NEW INSTRUMENTS AND METHODS OF MEASUREMENT

APPLICATION OF ATOMIC BEAMS IN SPECTROSCOPY *)

K. V. Meissner

1. INTRODUCTION

The solution of many problems of modern spectroscopy relating to fine structure, hyperfine structure, the Zeeman effect in hyperfine structure, and the Stark effect depends on the possibility of resolving very close lines. For the successful resolution of lines, on the one hand, spectroscopes of sufficiently high resolving power are needed, and on the other hand—sources of emission and absorption producing lines with such a small half-width that the individual components of a group of lines do not overlap.

Using a Fabry–Perot interferometer, it is not difficult to obtain a resolving power of several millions, provided that the metallic coatings of the interferometer plates have sufficiently high reflectivity and that the distances between the plates are sufficiently large. It is possible to obtain an apparatus width comparable with the natural width of a line. The inconvenience of the narrowing of the spectral region as the distance between the plates increases can be avoided by using Houston’s compound interferometer, consisting of two successive Fabry–Perot interferometers of different thicknesses, the distance between the plates in the thicker interferometer being a small multiple of the distance in the thinner one. In this case the resolving power of the combined instruments is practically the same as that of the thicker instrument, while the spectral region is that of the thinner etalon.

Thus the problem is reduced entirely to creating the required light source. In ordinary light sources the principal cause of the broadening of spectral lines is the Doppler effect, caused by the random motion of the emitting atoms or molecules. The Doppler half-width of a spectral line is given by the expression

\[ \Delta \nu = 2(2R \ln 2)^{1/2} \nu c^{-1}(T/M)^{1/2}\ \text{cm}^{-1} \]

) Reviews of Modern Physics*, 14, 68, 1942. Translation by M. Wolkenstein.

(\(R\)—the gas constant, \(c\)—the speed of light, \(T\)—the absolute temperature, \(M\)—the molecular weight). For example, for sodium at \(300^\circ\) K the half-width will be \(\Delta\nu = 4.3 \cdot 10^{-2}\ \text{cm}^{-1}\). In order to obtain a half-width 10 times smaller, which is necessary for resolving the hyperfine structure of the \(3^2 P_{1/2}\) term of Na, we must use a light source operating at a temperature below \(5^\circ\) K. Even a Schuler lamp, cooled with liquid hydrogen, cannot give such sharp lines.

Using an atomic beam, however, it is possible to reduce the width of a spectral line without resorting to low temperatures. An atomic beam, first investigated and described by L. Dunoyer[^1], is a stream of atoms moving approximately in one and the same direction. It is assumed that the density of the beam is so small that, at least over a short section of it, there are no collisions between atoms.

The principle of producing an atomic beam from metal vapors may briefly be reduced to the following. The metal is melted in a furnace tightly closed by a lid having a small opening, the “furnace aperture,” in the form of a round hole or slit. If the vapor pressure in the furnace is not too high, the atoms will fly out of the furnace aperture along rectilinear paths, forming a divergent beam of atomic rays. By attaching to the setup a so-called collimator chamber, it is possible to select a narrow beam from the divergent rays.

In the case of metal vapors that can easily be condensed on cooled walls, the collimator chamber consists of a metal vessel whose closing plate is placed opposite the furnace aperture and is provided with an aperture called the collimator aperture or image aperture. Only those atoms that are directed toward the image aperture will pass through the collimator and form an atomic beam. The directions of motion of the atoms will deviate only by small angles from the line connecting the centers of the image aperture and the furnace aperture, i.e. from the axis of the beam.

If the atoms are excited (by radiation or by collision with electrons) and if the radiation is observed with the aid of a spectroscope whose collimator axis is perpendicular to the beam axis, the Doppler width will decrease, since the components of the velocities of the radiating atoms in the direction of the line of observation decrease. The degree of increase in sharpness will depend on the features of the setup. An approximate estimate can be given on the basis of the following consideration.

Let \(o\) and \(i\) be the linear dimensions of the furnace and image apertures, determined by the intersection of the aperture profiles with the plane containing the beam axis and the mean line of observation (the axis of the spectroscope collimator). Then the greatest deviation of the path of an atom from the beam axis is given by the angle \(\alpha = (o+i)/2h\), where \(h\) is the distance between the apertures. The quantity reciprocal to \(\alpha\) is sometimes called the “collimation of the beam” \(C\). The component of the velocity of an atom \(v\) in the direction perpendicular to

of the beam axis, will have the greatest value \(v \sin a\), or, approximately, \(v/C\), and can easily be brought to a value ten to thirty times smaller than \(v\).

In a number of investigations circular apertures were used; in this case \(o\) and \(i\) are the diameters of the apertures. In many other investigations the apertures were slits arranged parallel to one another; their width was small in comparison with the length \(s\). If the line of observation is parallel to the extent of the slits, the angle of collimation \(a\) is determined by the expression

\[ a = (s_0 + s_i)/2h, \]

where \(s_0\) and \(s_i\) are, respectively, the lengths of the furnace and image slits. In the special case \(s_0 = s_i = s\) we obtain \(a = 1/C = s/h\).

From this rough consideration we may conclude that the Doppler width \(\Delta_B\), observed when a beam is used, will be approximately equal to \(\Delta_0/C\), where \(\Delta_0\) is the width of the line that would be observed for the gas in the furnace.

Introducing the “effective temperature” \(T_e\) of the beam, we can describe the decrease in the line width in another way as well. \(T_e\) may be defined as the temperature of a source producing lines of the same width as the atomic beam. Since the velocity is proportional to \(\sqrt{T}\), we have the relation \(T_e = T_0/C^2\), where \(T_0\) is the temperature of the furnace, and \(C\) is the collimation. The same reasoning can obviously be applied to the case of an absorbing beam.

This consideration of the widths of lines in an atomic beam is rough and gives only limiting values. A rigorous solution of the problem was given by Minkowski and Bruck\(^2\), who calculated in general form the intensity distribution observed when an atomic beam is used. Their results are important for investigations concerning the distribution of intensity in spectral lines obtained in experiments with an atomic beam. Since we shall not encounter these questions further on, we shall not go into details. It is only necessary to point out that the form of the intensity distribution in the beam differs from the distribution in the furnace and that the distribution changes depending on the sizes and shapes of the furnace and image apertures and on the distance between them.

An approximate expression for the effect of increasing the sharpness of emission lines in a beam as compared with furnace emission was derived for the case \(h/s > 1\), where \(s\) is the length of the slits of the image and furnace, and \(h\) is the distance between them. The ratio of the half-width for the beam \(\Delta_B\) to the half-width \(\Delta_0\) for the vapor in the furnace is approximately given by the formula

\[ \Delta_B/\Delta_0 = 0.41s/h, \]

which means that the width in the beam is approximately 2.5 times smaller than that determined from the rough estimate. In this consideration we neglect-

were influenced by the aperture of the spectrograph. Usually the corresponding correction is small, since relative apertures of \(f:20\) are used.

Although Dunoyer had already foreseen that atomic beams could be very useful in spectroscopy, only a few attempts were made in this direction. Dunoyer himself showed that excitation of the Na resonance beam was possible.\(^1\) Bogros\(^3\) observed by this method the fine structure of the Li 6708 Å resonance line, while Dobretsov and Terenin\(^4\) carried out a quantitative study of the hyperfine structure of the Na resonance line with the aid of resonance fluorescence of a sodium beam. An absorbing beam of mercury atoms was used by Schein\(^5\) and later by Bradsinas,\(^6\) but the systematic development of the method was undertaken only beginning in 1934 by Minkowski and Bruck,\(^14\) and also by Meissner and Luft,\(^15\) independently of one another. Bogros and Esclangon\(^16\) and R. A. Fisher\(^18\) excited a beam by a high-frequency discharge in the presence of argon at low pressure (which, however, prevented the development of an undisturbed atomic beam).

