NEW INSTRUMENTS AND METHODS OF MEASUREMENT

MOLECULAR BEAM TECHNIQUE*)

I. Estermann

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2. Definition of the concept of a molecular beam . . . . . . . . . . . . . . 90
3. Sources of molecular beams . . . . . . . . . . . . . . . . . . . . . . . 92
4. Focusing systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5. Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6. Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7. Experiments with molecular beams . . . . . . . . . . . . . . . . . . . . 117
8. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
9. Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

1. INTRODUCTION

The study of corpuscular rays has played, and continues to play, a very important role in the development of modern physics. Investigation of a stream of moving particles is one of the most direct and—at least in principle—one of the simplest methods of obtaining information of one kind or another about the properties of elementary particles. The study of beams of charged particles began in 1879, when William Crookes, in his experiments on electrical discharge in gases at low pressures, discovered the emission by the cathode of rectilinearly propagating radiation and established its corpuscular character. The conclusion later reached concerning the nature of these cathode rays, as a stream of free electrons, is a significant milestone on the path of development of modern physics. Beams of positively charged ions were first observed and investigated by E. Goldstein in 1894, after which the technique of experiments with beams of charged particles began to develop rapidly. Beams, however, consisting of neutral particles—namely, atoms or molecules moving in straight lines with thermal velocities—were known even before 1911, when Dunoyer first obtained them^12. However, the importance of molecular beams—

) Reviews of Modern Physics 18*, 300, 1946. Translation by I. S. Shapiro and B. Erozolimskii.

…as a scientific instrument was shown only by O. Stern^60 in 1919. Thanks to his work, the molecular-beam method became an exceptionally powerful tool for investigating many fundamental problems of modern physics.

2. DEFINITION OF THE CONCEPT OF A MOLECULAR BEAM

According to the kinetic theory, a gas is an aggregate of a very large number of particles moving in space along zigzag trajectories consisting of straight-line segments. The motion of a particle along each straight-line segment of its path ends with its collision with another particle or with the walls of the vessel containing the gas. If a vessel \(A\) (Fig. 1), filled with gas and having a small orifice \(O\) in one of its walls, is placed in an evacuated space \(B\), then the molecules inside \(A\) whose velocities are directed toward the orifice will fly out through it and will move rectilinearly in the vacuum. Since in a gas the molecules move in all possible directions, the molecules that have passed through \(O\) fill a solid angle \(2\pi\). If, however, in the vacuum, in the path of the particles that have flown out, a screen with an orifice \(C\) is placed, then from all the molecules that have effused through \(O\) there will be selected those whose velocity direction is close to the direction of the straight line joining \(O\) and \(C\), or, more precisely, those that fall within the solid angle determined by the sizes of the orifices \(O\) and \(C\) and the distance between them. If the orifice \(O\) is so small that it may be regarded as a point source, then the molecules that have passed through \(C\) form a molecular beam consisting of particles moving along nearly parallel trajectories. Since the paths of the particles do not intersect, collisions between them will occur only in those cases when some faster molecule overtakes another moving along the same trajectory. Such cases, however, are very rare, and we may therefore define a molecular beam as an aggregate of a large number of molecules moving in a high vacuum, practically without collisions, along nearly parallel straight-line trajectories.

Fig. 1. Diagram of an apparatus for obtaining a molecular beam.

Everything said above applies equally both to beams of flying molecules and to beams of flying atoms. We shall therefore not use the special term “atomic beams,” considering such beams to be “molecular,” but composed of monatomic molecules. It should be emphasized that molecular beams consist of neutral particles moving with thermal velocities corresponding to

temperature of the source*). The magnitudes of these velocities range from \(10^4\) to \(10^5\) cm/sec. However, the particles of which the molecular beam consists need not necessarily be stable molecules; the beam may also be formed by molecules that under ordinary conditions have a short lifetime—for example, free radicals, hydrogen or oxygen atoms, particles in metastable or excited states, etc.

It is also necessary to note the distinction between the concepts of a molecular beam and a gas jet. In the latter case the flow of particles has a hydrodynamic character. In a jet, many collisions occur between molecules, and the rectilinear portions of the trajectory of each individual particle are extremely short. Therefore the jet will not preserve its geometrical shape, determined by diaphragms (for example \(O\) and \(C\) in Fig. 1), because of turbulent spreading in all directions from the edges of the diaphragms. In a molecular beam, by contrast, collisions are practically absent, and almost every particle flies in a straight line through the entire apparatus. Nevertheless, there is no sharp boundary between these two cases. The first case may be realized at a sufficiently high pressure of the gas located behind diaphragm \(O\), whereas in order to obtain a molecular beam this pressure must be sufficiently low. These conditions will be discussed in more detail in Section Three.

An apparatus for experiments with molecular beams consists at least of the following four parts: 1) a source, usually a vessel filled with a gas or vapor and provided with an opening for the effusion of molecules, 2) a collimating system—in the simplest case, a diaphragm which, together with the opening of the source, determines the geometrical shape of the beam, 3) a detector for detecting the molecules of the beam, and 4) a vacuum housing, inside which all the remaining parts are placed and in which the pressure can be made so low that the mean free path of the beam molecules is many times greater than the distance between the source and the detector. In addition, depending on the nature of the problem under study, additional devices are used, such as sources of magnetic and electric fields, crystal gratings, velocity selectors, scatterers, etc. The effectiveness of the molecular-beam method for the investigation of a whole series of problems depends to a high degree on the intensity of the beam and on the sensitivity of the de-

*) Beams of neutral particles moving rectilinearly may also be obtained as a result of collisions of very fast ions (for example, \(\alpha\)-particles) with neutral molecules, owing to a redistribution of charges between ions and molecules. Beams of this kind, formed by particles possessing nonthermal velocities, are not called molecular beams.

detector. Therefore, in designing apparatus for work with molecular beams, special attention must be paid precisely to these factors. In the following sections these questions will be elucidated in greater detail.

3. SOURCES OF MOLECULAR BEAMS

To obtain molecular beams, the flow of gases or vapors through an aperture is used. If the substance under normal conditions is in the solid or liquid state, then, in order to obtain its vapor, it is usually placed in a small furnace, which can be heated to the appropriate temperature. Gases are simply led to the effusion aperture through a tube from a reservoir at room temperature. In some early investigations certain substances (silver, copper) were evaporated from the surface of a platinum incandescent filament. Owing to the fact that the evaporation method was used for obtaining molecular beams considerably earlier than beams of molecules of gaseous substances were obtained, the term “furnace,” by which the beam source is denoted, is often applied now even in those cases when the source not only is not heated, but is even cooled below room temperature. The general conditions for the formation of molecular beams are, of course, the same for vapors and for gases; therefore they will be discussed without any differentiation.

General questions

The problems of gas effusion were first investigated by Knudsen[^40], who showed that the laws governing the flow of gases through tubes and apertures depend on the ratio of the mean free path of the gas molecules to the diameter of the tube or aperture. For our discussion, the most important conclusion from these investigations is the law according to which the total gas flow, in the case when the diameter is small in comparison with the mean free path, depends only on the pressure difference, the dimensions of the tube or aperture, and the gas density \(\rho\), and does not depend on the coefficient of internal friction \(\eta\). This latter circumstance distinguishes the regime of molecular effusion from the regime of a hydrodynamic jet, whose flow depends essentially on the coefficient of internal friction.

The flow of gas penetrating into a vacuum through an “ideal” diaphragm, the wall thickness of which is small in comparison with the dimensions of the aperture, can be easily calculated on the basis of elementary kinetic theory. This flow is simply equal to the number of molecules \(N\) which strike a portion of wall of area \(a\), equal to the area of the diaphragm, per unit time, namely

\[ N = \frac{1}{4} n \bar{c} a, \tag{1} \]

where \(n\) is the number of molecules in \(\mathrm{cm}^3\), and \(\bar c\) is the mean velocity of a molecule. Using the relations \(n=\dfrac{p}{kT}\) and \(\bar c=(8kT/\pi m)^{1/2}\), where \(k=\dfrac{R}{N_0}\) is Boltzmann’s constant, and \(N_0\) is Avogadro’s number, one can express the number \(N\) as a function of the pressure \(p\), the absolute temperature \(T\), and the mass of the molecule \(m\), or the molecular weight \(M\). As a result we obtain:

\[ N=\frac{pa}{(2\pi mkT)^{1/2}}\ \mathrm{sec}^{-1}; \tag{2} \]

expressing this in grams, we obtain for the efflux of gas from an orifice:

\[ G=Nm=\frac{pam^{1/2}}{(2\pi kT)^{1/2}} =\frac{paM^{1/2}}{(2\pi RT)^{1/2}}\ \mathrm{g/sec}. \tag{2a} \]

or in moles

\[ Q=\frac{G}{M}=\frac{pa}{(2\pi MRT)^{1/2}}\ \mathrm{mol/sec}. \tag{2б} \]

Substituting numerical values of the universal constants and expressing \(p\) in mm of mercury, we obtain

\[ Q=5.83\cdot10^{-2}(MT)^{-1/2}pa\ \mathrm{mol/sec}. \tag{2в} \]

These equations are valid under the assumption that, during flight through the orifice, no collisions occur between the molecules. This means that the mean free path \(\lambda\) of the molecules in the furnace must be large in comparison with the diameter of the orifice \(d\). In most practical cases molecular effusion takes place under conditions where \(\lambda\) is equal to several \(d\); in earlier works this condition was simply denoted as \(\lambda \geq d\). A hydrodynamic gas jet is formed if the gas pressure in the furnace is so high that \(\lambda \ll d\). Thus it is clear that between the conditions under which the efflux of gas occurs according to the laws of molecular effusion and the conditions for the formation of a hydrodynamic jet, there is no sharp boundary; in intermediate regimes the flow depends on \(\eta\), \(\rho\), \(p\), and \(T\) in a considerably more complicated way. This case was investigated by Knudsen, and the principal result of this investigation is that the magnitude of the flow increases more slowly than \(p\).

Equations (1) and (2) express only the magnitude of the total gas flow through the orifice. However, the formation of a molecular beam depends essentially on the angular distribution of the molecules leaving the diaphragm. According to Knudsen’s work, the number of molecules \(dN\) emerging from an element of the surface of the diaphragm \(d\sigma\) and contained in an element of solid angle \(d\omega\) in a direction making an angle \(\vartheta\) with the normal to \(d\sigma\), is expressed as follows (see Fig. 2):

\[ dN=\mathrm{const.}\cos\vartheta\,d\omega\,d\sigma. \tag{3} \]

This equation is known as the cosine law of molecular effusion and has found full confirmation in experiment; it is analogous to Lambert’s law in optics.

The intensity of a molecular beam is defined as the number of molecules flying through a unit area of the transverse cross-section of the beam per second, or (which is the same thing) striking a unit area of the transverse section of the detector. From consideration of Fig. 3 it follows that the number of molecules leaving the surface element \(d\sigma\) of the source \(P\) and striking the area element \(d\sigma'\) of the transverse section of the beam or the surface of the detector \(P'\) is expressed as

\[ dN_{\sigma\sigma'}=\frac{n\bar c}{4}\cdot \frac{d\sigma\,d\sigma'\cos\theta_1\cos\theta_2} {r^2\displaystyle\int_0^{\pi/2}\!\!\int_0^{\pi/2}\cos\theta\sin\theta\,d\theta\,d\varphi}, \tag{4} \]

where \(\theta_1\) and \(\theta_2\) are the angles between the straight line joining \(d\sigma\) and \(d\sigma'\), and the corresponding normals. The intensity of the beam at \(P'\) can be calculated by integrating equation (4). In most apparatus for work with molecular beams \(\cos\theta_1=\cos\theta_2=1\), and as a result of integration over \(d\sigma\) (i.e. over the area of the furnace opening), taking equation (2в) into account, we obtain:

Fig. 2. Cosine law of effusion.

Fig. 3. Calculation of the intensity of a molecular beam.

\[ dN_{\sigma'}=I\,d\sigma'= \frac{5.83\cdot 10^{-2}ap\,d\sigma'}{\pi r^2(MT)^{1/2}} \ \text{moles/sec}, \tag{4a} \]

where \(a\) again denotes the area of the furnace opening, \(p\) is the pressure in the furnace, measured in mm Hg, \(r\) is the distance of the element \(d\sigma'\) from the furnace opening, and \(I\) is the intensity of the beam on the element \(d\sigma'\).

