FINE STRUCTURE OF THE HYDROGEN ATOM. I

W. E. Lamb and R. C. Retherford*)

A. INTRODUCTION

In August 1947 we published a preliminary communication1 on the use of microwaves for solving the problem of the fine structure of the hydrogen atom. A shift, amounting to about 1000 Mc, of the level \(2^2S_{1/2}\) relative to the position predicted by Dirac’s theory was reliably established. The accuracy of these first measurements was approximately 100 Mc. At that time we expected soon to publish a paper containing a detailed description of the experimental method and theory.

The apparatus with which the initial experiments were carried out was, naturally, very imperfect, and we were already engaged in developing a new, substantially improved version. At the first stage our aim was to attain an accuracy of 10 Mc, but later this value was reduced to 1 Mc. The program of work was extensive, and its execution encountered unforeseen difficulties, the overcoming of which required much time. As a consequence, the writing of the paper promised two years earlier was delayed.

At present we are occupied with carrying out the final measurements on our new apparatus. In order to attain increased accuracy, it proved necessary, in the analysis of the experimental data, to take into account a large number of weak instrumental and theoretical effects. It is obvious that setting forth all the details within a single paper would make it too cumbersome and would obscure the simple ideas on which the experiment is based. Therefore we intend to write a series of papers, of which the present paper is the first. It contains a brief history of the question, an analysis of the commu-

*) W. E. Lamb Jr. and R. C. Retherford, Phys. Rev. 79, 549 (1950). For a theoretical discussion of the problem see Ya. A. Smorodinskii, UFN 39, 325 (1949), and V. F. Weisskopf, UFN 41, 165 (1950). (Editor’s note.)

considerations that led to the choice of the method, estimates of the expected effects, a description of the apparatus constructed before May 1947, and the experimental results obtained with this apparatus. Since the quantitative estimates of the expected effects depend substantially on quantities whose values at the time when these estimates were first made were still unknown, we have considered it desirable to use modern data in order to facilitate the subsequent discussion. For this reason, some of the estimates refer to later versions of the apparatus. We hope that this will not cause the reader any undue difficulty.

The theory of the hydrogen atom and of the Zeeman effect for its fine and hyperfine structure is considered by us only insofar as is necessary for understanding the experimental data. In the present article we deal only with effects of the order of 10 Mc/sec and higher, leaving a more detailed discussion to the following article, where it will be truly necessary. The reduced-mass effect is in most cases ignored and is taken into account only in part in estimating the anomalous magnetic moment of the electron.

B. DISCUSSION OF THE PROBLEM

1. General considerations

The hydrogen atom is the simplest and, at the same time, the only atom for which exact theoretical calculations can be carried out. The theoretical discussion at present usually begins with the Dirac equation for the motion of an electron in the purely Coulomb field of a fixed point charge.

Corrections for the motion of the proton, its possible finite size, and the hyperfine structure due to its magnetic moment can be introduced into the calculations to a good approximation. This remains true also for effects associated with the interaction of the electron with the quantized electromagnetic field, which was discovered by Bethe³ after the deviations from Dirac’s theory had been found.

According to the Bohr theory of 1913, the energy levels of a hydrogen-like atom are given by the relation

\[ W_n=\frac{hcRZ^3}{n^2}, \tag{1} \]

where \(R\) is the Rydberg constant, \(h=2\pi\hbar\) is Planck’s constant, and \(c\) is the velocity of light in vacuum. For infinitely heavy nuclei \(R=109737.3\ \mathrm{cm}^{-1}\). The transition from the state \(n=3\) to the state \(n=2\) gives the red hydrogen line \(H_\alpha\). This line, as was

established in 1887 by Michelson and Morley,^4 is in fact a doublet. The explanation of its fine structure was given in 1916 by Sommerfeld,^3 proceeding from the relativistic dependence of the electron mass on its velocity. The two possible motions corresponding to the principal quantum number \(n=2\) differ in energy by the amount

\[ \Delta W_{2}=\frac{\alpha^{2}}{16}\,hcR\{1+O(\alpha^{3})\}, \tag{2} \]

where the fine-structure constant \(\alpha=\dfrac{1}{137.030}\) (Birge, 1941).

When quantum mechanics first arose, it was found that, although Bohr’s formula can also be obtained, an approximate allowance for relativistic effects gives a splitting \(8/3\) times greater than equation (2), which was, at least, in rough agreement with the observational results of that time. Only when the model of the rotating electron of Goudsmit–Uhlenbeck and Thomas was accepted in the theory did it again become possible to return to equation (2). Instead of two levels there were now three, as shown in Fig. 1. This theory indicated that an exact coincidence of the levels \(2\,{}^{2}S_{1/2}\) and \(2\,{}^{2}P_{1/2}\) should occur, but the calculations were not entirely free of ambiguity, since in the solution the ratio \(\dfrac{0}{0}\) appeared.

Fig. 1. Fine structure of the hydrogen levels of hydrogen with \(n=2\) according to Dirac’s theory.

Dirac’s theory of the electron (1928), which automatically endowed the electron with relativistic properties, spin, and magnetic moment, predicted the fine structure shown in Fig. 1 without the ambiguity mentioned above. The energy levels of a hydrogen-like atom are given in this theory by the expression

\[ W=mc^{2}\left[\left(1+\frac{(\alpha Z)^{2}}{\{\,n-|K|+(K^{2}-\alpha^{2}Z^{2})^{1/2}\,\}^{2}}\right)^{-1/2}-1\right], \tag{3} \]

where \(|K|=j+\dfrac{1}{2}\). In agreement with Fig. 1, levels having identical principal quantum numbers \(n\) and identical internal quantum numbers \(j\) are degenerate. Expansion of the right-hand side of formula (3) in powers of \(\alpha Z\) gives

\[ W_{nj}=-\frac{Z^{2}hcR}{n^{2}}-\frac{\alpha^{2}Z^{4}hcR}{n^{3}}\left(\frac{1}{j+\dfrac{1}{2}}-\frac{3}{4n}\right)+\ldots \tag{4} \]

According to this expression, the doublet splitting of the state \(n=2\) for hydrogen is equal to

\[ \Delta W_2=\frac{\alpha^2}{16}hcR, \tag{5} \]

which agrees with Sommerfeld’s formula, since the last term in expression (2) may, in the present approximation, be neglected completely.

Although the treatment of the hydrogen atom given by Dirac’s theory was consistent and satisfactory, the theory entailed certain strange considerations arising from the existence of states with negative energy. It therefore seemed highly desirable to subject the predictions of this theory concerning the fine structure to careful experimental verification. This was naturally carried out first of all by studying the spectrum of the hydrogen atom. Deviations from the theory could be attributed to one (or several) of the following circumstances: 1) an error in Dirac’s equation; 2) a deviation from the Coulomb law of attraction between the electron and the proton, possibly owing to the presence of short-range nonelectromagnetic forces\(^5\) or to vacuum-polarization effects in the theory of the positron\(^6\); 3) some finite and physically real difference in the infinite radiative shift of the frequencies of all spectral lines, predicted by the calculations made in 1930 by Oppenheimer\(^7\); or 4) effects as yet unexplained.

The work of Bechert and Meixner\(^8\) showed that hyperfine structure and reduced-mass effects cannot be responsible for any appreciable discrepancies.

2. Spectroscopic investigation of the \(H_\alpha\) line

The spectroscopic study of the \(H_\alpha\) doublet has a long history, beginning with the first resolution of the doublet structure in 1887 and continuing up to investigations still being carried out at the present time. For acquaintance with the early work we refer the reader to the review by Williams\(^9\), published in 1938. By 1940 the situation was quite unclear. On the one hand, the investigations of Houston\(^ {10}\) and Williams\(^9\) indicated fairly plausible discrepancies between theory and experiment. Pasternak\(^ {11}\) showed that these discrepancies could be explained if the level \(2\,^2S_{1/2}\) were raised above the level \(2\,^2P_{1/2}\) by approximately \(0.03\ \mathrm{cm}^{-1}\).

On the other hand, in 1940 Drinkwater, Richardson, and Williams\(^ {12}\) attributed all discrepancies to impurities in the source.

The theorists who tried to calculate the term shift proposed by Pasternak, starting from deviations from Coulomb’s law,

or else suffered failure because the predicted effects were too small, or fell into error[^13] because of the inadequacy of their theories. Their ardor was cooled to a considerable extent by the results of Drinkwater, Richardson, and Williams[^13], which confirmed Dirac’s theory.

Postwar measurements by Giulotto[^14] again led to the discovery of discrepancies, and recently Kuhn and Series[^15], using a discharge tube cooled by liquid hydrogen, found a shift of the \(2^2S_{1/2}\) level by \(0.043 \pm 0.006\ \mathrm{cm}^{-1}\).

The complex \(H_\alpha\) line has two principal peaks separated from one another by approximately \(0.317\ \mathrm{cm}^{-1}\). At the same time, in Williams’s work with deuterium the measured Doppler width of the lines is \(0.120\ \mathrm{cm}^{-1}\). Under these conditions, a truly rigorous spectroscopic test of the theory is extremely difficult. These difficulties are increased still further by the circumstance that the intensities of the components often deviate from the theoretically predicted values and vary depending on the discharge conditions. Only by using a source with an atomic beam can the Doppler effect be reduced enough to hope for an actual resolution of the components of the \(H_\alpha\) line. The natural width of the sharpest component is only about \(0.001\ \mathrm{cm}^{-1}\) (\(30\ \mathrm{MHz}\)), but according to estimates made by Mack[^16], the practically attainable width should be about ten times larger.

3. Possibility of Using Radio Waves in Hydrogen Spectroscopy

As early as 1928, Grotrian[^17] pointed out that the selection rules allow optical transitions with conservation of the principal quantum number \(n\), and that it should be possible to induce, by means of radio waves, such transitions between states with \(n = 2\), which correspond to the doublet splitting \(\Delta \nu = 0.365\ \mathrm{cm}^{-1}\), or to the wavelength \(\lambda = 2.74\ \mathrm{cm}\) (frequency \(10\,950\ \mathrm{MHz}\)).

Between 1932 and 1935 two German papers appeared[^18],[^19], in which an attempt was made to detect such transitions. At that time the experimenters had at their disposal, for investigations in the microwave range, only a spark oscillator with an extremely small output power.

The work had to be carried out with a continuous spectrum of radiation, using interferometric methods to isolate monochromatic radiation of the wavelength of interest. The radiation was passed through an absorption vessel containing a hydrogen discharge of the Wood type[^20], and the attenuation was measured as a function of wavelength. The first author, Betz, asserted that he had succeeded in observing absorption in the expected range of wavelengths.

waves \((\lambda = 3\ \mathrm{cm})\). He also observed selective absorption at wavelengths \(\lambda = 9\) and \(27\ \mathrm{cm}\), corresponding to transitions between the hyperfine-structure levels of the state with \(n = 3\). Three years later, working in the same laboratory, Haase repeated the experiments more carefully and failed in the attempt to find any absorption at wavelengths \(\lambda = 9\) and \(27\ \mathrm{cm}\). He did not investigate the region \(\lambda = 3\ \mathrm{cm}\). Haase estimated the expected selective absorption of energy by excited hydrogen atoms and came to the conclusion that it is too small to be detected. This question will be discussed in Appendix I from a somewhat more modern point of view. It seems strange that Haase gives no special justification for the positive result obtained by Betz.

4. Choice of method

After becoming acquainted with the work of Betz and Haase, we turned to clarifying to what extent modern microwave technique, which had undergone such considerable development during the war, might be suitable for carrying out a successful and reliable determination of the fine structure of the hydrogen atom by measuring the absorption of radiation of the corresponding wavelength in a Bude discharge.

In the course of considering this question, we arrived at a different method, which was in fact used by us. Although we did not manage to reach a definite conclusion about the possibility or impossibility of using the Bude tube method, the estimates we obtained left little hope of success. The corresponding calculations, revised in the light of modern data, are given in Appendix I. In view of the extreme roughness of the numerical estimates made by us, such a result by no means implies that, under properly chosen discharge conditions, detection of transitions between states of the hydrogen atom with \(n = 2\) would at present be completely impossible. However, from consideration of the perturbing influence of the electric field and of ionic collisions on the energy levels of the atom it is clear that, even if the indicated transitions could be detected, the results obtained in this way would be of greater significance for elucidating the conditions existing in a Bude discharge tube than for determining the fundamental properties of an isolated hydrogen atom.

Radio-frequency spectroscopy has two principal methods at its disposal. In one of them the substance under investigation absorbs, or in some other way acts upon, the radiation passing through it. In the other, the observed changes caused by the radiation occur in the substance itself. The first method is used in the usual

... variants of microwave spectroscopy^21, in which the radiation passes through a long column of absorbing gas. The second underlies the radio-frequency resonance method with molecular beams^22, when particles that have absorbed radiation are removed from the beam by an inhomogeneous magnetic field.

Since the first method did not seem to us sufficiently promising for the study of the fine structure of the hydrogen atom, we turned to studying the possibilities of the second method.

5. The atomic-beam method

In the case of the hydrogen atom, the \(2P\) state is destroyed and the atom passes into the \(1S\) state with emission of a photon (whose wavelength is \(\lambda = 1216\ \text{Å}\)) in \(1.595 \cdot 10^{-9}\) sec. During this time the atom can move only approximately \(1.3 \cdot 10^{-3}\) cm,

[Source of metastable H* atoms] — [Region of destruction of metastability by a radio-frequency field] — [Detector]

Fig. 2. Block diagram of the apparatus.

if one assumes that its velocity of motion is of the order of \(8 \cdot 10^{5}\) cm/sec. On the other hand, there is the possibility that the state \(2\,{}^{2}S_{1/2}\) will prove sufficiently metastable to make it possible to use beams of particles in this state. If, then, by means of a radio-frequency field or by some other means transitions to the \(2P\) state are induced, the subsequent transition to the \(1\,{}^{2}S_{1/2}\) state will occur so rapidly that the number of excited atoms in the beam will decrease. If, furthermore, it should prove possible to find a detector capable of responding selectively exclusively to excited hydrogen atoms, then the possibility would arise of measuring the difference of energies between the metastable level \(2\,{}^{2}S_{1/2}\) and the various \(2P\) states. The proposed experiment could then be represented by the block diagram shown in Fig. 2.

To carry out the projected program it was necessary to solve the following two problems: 1) the creation of a beam of atoms in the \(2\,{}^{2}S_{1/2}\) state, and 2) the detection of such atoms. The solution of these problems obviously depended on knowledge of the properties of metastable hydrogen atoms. Such atoms had never been...

were isolated in a properly arranged experiment, although they had been the subject of numerous speculations. The following section is devoted to a consideration of the properties of metastable hydrogen atoms.

6. Metastable Hydrogen Atoms

In 1924, Compton and Russell \(^{23}\), attempting to explain the great intensity of the Balmer lines in stellar spectra, noted that the lower of the two states with \(n=2\), obtained on the basis of Sommerfeld’s theory, should be metastable. In 1926, after the appearance of the theory of the spinning electron, Sommerfeld and Unsöld \(^{24}\) came to the conclusion that the state \(2\,{}^{2}S_{1/2}\) should still be metastable. On the other hand, Franck and Jordan \(^{25}\) advanced arguments casting doubt on the possibility of metastability of the level \(2\,{}^{2}S_{1/2}\).