II. DESIGN OF APPARATUS FOR OBTAINING ATOMIC BEAMS

1. Absorption method

Types of devices for the use of atomic beams in spectroscopy have been described in the literature. The simplest are devices for studying readily condensing vapors of metals with a low melting point. Jackson and Kuhn\(^7\) described such a simple apparatus in their first work, devoted to K and Na. It consists of a glass tube 36 cm long and 15 mm in diameter; its upper part is attached to an observation chamber provided with four windows arranged at right angles. A side tube connects the apparatus with a mercury vacuum pump. The lower part contains carefully distilled metal and is immersed to a depth of approximately 4 cm in an electrically heated bath of Wood’s alloy. A tightly fitted sheet of asbestos protects from overheating the upper part of the tube, in which all atoms condense except those moving in the region of collimation. The furnace apertures and openings are defined by the tube itself; the collimation reaches approximately \(24:1\). To study spectral lines, the light from a source producing a background for the absorption lines must pass through the observation chamber perpendicular to the beam.

A similar device, made of fused quartz, was used to study silver beams.\(^9\) For metals with high melting points, or for metals that act on glass or quartz, another arrangement is used. To study aluminum, Jackson and Kuhn\(^10\) constructed the apparatus shown in Fig. 1. A Pyrex glass bulb 10 cm in diameter is provided with three side tubes. Through one of them a small tantalum holder \(T\),

which can be heated electrically. The holder is made of a tantalum sheet \(0.3\) mm thick and \(5\) cm long; its cross-section has a V-shape, with a side dimension of \(5\) mm. The aperture at the vertex has a width of \(4\) mm. The image slit \(S\), parallel to the length of the holder, is at a distance of \(7\) cm and is cut in a nickel plate that completely separates the sphere from the observation chamber \(A\). The collimation can be varied from \(7:1\) to \(20:1\). Four pieces of aluminum, \(15\) mm long and \(2\) mm in diameter, are placed in the holder. This is sufficient to produce an atomic beam for \(90\) sec. at a heating current of \(90\) A. Since one exposure takes \(10\) sec., the capacity of the holder is sufficient. Observations are made through a side tube connected with \(A\) (perpendicular to the plane of the drawing).

Fig. 1. Jackson and Kuhn apparatus for an aluminum atomic beam.

Generally speaking, the devices described in II (3) can also be used for absorption experiments, especially in the case of metals with high melting points.

A very convenient installation for the study of absorption and fluorescence was described by Bogro in his dissertation on the physical properties of lithium vapors\(^3\).

2. Fluorescence method

For the excitation of resonance fluorescence in atomic beams, the same device as that described above may be used, with the difference that the observation chamber is provided with a window through which the exciting light is focused onto the atomic beam.

In order to obtain sufficient intensity of resonance fluorescence, it is necessary that the source of the exciting light produce sharp lines, as far as possible free from self-reversal. The most convenient light sources for these purposes are a Schuler lamp or an electrodeless tube with a high-frequency discharge.

3. Electron-impact method

The design of apparatus for studying atomic beams that permits excitation of the beam by electron impact is considerably more complicated than in the case of apparatus used in the absorption method.

The apparatus constructed by Minkowski and Bruck3 is shown in Fig. 2. The device producing the atomic beam is attached to the removable cover of a cylindrical bronze vessel 30 cm high and 18 cm in diameter. The electrodes for bombarding the beam are introduced through openings in the base. In the walls of the vessel there are two windows situated at an angle of 90° with respect to the aforementioned inlets. These windows make it possible to observe the atomic beam perpendicular to its axis.

Fig. 2. Apparatus constructed by Minkowski and Bruck for producing an atomic beam.

The tube \(R_1\), fastened in inlet 1, terminates in the furnace \(O\), surrounded by two concentric cylinders in order to reduce radiation losses. In the upper part of the furnace is placed the furnace slit, 1 cm long and of the usual width \(0.2\) mm. Inlet 2 contains a silver tube \(R_2\), to which are attached round disks \(P_1\) and \(P_2\), separated from one another by \(5\) cm. The lower disk is provided with image slits 1 cm long and 0.2 mm wide. By rotating the disk, different slits can be set in position. During exposure different slits must be used, since they narrow as a result of vapor condensation. In order to condense the vapor passing through the furnace slit in oblique directions, the furnace is surrounded by a copper cylinder \(R_4\).

The direction of observation is parallel to the slits of the furnace and of the image. Perpendicular to the beam axis and to the direction of observation, the beam is bombarded by electrons emitted by an oxidized wire of zigzag form, 15 cm long and covering an area \(1.3 \times 1.3\ \text{cm}^2\). A molybdenum grid of area \(1.6 \times 1.6\ \text{cm}^2\) is placed in front of the filament. The total emission is 500 mA, the grid voltage 200 V, and the current exciting the beam 200 mA. A mercury diffusion pump (20 l/sec) produces a vacuum better than \(10^{-6}\) mm Hg.

The apparatus constructed by Meissner and Luft does not differ in principle from the one described. The filament is a platinum sheet with an effective area of \(20 \times 20\ \text{mm}^2\); the grid consists of six nickel tubes through which water circulates. The copper anode is likewise provided with water cooling. The filament and the grid are arranged on concentric cylinders of radius about \(2.0\) cm. The grid

is located at a distance of approximately 2 mm from the filament. Such an arrangement provided satisfactory focusing of the electrons. The grid potential is about 500 V, the plate potential somewhat higher. The total emission current was at most 600 mA, the excitation current about 300 mA. In the upper part of the apparatus a silver holder, cooled by liquid air, is mounted.

A more recent design, proposed by W. Paul1 for beams of materials with high melting points, is shown in Fig. 3. Furnace \(O\) consists of a molybdenum cylinder 15 mm in diameter and 50 mm high, furnished for heating with a tungsten wire 0.4 mm in diameter and 220 cm long. Oxides of Be and Al are used for insulation. In this cylinder is fastened a cylindrical molybdenum holder, the cover of which is provided with a variable furnace slit \(S_1\), 8 mm long and of maximum width 2 mm. The volume is \(3.5\ \mathrm{mm}^3\). The furnace is provided with steatite tubes and an iron ring. It is surrounded by a nickel cylinder \(Z\), which reduces radiation losses. In the cover is situated the image slit \(S_2\), whose length is 8 mm, while its width can be varied from 1 to 6 mm. The base is attached to a glass tube 200 mm in diameter, closed in its upper part by a plate carrying a holder filled with liquid air. Here the atomic beam is condensed. Two side tubes contain a plate and the source of electrons for exciting the beam.

Fig. 3. Paul’s apparatus for substances with a high melting point.

The cathode is a nickel disk 2.5 cm in diameter, coated with oxides of the alkaline-earth metals. Indirect heating is used. Directly in front of the cathode there is a spatial tungsten grid \(G_1\). By means of a second grid \(G_2\) the required electron velocity is obtained. Cylinder \(L\) plays the role of a collecting electron lens. The anode, fastened in the tube on the opposite side, is water-cooled.

The total emission current is 2.2 A. By a suitable choice of the grid voltage it is possible to reduce the grid current to such an extent that the current used for excitation reaches 1.3 A. This high excitation current can, however, be used only for a period of from 1 to 2 hours. Therefore an excitation current of 0.5 A and a voltage between the plates of 500 V are usually used.

