EFFECT OF PRESSURE ON SPECTRAL LINES1

Henry Margenau and William W. Watson

Contents

1. Causes producing the broadening and displacement of spectral lines
   A. Damping of radiation. B. Doppler effect. C. Resonance between identical atoms. D. Broadening caused by forces of the van der Waals type. E. Broadening by extraneous factors that carry constant fields.
2. Experimental methods for determining the intensity distribution.
3. General method of calculating the intensity distribution under time-dependent perturbation.
4. Van der Waals interaction.
5. Resonance interaction.
6. The true distribution as a statistical distribution; velocity broadening.
7. Collision theory and its experimental confirmation.
8. Statistical theories.
9. Asymmetry and displacement.
10. Displacement of high series members of the alkali metals.

§ 1. Causes Producing the Broadening and Displacement of Spectral Lines

In the present article we shall consider the distribution of intensity in broadened spectral lines. This distribution will be denoted by the symbol \(I(\nu)\). The exact meaning of \(I(\nu)\) may be defined as follows: \(I(\nu)d\nu\) is proportional to the intensity in the frequency interval from \(\nu\) to \(\nu + d\nu\), lost by a light wave with a uniform spectral distribution in passing through an infinitely thin layer of absorbing atoms. Unless another normalization is specified, it is assumed that the coefficient of proportionality is chosen so that

\[ \int_0^\infty I(\nu)d\nu = 1. \]

Thus \(I(\nu)\) is an idealized quantity which is not obtained directly by experiment. It does not coinci-

gives, for example, the absorption intensity measured near the frequency \(\nu\), provided that absorption does not occur in a very thin layer of atoms. However, we shall see in § 2 how \(I(\nu)\) can be obtained from experimental data.

\(I(\nu)\) is proportional to the radiation intensity near the frequency \(\nu\) in a thin layer of atoms under analogous conditions. By analogous conditions is meant not only that the number and arrangement of the atoms must be unchanged, but also that the relative number of excited and unexcited atoms must correspond to a state of equilibrium with the radiation in the case of absorption. Since this condition is rarely fulfilled in actual experiments with radiation, the results presented here should not be applied without caution to the distribution of intensity in emission lines.

The theories considered here proceed from the assumption that the number of atoms in the excited state is so small that the interaction between excited atoms may be neglected. This is not always fulfilled when electrical methods are used to excite radiation, and the situation is often complicated still further by constant electric fields superposed on the fields surrounding the moving atoms.

Thus the question of the applicability of the arguments of the present article to emission must ultimately be decided separately in each particular case. The criteria for applicability to emission are as follows:

1. The number of excited atoms must be small in comparison with the number of unexcited atoms.

2. The exciting agent must act uniformly over an energy range greater than the width of the line. Thus, for example, if excitation is produced by illumination, the light must have constant intensity over the entire width of the fluorescence line.

3. If external fields are present, they must be taken into account in the proper way.

Theoretically one may, if desired, understand \(I(\nu)\) as the intensity obtained after the passage of a wave through a single absorbing atom. But since the matter concerns an individual atom, the absorption intensity at the frequency \(\nu\) is proportional to the radiation intensity of an excited atom at the frequency \(\nu\)¹). Therefore, in the theoretical description of phenomena occurring with a single atom, one may adopt the point of view of either absorption or emission, and we shall make use of this freedom where it proves convenient.

All frequencies discussed in this article are true (not angular) frequencies, measured in sec.\(^{-1}\).

In what follows the term “half-width” will often occur

¹) This follows from the universal relation between the Einstein coefficients \(A\) and \(B\).

distribution. Since the use of this term is ambiguous, we shall define the meaning of what we understand in this article by the half-width \(\Delta \nu_{\frac{1}{2}}\). \(\Delta \nu_{\frac{1}{2}}\) is that region of frequencies in which \(I(\nu)\) is greater than half of the maximum intensity \(I(\nu_0)\). Thus, if \(I(\nu)\) is symmetric with respect to its maximum \(I(\nu_0)\), then

\[ I\left(\nu_0 \pm \frac{1}{2}\Delta \nu_{\frac{1}{2}}\right) = \frac{1}{2} I(\nu_0). \]

This is the meaning usually assigned to \(\Delta \nu_{\frac{1}{2}}\) by experimentalists working in the region under consideration.

The classification of the various causes that influence the width and displacement of spectral lines is very difficult and cannot be carried out in an entirely unambiguous way. Individual agents interact with one another to some extent, so that it becomes difficult to separate them; and even theories that partly coincide in their basic ideas lead to disagreements among different investigators.

In the present article an attempt will be made to give a unified formulation of the theory, without adhering too strictly to time-honored distinctions, and it will be shown how various special theories—such as, for example, the theory of line broadening under the influence of collisions, the theory of resonance coupling, etc.—fit into a more general scheme.

Effects that are well understood and have been sufficiently treated in the literature1 will be discussed here only very briefly. A general account of work published up to 1932 may be found in the review by W. Weiskopf, and this review should be consulted for earlier works not mentioned in the present article. As for the theory, the problem of the width of spectral lines is usually presented in two aspects: from the point of view of classical physics and from the point of view of quantum mechanics. In those cases where such a distinction can be drawn clearly, attention will be paid to it, as well as to the difference in the results to which the two points of view lead.

We shall begin with a description of the various effects included in the range of phenomena under consideration.

A. Damping of radiation

The fact that a spectral line emitted by an isolated atom is not infinitely sharp is due to the process of damping of radiation. Its classical mechanism consists in the fact that, during oscillations and, consequently, during radiation, an electric

the charge continuously loses energy, and as a consequence the amplitude of oscillations decreases while the natural frequency \(\nu_0\) is preserved. However, a damped oscillation of this kind is not monochromatic; the frequency distribution in this case is determined by applying Fourier analysis to the electric moment as a function of time.

This gives

\[ I(\nu)=\frac{\dfrac{\gamma}{2\pi}}{(\nu_0-\nu)^2+\left(\dfrac{\gamma}{2}\right)^2}, \tag{1} \]

where

\[ \gamma=\left(\frac{4\pi}{3}\right)\left(\frac{e^2}{mc^2}\right)\nu_0^2\ \mathrm{sec}^{-1}, \]

and \(m\) and \(e\) are the mass and charge of the oscillating particle. Formula (1), obviously, is valid both for emission and for absorption, since the interaction between oscillators may here be neglected. The constants are chosen so that the total intensity is equal to unity, \(\int I(\nu)d\nu=1\).

In the case of absorption this quantity is proportional to the number of oscillators per unit volume, so that (1) must also be multiplied by this number.