In order to test the correctness of the Sommerfeld–Unsöld ideas on this question, Snoek, von Keussler, and others \(^{26}\) undertook a number of experimental investigations. They measured the relative intensity of the two principal groups of components of the absorption line \(H_{\alpha}\). If the state \(2\,{}^{2}S_{1/2}\) in the absorbing series were metastable to any appreciable degree, then the ratio of the intensities of the two peaks would have been altered. From the data they obtained, these authors concluded that the level \(2\,{}^{2}S_{1/2}\) is not metastable to any appreciable extent.

Rozhanskii and Van Vleck \(^{27}\) pointed out that the influence of an external electric field, even a weak one, leads to a mixing of the degenerate states \(2\,{}^{2}S_{1/2}\) and \(2\,{}^{2}P_{1/2}\), and stationary states are formed with wave functions

\[ \psi=\sqrt{2}\,[\psi(2\,{}^{2}S_{1/2})\pm\psi(2\,{}^{2}P_{1/2})]. \tag{6} \]

Both these states must decay with a constant of decay \(\dfrac{1}{2\tau_p}\) (where \(\tau_p\) is the lifetime of the state \(2\,{}^{2}P_{1/2}\)), so that they cannot be metastable. This conclusion, although correct in a certain sense, is not applicable to experimentally realizable conditions, when stationary states are absent. This was pointed out by Bethe \(^{36}\), who gave a systematic theory of the influence of an electric field on the fine structure of hydrogen. In the complete absence of a perturbing electric field, an atom in the state \(2\,{}^{2}S_{1/2}\) has a very long lifetime. (Bethe estimated that the lifetime of an isolated atom should in this case amount to several months, if relativistic effects are taken into account.) In the presence of a strong perturbing electric field, the result coincides with that indicated by Rozhanskii and Van Vleck. Increasing

...of an electric field from zero is accompanied by a change in the lifetime of the metastable atom from one limiting value to another. Bethe showed that the results of Fong-Keisler and Snook can be understood if one takes into account the cross sections for excitation of the various levels and the destruction of metastability by the perturbing electric field of the discharge. For an effective electric field of \(10\ \mathrm{V/cm}\) he found that the lifetime of the \(2^2S_{1/2}\) state should be approximately five times greater than the lifetime \(\tau_p\) of the \(P\) state. At high field strengths the lifetime varies inversely as the square of the field strength. For the typical conditions of Wood’s discharge tube, according to Bethe’s estimates, the lifetime should be approximately equal to \(2\tau_p\).

In 1940 Breit and Teller\(^{38}\) refined some of Bethe’s calculations and considered the question of the metastability of the \(2^2S_{1/2}\) state in connection with possible astrophysical applications. They showed that the mechanism of transition to the ground state by two-quantum radiation is more important than relativistic effects, and that the lifetime of an isolated metastable atom should therefore be of the order of \(\frac{1}{7}\ \mathrm{sec}\).

From the foregoing it was clear to us that the \(2^2S_{1/2}\) state could be appreciably metastable only under the condition that the perturbing electric field proved sufficiently weak. With an atomic beam length of \(6\ \mathrm{cm}\) and an atom velocity of \(8\cdot 10^5\ \mathrm{cm/sec}\), in order to ensure the survival of \(37\%\) of the atoms, it is necessary for the lifetime to be of the order of \(0.75\cdot 10^{-5}\ \mathrm{sec}\). In accordance with Bethe’s calculations, this required that the strength of the perturbing field should not exceed \(\frac{1}{3}\ \mathrm{V/cm}\). It would be by no means simple to prevent electrons and ions from reaching the detector in so weak a field without an extraordinary lengthening of the atomic beam used. In reality, of course, as we now know, the levels \(2^2S_{1/2}\) and \(2^2P_{1/2}\) are not degenerate. This considerably increases the stability of the \(2^2S_{1/2}\) state against destruction due to the Stark effect.

A simple generalization of Bethe’s calculations, given in Appendix II, shows that the lifetime of the \(2^2S_{1/2}\) state in an electric field of average strength is

\[ \tau_s = \tau_p \frac{\hbar^2\left(\omega^2+\frac{1}{4}\gamma^2\right)}{V^2}, \tag{7} \]

where \(V\) is the matrix element of the energy of the perturbing field and \(\hbar\omega\) is the splitting of the interacting levels. (Breit and Teller discussed the stabilizing action of the hyperfine-structure splitting and found that it leads to an increase of \(\tau_s\) by a factor...

tel, equal to four. Since the actual value of \(t_{\omega}\) is much larger than the hyperfine-structure splitting, the latter cannot substantially change the stability, and we shall here neglect the effect of the hyperfine structure.) According to equation (7), for

\[ \frac{\omega}{2\pi}=1000\ \text{MHz}, \]

the lifetime \(\tau_s\) will be approximately 400 times greater than in the case in which the degeneracy had not been removed*).

At that time, however, we did not take sufficiently seriously into account the possibility that the degeneracy might exist to as large an extent as it actually does, and we proposed to increase the stability of the \(2\,^2S_{1/2}\) state by applying a magnetic field producing a strong Zeeman splitting of the levels.

The presence of a magnetic field perpendicular to the atomic beam was also to serve to keep charged particles away from the detector. The third, and in fact the most important, function of the magnetic field will be discussed in Section 14.

7. Production of a beam of metastable hydrogen atoms

We considered a number of possible methods for obtaining a beam of hydrogen atoms excited to the metastable \(2\,^2S_{1/2}\) state. The simplest source would be a hydrogen discharge tube with a small aperture for the beam to emerge into vacuum. In the discharge, however, one is dealing with a mixture of molecular and atomic hydrogen, electrons and ions, and only a small fraction of excited atoms, as well as with the presence of Lyman and Balmer radiation of high intensity. The population of the \(2\,^2S_{1/2}\) state, estimated in Appendix I, would be about \(5\cdot 10^{10}\) atoms/cm\(^3\). The question was whether an appreciable number of these atoms could pass through the aperture before the metastable state was destroyed by the Holtsmark field produced by ions and electrons leaking at the same time through the same aperture. Even with the stabilizing influence of a magnetic field present, this did not seem realistic to us.

There was also the possibility of exciting some number of normal atoms optically to the \(3n\) state after they had left the discharge; 12% of the atoms excited in this way should pass\(^{39}\) into the \(2\,^2S_{1/2}\) state. Numerical estimates of the yield of metastable atoms showed the complete unsuitability of this method. Moreover, here one would have to deal with a very high intensity of ultraviolet—

*) This makes it necessary to reconsider the experiments of Snoek, Fón-Keissler, and others. Some details requiring consideration are indicated in Appendix I.

THE FINE STRUCTURE OF THE HYDROGEN ATOM

radiation, which would also reach the detector. This would have created additional difficulties, since almost all possible detectors of metastable atoms possess, at the same time, photosensitivity. After the successful realization of the experiments by another method, we attempted to use such a Wood tube as the source, but this attempt did not give a positive result.

Another method we considered consists in bombarding molecular hydrogen with electrons in the absence of a field. Metastable hydrogen atoms \(H^*\) can then arise as a result of the following process:

\[ \mathrm{H}_2 + e \to \mathrm{H} + \mathrm{H}^* + e'. \tag{8} \]

The potential-energy curves for various excited states of the molecule \(\mathrm{H}_2\), shown in Fig. 3, are based on the calculations of Hylleraas and James, Coolidge, and Present\({}^{30}\). A small segment of the curve corresponding to the ground state \({}^{1}\Sigma_g\) conventionally indicates the limits of the zero-point vibrations in the lower vibrational state. According to the Franck–Condon principle, electron bombardment will most effectively excite the state \({}^{3}\Sigma_g\) at energies smaller than the dissociation energy of the hydrogen molecule into \(\mathrm{H}+\mathrm{H}^*\). However, it should be expected that, for electrons whose energy is of the order of 15 eV, the yield of metastable atoms will be small. Such a “violation” of the Franck–Condon principle would be analogous to dissociation with ionization:

Fig. 3. Electronic energy levels of the hydrogen molecule as a function of internuclear distance.

\[ \mathrm{H}_2 + e \to \mathrm{H} + \mathrm{H}^+ + e' + e'', \tag{9} \]

observed by Bleakney\({}^{31}\) at an electron energy of 18 eV. It is seen from the figure that there exist repulsive states leading to dissociation into \(\mathrm{H}+\mathrm{H}^*\). Therefore metastable atoms should often be produced when molecular hydrogen is bombarded by high-energy electrons.

The difficulty of such a method for obtaining metastable atoms consists in the fact that the fragments arising as a result of the process of splitting

H + H* move in directions oriented chaotically with respect to the electron beam and, consequently, the detector can intercept only a small fraction of the metastable atoms formed. The background produced at the detector by ultraviolet photons reaching it, which arise when molecular hydrogen is bombarded by electrons with energies exceeding 11.5 ev, must obviously create additional difficulties.

In the original version of the apparatus, described below, attempts were made to obtain metastable hydrogen atoms by bombarding molecular hydrogen; these ended in failure. We now know \(^{32}\) that, if the conditions had been somewhat different, this method would have proved successful.

Fig. 4. Modified block diagram of the apparatus.

The third method, which was the one ultimately chosen by us, required the independent production of a beam of hydrogen atoms in the ground state and the simultaneous bombardment of this atomic beam by electrons whose energy would exceed by only a little the threshold value for excitation of the \(2\,{}^2S_{1/2}\) state, 10.2 ev. The proposed experiment could now be represented by the block diagram shown in Fig. 4.

8. Dissociation of molecular hydrogen

There is a whole series of methods for obtaining a beam of hydrogen atoms: 1) Wood’s tube, 2) microwave discharge, and 3) thermal dissociation in a tungsten furnace. No decisive arguments can be given for choosing the last of these. We were aware of the work carried out in 1923 by Olmstead and Compton \(^{33}\), who measured the critical potentials of atomic hydrogen obtained in a tungsten furnace, and we made extensive use of the advice of Doffendack \(^{34}\), who was a pioneer in investigations with such a furnace. On the other hand, the prewar work with a hydrogen atomic beam at Columbia \(^{22}\) had made very successful use of Wood’s tube as a source. It was felt, however, that in our case the ultraviolet radiation of the discharge would cause additional difficulties. The microwave-discharge method \(^{35}\) was rejected, since even slight scattering of radio-frequency power would introduce serious complications into the spectroscopic measurements.

If the existence of thermal equilibrium is assumed, then the expected degree of dissociation of hydrogen molecules in the reaction

\[ H_2 \to 2H \tag{10} \]

is given by the expression

\[ \frac{P^2(\mathrm{H})}{P(\mathrm{H}_2)}=K(T), \tag{11} \]

where \(P(\mathrm{H})\) and \(P(\mathrm{H}_2)\) are the partial pressures of atomic and molecular hydrogen, expressed in atmospheres, and \(K(T)\) is specified (also in atmospheres) as a function of the absolute temperature \(T\) by the equation

\[ \lg K=-\frac{21200}{T}+1.765\lg T-9.85\cdot 10^{-5}T-0.265, \tag{12} \]

as was found by Bonhoeffer\({}^{36}\) from Langmuir’s data. Setting

\[ P(\mathrm{H})=XP, \tag{13} \]

where \(P\) is the total pressure:

\[ P=P(\mathrm{H})+P(\mathrm{H}_2), \tag{14} \]

we obtain for the dissociation coefficient \(X\) the equation

\[ \frac{X^2}{1-X}=\frac{K(T)}{P}. \tag{15} \]

In Fig. 5 the dependence of \(X\) on \(T\) is shown for three values of the total pressure.

Fig. 5. Thermal dissociation of molecular hydrogen.

No direct measurements of the pressure inside the tungsten furnace could be made by us, but from the data on the dependence of the yield on the furnace temperature, obtained under conditions of successful performance of the experiment, it seems justified to assume that in the typical case it was about \(10^{-3}\) atmosphere and that at \(T=2500^\circ\mathrm{K}\) the dissociation amounted to about 64% of complete dissociation. At this temperature the most probable velocity of the hydrogen atoms in the beam is \(8\cdot 10^5\ \mathrm{cm/sec}\).

9. Excitation of hydrogen atoms by electron bombardment

The cross sections of various processes of excitation of hydrogen atoms by electron bombardment were calculated by Bethe\({}^{37}\) in the Born approximation, with exchange effects neglected. Although this approximation cannot be expected to be good for electron energies close to threshold, it is the only one available for our quantitative estimates. The cross sections are given by the expression

\[ \sigma_{nl}=\frac{8\pi hcR}{mv^3}\, |(nl|x|10)|^2\,[F_{nl}(y_{\max})-F_{nl}(y_{\min})], \tag{16} \]

where

\[ y=\left(1+\frac{n}{n+1}\right)^2\frac{Q}{h c R}, \]

\[ F_{30}(y)=-\frac{1}{5y^5}, \]

\[ F_{21}(y)=\ln\frac{y-1}{y}+\sum_{s=1}^{5}\frac{1}{s y^s}, \]

\[ F_{31}(y)=F_{21}(y)-\frac{4}{3y^6}+\frac{4}{7y^7}, \]

\[ Q_{\max}=\left(\sqrt{E}+\sqrt{E-E_{nl}+E_{10}}\right)^2, \]

\[ Q_{\min}=\left(\sqrt{E}-\sqrt{E-E_{nl}+E_{10}}\right)^2, \]

\(E\) is the electron energy, \(E_{nl}\) is the energy of the atomic state.

These cross sections are presented as functions of the electron energy in Fig. 6. The maximum cross section for excitation to the \(2s\) state \((\sigma_{\max}=2.2\cdot10^{-17}\,\text{cm}^2)\) is about one-ninth of the maximum cross section for excitation to the \(2p\) state, while for excitation to the \(3p\) state it is smaller than for excitation to the \(2s\) state by a factor close to 11.

Fig. 6. Cross sections for excitation of atomic hydrogen to various states by electron bombardment, calculated in the Born approximation without taking exchange effects into account.

Thus, the yield of atoms excited to the \(2s\) state as a result of cascade decay of the \(3p\) state may be disregarded in the calculation. The largest yield of atoms excited to the \(2s\) state is obtained at an electron energy of \(14.8\,\text{eV}\), but it does not fall to less than half the maximum value when the electron energy is decreased to \(11\,\text{eV}\).

Leaving aside the errors inherent in the Born approximation, especially serious for estimating excitation of the optically inaccessible \(2s\) state, is the neglect of electron exchange, and one may hope that, if exchange is taken into account, a much larger cross section will be obtained, with a maximum near

threshold. Since in actuality no calculations of exchange excitation, even in the Born approximation, seem to have been carried out for hydrogen, we used the value \(\sigma = 10^{-17}\ \text{cm}^3\) as a lower limit for the cross section, hoping that it would protect us from a gross miscalculation.

If the detector is photosensitive, then a signal with the same threshold energy, \(10.2\ \text{eV}\), should also arise, produced by Lyman photons emitted by atoms excited to the \(2p\) state. The presence of some residual amount of molecular hydrogen also leads to the emission of photons; the threshold energy for the latter is about \(11.5\ \text{eV}\). Only an insignificant fraction of these photons can reach the detector, but some background due to them is inevitable.

10. Recoil caused by bombardment

Usually, in experiments with atomic beams one deals with well-collimated beams. In our case it is necessary to bombard hydrogen atoms with electrons after they have left the source, since the destruction of metastable states caused by the Stark effect must be reduced to a minimum. Therefore it becomes necessary to choose between bombardment at right angles to the beam, along the direction of the beam, or against it. The last two methods of bombardment will not lead to a violation of homogeneity in the directions of motion of the atoms in the beam if the energy of the bombarding electrons only slightly exceeds the threshold. Such a choice would be natural for other atoms, but not for metastable hydrogen atoms. In this case it would prove necessary for the magnetic field to be directed parallel to the electron beam. If the electrons were sent against the atomic beam, then the metastable atoms would subsequently be forced to pass through an electric field which would destroy the metastable states. If the electrons were sent along the beam, then preventing them from reaching the detector would be a far from easy task. A method based on differences in time of flight would have to be rejected on grounds of intensity.