For lower voltages, which are more convenient, since the maximum of the excitation function is usually lower, it is not possible to obtain such a large current.

III. APPLICATIONS OF THE ATOMIC-BEAM METHOD

1. Distribution of Intensities in Lines

Minkowski and Bruck \(^{14}\) applied the atomic-beam method to the investigation of the red cadmium line \(6438\) Å. As the spectral instrument they used a Fabry–Perot interferometer in conjunction with a prism spectrograph. The focal length of the camera was \(50\ \mathrm{cm}\); the length of the etalon was \(11\ \mathrm{cm}\); the resolving power corresponding to a reflectivity of \(81\%\) was \(5\cdot 10^6\).

The intensity distribution was determined by photometering the interference pattern on a microphotometer. Intensity marks were obtained with the aid of a rotating sector or a step wedge.

It was established that the intensity distribution is asymmetric; the slope toward the lower frequencies is noticeably smaller. The half-width of the intensity curve was \(1.89\cdot 10^{-2}\ \mathrm{cm}^{-1}\) when the beam was excited by slow electrons (grid–cathode \(40\ \mathrm{V}\), anode–cathode \(50\ \mathrm{V}\)) and \(1.94\cdot 10^{-2}\ \mathrm{cm}^{-1}\) when excited by faster electrons (grid–cathode \(180\ \mathrm{V}\), anode–cathode \(445\ \mathrm{V}\)). This half-width is appreciably greater than the expected one, which can be calculated from the half-width determined by the instrument \(\left(4.55\cdot 10^{-3}\ \mathrm{cm}^{-1}\right)\) and the Doppler width \(\left(3.75\cdot 10^{-3}\ \mathrm{cm}^{-1}\right)\), and is equal to \(0.75\cdot 10^{-2}\ \mathrm{cm}^{-1}\).

The large half-width of the line and the observed asymmetry of the line intensity can be explained by isotope shifts. Cadmium consists of \(78\%\) even isotopes Cd 110, 112, 114, 116 and \(22\%\) odd isotopes 111 and 113. Since resolution of the pattern is impossible, only a rough estimate is accessible. Considering the whole area of the intensity curve as consisting of the areas of the individual isotopes, Minkowski and Bruck find that the shift toward the even isotopes should lie within the limits from \(3.5\cdot 10^{-3}\) to \(7\cdot 10^{-3}\ \mathrm{cm}^{-1}\).

The investigation was repeated by Meissner and Luft (unpublished) with the aid of a Fabry–Perot interferometer with a distance between the plates of \(18\ \mathrm{cm}\). The asymmetry of the intensity distribution was confirmed, as was the value of the half-width, \(1.9\cdot 10^{-2}\ \mathrm{cm}^{-1}\).

2. Fine Structures

Although at the beginning of the experiments with excited beams it seemed that the small intensities would make possible the investigation only of the strongest lines in the spectrum, in the end it proved possible to raise the exciting current to such a degree that the investigation of weak lines also became accessible. This is important, since there are many cases in which even multiplet fine structure is not resolved with ordinary light sources. We shall give two examples.

a) Fine structure of terms. Using somewhat longer slits and therefore allowing a somewhat greater line width, Meissner and Luft \(^{20}\) were able to excite a number of members of the subordinate series and to measure the absolute wavelengths and the splitting of the series lines.

The results were as follows. The \({}^{2}D\)-terms \(3^{2}D\) and \(4^{2}D\) could be resolved; the \({}^{2}D\)-splittings of the next two terms could be calculated from the distance between the strong components and from the \({}^{2}P\)-splittings. All resolved structures have the inverted order of terms; the lower value of \(J\) corresponds to the higher energy level.

Exact absolute values of the \(3^{2}P_{1/2,\,3/2}\)-terms were obtained from a calculation of the limits of similar series.

b) Fine structure of the \(3^{2}D\)-term of Mg I. The second case in which investigation of a multiplet by the atomic-beam method gave an unexpected result \(^{21}\) is the first member of the first subordinate series of magnesium
\[ \nu = 3sp\,{}^{3}P_{012} - 3sd\,{}^{3}D_{123} \]
with \(\lambda 3832\ \text{\AA}\). The \(D\)-structures of this line could not be resolved with ordinary light sources, for the splitting has a magnitude of several thousandths of an ångström. As a result of analysis of the line it turned out that the \(3^{3}D\)-term is a partially inverted term; the \(3D_{2}\)-term is lower than \(3D_{1}\). Only the weakest line of the complete triplet, \(3P_{2}—3D_{1}\), could not be observed, for its intensity is approximately \(1\%\) of the intensity of the strongest neighboring line. The partial inversion and the unexpectedly small splittings were explained by Pincherle \(^{22}\).

3. Hyperfine structure

The greatest progress was achieved in the investigation of the hyperfine structure of lines for which the splitting was too small to be resolved in the spectra of ordinary light sources. Both the absorption method and the emission method were successfully applied.

a) Absorption method. The absorption method can be applied only to resonance lines. Jackson and Kuhn studied in this way the hyperfine structure of the resonance lines of the elements Li, K, Na, Mg, Ag, and Al. In their first paper \(^{7}\), which also contains a description of the method, they considered potassium and sodium. Since the results obtained in this investigation were improved in subsequent ones \(^{11,12}\), there is no point in dwelling on its details.

In the second paper \(^{8}\), the isotope shift in the resonance lines of Mg was investigated. An article by P. Fischer \(^{39}\) is devoted to the same question. The third paper of Jackson and Kuhn \(^{9}\) concerns the hyperfine structure of the resonance lines of silver with \(\lambda 3281\) and \(3383\ \text{\AA}\). Each line proved to consist of four components; their positions and assign—

tion to the silver isotopes Ag(107) and Ag(109) are given below:

The most probable value of the nuclear \(I\) proved to be \(\frac{1}{2}\,h/2\pi\); this value is confirmed by investigation of the Zeeman effect (III, 4). With the aid of the Goudsmit formula, values of the nuclear magnetic moments were also obtained: \(-0.10\) nuclear magnetons for Ag(107) and \(-0.19\) nuc. magn. for Ag(109).

We shall next mention a paper on the hyperfine structure of aluminum, in which the structure of the resonance lines \(3^2P_{3/2,\;1/2}—4^2S_{1/2}\) with \(\lambda\) 3962 and 3944 Å and \(3^2P_{3/2,\;1/2}—3^2D_{3/2,\;1/2}\) with \(\lambda\) 3092.7; 3092.8 and 3082 Å is described. This paper contains detailed measurements of the intensity ratios of the components forming the indicated lines. The line \(3^2P_{1/2}—4^2S_{1/2}\) (3944 Å) has three components of approximately equal intensity with relative positions:

\[ -0.048,\quad 0.000,\quad +0.048\ cm^{-1}, \]

which shows that the two combining levels are split into two levels separated by \(0.048\ cm\). The line \(3^2P_{1/2}—3^2D_{3/2}\) has two components separated by \(0.0666\ cm^{-1}\); the ratio of the intensities of the longer-\(\lambda\) and shorter-\(\lambda\) components is \(1.21:1\). The observed splitting of this line is determined mainly by the splitting of the \(3^2P_{1/2}\) term and by the small unresolved structure of \(3^2D_{3/2}\); thus the splitting is somewhat larger than that observed for the 3944 Å line.