The half-width of the line according to (1) is equal to \(\gamma\). If \(\gamma\) is expressed not in terms of frequencies, but in terms of wavelengths, then \(\gamma\) becomes independent of the wavelength of the line,

\[ \gamma=\left(\frac{4\pi}{3}\right)\left(\frac{e^2}{mc^2}\right)\mathrm{cm} =1.17\cdot 10^{-4}\ \text{\AA}. \]

Experiment shows, however, that different spectral lines have different widths on the wavelength scale. Consequently, these classical considerations require revision.

In quantum mechanics the natural width of a line is explained by the fact that each of the two energy levels \(E_1\) and \(E_2\), between which the transition occurs, is not infinitely sharp, but has a finite width \(\Delta E_1\) and \(\Delta E_2\). According to the work of Weisskopf and Wigner, and also of Hylleraas\(^2\), the distribution of intensity within the line is given by the equation

\[ I(\nu)=\frac{\dfrac{\gamma}{2\pi}}{(\nu_{21}-\nu)^2+\left(\dfrac{\gamma}{2}\right)^2}, \tag{2} \]

formally identical with (1). Here \(\nu_{21}\) is the Bohr frequency

\[ \frac{E_2-E_1}{h}. \]

However, the half-width \(\gamma\) has a different meaning. It is obta...

is composed of the widths of two levels

\[ \gamma=\gamma_1+\gamma_2=\frac{\Delta E_1}{h}+\frac{\Delta E_2}{h}. \tag{3} \]

Each \(\Delta E\) can be computed from the uncertainty principle in the following way. If \(\Delta t_1\) is the mean time during which the atom is in the energy state \(E_1\), then

\[ \Delta E_1 \Delta t_1 \sim \frac{h}{2\pi}. \tag{4} \]

But \(\frac{1}{\Delta t_1}\) is the total number of spontaneous transitions per second when the atom is in the state \(E_1\). Let \(E_k\) be the lowest energy state into which the atom can pass from the state \(E_1\). According to the theory of radiation, the number of transitions from \(E_1\) to \(E_k\) in 1 sec. is equal to

\[ \left(\frac{8\pi^2 e^2}{mc^3}\right)\cdot \nu_{1k}^{\,2} f_{1k}, \]

where \(\nu_{1k}\) is the frequency and \(f_{1k}\) is the oscillator strength corresponding to the transition from \(E_1\) to \(E_k\).

Hence

\[ \frac{1}{\Delta t_1}=\frac{8\pi^2 e^2}{mc^3}\sum_k \nu_{1k}^{\,2} f_{1k}, \]

where the summation extends over all energy levels lower than \(E_1\). Thus from (4)

\[ \Delta E_1=\frac{4\pi e^2 h}{mc^3}\sum_k \nu_{1k}^{\,2} f_{1k}, \]

so that, as a consequence of (3),

\[ \gamma=\frac{4\pi e^2}{mc^3}\left(\sum_k \nu_{1k}^{\,2} f_{1k}+\sum_l \nu_{2l}^{\,2} f_{2l}\right), \tag{5} \]

where each state \(l\) has an energy less than \(E_2\).

If the line under consideration is a resonance line, then for the lower state downward transitions are impossible, as a result of which all \(f_{1k}\) are equal to zero. The second sum in (5) reduces only to a single term, \(\nu_{21}^{\,2} f_{21}\). For the Na \(D\)-lines, \(f_{21}\) is equal to \(1/3\), so that in this case (2) and (5) reduce to the classical formula (1). In general, however, \(\gamma\) will depend on the oscillator strengths for the various possible transitions and will not be one and the same on the wavelength scale for all spectral lines. For further details see Weisskopf’s work\(^{2}\).

Although the quantum-mechanical explanation of the natural width of lines does not use the idea of damping, the term “radiation damping” is still applied to this effect because of the similarity of the above results to the classical ones.

V. Doppler Effect

The thermal motion of gas atoms causes spectral lines to be blurred. Suppose that light from a source of constant intensity, emitting a continuous spectrum, passes through a gas. If all the atoms were at rest, they would absorb only the frequency \(\nu_0\). Atoms moving in the direction of propagation of the light with velocity \(v_x\) will, according to the Doppler principle, absorb the frequency

\[ \nu=\nu_0\left(1-\frac{v_x}{c}\right). \tag{6} \]

The relative number of such atoms in the interval \(dv_x\) is equal to

\[ \frac{dn}{n}=\left(\frac{M}{2\pi RT}\right)^{\frac12} e^{-\left(\frac{M}{2RT}\right)v_x^2}\,dv_x, \]

where \(M\) is the molecular weight of the gas.

This formula also represents that part of the total intensity which is absorbed at the frequency \(\nu\), related to \(v_x\) by equation (6). Thus, if we express \(v_x\) in terms of \(\nu\), we obtain

\[ I(\nu)=\left(\frac{Mc^2}{2\pi RT\nu_0^2}\right)^{\frac12} e^{-\left(\frac{Mc^2}{2RT\nu_0^2}\right)(\nu-\nu_0)^2}. \tag{7} \]

It follows from this that the Doppler effect causes absorption or emission of a line with an intensity distribution according to a Gaussian curve, and the half-width of the line, as is easily seen, will be

\[ \Delta \nu_{\frac12}=2(\lg 2)^{\frac12}\left(\frac{2RT}{Mc^2}\right)^{\frac12}\nu_0, \tag{8} \]

or, on the wavelength scale,

\[ \Delta \lambda_{\frac12}=2(\lg 2)^{\frac12}\left(\frac{2RT}{Mc^2}\right)^{\frac12}\lambda_0. \]

From this formula it is clear that, whereas the natural width of a line is approximately independent (on the wavelength scale) of the wavelength, the Doppler broadening decreases as one passes to shorter wavelengths. In the region of x-rays the Doppler effect can indeed be neglected in comparison with the natural width of the line.

These two broadening effects become equal in magnitude at wavelengths of about \(100\,\text{\AA}\). Therefore, in the optical region the natural width of a line under ordinary conditions is always masked by the Doppler effect. The distribution that results from both effects is, generally speaking, complicated\(^1\). It is symme—

\[ \text{} \]

\(^1\) For discussion and references see Weisskopf.

metrically with respect to \(\nu_0\). However, at the edges of the spectral line, where \((\nu-\nu_0)\) is considerably larger than the Doppler half-width, the intensities are still expressed by formula (2) and are not changed by the Doppler effect. The reason for this is that (7) is a very compressed distribution, which for large \((\nu-\nu_0)\) falls off much faster than the dispersion curve (2). Therefore it is possible to determine the natural width of a line, despite the presence of the Doppler effect, if measurements are made in the outer parts of the broadened line.