The choice of the first variant inevitably entails the appearance of changes in the direction of motion of the atoms due to recoil, perpendicular to the beam, and, what is considerably worse, an uncertainty in the magnitude of this recoil, as a result of which the initially well-collimated beam, under the influence of electron bombardment, must become diffuse. This means that the use of slits of width of the order of \(0.025\ \text{cm}\), as is usually adopted in work with atomic beams, becomes impractical.

On the other hand, there should have been no problem of the influence of small beam deviations in inhomogeneous magnetic fields, since the influence of radio-frequency fields, manifested in the removal of excitation from absorbing atoms, is incomparably easier to detect.

Fig. 7. Distribution of excited atoms by recoil angles. Curve a is the distribution over horizontal recoil angles \(\varphi\), obtained upon excitation of atomic hydrogen to the \(2s\) state by bombardment with electrons whose energy is close to threshold. The temperature of the atomic source (oven) is taken as \(2600^\circ\) K. The total probability is equal to unity if the angles are measured in radians. Curves b and c are the distributions over the horizontal recoil angles \(\varphi\) and the vertical recoil angles \(\chi\) for hydrogen atoms excited to the \(2s\) state by bombardment with electrons of energy \(13.6\) ev.

The distribution of excited atoms over recoil angles is calculated in Appendix III, and the results for a typical case are shown in Fig. 7. Even near threshold, the distribution curve over horizontal recoil angles covers (with probability exceeding half the maximum) the interval from \(4\) to \(8.7^\circ\), owing to the presence of a distribution of the atoms forming the beam over velocities.

At a bombarding-electron energy of \(13.6\) ev, the distribution curve over angles broadens, covering the interval of angles from \(3.6\) to \(10.8^\circ\), while the distribution over vertical recoil angles extends over the interval \(\pm 3.4^\circ\).

11. Detection of metastable hydrogen atoms

Hydrogen atoms excited as a result of electron bombardment to the metastable state \(2\,^2S_{1/2}\) are assumed to move toward the corresponding detector. We considered several possible methods of detection. Two of them had previously been used for detecting metastable atoms of other elements. In 1924 Webb\(^ {88}\) found that excited mercury atoms can eject electrons from metals. Later investigations by Oliphant\(^ {89}\) showed that the same is true in the case of metastable ato-

FINE STRUCTURE OF THE HYDROGEN ATOM

of helium atoms. The theory of this process was given by Massey[^49] and by Cobas and Lamb[^41].

If an excited atom comes close to the surface of a metal, then a case is energetically possible in which the atom returns to the normal state, while an electron released from the system receives the excitation energy. One of the possibilities for such a process is shown in Fig. 8. The energy conditions must then be such that the excitation energy of atom \(I\) exceeds the work function \(\varphi\) for an electron from the metal. In the case of helium the excitation energy is close to \(20\ \text{eV}\), and the condition is fulfilled for all metals. In the case of mercury, the excitation energy of the \(2\,{}^3P_0\) state is only \(4.68\ \text{eV}\), and the fulfillment of the condition becomes strongly dependent on the work function of the given metallic surface. Indeed, Sonkin[^42] showed in 1933 that the efficiency of detection of metastable mercury atoms is highly sensitive—up to a factor reaching 100 and more—to the presence of contamination on the surface. On the other hand, Dorrestein[^43] showed in 1942 that the efficiency of detection of metastable helium atoms on degassed platinum is about 40% and undergoes only comparatively small fluctuations during the day.

Fig. 8. Possible mechanism for the ejection of electrons from tungsten by metastable hydrogen atoms, caused by the Coulomb interaction between the atomic electron and the electrons of the metal.

No information, whatever, existed on the detection of metastable hydrogen atoms. However, in the case of these atoms the excitation energy—\(10.2\ \text{eV}\)—exceeds the work function from the metal sufficiently that one might expect just as high a detection efficiency as for helium. According to rough estimates of the type mentioned above, metastable hydrogen atoms moving with thermal velocities toward a metallic surface should, on average, cause the ejection of an electron before the atom reaches a distance of \(2\ \text{Å}\) from the surface. Since half of the ejected electrons are probably directed into the metal, one could expect an efficiency of the order of 50%.

Nevertheless, we did not have complete confidence in the possibility of detecting metastable hydrogen atoms by means of the mechanism described. These doubts arose in the following way. We also considered another detection method, based on Beuhl’s observation^44 that metastable mercury atoms, on striking hot molybdenum, are re-emitted from its surface in the form of ions. This process underlies the detection method based on the ionizing action of a surface, used in the case of molecular beams, when alkali atoms fall on an incandescent tungsten surface, the electrons being captured by the metal and ions evaporating from the surface as a result. Such a process may be represented by the scheme shown in Fig. 9, and may occur if the energy inequality is satisfied

\[ \varphi > I. \tag{17} \]

Fig. 9. Autoionization of a metastable hydrogen atom near a metallic surface.

In the case of hydrogen atoms in the \(2s\) state, the ionization potential is \(I = 3.4\) eV. Most surfaces that could remain unchanged under the conditions of the experiment should probably have had a work function \(\varphi > 3.4\) eV, and capture of the \(2s\)-electron by the surface should have been possible. Rough estimates of the mean distance at which the process under consideration can occur gave a value of \(5\) Å, i.e. a distance greater than that to which estimates led for the process of electron ejection. If the proton formed as a result of ionization can, with appreciable probability, be torn away from the surface, then this may serve as a method for detecting metastable hydrogen atoms. If, however, the proton is neutralized in some way that does not involve electron emission, then the process of electron capture may prove to be a very serious competitor of the reverse process of electron ejection. When our experiments proved successful, we found that electron emission does occur, but apparently not with as high an efficiency as might have been expected. We tried unsuccessfully to detect emission of positive ions from cold and un-degassed tungsten. Clearly, further investigations of the detection mechanism remain highly desirable.

We have briefly considered, but have not tried, methods based on the detection of Lyman alpha radiation with wavelength \(\lambda = 1216\) Å, arising when the quenching field is applied to the beam or to the bombarded region\(^{45}\). Such a method would be suitable with the use of counters. Another possibility for using counters, which we have not studied, might consist in using secondary multipliers for recording electrons ejected by metastable atoms.

12. Estimate of the yield

We are now in a position to estimate roughly the yield of metastable hydrogen atoms and the resulting electron current that should be expected under typical experimental conditions. Suppose that the atoms emerge from the aperture of a tungsten furnace. According to the kinetic theory of ideal gases, the number of hydrogen atoms emerging through a small aperture of area \(a\) into a solid angle \(\Omega\) at an angle \(\vartheta\) to the normal to the wall is equal to

\[ n_0 a v \Omega X \frac{\cos \vartheta}{4\pi}, \tag{18} \]

where \(n_0\) is the number of atoms and molecules per unit volume, \(X\) is the dissociation coefficient, and \(v\) is the mean velocity of the atoms in the furnace.

Only a few of these atoms will be able to reach the detector plate directly, since, taking into account the small recoil angles, they will undergo deflections. If, however, there were no such displacement and if on the way to the detector the atoms encountered no obstacles, then some fraction of the atoms emerging through the furnace slit would reach a detector of area \(A\), located at a distance \(R\) from the slit. In this case \(\Omega = A/R^2\) and \(\cos \vartheta \simeq 1\). Let \(f\) denote the fraction of atoms excited to the \(2s\) state when the atomic beam passes through the region subjected to electron bombardment. The excited atoms undergo deflections approximately sufficient for them to reach the displaced detector target. However, since there is a spread in the recoil angles, not all metastable atoms will necessarily be able to reach the detector. Only if the solid angle \(\Omega_1\), under which the detector is seen through the system slit from the center of the bombarded region, is greater than the solid angle \(\Omega_2\) of scattering of the beam caused by recoil, will the slit system not affect the metastable atoms. In other words, only a fraction \(\delta\) of them, which

we shall call the coefficient of “attenuation due to recoil,” will be able to reach the detector. One may then put

\[ \delta = \begin{cases} \dfrac{\Omega_1}{\Omega_2}, & \text{if } \Omega_1 < \Omega_2,\\[6pt] 1, & \text{if } \Omega_1 \geq \Omega_2 . \end{cases} \tag{19} \]

The number of electrons knocked out of the target by metastable atoms in 1 second is equal to

\[ S=\frac{\eta_0 a v X A f \delta \mu \eta}{4\pi R^2}, \tag{20} \]

where the factor \(\mu\) takes into account the fraction of metastable atoms that survive passage through the destructive electric field, and \(\eta\) is the efficiency of knocking electrons out of the metallic surface used as the target by metastable hydrogen atoms.

The efficiency of electron bombardment \(f\) may be estimated as follows. Suppose that the intensity of the electron beam is \(I\) electrons per second, and that its height is \(h\) and width \(w\).

The time during which an atom having velocity \(v\) crosses the bombarded region is \(w/v\). For an excitation process characterized by the cross section \(\sigma\), the probability of excitation per unit time is \(I\sigma/wh\), and, consequently, the probability of excitation of an atom passing through the bombarded region is

\[ f=\frac{I\sigma}{hv}=2.3\cdot 10^{-8} \tag{21} \]

for typical values of the electron current \(200\ \mu\mathrm{a}\), \(\sigma = 10^{-17}\ \mathrm{cm}^2\), \(h=1\ \mathrm{cm}\), \(v=8\cdot 10^5\ \mathrm{cm/sec}\). Thus, approximately one atom out of forty million is excited to the \(2s\) state while passing through the bombarded region.

For the detector signal we then obtain the final expression:

\[ S=\frac{\eta_0 a X A I \sigma \delta \mu \eta}{4\pi R^2 h}\ \frac{\text{electrons}}{\text{sec}}. \tag{22} \]

Many of the quantities needed to estimate the yield were not known at that time, and a certain degree of optimism was required in choosing relatively favorable values for them,

to predict the signal level accessible to measurement. Typical values were:

\[ \left. \begin{aligned} I&=1.87\cdot 10^{15}\ \frac{\text{electrons}}{\text{sec.}}\ (0.3\ \text{ma}),\\ n_0&=2.94\cdot 10^{15}\ \text{cm}^{-3}\ (10^{-3}\ \text{atm.},\ 2500^\circ\text{K}),\\ a&=3.1\cdot 10^{-3}\ \text{cm}^2,\\ A&=1.21\ \text{cm}^2,\\ R&=6.35\ \text{cm},\\ \sigma&=10^{-17}\ \text{cm}^3,\\ h&=1\ \text{cm},\\ X&=0.64,\\ \delta&=0.5,\\ \mu&=0.5,\\ \eta&=0.5. \end{aligned} \right\} \tag{23} \]

The values for \(a\), \(A\), \(R\), \(h\), \(I\), \(n_0\), and \(X\) were ultimately close to those actually realized. As explained above, the chosen value for \(\sigma\) was based on a very unreliable theory. The value \(\mu=\frac{1}{2}\) for the survival factor of metastable atoms was justified (see Section 16) provided that effects of destruction of the metastable state by the electric field were absent, including those that could have been due to contact potential differences or electric fields produced by charges deposited on contaminated protective surfaces.

The detector efficiency \(\eta=0.50\) was taken to be extremely large. The value of the attenuation coefficient due to recoil, \(\delta=\frac{1}{2}\), was obtained from consideration of the width of the angular-recoil distribution curve and of the areas of the apertures through which the deflected beam passed. These apertures could not have been made too large, since then the dimensions of the region in which the magnetic and radio-frequency fields had to be homogeneous would have become excessive. Likewise, the background level produced by photons arising in the bombarded region would have increased relative to the useful signal level, since the photons would not have been confined within a relatively small solid angle.

Under the assumptions stated above, the following value was obtained for the signal:

\[ S=3.26\cdot 10^{1}\ \frac{\text{electrons}}{\text{sec.}}=5.2\cdot 10^{-12}\ \text{ampere}. \tag{24} \]

This current is approximately \(5\cdot 10^{4}\) times greater than the limiting value \((10^{-16}\ a)\) that can be registered by means of an FP-54 electrometer tube and a sensitive galvanometer. Consequently, if one or even several of the estimates should prove too optimistic, the signal ought to be sufficiently strong for its detection and for use in accurate radio-frequency spectroscopy. As will be seen below, this estimate has in fact proved to be very optimistic.

13. Requirements on the Power of the Radio-Frequency Radiation

Metastable atoms are subjected to microwaves somewhere between the source and the detector. When the frequency is such that \(\hbar\omega\) is equal, or almost equal, to the energy difference between the Zeeman component of the level \(2\,{}^{3}S_{1/2}\) and the Zeeman component of the level \(2\,{}^{3}P\), the radiation can induce transitions to the nonmetastable level, and the detected signal will be attenuated. The degree of attenuation of the signal will depend on the intensity of the radiation, as well as on its frequency, the velocity of the atoms, and the length of the region filled by the radio-frequency field. We shall now estimate the power of the radio-frequency radiation required for appreciable destruction of the beam.

According to the quantum theory of radiation, the decay constant of the state \(2\,{}^{3}S_{1/2}\) with transition to one of the \(2p\)-states under the action of radio-frequency radiation is equal to*)

\[ \frac{1}{\tau_s} = \frac{2\pi e^2 \gamma S_0}{c\hbar^3}\, \frac{\left|((\mathbf e\mathbf r))\right|^2} {(\omega-\omega_0)^2+\left(\frac{\gamma}{2}\right)^2}, \tag{25} \]

where \(S_0\) is the energy-flux density of the incident radiation, having angular frequency \(\omega\) and electric polarization parallel to the unit vector \(\mathbf e\), \(\omega_0\) is the resonant angular frequency, and \(((\mathbf r))\) is the matrix element of the coordinate vector \(\mathbf r\) for the transition under consideration. As before, \(\gamma=\dfrac{1}{\tau_p}\) denotes the radiative damping constant for the \(2p\) states.

At resonance \(\omega=\omega_0\), and

\[ \frac{1}{\tau_s} = \frac{8\pi e^2 S_0}{c\hbar^3\gamma}\, \left|((\mathbf e\mathbf r))\right|^2. \tag{26} \]

The matrix element \(((\mathbf e\mathbf r))\) can be computed from Bethe’s equations\({}^{46}\), and for the transition from the state \(2\,{}^{3}S_{1/2}\) \((m=1/2)\)

*) The validity of equation (25) will be discussed in the following article.

in the state \(2^3P_{1/2}(m=-1/2)\), taking \(\mathbf e\) to be directed along the \(x\)-axis, we found

\[ \left|(|x|)\right|^2=3a_0^2. \tag{27} \]

The fraction of metastable atoms destroyed by the radio-frequency field is

\[ \varphi=1-e^{-\frac{l}{v\tau_s}}, \tag{28} \]

where \(l\) is the length of the region filled with the radio-frequency field, and \(v\) is the velocity of the atoms. For \(l=1\) cm and \(v=8\cdot 10^5\) cm/sec, the beam will be destroyed by 63% at

\[ \tau_s=\frac{l}{v}=1.25\cdot 10^{-6}\ \text{sec}. \tag{29} \]

According to equation (26), this corresponds to an energy-flux density at resonance of \(S_0=3.4\) mW/cm\(^3\). Since such a power flux is easy to obtain at any frequency, up to 30,000 MHz, we expected no difficulties from the standpoint of the power requirements for the radio-frequency radiation.