In most experiments molecular beams are subjected to the action of forces directed in a definite way, and the deflections caused by these forces are measured. Therefore one usually tries to give the beam cross-section the form of as narrow a rectangle as possible; the smaller side of it, parallel to the direction of the deflecting forces, we shall henceforth call the width of the beam. In some early experiments\(^{31}\) the required beam cross-section was obtained with the aid of

of a special rectangular slit located in front of the round opening of the oven; in later experiments the rectangular shape of the oven aperture (the slit) was used directly. As Stern indicated \(^{63}\), in this way the width of the beam can be made arbitrarily small without a noticeable decrease in intensity, if effects of the second order of smallness are not taken into account. According to equation (4a), the intensity of the molecular beam is proportional to the product of the pressure in the oven \(p\) and the area of the aperture \(a\). On the other hand, the magnitude of the pressure in the oven is limited by the requirement that the mean free path of the molecules inside the oven be sufficiently large in comparison with the dimensions of the aperture. In the case of a rectangular oven slit, this requirement refers to the width of the slit \(b\), and not to its length, since collisions of molecules in the direction of the length of the slit are comparatively few, occur, and do not cause the appearance of a “cloud” distorting the shape of the beam. Suppose that, for carrying out a certain experiment, the best conditions will be obtained if the mean free path is a certain number of times greater than the slit width, or \(\lambda/b = K\). A decrease of \(b\) by a factor of \(f\) will cause a decrease of the area \(a\), and therefore, in order to maintain constant intensity, it will be necessary to increase the pressure \(p\) by the same factor. Since the mean free path \(\lambda\) is inversely proportional to \(p\), both \(\lambda\) and \(b\) will be decreased by a factor of \(f\), so that their ratio \(K\) remains unchanged. Therefore an increase in pressure does not cause a change in the conditions of effusion or in the beam intensity, if the slit width is decreased by the same factor; or, in other words, the maximum beam intensity does not depend on the width of the oven slit.

Ovens

From the conditions set forth in the preceding paragraph, and also taking into account that the width of the oven slit usually ranges from \(10^{-2}\) to \(10^{-3}\) cm, it can be established that the pressure in the oven should not exceed \(1\) mm Hg (at a pressure of \(1\) mm Hg, \(\lambda\) is about \(10^{-2}\) mm). The substances used for obtaining molecular beams may possess the most varied physical and chemical properties. For some of them the pressure indicated above is reached at an oven temperature above \(1000^\circ\) C \(^{52}\); when working with certain gases, in order to obtain this pressure the “oven” must be cooled with liquid air. Moreover, as was already mentioned above, in some cases beams of unstable particles are used, for example hydrogen atoms \(^{68}\) or free radicals \(^{26,28}\). Hence it becomes clear that the construction of the source depends to a very great extent on the nature of the substance used for obtaining the beam. For gaseous substances the “oven” is usually a simple tube connected to a cylinder of gas and ending in a narrow slit. In order that the temperature of the effusing gas can be changed, heating devices are installed around the slit or

the target is connected to surfaces cooled by liquid air by means of heat-removing flexible copper rods. In order to ensure temperature equilibrium between the walls and the molecules of the beam, a piece of woven copper wire is inserted into the connecting tube near the heater. The gas pressure in the cylinder is maintained more or less constant and equal to a value satisfying the requirement \(\lambda \geqslant b\). When beams of unstable molecules are being studied, a reaction chamber is placed between the cylinder and the furnace target, for example a discharge tube for obtaining atomic hydrogen. In designing a furnace intended for the evaporation of solid or liquid substances, the choice of furnace material depends on the operating temperature and on the chemical properties of these substances. If the required vapor pressure is attained at relatively low temperatures, as, for example, in the case of Hg, Zn, Cd, H\(_2\)O, etc., the furnace may be made of copper, phosphor bronze, or glass; for higher temperatures steel, molybdenum, tantalum, Monel metal, etc., are used. The material of which the furnace is made must not enter into a chemical reaction or alloy with the substance used to obtain the molecular beam. Thus, for example, a furnace for cesium beams\({}^{25}\) must be made of Monel metal, despite the fact that the operating temperature does not exceed \(175^\circ\)C. It is also a very essential requirement that the surface of the substance loaded into the furnace be considerably larger than the area of the furnace slit, since the rate of increase of the amount of vapor must greatly exceed the value of the effusion rate through the furnace aperture, expressed by equation (26); it should be borne in mind that, owing to the presence of insignificant impurities on the surface of the evaporated substance, the evaporation rate may prove to be considerably less than the value known from tables\({}^{40}\). The heating device must be constructed so that the temperature of the target is higher than the temperature of the remaining parts of the furnace; otherwise the substance will condense at the edges of the target, and an unavoidable clogging of the target will occur. On the other hand, the vapor pressure inside the furnace is determined by the temperature of its coldest part. Therefore, if the furnace-temperature measurement serves to monitor the pressure, the temperature-measuring instrument (thermocouple) must be brought into contact with the coldest point of the furnace. Examples of practical furnace designs are given in Figs. 4 and 5. Figure 4 shows a furnace with an operating temperature of about \(200^\circ\)C, intended for chemically inert substances. The base of the furnace is a block of phosphor bronze drilled along its axis; at the front a tube with a slit device is screwed into it, and at the rear a container with the substance used to obtain the beam is screwed in or press-fitted. The heating device\({}^{37}\) is made in the form of a glass tube coated with platinum. The heater resistance must be several ohms. The construction of the furnace recently used to obtain cesium beams\({}^{25}\) is shown in Fig. 5. One of

One of the main difficulties encountered when working with alkali metals is the tendency of the latter to spread over the inner surface of the furnace, which leads to clogging of the slit. To overcome this difficulty, in the design shown in Fig. 5 partitions are installed between the evaporating metal and the slit, considerably lengthening the path between them. The furnace itself is a cylindrical block of Monel metal with an eccentric opening. The cesium metal, enclosed in a glass capsule, is placed inside a Monel sleeve, which serves as a partition; cesium vapor is led to the slit through the tube shown in the figure. The furnace is loaded through the rear opening, after which the head of the capsule is broken off already inside the furnace, in an atmosphere of nitrogen, and the rear opening is immediately closed with a stopper. A copper washer serves as a sealing gasket

Fig. 4. Furnace.
$O$ — furnace slit; $H$ — heater; $B$ — block of phosphor bronze; $S$ — tube with slit; $C$ — container with the substance being evaporated.

Fig. 5. Monel furnace for cesium.
$O$ — furnace slit; $T$ — connecting tube; $T\text{-}C$ — thermocouple; $W$ — copper gasket; $C$ — copper tube; $H_1$, $H_2$ — heaters; $M$ — Monel sleeve.

to prevent cesium vapor from leaking through the rear opening. The main heater consists of a copper cylinder with a ribbon winding of nichrome and mica insulation. On the front wall of the furnace there is an additional heater, serving to make the temperature of the slit slightly higher (by $5$–$10^\circ\mathrm{C}$) than the temperature of the remaining parts of the furnace. The thermocouple is sealed into the rear stopper. Owing to the large heat capacity of the furnace body, the temperature in it can be maintained constant for many hours, which gives good stability of the beam intensity.

Furnaces intended for high operating temperatures are distinguished by their small dimensions, owing to the presence of large heat losses by radiation. In addition, ordinary insulating materials cannot be used in such furnaces, since at high temperature they release large amounts of gas. For heating the furnace there may be used a device consisting of one or several tungsten spirals passed through cylindrical openings inside the furnace block. Such spirals, insulated with the aid of quartz bushings, make it possible to obtain very high temperatures; in this case heat is transferred to the furnace block exclusively by thermal radiation. If, however, the power required for heating the furnace is difficult to obtain

with the aid of thermal emission, then between the wall of the furnace and the tungsten spiral, fixed in the immediate vicinity of the furnace, a stream of electrons passes under high voltage (1000–2000 V). Such an electron current can liberate several hundred watts of thermal power. Fig. 6 shows such a furnace, with an operating temperature of about 1100° C, intended for obtaining bismuth beams^48. The heating spiral is installed with the calculation that the temperature of the slit should be higher than the temperature of the remaining parts of the furnace; the furnace itself is made of tungsten steel. A furnace with an even higher operating temperature (about 1250° C), with the aid of which indium beams were obtained, was an ordinary molybdenum block, cut along its diagonals, inside which indium was placed. No special loading aperture was provided, so that in order to recharge the furnace it was necessary to remove the slit faces. The power required to heat the furnace to a temperature of 1300° C was about 800 watts. Furnaces operating at high temperatures must be surrounded by water-cooled shields^31.

Fig. 6. Furnace heated by electron bombardment. \(O\)—the furnace slit; \(L\)—the cover; \(H\)—the heating spiral.

The slit arrangement of the furnace.

In deriving the equations for the intensity of a molecular beam it was assumed that the wall thickness of the source slit is infinitely small. In reality, however, the faces forming the slit have a finite thickness, and thus the slit is not an ideal aperture but a short channel with constant or variable cross-section. The effusion flux through such a channel \(N_c\), at a given pressure, is smaller than the effusion flux through an ideal aperture \(N_0\). Table I gives^9 values of the ratio

\[ W=\frac{N_c}{N_0}, \]

where \(N_c\) is the flux through a channel of length \(L\) with constant width \(b\), and \(N_0\) is the flux through an ideal slit of width \(b\), it being assumed that the height of the slit and of the channel is considerably greater than the width \(b\). Usually the slit width \(b\) is approximately 0.01 mm, while the effective thickness of the faces \(L\) may be several times greater than \(b\).

On considering Table I it may appear that the value of the molecular-beam intensity obtainable in practice is considerably smaller than the value calculated from equation (4). However, the length of the channel also manifests itself in the violation of the cosine law of the angular distribution of intensity toward an increase in the number of molecules flying out of the aperture in the forward direction, which leads to a partial compensation of the influence of the channel length on the intensity in the principal direction^10. In Fig. 7 is shown

Table I

Effect of channel length

\(L/b\)	\(W\)	\(L/b\)	\(W\)	\(L/b\)	\(W\)
0.0	1	1.2	0.6485	2.8	0.4712
0.1	0.9525	1.3	0.6321	3.0	0.4570
0.2	0.9096	1.4	0.6168	3.2	0.4439
0.3	0.8710	1.5	0.6024	3.4	0.4318
0.4	0.8362	1.6	0.5888	3.6	0.4205
0.5	0.8048	1.7	0.5760	3.8	0.4099
0.6	0.7763	1.8	0.5640	4.0	0.3999
0.7	0.7503	1.9	0.5525	5.0	0.3582
0.8	0.7265	2.0	0.5417	6.0	0.3260
0.9	0.7049	2.2	0.5215	7.0	0.3001
1.0	0.6848	2.4	0.5032	8.0	0.2789
1.1	0.6660	2.6	0.4865	9.0	0.2610
				10.0	0.2457

the angular distribution of the intensity of effusion from a short channel of circular cross section has been calculated for the case when \(b = 2r = L\). The solid curve represents the actual course of the distribution; the dotted line corresponds to the cosine distribution. For this case the number of molecules flying out in the direction of the channel axis is the same for the channel as for the orifice, whereas the total effusion flux from the channel amounts to only about half the flux from an ideal orifice.

As a rule, for practical application of the molecular-beam method the most essential point is to obtain the maximum possible beam intensity near the detector. However, the magnitude of this intensity depends not only on the effusion flux from the furnace in the direction of the beam, but also on the degree of “absorption” of the beam on its way to the detector, i.e., ultimately, on the magnitude of the residual pressure in the apparatus. Therefore, in view of the fact that, in general, a larger number of molecules flies out of an orifice than out of a channel, when comparing the efficiency of an ideal slit with the efficiency of one or another furnace channel it is necessary to provide that all molecules flying out of the furnace are in some way removed from the apparatus in order to avoid disturbing the vacuum. If the substance from whose molecules the beam is formed condenses readily, then the stated requirement is easily

Fig. 7. Angular distribution of molecules emerging from a short channel.

is carried out with the aid of large surfaces, cooled by water or air and placed near the furnace. If, however, the beam consists of gas molecules, their removal is effected by pumping, and then the pressure in the apparatus is determined by the ratio of the rates of effusion and pumping. It is clear that in this case it is desirable, while preserving the magnitude of the beam intensity, to reduce as far as possible the total magnitude of the effusion flux from the furnace; for this purpose, in many investigations sufficiently long channels were specially installed as the outlet device of the furnace[^20].

