WIDTH OF SPECTRAL LINES IN GASES

V. Weisskopf, Berlin*

I. General Part

In the present article the most important factors leading to the broadening of spectral lines in gases are compared. The main attention is concentrated on the theoretical side; the experimental material, generally speaking, is considered only in connection with the theory and makes no claim to completeness.

From the standpoint of modern theoretical ideas, one can without difficulty understand almost all cases of line broadening in gases, and moreover in an entirely unambiguous way. As causes of the broadening of spectral lines which, according to Bohr’s frequency condition, ought to be infinitely narrow, we shall name:

1. The damping of the virtual oscillator due to its own radiation; this is the cause of the natural width of lines.

2. The displacement of frequencies in the motion of the radiating atom according to the Doppler principle.

3. The mutual influence of atoms; here one distinguishes: a) Lorentz damping as a result of collisions, which in what follows, for brevity, we shall call impact damping; b) the broadening and shift of lines in the force fields of neighboring atoms. In this connection it is necessary to distinguish whether the interacting atoms are identical or belong to different elements. In the first case one speaks of c) broadening due to coupling between atoms.

The causes of broadening just cited for the most part act simultaneously. It is possible, however, in almost all cases to create conditions under which one of the causes of broadening is essential, or at least to indicate a frequency region within which only one of the causes acts.

In most cases it is necessary to consider separately absorption lines and emission lines. In both cases the thickness of the gas layer has a decisive influence on the form of the line. The magnitude of the absorption is determined by the absorption coefficient \(k\), a func—

* This article was written during the author’s stay at the Ukrainian Physico-Technical Institute in Kharkov.

WIDTH OF SPECTRAL LINES IN GASES

... frequency \(\omega\) (we use cyclic frequency, equal to the number of oscillations in \(2\pi\) sec.). When a path of one wavelength \(\lambda\) is traversed, the amplitude of a wave of frequency \(\omega\) decreases by a factor \(e^{-k(\omega)}\) (it should be borne in mind that the quantity \(\lambda=\dfrac{c}{\omega n}\) also depends on the refractive index \(n\), i.e., the measure of absorption is the product \(nk\)). The absorption \(A(\omega)d\omega\) by a layer of thickness \(l\) in the interval of frequencies between \(\omega\) and \(\omega+d\omega\) will be:

\[ A(\omega)d\omega=\left(1-e^{-\frac{4\pi nlk(\omega)}{\lambda_0}}\right)d\omega;\quad \lambda_0=\frac{c}{\omega} \tag{1} \]

for small \(l\), and, consequently, for an infinitely thin layer the intensity of the absorbed light is proportional to \(nk\).

It is expedient also to reduce the shape of the emission line to an infinitely thin layer, since a layer of finite thickness changes the shape of the line by secondary absorption. The dependence of intensity on frequency is some function \(I_E d\omega=I_E(\omega)d\omega\) for the interval between \(\omega\) and \(\omega+d\omega\).

In what follows, by the shape of an absorption or emission line (unless special reservations are made) we shall always mean the shape corresponding to an infinitely thin layer.

For the theory of the width of spectral lines, the absolute value of the intensity is often not so important as the ratio of the intensities of different frequencies. By the relative distribution of intensities of absorption or emission we understand a function of the frequencies \(\omega\) which, to within a constant factor, has the form \(nk\) or \(I_E\). The constant is usually chosen so that the total intensity, i.e. the integral over the entire line, is equal to unity. In most cases, broadening does not affect the total intensity of the lines, which is determined by the Einstein coefficients.

The shape of the emission line \(I_E\) depends on the conditions of excitation. If, for example, excitation is produced by illumination, then in the absence of all disturbing actions (Doppler effect, collisions) the emission line depends essentially on the shape of the illuminating line. Of interest is only that shape \(I_E\) which is due to thermal excitation or to an excitation equivalent to it, for example uniform excitation throughout the entire frequency interval of the line by a continuous spectrum or by electron collision, etc.

In very many cases the dependence of \(nk\) and \(I_E\) on frequency is the same; the two functions are then proportional to one another. This can be justified on the basis of Kirchhoff’s law, if it can be shown that the establishment of thermal equilibrium changes nothing in the course of these functions. For ordinary absorption experiments thermal equilibrium does not occur, so that Kirchhoff’s law is not directly applicable.

We assume that in the immediate vicinity of the line under consideration there are no other lines, so that questions rela-

overlaps with closely adjacent lines are excluded from consideration.

The functions \(I_E\) and \(uk\), or, more briefly, \(I(\omega)\), have a sharply expressed maximum at the point \(\omega_0\) and fall off rapidly on both sides of \(\omega_0\). The width of the \(\delta\)-line is determined by the equation:

\[ I(\omega_0 \pm \delta)=\frac{1}{2}I(\omega_0). \tag{2} \]

For an asymmetric line two widths are obtained—violet and red, corresponding to the sign \(+\) or \(-\) in (2). If the width of a line is expressed in wavelengths, we obtain:

\[ \delta^\lambda=\frac{2\pi c}{\omega_0^2}\delta. \tag{3} \]

The line width actually observed may be considerably greater than \(\delta\), which has been determined by us only for the form of the line reduced to an infinitely thin layer. Many authors understand by the line width the distance \(\delta'\) between two frequencies for which the intensity \(I(\omega)\) is half the intensity at the maximum. For a symmetric line, then, \(\delta'=2\delta\).

Fig. 1. Distribution of dispersion (—) and Doppler (– – –) with equal widths and equal total intensities.

The most frequently encountered line forms are the so-called dispersion form:

\[ I(\omega)=\frac{C\delta}{(\omega_0-\omega)^2+\delta^2} \tag{4} \]

and the Doppler form (Fig. 1):

\[ I(\omega)=\frac{C'}{\delta}e^{-\frac{\ln 2}{\delta^2}(\omega_0-\omega)^2}. \tag{5} \]

The constants \(C\) and \(C'\) are chosen in such a way that the total intensity does not depend explicitly on the width \(\delta\).

Experimental measurement of the form of a line can be carried out in two ways. The first method is direct photometry of the line obtained with the aid of a spectrograph of high resolving power. We shall call it the direct method. The second—an indirect method—can be applied when the course of the line form is in general known theoretically and it is necessary only to find the constants. In this case it is sufficient to measure the total absorption of light from some source with a known intensity distribution by a layer of the gas under investigation. Let \(I_E(\omega)\) be the distribution of intensity with frequency in the light source, \(k(\omega)\) the absorption coefficient of the absorbing gas, and \(l\) the tol-

the thickness of the layer in Fig. 1. Then the total absorption—the ratio of the absorbed intensity to the incident intensity—is equal to

\[ A=\frac{\displaystyle \int I_E(\omega)\left(1-e^{-\frac{4\pi k(\omega)}{\lambda_0}l}\right)d\omega} {\displaystyle \int I_E(\omega)\,d\omega}. \tag{6} \]

The quantity \(A\) is a function of \(l\) and of the constants determining the course of \(I_E\) and \(nk\). If some of the constants are known, the others can be determined by measuring \(A\) for different gas-layer thicknesses \(^{36,50,77,58}\).

When the light source has a continuous spectrum, so that over the range of frequencies of the absorbing line \(I_E=\mathrm{const}\), \(A\) is called the total absorption of the spectral line (Ladenburg and Reiche \(^{37}\)). On the other hand, one may take as the light source a luminous layer of exactly the same medium as the absorbing one; then the so-called linear absorption takes place, in which the absorption line has the form of the absorbed line.

The dependence of the total and linear absorption on the constants of the Doppler and dispersion distributions can be found in Ladenburg and Levy \(^{38}\).

Other, likewise indirect, methods of determining line widths are based on magneto-optical effects near a spectral line (Schott \(^{59}\)). The rotation of the plane of polarization in a magnetic field depends, for a given line shape, on its width. Since the assumptions about the shape of emission and absorption lines usually do not fully correspond to reality, these indirect methods inevitably contain sources of error capable of strongly distorting the result.

Two groups of line broadenings are distinguished. In Chapter II broadenings are considered that are inherent in the radiation of an isolated atom; they do not depend on the density of the gas. Chapter III contains a description of broadenings that depend on density and are caused by the mutual influence of atoms.

II. Natural line width and Doppler broadening

1. Natural line width

An electric dipole, performing harmonic oscillations, according to classical theory undergoes damping as a result of the radiation of energy. The energy of the oscillator decreases exponentially:

\[ E=E_0 e^{-\gamma t}; \tag{1} \]

where

\[ \gamma=\frac{2}{3}\cdot\frac{e^2}{mc^3}\cdot\omega_0^2, \tag{2} \]

\(\omega_0\) is the natural frequency of the oscillator.

This damping leads to the fact that the radiation is not strictly monochromatic. The expansion of the amplitude

\[ A(t)=A_0 e^{-\frac{\gamma}{2}t}\cos(\omega_0 t+\varphi) \]

gives for the coefficients of the Fourier integral:

\[ A(t)=A_0\int_{-\infty}^{+\infty} a(\omega)e^{i\omega t}\,d\omega, \]

\[ a(\omega)=\frac{A_0}{4\pi} \left[ \frac{e^{i\varphi}}{i(\omega-\omega_0)-\frac{\gamma}{2}} - \frac{e^{-i\varphi}}{i(\omega+\omega_0)+\frac{\gamma}{2}} \right]. \]

Near the middle of the line the second term may be neglected. Then for the distribution of the radiation intensity over different wavelengths one obtains:

\[ I_E(\omega)=\frac{\gamma}{2\pi}\cdot \frac{1}{(\omega_0-\omega)^2+\left(\frac{\gamma}{2}\right)^2}. \tag{3} \]

In order to calculate the natural shape of the absorption line, let us consider a set of identical oscillators with density \(N\) in \(1\ \mathrm{cm}^3\). Of course, this density must be sufficiently small that the interaction of the oscillators may be neglected. In these oscillators, forced oscillations are excited by incident light of frequency \(\omega\), with amplitude

\[ A_x(t)= \frac{e\frac{F}{m}}{\omega_0^2-\omega^2+i\gamma\omega}\, \sin(\omega t+\varphi_x) \approx \frac{\frac{eF}{2m\omega_0}}{\omega_0-\omega+\left(i\frac{\gamma}{2}\right)} \sin(\omega t+\varphi_x), \]

where \(F\sin(\omega t+\varphi_x)\) is the field strength at the place where the \(x\)-th oscillator is located. Here, as in what follows, we use approximations that are permissible only near \(\omega_0\). For the work \(W\) of the acting force in one second we have:

\[ W= -\frac{ \frac{e^2F^2}{2\omega_0 m}\cdot \omega \cdot \frac{\gamma}{2} }{ (\omega_0-\omega)^2+\left(\frac{\gamma}{2}\right)^2 }. \]

\(WNdx\) gives the energy absorbed by a unit surface of a plane layer of thickness \(dx\). Consequently, for the decrease of the energy flux \(S\) through this surface element we obtain:

\[ -\frac{dS}{dx}=NW, \]

but

\[ S=\frac{F^2c}{8\pi}. \]

where

\[ F = F_0 \cdot e^{-\frac{2\pi}{\lambda_0}\, nkx} \]

for a wave propagating in the direction \(x\); hence:

\[ nk(\omega)= \frac{\frac{2\pi \sigma^2}{\omega_0^2 n}\cdot N \cdot \frac{\gamma}{2}} {(\omega_0-\omega)^2+\left(\frac{\gamma}{2}\right)^2}. \tag{4} \]

\(nk(\omega)\) is proportional to \(I_{\xi}(\omega)\).

Thus the natural width \(\delta_n\) of a line, according to the classical theory, is equal to \(\frac{\gamma}{2}\). According to (1), \(\gamma\) determines the intensity of the loss of energy by radiation, so that the classical line width is proportional to the intensity of the line. If the width is expressed in Å:

\[ \delta_n^\lambda=\frac{2\pi c}{\omega^2}\cdot \frac{\gamma}{2} =0.6\cdot 10^{-4}\ \text{Å}, \tag{5} \]

then it proves to be independent of \(\lambda\) and the same for all spectral lines.