Fig. 10. Influence of radio-frequency saturation. The fraction \(\varphi\) of metastable atoms destroyed by the radio-frequency field at resonance, and the half-width of the resonance curve \(\Gamma\) as functions of the power of the radio-frequency radiation, according to equations (28) and (29).

From equations (28) and (25) it is clear that if the power of the radio-frequency radiation is sufficient to obtain complete destruction of the beam at resonance, then the effective resonance curve will be appreciably broadened as a result of radio-frequency saturation. The dependence of the degree of destruction of the beam \(\varphi\) and of the effective half-width of the resonance curve \(\Gamma\) on the power of the radio-frequency radiation is shown in Fig. 10.

14. Width of the Resonance Curves

According to equation (25), attenuation of the beam will occur at \(\omega=\omega_0\), and also for a band of frequencies close to \(\frac{\omega_0}{2\pi}\), whose width is equal to \(\frac{\gamma}{2\pi}\), i.e. 99.8 MHz. This value of the width follows directly from the uncertainty relation \(\Delta E \Delta t \simeq \hbar\) for the decay of the \(2^3P\)-state and cannot be reduced. In addition, as we shall see below, the hyperfine structure increases the line width by a factor that in some cases reaches two. The large width of the resonance curves constitutes a considerable fraction of the transition frequencies and presents the greatest difficulty for [[unclear: word beginning “дей…”]]

a relatively accurate test of Dirac’s theory. On the other hand, a favorable opportunity presents itself for a very thorough test of the theory of the radiative form of the Wigner–Weisskopf line, since the resonance curves have an incomparably greater natural width relative to the operating frequency than anywhere in atomic physics.

The most obvious way of determining the resonance frequency \(\omega_0/2\pi\) would be to measure the beam intensity as a function of the frequency of the microwave radiation, whose power would be kept constant. However, the latter condition is practically impossible to fulfill. Microwave oscillators that can be tuned over a range covering hundreds of megacycles are quite available, but the power they radiate depends on frequency. Even if this were not the case, the line (or waveguide) transmitting the radiation from the oscillator to the excitation region would be electrically long, and therefore the intensity of the radio-frequency field irradiating the atom would vary with frequency. One could resort to the aid of a regulator of the radio-frequency power level with a crystal probe or a bolometer located in the excitation region, but these devices are themselves sensitive to changes in frequency.

We therefore decided, in order to overcome the difficulty indicated above, to make use of the presence of a magnetic field, which was also necessary for other reasons. The frequency of the oscillator and the level of the power emitted by it were kept constant, but by varying the strength of the magnetic field the atomic energy levels were shifted in such a way that the difference between them passed through the resonance value.

In first approximation the atomic energy levels, and hence the frequencies, are linear functions of the magnetic-field strength, so that the resonance curves obtained in this way differ in form hardly at all from curves obtained by the usual method. In order, however, to interpret the results, it is necessary to know the magnetic-field strength and to rely on the theory of the Zeeman effect for the fine-structure levels of the hydrogen atom with \(n=2\). The results of this theory are considered in the next section.

15. The Zeeman Effect for the Fine Structure of the Hydrogen Atom

For the present it will be sufficient for us to consider an approximate theory of the Zeeman effect for the fine structure of the hydrogen atom, taking into account the interaction of the atom with an external magnetic field \(H\) by means of the perturbation energy

\[ \mathcal{H}'=\mu_0(\mathbf{L}\mathbf{H}+2\mathbf{S}\mathbf{H}), \tag{30} \]

where

\[ \mu_0=\frac{e\hbar}{2mc} \tag{31} \]

is the Bohr magneton, \(\mathbf L=\frac{1}{\hbar}[\mathbf r,\mathbf P]\) is the orbital angular momentum and \(\mathbf S\) is the spin angular momentum, measured in units of \(\hbar\). The splitting of the levels \(2\,{}^2S_{1/2}\) may be considered independently of the splitting of the levels \(2\,{}^2P\), since \(\mathscr H'\) contains no matrix elements coupling the \(s\)- and \(p\)-states. The solution of this problem is well known, and we simply write out the result from Bethe’s paper\({}^{47}\).

For the levels \(2\,{}^2S_{1/2}\), \(m_s=\pm \frac12\):

\[ E(2\,{}^2S_{1/2},\,m_s;\,H)=E(2\,{}^2S_{1/2})+2\mu_0 Hm_s. \tag{32} \]

For the levels \(2\,{}^2P_{3/2}\), \(m_j=\pm \frac32\):

\[ E\left(2\,{}^2P_{3/2},\,m_j=\pm \frac32;\,H\right) =E(2\,{}^2P_{3/2})+\frac34\,\mu_0Hm_j. \tag{33} \]

For the levels \(m_j=\pm \frac12\) of the states \(2\,{}^2P_{1/2},\,{}^2P_{3/2}\), the energies are given by the roots of the secular equation, namely:

\[ E=\frac12(E_+ + E_-)+\mu_0Hm_j \pm \]

\[ \pm \frac12\sqrt{(E_+-E_-)^2+\frac43\,\mu_0H(E_+-E_-)m_j+(\mu_0H)^2}, \tag{34} \]

where \(E_+\) and \(E_-\) are, respectively, the energies of the states \(2\,{}^2P_{3/2}\) and \(2\,{}^2P_{1/2}\) in the absence of the field.

If the energy is measured in units

\[ hf_1=\frac{2(E_+-E_-)}{3}\simeq 7300\ \text{MHz}, \tag{35} \]

taking as the origin of counting \(\frac13(2E_+ + E_-)\), and the magnetic field—in units

\[ H_1=\frac{2}{3\mu_0}(E_+-E_-)\simeq 5214\ \text{gauss}, \tag{36} \]

i.e. putting

\[ \mu_0H=\frac23(E_+-E_-)x \tag{37} \]

and

\[ E=\frac23(E_+-E_-)y+\frac13(2E_+ + E_-), \tag{38} \]

then we obtain

\[ y=-\frac14+m_jx\pm \frac12\sqrt{x^2+2m_jx+\frac94}. \tag{39} \]

In the same notation, the energies of the levels \(2^{2}P_{1/2},\ m_j=\pm \dfrac{3}{2}\) are given by the expression:

\[ y=\frac{1}{2}\pm 2x \tag{40} \]

and of the levels \(2^{2}S_{1/2},\ m_s=\pm \dfrac{1}{2}\):

\[ y=y_0-1\pm x, \tag{41} \]

where \(y_0\) measures the displacement of the level \(2^{2}S_{1/2}\) relative to the level \(2^{2}P_{1/2}\) in the absence of a magnetic field. A diagram of these energy levels is given in Fig. 11, showing the regions of anomalous

Fig. 11. Zeeman splitting of the fine structure of the levels of the hydrogen atom with \(n=2\), according to Dirac’s theory.

Fig. 12. Part of Fig. 11 on an enlarged scale, showing the crossings used for designating the Zeeman components of the energy levels.

Zeeman splitting, splitting in a field of average strength, and the Paschen–Back effect. Here, in accordance with the prediction of Dirac’s theory, \(y_0=0\). For brevity we shall denote these energy levels by single letters. The two metastable levels \(2^{2}S_{1/2},\ m_s=\dfrac{1}{2}\) and \(m_s=-\dfrac{1}{2}\) will be denoted, respectively, by \(\alpha\) and \(\beta\) (which corresponds to the traditional notation for Pauli spin wave functions). The levels \(2^{2}P\) will be denoted by the letters \(a, b, c, d, e, f\), as shown in Fig. 12,

on which, in enlarged form, the region of anomalous Zeeman splitting and splitting in a field of intermediate strength that is of greatest interest to us is shown.

The selection rules for electric dipole radiation in the region of anomalous Zeeman splitting are: \(\Delta m_j=0\)—for the electric vector of the radio wave parallel to the magnetic field, and \(\Delta m_j=\pm 1\)—for perpendicular polarization. Accordingly, for parallel polarization the allowed

Fig. 13. Expected resonance frequencies as functions of the magnetic-field strength for all allowed transitions leading from the metastable states

\[ \alpha\left(2^{2}S_{1/2},\ m=-\frac{1}{2}\right) \]

and

\[ \beta\left(2^{2}S_{1/2},\ m=-\frac{1}{2}\right) \]

into the nonmetastable states \(a, b, c, d\) \(\left(2^{2}P_{1/2}\right)\) and \(e, f\) \(\left(2^{2}P_{1/2}\right)\), according to Dirac’s theory.

transitions are \(ab, ae, \beta c, \beta f\), whereas for perpendicular polarization the transitions \(aa, ac, af, \beta b, \beta d, \beta e\) are allowed. The frequencies to be expected for these transitions are shown in Fig. 13 as functions of the magnetic-field strength.

In fact, since the fine-structure splitting

\[ \frac{\alpha^2 h c R}{16} \]

is equal to \(10950\) Mc/s and the displacement of the level \(s\Delta = 1000\) Mc/s, we now know that \(y_0 = 0.137\), and not zero, as Dirac’s theory predicts. On this basis the curves of the energy levels will be as shown in Fig. 14. The dependence of the corresponding transition frequencies on the magnetic-field strength is shown in Fig. 15.

Fig. 14. Zeeman splitting of energy levels. Same as in Fig. 12, but taking into account that the unperturbed level \(2\,^2S_{1/2}\) is raised above the level \(2\,^2P_{1/2}\) by 1000 Mc/s.

Fig. 15. Expected resonance frequencies as functions of the magnetic-field strength. Same as in Fig. 13, but taking into account that the unperturbed level \(2\,^2S_{1/2}\) is raised above the level \(2\,^2P_{1/2}\) by 1000 Mc/s.

16. Production of a polarized beam of atoms

From a comparison of Figs. 12 and 14 it is clear that the actually occurring displacement of the level \(2\,^2S_{1/2}\) leads to the difference noted above in the degree of stability of the two metastable states \(a\) and \(\beta\) in a magnetic field of the order of 540 gauss. For the destruction of the metastable state by an electric field \(\mathbf{E}\) we have

\[ \frac{1}{\tau_s} = \frac{\gamma V^2}{\hbar^2\left[\omega^2+\left(\frac{\gamma}{2}\right)^2\right]}, \tag{42} \]

where \(V = (|e\mathbf{E}\cdot \mathbf{r}|)\) is the matrix element of the perturbation energy \(e\mathbf{E}\cdot\mathbf{r}\), connecting the two interacting states, and \(\hbar\omega\) is the difference

energies of these states. For \(H=540\) gauss, \(\dfrac{\omega}{2\pi}\) is equal to \(2020\) MHz for transitions between the states \(a\) and \(f\), and to zero for transitions between the states \(\beta\) and \(e\). As a result, the destruction of the lower metastable state \(\beta\) will occur much more rapidly than the destruction of the upper metastable state \(a\); namely, the rates of destruction will differ by a factor

\[ 1+4\,\frac{\omega^2}{\gamma^2}\simeq 1630. \tag{43} \]

The principal perturbing electric field is produced by the motion of the atoms perpendicular to the magnetic field \(\mathbf H\); its strength is equal to

\[ \mathbf E=-\frac{1}{c}\,[\mathbf v,\mathbf H]. \tag{44} \]

Consequently, \(\mathbf E\) is perpendicular to \(\mathbf H\). If we introduce rectangular Cartesian coordinates with the \(z\)-axis directed along \(\mathbf H\), and the \(x\)-axis along \(\mathbf v\), then \(\mathbf E\) will be directed along the \(y\)-axis. The perturbation energy \(e\mathbf E\cdot\mathbf r\) has a matrix element coupling the states \(\beta\) and \(e\), but not the states \(\beta\) and \(f\). For \(v=8\cdot10^5\) cm/sec and \(H=540\) gauss, \(E=4.3\) V/cm.

The lifetime of the state \(\beta\) in a perturbing electric field of strength \(4.3\) V/cm, perpendicular to the magnetic field, obtained from Bethe’s calculation, as described in Section 6, is \(4.3\cdot10^{-8}\) sec, whereas the lifetime of the state \(a\) will be 1630 times greater, i.e., equal to \(7\cdot10^{-5}\) sec. This means that, in the presence of the dynamic electric field, almost all atoms excited to the state \(a\) can preserve their state of excitation during the time necessary for them to traverse the distance of 6 cm separating them from the detector.

At the same time, practically not a single atom excited to the lower metastable state \(\beta\) will preserve its excited state by the moment it reaches the detector. Consequently, the atomic beam, already at a small distance from the point where it is subjected to electron bombardment, will become polarized—almost all the excited atoms will have their electron spins oriented parallel to the magnetic field. Excited atoms with antiparallel electron spins will be filtered out as a result of the destructive action of the dynamic electric field. This method of obtaining a strongly polarized atomic beam may be compared with other known methods: the small spatial separation obtained in experiments on the deflection of an atomic beam, and the polarization of neutrons passing through strongly magnetized absorbers.

The situation described above is by no means limited to the critical value of the field strength \(H=540\) gauss, at which the levels \(\beta\) and \(e\) intersect, but extends over an entire interval of field strengths. With the apparatus to be described in the present article, the transitions \(\beta b\), \(\beta c\), \(\beta d\), \(\beta e\), \(\beta f\), shown in Fig. 15, were not observed.

The destruction of metastable states as a consequence of the Stark effect caused by the dynamic field also takes place at a magnetic-field strength close to that at which the levels \(\alpha\) and \(c\) intersect, i.e. close to \(H=4700\) gauss. This region is much wider than for the intersection of the levels \(\beta e\), since the dynamic electric fields are stronger, and the beam of metastable atoms should be strongly destroyed at field strengths exceeding 3000 gauss.

17. Influence of the hyperfine structure of the energy levels

The consideration, carried out in Section 15, of the energy levels corresponding to the states with \(n=2\) ignored the interaction of the magnetic moment of the nucleus (proton, deuteron) with the magnetic moment of the electron. Fortunately, although the influence of this interaction cannot in some respects be neglected, it can be determined for the purposes that interest us by means of perturbation theory. The magnetic moment of the nucleus

\[ \mu = g_I I \mu_0 \tag{45} \]

creates a magnetic field whose vector potential is

\[ \mathbf{A}=\frac{1}{r^3}[\mu,\mathbf{r}]. \tag{46} \]

Here \(g_I\) is the Landé factor for the nucleus (about \(\frac{5.6}{1836}\) for the proton and \(\frac{1.7}{1836}\) for the deuteron), \(\mu_0\) is the Bohr magneton (see equation (31)), and \(I\) is the nuclear-spin vector. According to the Dirac equation, the energy of interaction with the electron is \(e\alpha\cdot \mathbf{A}\), where \(\alpha_x,\alpha_y,\alpha_z\) are the Dirac matrices. The shift of the energy levels due to the hyperfine structure is small and can be calculated in the first approximation of perturbation theory by forming the average of \(\alpha\cdot\mathbf{A}\) over the unperturbed Dirac four-component wave function for the electron state. In forming this average, the two small components of the wave function must be computed near the nucleus with some caution in order to avoid the indeterminacy associated with the singularity of \(\mathbf{A}\) at the source. The theory\(^ {48}\) was developed by Bethe for the case of a vanishingly small external magnetic field. The results for the energies are given below.