From measurements of intensity, Jackson and Kuhn derive spin \(9/2\) for the nucleus Al\(^{27}\). This large value is not consistent with the value \(5/2\) obtained by Heyden and Ritschl\(^{23}\), who applied the interval rule to the hyperfine structure of the Al II lines. At the same time Millman and Kusch\(^{24}\) applied the magnetic-resonance method to a molecular beam and confirmed the value \(5/2\). This example shows that even careful intensity measurements can lead to doubtful results.

b) Absorption method. The absorption method was used by Bogro and Exlaigon\(^{16,17}\) to study Cd and Li and by Fisher\(^{18}\) for Mg. These authors used excitation by a high-frequency discharge in the presence of argon.

Excitation of the beam by electron impacts was used by Meissner and Luft for the investigation of the resonance lines of sodium\(^{15}\) and potassium\(^{23}\). In the work devoted to sodium, not only was the hyperfine splitting of the \(^{2}S_{1/2}\) term, first discovered by Schüler\(^{25}\), measured very accurately, but the splitting of the \(^{2}P_{1/2}\) term was also resolved. The splitting of the \(^{2}S_{1/2}\) term proved to be \(0.0594 \pm 0.0003\ cm^{-1}\),

For the \(^{2}P_{1/2}\) term, \(0.0058 \pm 0.0003\ \mathrm{cm}^{-1}\). Calculation of the total splitting of the \(^{2}P_{3/2}\) term gave \(0.0053\ \mathrm{cm}^{-1}\), but it was not possible to obtain resolution of the corresponding lines. These values were confirmed by Jackson and Kuhn\(^{12}\).

The value of the nuclear moment \(i = \dfrac{3}{2}\dfrac{h}{2\pi}\), obtained from Rabi’s magnetic measurements and from spectroscopic measurements of intensities, makes it possible to calculate the magnetic moment of the nucleus by the formulas of Fermi and Segrè and of Goudsmit. The \(S\)-splitting gives the value \(2.08\) nuclear magnetons; the \(P\)-splitting gives \(1.96\) nuclear magnetons.

The development of “radio-frequency spectroscopy” makes it possible to find the splitting of \(^{2}S_{1/2}\) with considerably higher accuracy. The investigation of Kusch and Millman\(^{26}\) gave for \(\mathrm{Na}^{23}\) the value \(\Delta\,^{2}S_{1/2}=0.059103\ \mathrm{cm}^{-1}\). On the other hand, the magnetic method of the molecular beam directly gives the factors \(g(i)\) with great accuracy and, for known values of \(i\), gives the magnetic moments of the nucleus without further considerations. For \(\mathrm{Na}^{23}\) \(\mu=2.215\) nuclear magnetons was found.

The second work of Meissner and Luft\(^{28}\) concerned an investigation of the resonance lines of \(\mathrm{K}^{39}\). This investigation was undertaken because the results of Jackson and Kuhn, obtained by the absorption method, were not in agreement with the results of Rabi and his collaborators concerning the sign of the nuclear magnetic moment. Jackson and Kuhn found from observations of intensity (the strong components toward the shorter wavelengths) that the fine levels show an inverted order of the terms and that therefore the magnetic moment of the nucleus is negative. The result obtained by Meissner and Luft was in agreement with Rabi’s result. They found the correct order of the terms. Later Jackson and Kuhn also confirmed this result\(^{11}\). The following line splittings were found:

\[ 7699\ \text{Å}:\Delta\nu_1=0.0163\ \mathrm{cm}^{-1}, \qquad 7665\ \text{Å}:\Delta\nu_2=0.0141\ \mathrm{cm}^{-1}. \]

The difference between these values shows that the splittings are caused by the splittings of the nuclear spin for the \(^{2}S\) and \(^{2}P\) levels. Starting from the value \(3/2\) for the nuclear spin, one can calculate from the center of gravity of the observed splittings the splitting of the terms themselves. It was found that \(\Delta\,^{2}S_{1/2}=0.0152\ \mathrm{cm}^{-1}\) and \(\Delta\,^{2}P_{1/2}=0.0033\ \mathrm{cm}^{-1}\), which corresponds to \(0.40\) nuclear magnetons and \(0.30\) nuclear magnetons (the value \(\Delta\,^{2}S_{1/2}\), known from radio-frequency spectroscopy, is \(0.015403\ \mathrm{cm}^{-1}\); Jackson and Kuhn\(^{11}\) find \(0.0153\ \mathrm{cm}^{-1}\)).

One of the most recent investigations of hyperfine structure with the aid of an excited atomic beam was the investigation by V. Paul\(^{19}\), devoted to the resonance line Be II (3130 Å). All investigators of the Be spectrum had been unable to find any trace of splitting or antisymmetry, since in all these investigations the light source was a Schuler tube with a hollow cathode, cooled by liquid air. Therefore we are dealing here with a Doppler half-width of the order of

0.1 cm\(^{-1}\) at 3130 Å. With the aid of the Be I \(\lambda 2349\) Å line, Mrozowski\(^{29}\) obtained a width of 0.21 cm\(^{-1}\) at 3130 Å. The lines studied by Mrozowski were unsplit and symmetrical.

Paul obtained a considerably smaller line width. Corresponding to a furnace temperature of 1500° K and collimation of 7.5 : 1, the expected Doppler width should be 0.015 cm\(^{-1}\); to this must be added the width determined by the apparatus, 0.008 cm\(^{-1}\), and thus the total width, 0.023 cm\(^{-1}\), is approximately seven times smaller than that obtained by Mrozowski when using a Scholer lamp.

Despite such a small width, Paul was unable to detect a splitting of the line. Using an interferometer with an interval of 40 mm, he could establish only that the stronger line of the fine-structure doublet showed a noticeable dissymmetry toward smaller \(\nu\). Estimating the width of the observed lines and comparing it with the calculated beam width, one can obtain limits for the splitting:

\[ 0.020 < \delta\nu < 0.040\ \text{cm}^{-1}. \]

According to Goudsmit,\(^{30}\) \(\delta\nu\) can be calculated for the \({}^{2}S_{1/2}\)-term. For \(i = {}^{3}/_{2}\) we obtain \(\delta\nu = 0.050\, g(i)\), where \(g(i)\) is the Landé factor for the nucleus. The splittings of the \({}^{2}P\)-terms are too small for them to have been observed.

Each fine-structure component is split into two hyperfine-structure components with an intensity ratio of 5 : 3. According to theoretical considerations, it must be assumed that the magnetic moment of the Be nucleus is negative; thus, the order of the \(F\)-terms must be reversed. Therefore we should expect that the weaker component will have the smaller wavelength.

The observed asymmetry is in agreement with these results of the theory. Moreover, the estimated limits of \(\delta\nu\) make it possible to give limits for the values of \(g\) for the nuclear moments

\[ -0.4 > g(i) > -0.8 \]

and limits of the nuclear moments

\[ -0.6 > \mu > -1.2. \]

These quantities are in agreement with the value \(g(i) = -0.893\), found by the resonance method.\(^{31}\) If we accept this value of \(g\), then the value \(i = {}^{5}/_{2}\) can be excluded, for in this case the hyperfine structure would have been resolved; \(i = {}^{1}/_{2}\) cannot be excluded, but \(i = {}^{3}/_{2}\) is the most probable value.

The last paper concerning hyperfine structure is devoted to the study of the isotope shift of magnesium lines, which the author

undertaken in 1937[^32]. With the aid of an excited beam and standards with spacings of 36, 42, and 60 mm, it proved possible to resolve the lines of the series \(3^1P — m^1D,\ m = 3, 4, 5\) into three approximately equally spaced lines which, according to their intensities, may be attributed to \(\mathrm{Mg}^{24,25,26}\). It was also possible to resolve one member of the sharp singlet series \(3^1P — 5^1S\). The theory was developed by Winti[^33].