In order to introduce the natural width of a line into quantum theory, two different paths were tried. The old quantum theory attempted to connect the damping of the oscillator, by means of the correspondence principle, with the probability of transition. The classical damping time according to (1) is equal to \(\tau=\frac{1}{\gamma}\). It was equated to the mean lifetime \(\tau\) of an atom in the upper level of the spectral line. Thus, while preserving the line form (3) and (4), instead of the classical quantity \(\gamma\) one took the reciprocal of the mean lifetime.

However, this result, on the basis of Dirac’s theory of light, is in contradiction with the conclusions of quantum mechanics. It follows from this theory that the line width is obtained on the basis of the following ideas (Sleter \(^{60}\), Weisskopf and Wigner \(^{71}\)): the levels of the atom are not taken to be infinitely sharp (see II). Therefore, as a consequence of the “smearing” of the terms, the transition from one term to another will correspond to a non-sharp spectral line, and we obtain a finite line width.

The “smearing” of the terms finds its explanation in Heisenberg’s uncertainty relation between energy and time. The finite lifetime \(\tau\) of a term causes an uncertainty of its energy of order \(\Delta T \sim \frac{1}{\tau}\). Therefore it is natural to assume that the distribution of the weights of the terms \(T\) (expressed in frequencies) around the center of gravity \(T_n\) of all terms is determined analogously to (3) and (4):

\[ G_n(T)=\frac{\gamma_n}{2\pi}\, \frac{1}{(T_n-T)^2+\left(\frac{\gamma_n}{2}\right)^2}. \]

The factor \(\frac{\gamma_n}{2\pi}\) is chosen in such a way that the sum of the weights of one level is equal to 1. \(\gamma_n\) must be equal to the reciprocal of the mean lifetime of the \(n\)-th term. Therefore

\[ \gamma_n = 3 \sum_m f_{nm}\frac{g_m}{g_n}\cdot \frac{2}{3}\frac{e^2\omega_{nm}^{\,2}}{mc^3}, \qquad \omega_{nm}=T_n-T_m, \tag{6} \]

where \(f_{nm}\) is equal to the number of dispersion electrons in the transition \(n\to m\), and \(g_n, g_m\) determine the statistical weight of the levels \(n\) and \(m\); \(\omega_{nm}\) is the Bohr frequency of the transition \(n\to m\). The summation must be carried out over all levels \(m\) lying below \(n\). Thus the specification of a level determines the energy of the atom incompletely; \(g_n(T)\,dT\) indicates the probability that the energy lies between \(T\) and \(T+dT\).

If, in a jump from a level \(n\) to a level \(n'\), the difference of the energies before and after the jump is emitted, then the width of the spectral line \(n\to n'\) is determined by the width of both levels. A simple argument shows that the probability of emitting a frequency \(\omega\) in the jump \(n\to n'\) is equal to

\[ I(\omega)=\int_0^\infty G_n(T)G_{n'}(T-\omega)\,dT, \]

\[ I(\omega)=\frac{\gamma_n+\gamma_{n'}}{2\pi}\cdot \frac{1}{(\omega_{nn'}-\omega)^2+\left(\frac{\gamma_n+\gamma_{n'}}{2}\right)^2}, \]

where \(\omega_{nn'}\) is the Bohr frequency of the transition \(n\to n'\). Thus we again obtain a dispersion distribution of width

\[ \delta_n=\frac{\gamma_n+\gamma_{n'}}{2}. \tag{7} \]

It is equal to the half-sum of the widths of the initial and final levels and can be calculated from atomic data on the basis of (6). A line with a small transition probability may (in contrast to the classical theory) be comparatively broad, provided that intense transitions are possible from one of its levels.

If we are dealing with a resonance line whose lower level (normal) \(n'\) has no width because of an infinite lifetime, while from its upper level \(n\) only one jump is possible, then on the basis of (6) we obtain:

\[ \delta_n=C\cdot\frac{\gamma}{2}, \qquad C=3f\ldots\frac{g_{n'}}{g_n}, \tag{8} \]

where \(C\) is the factor by which the classical width must be multiplied in order to obtain the quantum-theoretical one. For both sodium \(D\)-lines it is almost equal to 1.

The preceding arguments are valid only when the density of the light flux at the place where the atom is located is small. In view of the fact that light causes absorption and induced emission, the mean lifetime of the level

decreases, which leads to its broadening. This also applies to the normal level. However, the broadening of the normal level at normal densities of the luminous flux in the visible part of the spectrum is very small. It is related to the natural broadening of the line (in the absence of illumination) as the probability of a transition induced by the light wave is related to the probability of a spontaneous transition. It may therefore be neglected so long as the number of excited atoms is small in comparison with the number of unexcited ones, or so long as proportionality exists between the density of the luminous flux and the absorbed energy. The radiation density must reach the value

\[ \sigma(\omega)=\frac{2h\omega^3}{\pi c^3}, \]

in order to produce a broadening of the same order as that which is entailed by natural damping. This corresponds, for the visible part of black radiation, to a temperature of about 30 thousand degrees.

The ideas developed here permit further conclusions concerning the frequencies of successive jumps of an atom. Suppose an atom is at some level from which it can reach the normal state by two jumps (Fig. 2). If in the first jump \(AB\) such a frequency is emitted that the atom reaches a level lying below the center of gravity of term \(B\), then in the next jump the frequency \(BC\) must be less than the mean, since the atom cannot change its energy “within” the term. Therefore the frequencies of successive jumps are not independent of one another; they are connected in such a way that the broadening of the sum of all successively emitted frequencies is determined by the broadening of the initial level \(A\). This follows, moreover, also from the law of conservation of energy, which requires that the “smearing” of the energy at the beginning and at the end of the process be one and the same. At the beginning it is determined by the width of the level of the initial state; at the end the energy of the atom is strictly definite—the atom being in the normal state—and therefore the energy of the radiation—the sum of the emitted frequencies—must have the broadening of the initial state.

Fig. 2.

With the help of Dirac’s theory of radiation emission it can be shown that these arguments are in agreement with quantum mechanics. Here we shall indicate only briefly the essence of the method. The theory regards the atom and the emitted field as two quantum-mechanical systems in weak interaction with one another; the form of the interaction is borrowed from Maxwell’s theory. The emitted field is represented by an infinite number of independent oscillators, which correspond to the electromagnetic proper frequencies of the closed cavity within which the atom is enclosed. Owing to the coupling, the states cease to be stationary—

them. The probability of finding the atom at one of its levels will be a function of time of the form \(e^{-\gamma t}\). With the aid of perturbation theory, with certain simplifications, it is possible to compute with what probability the individual oscillators of the closed band will be excited, i.e. in what state the emitted field is after emission. This probability distribution over oscillators of different frequencies gives the natural width in agreement with the results indicated.

An exception is presented by cases in which, among successively arranged energy levels, there is a doublet (the difference is no greater than the natural width of the lines). Here the theory leads to results different from the picture of sharp levels. This occurs precisely for the harmonic oscillator.* (In real atoms the coincidence of two energy levels—within the required accuracy; a difference of levels smaller than the natural broadening is unlikely to occur.) The intuitive picture of sharp levels makes it possible to expect the width of the lines to be the greater, the higher the initial level, since the transition probability is proportional to the term number. Calculation, however, shows that the width is everywhere one and the same and coincides with the classical one.

The discrepancy between the intuitive picture and the results of the calculation arises because, in these special cases, each emitted quantum cannot be assigned to a definite transition of the atom; a definite quantum may be produced in all transitions whose energy difference, to within the line broadening, is equal to its frequency. Owing to this uncertainty, the resonance that arises requires a different mathematical treatment and leads to different results.

It is interesting to note that even with a very small anharmonicity of the oscillator one can again use the old picture, and therefore the line width in transitions from the upper levels would have to be considerably greater than for the harmonic oscillator. This circumstance could perhaps be tested experimentally.

2. Doppler broadening

If an atom moves rectilinearly relative to the observer with velocity \(v\) and in doing so emits light of frequency \(\omega_0\), then for the observer this frequency appears shifted by

\[ \Delta \omega = -\omega_0 \frac{\xi}{c}, \tag{9} \]

where \(\xi\) is the projection of \(v\) on the direction of observation. If we assume that all atoms of the gas emit strictly monochromatic light of frequency \(\omega_0\), then, as a result of thermal motion, there will be observed—

* In this case Pauli had already suspected in 1926 the necessity of a special treatment (see Handbuch der Physik, Geiger-Scheel, Bd. 23).

an broadened line. On the basis of Maxwell’s distribution law, the probability that \(\xi\) lies between \(\xi\) and \(\xi+d\xi\) is equal to

\[ \sqrt{\frac{m}{2\pi RT}}\, e^{-\frac{m\xi^2}{2RT}}\,d\xi, \tag{10} \]

where \(R\) is the gas constant and \(m\) is the molecular weight. The relative distribution of intensity in the line will therefore be

\[ I(\omega)=\mathrm{const}\, e^{-\frac{m}{2RT}\frac{c^2}{\omega_0^2}(\omega_0-\omega)^2}. \tag{11} \]

This is the so-called Doppler distribution, with width

\[ \delta_d=\sqrt{\ln 2}\,\frac{\omega_0}{c}\sqrt{\frac{2RT}{m}} . \tag{12} \]

Of course, exactly the same line shape is obtained also for an absorption line, if it is assumed that the atoms absorb a strictly definite frequency \(\omega_0\). In reality, the absorption and emission of atoms, owing to the natural width of the lines, are not strictly monochromatic; however, (11) is valid with a high degree of approximation, since the Doppler width in the visible and ultraviolet parts of the spectrum is considerably greater than the natural width. Thus, for example, for the heavy mercury atom the Doppler width for the line \(\lambda=2537\) at \(300^\circ\) abs. is \(\delta_d=1.34\cdot 10^{-3}\) Å, i.e. it exceeds the natural width by more than a factor of 20. It becomes of the order of the natural width only at a practically unattainable temperature of \(0.75^\circ\) abs.

If the Doppler width is expressed in wavelengths:

\[ \delta_d^\lambda=\sqrt{\ln 2}\,\frac{\lambda}{c}\sqrt{\frac{2RT}{m}}, \]

then it is seen that it decreases with wavelength. Since the natural width of lines does not depend on wavelength, for very short waves the ratio between the Doppler width and the natural width changes. Thus, the Doppler width for the line of wavelength \(100\) Å is already, at room temperature, equal to the natural width \(\delta_\lambda=0.6\cdot 10^{-4}\) Å (for mercury). Therefore, in the X-ray region we may, conversely, neglect the Doppler width in comparison with the natural width. On this basis, Ehrenberg, Mark, and Susich\(^ {9,1}\) attempted to measure the natural width of lines in the X-ray region, but without definite success, since the unresolved fine structure of the X-ray lines produced an apparent broadening.

The exact shape of a line broadened as a result of the Doppler effect is obtained if the “natural” width of the line is taken into account in the following way: the natural intensity distribution (3) of the emitting atom is shifted under motion

motion of the atom by \(\Delta=\dfrac{v}{c}\omega\), so that the middle of the line falls not at the frequency \(\omega_0\), but at \(\omega_0+\Delta\). Thus, for the distribution of intensities we obtain:

\[ \left. I(\omega)=\frac{\delta_n}{b\sqrt{\pi^3}} \int_{-\infty}^{+\infty} \frac{e^{-\frac{\Delta^2}{b^2}}}{(\omega-\omega_0-\Delta)^2+\delta_n^2}\,d\Delta, \right\} \tag{13} \]

where

\[ b=\frac{\omega_0}{c}\sqrt{\frac{2KT}{m}} = \frac{1}{\sqrt{\ln 2}}\,\delta_d, \]

\(\delta_d\) is the Doppler width, \(\delta_n\) the natural width, measured by the half-difference of frequencies whose intensities are equal and amount to half the maximum intensity (corresponding to the frequency \(\omega_0\)).

The constants are chosen so that the total intensity is equal to 1. The absorption line has the same form. In this case \(\Delta\) is the displacement of the frequencies of the incident light \(\omega\) relative to the atom of the absorbing gas.