For the state \(s\), the effective perturbation operator is

\[ w=\frac{16\pi}{3}\,g_I\mu_0^2|\psi(0)|^2\,\mathbf{I}\cdot\mathbf{S}, \tag{47} \]

or

\[ w=\frac{8}{3}\,g_I\frac{Z^3}{n^3}\alpha^2hcR\,\mathbf{I}\cdot\mathbf{S} \tag{48} \]

and gives a displacement of the energy levels equal to

\[ \frac{4}{3}\,g_I\frac{Z^3}{n^3}\alpha^2hcR \begin{cases} I, & \text{for } F=I+\dfrac{1}{2},\\[6pt] -(I+1), & \text{for } F=I-\dfrac{1}{2}. \end{cases} \tag{49} \]

The splitting of the two states is equal to

\[ \Delta w_n=\frac{8}{3}\,g_I\left(I+\frac{1}{2}\right)\frac{Z^3}{n^3}\alpha^2hcR. \tag{50} \]

For the ground state of the hydrogen atom this splitting is \(1416\) Mc/s (it increases to \(1420\) Mc/s for the anomalous value of the magnetic moment of the electron, as shown in the papers of Nafe, Nelson, and Rabi \(^{49}\)). In the case of interest to us, \(n=2\), the splitting is one eighth of this value, i.e. \(177\) Mc/s.

The theory of the Zeeman effect for this hyperfine structure in an arbitrary magnetic field is given by the well-known Breit–Rabi formula \(^{50}\) and is illustrated in Fig. 16. For our purposes it is sufficient to consider only the limiting case of a strong field, when \(\mathbf{I}\) and \(\mathbf{S}\) are not coupled to one another and precess independently about \(\mathbf{H}\), their projections being respectively equal to \(m_I\) and \(m_S\).

Fig. 16. Zeeman splitting of the hyperfine structure of the level \(2^2S_{1/2}\). The splitting in the absence of a field is chosen so that it is equal to one eighth of that measured by Nafe and Nelson for the ground state. The dashed lines show the energy levels that are obtained if the hyperfine structure is ignored.

Then in equation (48) \(\mathbf{I}\cdot\mathbf{S}\) may be replaced by \(m_I\cdot m_S\), and for the energy of the perturbation responsible for the appearance of the hyperfine structure we obtain the expression

\[ w=\frac{8}{3}\,g_I\frac{Z^3}{n^3}\alpha^2hcR\,m_I m_S. \tag{51} \]

The state \(a\), for which \(m_S=+\dfrac{1}{2}\), is split into \(2I+1\) states. For hydrogen \(I=\dfrac{1}{2}\) and there exist two states,

separated by an interval \(\frac{1}{2}\Delta w\), or \(88\) Mc/s. The state \(\beta\) \((m_S=-\frac{1}{2})\) is split in exactly the same way, but the levels are arranged in the reverse order owing to the change in sign of \(m_S\). In interpreting more accurate measurements, which will be described in the following paper, it will, of course, become necessary to discuss the validity of this approximation for a strong field. Within the framework of the present article the error is small, since the dimensionless parameter

\[ x'=\frac{(g_J-g_I)\mu_0 H}{\Delta w} \tag{52} \]

reaches unity already at a magnetic-field strength of \(63.6\) gauss, whereas almost all the data described here were obtained in fields of much greater strength.

For other states, different from \(s\), Bethe assumed that the fine-structure splitting is large in comparison with the hyperfine-structure splitting and derived for the perturbation operator the equation

\[ w=2\mu_0^{\,2}g_I\left(r^{-3}\right)_{\mathrm{cp}} \frac{L(L+1)}{J(J+1)}\,\mathbf I\cdot\mathbf J = g_I\frac{Z^3}{n^3}\, \frac{a^2hcR\,\mathbf I\cdot\mathbf J} {\left(L+\frac{1}{2}\right)J(J+1)} . \tag{53} \]

In the absence of a magnetic field

\[ \mathbf I\cdot\mathbf J= \frac{1}{2}\,[F(F+1)-I(I+1)-J(J+1)] \tag{54} \]

and

\[ (\mathbf I\cdot\mathbf J)_{F=|I-J|}-(\mathbf I\cdot\mathbf J)_{F=I+J} = \begin{cases} I(2J+1), & J>I,\\ J(2I+1), & J<I, \end{cases} \]

so that for the hydrogen atom the state \(P_{1/2}\) is split into two states with quantum numbers of the resultant angular momentum \(F=1,0\), and the state \(P_{3/2}\) is split into two states with \(F=2,1\). The magnitudes of the splittings are respectively equal to \(1/3\) and \(2/15\) of the splitting of the corresponding \(2S_{1/2}\) states.

In the presence of a magnetic field not strong enough to destroy the coupling between \(\mathbf L\) and \(\mathbf S\), one may continue to use equation (53). However, to the Hamiltonian there should be added the energy of orientation of the vectors \(\mathbf I\) and \(\mathbf J\) in the magnetic field:

\[ \mu_0(g_J\mathbf J-g_I\mathbf I)\cdot\mathbf H, \]

and the problem becomes mathematically identical with the problem of the Zeeman effect for an \(S\) state, so that the Breit–Rabi formula can be used if \(I\) or \(J\) is equal to \(\frac{1}{2}\).

We obtain a simplification if we consider a magnetic field sufficiently strong to destroy the coupling of I with J, but not strong enough to destroy the coupling of L with S. Then \(\mathbf{I}\cdot \mathbf{J}\) may be replaced by \(m_I \cdot m_J\), and the perturbation energy responsible for the presence of hyperfine structure is equal to

\[ w=-\frac{g_I Z^3 2hcR}{n^3J(J+1)\left(L+\frac{1}{2}\right)}\,m_I m_J+\mu_0 H\left(g_J m_J-g_I m_I\right). \tag{55} \]

As a result of the presence of hyperfine structure, turning again for simplicity to the case \(I=\frac{1}{2}\), each state proves to be split into two states, displaced equally on the energy scale relative to the unperturbed state.

In the approximation considered here, the splitting of the hyperfine structure depends on the magnetic-field strength only because of the presence in expression (55) of the term \(-\mu_0 H g_I m_I\). The expected splitting should be:

\[ \frac{1}{2}\Delta w=88\ \text{MHz}\quad \text{for the state } 2\,{}^3S_{1/2}, \]

\[ \frac{1}{6}\Delta w=29\ \text{MHz}\quad \text{for the state } 2\,{}^3P_{1/2}, \]

\[ \frac{1}{30}\Delta w=6\ \text{MHz}\quad \text{for the state } 2P_{3/2},\ m_J=\pm\frac{1}{2} \]

and

\[ \frac{1}{10}\Delta w=9\ \text{MHz}\quad \text{for the state } 2P_{3/2},\ m_J=\pm\frac{3}{2}. \]

In the case of deuterium, where \(I=1\), the picture becomes more complicated, but the magnitude of the splitting is very small because the nuclear magnetic moment is smaller.

The splitting of energy levels caused by hyperfine structure leads to a complication of the observed resonance curves. In most cases the natural width of the curves exceeds the hyperfine-structure splitting, and the resulting curve is formed by two more-or-less well-resolved resonance curves of the Wigner–Weisskopf type.

As an illustration we use here the case of transitions from the state \(a(2^2S_{1/2},\ m_s=\frac{1}{2})\) to the states \(e(2^2P_{1/2},\ m_j=\frac{1}{2})\) and \(f(2^2P_{1/2},\ m_j=-\frac{1}{2})\).

The observed transitions belong to the type of electric dipole transitions and, since the magnetic field destroys the coupling of the nuclear—

of angular momentum with other vectors of the momentum, the selection rules will be \(\Delta m_I=0\).

The allowed transitions are shown in Fig. 17. It should be noted that, owing to the difference in sign of the product \(m_I \cdot m_J\), the distance between the two peaks is equal to \(\frac{4}{3}\Delta \omega\), or 117 Mc, for the transition \(af\), and only \(\frac{2}{3}\Delta \omega\), or 58 Mc, for the transition \(ae\).

Fig. 17. Hyperfine structure of the levels \(2^2S_{1/2}\left(m=-\frac{1}{2}\right)\) and \(2^2P_{1/2}\) in a strong magnetic field. The allowed transitions are indicated by arrows. The peaks of the resonance curve are separated, respectively, by intervals of 117 and 59 Mc. This interval is comparable with the radiation width of 100 Mc. For simplicity it is assumed that the components of the hyperfine structure are shifted by the same amount relative to the levels shown by the dashed line and corresponding to the complete Back–Goudsmit decoupling.

The natural width of an isolated transition is 100 Mc. When two such curves are superposed on one another, the resulting complex curve has two peaks only in the

case where the peaks are separated by a distance exceeding half the natural width, i.e. 50 Mc/s. The theoretical form of these curves is shown in Fig. 18, a and b. Even in the case of the \(af\) transition, where the distance between the peaks is 117 Mc/s, the presence of hyperfine structure leads only to a flattening of the top of the resonance curve, if there are any other causes leading to its broadening.

Fig. 18. Ideal theoretical resonance curves showing the influence of hyperfine structure for the \(ae\) and \(af\) transitions. If the resonance curves are plotted as functions of the magnetic-field intensity, then the resonance curve for the \(af\) transition becomes narrower than the curve for the \(ae\) transition.

In the measurements described in the present article, the observed peaks were noticeably broadened, in comparison with the expected ones, as a result of the inhomogeneity of the magnetic field.

B. APPARATUS

18. Preliminary Attempts

In view of the fact that the production and detection of metastable hydrogen atoms appeared theoretically possible, but had not been experimentally tested, it was evidently necessary first of all to achieve the simultaneous realization of both processes. The arrangement of the initial apparatus intended for this purpose was made as simple as possible. It consisted of a source and a detector of metastable atoms, enclosed in glass bulbs connected by a glass tube 150 mm long and 12.5 mm in diameter. A magnetic field could be applied either to the source, to the detector, or to the connecting tube. It was produced by permanent magnets of the type used in K-band magnetrons. It was assumed that later the connecting tube would pass through the broad wall of an X-band waveguide (10 × 22.5 mm), so that the metastable atoms could be acted upon by a radiation field having a frequency of about 10,000 MHz.

We chose the second of the methods considered in Section 7 for producing metastable hydrogen atoms, consisting in the direct excitation of molecules by electron bombardment, since it seemed to us the simplest. The electron gun consisted of an oxidized cathode, a grid, and an anode.

The last two electrodes were maintained at the same positive potential relative to the cathode, in order to create a region comparatively free of electric field in which metastable atoms could be formed. This device provided a stream of electrons with energies up to 50 eV.

For the reasons discussed in Section 11, a method of detection was chosen using electrons knocked out of a metal by a beam of metastable atoms. The atoms were directed onto a tungsten target, and the electrons knocked out by them were collected by a positive electrode. This method had the disadvantage that electrons knocked out from the surface by any other means—for example, as a result of the photoelectric effect—created a background whose magnitude had to be appropriately limited.

However, the photoelectric current forming the background made it possible to study the electron-bombardment regime by obtaining excitation curves for hydrogen and helium.

Hydrogen entered the region bombarded by electrons, and if metastable atoms arose, then some of them, after passing through the connecting tube, should have reached the detector. Unfortunately, electron bombardment of molecular hydrogen entails various kinds of effects that are difficult to distinguish ...

from those expected in the case of metastable hydrogen atoms. By means of magnetic fields and an electric field, produced by a pair of auxiliary electrodes introduced for this purpose into the connecting tube, it was shown that these phenomena could be attributed to photons, electrons, positive ions, and charges collecting on the glass walls. The last effect was especially unpleasant.

By the usual method of investigation at this stage, rough excitation curves were obtained as functions of the energy of the bombarding electrons, with the corresponding corrections for the presence of extraneous ions, electrons, and the influence of charges on the glass walls. Although at that time there was no possibility of confirming the presence of metastable atoms, it was evident from the magnitude of the photon background and the low potential of its occurrence that the chosen excitation process was not as favorable as had been expected. It became clear that a two-stage method, consisting of preliminary dissociation of the molecules and subsequent excitation of the atoms by electron bombardment, would be less subject to these difficulties.

19. Working model

When it became clear that a radical change of the apparatus was inevitable, a new apparatus was built, made entirely of metal, with the introduction of all the improvements whose necessity had become apparent by that time. Nevertheless, before full-fledged results could be obtained, numerous changes had to be made in it.

The general scheme of the apparatus is shown in Fig. 19, which represents a horizontal cross-section along the axis of coaxial circular cylinders forming the right and left chambers. The rectangular region \(J\) is a cross-section of a rectangular waveguide of the \(X\)-band \((10 \times 22.5\ \mathrm{mm})\), passing through the apparatus in the vertical direction. The magnetic field is directed perpendicular both to the waveguide and to both chambers. The pole pieces of the magnet are mounted in recesses in the apparatus having the shape of a truncated cone. With the exception of a certain amount of ferromagnetic materials, such as steel and Kovar, near the glass joints, the outer shell of the apparatus was made mainly of copper.

The right chamber contains the detecting electrodes; in the left chamber are placed the hydrogen dissociator and the electron gun. In the intermediate chamber \(J\) a radio-frequency field is produced. Hydrogen atoms, after leaving the dissociator \(A\), enter the region \(D\), in which a fraction of them is excited to the \(2s\) state as a result of bombardment of the atomic beam by electrons.

After changing, as a result of recoil, the direction of their motion somewhat, the atoms pass through slits \(I\) and the region \(J\), filled with radio-frequency radiation, and strike the detecting target \(L\).

In accordance with the considerations discussed in Section 10 and Appendix III, in choosing the relative

Fig. 19. Cross section of the apparatus. \(A\)—tungsten furnace of the hydrogen dissociator, \(B\)—screens, \(C\)—anode of the electron gun, \(D\)—bombarded region, \(E\)—accelerating grid of the electron gun, \(F\)—control grid; \(G\)—cathode of the electron gun, \(H\)—cathode heating, \(I\)—slits, \(J\)—waveguide, \(K\)—wires serving for controlled destruction of the metastable state of atoms by a constant electric field and at the same time constituting a transmission line for microwave radiation, \(L\)—target of the detector of metastable atoms, \(M\)—electrode collecting electrons.

position of the hydrogen dissociator, the electron gun, and the slits \(I\), the recoil angle was taken into account. As such, the mean value \(\psi\) for electrons with energy \(13.6\) eV was taken; it was approximately \(8^\circ\) (see Appendix III).

20. Detector

The detecting electrodes consisted simply of a pair of tungsten plates \(L\) and \(M\). As was described in Section 11, metastable atoms striking the plate \(L\) knocked electrons out of it, which were then collected on the plate \(M\), whose potential was maintained at 3 or 4 V above the potential of plate \(L\). The electron current was measured by means of a standard electrometric circuit\(^ {53}\) with an FP-54 tube. The input resistan-

was about 95,000 megohms. The current sensitivity was about \(1.5\cdot 10^{-16}\) A/mm. It was found that cleaning the target—which at that time was a tungsten ribbon \(0.1\) mm thick and \(6\) mm wide—by annealing it at a temperature of \(1000^\circ\)C or higher had no stable effect. Therefore, without any noticeable loss of sensitivity, an unannealed target was used. The detector response to metastable atoms proved not to depend too strongly on the collector voltage when the latter was varied from 2 to 4 V, but fell sharply when it was reduced to zero, and became vanishingly small at a negative collector voltage of the order of 1 V.

Since the surface of the target was subjected only to rough cleaning during its preparation, it was beyond doubt that it was covered with a layer of various kinds of contamination. In a number of cases, after operation for some time, this layer of contamination became visible; however, its presence apparently did not have a noticeable effect on the detector efficiency.

