 2s = \frac{\mu l^2}{3kT}\frac{dH}{dx}, \]

to determine \(\mu\), which turned out to agree with the theoretical value

\[ \frac{he}{4\pi m}. \]

The authors estimate the experimental errors at \(10\%\). These results may be regarded as one of the most brilliant confirmations of the quantum theory.

Unfortunately, the difficulties of the experiment apparently proved so great that, apart from silver,\(^{2)}\) the authors did not succeed in investigating any other substance.\(^{3)}\)

\(^{1)}\) We shall show this for a circular orbit. \(\mu = Si = \dfrac{Se}{\tau} = \dfrac{\pi a^2 e}{\tau}\); \(M = mva = \dfrac{2\pi a}{\tau}ma = \dfrac{2\pi a^2 m}{\tau}\), whence \(\mu = M\dfrac{e}{2m}\).

\(^{2)}\) For silver, according to Bohr, the magnetic moment is created by a single outer electron, for which the quantum number \(n=1\).

\(^{3)}\) Recently there appeared a work of Stern and Gerlach\(^{21)}\), in which the experiments with silver were repeated, and new ones were made with copper, lead, and bismuth. For copper, in agreement with the theory, the same \(\mu\) was obtained as for silver. For bismuth and lead, splitting was likewise obtained, but the numerical results are unclear. It is surprising that, in bismuth, the beam is not split into two lying on both sides of the undeflected beam, but into two of which one coincides with the undeflected beam and the other is deflected.

CONCLUSION.

We have seen that not even 10 years have passed since studies of phenomena in the molecular beam first appeared. Thus this is one of the youngest branches of modern physics. We have seen that the molecular-beam method can serve for the direct determination of a number of atomic and molecular constants, such as the velocities of molecules, mean free paths, masses, the radii of atoms and molecules, and their magnetic moments. Moreover, it can make it possible to investigate phenomena connected with quantum theory; it is evidently destined to resolve the question of the mechanism of vapor condensation and, in particular, the mechanism of crystal growth. This is one of the methods of modern physics richest in prospects.

However, on the other hand, we have seen that it has not been possible to go further than preliminary results with this method. It contains such great experimental difficulties, such a new technique of measurement, that it does not yet yield to the efforts of experimentalists.

The impression is that the method is awaiting some new, ingenious idea in order to become easy, accessible, and precise; after that, its application will rapidly develop in the most diverse branches of physics, which will place it on a par with the principal methods of modern physics, such as X-ray analysis or the method of bombarding atoms with electrons.

But the appearance of such a new idea must evidently be preceded by persistent and careful investigations of the phenomena occurring in the beam, chiefly its condensation.

LITERATURE.

1) L. Dunoyer. C. Rend. 152, p. 592, 1911.
2) R. Wood. Phil. Mag. 30, p. 300, 1915.
3) R. Wood. Phil. Mag. 32, p. 364, 1916.
4) M. Knudsen. Ann. d. Phys. 48, p. 1113, 1915.
5) M. Knudsen. Ann. d. Phys. 50, p. 472, 1916.
6) M. Volmer. Zeitschr. f. Phys. 5, p. 31, 1921.
7) R. Gross und M. Volmer. Zeitschr. f. Phys. 5, p. 188, 1921.
8) M. Volmer und J. Estermann. Zeitschr. f. Phys. 7, p. 13, 1921.
9) M. Volmer. Zeitschr. f. Phys. 9, 1922 p. 193.
10) Langmuir. Phys. Rev. 8, p. 149, 1916.
11) Ya. I. Frenkel. ZS. f. Phys. 26, p. 117, 1924.
12) N. Semenov and Yu. Khariton. ZS. f. Phys. 25, 1924.
13) O. Stern. ZS. f. Phys. 2, p. 49, 1920.
14) O. Stern. ZS. f. Phys. 3, p. 417, 1920.
15) M. Born. Phys. ZS. p. 578, 1920.

16) P. Kapitza and X. Semenoff. Collection published in Berlin by the Scientific-Technical Department of the Supreme Council of the National Economy, 1922.

17) O. Stern. ZS. f. Phys., vol. 7, p. 149, 1921.

18) W. Gerlach and O. Stern. ZS. f. Phys., vol. 8, p. 110, 1921.

19) W. Gerlach and O. Stern. ZS. f. Phys., vol. 9, p. 349, 1922.

20) W. Gerlach and O. Stern. ZS. f. Phys., vol. 9, p. 353, 1922.

21) W. Gerlach and O. Stern. Annalen d. Phys., 1924.

1. Since the metal molecules are emitted by the wire uniformly in all directions, the flux density, i.e. the number of molecules falling on a unit surface of the plate, will be greater on that part of the plate which is opposite the wire, i.e. at the smallest distance from it, than at the edges. ↩↩↩↩↩