In general, expression (13) represents a combination of the natural intensity distribution with the Doppler phenomenon and is therefore also applicable to the case of the simultaneous action of the Doppler phenomenon and damping caused by collisions; broadenings due to these two phenomena may already be of the same order in the visible part of the spectrum. Unfortunately, the integral (13) cannot be taken in finite form. In the limiting cases \(\delta_n\ll b\) or \(\delta_n\gg b\), \(I(\omega)\) passes, of course, into the expression for the Doppler intensity distribution or into the natural one (also called dispersion). It should be noted that in the case \(|\omega-\omega_0|\ll b\) (i.e. for the wings of the lines) the integral can be represented in the form of a well-convergent series.

Denoting

\[ \omega-\omega_0=\Delta\omega, \]

we obtain:

\[ I(\Delta\omega)= \frac{\mathrm{const}}{(\Delta\omega)^2+\delta_n^2} \int_{-\infty}^{\infty} \left( 1+\frac{3\Delta^2}{(\Delta\omega)^2+\delta_n^2}+\cdots \right) e^{-\frac{\Delta^2}{b^2}}\,d\Delta \]

and after integration

\[ I(\Delta\omega)= \frac{\mathrm{const}}{(\Delta\omega)^2+\delta_n^2} \left( 1+\frac{3}{2}\frac{b^2}{(\Delta\omega)^2+\delta_n^2}+\cdots \right). \tag{14} \]

For frequencies so far removed from the middle of the line that \((\Delta\omega)^2+\delta_n^2\gg b^2\), the dispersion distribution holds, and the Doppler phenomenon is quite imperceptible. The deviation from the dispersion distribution for \(\Delta\omega\sim 12\,b\) is less than \(1\%\).

The simultaneous phenomenon of the dispersion intensity distribution and the Doppler distribution (13) was first taken into account

Fochtom \(^{63}\). And in the more recent literature they refer to his formula, although, despite its generality, it is of little use for practical application. He computes the complex refractive index of a gas, and in order to obtain explicit functions he is compelled to restrict himself to small densities. But precisely here it is permissible to compute absorption and emission as the superposition of the actions of the individual atoms. This simple calculation was carried out by Majerstein \(^{41}\), Reiche \(^{53}\), and Zemański \(^{7}\). A very detailed discussion of integral (13)* may be found in Heitler \(^{42}\) and Hanneke \(^{54}\).

3. Measurements of the natural width and of the broadening due to the Doppler effect

The measurement of the natural width in the visible region is made difficult by the fact that it is always covered by a considerably greater Doppler broadening. Therefore Minkowski \(^{4}\) attempted to measure the intensity distribution of the sodium \(D\)-lines at the edges (“wings”) of the lines, where the distribution is predominantly dispersive. In order to obtain appreciable absorption for frequencies far from the middle of the line, he used an absorption tube 1 m long. The use of large vapor densities would entail the combined action of other causes of broadening as well (impact damping and broadening due to the interaction of identical atoms).

If the dimensions of the tube are chosen so that those frequencies for which the Doppler phenomenon is still noticeable are completely absorbed, then the absorption of the remaining frequencies is determined only by natural damping. This damping can be calculated either directly from measurement of the line on the basis of (3), or from the total absorption according to (6) (Chapter I), which under these circumstances is in no way different from the total absorption without the Doppler phenomenon. Minkowski obtained for both \(D\)-lines \(\delta_n=0.63\cdot 10^8\), which is in complete agreement with formula (8), according to which \(\delta_n=0.64\cdot 10^8\).

Minkowski’s measurements were made at pressures between 0.0046 mm and 0.01 mm. At lower pressures an accurate measurement of the line shape was impossible because of the small dispersion of the spectral apparatus; at higher pressures a deviation from the indicated result was observed, the width increasing with pressure. Minkowski explained this by the interaction of atoms; we shall show below that this is the result of impact damping.

In an analogous manner, Schütz \(^{58}\) and Weynter \(^{76}\) measured the natural width of the \(D\)-lines on their “wings” by means of magneto-optical phenomena and obtained, at sufficiently low pressures, agreement with the theory. Then Schütz \(^{59}\) found that the natu-

* A table of numerical values of the integral may be found in M. Born, “Optik,” 1933, p. 486.

the natural width of the blue cesium line is equal to 0.24 of the width of the classical oscillator line; the comparatively large width is caused, despite the small value of \(f\) \((f=2.7\cdot 10^{-5})\), by the intense transitions from the upper level \(7^{2}P—7^{2}S\) and \(7^{2}P—5^{2}D\).

In order to test the suitability of the formula derived from quantum theory, it is necessary to take transitions which do not lead to the normal level, and thereby to verify the influence of the widths of both levels. This was accomplished by Unsöld \(^{68}\) on the Balmer lines in the solar spectrum. In some parts of the solar atmosphere the pressure is so small that other broadenings, apart from natural and Doppler broadening, play no role; the Doppler phenomenon does not interfere because of the long absorption path, for the same reason as in Mankovsky’s case. The width found lies, within the limits of error, in agreement with (7). Recently Schütz \(^{60}\) investigated the influence of the lower level on the width of the emission lines of neon. The resonance line of neon \(\lambda=736\ \text{\AA}\) is obtained as a result of the transition \(s_{2}—p_{0}\), with \(cf \sim \dfrac{1}{2}\); the lifetime of the resonance level \(s_{2}\), because of the large transition frequency, is very short in comparison with the lifetime of the levels corresponding to the visible part of the spectrum. Therefore Schütz investigated the intensity distribution for the lines of the visible part \(p_i—s_i\), where all \(s_i\), except the resonance level \(s_{2}\), are metastable states. All the lines give a Doppler distribution, and only the lines \(p_i—s_{2}\) have a dispersion distribution, due to the large width of the level \(s_{2}\).

The first measurements of the Doppler width belong to Michelson \(^{47}\). They were treated by Schenrock \(^{56}\) and found to be in agreement with the theory. These measurements, as well as the measurements of Buisson and Fabry \(^{5}\), indicate the proportionality of the width to the quantity \(\sqrt{T/m}\), since other causes of broadening can be avoided.

At considerable pressures, collisional broadenings are detected first of all, and in hydrogen lines—broadenings due to the Stark effect.

Doppler broadening follows laws quite different from those in (11) in the case when the motion of the atoms is not thermal. Thus, for example, Bleakney and Frank \(^{3}\) observed a strong broadening of the Balmer lines (of the hydrogen atom) when \(H_{2}\) molecules were excited by electron collisions. Owing to the collisions, the molecule dissociates into an excited and an unexcited atom, whose kinetic energy is equal to the difference between the collision energy and the work of dissociation. The velocity of the excited atom may turn out to be considerably greater than the velocity of thermal motion, so that the emission line is extraordinarily broadened owing to the Doppler phenomenon. The experimental results of Bleakney and Frank confirm this interpretation, although its quantitative verification is impossible, since the energy of the electron after passing through the gas is rather indefinite. The authors obtained a width independent of temperature and equal to

\(\delta = 0.12\ \text{Å}\) (mean velocity \(40\ \text{V}\), gas pressure \(0.06\ \text{mm}\)), whereas the “thermal” Doppler width was \(0.04\ \text{Å}\). At lower electron energy a certain dependence on temperature was noticeable, since thermal motion under such conditions already makes itself felt.

The shape of the line can be determined with greater accuracy if the dissociation of the molecules is caused not by electron impact but by light, because in this case the energy acquired by the molecule is known exactly and is equal to \(h\nu\). The excited products of dissociation all have the same velocity \(v\), so that all displacements between \(0\) and \(\omega_0 \frac{v}{c}\) are represented with equal probability. The line has the form of a quadrilateral of width \(\omega_0 \frac{v}{c}\), which is further overlapped by the thermal Doppler width. This case was investigated by Guth and Franck\({}^{30}\) in NaJ vapor. The molecule was dissociated into an iodine atom and an excited sodium atom by means of ultraviolet light; the sodium line gave an anomalous width. The shape of the line could not be measured.

III. Molecular Interactions

1. Introduction

In this chapter we shall consider broadenings caused by the interaction of atoms; in contrast to the broadenings considered earlier, they depend on the density of the gas. The spectrum of radiation of an atom is altered by the force fields of neighboring atoms; the frequency and amplitude are therefore no longer constant in time. Let \(A(t)\) and \(\omega_0(t)\) be the amplitude and the proper frequency of the atom as functions of time; then the oscillation of the oscillator in the atom that causes this radiation will be:

\[ A(t)\cdot e^{\,i\int_0^t \omega_0(t')\,dt'} \tag{1} \]

The relative distribution of intensities \(I_E(\omega)\) in the radiation is determined by the Fourier expansion of this oscillation:

\[ I_E(\omega)=\mathrm{const}\left|\int_0^T A(t)e^{\,i\int_0^t \omega_0(t')\,dt'}e^{-i\omega t}\,dt\right|^2 . \tag{2} \]

If the time interval \(T\) over which we examine the functions \(A(t)\) and \(\omega_0(t)\) is chosen sufficiently large, so that the atom experiences almost all possible influences in this gas, then the intensity distribution (2) will be the same on the average for all atoms and, thus, will represent the most general expression for the shape of the emission line.

Let us now consider an absorption line. We assert that

it has the same form (2), if the interactions of the atoms are the same as in emission. In fact, let us represent these interactions as replaced by some external forces acting on the oscillator, the influence of which does not depend on the temperature, and then establish thermodynamic equilibrium. Then the line shape determined by these external forces must be the same for absorption and emission. Thus the shape of the absorption lines can always be reduced to the shape of the emission lines obtained when the atoms are acted upon by the same force field.

To establish the shape of the line, it is therefore necessary to determine the change in the natural frequency \(\omega_0\) and in the amplitude \(A\) under the influence of the force fields of neighboring atoms. Here it is important to note that the line shape is determined not only by the various values which the natural frequency \(\omega_0\) assumes in the course of time. It would be erroneous to suppose, for example, that the intensity \(I(\omega)\,d\omega\) is proportional to the probability of finding \(\omega_0\) between \(\omega\) and \(\omega+d\omega\), or proportional to the square of the amplitude of the oscillation when \(\omega_0\) lay between \(\omega\) and \(\omega+d\omega\). The actual line shape, which is obtained on the basis of the Fourier expansion (2) of the oscillatory process (1), may contain frequencies which are not natural frequencies of the radiating oscillator.

If, however, \(\omega_0\) changes so slowly that the product of the change \(\Delta\omega_0\) by the time \(\Delta\tau\) during which it occurs, \(\Delta\omega_0\Delta\tau\), is small in comparison with 1, then the Fourier expansion gives, in the main, only those frequency values which \(\omega_0\) assumed, with an intensity corresponding to their statistical weight and to the square of their amplitude. We shall therefore define, in addition to the intensity distribution (2), also the distribution of statistical weights \(H(\omega_0)\).

\(H(\omega_0)d\omega_0\) is the relative probability that the natural frequency lies between \(\omega_0\) and \(\omega_0+d\omega_0\), multiplied by the mean square of the amplitude in this frequency region. Generally speaking, the actual line shape \(I(\omega)\) is quite different from \(H(\omega_0)\); only if the condition \(\Delta\omega_0\Delta\tau \ll 1\) is fulfilled is the relation \(I(\omega)\sim H(\omega_0)\) valid. \(H(\omega_0)\) therefore gives the true distribution of intensity for an infinitely slow motion of the atoms. \(H(\omega_0)\) essentially does not depend on the temperature, since it is immaterial whether the different values of \(\omega_0\) follow one another rapidly or slowly. A weak temperature dependence can arise only insofar as, in collisions with large momenta, large frequency shifts are obtained. \(I(\omega)\), on the contrary, obviously in general depends strongly on the temperature. The more rapidly the changes of \(\omega_0\) proceed, the broader the intensity distribution.

In most cases it is easier to determine \(H(\omega_0)\) theoretically. Then one can already estimate and introduce the changes caused by the motion of the atoms.