21. Hydrogen Dissociator

Atomic hydrogen was obtained as a result of thermal dissociation of molecular hydrogen, as was discussed in Section 8. Details of the apparatus are shown in Fig. 20. A thin-walled tungsten cylinder \(1.6\) mm in diameter was made, as shown, from a tungsten sheet \(0.1\) mm thick. In the middle of the cylinder, parallel to its axis, a slit \(0.2\times1.5\) mm in size was cut. The cylinder was tightly inserted into an opening made in the molybdenum end of the water-cooling tube. Molecular hydrogen was introduced through a tube located inside the water-cooling channel. By passing an electric current through the tungsten tube, the latter could be heated, in its central part, to a temperature sufficient to obtain the required degree of dissociation. The atoms thus formed flew out through the small aperture, which served as the source. Naturally, a significant amount of molecular hydrogen leaked through at the ends of the cylinder.

The operating temperature in the central part of the cylinder was usually close to \(2500^\circ\)K. Higher temperatures, although they could be reached and were desirable, led to excessively rapid destruction of the tungsten cylinders. We have no reliable data on the degree of dissociation obtained by this method, but on the basis of estimates made in Section-

*8, the degree of dissociation should be close to 64%. The inconveniences associated with working at such a comparatively low degree of dissociation were compensated by the general convenience of the method.

Fig. 20. Details of the construction of the hydrogen dissociator and tungsten furnace.

It was found that alternating current can be successfully used for heating the hydrogen dissociator. Under typical operating conditions the current was about 80 A and the voltage drop about 2 V.

22. Electron gun

The electron gun proved to be the most difficult part of the apparatus, and much effort was required to determine how it should be constructed in order to perform its functions well. Since its improvement for the purpose of a more accurate determination of the displacement of the \(2S\) level required considerably more work, this question will be considered in greater detail in Part II.

On the basis of the considerations given in Sections 6–10, it is entirely possible to establish the properties of the electron gun that are best for the purposes of the present experiment. The electric field in the bombarded region, including the field produced by the space charge, must be as weak as possible. The energy of the bombarding electrons must be close to the threshold value \(10.2\ \text{eV}\), (a) in order to reduce the spread of recoil angles and (b) to eliminate background-producing effects,

THE FINE STRUCTURE OF THE HYDROGEN ATOM

associated with the presence of molecular hydrogen and arising when the energy of the bombarding electrons starts at 11.5 eV.

In practice, however, in order to obtain a satisfactory signal level it proved necessary to depart substantially from these requirements.

The electron gun we used consisted of a cathode, a control grid, an accelerating grid, and an anode, as shown in Fig. 19. The cathode \(G\) was oxide-coated and indirectly heated. The cathode and the control grid \(F\) were completely enclosed inside the accelerating electrode \(E\), which had a mesh structure only on the side facing the anode. Such an arrangement served to shield the bombarded region \(D\), in which excitation of the atoms occurred, from the electric field created between the accelerating electrode and the cathode. The shields \(B\) prevented radiation and evaporating material from the hydrogen dissociator from entering the excitation region and the detector. In the initial models of this type of electron gun the anode \(C\) was absent; it was assumed that the outer shell of the instrument would serve the role of the anode. With such an arrangement, however, it was not possible to detect metastable atoms with certainty. This could be attributed either to a shortage of atomic hydrogen or to defects in the detecting circuit.

The question of the presence or absence of hydrogen atoms was resolved by an independent method. For this purpose the hydrogen emerging from the dissociator was directed onto a layer of soot of yellow molybdenum oxide. The presence of hydrogen atoms was clearly revealed by the rapid transformation of the yellow oxide into blue. In the absence of better control, the detector’s high sensitivity to photons could be regarded as direct evidence that it should also be sensitive to metastable hydrogen atoms. Thus there were indications that a possible source of the difficulties was the electron gun. It was found that the influence of the space charge formed in the region between the accelerating electrode and the anode had been substantially underestimated. At the low electron energies used, \(10\text{—}20\ \mathrm{eV}\), a current with a density of about \(0.2\ \mathrm{mA/cm^2}\) produces a large potential drop between the accelerating grid and the anode, strongly dependent on the distance separating the electrodes. The introduction of a separate anode reduced this distance from 3 to approximately 1 cm or less. After this had been done, an effect was immediately detected which had to be regarded as produced by metastable hydrogen atoms.

It was found that the most advantageous operating conditions are: accelerating-grid voltage about 13.5 V and anode bias plus 3 V relative to the accelerating grid. The magnitude of the anode current

was maintained at about 0.3 ma by regulating the voltage on the control grid. These conditions were subjected to considerable changes, the grounds for which are considered in Section 28. In general, the choice of operating regime was dictated by the need for a compromise between instability and signal level.

The cathode was made of nickel powder fused onto a molybdenum base and coated with the usual triple mixture of barium, strontium, and calcium carbonates. The cathode area was approximately 1.5 cm². The control grid was made of tungsten wire 0.05 mm in diameter (6.4 turns per centimeter); the accelerating grid, made of the same wire, had 10 turns per centimeter. The cathode and accelerating control grids were each located at a distance of approximately 0.8 mm. The distance from the accelerating grid to the anode was about 9 mm.

23. Radio-frequency and constant destructive electric fields

Metastable atoms were subjected to the action of radio-frequency or constant electric fields in the region of waveguide \(J\) (Fig. 19). The atoms entered this region and left it through slits \(I\) (4.7 mm wide and 16 mm high). Wires \(K\), introduced into the apparatus long before we turned to the use of a radio-frequency field, performed auxiliary functions for the identification of various kinds of particles—electrons, ions, and photons—which at one time were being detected. Subsequently it turned out that the presence of these wires is of great importance for the identification of metastable atoms, because the electric field formed between the wires as a result of applying a constant voltage to them produces a Stark effect, causing mixing of the \(2\,{}^{4}S_{1/2}\) and \(2\,{}^{2}P_{1/2}\) states, which leads to the destruction of the metastable states of the atoms and to the transition of the atoms into the \(1S\) states, thereby causing a decrease in the measured detector current. In view of the difficulties encountered in maintaining the number of metastable atoms in the beam at a sufficient level, the use of this possibility proved necessary during the measurement process, and not only in the preparation of the apparatus, as had initially been supposed. Finally, transitions to the \(2\,{}^{2}P_{1/2}\) level occur at frequencies from 1500 to 6000 MHz. For frequencies in this range, the wires \(K\) were used as a two-wire transmission line.

The wires were located near the straight lines joining the ends of the slits, but did not shield the latter. The electrostatic field produced by the wires when a voltage of 25 V or more was applied to them was quite sufficient for destruction.

practically all the metastable atoms in the transmitted beam. Since the efficiency of destroying the atoms is determined precisely by the field strength, the required voltage depends strongly on the configuration of the various electrodes and conductors. In view of the fact that the configuration of the radio-frequency field was rather complicated and we made no attempt to terminate the line with the corresponding load, it was difficult to estimate the magnitude of the required power of the radio-frequency radiation. It must, of course, have exceeded the value obtained in the estimates considered in Section 13. In practice it was found that the generator had to deliver a power of about 10 watts or more; the required power depends on the frequency, mainly because of the frequency dependence of the load impedance.

Radio-frequency radiation in the frequency range from 3000 to 10,000 MHz was obtained directly from klystrons 2K41, 2K44, or 2K39. Frequencies close to 1200 MHz were obtained by doubling the frequency from a 2K44 klystron with the aid of a 1N23 crystal multiplier. To obtain frequencies in the range from 1500 to 2600 MHz, a 2C40 lighthouse oscillator was used. The frequency stability attainable when using a conventionally regulated power supply was found to be sufficient. The frequency or, more precisely, the wavelength of the radiation was measured with a coaxial cavity wavemeter.

24. Magnetic Field

The magnetic field was produced by a small electromagnet. The conical pole pieces of the electromagnet were 5 cm in diameter, tapering at the end to 2.5 cm in diameter. In these experiments the gap between the poles was 25 mm. With this magnet it was easily possible to obtain a magnetic field strength of 3000 gauss or more. The magnet was fed from a rectifier through a filter with a stabilizer.

The field strength was calibrated by moving the detector electrodes aside and introducing, through slit \(I\) into the center of cavity \(J\), which was filled with radio-frequency radiation, a small test coil. To obtain the calibration curve for the dependence of the magnetic-field strength \(H\) on the current feeding the electromagnet, under the experimental conditions the usual demagnetization procedure and the magnetization-curve method were used. The calibration was carried out with an accuracy of approximately 0.5%. Such accuracy was more than sufficient, since it was found that in the region where the radio-frequency field acted on the beam of metastable atoms, the inhomogeneity of the magnetic field lay within approximately 33 gauss at a mean field strength of 1000 gauss.

25. Gas supply and pumps

In the experiments described, hydrogen was used in the form of a technically purified gas. According to the firm’s data, it contained not less than 99.7% hydrogen; the remainder consisted of oxygen in the form of water vapor.

The gas was stored in three one-liter glass vessels and was admitted into the hydrogen dissociator through a permanent gas leak made by sealing an unpolished piece of tungsten wire 0.5 mm in diameter into a thin Pyrex capillary. It was usually necessary to make three or four such leaks before obtaining one that passed the desired amount of gas. This amount was approximately \(5 \cdot 10^{-6}\) l of air per second at a pressure difference of 1 atm. The rate of admission of gas into the dissociator could be regulated by changing the pressure in the vessel from which the leakage occurred. The decrease in the gas pressure in the vessel as a result of its outflow into the apparatus was an undesirable effect, but its influence on the amount of gas leaking in was small in comparison with the fluctuations and changes caused by other factors.

Because of the insufficient magnitude of the signal obtained, the admission of gas into the dissociator had to be increased until the working pressure in the apparatus reached \(10^{-3}\) mm Hg. At somewhat higher pressures the diffusion pumps already ceased pumping. The pumping speed was evidently not in agreement with the leakage rate, but not to such an extent as to prevent the experiment from being carried out. The apparatus was pumped through two 25-mm pipelines (not shown in Fig. 19) by means of three-stage fractionating pumps. The low vacuum was provided by a mechanical pump.

G. OBSERVATIONS

26. Magnitude of the signal

The detector current forming the signal is a stream of electrons ejected from the detector plate by metastable atoms. The background is the current produced by photoelectrons arising, primarily, as a result of excitation to molecular states of the hydrogen molecules present in the apparatus and, to some extent, as a result of excitation of hydrogen atoms to atomic states different from \(2^2S_{1/2}\).

At the energy of the bombarding electrons used in the experiments described, the background level was three or more times higher than the signal level. A magnetic field with an intensity of 100 gauss or more protected the detector from the incidence upon it of ions or electrons from the electron gun.

The signal was observed as the deflection of a galvanometer obtained when a constant voltage, sufficiently strong to destroy practically all metastable atoms, was applied to the wires \(K\) (shown in Fig. 19), as described in Section 23. This deflection was usually from 20 to 50 cm on the galvanometer scale and in individual cases reached 80 cm, which corresponded to a current of \(1.2 \cdot 10^{-13}\) A.

This value is approximately 40 times smaller than the value obtained as a result of the estimates considered in Section 12. Some unknown part of this discrepancy may be attributed to the destruction of metastable states by stray electric fields in the electron gun, produced by charges that collect on nonconducting deposits deposited on metallic surfaces (see Section 28). At times this circumstance led to the practically complete disappearance of the signal. On the other hand, the strongest signal was obtained in those cases when the metallic surfaces had been carefully cleaned of all deposits. It seems unlikely that the amount of insulating material then remaining in the apparatus would have been sufficient to ensure such intense destruction of the beam of metastable atoms.

The estimates of all quantities entering equation (20), except for the detector efficiency \(\eta\), were, in all probability, underestimated, so that \(S\) represents a lower bound for the expected signal. The discrepancy with the observed signal can be eliminated if one takes \(\eta = \frac{1}{80}\) instead of \(\frac{1}{2}\), as was done in Section 12. Possible grounds for so small a detector efficiency were indicated in Section 11.

27. Dependence of the Magnitude of the Signal on the Magnetic-Field Strength

The observations covered the interval of changes in magnetic-field strength from 50 to 3000 gauss. At each value of the magnetic-field strength, some fraction of the metastable atoms is lost from the beam as a result of destruction of the metastable states by a dynamic electric field whose strength is determined by equation (44). The magnitude of these losses increases with increasing magnetic-field strength, as is clear from the discussion carried out in Sections 6 and 16. It was found that the signal generally weakens as the field strength is increased within the investigated interval and becomes extremely weak at the strongest fields, in qualitative agreement with theory. The design features

the apparatus did not permit a thorough investigation of this phenomenon. In addition to the unsuitable geometry, interaction effects in the electron beam made the results difficult to interpret.

28. Signal stability

In Section 26 it was mentioned that the magnitude of the signal was subject to appreciable changes. Some of these were explained in Section 27 as being caused by the magnetic field, but along with this there were also uncontrolled changes produced mainly by the electron beam. Some of these latter changes were due to the cathode, which had to be replaced every three or four days because of loss of emission. Cathode poisoning was presumably caused by oil vapors diffusing from the pump, or by water vapor entering the apparatus with the hydrogen. So long as the emission was sufficient, the drop in current could be corrected by adjusting the voltage on the control grid.

Together with loss of emission by the cathode, changes were observed, as mentioned in Section 22, in the optimum operating values of the current and voltage in the electron beam. Corresponding to this was a decrease in the magnitude of the signal, which toward the end of the measurements became appreciably weaker than at their beginning. The cause of this remained unclear at the time, but now we know that such a decrease in signal magnitude is due to the formation of charged nonconducting layers on the grids and anode. Responsible for this is the penetration into the apparatus of oil vapor from the pump and, possibly, grease vapor. Along with this, dissociation as a result of electron bombardment also takes place; alkaline-earth oxides evaporating from the cathode surface may also make their own contribution to this effect. Charges accumulating on these insulating layers increase the stray electric fields, which have a destructive action on metastable atoms and, consequently, cause a weakening of the signal. They also exert a large influence on the electronic operating conditions of the electron beam. This circumstance will be considered in Part II.

In order to obtain a sufficiently strong signal, it proved necessary to increase the hydrogen pressure in the source as much as possible without the increase of pressure in the apparatus leading to any noticeable destruction of the metastable states as a result of interatomic collisions. This, unfortunately, turned out to be associated with the appearance of numerous short-period fluctuations, mainly in the background level, which, as noted in Section 26, was already much greater than the signal.

Fluctuations of the background level were superimposed on fluctuations in the beam

metastable atoms and constituted a serious limitation on the accuracy of the measurements.

Changes in the detection efficiency could also be a conceivable cause of both long- and short-period changes in the magnitude of the signal. However, as a rule they could with equal justification be attributed to other causes.

29. Procedure

In Section 14 it was shown that it would be very difficult to observe the shape of the resonance curve while keeping the magnetic-field strength constant and varying the frequency. The inverse procedure was such that the oscillator frequency was kept at a constant, predetermined value, and the magnetic-field strength was varied. The procedure consisted simply in demagnetizing the magnet and then increasing the magnetic-field strength stepwise; at each value of it the increase in the signal produced by interrupting the radio-frequency radiation entering the waveguide was determined. The resonance curves obtained in this way at different frequencies are shown in Figs. 21–24. In these figures the galvanometer deflections, obtained by interrupting the radio-frequency radiation, are plotted as functions of the magnetic-field strength.