As a result of repeated measurements of the dependence of the intensity of molecular beams on the pressure in the furnace, it has been found that at sufficiently low pressures ($\lambda > 3b$ for the case of beams of $\mathrm{H_2O}$ molecules) the magnitude of the intensity increases proportionally to $p$, whereas at higher pressures the increase in intensity slows down*). At the same time, with increasing pressure the transverse cross section of the beam becomes larger than the cross section determined by the geometrical dimensions of the slits. This effect was explained by the formation of a “cloud” of molecules immediately in front of the furnace slit[^38,^41]. This cloud, occupying a volume with a cross section larger than that of the slit, is formed as a result of collisions of molecules with one another inside the slit or in its immediate vicinity. The “cloud” serves, as it were, as the “source of the beam.” Therefore the intensity of the molecular beam is already determined not by the magnitude of the total effusion flux, but by the “brightness” of the cloud, which is less than the “brightness” of an ideal slit. Using this terminology of radiation theory, one may say that the magnitude of the total radiation flux of the source increases with increasing pressure in the furnace faster than the “brightness,” the magnitude of which determines the intensity of the beam. In addition, the formation of the cloud causes a distortion of the velocity distribution of the molecules in the beam (see Section 7). As a consequence of all this, it is usually recommended to work at a pressure in the furnace considerably lower than the pressure corresponding to the maximum value of the total effusion flux.

The jaws forming the slit are usually made of the same material as the furnace itself and are either fastened by means of “dovetails” or screwed to the carefully ground surface of the front wall of the furnace, as shown in Figs. 4–6. To obtain a slit as close as possible to an ideal aperture, the edges of the jaws are ground back like a knife blade; if the slit is to be given the form of a channel, the jaws are made with rectangular end faces. For beams of gas molecules, slits with glass jaws have also been used.

*) Johnson[^33] reported that he had found significant deviations from the indicated dependence; however, this should be ascribed to the fact that, owing to the high pressure inside the furnace, his apparatus had a jet regime rather than a molecular-beam regime.

4. FOCUSING SYSTEMS

The focusing of molecular beams which, as was indicated above, consist of neutral particles, cannot be carried out with the aid of “lenses” suitable for charged particles. Therefore the optics of molecular beams is the optics of a chamber with an aperture. The simplest focusing system is a single collimating slit, which together with the furnace slit determines the shape of the beam, as shown in Fig. 8. In this case the beam cross section consists of a central region of constant intensity (the “light” region) and edge regions (“penumbra”), in which the intensity decreases according to a linear law from its maximum value to zero. Thus the intensity distribution in the beam cross section is trapezoidal. Formation of a beam with the above-mentioned ideal intensity distribution over the cross section requires a very precise alignment (parallelism) of both slits, which often constitutes the most difficult problem in the construction of the apparatus, especially when using narrow beams of great length. In some installations the slit diaphragms are fixed in rigid mounts, preserving once and for all the selected arrangement; if this is inadmissible, then one of the slits is made movable, so that it is possible to adjust its position during the experiment itself. In this case the final setting of the movable diaphragm is made after checking the intensity distribution over the beam cross section.

Fig. 8. Formation of a beam in the case of two slits. \(O\) — furnace slit; \(C\) — collimating slit; \(S\) — mixing; \(I_0\) — maximum intensity.

Fig. 9. Additional diaphragm serving as a source. \(O\) — furnace slit; \(C\) — collimator slit; \(F\) — additional slit.

In cases where the beam source is subjected to heating or cooling, it is impossible to rigidly fasten both the furnace slit and the collimator slit, since the mutual arrangement of these slits changes when the temperature changes. It is considerably easier to achieve rigid focusing of the beam if an additional diaphragm is used (Fig. 9). This additional diaphragm is at room temperature and its position remains unchanged when the temperature of the furnace changes, so that the focusing of the beam is not disturbed, provided only that the additional diaphragm is sufficiently rigidly fastened relative to the collimating device. The use of an additional diaphragm is also expedient because in this way the effect of increasing the effective width of the furnace slit is eliminated,

occurring, as was indicated in the preceding section, as a result of the formation of a molecular cloud. If the arrangement of the additional diaphragm permits adjustment of its position, then the aperture in the furnace is made somewhat wider than the slit of the additional diaphragm, so that adjustment of the latter is not too difficult and is not disturbed by small displacements during the experiment. If, as the rigid frame on which the slit devices are mounted, a long glass plate is used^25 (Fig. 10), then the relative displacement of slits located at a distance of one meter from one another can be brought to a value considerably less than 0.001 mm. The horizontality of the slit setting is checked with the aid of a small sensitive level fixed on the frame carrying the slit system.

Fig. 10. Slit system assembled on a glass plate.

The need for an additional diaphragm may also be dictated by the following considerations. In contrast to ordinary light rays, a molecular beam, passing through the apparatus, is scattered because of the presence in it of residual gas. The presence of this gas is partly due to the imperfection of the pumps, small leaks, etc., partly to the evolution of gas by the hot walls of the furnace and of the apparatus, which is also heated by radiation from the furnace, and partly to the very process of the molecules escaping from the slit of the furnace (see Section 3). Whereas the first group of causes can at present be eliminated to a considerable extent by the use of modern pumps and good sealing of the apparatus, the other two groups nevertheless cause, in the immediate vicinity of the furnace, some increase in the gas pressure, for the reduction of which special measures must be taken. The furnace must be surrounded by cooled surfaces and screens, especially in cases where the working temperature is high, and by special water-cooled jackets if a readily condensable substance is used to form the beam. In the case where the beam is formed by a gaseous substance, the latter cause of the appearance of residual pressure in the chamber can be eliminated only with the aid of additional pumping by a pump; moreover, the pressure established in the chamber will be determined, as was already indicated above, by the equilibrium between effusion and pumping.

The additional diaphragm makes it possible to divide the whole apparatus into two parts^29. One we shall call the furnace chamber, and the other the collimator chamber (Fig. 11). Each chamber is connected to its own pump. If, for the reasons indicated above, the residual gas pressure in the immediate vicinity of the furnace (i.e., in the furnace chamber)

is equal to \(p\), then the pressure in the collimator chamber can be reduced, in comparison with \(p\), by a factor \(R\), where \(R = W_p/W_F\), if \(W_F\) denotes the rate of passage of gas through the additional diaphragm, and \(W_p\) the effective pumping speed of the pump of the collimator chamber. In order to make the factor \(R\) as large as possible, it is necessary that the additional diaphragm have a large resistance for the residual gas passing through it; this can be achieved by lengthening the channel of this diaphragm. The width of the additional diaphragm may be made greater than the width of the furnace aperture, so that the latter continues to serve as the source; with such a relation of dimensions, adjustment of the entire focusing device is facilitated. If, however, the furnace aperture does not serve as the source, then the focusing system consists of the following parts: the furnace aperture, the additional diaphragm—the channel through the partition between the furnace and collimator chambers, a separate diaphragm serving as the source, and the collimator slit. With such a composition of the focusing system, the shape of the beam is determined by the last two elements.

Fig. 11. Apparatus for obtaining beams of \(H_2\), divided into the furnace chamber \(A\) and the collimator chamber \(B\). \(O\)—furnace slit; \(F\)—additional slit; \(C\)—collimator slit; \(M\)—manometers; \(P\)—magnet poles; \(S_1, S_2\)—adjusting screws.

The decrease in the intensity of a molecular beam as a result of scattering by molecules of the residual gas at a distance \(l\) from the source is expressed by the formula

\[ I = I_0 e^{-\frac{l}{\lambda}} = I_0 \exp\left[-\frac{l_0}{\lambda_0}\right]\cdot \exp\left[-\frac{l_c}{\lambda_c}\right], \ldots , \tag{5} \]

where \(\lambda_0\) and \(\lambda_c\) are the mean free paths of the beam molecules in the scattering gas in the furnace and collimator chambers, respectively, and \(l_0\) and \(l_c\) are the path lengths of the beam in these chambers. Since the pressure in the furnace chamber is always greater than the pressure in the collimator chamber, \(\lambda_c > \lambda_0\). Therefore the quantity \(l_0\) should be made as small as possible, especially in cases where the beam is formed by gas molecules, and the quantity \(\lambda_0\) is inversely proportional to the amount of gas issuing from the furnace aperture. However, reduction of the distance between the furnace aperture and the separating channel between the chambers is limited by the circumstance that molecules emerging from the furnace aperture, reflec-

are reflected back according to a cosine law from the separating wall and thereby cause an additional effective pressure in the oven chamber, which rapidly increases as \(l_0\) decreases. As a result, for \(l_0\) there exists a certain optimum value, and in order to attain the maximum beam intensity the geometrical dimensions of the focusing system, analogous to that shown in Fig. 12, must be chosen in a quite definite manner.

It also follows from equation (5) that, in order to obtain a beam of sufficient intensity, the mean free paths of the beam molecules \(\lambda\) must be several times greater than the paths \(l\) traversed by the beam. It is necessary, moreover, to note that in a narrow beam a molecule deflected as a result of a collision by only a few angular seconds will leave the beam. The values of \(\lambda\) that must be substituted into equation (5) are therefore several times smaller than the values calculated at the given pressure from the coefficients of viscosity and thermal conductivity[^37]. Experiments with molecular beams provide a full possibility of measuring the true value of the mean free path of gas molecules.

Fig. 12. Channel serving as an additional diaphragm.

5. DETECTORS

The considerations concerning the design of sources and collimating systems, presented in the preceding sections, apply, to one degree or another, to the technique of obtaining any molecular beams, irrespective of their nature. As for the technique of registering beams, here the question depends essentially on the chemical and physical properties of the substance whose molecules form the beam. It is scarcely necessary to point out once more that the sensitivity of a method using a molecular beam depends to the same extent on the beam intensity as on the sensitivity of the detector, i.e. on the minimum number of particles that can be detected by the detector. Sometimes the concept of the “effective intensity” of a beam is even introduced in units of detector readings. Since the intensity of a molecular beam at a distance \(r\), according to equation (4a), is inversely proportional to \(r^2\), the sensitivity of the detector in each given case determines the maximum permissible length of the beam. In this we again find an analogy with optical experiments.

Just as in optics, in molecular-beam technique there exist detectors of the integral type, analogous to the photographic plate, and detectors with direct registration, corresponding to photocells, thermopiles, or bolometers in optics. We shall begin our survey with integral detectors.

Condensation detectors

In the first experiments with molecular beams^12, a glass plate was placed in the path of the beam molecules. The molecules striking it formed a deposit (“image”), the shape of which corresponded to the cross-section of the beam. A carefully polished metal plate may also serve as a detector “target.” If the exposure is sufficiently long, this deposit becomes visible. If, for some reason, the exposure must be interrupted before the deposit forms a visible image, then it is often possible to “develop” this invisible image by means of processes analogous to the development of a latent image on a photographic plate.

It is clear that this method is applicable only when the substance whose molecules make up the beam condenses at the temperature of the detector plate. Thus, for example, for beams of silver atoms used in the classical experiments^31,60, this requirement was satisfied at room temperature of the plate. In the general case there is always a certain maximum temperature of the plate at which condensation of the substance on its surface still occurs. One might expect that this critical condensation temperature is determined only by the ratio of the saturated vapor pressure of the beam substance at the given plate temperature to the pressure in the beam near the plate. However, there is no such direct relation. It is true that after a deposit has already formed on the surface of the plate, the condition for its further growth consists only in the effective pressure in the beam near the detector being greater than the saturated vapor pressure of the substance under investigation; however, for the formation of the first traces of deposit the temperature of the plate must be considerably lower, and its value depends not only on the beam substance and the plate material, but also on the intensity of the beam and its geometrical dimensions, especially on its width^16,17. Under certain experimental conditions, not only complete condensation on the plate or complete reflection of the beam from it is possible, but also partial condensation. In this case only the central part of the beam forms a deposit on the plate, while from the edges of the beam, as well as from scattered molecules, no traces remain on the plate. This is possible if the temperature of the plate is low enough for the beam with the greater intensity to condense, but too high for condensation of parts of the beam with lesser intensity, regardless of the duration of the exposure. This phenomenon is directly connected with the existence of an intermediate phase in the condensation process^44. The molecules striking the plate do not immediately settle on it, forming particles of the solid phase, but continue to move along its surface in the form of a two-dimensional gas. Such motion of the molecules continues until they either leave the surface

plate, or, on encountering other molecules, form complexes of larger size, which then settle on it permanently^67. Therefore very thin deposits are not homogeneous layers, but consist of many individual crystallites, each of which is an accumulation of a large number of atoms or molecules (Fig. 13). This was directly confirmed by investigations of the structure of deposits with the aid of the ultramicroscope^15. Study of the structure of deposits indicates the exceptional sensitivity of the condensation method of detection. In the classical experiments (silver on glass), deposits whose thickness exceeded a diatomic layer were already directly visible, while deposits of lesser thickness could be detected by development^14.