4. Hyperfine-Structure Zeeman Effect

The Zeeman effect of hyperfine structure is very important for determining the nuclear moment. In weak fields each fine level is split into \(2j + 1\) components; in strong fields, however, each component of the anomalous Zeeman effect of the fine structure consists of \(2i + 1\) hyperfine components (the Back–Goudsmit effect), having a completely uniform arrangement and intensities. If it is possible to resolve this hyperfine structure sufficiently so that the number of components can be counted accurately, then the value of \(i\) of the nuclear spin can be found with greater precision. This is a great advantage in comparison with the method using intensity ratios.

This method has been applied several times in those cases where the separations were large enough to give resolved components with ordinary light sources. Experiments with atomic beams were carried out by Jackson and Kuhn for Li, Na, and K. They used the absorption method and employed a compound interferometer as the resolving instrument. In the case of potassium[^11] they were able to separate the hyperfine structure of \(\mathrm{K}^{39}\) and \(\mathrm{K}^{41}\). The collimation was very high, namely \(35:1\). In order to obtain sufficient absorption, three atomic beams in succession were used. The Zeeman effect in \(\mathrm{K}^{39}\) was studied with one beam and a collimation of \(25:1\). The intensity ratios found for \(\mathrm{K}^{39}\) in two resonance lines, 7664 and 7694 Å, were respectively 1.44 and 1.45. With these values the value \(i = 3/2\) is in agreement; the calculated values for \(i = 3/2\) are 1.40. The stronger components of the hyperfine structure have greater wavelengths; consequently the order of the terms is normal, in agreement with the experimental results of Rabi and, Meissner and Luft (III, 3, b).

In order to establish the value \(3/2\) precisely, the Zeeman effect of the \(\pi\)-components for \(\mathrm{K}^{39}\) was studied in a field of intensity 730 gauss. Each of the two \(\pi\)-components proved to consist of four lines; their positions were: \(-0.0172,\ -0.0136,\ -0.0100,\ -0.0063,\ +0.00180,\ +0.0148,\ +0.0106,\ +0.0056\ \mathrm{cm}^{-1}\). The value of \(i\) for \(\mathrm{K}^{41}\) could be found only from the intensity ratio. It proved to be \(3/2\). Calculation of the magnetic moment gives 0.22 nuclear magneton. The ratio of the moments \(\mu_{39}/\mu_{41} = 1.77 \pm 0.05\) was established with great accuracy.

In a subsequent paper[^12], Jackson and Kuhn investigated the Zeeman effect of the hyperfine structure of Na over a wide interval of field-strength values, using a compound interferometer consisting of two successive etalons of size 2 and 8 cm. The spectral range was \(0.25\ \mathrm{cm}^{-1}\), and it was possible to resolve two lines separated by \(0.003\ \mathrm{cm}^{-1}\). The \(D_1\) line was observed at 11 different field strengths from 790 to 2090 gauss, and the \(D_2\) line at field strengths from 1060 to 3060 gauss. Only the \(\pi\)-components were investigated.

The observations proved to be in agreement with the theory (Heisenberg and Jordan, 1926; Darwin, 1927; Goudsmit and Bacher, 1930).

Figure 4 shows the theoretical splitting of the hyperfine-structure levels. Figure 5 (Figs. 1, 2, and 3, reference[^12]) gives reproductions of the interference patterns obtained at different field strengths. The drawings clearly show the development of the splitting and directly give the value \(i = 3/2\), for at a sufficiently large field strength each hyperfine component consists of four lines.

Fig. 4. Term diagram of the Zeeman effect of the sodium line \(3^2S_{1/2} — 3^2P_{3/2}\).

The last paper by Jackson and Kuhn that we shall mention here is devoted to the hyperfine structure and the Zeeman effect in the resonance line of Li2. This investigation was carried out with a multiple atomic beam and collimation from \(1/30\) to \(1/40\), which gave absorption lines thirty to forty times narrower than the partial Doppler width. A compound interferometer (successive 5-millimeter and 5-centimeter etalons) gave a spectral range of \(1\ \mathrm{cm}^{-1}\) and a resolution of about \(0.004\ \mathrm{cm}^{-1}\). The distance between neighboring emission lines forming the background for the absorption lines was about \(1/3\) of the order of the 5-mm etalon. Investigation of the Zeeman effect gave the value \(i = 3/2\). The two Zeeman \(\pi\)-components proved to consist of four resolved lines, the distance between which is about \(0.005\ \mathrm{cm}^{-1}\). Knowing \(i\), one can calculate the hyperfine splitting of the \(^{2}S\)- and \(^{2}P\)-terms participating in the transition. The measured values of the line splittings are as follows:

\[ 2^2S_{1/2} — 2^2P_{3/2}\quad \Delta \nu_1 = 0.0270 \pm 0.0001\ \mathrm{cm}^{-1}, \]

\[ 2^2S_{1/2} — 2^2P_{1/2}\quad \Delta \nu_2 = 0.0280 \pm 0.0002\ \mathrm{cm}^{-1}. \]

1

Labels: 1840, 1450, 1200, 1060, 930, 0; b; center of gravity.

Graph labels: gauss; 2320, 1570, 1060; blackening; \(a, m\).

2

Labels: gauss; 3080, 2360, 2000, 1840, 1570, 1330, 1060, 0; a; center of gravity.

Graph labels: 1570, 1060, 830; blackening; \(a, m\).

3

Labels: \(S_{1/2}-P_{3/2}\); \(S_{1/2}-P_{1/2}\); c; \(A_1\), \(A_2\), \((B)\), \(B_2\); \(\nu \to\).

Graph labels: \(S_{1/2}-P_{1/2}\); \(S_{1/2}-P_{3/2}\); \(A_1\), \(A_2\), \((B)\), \(B_2\).

Fig. 5. Interference patterns of sodium lines.

Calculation gives the hyperfine splittings of the \(S_{1/2}\)- and \(P_{1/2}\)-terms:

\[ \Delta S_{1/2}=\frac{1}{2}(\Delta\nu_1+\Delta\nu_1)=0.0275\pm0.0003\ \text{cm}^{-1}, \]

\[ \Delta P_{1/2}=\frac{3}{2}(\Delta\nu_2-\Delta\nu_1)=0.0015\pm0.0009\ \text{cm}^{-1}. \]

The value of \(\Delta S_{1/2}\) is very close to that found by Fox and Rabi\(^{34}\), \(\Delta S_{1/2}=0.0267\pm0.0003\ \text{cm}^{-1}\).

The nuclear magnetic moment could be calculated from the splitting of \(^{2}S\). The calculation gives \(\mu=3.25\) nuclear magnetons. The very same value (3.250 nuclear magnetons) is obtained directly by means of the method of magnetic resonance in a molecular beam\(^{35}\).

5. Application of the atomic-beam method to the study of the inverse Stark effect

The atomic-beam method is very convenient for investigating the inverse Stark effect, first studied by Ladenburg\(^{38}\) in the case of the sodium \(D\)-lines. The high vacuum required for producing an atomic beam is very favorable for maintaining a strong electric field.

This method was developed by Kopfermann and his collaborators\(^{36,37,19}\). The apparatus used by Paul\(^{39}\) was the device described above (II, 3). The device for exciting the beam by electron impacts was removed and replaced by a condenser consisting of two flat plates with a separation of 1 mm (Fig. 6). Polished nickel plates (area \(15\times20\ \text{mm}^2\)) were attached to amber plates. One of them was fastened to a copper plate provided with the image slit of the apparatus for producing the atomic beam. By means of four screws \(L_1\), the plates were set parallel. By means of screws \(L_2\), connecting one of the amber plates with the plate having the image slit, the condenser could be displaced in the horizontal direction. All adjustment could be carried out from outside.