We now turn to the consideration of those causes of broadening that are produced by the action on the absorbing atom of its neighbors. Whereas the two preceding effects were completely independent of the density of the gas in which absorption occurs, the effects considered below depend strongly on it.

C. Resonance between identical atoms

Let us first suppose that the gas is monatomic and that all its atoms are of one and the same kind. The simplest classical description of the process of absorption of light by such an aggregate is obtained if we replace each atom by a certain oscillator with a single natural frequency \(\nu_0\), and then introduce interaction forces between all pairs of oscillators. These interaction forces are the same as those acting between two dipoles, i.e. they have the form

\[ F_{ij}=\frac{\mathrm{const}}{r_{ij}^{4}}, \tag{9} \]

where \(r_{ij}\) is the distance between the \(i\)-th and \(j\)-th oscillators.

As a result of this interaction the total aggregate of atoms will have not a single natural frequency \(\nu_0\), but a very large number of natural frequencies distributed about \(\nu_0\). If there were no thermal motion, these frequencies would be discrete; in actuality they form a continuum, and this leads to a broadening of the line. Broadening arising for this reason is often referred to as being due to coupling.

Upon a somewhat more detailed consideration of this process from the point of view of quantum mechanics, it appears in a different aspect. The atoms of the gas, so long as they are in the normal state, have a symmetric distribution of charge and do not interact as dipoles do. However, as will be shown in § 4, forces of the type (9) appear between two identical atoms, one of which is excited. These forces are the result of a continuous exchange of light quanta between the two partners, i.e. they owe their origin to optical resonance. Thus in reality the situation is much more complicated than it might have seemed on the basis of the simple picture of the preceding paragraph. Taking into account the true origin of the interaction forces, the broadening under consideration is sometimes treated as resonanс-

extension. In order to avoid confusion, we emphasize that there are no physical differences between the broadening produced by the resonance of identical atoms and the broadening due to dipole interaction. This question will be considered in detail in § 4 and the following sections.

D. Broadening Caused by Forces of the van der Waals Type

In many experiments the absorption line, which at very small gas pressures ought to be sharp, is distorted because the absorbing atom is under the influence of foreign atoms or molecules. For the present we shall suppose that the latter have spherical symmetry and, consequently, have no permanent polarity. The presence of foreign perturbing particles will change the absorption properties of the atom by a continuous-in-time change in the position of its energy levels. In general, the higher levels will be distorted more strongly than the normal ones, so that changes in the energy differences will result. The interaction between the lowest state of the absorbing atom and a foreign perturbing atom is described by introducing the well-known van der Waals forces between neutral particles; in a similar way the interaction between a higher energy state and an unexcited perturbing atom may also be described. The broadening effect resulting from the action of these forces gives rise to a very curious asymmetry in the spectral line and can be followed experimentally up to very high gas pressures. In a detailed discussion of this effect (§§ 4, 8, 9) it will always be assumed that the number of foreign atoms or molecules surrounding the optically active atom is considerably greater than the number of its neighbors of the same kind, so that resonance broadening may be neglected. In most experiments the broadening effect of this type is considerably greater than the effect of damping radiation and the Doppler effect, so that the latter may likewise be neglected.

Fig. 1. Schematic course of the potential-energy curves in the case of van der Waals broadening.

A qualitative description³ of the appearance of a line broadened as a result of the presence of a foreign gas can be given with the aid of Fig. 1, in which the energies of the upper and lower states of the radiating atom (for example Na) are plotted (schematically) as functions of \(R_1\)—the distance separating the Na atom from the perturbing molecule. At present we are interested only in the extreme ascending parts of these curves, since their inner parts are of significance only for encounters

at small distances, which, generally speaking, are less probable than others. If all optical transitions took place at infinite separation of the atoms, the line would be sharp and would have the normal energy \(E\), represented by the arrow \(a\). But the average length of the arrow is less than \(a\); on the average the frequencies of the spectral line are less than the normal frequency \(\nu_0\), so that the line must be shifted toward the red region. Moreover, the intensity inside the line is considerably greater on the red side of \(\nu_0\) and smaller on the violet side, since transitions such as \(d\) are rare.

Fig. 2. Microphotometric curve of the contour of the \(D_2\) line, broadened by argon. Small maximum: curve for an argon pressure of 1.85 atm; large maximum: curve for an argon pressure of 17.8 atm. The standard line (Ne, \(\lambda\) 5881.896) is visible on the left (the lower curve is slightly out of focus in reproduction).

All these characteristic features are visible in the microphotograph reproduced in Fig. 2. It shows the \(D_2\)-line, broadened in one case by the action of 1.85 atm, and in the other by 17.8 atm of argon pressure. The shift and asymmetry are clearly visible.

The greater part of the present article will be devoted to the effects \(C\) and \(D\). But before turning to their detailed discussion, we must touch on other types of broadening.

E. Broadening by external perturbing factors carrying constant fields

The fields of interest to us here may be produced by ions, dipoles, or multipoles. There is no complete theory of broadening under the action of these agents. Debye \(^{4}\) and Holtsmark \(^{5}\) calculated the width of a line under the following simplifying assumptions:

1. Broadening may be regarded as a displacement of energy levels due to the Stark effect.
2. The effective field is taken to be the field produced by all perturbing factors at the center of the radiating atom.
3. The field changes infinitely slowly in comparison with the time of emission or absorption.

Assumption (2) is doubtful, because the radiating electron is usually located near the periphery of the atom, and the field is extremely nonuniform. However, a more detailed consideration would encounter all the difficulties of the theory of the Stark effect in a nonuniform field. Assumption (3) is not plausible when the density of the perturbing agents is large and the line is broad.

The results of the theory have been subjected to quantitative experimental verification only for those cases in which the perturbation was produced by ions6. We shall therefore confine our consideration to this case. Debye and Goldsmark showed that the intensity distribution, which the latter author represented graphically, has the half-width

\[ \Delta \nu_{\frac{1}{2}}=\operatorname{const} n_1^{\frac{2}{3}}, \]

where \(n_1\) is the number of perturbing particles per unit volume. The constant is very simply related to the width due to the Stark effect in a field of unit strength. Since we do not intend to return to the broadening produced by ions, we shall now consider the experimental material relating to it.