Fig. 13. Structure of thin deposits.

The condensation method of recording has a great advantage over all others in view of its simplicity; this method can be applied to beams consisting of molecules of any substances, except gaseous ones, provided only that sufficient cooling of the plate is ensured. Examination of the exposed plate is carried out either outside the apparatus or inside it; in the latter case a special prism and a microscope with slight magnification are used. The method of examining the plate in the apparatus itself not only makes it possible to observe the process of deposit formation, but is also indispensable when the image in air is spoiled as a result of evaporation of the deposit. If the beam intensity is so small that no visible image is formed, then after exposure the plate is subjected to development. The developer for metallic deposits on glass consists of a solution of 1–2 percent hydroquinone in water, to which a small amount of gum arabic is added as a protective colloid and a few drops of a 1 percent solution of lapis. The “development” of the image is based on the fact that atomic silver, which is formed in this solution, is deposited especially well on those places of the plate on which there is already a film of metal formed by condensation of the beam. Usually the image becomes visible after

several minutes after immersing the plate in the developer. There is also another method of development, possessing even greater sensitivity, for which it is not necessary to remove the plate from the apparatus; in this method one uses the property of vapors to condense on those small crystals of the deposit of which the invisible image consists. For this purpose, vapors of water, mercury, or cadmium, or tobacco smoke, can be used successfully.

The principal drawback of the condensation method of recording is that, despite all attempts, it has not proved possible to use it for quantitative measurements of intensity. Except in cases where very thick deposits are obtained, the relation between the thickness of the deposit and the beam intensity is unknown and, in all probability, very complicated because of the complexity of the condensation process. The most successful attempts are those using, as a measure of the beam intensity, the time of appearance of the image on the plate,^38 and although such a method is, of course, unsuitable for absolute measurements of intensity, it nevertheless can be used for determining relative intensities. This method is often used in determining regions of equal intensity in beams that have passed through deflecting fields.^17

Chemical detectors

The method of recording considered below is suitable only for such beams as consist of chemically active molecules capable of producing noticeable changes on the surface of the detector. Thus, for example, hydrogen atoms can be detected by means of the reduction reaction on a “target” of MoO$_3$ or WO$_3$,^34,68 and atomic oxygen by means of the oxidation reaction of PbO to PbO$_2$.^42 In all the cases listed, a change in the coloration occurs in those regions of the detector target on which the beam has fallen. The method of chemical recording also proves useful in other cases, in particular for beams of unstable molecules, for example, of free radicals.

The integral methods of recording described in the last two sections played a particularly important role at the early stage of development of molecular-beam techniques because of their simplicity and good sensitivity. However, as has already been indicated, with these methods even semiquantitative measurements are difficult to carry out, and therefore at the present time they have to a considerable extent been supplanted by methods of direct recording. This is due to the fact that contemporary physics requires from experiments with molecular beams exact knowledge of the distribution of intensity in the beam.

The numerous direct-counting detectors that have found exceptionally wide application in work with molecular beams can be divided into a group of detectors with surface ionization and a group of detectors of the manometric type. Recorders of the first group pri-

change for atoms with low ionization potentials, for example for alkali metals, Ba, Ca and In, or for beams of molecules containing atoms of these substances; the second group is used in those cases when the substances forming the beam are gases or vapors with a low condensation temperature. In special cases direct-counting detectors of other types were employed; it is possible that some of them will find wider application.

Detectors with Surface Ionization

If an atom of an alkali metal, say cesium, encounters on its path the hot surface of a tungsten filament, it may give up to it one electron and rebound from this surface already in the form of a positive cesium ion[^46]. These ions can be collected on a special “collector” electrode; for this purpose it must have a negative potential relative to the tungsten filament. The magnitude of the ion current flowing as a result gives directly the number of atoms arriving per second at the detector filament.

The mechanism of surface ionization, in brief, is as follows[^36],[^53]. If \(I\)—the ionization potential of an atom incident on the surface of the detector filament—is less than the binding energy of an electron on the surface of the filament (the work function \(\Phi\)), then there is a definite probability that, when the atom strikes the surface of the filament, it will lose its valence electron and, if the temperature of this surface is sufficiently high, will rebound from it already in the form of a positive ion. The ratio of the number of ions \(n^+\) to the number of neutral atoms \(n\) leaving the surface is equal to

\[ n^+/n = \exp(\Phi - I)/kT, \tag{6} \]

where \(k = 8.5 \cdot 10^{-4}\) electron-volt per degree is Boltzmann’s constant, and \(I\) and \(\Phi\) are measured in the same units. At \(T = 1200^\circ\mathrm{K}\), \(kT \simeq 0.1\) electron-volt, and if \(\Phi - I = 0.5\) electron-volt, then \(\frac{n^+}{n} = e^5 = 148\), which means that practically every atom reaching the detector is ionized. For the surface of tungsten free of thorium impurity, the work function \(\Phi\) is equal to 4.48 electron-volts, and therefore the above relation between \(\Phi\) and \(I\) holds for cesium, for which \(I\) is equal to 3.9 electron-volts. For K and Rb, for which \(I = 4.1\) electron-volts, \(\frac{n^+}{n} = e^{3.8} = 45\), which also means practically one-hundred-percent ionization; however, for example, for Na (\(I = 5.13\ \mathrm{eV}\)) and Li (\(I = 5.37\ \mathrm{eV}\)) a wire of pure tungsten can no longer serve as a detector. By oxidizing the surface of tungsten, it is possible to increase the work function considerably and thus to register atoms whose ionization potentials reach six electron-volts[^45]. For oxidation the tungsten wire is heated to \(1500^\circ\mathrm{K}\) and placed for several minutes

into an oxygen atmosphere at a pressure of 0.1 mm Hg. During operation of the detector, the temperature of the wire must in no case exceed 1600° K, since at this temperature the oxide layer begins to evaporate rapidly. However, even at lower operating temperatures the oxide layer on the filament must be restored from time to time.

It is, of course, very important that the tungsten used for making the detector contain no impurities that reduce the magnitude of the electron work function from the surface; thorium is especially harmful, since its impurities reduce the work function to 2.6 electron-volts.

A detector with surface ionization can register not only atoms of the elements mentioned, but also various molecules containing any of these atoms. In practice1 the detector filament is a short piece of tungsten wire, stretched with the aid of a small spring and heated to a temperature of 1200–1500° K. The filament is surrounded by a cylinder of sheet nickel, with two diametrically opposite rectangular openings through which the beam is passed. The filament diameter may vary from 0.01 to 0.1 mm. The ion current flowing between the filament and the collecting cylinder is usually measured by means of a single-tube amplifier (assembled with an FP-54 tube or the like). The circuit diagram of the most frequently used amplifier is shown in Fig. 14. In order for the detector to have high sensitivity, the leakage resistance of the grid of the amplifying tube must be not less than \(10^{11}\) ohms; for this it is necessary that the insulation of the collecting electrode be of high quality. The total sensitivity of such a circuit can be brought up to \(4 \cdot 10^{-16}\) ampere per millimeter of galvanometer deflection. This means that the galvanometer pointer deflects by 1 mm of the scale if 2500 atoms per second strike the detector filament. The high sensitivity of this method makes it possible to use long beams and very thin detector filaments, which is highly important for investigating the intensity distribution of a molecular beam subjected to deflection in weak fields. As a numerical example, let us consider the case in which a beam of cesium atoms, emerging from the slit of a source \(3 \cdot 10^{-1}\) cm high and \(2 \cdot 10^{-3}\) cm wide, strikes a detector filament \(2 \cdot 10^{-3}\) cm in diameter with an effective length of \(3 \cdot 10^{-1}\) cm.

Fig. 14. Circuit of a detector with surface ionization. \(F\)—filament; \(C\)—ion collector; \(R_1\)—grid leakage resistance; \(R_2, R_3, R_4, R_5, R_6\)—resistances; \(P_1, P_2\)—potentiometers; \(M\)—milliammeters; \(A\)—Ayrton shunt; \(G\)—galvanometer.

If the distance between the detector and the furnace slit is \(200\ \text{cm}\), the pressure in the furnace is \(0.1\ \text{mm}\), and the furnace temperature is \(450^\circ\text{K}\), then the number of moles reaching the detector per second, in accordance with equation (4a), is equal to

\[ N_{\sigma}=\frac{5.83\cdot 10^{-2}\times 6\cdot 10^{-4}\times 10^{-1}\times 6\cdot 10^{-4}} {\pi\cdot 4\cdot 10^{-4}(132\times 450)^{1/2}} =7.6\cdot 10^{-17}\ \text{mole/sec}. \]

This quantity corresponds to an ionic current in the detector equal to \(7.35\cdot 10^{-12}\) ampere, or to a galvanometer deflection of \(2\cdot 10^{4}\ \text{mm}\), if the amplifier whose circuit was given above is operating. To measure the intensity distribution in the beam, it is necessary to be able to move the detector filament parallel to itself in a direction perpendicular to the beam. The detector filament may be mounted on a rotating support with some eccentricity relative to its axis, or some mechanical device with micrometer screws, slides, and levers may be used. Recently a very successful and original design was applied for this purpose\(^{25}\): the detector device was fastened to the free end of a Bourdon tube taken from an ordinary manometer (Fig. 15); a change in pressure inside the tube by \(1\ \text{cm Hg}\) causes a displacement of the free end by almost exactly one hundredth of a millimeter, and since the pressure change can be controlled with an accuracy of up to a few tenths of a millimeter Hg, the accuracy with which the displacement of the detector filament is fixed exceeds \(1/1000\ \text{mm}\). This method permits remote adjustment of the detector position and has proved in practice to be very accurate and easily realizable.

Fig. 15. Detector with a surface ionizer. \(B\) — Bourdon tube; \(F\) — incandescent filament; \(A\) — anode cylinder; \(Q\) — quartz insulator.

Manometric Detectors

If a beam of molecules of a substance that does not condense under the experimental conditions is passed through a narrow slit into a chamber closed on all sides, then the pressure in this vessel will increase until equilibrium is established between the molecules of the beam entering the slit and the gas molecules leaving through the same slit\(^{37}\). This increase in pressure in the chamber is due to the fact that, whereas the molecules entering the vessel (the beam molecules) have velocities directed in one direction, the velocity distribution of the molecules leaving the vessel obeys the laws of gas effusion, since the beam molecules that have entered the vessel, being reflected from the walls according to the cosine law, acquire velocities in all directions. The number of molecules \(n_i\) entering the slit can be calculated from the equations

(4 and 4a), so that

\[ n_i=\frac{P_0 a a'}{(2\pi m k T_0)^{1/2}\pi r^2}, \]

where \(P_0\) is the pressure in the furnace, \(a\) is the area of the furnace slit, \(a'\) is the area of the detector slit, and \(r\) is the distance between the furnace and detector slits. The number of molecules leaving the detector slit, in accordance with equation (2), is equal to:

\[ n_e=\frac{P_D a'}{(2\pi m k T')^{1/2}}, \]

where \(P_D\) is the pressure in the detector chamber corresponding to equilibrium, and \(T'\) is the temperature in the chamber. Thus, the pressure at which equilibrium is established can be found from the condition \(n_i=n_e\), namely,

\[ P_D=\left(P_0 a/\pi r^2\right)\cdot (T'/T_0)^{1/2}. \tag{7} \]

If, as an example, one takes a helium beam and the values \(a=10^{-3}\ \mathrm{cm}^2\), \(r=20\ \mathrm{cm}\), \(P_0=1\ \mathrm{mm\ Hg}\), then the value of \(P_D\) is found to be approximately \(10^{-6}\ \mathrm{mm\ Hg}\). It is possible, however, to increase \(P_D\) considerably if the detector slit is replaced by a long narrow channel. In this case the number \(n_i\) will not change, while \(n_e\) will decrease by a factor of \(K\), where

\[ K=\frac{l}{b}\,\frac{1}{0.5+2.3\log(2a/b)}, \]

which is the ratio of the resistance to gas flow of a channel of width \(b\), height \(a\), and length \(l\), to the resistance of a slit with the same cross-sectional area[^59]. In practice it is possible to make \(K\) approximately equal to 50, so that the pressure in the chamber \(P_D\), in our example, can be brought up to \(5\cdot 10^{-5}\ \mathrm{mm\ Hg}\).