Fig. 6. Apparatus for observing the Stark effect.

Using the same apparatus, Engel and Kopfermann\(^{36}\) investigated singlet lines and were able to measure very small shifts caused by the Stark effect. Paul\(^{19}\) investigated the resonance multiplet Cr, \(^{7}S_3—{}^{7}P_{2,3,4}\) (\(\lambda\ 4254,\ 4275,\ 4290\ \text{Å}\)). A continuous background for the absorption lines was produced by means of a hollow cathode, cooled with water and provided with a chromium cylinder.

At field strengths of 206, 234, and 275 kV/cm it was possible to observe red shifts of the lines of the order of \(2 \cdot 10^{-3}\) Å. These shifts were proportional to the square of the field strength. A weak broadening of the lines could also be observed.

Since the author was unable to obtain papers\(^{36,37}\), we cannot give details of the results of these new investigations. We must therefore confine ourselves to citing these papers.

6. Further possible applications of the beam method

It is quite obvious that the beam method can also be applied to the investigation of band spectra. However, in order to obtain satisfactory intensity or absorption, it is necessary to use several molecular beams in succession. In this way the thickness of the emitting or absorbing medium can be increased without increasing the width of the lines.

We shall further point out that a light source containing a beam can be very useful for precision measurements of the intensities of spectral lines. This source makes it possible to eliminate the effect of self-absorption, which is achieved by using different beam depths. This can easily be done by suitably choosing the length of the image slit.

Of great importance is the development of atomic-beam apparatus for studying the spectra of gases for which the condensation method is inapplicable. This problem is addressed in the papers of Williams\(^{40}\) and Mack\(^{41}\).

LITERATURE

1. Dupouet, C. R., 152, 592, 1911; 157, 1060, 1913; Le Radium, 8, 142, 1911; 10, 400, 1913.
2. R. Minkowski u. H. Bruck, Z. Physik, 95, 274, 1935.
3. A. Borgos, C. R., 183, 124, 1926; Ann. de Physique, 17, 199, 1932.
4. L. Dobrezova, A. Terenin, Naturwiss., 16, 656, 1928.
5. M. Schein, Helv. Phys. Acta, 2 supplement 1, 1929; Ann. d. Physik, 85, 257, 1928.
6. P. Brazdzinnas, Ann. d. Physik, 6, 739, 1930.
7. D. Jackson a. H. Kuhn, Nature, 134, 25, 1934; Proc. Roy. Soc. A., 148, 335, 1935.
8. D. Jackson a. H. Kuhn, Proc. Roy. Soc. A., 154, 679, 1936.
9. D. Jackson a. H. Kuhn, Proc. Roy. Soc. A., 158, 372, 1937.
10. D. Jackson a. H. Kuhn, Proc. Roy. Soc. A., 164, 48, 1938.
11. D. Jackson a. H. Kuhn, Proc. Roy. Soc. A., 165, 303, 1938.
12. D. Jackson a. H. Kuhn, Proc. Roy. Soc. A., 167, 205, 1938.
13. D. Jackson a. H. Kuhn, Proc. Roy. Soc. A., 173, 278, 1939.
14. R. Minkowski u. H. Bruck, Z. Physik, 95, 284, 1935.
15. K. Meissner u. K. Luft, Ann. d. Physik, 28, 667, 1937.
16. A. Borgos et F. Esclangon, C. R., 195, 369, 1932.
17. F. Esclangon, Ann. de Physique, 1, 267, 1934.
18. R. Fisher, Phys. Rev., 51, 381, 1937.

1. W. Paul, Z. Physik, 117, 774, 1941.
2. K. Meissner and K. Luft, Ann. d. Physik, 29, 698, 1937.
3. K. Meissner, Ann. d. Physik, 31, 518, 1938.
4. L. Pincherle, Phys. Rev., 58, 251, 1940.
5. M. Heyden and R. Ritschl, Z. Physik, 108, 739, 1938.
6. S. Millman and P. Kusch, Phys. Rev., 58, 438, 1940.
7. H. Schueler, Naturwiss., 16, 512, 1928.
8. S. Millman and P. Kusch, Phys. Rev., 58, 438, 1940.
9. P. Kusch, S. Millman and I. Rabi, Phys. Rev., 55, 1176, 1939.
10. K. Meissner and K. Luft, Z. Physik, 106, 362, 1937.
11. S. Mrozowski, Phys. Rev., 55, 793, 1939.
12. S. Goudsmit, Phys. Rev., 43, 636, 1933.
13. P. Kusch, S. Millman and I. Rabi, Phys. Rev., 55, 666, 1939.
14. K. Meissner, Ann. d. Phys., 31, 505, 1938.
15. J. Vinti, Phys. Rev., 56, 1120, 1930.
16. M. Fox and I. Rabi, Phys. Rev., 48, 746, 1935.
17. I. Rabi, S. Millman and P. Kusch, Phys. Rev., 55, 526, 1939.
18. L. Jenckel and H. Kopfermann, Z. Physik, 117, 145, 1941.
19. H. Kopfermann and Ch. Otzen, Z. Physik, 117, 156, 1941.
20. R. Ladenburg, Z. Physik, 28, 51, 1924.
21. R. Fisher, Rev. Mod. Phys., 14, 79, 1942.
22. W. E. Williams, Rev. Mod. Phys., 14, 94, 1942.
23. J. Mack and Barkofsky, Rev. Mod. Phys., 14, 82, 1942.

TRANSLATOR’S ADDENDUM

Meissner’s article is his report at the conference on spectroscopy convened at the University of Chicago in June 1942. All the conference reports were published in Rev. Mod. Phys., 14, Nos. 2–3, April—July 1942. The conference dealt exclusively with questions of theoretical spectroscopy. In his opening address, the chairman of the physical section of the conference, R. Mulliken, emphasized the great importance of theoretical research and the necessity of continuing it even in wartime. Among other very interesting sections of the conference, a number of reports were devoted to atomic-beam spectra. In addition to Meissner’s review given here1, the following were reported: “Light sources with an atomic beam and their application to the analysis of the structure of the Mg I resonance line” by R. Fisher2, “Atomic-beam apparatus for the study of atomic spectra of gases, especially hydrogen” by Mack and Barkofsky3, and “An attempt to excite an atomic beam of monatomic hydrogen” by Williams4.

In the first of these works, a comparison was made of two methods of exciting an atomic beam. In the first method, developed by Esclangon5 and applied by Carpenter and Fisher6,7, excitation of the atomic beam is achieved by means of a high-frequency electrodeless ring discharge in argon located in the chamber through which the atomic beam passes. The concentration of argon is so small that the beam is not noticeably scattered along its path.

in the chamber. The mean free path of argon under these conditions is several centimeters.

A light source based on this principle proved to be very intense—the resonance lines Na and Mg were obtained with exposures of the order of 1 min when using an optical system that included a Fabry–Perot interferometer and had an aperture ratio \(f : 25\). The resonance lines Mg II were obtained in 10 min. The excitation of the beam’s luminosity is produced in this source by impacts of randomly moving electrons created in the discharge.

Using this source, Fischer obtained, for the resonance line Mg I, results different from those obtained by Jackson and Kuhn\(^{8}\) by the absorption method (cf. Meissner’s article). For a final resolution of the question Fischer constructed a light source of another type, based on the principle described by Meissner. In this source the excitation is likewise produced by electron impacts, but the electrons are directed—they move in the form of an electron beam from an oxide heated emitter through a series of diaphragms. Such a source using an electron gun has considerable advantages. It gives lines approximately five times sharper than a source with a ring discharge. However, the intensities in the source with the electron gun are 20–30 times lower. Therefore a source with a ring discharge is expediently used in solving such problems as allow a line width of the order of \(0.02\ \mathrm{cm}^{-1}\).