It has long been known that, in a spark discharge between metallic electrodes in a gas, most spectral lines of both the material of the two electrodes and of the gas are broadened, often with a displacement of the center of gravity of the line toward the red region, owing to intermolecular fields. The lines of atomic hydrogen, as well as the lines of other elements that exhibit a Stark effect of the first order, are noticeably broadened by interatomic fields even under comparatively weak electrical excitation. Goldsmark and Trumpy7 investigated the arc lines Li, Ag, Cu, and Ni, which exhibit a first-order Stark effect of known magnitude, so that from the observed broadening of the lines it was possible to determine the value of the mean intermolecular electric field. It turned out that the field strength increases with the current in the arc according to the law \(n^{\frac{2}{3}}\) for ions, and reaches \(30\,000\ \mathrm{V/cm}\) for \(i=17\ \mathrm{A}\). At this current strength the Li \(4132\ \text{\AA}\) line \(3P—6D\) has a width greater than \(8\ \text{\AA}\)1.

Interesting observations on the broadening of the Balmer lines of hydrogen due to the interatomic Stark effect were carried out by Merton8, Gilbert9, and Finkelnburg10. Gilbert found that in a condensed discharge in hydrogen at a pressure of \(250\ \mathrm{mm}\), the lines \(H_{\beta}\), \(H_{\gamma}\), and \(H_{\delta}\) broaden symmetrically by approximately \(60\ \text{\AA}\), while the \(H_{\beta}\) line shows an incipient splitting into two groups of Stark components. In a volume discharge in hydrogen at various pressures up to \(30\ \mathrm{atm}\), between electrodes separated from one another by \(1\ \mathrm{mm}\), Finkelnburg found that with increasing pressure the broadening of the Bal—

1 The first-order Stark effect appears only for terms with high orbital quantum numbers, when the valence electron is in a hydrogen-like orbit. Under ordinary excitation conditions, lines involving these terms will always be somewhat broadened owing to intermolecular fields. Therefore the series to which this Li line belongs is called diffuse.

of the Balmer lines increases considerably; the lines close to the beginning of the series begin to merge with the continuous background, and, finally, at 30 atm only H_\(\alpha\) remains as a broad maximum of intensity in the continuous spectrum. At 2 atm the widths of the H_\(\alpha\), H_\(\beta\), and H_\(\gamma\) lines (the last two being modified so as to have the same maximum intensity as H_\(\alpha\)) are respectively 550, 1680, and 2650 cm\(^{-1}\), i.e., they are almost exactly in the ratio \(1:3:5\). This rule, together with other facts, points to the interatomic Stark effect as the principal cause of the broadening. If, with increasing pressure, a line becomes so broad that it merges with the continuous spectrum, the corresponding higher state of the atom can no longer be exactly quantized. At a pressure of 30 atm the orbits with \(n=3\) practically no longer exist.

Comparing his results with those of Raush von Traubenberg, concerning the quenching of the higher members of the Balmer series under the application of a strong external electric field, Finkelnburg concluded that at 1 atm the mean interatomic field must be equal to \(2\cdot10^{5}\) V/cm. At 5 atm this mean field strength is approximately \(5\cdot10^{5}\) V/cm, whereas at 30 atm it has a value of about \(2\cdot10^{6}\) V/cm. Since H\(_2\) has no dipole moment, only ions and quadrupoles can create these fields. At 1 atm the mean interatomic field produced by quadrupole moments is considerably less than \(10^{3}\) V/cm, but at 100% ionization the mean field of the ions, according to Debye and Holtsmark, should be equal to \(1.3\cdot10^{6}\) V/cm. Thus the observed \(2\cdot10^{5}\) V/cm indicates 15% ionization along the path of the discharge. According to the law that the ionic field strength increases proportionally to \(n^{2/3}\), and under the condition of an equal degree of ionization at all pressures, we obtain that the field strength at a pressure of 30 atm should increase by \(30^{2/3}\simeq 10\) times, which again gives approximately \(2\cdot10^{6}\) V/cm. At 1 atm the H_\(\beta\) line, as it turns out, has an asymmetric contour in accordance with the calculated reduced intensity of the long-wavelength components of the Stark splitting pattern in a strong field.

The reversal of the metal lines in Finkelnburg’s spectrograms is obtained as a consequence of the partial pressure of the metal vapor. He showed, however, that the partial pressure of the metal vapor along the path of the discharge depends on the total pressure in the chamber.

Knauss and Brian \(^{11}\) recently found that in the spectrum produced when a strong current passes through a narrow stream of Hg vapor, some lines have a width of up to 100 Å, whereas others merge completely with the continuous background. Such a large broadening of the lines can be caused only by very strong interatomic fields.

The absorption spectra of solutions of Hg atoms in various solvents, as was shown by Reichardt and Bonhoeffer \(^{12}\), give

the line 2537 Å, split into two broad components. This splitting is attributed to the action of a strong interatomic electric field. For Hg in H₂O, for example, they concluded from the magnitude of the splitting that the effective mean field strength is equal to \(33\cdot 10^6\) V/cm—a value of the same order of magnitude as in Finkelnburg’s experiments for a gas discharge at very high pressures.

§ 2. Experimental Methods for Determining the Intensity Distribution

Before proceeding to a discussion of the individual types of broadening under the influence of pressure, we must give a very brief survey of the methods for the experimental determination of the intensity distribution over a broadened spectral line. A direct determination of this distribution can be made by the method of photographic photometry. If both the intensity of the source of continuous radiation and the sensitivity of the photographic plate do not change appreciably over the entire frequency interval under consideration, and if the absorption intensity and exposure time are chosen so that the record lies wholly on the straight-line part of the characteristic curve of the emulsion, then the photometric curve directly gives the distribution \(I(\nu)\) considered in the preceding paragraph. Indeed, if \(i(\nu)\) denotes the intensity of the transmitted light, \(i_0\) the intensity of the light incident on the photographic plate in the absence of absorption, and \(l\) the absorption per unit length, then

\[ i(\nu)=i_0 e^{-I(\nu)l}. \tag{1} \]

This means that \(\ln i=\text{const}-I(\nu)l\). But for a photographic plate on the straight-line part of the characteristic curve, \(\ln i\) is proportional to the density \(D\). Thus \(D\sim -I(\nu)\).

Generally speaking, however, the photographic plate must be calibrated. Apparently the best method is the use of a stepped filter having several steps of known transparency, placed either immediately in front of the slit of the spectrograph or on the cassette exactly at one of the edges of the absorption line. The first arrangement requires uniform illumination of the whole slit, which is a difficult task when using any ordinary source of continuous radiation. Therefore the second arrangement should be preferred, the steps of the filter being placed parallel to, and sufficiently far from, the absorption line so that the filter does not affect the absorption. With the aid of the blackening marks obtained in this way, the photometric curve can easily be transformed into the true absorption curve \(I(\nu)\).