In modern investigations the accuracy of measuring the beam intensity must be such that the error does not exceed one thousandth of the measured quantity; the intensities of the deflected parts of the beam, often amounting to no more than \(1\%\) of the original beam, must likewise be measured with a high degree of accuracy. Therefore the manometer for measuring \(P_D\) must have a sensitivity down to \(10^{-8}\)—\(10^{-9}\ \mathrm{mm\ Hg}\). At the same time, the absolute value of the pressure in the manometric chamber, which is of the same order as the pressure in the collimator chamber, i.e. about \(10^{-6}\ \mathrm{mm\ Hg}\), fluctuates within several percent owing to the imperfection of the pump. Thus, the manometric system must register pressure changes of the order of \(10^{-9}\ \mathrm{mm\ Hg}\) at an absolute pressure from \(10^{-5}\) to \(10^{-6}\ \mathrm{mm\ Hg}\), and moreover in the presence of pressure fluctuations in the surrounding medium. This problem can be solved by installing a second detector with a separate manometer, the entrance aperture of the slit or channel of the second detector chamber being directed so that the molecular beam does not enter it. Then the influence of pressure fluctuations in the installation, identical ...

acting on both manometric systems, can be eliminated by means of a compensation circuit^37.

As high-sensitivity manometers, both ionization instruments and hot-filament instruments (Pirani manometers) are used. Manometers of the first group are more sensitive to heavy atoms with low ionization potentials (Hg); Pirani manometers are used mainly for light gases, such as, for example, He, H$_2$, etc. Effective compensation is considerably easier to accomplish with hot-filament manometers. Therefore these manometers are used much more often than others, and only they will be discussed below.

Before Pirani manometers were applied in experiments with molecular beams, it was known from the literature that they have a sensitivity to pressures of the order of $10^{-5}$ mm Hg, so that the use of these manometers for investigations of molecular beams encountered the necessity of increasing their sensitivity by at least a factor of 1000. In order to understand how this was achieved, it is useful to examine, at least in an approximate theory, this instrument^13, ^37. The principle used in Pirani manometers consists in the fact that the temperature of a filament heated by a source of constant power depends on the pressure of the surrounding gas. By recording the change in the temperature of the filament through changes in its ohmic resistance, we can thus judge changes in the pressure around the filament. The filament temperature corresponding to thermal equilibrium is determined by the condition of equality between the energy liberated by the heating current and the energy of the heat losses, consisting of: a) losses due to thermal conduction of the surrounding gas, b) losses to radiation, and c) losses due to thermal conduction at the ends of the filament. Only the first part of the heat losses depends on the gas pressure (namely, at sufficiently low pressures, which are precisely those occurring in experiments with molecular beams, it is proportional to $p$). Therefore, to increase the sensitivity of the manometer it is necessary to arrange that the heat losses due to thermal conduction of the surrounding gas be as large as possible and, in any case, exceed all the other types of losses by many times. The fraction of the thermal energy lost at the ends of the filament can be made negligibly small if the length of the filament is large in comparison with its transverse dimensions, and the filament itself is made in the form of a thin ribbon in order to increase its surface. The energy lost because of the thermal conduction of the surrounding gas is equal to

\[ E_C=\frac{1}{4}\,\bar{n}c\bar{u}(T-T')a(C_v/N_A), \tag{8} \]

where $a$ is a numerical coefficient, $C_v$ is the molar heat capacity, $N_A$ is Avogadro’s number, $T$ is the temperature of the filament, $T'$ is the temperature of the chamber walls, and $a$ is the surface area of the filament. Finally, the energy lost to radiation is equal to

\[ E_R=\varepsilon\sigma(T^4-T'^4)a, \tag{9} \]

where \(\varepsilon\) is the emissivity, and \(\sigma\) is the radiation constant. Neglecting heat losses at the ends of the filament, we can write the condition of thermal equilibrium, in the case of a constant heating energy \(E\), in the following form:

\[ E=E_R+E_C=a\left[\varepsilon A\left(T^4-T'^4\right)+Bap\left(T-T'\right)\right]. \tag{10} \]

Here \(A\) and \(B\) are constants that can be obtained from equations (8), (9), and (2). It follows from equation (10) that, since in order to increase the sensitivity of the manometer \(E_C\) must be made considerably larger than \(E_R\), the coefficient \(a\) of the filament material should be as large as possible, while \(\varepsilon\) should be as small as possible. This requirement is not always easy to satisfy, since well-polished surfaces have not only low emissivity \(\varepsilon\), but, as a rule, also a small coefficient \(a\). In addition, the operating temperature of the filament must be low, and the temperature difference between the filament and the chamber walls as large as possible. For this purpose the entire manometer is immersed in a vessel with liquid air, so that the temperature of the heated filament proves to be below room temperature. A good material for making the heating filament is nickel, which has a very large temperature coefficient of resistance. Nickel is sufficiently soft, which makes it possible easily to roll nickel wire into a thin ribbon; in practice ribbons about \(0.3\) micron thick have been used. Figures 16 and 17 show two manometers of this type. The first manometer\(^{37}\), made of glass, is fastened outside the vacuum casing of the experimental apparatus and is connected with the detector slit by means of a thin tube; the second\(^{29}\) is made in a metal case, is placed inside the apparatus, and is cooled owing to the thermal conductivity of the metal by means of liquid air.

Fig. 16. Pirani manometer (glass).

Fig. 17. Pirani manometer with small volume. \(S\) — detector slit; \(I\) — insulator.

The use of a compensation circuit reduces not only the influence of pressure fluctuations in the collimator chamber, but also the influence of temperature fluctuations, and also considerably reduces the “creep” of the zero during heating of the manometers, until thermal equilibrium has been reached. To implement compensation, both manometers are placed in one temperature bath, or else, in the case of a metal construction, are simply embedded in one copper block, and their filaments are connected into opposite arms of a Wheatstone bridge. The other two arms of the bridge are usually kept constant during the experiment, and thus the readings of the galvanometer connected in the bridge diagonal are attributed entirely to changes

the resistance of the filament of the manometer registering the beam. For continuous monitoring of the zero setting along the path of the molecular beam, a movable shutter is mounted, and readings are taken at equal time intervals, alternately with the shutter open and with it closed. In order to make it possible to measure simultaneously both the deflections of the main beam and the deflections of individual molecules, the galvanometer is provided with an Ayrton shunt.

The time required for equilibrium to be established in the manometer depends on three factors. It increases with the volume of the detector chamber, including the volume of the manometer itself, and with the increase of the flow resistance of the detector slit or channel, and decreases as the temperature of the filament increases. To obtain large \(K\) (\(\sim 50\)), the volume of the manometer must be very small (\(\sim 1\ \mathrm{cm}^3\)); in this case, the increase in detector sensitivity, which, according to equation (10), can be achieved by lowering the filament temperature, is in practice limited by the simultaneous increase in the time required for the readings to become established. It should be noted that this equilibrium-establishment time is also proportional to \(\sqrt{M}\) of the gas, and the effect of absorption of the gas by the walls of the manometer may increase this time so much that the use of the manometer becomes impossible. As a reasonable limit one should adopt the requirement that the time during which a reading is established equal to \(99\%\) of the equilibrium value should not exceed one minute. The best operating conditions are usually determined by direct selection in experiment.

In apparatus for the study of molecular beams, provision is usually made for the possibility of moving the detector slit in the plane of the beam cross section; in this case the detector is moved either together with the manometer or, if a flexible connection between them is provided, without the manometer. A system for displacing the detector can be implemented with the aid of various kinds of hinges, micrometer screws, etc.

Thermal detectors

This group includes various devices that use the thermal energy of the molecular beam for its registration. None of these devices has found wide application; however, they have been used in certain cases for the investigation of specific problems when more advanced methods proved unsuitable. Among them one should mention the radiometer\(^{8, 49}\), which can be constructed with a view to a short equilibrium-establishment time and a sensitivity of the order of \(0.1\) radian of deflection per microdyne. A molecular beam \(1\ \mathrm{cm}\) high has, at a distance of \(10\ \mathrm{cm}\) from the source, under the best operating conditions in the furnace, a pressure of the order of \(10\ \mathrm{dyn}^{-3}/\mathrm{cm}^2\) on the detector. Thus the radiometer can be used as a direct-counting detector for short beams, but the sensitivity of this method

is considerably inferior to the sensitivity of the methods described in the preceding sections.

Thermoelements and bolometers have too low a sensitivity for recording such small portions of energy as are possessed by the molecules of a beam; however, they may be used in cases where, when the beam interacts with the surface of the detector “target,” heat is released by some chemical reaction. Thus, for example, this occurs for beams of atomic hydrogen or oxygen, which, upon striking the detector surface, release chemical thermal energy of the order of \(10^5\) cal/mole, whereas the kinetic energy of such beams is only \(\sim 10^3\) cal/mole. In the same way beams consisting of free radicals may be registered. Finally, as a kind of bolometer intended for detecting condensing substances, a narrow strip of nickel was used; in this case the heat released upon sublimation, amounting to about \(10^4\) cal/mole, was utilized.\(^{22}\)

Detectors with Space Charge

All the detectors described so far are suitable for measuring molecular beams formed only by definite substances; detectors with surface ionization and manometers with a heated filament have one and the same range of application. However, there exists a whole series of substances for which all the described methods of detection prove unsuitable. We shall now describe a method, not yet developed in detail, but promising to become a universal means of measuring beams. This method is based on the property of positive ions to destroy a cloud of electronic space charge.\(^{35}\) If a heated tungsten filament emitting electrons is placed along the axis of a positively charged anodic cylinder (Fig. 18), then, at a given voltage and a sufficiently high filament temperature, the magnitude of the electron current will be limited owing to the formation around the filament of a cloud of negative space charge. If gas molecules enter the cylinder from outside, they are ionized as a result of collisions with fast electrons; the positive ions formed thereby partially neutralize the negative charge of the space cloud around the filament, and the anodic current consequently increases. This effect, produced by the gas ions, is considerably enhanced by the circumstance that the ions do not reach the cathode at once, but move along the filament

Fig. 18. Detector with space charge (Kingdon camera). \(F\)—heated filament; \(A\)—anodic cylinder; \(S\)—detector slit.

along spiral trajectories. The effect can be further enhanced if the anode cylinder is completely closed off at both ends and only two openings are left, through which pass the filament and the slit through which the beam molecules enter. The point is that, just as in the case of the manometric detectors discussed above, the entry of beam molecules into the resulting “chamber” causes an increase of pressure in it (if one assumes that no condensation occurs on the walls of the cylinder), which promotes the destruction of the space charge. For ionization of the molecules entering the “chamber” it is only necessary that the voltage applied to the anode cylinder be greater than the ionization potential of these molecules. Under optimum conditions, mercury vapor at a pressure of \(10^{-8}\) mm can cause an increase in the anode current, owing to partial destruction of the space electron cloud, by \(0.15\) mA. By using a second such chamber and a compensation circuit, we can, first, get rid of the initial anode current and, second, eliminate the influence of fluctuating pressure changes in the apparatus.

Detectors with space charge were first used for beams of mercury atoms, for which satisfactory methods of recording had previously not existed at all; the detector proved to be very sensitive, capable of indicating the presence of mercury vapor at a pressure of the order of \(10^{-11}\) mm Hg in the chamber. This method, in all probability, will be quite universal, since the molecules and atoms of any substances are ionized under the action of a stream of electrons. If the substance of the beam condenses at room temperature, then the temperature of the chamber walls can be raised above the critical condensation temperature by using the thermal effect of electron bombardment. Successful experiments have been carried out with beams of molecules of organic substances, including benzophenone\(^{23}\), and with some methyl salts of halides\(^{32}\). Nevertheless, it should be emphasized that further substantial technical development of this method is still required. In any case, the results obtained so far are very promising.