As a result of experiments with a source containing an electron gun, Fischer established that the true structure of the resonance line Mg I \(\lambda 2852\) Å differs both from that obtained by Jackson and Kuhn and from that obtained with the aid of a source with a ring discharge. The line was resolved into three components with an intensity ratio \(7:1:1\), corresponding to the magnesium isotopes \(24, 25,\) and \(26\).

Mack and Barkovsky\(^{3}\) point out that none of the sources and methods described by Meissner makes it possible to study the fine and hyperfine structure of spectra of substances gaseous under ordinary conditions. Meanwhile, investigation of the simplest gaseous element—hydrogen \(H^{1}\)—is of exceptional interest. It is here that the fundamental propositions of modern quantum mechanics in its relativistic form can be tested. Dirac’s theory makes it possible to predict the positions and intensities of the components of each line belonging to a system consisting of an electron and a fixed point charge. Precisely hydrogen can, to a high degree of accuracy, be regarded as such a system. Consequently, a detailed study of the hydrogen spectrum provides a kind of test of Dirac’s theory.

The most convenient for investigation is the \(\alpha\)-line of hydrogen, \(\lambda 6562.79\) Å. In this region of the spectrum silver has a high reflection coefficient (which is essential for the use of the Fabry–Perot interferometer);

the $\alpha$-line can be photographed. At the same time the $\alpha$-line is especially simple. A whole series of interferometric works was devoted to its study. However, in a number of cases they led to discrepant results. The most advanced spectroscopic technique at the present time, with an atomic beam, has not once been successfully applied to the study of hydrogen. Therefore Mack and Barkovsky analyzed in detail the conditions for the successful use of an apparatus with an atomic beam and constructed an apparatus satisfying these conditions.

First of all the authors consider what the geometrical conditions in the apparatus must be, under which precisely the atoms moving in the beam, and not those scattered in the chamber, will be excited. The basic scheme for the calculation is given in Figs. 7 and 8. The atomic beam propagates in the direction $x$, the electrons exciting the radiation—in the direction $y$, and the emitted light is observed in the direction $z$. The atoms of a monatomic gas, at pressure $p_1$, temperature $T_1$, and density $n_1$, flow out through the first slit of width $Z_1$. After passing through a series of slit diaphragms, the atoms enter through the slit numbered $k-1$ into the last chamber, in which there exist pressure $p_k$, temperature $T_k$, and

Fig. 7.

Fig. 8.

density $n_k$. The number $n(x)$ of atoms located in the beam and occurring per unit volume can be calculated. It is equal to

\[ n(x)=\frac{n_1\Omega}{4\pi}, \tag{1} \]

where $\Omega$ is the solid angle $y_1 z_1/x^2$ under which the first slit is seen from a distance $x$ from it (cf. Fig. 8).

APPLICATION OF ATOMIC BEAMS IN SPECTROSCOPY

When the beam is bombarded by an electron beam moving in the direction \(y\) and having a cross section \(z_e\) along the \(z\)-axis, we shall obtain a definite number of excited atoms of the beam visible in the direction \(z\). If \(z_e\) is so small that each electron passes through the entire cross section of the beam, then the number \(b\) of excited atoms of the beam visible per unit cross section is equal to

\[ b=f n(x) z_e=f_e n_1 z_e \frac{\Omega}{4\pi}, \tag{2} \]

where \(f\) is the fraction of atoms excited by the electrons.

The scattered, “wandering” gas in the \(k\)-th chamber is under the conditions \(p_k, T_k, n_k\). The number of excited atoms visible per unit cross section is equal to

\[ s=f n_k z_e, \tag{3} \]

and we obtain

\[ \frac{s}{b}=\frac{4\pi}{\Omega}\frac{n_k}{n_1} =\frac{4\pi}{\Omega}\frac{p_k}{p_1}\frac{T_1}{T_k}, \tag{4} \]

or, in the case \(T_1 \approx T_k\),

\[ \frac{s}{b}=\frac{4\pi}{\Omega}\frac{p_k}{p_1}. \tag{5} \]

In order that the intensity emitted by the “wandering” atoms be much less than the intensity emitted by the atoms of the beam, it is necessary that the condition

\[ \frac{p_k}{p_1}\ll \frac{4\pi}{\Omega} \tag{6} \]

be satisfied.

The apparatus must satisfy this requirement in order that it be possible to obtain a sharpness of the lines corresponding to the atomic beam. Obviously, in the case of non-gaseous substances the requirements are different. They reduce only to ensuring that \(p_k\) be small and not affect the collimation of the beam. In this case the pressure \(p_k\) is created by an external gas, the lines of which do not overlap those being investigated; the “wandering” atoms of the metal vapor immediately condense on the walls.

The authors further analyze the influence of the Doppler effect in the apparatus on the spectral pattern obtained. The Doppler effect for a line emitted by an atomic beam excited by electron impacts may be decomposed into five different effects. First, the presence of a transverse component in the initial velocity of the atom, causing a broadening of the order

\[ \delta_1=\frac{\Delta \nu}{\nu}=\frac{v}{c}\frac{z-z_1}{x}, \tag{7} \]

where \(z_1\) is the coordinate of the point in the first slit from which the atom emerges; \(z, x\) are the coordinates of the point at which the atom emits; \(v\) is the velocity of the atom; \(c\) is the speed of light. In the apparatus of Mack and Barkovsky \((\delta_1)\sim 2.8\cdot 10^{-7}\).

Second, the Doppler shift arises as a result of the change in the atom’s velocity vector upon collision with the exciting electron:

\[ \delta_2=\frac{\Delta \nu}{\nu}=\frac{p_z}{Mc}, \tag{8} \]

where \(p_z\) is the projection of the momentum acquired by the atom onto the \(z\)-axis, and \(M\) is the mass of the atom. A detailed analysis, based on the quantum-mechanical theory of collisions\(^9\), shows that faster electrons produce a smaller half-width and that, in general, the line width determined by this factor is approximately inversely proportional to the square root of the bombardment voltage. The order of magnitude of \(\delta_2\) under the conditions used by the authors is \(\sim 10^{-6}\).

Third, atoms acquire a recoil velocity upon emission of a quantum, which causes a red shift of the line. For the \(\alpha\)-line of \(\mathrm{H}^1\),

\[ \delta_3=-\frac{h}{2\lambda Mc}=-1.0\cdot 10^{-9}. \tag{9} \]

Fourth, it is necessary to take into account the difference in the directions of the light rays entering the optical system. Since rays entering the system make different angles with the direction of motion of the atom, a red shift and broadening of the line of the order of \(10^{-7}\) arise in the authors’ apparatus. Finally, fifth, one must take into account a certain inaccuracy in the setup: the optical axis of the system makes an angle with the beam that differs somewhat from a right angle. This produces a shift and broadening of the line. The authors estimate it in their setup as \(+8\cdot 10^{-8}\). Comparing all these factors, Mack and Barkovsky come to the conclusion that it is possible to reduce the amount of broadening so much that the electronic structure of the hydrogen \(\alpha\)-line can be fully resolved. For deuterium the order of all the quantities is the same, and the conditions do not change. Let us give the separations and half-widths of the components of the hydrogen \(\alpha\)-line, obtained on the basis of theoretical calculation\(^ {10}\).

Here \(\alpha\) is the fine-structure constant, equal to \(\dfrac{e^2}{\hbar c}=\dfrac{1}{137.3}\).