The ordinates of this curve are proportional to the quantity \(nk\), known

as the absorption index. The latter is determined from the equation

\[ i(\nu)=i_0 e^{-4\pi n k \frac{l}{\lambda}}, \tag{2} \]

so that

\[ n k=\left(\frac{\lambda}{4\pi l}\right)\ln\frac{i_0}{i}. \tag{3} \]

The true absorption index, if desired, can be calculated for each value of \(\nu\) from the ratio \(\frac{i}{i_0}\). For the purposes of many investigations, in which only the relative change of intensity across a broadened line is essential, knowledge of \(I(\nu)\) is equivalent to knowledge of \(n k(\nu)\).

In order to obtain the complete absorption-distribution curve, it is not necessary for the absorption at the middle of the line to be complete. The experiment should usually be arranged so that, for every gas pressure in the absorbing tube, the total absorption at the line maximum would amount to approximately \(75 I_0\). The position of the absorption maximum is given directly by the readings of the photometer, and, in passing to the true absorption curve, the position of this maximum is not shifted with respect to the photometer readings for standard lines.

It is difficult to give a simple numerical expression for the asymmetry in a spectral line broadened by pressure. One may, for example, give the ratio of the area under the absorption curve on the high-frequency side of the maximum to the corresponding area on the low-frequency side \(^{13}\), or one may determine the ratio of the red “half” to the violet “half” of the line half-width \(^{14}\). It is obvious that neither of these ratios gives sufficient information about the exact contour of the absorption line.

This method of directly determining the shift and asymmetry of an absorption line broadened by pressure has been used by many investigators. A more detailed account may be found in the papers of Füchtbauer and his collaborators \(^{15}\) and in Margenau and Watson \(^{16}\).

The oscillator strengths and transition probabilities can be calculated from the magnitude of the total absorption \(\int_0^\infty (n k)\,d\nu\), obtained by graphical integration of the curve \((n k,\nu)\). According to radiation theory, the oscillator strength for a given kind of atom is equal to

\[ f=\frac{4\nu m}{n_e e^2}\int_0^\infty n k\,d\nu, \]

whereas the transition probability \(A\) is calculated by the formula

\[ A=\frac{4\pi}{cn_1\eta}\int_0^\infty n k\,d\nu . \]

From such measurements it was determined\({}^{17}\) that the oscillator strength for the resonance lines of the alkali metals is approximately equal to unity, whereas for the Hg 2537 Å line the value is about \(1/35\).\({}^{18}\) It was found that the total absorption of this Hg line decreases with increasing pressure of the foreign gas,\({}^{18}\) so that \(f\) becomes approximately equal to \(1/100\) when, for example, the line is broadened by a pressure of 36 atm of \(\mathrm{CO}_2\). The value \(1/35\) is calculated by extrapolating to zero density of the foreign gas. Trumpy\({}^{19}\) used this method to investigate the exact course of the rapid decrease of the transition probability with increasing quantum number in the principal series of Na. Weibull\({}^{20}\) made similar measurements for the lines of the principal series of Cs.

Other, less direct methods are also used for measuring the half-width and the intensity distribution in spectral lines. In these methods certain theoretical expressions are assumed for the line contour, containing unknown constants, which are determined from measurements of the total absorption.

Ladenburg and Levy,\({}^{21}\) for example, considered the case in which the light source does not give a continuous spectrum, but is a column of excited gas absorbing a line included in the spectrum of the light source. These authors gave a detailed dependence of the absorption on the constants of the Doppler and dispersion distribution. Measurement of the rotation of the plane of polarization and simultaneous study of the spectral line in magneto-optical experiments can also lead to determination of the magnitude \(f\) and of the line width.\({}^{22}\) The relation between the line width and the rotation of the plane of polarization depends, however, on the character of the expressions adopted by us for the intensity distribution in the line.

§ 3. General method for calculating the intensity distribution under a time-dependent perturbation

In the effects outlined in § 1 C, D, and E there is a common feature: the frequency of the radiating atom is shifted under the influence of external actions, and these actions change with time. The classical method of calculation under such conditions is clear. The active atom is replaced by an oscillator with natural frequency \(\nu_0\). Under the action of moving neighbors this frequency becomes \(\nu=\nu_0+\Delta\nu(t)\). The amplitude of the oscillations, generally speaking, is also subjected

disturbances and becomes a function of time \(A(t)\). Thus the electric moment, written in complex form, will at some \(t\) be

\[ M=A(t)\exp\left[2\pi i\int_0^t \nu(t)\,dt\right]. \tag{1} \]

The amplitude \(J(\nu')\), corresponding to the frequency \(\nu'\), is found by expanding (1) in a Fourier integral

\[ M(t)=\int_0^\infty J(\nu') e^{2\pi i\nu't}\,d\nu'. \tag{2} \]

Then the amplitude will be

\[ J(\nu')=\int_{-\infty}^{\infty} M(t)e^{-2\pi i\nu t}\,dt, \tag{3} \]

or, taking (1) into account,

\[ J(\nu')=\int_{-\infty}^{\infty} A(t)\exp\left\{2\pi i\left[\int_0^t \nu(\tau)\,d\tau-\nu't\right]\right\}\,dt. \tag{4} \]

The intensity at the frequency \(\nu'\) is finally expressed in terms of the amplitude as

\[ I(\nu')=\left|J(\nu')\right|^2. \tag{5} \]

It is difficult to calculate the exact time dependence of the amplitude \(A\) of the oscillator. It is usually assumed that \(A\) is constant over certain definite time intervals and vanishes outside these intervals.

Although the treatment of the problem from the point of view of quantum mechanics is undoubtedly different, Lenz \(^{23}\) pointed out as early as 1924 that, by virtue of the correspondence principle, the final results must be the same.

Weisskopf \(^{24}\) showed that formulas (4) and (5), under certain simplifying conditions, are also valid in quantum mechanics. His proof uses the Kramers–Wentzel–Brillouin approximation. It is possible to justify (4) and (5) without applying this method. In what follows we shall give a simple proof in which it is assumed that the perturbation acts only on the energy of the radiating atom, but not on the charge distribution \(^{1}\).

In classical terminology this is equivalent to the assumption that the amplitude of the oscillations \(A\) is constant.

\(^{1}\) More precisely: the matrix elements between different atomic states do not change appreciably.

The act of emission of a photon may be regarded as a measurement of the energy of the atom in the excited state, provided that the energy of the lower state is known. The number of photons emitted with frequency \(\nu'\), i.e. \(I(\nu')\), is thus proportional to the probability that the excited state has an energy \(E'=h\nu'\) greater than the lower state, whose energy we take to be zero. This probability is determined by representing the state function of the excited state as a function only of the energy and then selecting the coefficient at the energy \(E'\). Its square represents the probability, and hence also the required intensity.