The intensity distribution (2) was obtained under the assumption

classical theory of light. We shall show below that quantum mechanics, under certain assumptions*1, also leads to the same Fourier expansion. When the atom moves in the region of influence of other atoms, the terms change their values. The natural frequency \(\omega_0\) of the atom at the time \(t\) is then given by the difference of the two terms corresponding to the spectral line, on the assumption that the changes of the terms are sufficiently slow in comparison with the light oscillations, as is the case at normal temperatures. Further, we assume that, for a sufficiently small change of the terms, the amplitude \(A\) does not change or changes only adiabatically, i.e., that there are no transfers of excitation energy or other nonadiabatic processes.

In this case we must show that the radiation of the atom has the intensity distribution (2). The Schrödinger equation for the translational motion of an atom in the field of action of neighboring atoms is

\[ \Delta \psi_n(Er)+\frac{8\pi^2 m}{h^2}\bigl(E-V_n(r)\bigr)\psi_n(E,r)=0, \tag{3} \]

if the other atoms are assumed to be at rest. \(\psi_n(E,r)\) depends on the coordinates of the center of gravity \(r\), on the energy of motion \(E\), and on the state \(n\) of excitation of the electron. In the same way, the potential \(V(r)\) depends on \(n\). The intensity of radiation in the transition \(n\to n'\), with a simultaneous change of kinetic energy \(E\) to \(E'\), is proportional to

\[ \left|\int q\,\varphi_n(q)\psi_n(E,r)\,\overline{\varphi_{n'}(q)}\,\psi_{n'}(E',r)\,dq\,dr\right|^2, \]

where \(\varphi_n(q)\) is the eigenfunction of the electron of the atom in the state \(n\), and \(q\) denotes the coordinates of the electron.

The frequency of the radiation in this transition is

\[ \omega=\omega_{nn'}+\frac{2\pi(E-E')}{h}, \tag{4} \]

where \(\omega_{nn'}\) is the Bohr frequency of the atomic transition \(n\to n'\). We assume that the matrix element of the electron

\[ A_{nn'}=\int q\,\varphi_n(q)\varphi_{n'}(q)\,dq \]

does not depend on \(r\) even in places where the potential is changed. This assumption is equivalent to the earlier assumption of the constancy of the amplitude \(A\). Then the intensity is given by the expression:

\[ I(\omega)=\mathrm{const}\,\left|A_{nn'}\int \psi_n(E',r)\psi_{n'}(E',r)\,dr\right|^2, \tag{5} \]

where \(\omega\) is determined from (4). We shall now show that (5) is identical with (2).

For simplicity, let us consider the Schrödinger equation (3) as one-dimensional and, moreover, assume that the atom passes only through the outer layers of neighboring atoms, which is most often the case. Then (3) can be solved by the approximate method [[unclear: line cut off]].

Wenzel–Brillouin, and we obtain:

\[ \psi_n(E,x)=\frac{\mathrm{const}}{\sqrt{p_n}}\,e^{-\frac{2\pi i}{h}\int_0^x p_n\,dx}; \qquad p_n=\sqrt{2m\,[E-V_n(x)]}, \]

where \(x\) is the coordinate of the atom’s center of gravity.

Equation (5) then gives:

\[ I(\omega)=\mathrm{const}\left| \int \frac{1}{\sqrt{p_n p_{n'}}}\, e^{\frac{2\pi i}{h}\int_0^x (p_{n'}-p_n)\,dx}\,dx \right|^2 . \]

This is identical with (2), if one puts \(dx=vdt\), \((p_{n'}-p_n)v\sim T_{n'}-T_n\), and \(\sqrt{pp'}\sim mv\), which holds under our assumptions, with \(T=E-V_n\), \(v=p/m\), and, moreover, if one takes into account that

\[ \omega_0(t)=\omega_{nn'}+\frac{2\pi}{h}\,[V_n(x)-V_{n'}(x)], \]

where \(x\) is the position of the atom at the moment \(t\).

From this calculation it is clear that the deviation from Bohr’s frequency rule, which entails the distribution of intensities (2), corresponds to the transformation of light energy into kinetic energy. After the emission or absorption of a frequency not equal to the difference of atomic terms, the atom is in another state of motion: this process is completely analogous to the appearance of band spectra of diatomic molecules with a simultaneous change of the electronic and vibrational states. The set of terms there consists of electronic terms, upon which a continuous spectrum of nuclear vibrations is superposed on both sides. In the same way, the broadening of lines obtained on the basis of (2), caused, for example, by collisional damping, arises because a continuous spectrum of translational motion of the atom is superposed on both sides of the electronic term. The decrease of intensity in a broadened line on both sides of the Bohr frequency corresponds, in the analogous band spectrum following from the Franck–Condon principle, \({}^{12}\) to the fact that the intensities of bands are the weaker, the greater the quantum jump of the nuclear vibration, i.e. the greater the transfer of energy between the light and the vibrational energy of the nucleus.

It should be pointed out that Ollenberg \({}^{51}\) investigated the possibility of line broadening due to the recoil or absorption of energy by an atom passing by. He justified this by the open continuous bands appearing on both sides of the absorption line when a noble gas is added under sufficiently high pressure. In some noble gases this spectrum even showed a banded structure, which Ollenberg ascribed to the formation of molecules of mercury—noble gas. According to the above, the broadenings considered here (especially

impact damping) also belong to the category considered by Olsdenberg.

The phenomena observed by him constitute, as it were, a transition between broadenings due to the fields of neighboring atoms (impact damping) and bands of electronic oscillations. In impact damping the atom is under the influence of a neighboring atom only for a short time; in Olsdenberg’s experiments it is partly delayed and executes certain oscillations leading to a band structure; finally, in diatomic molecules the bond is very strong and therefore the spectrum is sharp and discrete.

Below we shall first consider impact damping, which can be derived from very general ideas about the nature of interactions between atoms. Then we shall consider the changes following from exact knowledge of the potential fields (asymmetry and displacement), next—the influence of identical atoms on one another, causing broadening as a result of interaction, and, finally, the influence of molecular electric fields, leading to broadening as a result of the Stark effect.

2. Impact damping

The strongest interaction of gas atoms takes place during collision. As is well known, it is impossible to establish exactly the time of impact, since the influence begins to be felt already at large distances. We shall, however, assume in what follows that the gas density is so small that the time during which an atom flies freely and is not subjected to extraneous influences is considerably greater than the time during which it is under the influence of neighboring atoms. We shall take the collision time to be small in comparison with the time of flight, and the radius of impact to be small in comparison with the mean free path.

The effects of collisions on the radiation of an atom are varied: thus, a collision may stop the radiation. Experimental proof of this phenomenon is the quenching of resonance fluorescence upon the addition of a gas, where the cessation of the oscillations of the luminous atoms in the presence of a foreign gas is clearly visible. Thus, as T. A. Lorentz has already shown, the broadening of the line results. The Fourier expansion for an oscillation of frequency \(\omega_0\), which after a time \(\tau\) ceases, gives the intensity distribution

\[ I(\omega)=\mathrm{const}\left|\int_{0}^{t_0+\tau} e^{i(\omega_0-\omega)t}\,dt\right|^2 =\frac{2\,\mathrm{const}}{(\omega_0-\omega)^2}\{1-\cos[(\omega_0-\omega)\tau]\}. \]

\(\tau\) is the time between two collisions; the probability that it lies between \(\tau\) and \(\tau+d\tau\) is equal to \(\frac{1}{\tau_0}e^{-\tau/\tau_0}\, \(d\tau\), where \(\tau_0\) is the mean time between two collisions.

Thus the intensity distribution is expressed as follows:

\[ I(\omega)=\mathrm{const}\int_{0}^{\infty} \frac{e^{-\tau/\tau_{0}}}{(\omega_{0}-\omega)^{2}} \left[1-\cos(\omega_{0}-\omega)\tau\right]\,d\tau = \mathrm{const}\, \frac{1}{(\omega_{0}-\omega)^{2}+\left(\frac{1}{\tau_{0}}\right)^{2}}. \tag{6} \]

We obtain a dispersion distribution with width

\[ \delta_{s}=\frac{1}{\tau_{0}}, \]

so that the width is equal (in cyclic frequencies) to the number of collisions per second.

The calculation will be more accurate if one takes into account that the amplitude of the oscillation decreases as a result of natural damping. Then one obtains:

\[ I(\omega)=\mathrm{const}\, \frac{(\delta_{n}+\delta_{s})^{2}}{\pi}\, \frac{1}{(\omega-\omega_{0})^{2}+(\delta_{s}+\delta_{n})^{2}}. \]

Thus the natural width and the collisional damping add together.

The simultaneous influence of the Doppler phenomenon was discussed in the preceding chapter. The intensity distribution is determined by the integral*

\[ I(\omega)=\mathrm{const}\int \frac{e^{-\Delta^{2}/b^{2}}\,d\Delta} {(\omega-\omega_{0}-\Delta)^{2}+(\delta_{s}+\delta_{n})^{2}}, \tag{7} \]

where

\[ b=\frac{\omega_{0}}{c}\sqrt{\frac{2kT}{m}}. \]

For \(|\omega-\omega_{0}|\gg b\), only the shape of the line due to collisional damping is significant (Ch. II, § 2). The mean number of collisions per second \(\frac{1}{\tau_{0}}\) is calculated from the kinetic theory of gases under the assumption that the pressure of the foreign gas is considerably

* Strictly speaking, the Doppler effect should already have been taken into account in the function \(\omega_{0}(t)\) in (2), since the natural frequency after each collision, when the direction of motion of the atom changes, changes its value for a stationary observer. These changes of frequency should give, in the Fourier expansion, an additional broadening in addition to the usual Doppler broadening (this possibility was noted to us by F. Gouterman). Calculation shows that this additional broadening is always small in comparison with collisional broadening. The point is that the Doppler effect changes after a collision only the frequency, but not the phase. This additional broadening therefore has only theoretical significance, since the phase changes in collisions that change the direction of motion in any case, and this causes a considerably larger broadening.

above the pressure of the main gas, so that the broadening between two identical radiating atoms may be neglected:

\[ \frac{1}{\tau_0}=\pi \rho^2 Nv=2\rho^2N\sqrt{2\pi kT\,\frac{m_1+m_2}{m_1m_2}}. \tag{8} \]

Here \(v\) is the relative velocity of the colliding atoms, \(m_1\) is the mass of the luminous atom, and \(m_2\) that of the foreign atom. If there is no foreign gas, one should put \(m_1=m_2\). \(N\) is the number of atoms in cubic centimeters and \(\rho\) is the optical diameter characteristic of this process. It is equal to the distance between the centers of the atoms at the time of collision, i.e. to the sum of the radii of both particles.

Little can be said theoretically about the collision diameter, since we do not know the mechanism of quenching. In any case, it must at least be equal to the effective diameter corresponding to the quenching of resonance fluorescence. It may be larger, for, as we shall presently see, other effects of collisions are also possible. In most cases it is considerably larger than the gas-kinetic diameter, which, according to Stern and Volmer\(^{65}\), is already accounted for at least by the fact that the “optical radius” is determined by the radius of an atom in the excited state, which is always greater than the normal one.

That broadening caused by collisions is not produced solely by the cessation of oscillations upon impact became clear from the moment it was discovered upon adding gases that do not quench resonance fluorescence. Thus, for example, the \(D\)-lines of sodium are broadened to the same extent upon addition of nitrogen and helium, whereas nitrogen strongly quenches the resonance fluorescence of the \(D\)-lines, while helium does not quench it at all.

The mechanism of collisional damping in non-quenching gases may be represented as follows (Lenz\(^{40}\), Kallmann and London\(^{35}\); for a quantum-theoretical calculation see Weisskopf\(^{71}\)): the natural frequency of the atom, which in the unperturbed state has the value \(\omega_0\), changes during the collision time \(\Delta \tau\), but after the collision again assumes the original value \(\omega_0\). The change, however, is so strong that the phase after the collision no longer coincides with the phase that the oscillation would have had if it had not been perturbed. The phase shift is given by the expression

\[ \theta=\int_{\Delta \tau}\Delta\omega_0(t)\,dt. \]

The change \(\Delta\omega_0\) as a function of time must be integrated over the collision time. If one now expands this oscillation, perturbed during the collision, in a Fourier series, then the magnitude of the shift over the short interval \(\Delta\tau\) will be insignificant; but if the free-path time \(\tau \ll \Delta\tau\), then the phase shift will play a large role. The result of the Fourier expansion will then be the same as in the expansion of a purely monochro-

matical oscillation with frequency \(\omega_0\), which after each free path changes its phase by a jump. This expansion gives, as is easy to see, precisely the same result, the same width and shape of the line, as the previously considered interrupted oscillation.