6. VACUUM SYSTEM

It is quite obvious that, in order to preserve the intensity and geometrical shape of a molecular beam, it is necessary that the scattering of the molecules composing it in the residual gas in the apparatus be insignificant. Therefore it is extremely important to maintain inside the apparatus such a low pressure that the mean free path \(\lambda\) of the beam molecules is several times greater than the actual length of the beam \(l\). (If \(\lambda = l\), then only \(36\%\), or \(1/e\), of the molecules leaving the source will reach the detector.) As was already indicated in Section 3, the pressure inside the apparatus increases as a result of its being filled with the beam molecules; consequently, it is necessary to ensure continuous removal of these molecules. If the substance whose molecules form

beam, is readily condensed, then for this purpose it is very expedient to use large traps cooled by liquid air (1 cm² of cooled surface, in the case of condensable vapors, gives an effectiveness corresponding to a pumping speed of about 11 liters per second). If, however, the substance of whose molecules the beam consists is gaseous, pumps with high pumping speeds are necessary; moreover, the cross sections of all connecting tubes, vapor traps, etc., must be sufficiently large, so that the effective pumping speed from the apparatus may amount to from 10 to 100 liters per second. If one recalls that the length of the molecular beam, which in Stern’s early experiments was only 6 cm, can now be more than two meters, then one can understand that this has become possible only thanks to the rapid progress of vacuum technique. The general principles of vacuum technique apply to the construction of the vacuum jacket of an apparatus for producing a molecular beam; a specific feature is the division of the apparatus into a furnace chamber and a collimator chamber, which are pumped by separate pumps, so that it is always possible to maintain a higher vacuum in the second chamber than in the first. Therefore, as was already mentioned earlier, the length of the beam in the furnace chamber must be made as short as possible, and the two chambers must be connected by a long narrow channel with a large resistance to the flow. We shall confine ourselves to these general considerations; it is difficult to give any concise survey of specific designs, since in almost every case of investigating a definite problem the design of the apparatus was developed anew.

7. EXPERIMENTS WITH MOLECULAR BEAMS

It is not the purpose of this article to describe the details of the experimental problems solved by means of the molecular-beam method. Therefore we shall dwell only on the principal characteristic features inherent in work with molecular beams. In the main, the molecular-beam method makes it possible to investigate the effects of the action of physical forces on individual atoms or molecules; moreover, these effects are observed directly, and the method itself is, at least in principle, extremely simple. The aims pursued in work with molecular beams may be divided into two groups: a) direct experimental verification of theoretical propositions of fundamental importance, and b) measurements of various parameters of atoms or molecules. In some cases these parameters cannot be measured by any other method; in other cases the molecular-beam method possesses greater sensitivity and gives more accurate results.

Among the problems of the first group solved by means of the molecular-beam method are the confirmation of the rectilinear motion of gas molecules¹²; the measurement of the velocity distribution of molecules in a beam⁸, ⁴³, ⁶⁰;

direct demonstration of the spatial quantization of molecules (the Stern–Gerlach experiment) ^31; verification of the wave properties of particles of matter ^19 and confirmation of the de Broglie equation \(\lambda = \frac{h}{mv}\) ^20, as well as many other most important problems. To the second group belong measurements of the electric and magnetic moments of atoms and molecules, of the effective cross sections of molecular collisions, and the exceptionally important measurements of the spin and magnetic moments of nuclei.

Experiments with molecular beams are in many respects analogous to optical experiments. This is already evident from consideration of the scheme for obtaining a beam by means of a source and a system of collimating slits. However, this analogy becomes still more complete from the point of view of de Broglie’s theory, which connects with the motion of particles of mass \(m\) and velocity \(v\) the existence of a wave of length \(\lambda = \frac{h}{mv}\). We may speak of the “geometrical optics” of a molecular beam when the wave properties of the particles are inessential, and of the “wave mechanics” of a molecular beam if it is necessary to take these wave properties into account.

An ordinary molecular beam consists of molecules with a wide range of velocities and therefore corresponds to white light. The spectrum of velocities in the beam is expressed by the corrected Maxwell distribution law ^61

\[ dN/N_0 = (2v^3/a^3)\exp(-v^2/a^2)\,d(v/a), \tag{11} \]

where \(dN/N_0\) is the fraction of beam molecules whose velocities lie between \(v\) and \(v + dv\), and \(a = (2RT/M)^{1/2}\) is the most probable velocity. (The “correction” of Maxwell’s law for the case of a beam consists in replacing \(v^2\) by \(v^3\). This is necessary, since the probability that a molecule will fly out of the furnace slit is proportional to \(v\).)

Monochromator

Experiments in optics are considerably simplified when monochromatic light is used. The same is in principle true also in the technique of molecular beams. Of course, there are no sources that would give a monochromatic molecular beam, i.e., a beam all of whose molecules would have identical velocities. Monochromatization of molecular beams can be carried out either by sorting molecules according to velocities with the aid of special mechanical devices ^20, or by spreading out the beam with the aid of some deflecting forces into a “spectrum” and isolating a definite portion of this spectrum with the aid of a special slit ^20, ^55. The second method is entirely analogous to optical monochromatization and will be discussed in the corresponding section. Here we shall consider in more detail the first method, which has much in common with Fizeau’s experiment for determining the speed of light.

Let there be two disks, each of which has at its edge a narrow short slit directed along the radius, rotating on a common axis. Such a device is placed either in front of or behind the collimating slit of the apparatus for obtaining a molecular beam. If the slit in the second disk is displaced relative to the slit in the first disk by an angle \(\theta\) (in the direction opposite to the direction of rotation), then only certain molecules will be able to fly through both slits, namely molecules possessing such velocities that during the time \(\Delta t\) in which they traverse the distance \(l\) between the two disks, the disks turn through the angle \(\theta\). If the velocity of the molecules is equal to \(v\), and the angular velocity of rotation of the disks is \(\omega\), then we obtain the equation

\[ \Delta t=\frac{l}{v}=\frac{\theta}{\omega}. \tag{12} \]

If the width of the slits, measured in radians, is equal to \(\gamma\), then the velocities of the molecules that can fly through both slits lie between \(v_1\) and \(v_2\), equal to

\[ v_1=\frac{\omega l}{\theta-\gamma}\quad \text{and}\quad v_2=\frac{\omega l}{\theta+\gamma}. \tag{13} \]

To select molecules having velocities within the limits from \(v_1\) to \(v_2\), it is necessary to choose the values of \(\gamma\) and \(l\) so that they satisfy the relations

\[ \omega=2\pi\nu=\frac{2\gamma(v_1v_2)}{l(v_1-v_2)} \tag{14} \]

and

\[ \theta=\gamma(v_1+v_2)/(v_1-v_2). \tag{15} \]

In practice both disks are provided with a large number of radial slits, separated from one another by an equal number of angular degrees, and the disks are mounted in such a way that all the slits of one disk are parallel to the corresponding slits of the other. Let us number all the slits, beginning from some radius, and consider the molecules that fly through slit \(i\) on the first disk and slit \(i+1\) on the second; for such molecules \(\theta=2\pi/n\). The fraction of the total number of molecules of the beam that pass through the slits of the first disk is \(\gamma/(2\pi/n)\); therefore it is important that the number of slits \(n\) be as large as possible.

It should be noted that, besides the molecules mentioned, slower molecules will also pass through both disks, namely molecules that enter the \((i+2),\ldots,(i+k)\) slits of the second disk. For such molecules \(\theta=k\,2\pi/n\). They correspond to a spectrum of “higher order.” If, under the conditions of the experiment, only a spectrum of a quite definite order is needed, additional disks are installed for this purpose. By choosing a spectrum of higher order, we can increase

reducing the resolving power of this molecular spectrograph without diminishing the intensity of the beam.

As a numerical example, let us take data from an actual experimental setup[^20]: \(l = 3.1\ \text{cm}\), \(n = 408\), \(\gamma = 2 \cdot 10^{-3}\), \(\nu = 133\ \text{rev/sec}\), \(v_1 = 1.93 \cdot 10^5\ \text{cm/sec}\), and \(v_2 = 1.49 \cdot 10^5\ \text{cm/sec}\).

Reflection

The very first investigations of the laws of reflection of molecules confirmed the validity of the cosine law even for well-polished surfaces[^40]. From the standpoint of de Broglie’s theory, this fact is not unexpected, since the order of magnitude of the wavelengths associated, according to this theory, with the motion of the molecules of a beam fluctuates around \(10^{-8}\ \text{cm}\), so that even the most carefully polished surface is rough for them. It is known, however, that even from rough surfaces light can be reflected according to the mirror law if the ray falls on the surface at a small angle to it. Corresponding experiments were carried out with the reflection of \(H_2\) beams from mirror-like metallic surfaces at angles from 0.5 to \(3 \cdot 10^{-3}\) radians[^37]. The results of these experiments fully confirmed the expected effect. The intensity of the reflected beam (the reflection coefficient) increased rapidly as the angle at which the beam fell on the reflecting surface decreased, from \(0.75\%\) at an angle of \(2.25 \cdot 10^{-3}\) radians to \(6\%\) at \(1 \cdot 10^{-3}\) radians. With a decrease in the temperature of the beam and, consequently, with an increase in the mean de Broglie wavelength, the reflection coefficient also increased. These experiments gave the first evidence of the existence of wave properties of molecular beams. However, considerably more complete information about the wave properties of moving particles was obtained as a result of the experiments described below.

Refraction

The deflection of a molecular beam in a force field is analogous to the refraction of a ray in geometrical optics. We shall distinguish two cases: the motion of particles in a space with a continuously varying refractive index \(n\), and the transition of a particle from a region with refractive index \(n_1\) into a region with refractive index \(n_2\). The first case occurs most often in experiments with molecular beams; the second is most often considered in geometrical optics, although it too has been realized in some experiments with beams. As an example of the first case, we may consider the motion of a particle in the field of the force of gravity. The trajectory of such a particle can easily be calculated with the aid of the usual equations of mechanics, but we may also investigate this question by the methods of geometrical optics. In such a treatment, the relation between the refractive index, on the one hand, and the potential energy of the particle in the po-

le \(U\) and the total energy of the particle \(E\), on the other hand, is expressed by the equation

\[ n=\left[(E-U)/E\right]^{1/2}, \]

and the path of the particle between two points \(P_1\) and \(P_2\) can be calculated from Fermat’s principle:

\[ \delta \int_{P_1}^{P_2} n\,ds=0. \]

In a gravitational field \(U=mgh\), and consequently the refractive index varies proportionally to \(h^{1/2}\).

Let us take the conditions of a real experiment\(^{25}\), in which a molecule, possessing velocity \(v\), flies through two slits \(s_1, s_2\), placed on one horizontal line at a distance \(l\) from one another (Fig. 19). The trajectory of the molecule will then be a parabola, and the displacement of the molecule along the vertical \(s\) at a distance \(l\) from \(s_2\) is equal to

\[ s=g(l^2/v^2), \]

where \(g\) is the acceleration due to gravity.

Fig. 19. Deflection of a molecular beam under the action of gravity.

However, the velocities of the molecules in the beam are not the same; moreover, we have assumed up to now that the distribution of these velocities corresponds to the corrected Maxwell law. With the aid of a detector \(D\), located at a distance \(l\) from \(s_2\) and moved in the vertical plane, we can measure the intensity distribution, which will correspond to the distribution of molecular velocities in the beam (if the slits and the dimensions of the detector are considered infinitely small). Such an experimental arrangement, a kind of molecular spectrograph, was used to measure the actual distribution of molecular velocities in the beam. It turned out that this distribution does not always coincide with Maxwell’s law. The same arrangement, obviously, can also serve as a monochromator; indeed, a slit displaced by \(s\) from the line of the slits \(s_1\) and \(s_2\) and separated from the second slit by \(l\), will select molecules having velocities \(v=l(g/s)^{1/2}\).

Owing to the finite width of the slits, the observed pattern of the intensity distribution is in reality more complicated. If it is assumed that the width of both slits is \(b\), then the intensity distribution in the undeflected beam would be a symmetric trapezoid, and the distribution in the deflected beam is represented by the curve shown in Fig. 20, which can be calculated from the equation

\[ \frac{I}{I_0} = \frac{s_x}{b} \left\{ \frac{s}{s_x}\exp\left(-\frac{s_x}{s}\right) - \frac{s-b}{s_x}\exp\left[-\frac{s_x}{s-b}\right] - \frac{s-2b}{s_x}\exp\left[-\frac{s_x}{s-2b}\right] + \frac{s-3b}{s_x}\exp\left[-\frac{s_x}{s-3b}\right] \right\}. \tag{16} \]

In this equation \(s_a\) is the deflection of the molecules possessing the most probable velocity \(a\), \(I_0\) is the maximum intensity of the undeflected beam, and \(I\) is the intensity at a distance \(s\) from the upper edge of the undeflected beam. To obtain equation (16) one must first consider the intensity distribution of an infinitely small beam of molecules, which are contained on the segment \(ds\) of the trapezoid representing the intensity distribution of the undeflected beam, and then integrate the result over the entire trapezoid[^64]. If the intensity distribution in the undeflected beam is not trapezoidal, then, as a result of the integration, one obtains, of course, an equation different from equation (16).