If the atom is in an energy state with energy \(E'\), then its \(\Psi\)-function is

\[ \Psi=\psi(q)e^{2\pi \frac{E'}{h}t}, \tag{6} \]

where \(q\) denotes the coordinates of position and \(E'\) does not depend on \(t\). Suppose now that at \(t=0\) a perturbation \(\varepsilon(t)\) arises. The state function will then, according to Schrödinger’s equation,

\[ \frac{h}{2\pi i}\left(\frac{d}{dt}\right)\Psi=\varepsilon(t)\Psi, \]

whose solution will be

\[ \psi(q)\exp\left[\frac{2\pi i}{h}\int_0^t \varepsilon(\tau)\,d\tau\right]. \tag{7} \]

This expression may be represented in the form of a function of type (6), i.e. we must apply Fourier analysis to (7). If we put \(\frac{E'}{h}=\nu'\) and \(\frac{\varepsilon}{h}=\nu\), then the probability amplitude will be

\[ a(\nu')=\int_{-\infty}^{\infty} \exp\left\{2\pi i\left[\int_0^t \nu(\tau)\,d\tau-\nu't\right]\right\}\,dt, \]

which is proportional to \(J(\nu')\) in (4), if \(A(t)\) is regarded as constant. Under these conditions the classical method of calculating the intensity distribution becomes correct, provided that the classical frequency \(\nu(t)\) is replaced by \(\frac{\varepsilon(t)}{h}\), where \(\varepsilon\) is the perturbing energy as a function of time. If the energy of the lower state also varies with time, then \(\varepsilon(t)\) must be interpreted as the difference of the energies between the upper and lower states, as is shown by consideration of the preceding derivations.

Thus our problem is reduced to the determination of \(\varepsilon\). It will depend on the spatial arrangement of all the atoms of the gas and, owing to their motion, on time. The following 2 para-

graph will be devoted to the principal perturbations that create the width of spectral lines. As a definition one may write

\[ \varepsilon = E_2 - E_1 + \Delta \varepsilon_2 - \Delta \varepsilon_1 , \tag{8} \]

where \(E_2\) and \(E_1\) are the proper energies of the upper and lower states, and \(\Delta \varepsilon_1\) and \(\Delta \varepsilon_2\) are the perturbations of these energies by neighboring atoms. Accordingly, we have

\[ \nu = \nu_0 + \Delta \nu , \tag{9} \]

where

\[ \nu_0 = \frac{1}{h}(E_2 - E_1) \quad \text{and} \quad \Delta \nu = \frac{1}{h}(\Delta \varepsilon_2 - \Delta \varepsilon_1). \]

§ 4. Van der Waals interaction

We shall first return to the question of broadening under the influence of foreign gases, in which the bond between atoms is reduced mainly to forces of the van der Waals type. The quantum-mechanical meaning of these forces was first investigated by F. London \(^{25}\), who also gave a formula for their calculation. The two quantities to be determined are \(\Delta \varepsilon_1\) and \(\Delta \varepsilon_2\). The first of them is the additional energy acquired by the normal state of the emitting atom owing to the presence of all perturbing agents; the second is the same additional energy of the excited state. We shall now consider only \(\Delta \varepsilon'_1\) and \(\Delta \varepsilon'_2\)—the additional energies obtained from the presence of only one perturbing atom.

London’s formula for the perturbation energy between an atom in state \(k\) and another (different from it) atom in state \(l\) has the form

\[ \Delta \varepsilon'_{kl} = \]

\[ = -\frac{1}{R^6}\frac{3}{2m^2}\left(\frac{he}{2\pi}\right)^4 \sum_{k'l'} \frac{f_{kk'} g_{ll'}}{(E_{k'}-E_k)(F_{l'}-F_l)(E_{k'}+F_{l'}-E_k-F_l)} + S, \tag{1} \]

where \(R\) is the distance between atoms; \(f_{kk'}\) is the oscillator strength corresponding to the transition \(k \to k'\) for the first atom; \(g_{ll'}\) is the same for the second. \(E\) and \(F\) are the energies of the various states of the two atoms. States for which \(k'=k\) or \(l'=l\) are not included in the sum. The first term in (1) represents the first term of the expansion of \(\Delta \varepsilon'_{kl}\) in decreasing powers of \(R^2\); thus the residual term \(S\) begins with a constant divided by \(R^8\). If we take \(k\) to refer to the ground state of the emitting atom and \(l\) to the ground state of the perturbing atom, then equation (1) gives \(\Delta \varepsilon'_1\). Since every \(E_{k'} > E_k\) and every \(F_{l'} > F_l\), the term proportional to \(R^{-6}\) will always be negative. The energy of the ground state, therefore, decreases for such values of \(R\) for which \(S\) may be neglected.

In considering \(\Delta \varepsilon_2'\) we shall assume that the atom emitting this line is in an excited \(P\)-state. Most experiments on line broadening were carried out on resonance lines or on higher members of the principal series; consequently, this assumption is apparently justified. Then, if we take \(k\) to correspond to this \(P\)-state, and \(l\) to the ground state of the perturbing atom, formula (1) will not give \(\Delta \varepsilon_2'\) exactly; however, (1) corresponds to the average over all orientations of the excited atom. The energy of the instantaneous interaction depends on the magnetic quantum number of the \(P\)-state\({}^{26}\) and may indeed have opposite signs for different values of this quantum number. Thus these forces are not strictly central and they cannot be classified as polarization forces. For simplicity we shall ignore this difficulty and use (1) to calculate \(\Delta \varepsilon_2'\). In fact, by doing so we ignore a possible cause of broadening, since we replace \(\Delta \varepsilon_2'\) by a sharp mean value. But neither experiment nor theory is at present refined enough to confirm these details by investigation.

\(\Delta \varepsilon_2'\) is not necessarily negative. One may be sure that \(f_{kk'}'\) always has the same sign as the corresponding energy difference \(E_{k'}'-E_k'\); consequently the sign of \(\Delta \varepsilon_2'\) (discarding \(S\)) depends on the different values of \((E_{k'}'-E_k+F_{l'}'-F_l)\). If these are substantially positive, then \(\Delta \varepsilon_2'\) will be negative. In general this will be the case in which there is only one transition caused by a negative \(E_{k'}'-E_k\) (namely, if \(k'\) denotes the normal state), and all \(F_{l'}'-F_l>0\). Thus, if the emitting atom is an alkali-metal atom and the perturbation is produced by a noble gas, the only negative \(E_{k'}'-E_k\) reaches \(2\text{--}3\ \mathrm{V}\), whereas the smallest \(F_{l'}'-F_l\) is of the order of \(10\ \mathrm{V}\) (the transition from the normal ground state to the lowest excited state). This situation is quite general when the emitter is a metal atom and the perturbing factor is a gas.