The fact that the phase changes quite strongly despite the short collision time is seen from the following approximate calculation. The collision time, at normal temperatures and effective cross sections, is approximately \(10^{-13}\) sec. Since the visible frequencies are approximately equal to \(10^{15}\), an average frequency shift of \(1\%\) is already sufficient to change the phase by \(\pi\).

The collision radius that is significant for this mechanism then gives the distances at which two atoms must fly past one another in order to experience a still appreciable phase shift. It is easy to understand that here, too, the radii are larger than the gas-kinetic ones. If the change in frequency \(\Delta\omega_0(r)\) is given as a function of the distance \(r\) between the colliding atoms, and if we assume that the change in direction during the flight near the edge of the sphere of appreciable action may be neglected, then the optical diameter \(\rho\) is determined by the condition that, in flying along a straight line at a distance \(\rho\) from the neighboring atom, the phase changes by \(\sim 1\). Consequently, it must be:

\[ \theta=\int_{-\infty}^{+\infty}\Delta\omega_0(r)\,dt\sim 1 \qquad r=\sqrt{v^2t^2+\rho^2}, \]

where \(v\) is the mean relative velocity of the colliding particles. The integral may have infinite limits, since outside the collision time no change occurs. Thus, in order to calculate \(\rho\), it is necessary to know only the function \(\Delta\omega_0(r)\), which can be approximately determined.

We shall first consider the collision of two homogeneous atoms. For this one may compute, by the classical theory, the mean changes in the frequencies of two identical oscillators coupled only by the dipole interaction. We have:

\[ |\Delta\omega_0|=\frac{K}{r^3},\qquad K=\frac{1}{4}\frac{e^2}{m\omega_0}f \tag{9} \]

(in order to introduce a correction into the classical calculation, one should multiply by the quantity \(f\) of the corresponding line; \(m\) is the mass of the electron).

Then from

\[ \theta=\int\frac{K\,dt}{\sqrt{(v^2t^2+\rho^2)^3}}\sim 1 \]

we obtain

\[ \rho\simeq \sqrt{\frac{2K}{v}}. \tag{10} \]

If one substitutes here, for example, the values for the sodium \(D_1\)-line at \(500^\circ\), then the optical diameter of the collision is found to be \(3.5\cdot 10^{-7}\ \mathrm{cm}\). The Hg—2537 line gives, at room temperature, \(1.43\cdot 10^{-7}\ \mathrm{cm}\). These considerable values of the effective

radii is due to the resonance action of identical atoms. The effective cross section \(\pi \rho^2\) is inversely proportional to the velocity. But the broadening due to collisional damping according to (8) is proportional to the product \(\pi \rho^2 v\), so that, since the damping is in fact caused only by the dipole interaction of identical atoms, the line width does not depend on temperature, but only on density. We then obtain from (8) and (10) for the magnitude of the line broadening in a homogeneous gas:

\[ \delta_s=\frac{\pi}{2}\frac{e^2}{m\omega_0}fN. \tag{11} \]

Since in § 4 we shall show that the so-called coupling width between identical atoms changes only insignificantly the width due to collisional damping, we may regard expression (11) as the actual width at gas density \(N\), since only dipole forces are essential for the interaction. For not too small values of \(f\), these forces will be predominant.

The influence of a collision with a foreign atom can no longer be calculated so exactly. One may form a clear idea of the course of \(\Delta \omega_{ik}(r)\), following Jablonski\(^{32}\), by drawing the “Franck potential curves” of the luminous atom, i.e. the values of its terms as functions of the distance \(r\) from the perturbing foreign atom (Fig. 3). The difference of the two potentials then determines the proper frequency \(\omega_0\), if the impact may be regarded as an adiabatic process, which occurs in collisions that do not cause quenching. Since the upper term is probably more strongly affected than the lower, we may, in particular, for the outer regions of the sphere of influence regard the course of the upper curve as decisive for \(\Delta \omega\). In the outer region the curve is lowered or raised owing to polarization action, which may be approximately estimated according to London\(^{44}\). Let us consider two unlike atoms I and II, of which the first is excited. The change in potential is equal to

\[ \Delta E=-\frac{C}{r^6}. \tag{12} \]

Fig. 3.

Assuming that from the excited state of atom I only one transition—to the normal state—is possible, we obtain approximately for the upper and lower limits:

\[ \begin{aligned} C_{\max} &= \frac{3}{2}\alpha_2 \Delta F(\alpha_1+\alpha'),\\ C_{\min} &= \frac{3}{2}\alpha_2 \Delta F\left(\alpha_1\frac{\Delta E'}{\Delta F+\Delta E'}+\alpha'\right),\\ \alpha' &= \frac{e^2}{m}\left(\frac{h}{2\pi}\right)^2 \frac{f_0\Delta F}{(\Delta F-\Delta E_0)(\Delta E_0)^2}. \end{aligned} \tag{12a} \]

Here denote: \(z_1\) is the polarizability of the excited atom I, \(z_2\) is the polarizability of the excited atom II, \(f_0\) and \(\Delta F_0\) are the value of \(f\) and the energy difference corresponding to the line excited in atom I, \(\Delta F\) is the mean excitation energy of atom II, \(-\Delta E'\) is the energy difference between the excitation state I and the nearest higher term with which the preceding one combines; \(\Delta E''\) is the difference between the ionization energy and the excitation state I.

These expressions are obtained from the well-known exact formula (London):

\[ C=\frac{3}{nv}\left(\frac{he}{2\pi}\right)^4 \sum_{k'e'} \frac{f_{kk'}' f_{ee'}'} {(E_{k'}'-E_k')(\bar F_{e'}'-F_e')(E_{k'}'+F_{e'}'-E_k'-F_e')}, \]

where \(E_k\) and \(F_e\) are the energies of the states of atoms I and II, and \(f_{kk'}\) and \(f_{ee'}\) denote the \(f\)-values of transitions to states with energies \(E_{k'}\) and \(F_{e'}\). The radius of action \(\rho\) is determined from

\[ C\cdot \int_{-\infty}^{+\infty}\frac{dt}{(v^2t^2+\rho^2)^3}\sim 1. \]

Thus:

\[ \rho \simeq \sqrt[5]{\frac{C}{v}}. \tag{12h} \]

Calculations of \(\rho\) for various foreign gases on the basis of this formula are given in the following paragraph for comparison with the experimental material. They lie between 4 and 10 Å, i.e., they have a quite acceptable magnitude, which, moreover, depends only weakly on the choice of \(\alpha\) and \(V\), owing to the fifth root. The principal shortcoming of this estimate consists above all in the fact that London’s approximation is valid here only qualitatively. At these small distances the electron clouds already overlap noticeably and, of course, produce additional displacements of terms, which considerably change \(\rho\) (increase it). Nevertheless, the summary in the following paragraph will show that the considerations indicated qualitatively reflect the actual relations correctly.

We shall also have to assume, in the case of quenching collisions, that the frequency of the radiating atom changes even before the oscillation has stopped. Little can be said, however, about the magnitude of the change. London’s formula is inapplicable here, since the process is probably nonadiabatic. The effective quenching diameter can therefore be only the lower bound of the optical diameter, which is also consistent with the experimental data.

3. Experimental material relating to impact damping

It has long been known that spectral lines possess a width depending on the pressure of the gas. This was established by measurements of infrared bands \({}^{8,1,23}\). A line with a large

a long wavelength, and therefore are well suited for these investigations, because the ratio of the Doppler width to the impact broadening is proportional to the frequency, and therefore at atmospheric pressures in the infrared region the Doppler effect can be neglected in comparison with impact damping. Quantitative data on the magnitude of the width cannot be obtained from these works. The measurements reveal, with greater or lesser accuracy, proportionality between width and pressure.

Recently Becker\(^{2}\), Kussmann\(^{31}\), and Wilmer\(^{78}\) have investigated impact broadening in infrared bands. Wilmer—on CO\(_2\), Becker and Kussmann—on HCl. They investigated the dependence on admixture of various foreign gases and obtained specific effects. Whereas Kussmann and Wilmer, by an indirect route, determined only the change in absorption, Becker, on the HCl band \(3.46\,\mu\), was able directly to show the dispersion form of the line. From his measurements at high gas pressure—without admixture of a foreign gas (\(4\)—\(12\,atm\))—the effective radius is obtained equal to \(13.6\,\text{\AA}\). The author himself considers it necessary, along with impact damping, to introduce also a width caused by coupling, and therefore obtains a somewhat smaller diameter, which in our opinion is unnecessary (see § 5). Owing to the especially small value of \(\gamma\) for the infrared band, the theoretical formula (11) is no longer applicable here, since the extent of the molecule is probably considerably greater than the calculated cross section, so that frequency perturbations by deformation of the electron shell are considerably greater than the dipole interaction (see below for Hg). The optical effective cross section between identical molecules in this case is insignificantly greater than between unlike ones, which Lazarev\(^{39}\) also found experimentally.

A rigorous proof that the observed broadening is caused precisely by collisions and therefore depends only on the number of collisions was given by Orthmann\(^{52}\), observing the broadening of the Hg \(2537\,\text{\AA}\) line by addition of H\(_2\), showing that an increase in temperature from \(T_1\) to \(T_2\) gives the same broadening as an increase in pressure by \(\sqrt{T_2/T_1}\) times. The effective cross section is thereby obtained independent of temperature. He used an indirect method for measuring the width. Kunze\(^{33}\) succeeded in tracing the proportionality between the width due to impact damping and the pressure on the Hg line up to \(50\,atm\).

The first quantitative data relating to the width, and the optical collision diameters obtained from it, can be found in the works of Füchtbauer and his collaborators. They investigated the resonance lines Na\(^{16,17}\), Cs\(^{14}\), and Hg\(^{15}\) at very high pressures of a foreign gas, from 3 to \(50\,atm\), in order to cover the Doppler width. The width obtained in this case is of the order of magnitude of one \(\text{\AA}\), and therefore is easily measured directly. Minkowski\(^{48,50}\) measured the width of absorption lines

at considerably lower pressures of the foreign gas. He worked with pressures approximately equal to 100 mm, and avoided the Doppler width by using, as he did in his measurements of the natural width of the line\({}^{49}\), very thick layers of gas. In this case only the wings of the absorption line were measured, from which it is easy to calculate the width if the shape of the line is known. The observed asymmetries will be discussed in the following paragraph*. Indirect methods for determining the width of Hg lines were used by Zemansky\({}^{70}\) and Kuhn\({}^{33}\), and by Schutz\({}^{58}\) for Na. Zemansky and Kuhn measure the total absorption of a broadened line; Schutz uses rotation of the plane of polarization in a magnetic field to determine the width.

In Tables 1a and 1b below, the optical diameters, obtained from observations of line widths, of the best-studied vapors Na and Hg with several foreign gases are compared. They are calculated from the formula:

\[ \rho^{2}=\frac{\delta_{\xi}}{2N\sqrt{2\pi kT}}\cdot \sqrt{\frac{m_{1}m_{2}}{m_{1}+m_{2}}}, \]

where \(\delta_{\xi}\) is the width expressed in units of cyclic frequency.