Fig. 20. Intensity distribution in a deflected beam.

Deflection in a magnetic field takes place in a completely analogous way. For a molecule possessing a magnetic moment \(\mu\), the potential energy is

\[ U=\Delta E_m=-\mu_H H, \tag{17} \]

where \(\mu_H\) is the component of the magnetic moment in the direction of the field \(H\). Thus, the refractive index is equal to

\[ n=[(E+\mu_H H)/E]^{1/2}. \tag{18} \]

If \(dH/dz\) is the field gradient in the direction perpendicular to the beam, then the deflecting force acting on the molecule is equal to

\[ F_m=\mu_H(dH/dz), \tag{19} \]

and the acceleration of the molecule in the direction of the field gradient is

\[ a_m=(\mu_H/m)\cdot(dH/dz), \tag{20} \]

where \(m\) is the mass of the molecule.

According to the theory of space quantization, \(\mu_H\) can have only discrete values

\[ \mu_H=g_H m_j\mu_B, \tag{21} \]

where \(\mu_B\) is the Bohr magneton,

\[ \mu_B=\frac{eh}{4\pi mc}, \]

\(g_H\) is the Landé factor, and \(m_j\) is the magnetic quantum number, related to the internal quantum number of the atom \(l\) by the relation

\[ m_j=l,\ l-1,\ l-2,\ldots,\ -l. \]

A beam of atoms in states with quantum number \(l\), passing through an inhomogeneous magnetic field, will therefore split into \(2l+1\) beams, which will be deflected by the field in different ways depending on the value of \(m_j\) for their atoms. Classi-

…ical Stern–Gerlach experiment was performed in order to test the theory of spatial quantization. The inhomogeneous magnetic field was produced by means of two pole pieces, whose shape is shown in Fig. 21. The beam consisted of silver atoms with \(I=1/2\) and \(g_k=2\). In this case \(m_j\) is equal to \(\pm 1/2\), and

\[ a_m=\pm\left(\frac{\mu_B}{m}\right)\cdot\left(\frac{dH}{dz}\right), \]

which means that the beam is split into two beams deflected in opposite directions.

Fig. 21. Pole pieces in the Stern–Gerlach experiment.

A large number of similar experiments have been carried out. In order to estimate the magnitude of the deflection, let us take a numerical example. Let the gradient of the magnetic field between the poles shown in Fig. 21, at the place where the beam passes, be \(dH/dz=5\cdot 10^4\) gauss/cm; the length of the region of space filled with the magnetic field is about \(10\) cm; \(\mu_B\approx 10^{-20}\) electromagnetic units. For \(K\)-atoms \(I=1/2\), \(g_k=2\), \(m=6.7\cdot 10^{-23}\) g, and the most probable velocity (at an oven temperature of \(450^\circ\) K)

\[ a=\left(\frac{2RT}{M}\right)^{1/2}=4.33\cdot 10^4\ \text{cm/sec}. \]

Then at the ends of the pole pieces

\[ s_a=\frac{1}{2}a_m t^2=\pm\left(\frac{\mu_B}{m}\right)\cdot\left(\frac{dH}{dz}\right)\left(\frac{l^2}{a^2}\right)\approx 0.83\ \text{mm}. \tag{22} \]

The distribution of intensity in each of the deflected beams can be calculated from equation (16); as in the case of deflection of a beam in a gravitational field, it depends on the law of the distribution of velocities in the undeflected beam. Therefore deflection in a magnetic field may likewise be used either for beam spectroscopy\({}^{11}\) or for purposes of monochromatizing them\({}^{55}\); in the latter case, a slit displaced by an amount \(s\) from the axis of the undeflected beam will select molecules whose velocities lie within limits expressed by an equation analogous to equation (13).

It should be added that the splitting of the beam described above corresponds to the case of double refraction (or multiple refraction) in optics, and that each of the separated beams is “polarized,” i.e., the vectors of the mechanical moments of the atoms in each beam have a definite direction or precess about the direction of the field in such a way that the angle between the vectors \(I\) and \(H\) remains constant.

The same experimental arrangement can be used to measure the magnetic moments of atomic nuclei. However, the magnitudes of the latter are of order \(10^{-3}\mu_B\), while the greatest value of the field gradient \(dH/dz\) that can be obtained in practice is about \(5\cdot 10^5\) electromagnetic units; thus the beam deflections arising from the interaction of the nuclear moments with the field are considerably smaller than those obtained in the experiments described…

above. Moreover, the direct application of this method in such a simple form for measuring nuclear moments is possible only in the case where the magnetic moments of the electron shells of the atoms are equal to zero, for example, in the case of molecules of the type \(H_2\), \(HD\), and \(D_2\). With the aid of experiments of this kind, numerical values of the magnetic moments of the proton and the deuteron were obtained for the first time.

The optical case of the passage of a ray from a homogeneous medium I with refractive index \(n_1\) into medium II with refractive index \(n_2\) was also realized in experiments on the magnetic refraction of molecular beams\(^{54}\). Let a beam \(XO\) of \(K\)-atoms, having velocity \(v\), pass from region \(A\), in which the magnetic field is zero, into region \(B\), where there is a homogeneous field of intensity \(H\). In accordance with Snell’s law,

Fig. 22. Refraction of a molecular beam.

\[ \frac{\sin\theta}{\sin\theta'}=\frac{n_2}{n_1} =\left(\frac{E \mp \mu_b H}{E}\right)^{\frac12}\simeq \left(1\mp\frac{\mu_b H}{\frac12 mv^2}\right)^{\frac12}, \tag{23} \]

where \(\theta\) and \(\theta'\) are the angles between the normal to the interface of regions \(A\) and \(B\) and the directions of the incident and refracted beams (Fig. 22).

The deviation of the beam from its initial direction \(\delta\), which is always sufficiently small that one may approximately put \(\cos\delta=1\), \(\sin\delta=\delta\), is expressed as follows:

\[ \delta=\pm(\theta'-\theta)=\pm\frac{\mu_b H}{mv^2}\,\operatorname{tg}\theta . \tag{24} \]

Thus, the initial beam is split into two polarized beams deflected in opposite directions. If the state of the atoms of the beam is characterized by the value of the internal quantum number \(J\), then upon passing into the region with the magnetic field the beam splits into \(2J+1\) separate beams.

The refraction of beams in an electric field\(^{18,21,69}\) may be obtained by analogous methods. To obtain refraction in a “continuous medium,” the beam is passed through an inhomogeneous field, for example, near and parallel to the axis of a cylindrical capacitor with electrode radii \(r_1\) and \(r_2\). If the distance of the beam from the cylinder axis is \(r\), and the potential difference on the capacitor is \(V\), then the beam passes through a field with intensity \(E=\dfrac{V}{r\ln r_1/r_2}\) and with gradient \(\dfrac{dE}{dr}=V\cdot\dfrac{1}{r^2\ln r_1/r_2}\). Other field sources were also constructed in which the deflecting force retains a constant value over some length\(^{50}\). To realize the case of refraction at the boundary of two media, the beam is intro-

is introduced at some angle into the space between the plates of a plane capacitor; for the calculation, the equations we have given for deflection in a magnetic field are valid. Atoms do not possess an intrinsic electric moment; therefore, in experiments with deflections in an electric field we measure the electric moment of atoms induced by the external field, i.e., the degree of polarizability of the atoms. In the case of complex molecules, the deflection is determined both by the polarization of the molecule in the external field and by the molecule’s intrinsic electric moment \(\mu_E\), if such exists. However, because of the thermal rotation of such molecules, the picture of spatial quantization is considerably more complicated than in the case of atoms, and the interpretation of the observed deflections in fields is very difficult.

Diffraction

Let us consider the motion of the molecules of a beam from the standpoint of wave mechanics. In accordance with de Broglie’s equation, a particle of mass \(m\), flying with velocity \(v\), corresponds to a wave of length \(\lambda = \dfrac{h}{mv}\). For light atoms or molecules (H, He, \(\mathrm{H}_2\)) having thermal velocities, the order of magnitude of \(\lambda\) is \(10^{-8}\) cm, and for other molecular beams it is still smaller. Therefore, on ordinary slits, whose width is about \(10^{-3}\) cm, there is no possibility of observing diffraction phenomena for waves of such small length. These considerations suggested the idea that, in order to observe particle diffraction, a crystal should be used as the diffraction grating. The diffraction patterns obtained with beams of H, He, and \(\mathrm{H}_2\) reflected from the surface of an LiF crystal\({}^{19,20,34}\) agree fully with the pattern expected for the diffraction of waves by a spatial grating. Using a monochromator analogous to that described above to select particles with a definite velocity \(v\), and determining the wavelength \(\lambda\) from the measured angles of the diffraction maxima, it was possible to verify the validity of de Broglie’s equation to an accuracy of up to 1%. The results of experiments on the diffraction of molecular beams give considerably more than the first experiments on electron diffraction, since they show that de Broglie’s equation is valid not only for elementary particles, but also for atoms and even for complex molecules, and, in addition, they make it possible to verify directly the quantitative dependence of wavelength on mass.

The technique for carrying out the experiment on the diffraction of He atoms and \(\mathrm{H}_2\) molecules is, in brief, as follows: the beam, emitted by a heated or cooled furnace, is directed onto a fresh cut of an LiF crystal at an angle of 10 to 20° to the surface of the cut, and the space around the crystal is investigated with the aid of a detector slit connected to a Pirani manometer. Using a slit selector together with the detector slit, we obtain a “monochromatic” beam, since

the selector passes atoms with a definite wavelength. In a “spectrograph” with two crystals,^20 the monochromatization of the beams was carried out by the first crystal, and the analysis by the second. Of the large number of other crystalline substances, only NaCl makes it possible to obtain diffraction patterns; however, these too are obtained with lower intensity than in the case of LiF crystals. In experiments with atomic hydrogen^34 the diffracted atoms were recorded by means of chemical detectors.

Diffraction phenomena of a molecular beam can also be detected in experiments on scattering the beam in gases. Whereas scattering from a purely classical point of view is regarded as the result of elastic collisions occurring between ball-molecules, from the point of view of quantum mechanics this phenomenon is the result of diffraction of de Broglie waves. Many scattering experiments have been carried out with the aid of molecular-beam technique; their results confirmed the validity of the quantum-mechanical point of view and made it possible to determine the effective scattering cross section more accurately than could be done by any other method.^27,39,57

Nuclear Spins and Magnetic Moments of Atoms

Although the questions to which this section is devoted have already been touched upon in some form above, the use of the method of molecular beams for solving nuclear problems^5 is of such great importance that it deserves a more detailed description even in an article devoted chiefly to the technical side of the matter.

If the quantum number determining the spin of the nucleus is denoted by \(I\), and the quantum number of the total mechanical moment of the electron shell by \(J\), then a monochromatic beam of atoms in a state with quantum numbers \(I\) and \(J\) will be split in a magnetic field into \(Z = (2I + 1)(2J + 1)\) beams, according to the number of different quantum states.^4 The magnetic energy of an atom in one of such states, and consequently also the refractive index of a beam of such atoms in an inhomogeneous field, depend not only on the field gradient \(dH/dz\), but also in a very complicated way on the magnitude \(H\) itself. In weak fields all \(Z\) energy levels of atoms in the magnetic field follow one another through almost equal energy intervals, and, by counting the number of deflected beams, we can determine the spin quantum number of the nuclei \(I\). In an experiment with Na atoms^55 \(\left(J = \dfrac{1}{2}\right)\) the beam was monochromatized by means of a strong magnetic field, in which it was split into two beams with different values of \(m_j\); after this, the atoms that had passed through the selector slit were passed through a long region of a weak inhomogeneous field; as a result the beam was split into \(2I + 1 = 4\) beams, whence \(I = 3/2\). The magnetic moments of nuclei can

can be calculated from the distances between the maxima of the separated beams on the basis of the theory of the hyperfine structure of spectra.

An analysis of the dependence of all \(Z\)-levels of the magnetic energy on \(H\) shows that the energy of some levels, at certain values of \(H\), passes through zero. By measuring these values of \(H\), one can calculate both the nuclear spins and the structure of the hyperfine structure of the spectrum. This method, known as the method of zero moments, does not require preliminary monochromatization of the beam\(^{51}\). The detector is placed on the axis of the undeflected beam at the far end of a weak inhomogeneous magnetic field. The magnitude of the field is increased monotonically, and the values of \(H\) are noted at which the detector registers peaks of “zero beams.” From the known structure of the hyperfine structure it would be possible to calculate also the magnetic moment of the nucleus, if only the wave function of the given atom were known. However, for all atoms except H and D, the wave function is known only very approximately.