There exists, however, a case where \(\Delta \varepsilon_2'>0\). It is, for example, positive\({}^{1)}\) in the case of the interaction of an excited sodium atom with a normal potassium atom\({}^{26}\), for which the predominant term in the summation (1) is the one associated with two resonance transitions, and it is positive. From these considerations \(\varepsilon\) (see 3,8)\({}^{2)}\) is greater than \(E_2-E_1\), and the spectral line should have been shifted toward the violet side. The experiments clarifying this, however, have not been carried out. In general there is a shift toward the red end, which in most cases excludes \(-\Delta \varepsilon_2'>-\Delta \varepsilon_1'\).

For calculations, formula (1) can be simplified. Suppose, for example, that the substance producing the broadening is bla—

\({}^{1)}\) Because \(E_{k'}-E_k\sim -2.1\ \mathrm{V}\) for Na, \(F_{e'}-F_e\sim +1.6\ \mathrm{V}\) for K.
\({}^{2)}\) In references to equations of preceding paragraphs the first number indicates the paragraph.

city gas. The energy differences \(F_{l'}-F_l\) will then be grouped around the ionization energy; the smallest will be the resonance energy, which is relatively large for noble gases. The greatest value of the energy is, evidently, infinity, since we must not exclude transitions to the continuous spectrum from the summation in (1), but the value of \(f\) for these transitions falls rapidly beyond the ionization limit. Thus an estimate of \(\Delta\varepsilon'\) can be obtained by replacing all terms \(F_{l'}-F_l\) by some mean energy difference \(\overline F\), which we take to be equal to the ionization energy. If, in doing so, we use the well-known formula for the polarizability \(\alpha\), then in formula (1) one can get rid of the summation over \(l'\), and as a result one obtains \({}^{27}\)

\[ \Delta\varepsilon'=-\frac{1}{R^6}\frac{3}{2m}\left(\frac{he}{2\pi}\right)^2 \alpha F\sum_{k'}\frac{f_{kk'}}{(E_{k'}-E_k)(\overline F+E_{k'}-E_k)}+S, \tag{2} \]

where \(\alpha\) and \(\overline F\) refer, of course, to the gas. This formula is applicable when the value is known for the sharpest lines of the emitting atom. For non-noble gases, where there is a large scatter in the quantities \(F_{l'}-F_l\), its suitability is extremely doubtful; however, it can still be used to determine the order of magnitude of the effect.

Numerical calculations on the basis of (2), despite the uncertainty already mentioned, show that \(|\Delta\varepsilon_2'|>|\Delta\varepsilon_1'|\) and that \(S\) may be neglected; however, \(\Delta\varepsilon_1'\) cannot be discarded in comparison with \(\Delta\varepsilon_2'\). The order of magnitude of these energies is several millivolts at a distance of about \(5\ \text{Å}\).

Next we must consider the role of \(S\). The first term in (1) corresponds to the dipole interaction of the two atoms under consideration. \(S\) contains the effects of higher multipoles.\({}^{1)}\) The quantity \(S\) may be neglected when the extent of the charge distribution is considerably smaller than the distance between the atoms. Roughly speaking, \(S\) becomes at all noticeable for \(\Delta\varepsilon_2'\) at values of \(R\) less than \(10\ \text{Å}\); for \(\Delta\varepsilon_1'\) it can still be neglected for somewhat smaller distances. The total effect of \(S\) decreases \(\Delta\varepsilon'\), i.e., increases the attractive forces. At distances \(\sim 5\ \text{Å}\), formula (1) ceases completely to be valid, and then exchange forces of the valence type appear, producing either strong repulsion or (in the case uninteresting from the point of view of the present article, since the atomic line then cannot be emitted) a chemical compound. It follows from this that the effect of \(S\) can be considered only qualitatively.

Until now we have considered the interaction between two separate atoms \(\Delta\varepsilon'\). We now turn to the consideration of the com-

\({}^{1)}\) They exist even if there are no permanent multipoles.

of an aggregate consisting of one radiating atom and a large number of perturbing ones. This transition will be simple, since the forces expressed by formula (1) possess a very essential property of additivity[^29]. Taking this into account and omitting \(S\), (3, 8) and (3, 9) may be written as follows:

\[ \varepsilon = E_2 - E_1 + a \sum_i \frac{1}{R_i^6}, \tag{3} \]

and

\[ \nu = \nu_0 + b \sum_i \frac{1}{R_i^6}, \tag{4} \]

where the summation extends over all perturbing atoms, \(a\) is the difference of the coefficients of \(\frac{1}{R^6}\) in (1) for the excited and normal states of the radiating atom, and \(b = \frac{a}{h}\); \(b\) is in general negative and has the order of magnitude \(10^{-32}\) or \(10^{-31}\ \mathrm{cm^6\,sec^{-1}}\).

§ 5. Resonance interaction

Let us again consider the perturbation energy which occurs when there are only 2 atoms. \(\Delta \varepsilon_1'\), which, as in the preceding paragraph, refers to the additional energy if both atoms are in the normal state, is of the van der Waals type. The van der Waals forces in the case of metal atoms are large; however, at the same time they are small in comparison with the resonance forces, which appear in \(\Delta \varepsilon_2'\). Therefore we shall completely discard \(\Delta \varepsilon_1\). This is justified, since we are dealing with large distances (low pressure), because the resonance perturbation is proportional to \(\frac{1}{R^3}\), while \(\Delta \varepsilon_1 \sim \frac{1}{R^6}\).

\(\Delta \varepsilon_2'\) depends on the orientation of the atom, i.e., on the magnetic quantum number \(m\) of the excited atom. Indeed, we have

\[ \Delta \varepsilon_2' = \gamma \frac{e^2 h f_{12}}{8\pi^2 m \nu_0}\frac{1}{R^3}, \tag{1} \]

where the numerical factor \(\gamma\) takes the value \(-2\) if \(m = 0\), and \(+1\) if \(m = +1\)1; \(f_{12}\) is the value of \(f\) corresponding to the transition from the normal to the excited state; \(\nu_0\) is the frequency of the spectral line.