TABLE 1a

Optical diameters of the Hg line 2537 with respect to foreign gases in Å

Impurity	H₂	He	Ar	Ne	N₂	O₂	CO₂	CO
Füchtbauer, Joos and Dinkelacker	5.27	—	9.44	—	8.05	8.07	11.2
Zemansky	4.95	3.88	7.85	—	7.15	—	—	6.68
Kuhn	—	5.1	8.8	6.1	—	—	—	—
Theoretical, on the basis of (12b), max	—	5.0	8.3	6.7	7.9	—	9.1	—
Theoretical, on the basis of (12b), min	—	3.2	5.4	4.4	5.1	—	6.0	—
Quenching diameter according to Stuart	5.2	—	—	—	—	7.7	—	6.9

For comparison, there are also given, for strongly quenching gases, i.e. H₂, O₂, and CO, for Hg, H₂ and N₂ for Na, the effective quenching radii of resonance fluorescence on the basis of

* I express my gratitude to Prof. Minkowski for a detailed communication of his experimental data.

WIDTH OF SPECTRAL LINES IN GASES

measurements of Stuart[^66] and Mannkopff[^44]; for non-quenching gases the effective diameters theoretically calculated on the basis of (12b) are given. The polarizabilities of the gases and the values of the energies

TABLE 1b

Optical diameters of the sodium Na-D line relative to various gases, in Å.

Source	H₂	He	Ar	Ne	N₂
Schutz	5.8	5.6	9.0	6.15	8.5
Linkovsky	—	—	7.9	—	7.2
Theor. on the basis of (12), max	—	6.3	10.0	7.9	—
Theor. on the basis of (12), min	—	4.8	7.6	6.0	—
Quenching diameter according to Mannkopff	4.5	—	—	—	8.2

are compared in Table 2. The polarizabilities of excited atoms may be calculated from the corresponding second-order Stark-effect lines:

\[ \Delta=\frac{\alpha}{2h}F^2, \]

where \(\Delta\) is the frequency displacement in the field \(F\). The values of the Stark effect given in Table 2 are taken from the works of Ladenburg[^36] and Brazhunas[^4].

TABLE 2

	He*	Ne	Ar	Na	Hg	N₂	O₂	CO₂	CO
Polarizability \(\alpha \cdot 10^{24}\)	0.20	0.39	1.63	96*	20*	1.74	1.57	2.7	1.9
Minimum excitation energy in V	1.98	16.5	11.5	2.1	4.7	6.5	8	—	6.1
Ionization energy	24.6	21.5	15.6	5.1	10.4	17	13	14.3	10

We see that the experimental results differ so strongly from one another that one cannot expect a better agreement of experiment with theory. The measurements of Füchtbauer and his collaborators, in particular the measurements with Na, were made earlier. The high pressure could have affected the reliability of the results; therefore they should be assigned less weight. The theoretical values obtained from quenching (12b) are smaller than the experimental ones, which is entirely accommodated within the framework of the theory. The quantitative ratios of the quantities

is displayed correctly everywhere. There is an obvious dependence of the diameter on the molecular weight of the foreign gas: a larger molecular weight entails a larger optical radius. On the basis of our theoretical ideas, this circumstance is explained for non-quenching gases by the fact that 1) the polarizability varies in the same way, and 2) the relative velocity is smaller at a higher weight, which, according to (12b), causes a larger radius.

There is a considerably smaller number of experimental works on the broadening with increasing density of the principal gas. That in this case the broadening is considerably stronger than when a foreign gas is added was established by Gerlach and Schütz^13. This width is very often erroneously explained by the resonance coupling of identical atoms^2,13. In § 5 it will be shown that resonance coupling changes the damping due to collisions only insignificantly. True, the increased optical diameter is also conditioned by a kind of resonance phenomenon.

Minkowski^49 noted, when measuring the natural width of the Na—\(D_1\) line, that at pressures beginning approximately from \(1/100\) mm and higher the line is broader than would be required by natural damping alone. If, on the basis of formula (11), one calculates the width of the collisional damping at \(1/100\) mm pressure (\(N=1.6\cdot 10^{14}\)), then one obtains \(\delta_s=0.69\cdot 10^7\ \mathrm{sec}^{-1}\). This width therefore reaches just the order of magnitude of the natural line width (\(\delta_n=3.2\cdot 10^7\ \mathrm{sec}^{-1}\)). The same observations were made by Schütz^58 and Beingerov^5, who measured the width indirectly by means of magnetic rotation. In Schütz, deviations from natural damping were already observed at \(4\cdot 10^{-3}\) mm; if one takes into account observational errors, it may be considered that the observed deviations are explained by collisional damping, the magnitude of which according to (11) is given only approximately. In Minkowski’s unpublished photographs, which he most kindly made available to me, the resonance line of cesium at 5 mm vapor pressure has a width of approximately \(\delta_s\sim 0.8\cdot 10^{10}\), which agrees very well with the theoretical value \(\delta_s\sim 1\cdot 10^{10}\) according to (11).

The broadening of the mercury line was measured by Ortmann and Pringsheim^53. They found that increasing the mercury vapor pressure from 0.0001 to 7.3 mm produces the same broadening as adding a Ne—He mixture under a pressure of 250 mm. From this pressure difference one can calculate the ratio of the optical diameters. The diameter of the atoms of the mixture may be taken from Table 1. We thus obtain, for the effective diameter, \(\rho\sim 50\ \mathring{\mathrm{A}}\), whereas formula (11) gives, for the temperature used by Ortmann and Pringsheim, only \(14.3\ \mathring{\mathrm{A}}\). This discrepancy indicates that, along with the dipole action, other causes must also produce considerable broadening; it is possibly connected with the small value of \(f\) for the Hg line. The model used for deriving (11) replaces the whole atom by an oscillator, whose inertia at \(f=\frac{1}{35}\)

very small amplitudes, which are probably considerably smaller than the dimensions of an excited mercury atom. The width will then no longer depend on temperature, which, perhaps, is accessible to experimental verification. For the Na-\(D_1\) line with \(f=\frac14\), it appears more permissible to replace the atom by an oscillator.

Concerning the broadening of the Na and Hg lines (in Na and Hg vapors) there are also the works of Trumpy\(^{67}\), carried out by him under the influence of Holtsmark’s theory\(^{28}\); he found there the required proportionality to the square root of the density, in contradiction with formula (11). We shall show in Sec. 5 that the width due to coupling should have been insignificant in comparison with collisional damping and should not be found to depend on \(\sqrt{N}\). The broadenings observed by Trumpy, moreover, are considerably larger than those of other authors. A high pressure of foreign gas greatly interferes with his measurements. We therefore think that Trumpy’s measurements do not testify against the fact that the broadening in the gas itself is due to collisions. The same is true for the measurements of the width by Harrison and Slater\(^{29}\) on high-frequency lines of the principal series of Na, who likewise supposed that they had found a dependence on \(\sqrt{N}\). A detailed criticism of Trumpy’s measurements may be found in Kuhn, Pollanyi, and Fortrat\(^{21}\), and of the works of Harrison and Slater in Minkowski\(^{50}\).*

Entirely outside the framework of existing theories are Weibel’s measurements\(^{77}\) on the high-frequency lines of the principal series of cesium. He finds, at vapor pressures from 10 to 30 mm, broadenings which are approximately 200 times greater than the broadening in oxygen at the same pressure; from this there follows for the optical radius a value greater than 100 Å. Since the values of \(f\) for these lines are smaller than \(10^{-4}\), dipole interactions cannot occur, so that Holtsmark’s theory is inapplicable, although a pressure dependence on \(\sqrt{N}\) was found. Owing to the large polarizability of cesium \((\alpha = 54\cdot 10^{-24})\), it is true that the width due to van der Waals forces will be especially large (formula 12b); it cannot, however, reach the observed value, since the forces rapidly decrease with \(\frac{1}{r}\). One might perhaps invoke, as an explanation, the formation of Cs\(_2\) quasi-molecules, which, because of their large polarizability, must often and strongly shift the energy of the normal state. R. Kuhn (“Zs. Physik” 78, 782, 1932) observed similar broadenings on the first lines of Cs. However, the line shape he found (sharp edges) differs greatly from that found by Weibel.

* Since Ornstein and Prins measured the broadenings of oblique images, by comparison the broadenings in the vapors themselves and in the foreign gas can be strongly distorted as a result of insignificant differences in the shape of the lines in these two cases (asymmetry), which can, owing to the superposition of closely spaced fine-structure components, strongly affect the overall absorption. I owe this remark to Prof. Minkowski.

4. Asymmetries and Shifts

In the preceding paragraphs only the change of phase and the cessation of oscillations caused by collisions and producing symmetrical broadening were taken into account. Below we shall investigate the influence of the frequencies altered by collisions on the shape of the line.

In general, these frequencies can be noticeable only when the collision time \(\Delta\tau\) is not too small in comparison with the duration of the free path \(\tau\). Otherwise the intensity radiated during \(\Delta\tau\) at the frequencies is too small in comparison with the total intensity, so that the altered frequencies can manifest themselves only through a change of phase. In this paragraph we shall consider only the actions of atoms of a foreign gas. The influences of identical atoms will be considered in the next paragraph, as will the width due to coupling.

If approximate values of the quantities are substituted into London’s formula (12), then, for distances between atoms of the order of \(10^{-7}\,\mathrm{cm}\), one already obtains a measurable frequency shift, for example \(10^9\,\mathrm{sec}^{-1}\). In order for this to be noticeable in the shape of the line, the free path, for an effective diameter of about \(10^{-7}\,\mathrm{cm}\), must be of the same order of magnitude. This occurs at temperatures approximately \(300^\circ\), already at pressures above \(1\) thousand mm. We must therefore expect, at several atmospheres of pressure of a foreign gas, deviations from the line shape due to impact damping.

In order to compute theoretically the correct form of the lines, it would be necessary: 1) to know exactly the character of the action of the atoms, in order to determine \(\omega_0(t)\) and \(A(t)\); 2) to decompose the oscillation thus obtained according to (2). Since this is impossible, we shall content ourselves with a qualitative determination of the frequencies radiated during collisions. We assume that they appear in the spectrum with an intensity proportional to the duration of their existence and with a diffuseness depending on their short emission time (we determine, consequently, only the function \(H(\omega_0)\) instead of \(I(\omega)\), § 1).

For this purpose we shall consider, together with Jablonski\(^{32*}\), Fig. 3, which represents the qualitative course of the potential curves for Na or Ng relative to atoms of a foreign gas.

The polarization actions probably cause here a lowering of the excited term, since in the quadratic Stark effect for both gases it is shifted to the red side. We see from the figure that all collisions whose minimum distance \(r_0\) is not less than the segment \(OA\) produce a frequency shift to the red side. Since passages with large \(r\) are more frequent,

* In the cited work Jablonski assumes that the line shape determined in this way already gives the full collision width. We have seen, however, in § 2 that true impact damping cannot yet be contained here.

than with small \(r\), then, with any noticeable lowering of the potential curve, the red shift is more pronounced than the violet one.

Deviations to the red side can at most be equal to the depth of the polarization depression of the upper potential curve. This depth is of the order of several millivolts. The limit for deviations to the violet side is given, for an emission line, by the value \(\frac{3}{2}kT\); above this the atom cannot rise along the potential curve. For an absorption line the deviation is theoretically not limited, since the upper curve rises earlier than the lower one and can very quickly attain rather large values. Both deviations, however, are always very improbable, since they occur when the atoms are brought very close together; long-wave deviations, if they are more or less noticeable, may in any case be neglected.

Owing to our insufficient knowledge of the potential curves, nothing definite can be said about the origin of the line shape. Therefore below it will only be shown how, with the aid of plausible assumptions, the experimental results may be interpreted. Margenau\({}^{60}\) made attempts to determine the shifts quantitatively by means of a more exact estimate of the polarization forces and arrived at the same results.

The measurements of Minkowski\({}^{50}\) of the \(D\)-lines with the addition of \(\mathrm{N}_2\), \(\mathrm{H}_2\), Ar, N, and certain hydrocarbons require special consideration. He observed asymmetry already at pressures of the foreign gas of \(10\)—\(200\) mm (the vapor’s own pressure \(10^{-3}\)—\(10^{-4}\) mm), at which the ratio \(\frac{\Delta \tau}{\tau}\) is of the order of \(10^{-2}\). This does not contradict our ideas, but is based only on a peculiarity of his method. To avoid the Doppler effect, he uses for absorption measurements very thick layers of gas, which completely absorb the inner part of the spectral line, so that only the edges at a distance of several tenths of an Å from the middle are measured. There, however, the intensity is so small in comparison with the middle of the line that even the addition of \(1/100\) of the total intensity can already be noticed. Absorption in thick layers is therefore extraordinarily sensitive to large frequency shifts having small intensity.