The half-width of the \(H^1\) lines obtained in the apparatus of Mak and Barkovsky is approximately 10 times greater than the natural half-width, but the distances between the split components are 2–12 times greater than the obtained half-width. Consequently, the structure of the \(H_\alpha\) line can indeed be studied with this apparatus. Mak and Barkovsky limited themselves in their report to the analysis presented and to a description of the apparatus. The authors give no research results. Nevertheless, their report is of interest in many respects.

In a short note by Williams\(^{4}\), the conditions for exciting an atomic beam by a high-frequency discharge at very low pressures \((10^{-5}—10^{-6}\ \mathrm{mm})\) are considered. Williams points out that the possibility of creating such light sources is essential for metrology. In particular, the lines of pure even isotopes of heavy elements (for example, \(\mathrm{Hg}^{198}\))\(^{11}\), excited at low temperature, can provide a standard with considerably greater symmetry and homogeneity than the contemporary standard—the red cadmium line. In this connection, we shall point to the work of B. L. Ponizovsky\(^{12}\), who studied in detail the conditions determining the choice of a wavelength standard and proposed as a new standard the infrared krypton line \(\lambda 9751,\ 759\ \text{\AA}\).

The reports devoted to atomic beams at the conference in Chicago show that this field is of very great interest for the theory and practice of spectroscopy. The use of atomic beams creates exceptional possibilities for studying the details of spectral structure, for determining nuclear spin and magnetic moment, and for developing a new metrological standard.

In conclusion, let us say a few words about “radio-frequency spectroscopy,” repeatedly mentioned by Meissner in his review.

Every system having angular momentum

\[ \vec{j}\,\frac{h}{2\pi}, \]

thereby has a magnetic moment

\[ \vec{\mu}=\vec{j}\,\frac{e h}{4\pi m c}. \tag{10} \]

The factor \(\frac{e h}{4\pi m c}\), where \(e\) and \(m\) are the charge and mass of the electron, is the Bohr magneton. For the magnetic moments of nuclei one uses a quantity 1860 times smaller—the nuclear magneton \(\frac{e h}{4\pi M c}\), where \(M\) is the mass of the proton. A system with moment \(\vec{j}\), placed in a magnetic field \(H\), precesses about the direction of the field. The projections of the angular momentum \(\vec{j}\) on the direction of the field have \(2j+1\) values \(j,\ j-1,\ldots,1,0,-1\ldots,-j\). As is known, this spatial quantization is directly demonstrated with the aid of the Stern–Gerlach experiment, in which

an atomic beam is passed through an inhomogeneous magnetic field, which causes deflections of the atoms.

The determination of nuclear moments by this method2 is difficult, for nuclear magnets, as we have already said, are very small. Nevertheless, in a number of studies based on the measurement of small deflections of molecules with zero electronic moment, it proved possible to measure the nuclear element of hydrogen3. It turned out to be 2.5 times larger than the nuclear magneton4. The method was improved by Rabi5, who found the proton moment to be equal to \(3.25 \pm 10^0\%\) nuclear magnetons.

In 1937 Rabi proposed a new, very elegant method for determining the sign and magnitude of the magnetic moment of the nucleus in the case when the nuclear spin is known6. This method also formed the basis of radio-frequency spectroscopy7.

Let us imagine an atomic beam propagating in a strong inhomogeneous magnetic field and therefore being deflected in it. The deflections of individual particles of the beam will differ depending on their effective moments and velocities.

Next, the beam passes through a strong homogeneous magnetic field, and then enters a second inhomogeneous field, entirely similar to the first but with the opposite direction of inhomogeneity. The deflection of the particles will be equal and opposite to the deflection in the first field, and a particle detector placed at the end of the beam will show the same number of particles as would have been obtained in the absence of inhomogeneous fields. Now let us superpose on the region of the intermediate homogeneous field an additional weak alternating field directed perpendicular to the homogeneous one. This alternating field will cause substantial changes in the behavior of the deflected particles. The effective moments of the particles in the homogeneous field \(H\) are different, for different magnetic quantum numbers \(m\) correspond to different angles of precession. The perpendicular alternating field \(H_1\), oscillating with frequency \(\nu\), creates the possibility of transitions between states with different \(m\), for example between \(m_1\) and \(m_2\). The probability of such a transition will, generally speaking, be small, so long as the field frequency \(\nu\) does not coincide with the Larmor precession frequency

\[ \nu_L = gH(e/2Mc), \tag{11} \]

where \(g\) is the gyromagnetic factor, depending on the moment of the particle. The frequencies \(\nu\) are of the order of radio frequencies—several kilocycles. As a result of transitions between different \(m\)’s, deflections will appear in the second inhomogeneous field of the apparatus described which do not exactly compensate the deflections in the first field, and the detector will show a decrease in the number of particles. By choosing the frequency \(\nu\) of the field \(H_1\) so that this decrease becomes maximal, we find the Larmor precession frequency \(\nu_L\), with which the field \(H_1\) resonates. The whole apparatus is, in essence, analogous to a certain polarization

system—the first inhomogeneous magnetic field plays the role of a polarizer, the second that of an analyzer.

Such is the basic scheme of the experiments of Rabi and his collaborators. Studying chiefly molecular beams and, in particular, comparing the results for H\(_2\), HD, and D\(_2\), Rabi determined the nuclear magnetic moments for H and D\(^ {19}\). They proved to be equal to

\[ \mu_{\mathrm{H}} = 2.785 \pm 0.02 \ \text{nuclear magnetons}, \]

\[ \mu_{\mathrm{D}} = 0.855 \pm 0.006 \ \text{nuclear magnetons}. \]

The method of radio-frequency spectra, as applied to atoms, makes it possible to determine very accurately the separations between the components of the hyperfine structure. These separations fall precisely in the radio-frequency region. We have already seen in Meissner’s review what difficulties are involved in the optical investigation of hyperfine structure. The point is that the lifetime of the individual hyperfine components of the ground state of an atom is very long and, correspondingly, the intensity of spontaneous radiation is small. However, when the atom is irradiated with electromagnetic waves of its own frequency, it will absorb or emit a quantum of this frequency. Such a change of state will cause a reorientation of the atom in the magnetic field and therefore can be detected by the magnetic-resonance method described. The separations between the hyperfine components are thus determined with very great accuracy\(^ {20}\). Combining magnetic-resonance studies of molecular and atomic beams, Millman and Kusch\(^ {21}\) were able to determine quite accurately the nuclear moments of a number of atoms.

Further details concerning the study of molecular and atomic beams will be found by the reader in the informative review by Bessey and Simpson\(^ {22}\). There also are given a summary of values of nuclear magnetic moments and an extensive bibliography.

LITERATURE

1. K. Meissner, Rev. Mod. Phys., 14, 63, 1942.
2. R. Fisher, Rev. Mod. Phys., 14, 79, 1942.
3. J. Mack a, E. Barkofsky, Rev. Mod. Phys., 14, 82, 1942.
4. W. E. Williams, Rev. Mod. Phys., 14, 94, 1942.
5. E. Schlangon, Ann. d. Physique, 1, 268, 1934.
6. R. Fischer a. B. Carpenter, Phys. Rev., 49A, 417, 1936.
7. R. Fisher, Phys. Rev., 51A, 381, 1937.
8. D. Jackson a. H. Kuhn, Proc. Roy. Soc. A., 154, 679, 1936.
9. H. Bethe, Ann. d. Phys., 5, 325, 1930.
10. L. Bethe, Quantum Mechanics of the Simplest Systems, ONTI, 1935, §§ 9, 10, 41–44.
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