The variability of the electron gun, considered in the preceding section, made it necessary to find a new optimum operating mode before each series of measurements. For this, two or three hours of operation were usually sufficient.

In view of the difficult experimental conditions, in order to increase the observed effect radio-frequency radiation of excessive power was used.

In the main, the influence of long- and short-period fluctuations and changes in the magnitude of the signal could be eliminated by measuring the signal at each value of the magnetic-field strength, using a constant destroying electric field, and expressing the degree of destruction of the beam of metastable atoms by the radio-frequency field as the ratio of the destruction of the beam by the radio-frequency field to its destruction by the constant field. These quantities, when plotted as functions of \(H\), would give a more exact form of the resonance curve. However, it was very difficult to keep the apparatus in operating condition for such a long time as was necessary to obtain a resonance curve by the method described above, performing only a few observations of the full signal. This made the use of the more exact method practically infeasible. Nevertheless, the resonance peaks were sufficiently well defined that changes in the magnitude of the signal with changes in the magnetic-field strength did not entail a large error.

D. Analysis of Data and Results

30. Shape and width of the resonance curves

Several resonance curves, typical of the best measurements, are shown in Figs. 21–24, illustrating examples of all the observed transitions. It should be noted that these are precisely transitions from the upper metastable level \(\alpha\). In accordance with the considerations carried out in Section 16, no other transitions were observed.

Fig. 21. Observed resonance curve.

Fig. 22. Observed resonance curve.

Comparison of the experimental resonance curves for hydrogen, shown in Fig. 24, with the theoretical curve depicted in Fig. 18, \(a\), shows that the expected partial resolution of the hyperfine structure for the transition \(\alpha f\) is not observed at all. Likewise, no flattening of the top of the resonance curve is found for the transition \(\alpha f\) shown in Fig. 18, \(b\). These discrepancies with the theoretical predictions may be due to the inhomogeneity of the magnetic field and to the phenomenon of radio-frequency saturation. The inhomogeneity of the magnetic field, amounting in the region \(J\) to 33 gauss per 1000 gauss, should lead to a smearing of the hyperfine structure and to an increase in the theoretically expected width of the resonance peak. As was established in Section 13, radio-frequency saturation also leads to an increase in the half-width of the resonance peak and to a corresponding deterioration in the resolution of the hyperfine structure. In addition, a cause of the broadening of the resonance peak may be leakage of the radio-frequency field from the waveguide through comparatively large-

Fig. 23. Observed resonance curve.

Fig. 24. Observed resonance curves.
The curves obtained in the case of hydrogen and deuterium are similar.

aperture inside the bombarded region. The destroyed metastable atoms are in this region in a magnetic field that is somewhat weaker than the field inside the waveguide. Consequently, it is to be expected that the half-widths of the resonance peaks will appreciably exceed the calculated values. This is seen from the table given here. In obtaining the calculated values, only the radiation width of \(100\) Mc/s and the unresolved hyperfine structure of the \(S\) and \(P\) levels were taken into account. The expected Doppler broadening differences between hydrogen and deuterium are not found, possibly because of the strong saturation effect in the latter case.

Half-widths of the resonance peaks

Substance	Transition	Half-width of peak in Mc/s	Half-width of peak in Mc/s
Substance	Transition	observed	calculated
H	\(af\)	410	219
H	\(ae\)	320	159
D	\(af\)	400	128
D	\(ae\)	320	114

Fig. 25. Summary of data obtained in measurements of resonance peaks. The solid lines correspond to Dirac theory, as in Fig. 13; the dashed lines correspond to the displaced curves of Fig. 15, drawn allowing for the fact that the level \(2\,^{2}S_{1/2}\) is raised by \(1000\) Mc/s. The agreement of the dashed curves with the experimental points clearly testifies to the reality of such a displacement of the level \(2\,^{2}S_{1/2}\).

31. Results

The resonance values of the magnetic-field strength were found simply from the position of the observed peak—in the case of sharply expressed peaks—and by averaging the fields corresponding to the half-height of the peak—for broadened peaks. No corrections for mutual overlap of peaks or for other factors were introduced. The resonance values of the magnetic-field strength obtained in this way at various frequencies are shown in Fig. 25. The theoretical curves for the Zeeman effect, calculated on the assumption of the correctness of Dirac’s theory, are shown by solid lines. For comparison with the experimentally obtained points, the theoretical curves were shifted downward by \(1000\) Mc/s—in the case of the transition to the

level \(2^2P_{3/2}\)—and higher by \(1000\) Mc/s—in the case of the transition to the level \(2^2P_{1/2}\).

In view of the uncertainty introduced by the inhomogeneity of the magnetic field and by the displacement of the peaks due to the decrease in the signal with increasing magnetic-field strength, no attempts were made to obtain better agreement between the shifted curves and the experimental points.

The results show beyond doubt that, contrary to Dirac’s theory but in substantial agreement with Pasternack’s hypothesis, the level \(2^2S_{1/2}\) lies above the level \(2^2P_{1/2}\) by approximately \(1000\) Mc/s \((0.33\ \mathrm{cm}^{-1})\), or about \(9\%\) of the doublet splitting. Within the accuracy of these results, no discrepancy has been found between the observed doublet splitting of the \(P\)-levels and that predicted by Dirac’s theory.

A comparison of the resonance curves for hydrogen and deuterium shows that, to this accuracy, the displacement of the levels \(2^2S_{1/2}—2^2P_{1/2}\) for deuterium is the same as for hydrogen.

APPENDIX I

Conditions in the Wood discharge

The absorption of radio waves by excited hydrogen atoms in a Wood discharge tube depends on the population of the various states. The latter, in turn, depends on the relation between the processes of formation and destruction of excited atoms. This entails an extensive program of investigations necessary for carrying out quantitative calculations. Here we shall have to content ourselves with the crudest estimates. Only the states \(n=1\) and \(n=2\) of atomic hydrogen will be considered.

We assume that the \(2p\) state is destroyed by transition to the \(1s\) state with a decay constant corresponding to the natural lifetime \(\tau_p=1.6\cdot 10^{-9}\) sec, and that excitation of the \(2p\) state occurs primarily under the action of two causes: 1) electron bombardment and 2) absorption of Lyman resonance radiation emitted by other atoms. Since the absorption coefficient for this radiation is very large, a quantum of resonance radiation can, on the average, leave the tube only after undergoing a large number of acts of absorption and reemission. This number has been estimated\(^{53}\) as lying, for typical discharge conditions, between 500 and 1000. As a result, the effective decay constant of the \(2p\) states becomes much smaller than that corresponding to the natural lifetime, and the population of the \(2p\) states correspondingly increases.

For the \(2^2S_{1/2}\) state the situation is substantially different. As indicated in Section 9, the probability of excitation from the \(1^2S_{1/2}\) state as a result of electron bombardment is only

about one tenth of the probability of excitation of the \(2p\) states. Capture of resonance radiation here plays no role; for the state \(2\,{}^3S_{1/2}\), optical excitation from the ground state is impossible. However, the state \(2\,{}^3S_{1/2}\) under certain conditions may be metastable, and, despite the small probability of excitation, its population may prove to be high. As shown in Section 6, the decay constant for the state \(2\,{}^3S_{1/2}\) increases with an increase in the strength of the electric field acting on the atom, and decreases with the growth of the splitting of the states \(2\,{}^3S_{1/2}\) and \(2\,{}^3P_{1/2}\). If this splitting is zero, as follows from Dirac’s theory, then the lifetime of the state \(2\,{}^3S_{1/2}\) under typical conditions of a glow discharge exceeds the natural lifetime only by a few times; but if the splitting is \(1000\) Mc/sec, then the lifetime increases approximately to \(900\,\tau\), and the population of the state \(2\,{}^3S_{1/2}\) correspondingly increases. Let us consider transitions between the states \(2\,{}^3S_{1/2}\) and \(2\,{}^3P_{3/2}\), induced by radio waves. If these states are populated in accordance with their statistical weights (equal distribution), then no appreciable net absorption of radio waves will be observed, since induced emission will exactly compensate induced absorption. (Spontaneous transitions between the states \(2\,{}^3P_{3/2}\) and \(2\,{}^3S_{1/2}\) occur negligibly rarely.) If the population of the state \(2\,{}^3S_{1/2}\) exceeds the population of the state \(2\,{}^3P_{3/2}\), then net absorption of radio-frequency radiation should be observed. On the other hand, if the population of the state \(2\,{}^3P_{3/2}\) is higher than the population of the state \(2\,{}^3S_{1/2}\), then net induced emission (negative absorption) will take place.

On the basis of the preceding consideration alone, one might have expected that the \(2p\) levels would be approximately five times more populated than the \(2s\) levels. In that case one would have expected negative absorption and, as estimated below, a very considerable one. It is generally recognized that estimates of tube population are crude. However, it seems that, if there were no mechanism providing an effective coupling between the states \(2\,{}^3S_{1/2}\) and \(2\,{}^3P_{1/2}\), then a strict equal distribution could occur only by chance.

As indicated above and as considered quantitatively in Section 6, the coupling between these two states is produced by the Stark effect. If there were no capture of resonance radiation, then the population of the state \(2\,{}^3P_{1/2}\) would be insufficient for establishing an equal distribution by means of this mechanism. However, on the basis of the numerical estimates given above and in Appendix II, it seems quite possible that an equal distribution can be established and that the absorption of radio-frequency radiation will be appreciably reduced.

If equipartition has not been achieved, then pure absorption or pure emission takes place. In order to turn these qualitative considerations into a quantitative estimate, we shall calculate the nonequilibrium fraction, induced by the radio-frequency radiation, of transitions from the state \(2\,^2S_{1/2}\) to the state \(2\,^2P_{3/2}\), ignoring the reverse transitions, which may almost annul the whole effect or even change its sign. Although the result of such a calculation may be an overestimate of the expected “absorption,” it will provide a convenient basis for discussion.

We proceed to estimate the absorption coefficient. In a typical Wood discharge tube the pressure may be taken equal to \(0.15\) mm Hg, which corresponds to the density

\[ n_{\mathrm H}=\frac{0.15\times 2.687\times 10^{19}}{760}=5.3\cdot 10^{15} \tag{56} \]

hydrogen atoms in \(1\ \mathrm{cm}^3\).

The number of atoms per unit volume excited during \(1\) sec. to the state \(2\,^2S_{1/2}\) will be equal to

\[ J\sigma n_{\mathrm H}, \]

where \(eJ\) is the electron-current density and \(\sigma\) is the excitation cross section. If we equate this quantity to the assumed number of atoms per unit volume returning during \(1\) sec. to the ground state,

\[ \frac{n^*}{900\,\tau_p}, \]

then we obtain

\[ n^*=900\,J\sigma\tau_p n_{\mathrm H}. \tag{57} \]

Taking an electron-current density corresponding to \(0.1\ \mathrm{a}/\mathrm{cm}^2\):

\[ J=\frac{0.1}{1.602\cdot 10^{-19}}=6.24\cdot 10^{17}\ \frac{\text{electrons}}{\mathrm{cm}^2\,\mathrm{sec}} \]

and

\[ \sigma=10^{-17}\ \mathrm{cm}^2 \]

(see Section 9), we find the concentration of atoms excited to the state \(2\,^2S_{1/2}\):

\[ n^*=4.7\cdot 10^{10}\ \mathrm{cm}^{-3}. \tag{58} \]

Let us now consider the absorption of radio waves by these excited hydrogen atoms, taking into account, as stated above, only transitions to the state \(2\,^2P_{3/2}\). Just as in Section 13, the probability of a transition upon absorption of radiation is taken equal to

\[ \frac{1}{\tau_{\mathrm{ind}}} = \frac{2\pi e^2 S_0\gamma}{c\hbar^2}\, \frac{|(\mathbf e,\mathbf r)_{12}|^2}{(\omega-\omega_0)^2+\left(\dfrac{\gamma}{2}\right)^2}. \tag{59} \]

where \(S_0\) is the energy density of the incident radiation having angular frequency \(\omega\) and electric-polarization vector parallel to the unit vector \(\mathbf e\); \(\omega_0\) is the resonant angular frequency; \((|\mathbf r|)\) is the matrix element of the coordinate vector \(\mathbf r\) of the atomic electron for the transition from the state \(2^2S_{1/2}\) to the state \(2^2P_{3/2}\);

\[ \gamma=\frac{1}{\tau_p} \]

is the quantity inverse to the lifetime of the state \(2^2P_{3/2}\). For some estimates, instead of the transition probability it is more convenient to use the cross section \(\sigma_{\mathrm{ind}}\) for absorption of a microwave photon. This quantity is determined by the equation

\[ J_p\sigma_{\mathrm{ind}}=\frac{1}{\tau_{\mathrm{ind}}}, \tag{60} \]

where \(J_p\) is the flux density of such photons. Since \(J_p=\dfrac{S_0}{\hbar\omega_0}\), we have

\[ \sigma_{\mathrm{ind}}=\frac{\hbar\omega_0}{S_0\tau_{\mathrm{ind}}}, \]

or

\[ \sigma_{\mathrm{ind}}=\frac{2\pi e^2}{\hbar c}\, \frac{|((\mathbf e,\mathbf r))|^2\omega_0\gamma} {(\omega-\omega_0)^2+\left(\dfrac{\gamma}{2}\right)^2}. \tag{61} \]

For the sum of all transitions to the sublevels of the state \(2^2P_{3/2}\), one should take \(|((\mathbf e,\mathbf r))|^2\) equal to two thirds of the value obtained by summing over all sublevels of the states \(2^2P_{1/2}\) and \(2^2P_{3/2}\).

The latter value is obtained in calculating \(\sigma_{\mathrm{ind}}\) for transitions from the state \(2s\) to the state \(2p\) without taking the electron spin into account. Hence\({}^{53}\)

\[ |((\mathbf e,\mathbf r))|^2 \longrightarrow \frac{2}{3}\,\frac{27a^2}{3}=6a_0^2. \tag{62} \]

Under resonance conditions

\[ \omega=\omega_0=2\pi\cdot 10\,950\cdot 10^6\ \mathrm{sec}^{-1}, \tag{63} \]

and using the value

\[ \gamma=\frac{1}{1.595\cdot 10^{-9}}=6.25\cdot 10^8\ \mathrm{sec}^{-1}, \tag{64} \]

we find

\[ \sigma_{\mathrm{ind}}=3.4\cdot 10^{-15}\ \mathrm{cm}^2. \tag{65} \]

For a concentration of atoms excited to the state \(2^2S_{1/2}\), \(n^*=4.7\cdot 10^{10}\) atoms/\(\mathrm{cm}^3\), the absorption coefficient should then be equal to \(\mu=1.6\cdot 10^{-4}\ \mathrm{cm}^{-1}\). By the standards of present-day microwave spectroscopy this is a large absorption coefficient, but the width of the resonance peak \(\dfrac{\gamma}{2\pi}\), \(100\) Mc/s, is many times greater than the widths usually encountered in this region. As indicated above, it appears possible that

this absorption is almost completely compensated by the reverse transitions. However, in view of the extreme crudeness of the numerical estimates, it is not excluded that there may be some departure from equipopulation, and that absorption or induced radiation may be detected. It is therefore highly desirable that the investigation of these effects be carried out under conditions of such a discharge as do not favor the establishment of equipopulation.