The deviations from the line shape obtained by him, caused by collisional damping, are in full agreement with theoretical expectations. On the red side he found such asymmetries at distances of several tenths of a millivolt from the middle of the line, which precisely coincide with the order of magnitude of the polarization energies. Shifts to the violet side appear only for two foreign gases, \(\mathrm{H}_2\) and He, and are very weak. These gases, according to Table 1, have weak polarizability. Consequently, the excess of the violet shift is understandable. The magnitude of the red shifts—

calculation in the order: Ne, N₂, Ar, hydrocarbons, which, according to Table 1, corresponds to the magnitude of polarizability, with the exception of N₂, which should have acted more strongly than argon. But N₂ is a quenching gas, and it is understandable that its frequency displacement during a collision cannot be fully manifested, since the oscillations cease. The relation between the red displacement and the molecular weight of the foreign gas, established by Minkowski, can be explained only by the already mentioned parallelism between polarizability and molecular weight.

We shall now consider the measurements of Füchtbauer and his collaborators. Here the absorption coefficient was measured directly in thin layers of gas; moreover, at pressures which, in accordance with our expectation, always lay above several atmospheres, asymmetries were observed in the distribution of intensities in broadened lines. Asymmetries, amounting here also to several tenths of an Å, lay on the red side, with the exception of H₂, which either broadens symmetrically or causes a weak displacement toward the violet side. Füchtbauer and Goffin¹⁴ found in cesium at a nitrogen pressure of 3.2 atm a red displacement; Füchtbauer and Schell¹⁵—in the D-lines at 2 atm of nitrogen; likewise Füchtbauer and Meier¹⁶—at 3.5 atm N₂. The most accurate quantitative data are found in the work of Füchtbauer, Joos, and Dinkelacker¹⁹ on the distribution of intensities in the Hg—2537 line upon addition of H₂, N₂, Ar, O₂, and CO₂ at pressures beginning with 10 atm and above. At 10 atm and 500° abs., the mean distance between two atoms is already approximately \(2 \cdot 10^{-7}\) cm. We must therefore assume that the Hg atom, over a large part of its path, is under the influence of polarization forces.

The red displacements do not exhibit a parallel course with polarizability. Probably one can no longer expect even approximate applicability of London’s formula (12) here, since at such small interatomic distances the Hg atom is subjected simultaneously to the influence of several foreign atoms. In addition, in the majority of cases the electron shells strongly penetrate one another and thus produce new frequency displacements which no longer have anything in common with polarizability. The small asymmetry upon addition of H₂ depends, probably, both on the small electron shell and on the small polarizability.

To the same category belongs the asymmetric broadening of the ultraviolet resonance line of helium 585 Å found by Hopfield²⁴. The line is asymmetrically broadened toward the side of high frequencies. Theoretical interpretation was found by Weizel²⁵, who succeeded in showing, from an analysis of the terms arising when two helium atoms approach one another, that the term combining with the ground state leads to repulsion and therefore rises strongly when the atoms approach.

Furthermore, in connection with this, one must mention the asymmetric broadenings of mercury lines already cited in § 1 upon...

with a pressure of the gas, either its own or foreign (usually 20–200 mm Hg, more than 1 atm of foreign gas), observed by Oldenberg in absorption, which in argon even revealed a banded structure. Here, likewise, the resonant level of the Hg atom is shifted in a complicated way by weakly bound foreign atoms. On the contrary, the considerably larger broadenings found by Oldenberg in the fluorescence spectrum of this mixture of gases cannot be based on adiabatic shifts of terms, but on other modes of energy transfer. On the red side of the line they produce a continuous spectrum up to 100 Å in length.

Summing up, one may establish that the observed shifts of lines and asymmetries can in the main be interpreted theoretically without resorting to rather implausible assumptions about the influence of foreign atoms on the terms.

Up to now we have considered chiefly the interaction of two atoms. This is sufficient to establish the shape of a line under the assumptions of low density and a small sphere of action. In particular, the second assumption is no longer fulfilled: 1) when identical atoms act upon one another, since dipole forces decrease only as \(1/r^3\)—the simple consideration of only two atoms in items 1 and 2 is therefore permissible only when it can be shown that the influence of all identical atoms on one another (the width of the coupling) contributes nothing essential; 2) if the atoms carry charges, dipole or quadrupole moments, and the terms of the emitting atom are sensitive to electric fields. Both cases will therefore be considered separately in the following sections.

5. Width due to coupling

In a homogeneous gas, upon absorption or emission of a resonance line, coupling is established between the atoms, based on the fact that the radiation is repeatedly absorbed and again emitted until a stationary state of equilibrium is established, in which all atoms oscillate coherently. In order to calculate this state of oscillation, the gas atoms are replaced by oscillators which, in the case of absorption, are excited by a single light wave, and in the case of emission acquire, owing to thermal excitation, impulses distributed statistically in magnitude and direction. These oscillators are coupled by means of dipole fields.

What factors play a role here is most clearly seen in the simplest example of two coupled oscillators\({}^{20}\). A system of two identical oscillators, the distance between which is small compared with the wavelength \(c/\omega_0\), has two natural oscillations \(\omega_0 \pm \Delta\). One is a parallel oscillation, the other an antiparallel one. The natural damping in the parallel oscillation is twice as large as that of an isolated oscillator; in

antiparallel oscillation, the damping increases still more, and will acquire a quadrupole value. For the radiation itself, therefore, only the parallel oscillation has significance. From this we conclude that, when the number of oscillators is large, the following effects take place: a shift of the line toward the red, since oscillations that have strong damping, and therefore also strong absorption or emission, are displaced toward the side of longer waves; broadening: 1) owing to the different magnitudes of the displacements \(\Delta\) of the individual natural oscillations, 2) owing to their sometimes considerable, increased damping, which must produce a large natural width of the line*.

An exact calculation of the oscillatory state of many oscillators encounters insurmountable difficulties. At best one can make general statements concerning the distribution of the natural oscillations. Holtsmark\(^{23}\) calculated, for example, the mean square \(\overline{\Delta^2}\) of the deviation from the unperturbed frequency of the oscillator. This quantity, however, characterizes the width of the lines only when the form of the line and the intensities of the individual natural oscillations in the radiation are known. Holtsmark, however, arbitrarily assumes that the line form is a Gaussian error curve and that all oscillations have the same intensity. He then obtains, after averaging over all possible distributions of the oscillators in space, for the width:

\[ \delta_{\mathrm{av}} = \sqrt{ \ln 2\,\frac{16\pi}{15}\, \frac{e^2}{2m_0 r_0^3} }\,\sqrt{N}. \]

The dependence on the square root of the density and on the minimum distance of the oscillators \(r_0\), i.e. on the radius of the atom, seems strange. The latter is explained by the incorrectness of the assumption of a Gaussian distribution, in which close passages with a large change of frequency (in reality rare) have too great a weight. (How little the mean square of the frequency shift and the line width are connected is seen from the dispersion distribution, for which \(\Delta^2\) is infinite despite the finite width of the lines.)

Frenkel\(^{13}\), who transferred Holtsmark’s calculation into quantum mechanics, avoids one error by taking into account the intensities of the natural oscillations. However, he too assumes a Gaussian distribution and obtains the same dependence on \(r_0\) and \(N\). Further, L. Schott-Mensing\(^{46}\) showed that a somewhat more accurate averaging over the positions of the atoms gives a direct proportionality between the width and \(N\) (the same author still earlier\(^{45}\), considering the question of the coupling of two atoms from the point of view of the old quantum theory, found, when applying his results to a gas, a proportionality between the width and \(N\)).

* These effects depend essentially on the arrangement of the atoms. With a regular arrangement, for example in a crystalline lattice, they are entirely eliminated by interference. A large ideal crystal has an infinitely narrow absorption line;

All these works* say little about the actual broadening due to coupling as a consequence of the arbitrarily chosen line shape. There is, however, a simple method1 for determining the line shape in absorption. The oscillatory state of a gas under the action of a plane light wave is known from the theory of dispersion. This theory gives, for the complex refractive index \(n=n(1+ik)\) (\(n\) is the ordinary refractive index, \(k\) the absorption coefficient),

\[ \frac{n^{2}-1}{n^{2}+2}=\frac{4\pi}{3}N\alpha , \tag{13} \]

where \(\alpha\) is the polarizability of an individual atom for the frequency \(\omega\). According to the theory of forced oscillations, we have:

\[ N\alpha=\frac{A}{(\omega_{0}-\omega)-i\gamma},\qquad A=\frac{1}{2}\frac{e^{2}N}{\omega_{0}m}\cdot f , \tag{14} \]

where \(\omega_{0}\) is the natural frequency, \(\gamma\) the damping constant of the oscillators represented in place of the atoms, and \(f\) the number of dispersion electrons belonging to the corresponding transition. This formula already includes the interactions of the atoms, since they arise from the dipole influences of the oscillators replacing the atoms. This is precisely what distinguishes the Lorentz–Lorenz formula (13) from the simpler formula

\[ n^{2}-1=4\pi N\alpha , \]

which merely equates the macroscopic polarizability

\[ \frac{\varepsilon-1}{4\pi} \]

to the sum of the atomic polarizabilities (\(\varepsilon=n^{2}\) is the dielectric constant).

The gas absorption coefficient \(nk\) is easily obtained from equation (13), which gives explicitly the form of the broadened absorption line. If we introduce the quantities of zero dimensionality

\[ b=\frac{4\pi A}{\gamma},\qquad \Delta=\frac{\omega-\omega_{0}+\dfrac{4\pi A}{3}}{\gamma} \tag{15} \]

and thus express all frequencies in units of the damping constant \(\gamma\), then one obtains

\[ nk=\sqrt[4]{\frac{(\Delta-b)^{2}+1}{\Delta^{2}+1}}\, \sin\frac{1}{2}\operatorname{arctg}\frac{b}{\Delta(\Delta-b)+1} \tag{16} \]

for the shape of the absorption line, which depends only on the single parameter \(b\). For \(b\ll 1\), (16) takes the form:

\[ nk=\frac{1}{2}\frac{b}{\Delta^{2}+1} =\frac{2\pi A\gamma}{(\omega_{0}-\omega)^{2}+\gamma^{2}} . \]

corresponding to the ordinary dispersion distribution (4) of Chap. II. The condition is satisfied if

\[ \frac{2\pi e^{2}}{\omega_{0}m}\, fN \ll \gamma . \tag{17} \]

\(\gamma\) consists of the natural damping \(\delta_n\), and the collisional damping \(\delta_s\). At very low density \(\delta_n \gg \delta_s\). Substituting \(\delta_n\) from (2) and (8) of Chap. 2, we obtain the condition under which (17) is applicable:

\[ \frac{1}{8\pi^{2}}\,\frac{g_n}{g_n'}\, Z \ll 1, \tag{18} \]

where \(Z\) is the number of atoms in a cube with edge equal to the wavelength. For the \(D_1\)-line, in order for condition (18) to be fulfilled at \(500^\circ\) abs., the pressure must be \(p \ll 0.5\cdot 10^{-2}\) mm. If inequality (18) is not satisfied, then we have merging and broadening of lines, which we must, in essence, regard as the width of the bond.

Thus, according to our views, the width of the bond is completely contained in the course of the absorption coefficient obtained by the Lorentz–Lorenz formula.

Fig. 4.