Reorientation

The study of this effect\(^{80}\), which, by analogy with optics, may be called a change in the direction of polarization of the beam, proved to be an extremely powerful stimulus for new applications of the molecular-beam method. If a beam of atoms, selected by a field, with a definite spatial orientation of the moment, i.e., a polarized beam, is passed through a second magnetic field, then the atoms of the beam may remain in the former quantum state (preserve their orientation), or may “jump” into some other one. Such reorientation of atoms may be caused by abrupt changes of \(H\); however, the condition most favorable for the appearance of reorientation is the passage of the beam through such a homogeneous field varying in time that the frequency of its variable component is equal to the Larmor frequency of precession of the atoms caused by the constant component of this field. The use of the phenomenon of reorientation for measuring the magnetic moments of nuclei is one of the most elegant applications of molecular-beam technique\(^{56}\).

First the beam is split into several polarized beams in a strong inhomogeneous field. If, following this field, a second field is produced with an oppositely directed gradient, then the diverged beams will again converge at a definite point, at which the detector is placed. Such refocusing will occur only for those atoms which, in passing through both fields, remain in the former quantum state. Let us impose, in the short space between the two indicated fields, a constant homogeneous magnetic field of strength \(H\) and an alternating field varying in time with frequency \(\nu_f\). Then the Larmor precession of the atoms of the beam in the central field will occur with frequency \(\nu_L = \frac{\mu H}{h}\). If po-

choose such a value of \(H\) that \(\nu_L\) will be equal to the frequency \(\nu_f\) of the alternating field, then a sharply expressed resonance will arise, which will produce the phenomenon described above of reorientation of atoms with magnetic moment \(\mu\). Such reoriented atoms will no longer be refocused by the second inhomogeneous field and therefore will not reach the detector. Consequently, the indicated resonance will cause a decrease in the intensity at the detector. By observing the minimal peaks in the detector readings and noting the corresponding values of the Larmor frequency, we can calculate the magnitude of the magnetic moment \(\mu\).

Beam balancing

The technique of this method, which is only now being developed, consists in compensating the acceleration due to the force of gravity acting on the molecules by means of magnetic, electric, or other forces.^65 A specific feature of this method is its “null” character, since molecules for which complete compensation has been achieved are not deflected over the entire length of their path. In this case, for the quantitative evaluation of the results of the experiment, exact knowledge of the velocity distribution of the molecules in the beam becomes immaterial, which makes it possible to avoid the most serious difficulty causing uncertainty in the results of many experiments with molecular beams. Indeed, as was indicated, the actual velocity distribution in molecular beams does not always coincide with the theoretical law by means of which equations for calculating the intensity distribution, such as equation (16), were obtained. These deviations from theory are most serious for slow molecules and in many cases strongly affect the evaluation of experimental results. Therefore it should be expected that the balancing technique will be very useful in solving problems requiring measurements of high accuracy.

8. CONCLUDING REMARKS

We have attempted to give a survey of the methods and technical procedures used in obtaining molecular beams and in experimenting with them. Of course, it is impossible within the scope of this article to give a complete description of the numerous designs and individual problems encountered in work with molecular beams. Physical laboratory technique has not yet attained such perfection that experimental apparatus can be assembled from certain standard parts and units. On the contrary, at the present time, in order to solve each problem, one must design anew a whole series of parts of the apparatus. Therefore we have tried only to emphasize what is characteristic in one method or another and to indicate the basic principles of the individual components of experimental apparatus, wishing to convey to the reader such

information that will be useful to him both for a better mastery of the method and in the direct solution of special design problems.

In considering the various applications of the molecular-beam technique, we have also tried to single out only the basic outlines of one method or another and have emphasized the analogy with “light optics.” The description of the various experimental problems has, as far as possible, been given briefly, with an indication of several typical examples for each case of application; moreover, attention has always been drawn to the physical essence of the phenomena on which they are based.

There is one more general feature of experiments with molecular beams that should be mentioned, since it illustrates well the advantages of this method over others. In determining, for example, magnetic moments, we measure the energy of an atom or molecule in a magnetic field. The same can be done by means of an optical method, studying the Zeeman splitting \(\Delta \nu\) (or, in the case of measuring nuclear moments, studying the hyperfine structure of the optical spectrum). However, if \(\Delta \nu = \dfrac{\mu H}{hc}\) is very small, then the spectral lines cannot be resolved because of the natural width of the lines. At the same time, two parts of a molecular beam, deflected in different ways, can always be separated, provided only that the detector is moved to a sufficiently large distance. Considering this question from another side, we may apply the uncertainty principle in the form \(\Delta E \cdot \Delta t \simeq h\), where \(\Delta E\) is the smallest energy difference that can be measured in the time \(\Delta t\). The lifetime of atoms in optically excited states \(\Delta t\) is of the order of \(10^{-8}\) sec.; consequently, the smallest energy difference that can be measured by the optical method is \(\Delta E = 10^{-19}\) erg. The “lifetime” of a molecular beam in a field of length, say, \(10\) cm is of the order of \(10^{-3}\) sec.; hence, in principle, it is possible to measure energy differences of the order of \(10^{-24}\) erg. This comparison shows that in many cases the molecular-beam method can possess greater sensitivity than any optical method.

9. LITERATURE

The question of the molecular-beam technique has been considered in two books\(^{1,2}\) and in several review articles, some of which are listed below\(^{3-6}\). Most of the leading work carried out in Stern’s laboratory in Hamburg was described in a series of articles in the journal Zeitschrift für Physik between 1926 and 1933 under the title “Investigation of the molecular-beam method.” Of the numerous works, only those are cited below in which either for the first time, or in very great detail, certain essential technical innovations were described, as well as articles in which the physical essence of the phenomena was examined,

\(^9\) Uspekhi, Vol. XXXII, No. 1

laid as the basis of these technical methods. A very detailed list of works covering the period from 1930 to 1941 is contained in $^3$.

A. Works of a General Character

1. R. G. J. Fraser, Molecular Rays, Cambridge University Press, Cambridge (1931).

2. R. G. J. Fraser, Molecular Beams, Methuen and Company, Ltd., London (1937).

3. W. H. Bessey and O. C. Simpson, Recent work in molecular beams, Chem. Rev. 30, 239 (1942).

4. I. Estermann, Atom- und Molekularstrahlen. Handwörterbuch der Naturwissenschaften, G. Fischer Jena (1931), 1, 536.

5. D. R. Hamilton, Molecular beams and nuclear moments, Am. J. Phys. 9, 319 (1944).

6. W. H. Rodebush, Molecular rays, Rev. Mod. Phys. 3, 392 (1931).

B. Special Articles

1. G. Breit and I. I. Rabi, Phys. Rev. 38, 2032 (1931).

2. J. L. Costa, H. D. Smyth and K. T. Compton, Phys. Rev. 30, 349 (1927).

3. P. Clausing, Physica 9, 65 (1929).

4. P. Clausing, Zschr. f. Physik 63, 471 (1930).

5. V. W. Cohen und A. Ellet, Phys. Rev. 52, 502 (1937).

6. L. Dunoyer, Comptes rendus 152, 534 (1911); Le Radium 8, 142 (1911); 10, 400 (1913).

7. Ellet and R. M. Zabel, Phys. Rev. 37, 1102 (1931).

8. I. Estermann and O. Stern, Zschr. f. physik. Chemie 106, 399 (1923).

9. I. Estermann, Zschr. f. physik. Chemie 106, 403 (1923).

10. I. Estermann, Zschr. f. Electrochemie 31, 441 (1925).

11. I. Estermann, Zschr. f. Physik 33, 320 (1925).

12. I. Estermann, Zschr. f. physik. Chemie 1, 161 (1923).

13. I. Estermann and O. Stern, Zschr. f. Physik 61, 95 (1930).

14. Estermann, R. Frisch and O. Stern, Zschr. f. Physik 73, 348 (1931).

15. I. Estermann and R. G. J. Fraser, J. Chem. Phys. 1, 390 (1933).

16. I. Estermann and M. Wohlwill, Zschr. f. physik. Chemie 20, 195 (1933).

17. I. Estermann and O. Stern, Zschr. f. Physik 85, 135 (1933).

18. I. Estermann, O. C. Simpson and O. Stern, Phys. Rev. 52, 535 (1937).

19. I. Estermann, O. C. Simpson and O. Stern, Phys. Rev. 53, 947 (1938); 67, 346 (1944).

20. R. G. J. Fraser, Trans. Faraday Soc. 30, 182 (1934).

21. R. G. J. Fraser and L. F. Broadway, Proc. Roy. Soc. A141, 626 (1933).

22. R. G. J. Fraser and T. N. Jewitt, Proc. Roy. Soc. A160, 563 (1937).

23. R. Frisch and O. Stern, Zschr. f. Physik 85, 4 (1933).

24. R. Frisch, T. E. Phipps, E. Segrè and O. Stern, Nature 130, 829 (1932); Zschr. f. Physik 73, 185 (1931); 80, 610 (1933).

25. W. Gerlach und O. Stern, Ann. d. Physik 74, 673 (1924).

26. T. N. Jewitt, Phys. Rev. 46, 616 (1934).

27. T. H. Johnson, Phys. Rev. 31, 103 (1928).

1. T. H. Johnson, J. Frank. Inst. 210, 135 (1930).
2. K. H. Kingdon, Phys. Rev. 21, 408 (1923).
3. K. H. Kingdon, Phys. Rev. 23, 778 (1924).
4. F. Knauer and O. Stern, Zschr. f. Physik 53, 766 (1929).
5. F. Knauer and O. Stern, Zschr. f. Physik 39, 764 (1926).
6. F. Knauer, Zschr. f. Physik 80, 80 (1933).
7. M. Knudsen, The Kinetic Theory of Gases, Methuen and Company, Ltd., London (1934).
8. Kratzenstein, Zschr. f. Physik 93, 279 (1935).
9. E. O. Kurt and T. E. Phipps, Phys. Rev. 34, 1357 (1929).
10. B. Lammert, Zschr. f. Physik 56, 244 (1929).
11. I. Langmuir, Phys. Rev. 8, 149 (1916).
12. I. Langmuir and K. H. Kingdon, Phys. Rev. 24, 510 (1924); 34, 133 (1929).
13. I. Langmuir and K. H. Kingdon, Proc. Roy. Soc. 21, 380 (1923).
14. A. Leu, Zschr. f. Physik 41, 551 (1927).
15. A. Leu, Zschr. f. Physik 49, 498 (1928).
16. H. Mayer, Zschr. f. Physik 52, 235 (1928); 58, 373 (1929).
17. E. McMillan, Phys. Rev. 38, 1568 (1931).
18. S. Millman, Phys. Rev. 47, 739 (1935).
19. S. Millman, I. I. Rabi and J. R. Zacharias, Phys. Rev. 53, 384 (1938).
20. T. E. Phipps and M. J. Copley, Phys. Rev. 45, 344 (1934).
21. I. I. Rabi, Nature 123, 163 (1929); Zschr. f. Physik 54, 190 (1929).
22. I. I. Rabi and V. W. Cohen, Phys. Rev. 43, 582 (1933).
23. I. I. Rabi, S. Millman, P. Kusch and J. R. Zacharias, Phys. Rev. 55, 526 (1929).
24. S. Rosin and I. I. Rabi, Phys. Rev. 48, 373 (1935).
25. H. Scheffers, Physik. Zschr. 37, 220 (1936); 41, 399 (1940).
26. M. V. Smoluchowski, Ann. d. Physik 33, 1559 (1910).
27. O. Stern, Zschr. f. Physik 2, 49 (1920).
28. O. Stern, Zschr. f. Physik 3, 417 (1920).
29. O. Stern, Zschr. f. Physik 7, 249 (1921).
30. O. Stern, Zschr. f. Physik 39, 751 (1926).
31. O. Stern, Zschr. f. Physik 41, 563 (1927).
32. O. Stern, Phys. Rev. 51, 852 (1937).
33. J. B. Taylor, Zschr. f. Physik 57, 242 (1929).
34. M. Volmer and I. Estermann, Zschr. f. Physik 7, 13 (1921).
35. E. Wrede, Zschr. f. Physik 41, 569 (1927).
36. E. Wrede, Zschr. f. Physik 44, 261 (1927).

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