As a complicating factor there will appear here a large, frequency-dependent background, produced by the absorption of microwaves by electrons. Gaase[^19] found good agreement with the equation

\[ \mu_{\text{electr}}=\frac{2e^{2}z\lambda^{2}N_{e}}{\pi mc^{3}}, \tag{66} \]

derived by Stuart[^54] for the absorption coefficient of radiation of wavelength \(\lambda\). Here \(z\) is the number of collisions with gas molecules experienced by an electron in 1 sec., and \(N_e\) is the number of free electrons in \(1\ \text{cm}^3\). For typical conditions:

\[ \begin{gathered} z=0.95\cdot 10^{3}\ \text{sec.}^{-1},\\ N_e=2\cdot 10^{3}\ \text{cm}^{-3},\\ \lambda=3\ \text{cm} \end{gathered} \]

we have

\[ \mu_{\text{electr}}=1.03\cdot 10^{-4}\ \text{cm}^{-1}. \tag{67} \]

APPENDIX II

Destruction of Metastable Hydrogen Atoms by an Electric Field[^55]

We intend here to generalize the theory of the influence of a homogeneous electric field on the lifetime of the \(2^{2}S_{1/2}\) level, given by Bethe[^26], since in Bethe’s calculations the removal of the degeneracy of the states \(2^{2}S_{1/2}\) and \(2^{2}P_{1/2}\) was not taken into account.

We shall consider two excited levels with probability amplitudes \(a\) and \(b\). State \(a\) is metastable in the absence of an external electric field and has a small decay constant \(\gamma_a\) (a two-quantum transition to the ground state with lifetime \(\frac{1}{7}\) sec.). Level \(b\) is situated energetically above level \(a\) by the amount \(E_b-E_a=\hbar\omega\) (\(\omega\) may be negative) and is destroyed by transition to the ground state with emission of resonance radiation. The corresponding decay constant is

\[ \gamma_b=\frac{1}{\tau_p}. \]

The perturbation-theory equations then have the form

\[ \left. \begin{aligned} i\hbar \dot a &= V^{*}e^{-i\omega t}b-\frac{1}{2}i\hbar\gamma_a a,\\ i\hbar \dot b &= Ve^{-i\omega t}a-\frac{1}{2}i\hbar\gamma_b b, \end{aligned} \right\} \tag{68} \]

where \(V=(b|e\mathbf E,\mathbf r|a)\) is the matrix element of the energy of the perturbing electric field for the transition \(a\to b\). Decay is treated by the phenomenological introduction of a damping term; this can be justified by writing equations of the Wigner–Weisskopf type\({}^{56}\). The general solution of equations (68) has the form

\[ a=A_1e^{\mu_1t}+A_2e^{\mu_2t}, \]

\[ b=-\frac{\hbar}{iV^{*}}\left[\left(\mu_1+\frac{1}{2}\gamma_a\right)A_1e^{(\mu_1+i\omega)t} +\left(\mu_2+\frac{1}{2}\gamma_a\right)A_2e^{(\mu_2+i\omega)t}\right], \tag{69} \]

where \(\mu_1\) and \(\mu_2\) are the roots of the quadratic equation

\[ \left(\mu+\frac{1}{2}\gamma_a\right) \left(\mu+i\omega+\frac{1}{2}\gamma_b\right) +\frac{|V|^2}{\hbar^2}=0. \tag{70} \]

In most applications the term \(\gamma_a\) may be discarded. Then, for weak electric fields,

\[ \left. \begin{aligned} \mu_1 &\simeq -i\omega-\frac{1}{2}\gamma_b,\\ \mu_2 &\simeq -\frac{|V|^2}{\hbar^2\left(i\omega+\frac{1}{2}\gamma_b\right)}. \end{aligned} \right\} \tag{71} \]

Excitation of the metastable state corresponds to the initial conditions:

\[ a=1,\quad b=0\quad \text{at } t=0. \tag{72} \]

This gives

\[ \left. \begin{aligned} A_1+A_2&=1,\\ \mu_1A_1+\mu_2A_2&=0, \end{aligned} \right\} \tag{73} \]

whence

\[ A_1=\frac{\mu_2}{\mu_2-\mu_1},\quad A_2=-\frac{\mu_1}{\mu_2-\mu_1}. \tag{74} \]

The probability that the level \(a\) will remain occupied after the lapse of time \(t\) is equal to

\[ |a|^2=\left|A_1e^{\mu_1t}+A_2e^{\mu_2t}\right|^2. \tag{75} \]

The first term has a small coefficient and decays rapidly. Since \(|A_2| \simeq 1\), the effective decay constant is

\[ \gamma_{\text{Stark}}=\mu_2+\mu_2^*= \frac{\gamma_b |V|^2}{\hbar^2\left(\omega^2+\frac{1}{4}\gamma_b^2\right)} . \tag{76} \]

For \(\omega=0\) this leads to the result obtained by Bethe. In the case of several widely separated levels \(b\), none of which is coupled too strongly with level \(a\), the decay constants are additive.

In the limit \(|V|^2 \to 0\) the exact solution gives

\[ \mu_2' + \mu_2^* = \gamma_a, \]

whereas in the limit

\[ \frac{|V|^2}{\hbar^2\left(\omega^2+\frac{1}{4}\gamma_b^2\right)} \to \infty \]

we have

\[ \mu_i' + \mu_i^* = \frac{\gamma_b}{2} \]

for \(i=1,2\), i.e. the expected result. The approximate expression (76) is sufficient for most purposes.

In discussing the question of level populations in Appendix I, a problem arose concerning the mechanism by which equipartition between the states \(2^2P_{1/2}\) and \(2^2S_{1/2}\) can be established. Estimates showed that without such a mechanism the population of the \(2^2P_{1/2}\) level may be considerably (up to 12 times) greater than the population of the \(2^2S_{1/2}\) level. Let us now consider an atom excited by resonance radiation to the state \(2^2P_{1/2}\). Very quickly such an atom will make a transition to the state \(1^2S_{1/2}\), accompanied by reemission of resonance radiation. However, owing to the coupling between the states \(2^2P_{1/2}\) and \(2^2S_{1/2}\) caused by the Stark effect, there is a certain small probability that, instead of a transition to the ground state, a transition to the long-lived state \(2^2S_{1/2}\) will occur. This probability can be calculated from the solution of equation (68) with the initial conditions:

\[ a=0,\qquad b=1 \quad \text{for } t=0. \tag{77} \]

Since the term containing \(\mu_1\) in the exponent decays rapidly, the probability of transition to the metastable state is

\[ |a|^2=|A_2'|^2 \simeq \frac{V^2}{\hbar^2\left(\omega^2+\frac{1}{4}\gamma_b^2\right)^2} = \frac{\gamma_{\text{Stark}}}{\gamma_b} \simeq \frac{1}{800}. \tag{78} \]

Since resonance radiation undergoes, on average, from 400 to 1000 acts of absorption and reemission before it succeeds in leaving the discharge region, there is a high probability

that the long-lived state \(2^2S_{1/2}\) will be populated before radiation is emitted. Consequently, such a mechanism for establishing an equilibrium distribution between the states \(2^2P_{1/2}\) and \(2^2S_{1/2}\) apparently is plausible.

APPENDIX III

Distribution over recoil angles

As was noted in Section 10, hydrogen atoms excited to the metastable state as a result of electron bombardment experience recoil, measured by the angle \(\theta\), which indicates the change in the direction of their motion. Our purpose here will be to find the distribution of the excited atoms over recoil angles. Let \(M\) be the mass of the atom, \(\mathbf{V}_0\) its initial velocity, \(\mathbf{V}_1\) its final velocity, and let the velocity of the bombarding electron of mass \(m\) before the collision be \(\mathbf{v}_0\), and after the collision \(\mathbf{v}_1\). From the law of conservation of momentum in the collision,

\[ M\mathbf{V}_0 + m\mathbf{v}_0 = M\mathbf{V}_1 + m\mathbf{v}_1 . \tag{79} \]

The law of conservation of energy gives, approximately,

\[ \frac{1}{2}mv_0^2 - \frac{1}{2}mv_1^2 = \frac{3}{4}hcR, \tag{80} \]

where we neglect the change in the kinetic energy of the atom.

Conservation of momentum can be represented graphically as shown in Fig. 26, where \(\varepsilon\) is the angle of deflection of the electron scattered as a result of the inelastic collision. The circle in

Fig. 26. For the calculation of the distribution over recoil angles in the excitation of hydrogen atoms to the state \(2^2S_{1/2}\) by bombardment with electrons of energy \(13.6\ \mathrm{eV}\).

Fig. 26 conventionally represents the sphere formed by the ends of the vectors \(m\mathbf{v}_1\); it should be remembered that the vector \(\mathbf{V}_1\) may therefore lie outside the plane determined by the vectors \(\mathbf{V}_0\) and \(\mathbf{v}_0\). For given values of \(V_0\) and \(v_0\), the angle \(\theta\) may have different values owing to the distribution over the angles \(\varepsilon\). For simplicity, let us assume that the distribution in \(\varepsilon\) is spherically symmetric. Such an assumption is not so bad near threshold, where \(v_1\) is relatively small and the wave function of the scattered electron corresponds, essentially, to zero orbital angular momentum.

Taking into account the conditions of the experiment, we were interested separately in the horizontal \((\psi)\) and vertical \((\chi)\) recoil angles. Approximately one may put

\[ \tg \psi=\frac{m v_0-m v_1'}{M V_0}, \tag{81} \]

where \(v_1'\) is the magnitude of the projection of the vector \(\mathbf v_1\) on the direction \(\mathbf v_0\). In an analogous way,

\[ \tg \chi=\frac{m v_1''}{M V_0}, \tag{82} \]

where \(v_1''\) is the magnitude of the projection of \(\mathbf v_1\) on the perpendicular to the horizontal plane.

Since the angles \(\psi\) and \(\chi\) are very small, the tangents may be replaced by the angles, expressed in radian measure. Under the assumption of a spherically symmetric distribution for \(\mathbf v_1\), the probability of the distribution with respect to \(v_1'\) is the same within the limits from \(v_1\) to \(-v_1\); the same*) holds also for \(v_1''\). Hence the probability distribution with respect to \(\psi\) is the same for the interval of angles \(\psi_1 \leq \psi \leq \psi_2\), where

\[ \psi_1=\frac{m v_0-m v_1}{M V_0} \quad \text{and} \quad \psi_2=\frac{m v_0+m v_1}{M V_0}, \tag{83} \]

i.e. we may write

\[ P(V_0,\psi)\,d\psi= \begin{cases} \dfrac{M V_0}{2m v_1}\,d\psi & \text{for } \psi_1 \leq \psi \leq \psi_2,\\[6pt] 0 & \text{in all other cases,} \end{cases} \tag{84} \]

which satisfies the normalization condition

\[ \int_{\psi_1}^{\psi_2} P(V_0,\psi)\,d\psi=1. \tag{85} \]

Analogously,

\[ P(V_0,\chi)\,d\chi= \begin{cases} \dfrac{M V_0}{2m v_1}\,d\chi & \text{for } -\chi_1 \leq \chi \leq +\chi_1,\\[6pt] 0 & \text{in all other cases,} \end{cases} \tag{86} \]

where

\[ \chi_1=\frac{m v_1}{M V_0}. \tag{87} \]

Expressions (84) and (86) are valid for specified initial velocities \(\mathbf v_0\) and \(\mathbf V_0\) of the bombarding electrons and hydrogen atoms. The bombarding electrons may be assumed monoenergetic. The distribution of the atoms with respect to velocities, however, leads to additional broadening of the distribution with respect to scattering angles. Assuming that in the furnace serving as the source of the atomic beam,

*) We neglect correlations between \(\psi\) and \(\chi\).

there is thermal equilibrium corresponding to the temperature \(T\), then for the distribution of the atoms in the beam by velocities we have the expression

\[ N(V_0)\,dV_0=AV_0^3 e^{-\frac{V_0^2}{U^2}}, \tag{88} \]

where \(A\) is determined from the normalization condition \((A=2U^{-4})\), and the parameter \(U\) from the relation \(\frac{1}{2}mU=kT\). The most probable velocity of the atoms forming the beam is \(\sqrt{\frac{3}{2}U}\).

Taking into account the distribution of the atoms by velocities, we obtain for the distribution of the excited atoms by angles \(\psi\):

\[ P(\psi)=\int_0^\infty N(V_0)P(V_0,\psi)\,dV_0 = \frac{M}{2mv_1}\int_{V_a}^{V_b}N(V_0)V_0\,dV_0, \tag{89} \]

where

\[ V_a=\frac{mv_0-mv_1}{M\psi};\qquad V_b=\frac{mv_0+mv_1}{M\psi}. \tag{90} \]

Making the substitution \(y=\frac{V_0}{U}\), we obtain

\[ P(\psi)=\frac{MU}{mv_1}\int_{y_1}^{y_2}y^4e^{-y^2}\,dy, \tag{91} \]

where

\[ y_1=\frac{V_a}{U}\quad \text{and}\quad y_2=\frac{V_b}{U}. \tag{92} \]

Similarly,

\[ P(\chi)=\frac{MU}{mv_1}\int_0^{y_3}y^4e^{-y^2}\,dy, \tag{93} \]

where

\[ y_3=\frac{mv_1}{M\chi}. \tag{94} \]

Electron bombardment with energies close to threshold

These expressions are substantially simplified at the threshold value of the energy of the bombarding electrons, when \(v_1\to 0\). In this case there is no vertical recoil, and the distribution over horizontal angles takes the form

\[ P(\psi)=2\left(\frac{mv_0}{MU}\right)^4\psi^{-5}e^{-\left(\frac{mv_0}{MU\psi}\right)^2}. \tag{95} \]

This function is shown in Fig. 7 for a furnace temperature \(T = 2600^\circ \mathrm{K}\) \(\left(U = 6.55 \cdot 10^5\ \text{cm/sec}\ \text{and the most probable speed of motion of the atoms in the beam } \sqrt{\frac{3}{2}}\,U = 8.03 \cdot 10^5\ \text{cm/sec}\right)\).

Bombardment by electrons with energy 13.6 ev

At this energy \(v_0 = \alpha c\) and

\[ v_1 = \frac{1}{2}v_0 = \frac{1}{2}\alpha c, \]

where \(\alpha\) is the fine-structure constant.

Thus,

\[ y_1 = \frac{1}{3}y_2 = \frac{m\alpha c}{2\mu U\psi} \quad \text{and} \quad y_4 = \frac{m\alpha c}{2\mu U\chi}. \]

The values of the integrals (91) and (93) were found by numerical integration, and the corresponding graphs for \(P(\psi)\) and \(P(\chi)\) are given in Fig. 7.

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2. H. A. Bethe, Phys. Rev. 72, 339 (1947). A relativistic calculation of the term shift was made by N. M. Kroll and W. E. Lamb Jr., Phys. Rev. 75, 388 (1949) and J. B. French and V. F. Weisskopf, Phys. Rev. 75, 1240 (1949). Different results were obtained by A. D. Galanin, JETP 19, 521 (1949); Y. Nambu, Prog. Theor. Phys. 4, 82 (1949); O. Hara and T. Tokano, Progr. Theor. Phys. 4, 103 (1949); Fukuda, Miyamoto and Tomonaga, Progr. Theor. Phys. 4, 121 (1949).

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15. P. Caldirola, Nuovo Cimento 5, 339 (1948), considered the influence of the level shift on the destruction of the state \(2^2S_{1/2}\) as a consequence of the Stark effect, but his calculations contain the same error already mentioned in section 26 as does work 27.

16. See, for example, G. Wentzel, Handbuch der Physik (1933), second edition, vol. 24/1, p. 752.

1. Reference number as printed in the original. ↩