The form of the absorption lines is determined exclusively by the parameter \(b\). It is noteworthy that \(b\), at high pressures, tends to a limiting value, the same for all gases and independent of temperature, since the collisional damping is caused only by dipole interaction. According to (11), the part of the damping caused by collisions \(\gamma_2\) is, in this case:

\[ \delta_s=\frac{\pi}{2}\,\frac{e^{2}}{m\omega_{0}}\, fN . \]

At higher pressures one may neglect \(\delta_n\) in comparison with \(\delta_s\), and we obtain

\[ b=4 \quad \text{for } \delta_s \gg \delta_n . \]

The corresponding form of the line is shown in Fig. 4; it does not depend on temperature, since (11) holds. The dependence on pressure consists only in a change of scale, namely the unit for \(\gamma\) increases proportionally to the density. The asymmetry of the line shape corresponds exactly to the asymmetry, considered in the preceding section, of the line shape caused by the force fields of neighboring atoms. For comparison, the dotted line is drawn without allowance for the bond. It should be noted that the difference between the two curves near their maximum has no physical significance. \(nk\) is there greater than or equal to 1, so that in a layer of normal thickness the entire region of frequencies is completely

is absorbed. On the contrary, for layers with a thickness comparable with the wavelength, the Lorentz–Lorenz formula is no longer valid. The influence of the coupling can therefore be found only at the edges of the line, where, however, it is not so strongly expressed as at the center. There can therefore be no question of an actual broadening of the coupling independent of the damping due to collisions.

Taking account of the Doppler effect modifies the preceding discussion as follows:

1) At low pressures the Doppler effect diminishes the coupling. The deviation from the usual form of the line (7), broadened by damping and by the Doppler effect, occurs only at densities which no longer satisfy the condition *

\[ \frac{1}{8\pi^{2}}\frac{q_{0}}{q_{n}}Z \ll \frac{\delta_{0}}{\delta_{n}} . \tag{19} \]

On the right stands not 1, as in (18), but the ratio of the Doppler width to the natural width, which for the \(D\)-line at \(500^\circ\) is approximately 200. In order that (7) should hold, it must therefore be \(p \ll 4\ \mathrm{mm}\).

2) The Doppler effect modifies the form of the broadened line in a very complicated way. However, it appears only in the region of frequencies lying, approximately, at sevenfold distances to the right and to the left of \(\omega\). At a greater distance the change is less than 1%. If, therefore, the total width of the line is approximately ten times greater than the Doppler width, then the entire frequency region altered by the Doppler effect lies within the region where \(nh \sim 1\), so that the observed line shape can nevertheless be represented by the course of absorption in Fig. 4, which may be applied under the condition

\[ \delta_{0}<\delta_{s} \quad \text{or, by (12),} \quad \frac{\omega_{0}}{c}\sqrt{\frac{5kT}{\mu}} < \frac{\pi}{2}\,\frac{e^{2}}{m\omega_{0}}\,fN, \tag{20} \]

where \(\mu\) is the molecular weight, and \(m\) is the mass of the electron.

For Na at \(500^\circ\) abs. it must be \(p>20\ \mathrm{mm}\); for Hg at \(300^\circ\), owing to the small value of \(f\), one obtains even \(p>75\ \mathrm{mm}\); if, however, one takes for Hg the value measured by Orthmann and Pringsheim (see item 3), then \(p>8\ \mathrm{mm}\). For densities satisfying neither (19) nor (20), the course of absorption cannot be written explicitly. We may, however, quite well assume that in this intermediate region the coupling does not substantially change the form of the line.

The universal line form in Fig. 4 is, however, correct only insofar as for collisional damping no additional actions are of importance, i.e., insofar as equation (12) is valid. Thus it represents the line form in a pure gas of oscillators. But for Hg and Cs, for example, the width according to Orthmann and Pringsheim is greater than (10), from which one may conclude that there are still other factors causing damping. In this way the value of \(b\) can only be diminished, which still more weakens the action of the coupling.

\[ {}^{*}\ \delta_{0}\ \text{is the Doppler width divided by } \sqrt{2}. \]

The purpose of this subsection was to show that the broadening of absorption lines due to an increase in the gas’s own pressure cannot be based on the bonding of atoms, but occurs because of the optical cross section that is strongly increased owing to resonance.

What has been said is, generally speaking, not valid for the shape of emission lines.* It nevertheless seems admissible that the shape of the emission line does not differ very greatly from that of the absorption line, so that it may be assumed that bonding is not of essential importance in emission either.

6. Broadening due to the Stark effect

The electric fields of the charges, dipole and quadrupole moments of the atoms of a gas shift terms as a consequence of the Stark effect and thus can cause broadening.^63 It is impossible to follow the perturbation of the frequency in time; one can only determine the probability (statistical weight) of the occurrence of a field of a given intensity and the frequency shift associated with it. We therefore determine only \(H(\omega_0)\), and not \(I(\omega)\). The difference between \(H(\omega_0)\) and the actual distribution of intensity \(I(\omega)\) will be of the order of magnitude of the width due to collisions, since the additional width depends essentially on the Fourier decomposition of the strong frequency changes that occur in collisions. If, therefore, the broadening due to collisions is small compared with the width of \(H(\omega_0)\), then \(H(\omega_0)\) may be regarded as determining the line shape.

The question arises whether, in fields that vary strongly in space and time, the term shifts obtained correspond to the value of the Stark effect in constant fields. Since the changes in time are, after all, still slow compared with the frequencies of light, the main problem consists in considering the question of spatial inhomogeneities, whose magnitude is probably of the order of the dimensions of the atom. Since the Stark effect in inhomogeneous fields has not yet been investigated,^64 we shall assume, together with Holtsmark, that for the splitting of terms the decisive importance is the field strength at the center of the atom.

Debye^7 gave a simple method for estimating the mean intensity of the intermolecular field \(\overline{F}\) and, in particular, its dependence on pressure. We distinguish 3 cases: 1) the molecules are charged (ions), 2) they carry dipole moments \(\mu\), 3) they have a quadrupole moment \(\Theta\). The mean field strength to be calculated has the dimension \(e^{1/2}\,\mathrm{cm}^{1/2}\,\mathrm{sec}^{-1}\). In case 1), \(\overline{F}\) must be found from the charge of an \(e\)-ion with dimension \(e^{1/2}\,\mathrm{cm}^{3/2}\,\mathrm{sec}^{-1}\) and from the number \(N\) of gas atoms in cubic centimeters. From these quantities one—

* Except for the case when emission is produced by light (resonance fluorescence). The emission line then has, if the Doppler effect is not taken into account, the same shape as the absorption line.

the field strength can be composed only in the combination \(eN^{2/3}\). We may therefore assume that the mean field strength has the following form:

\[ F=C_1 eN^{2/3}. \]

For case 2) we obtain in the same way

\[ F=C_2\mu N \]

and for quadrupole moments

\[ F=C_3\Theta N^{4/3}. \]

The values of the constants can be given only by an exact calculation, carried out by Holtsmark\(^{25,26,27}\). He obtains:

\[ C_1=2.60,\quad C_2=4.54,\quad C_3=8.26. \]

We give here the results of the calculation: the probability \(W(f)\,df\) of finding in the gas a reduced field strength \(f=\dfrac{F}{\bar F}\) with absolute value \((f)\) is given by

\[ W(f)=\frac{2}{\pi f}\int_0^\infty v\,dv\,\sin vf\,e^{-\left(\frac{v}{f}\right)^{3/k}}, \]

\(k=2,3,4\) for ions, dipoles, and quadrupoles. The integral is expressible explicitly only for dipoles and gives:

\[ W(f)=\frac{4}{\pi}\frac{f^2}{(f^2+1)^2}. \tag{21} \]

A graphical representation of the functions for the two other cases may be found in Holtsmark\(^{27}\).

The form of the broadened line depends on the character of the influence of the field on the frequency. The linear Stark effect gives a symmetric line, the quadratic one a one-sided broadened line. If it is assumed that the field strength \(F\) gives a splitting with distribution of intensities \(I(F,\omega)\), then the resulting line shape is determined by the expression:

\[ J(\omega)=\int_0^\infty I(F,\omega)W(F)\,dF. \tag{22} \]

Below we shall consider only the line shape due to the symmetric Stark effect. Asymmetric broadenings have not been measured experimentally directly.

Holtsmark simply assumes for the linear Stark effect that, under the field \(F\), the line is symmetrically stretched into a broad band, the total amount of light remaining constant. The width of the \(\Delta\)-band must be equal to the maximum Stark splitting \(\Delta=s\cdot F\). Thus a crude averaging over the actual splittings is obtained. From (21) and (22) ...

is obtained then, for broadenings due to dipoles, the dispersion distribution:

\[ J(\omega)=\frac{\delta_{\mathrm{Cl}}}{\pi}\, \frac{1}{(\omega-\omega_0)^2+\delta_{\mathrm{Cl}}^{\,2}}, \]

where the width \(\delta_{\mathrm{Cl}}\) is the maximal Stark splitting in the mean field \(F\):

\[ \delta_{\mathrm{Cl}}=s\cdot F=4.54\cdot s\cdot \mu\cdot N . \]

Ions give an analogous line form, not expressible explicitly mathematically, whose width corresponds to Stark broadening at \(1.25\,\overline F\):

\[ \delta_{\mathrm{Cl}}=3.25\,s\cdot e\cdot N^{\frac{2}{3}} . \]

For quadrupoles:

\[ \delta_{\mathrm{Cl}}=5.52\,s\cdot \Theta\cdot N^{\frac{2}{3}} . \]

For the experimental proof of broadening by means of the Stark effect, hydrogen is best suited, owing to its strong linear Stark effect. The splitting of the \(H_\alpha\) line is equal to \(1\ \text{\AA}\) at 15 electrostatic units. The majority of the lines of other elements do not have a linear Stark effect. Only strongly hydrogen-like terms exhibit linear splitting in light elements. These are terms of higher quantum numbers, beginning, approximately, with \(D\)-terms. Since at normal field strengths effects of higher orders may be neglected in comparison with the linear ones, the lines originating from \(D\)- or higher terms are more strongly broadened than the transitions \(P—S\). Hence the expression “diffuse” series for the series \(D—S\) and \(D—P\).

Holtsmark calculated the broadening of the \(H_\alpha\) line caused by the quadrupole moment of the molecule \(\mathrm{H}_2\) \((3.2\cdot 10^{-26})^{25,26}\), and found the width at different pressures to be in agreement with the measurements of Michelson\({}^{46}\). The broadening is approximately \(0.1\ \text{\AA}\) at a pressure of \(200\ \mathrm{mm}\) and \(300^\circ\) abs. Very considerable broadenings are observed in emission lines in a voltaic arc, where the molecular electric field is excited by ions. By this method Gebert\({}^{21}\) obtained, for the Balmer lines, widths up to \(60\ \text{\AA}\).

Holtsmark and Trumpy\({}^{29}\) investigated in the same way the lines of other elements which also exhibit the linear Stark effect, namely He, Li, Ag, Cu, and Ni. The magnitude of the Stark effect of the lines of these elements was known, so that the authors could draw conclusions about the mean molecular electric field from the measured width. From the widths of the most diverse lines there was always obtained, approximately, the same field strength, lying between 10 and 50 thousand V/cm, at \(10—20\ \text{\AA}\), which corresponds approximately to \(10^{16}\) ions in \(1\ \mathrm{cm}^3\). The line width was of the order of \(1\ \text{\AA}\).

If the molecular-electrical broadening by means of the linear Stark effect is of substantial importance in the cases considered, then one may nevertheless be certain that molecular electric fields have almost no influence on the resonance lines of higher elements. Both the Na \(D\)-lines and the Hg line 2537, for example, have no linear Stark effect, and the quadratic one amounts to \(1{,}06 \cdot 10^{[[unclear: exponent]]} F^2\) Å for Na and \(5{,}5 \cdot 10^{-14} F^2\) and \(1{,}9 \cdot 10^{-14} F^2\) Å for Hg \(\delta\)- and \(\pi\)-components, where \(F\) is the field strength measured in V/cm. This gives, even in a highly compressed gas \((d = 3{,}10^{-5})\), with atomic fields \((F \sim 10^5\ \mathrm{V/cm})\), a broadening and displacement incomparably smaller than the natural line width.

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1. Recently Mrowka[^31] calculated the mean-square displacement \(\Delta^{2}\) for atomic gases H on the basis of wave mechanics, but he takes Gauss’s law as the line-width expression. The frequency shifts here are caused chiefly by exchange forces (Austauschkräfte) between H atoms, so that his result is, for this reason also, inapplicable to a real gas. ↩