In deriving (1) we proceeded as follows. If one of the atoms is in the excited state \(\psi'\), and the other in the normal state \(\psi\), then the combined state is a superposition of two functions

\[ \psi_1 = \psi'(1)\psi(2) \quad \text{and} \quad \psi_2 = \psi(1)\psi_2', \tag{2} \]

since we do not know which of the two atoms is excited. Consequently, the complete function of the unperturbed state is

\[ \Psi=c_1\psi_1+c_2\psi_2 . \tag{2} \]

with two possible values of \(c_1\) and \(c_2\). The perturbing energy is

\[ V=-\frac{e^2}{R^3}(2z_1z_2-x_1x_2-y_1y_2), \tag{3} \]

where \(x_1\) is the coordinate of the electron in the first atom, measured from its nucleus, \(x_2\) is the coordinate of the electron in the second atom, etc. Let \(V_{12}\) be the matrix element of expression (3) between \(\psi_1\) and \(\psi_2\). Its value, as is easy to see, is

\[ V_{12}=V_{21}= \begin{cases} -\dfrac{2}{3}\dfrac{e^2}{R^3}|r_{12}|^2, & \text{if } m=0,\\[6pt] +\dfrac{1}{3}\dfrac{e^2}{R^3}|r_{12}|^2, & \text{if } m=\pm1. \end{cases} \tag{4} \]

\[ V_{11}=V_{22}=0 \]

\(r_{12}\) is the radial matrix element between the normal and excited states. According to the well-known rule, the perturbed energy \(\Delta\varepsilon_2'\) (which we shall temporarily denote by \(\Delta\)) is found by solving the system of equations

\[ \sum_j c_j\left(V_{kj}-\delta_{kj}\Delta\right)=0, \tag{5} \]

and this leads to

\[ \left| \begin{matrix} -\Delta & V_{12}\\ V_{12} & -\Delta \end{matrix} \right|=0. \tag{6} \]

Consequently,

\[ \Delta=\pm V_{12}. \tag{7} \]

Substituting this value into (5), we arrive at two systems with:

\[ c_1=c_2 \quad \text{and} \quad c_1=-c_2, \tag{8} \]

showing that the combination (2) is either symmetric or antisymmetric. However, from the states (2) the system can pass (under dipole radiation) into a state in which neither atom is excited, namely: \(\Psi_0=\psi(1)\psi(2)\).

If we calculate the probabilities of transition from \(\Psi_0\) to both states \(\Psi\), it turns out that the probability of transition to the antisymmetric \(\Psi\) vanishes. Thus the second possibility in (8) is excluded and only

\[ \Delta\varepsilon_2'=+V_{12}. \tag{9} \]

remains. Substitution of (4) into (9) leads to formula (1), if we remember that

\[ |r_{12}|^2=\frac{3h}{8\pi^2 m\nu_0}f_{12}. \]

The interaction energy can have both signs, and the choice depends on the value of \(m\) [and not on the two possible solutions of (6)]. Averaging over all values of \(m\) gives 0. The classical interpretation of this fact is that, on the average, \(\Delta\varepsilon_2'\) is almost the same as

is often positive as well as negative. Indeed, it can be shown that, if one uses the classical method of averaging over angles, then \(\Delta \varepsilon'_2\) is obtained equal to \(\dfrac{e^2 h f_{12}}{8\pi^2 m \nu_0 R^3}\), multiplied by a function of the angle whose maximum value is unity and whose average is 0. Thus equation (1) can be interpreted analogously to the classical interaction of two permanent dipoles with equal moments

\[ \mu=\left(\frac{e^2 h f_{12}}{8\pi^2 m \nu_0}\right)^{\frac12}. \]

However, this analogy fails when the number of interacting particles is greater than two. The forces represented by equation (1) are not additive.

In order to see this, let us return to the derivation of (1). Suppose that there are \(n\) atoms, one of which is excited. Then we have \(n\) functions

\[ \begin{aligned} \psi_1&=\psi'(1)\psi(2)\ldots\psi(n),\\ \psi_2&=\psi(1)\psi'(2)\ldots\psi(n),\\ &\ldots\ldots\ldots\ldots,\\ \psi_n&=\psi(1)\psi(2)\ldots\psi'(n), \end{aligned} \]

and instead of (2)—\(n\) linear combinations of the type

\[ \Psi=c_1\psi_1+c_2\psi_2+\ldots+c_n\psi_n . \]

\(V\) is now more complicated; it includes the angles between the various radius-vectors connecting the atoms. Instead of (6) we obtain the secular equation

\[ \left| \begin{array}{cccc} V_{11}-\Delta & V_{12} & \ldots & V_{1n}\\ V_{21} & V_{22}-\Delta & \ldots & V_{2n}\\ \ldots & \ldots & \ldots & \ldots\\ V_{n1} & V_{n2} & \ldots & V_{nn}-\Delta \end{array} \right|=0. \tag{10} \]

The solutions \(\Delta\) of this equation are irrational functions of \(V_{ij}\). When (10) is solved, a number of coefficients can be determined by substituting into (5). To each series there must be assigned a weight proportional to the oscillator strength corresponding to the transition to the unexcited state \(\Psi_0=\psi(1)\psi(2)\ldots\psi(n)\), which is also the weight of the corresponding \(\Delta\). Only antisymmetric states will have zero weight. Finally, to each eigenvalue \(\Delta\) one may assign three different quantities, depending on the three possibilities for \(m\).

Our aim was only to survey the method briefly and to point out the difficulties contained in it. The problem has not been solved, although various attempts have been made in this direction[^28]. This was also considered by Weisskopf[^2], who pointed out that the use of the curve errors with the distribution of the solutions \(\Delta\) of equation (10) is incorrect. To solve (10) is, obviously, impossible if \(n\) is large. However, it can be shown that the sum of all \(\Delta\)'s is zero. From this circumstance Holtsmark and Frenkel concluded that the distribution of \(\Delta\) is symmetric with respect to \(\Delta=0\). This, of course, is true, but it gives no conclusions concerning the distribution of frequencies inside the broadened line, since \(\Delta\) does not possess equal weight.

From these considerations we shall apply the theory of resonance interaction only to those cases of line broadening in which-

...can be neglected in comparison with the simultaneous action of several perturbing atoms, so that we may confine ourselves to the use of equation (1).

This is valid at low pressures, if attention is paid only to the edge of the line\(^{1}\). Under these simplifying conditions \((\Delta \varepsilon_2 = \Delta \varepsilon'_2;\ \Delta \varepsilon_1 = 0)\), taking into account (3, 9) and (1), we may write

\[ \nu = \nu_0 \pm \frac{B}{R^3}, \qquad \text{where} \qquad B = \frac{e^2 f_{12}}{8\pi^2 m \nu_0}. \tag{11} \]

The factor \(\gamma\) here has been replaced by its “classical value” \(\pm 1\).

(To be continued.)

\(^{1}\) At the edges of the line the frequency is displaced, becomes larger, which is caused by close encounters. For close encounters the probability of a double (or higher-order) collision is small.

1. The symbol \(m\) occurring in formulas like (1) refers, of course, to the mass of the electron. ↩↩↩