DISPERSION IN ELECTRICALLY EXCITED GASES

Rudolf Ladenburg, Princeton *

I. General part. II. Transition probability and lifetime of the energy levels of neon. III. Number of atoms in excited states, in the positive column, statistical equilibrium between different levels, relation to the electron temperature. IV. Influence of strong currents on the dispersion of excited gases and the phenomenon of negative dispersion. V. Statistical equilibrium between the \(s\)- and \(p\)-states of neon.

I. General Part

Near the absorption frequency of a rarefied gas (but outside the absorption region) the refractive index is determined by the expression **:

\[ n - 1 = \frac{e^2}{2\pi m}\cdot \frac{F_{kj}}{\nu_{kj}^2 - \nu^2}. \]

Therefore measurements of anomalous dispersion in this region make it possible to calculate the quantity \(F_{kj}\), which, according to quantum theory, is equal to:

\[ F_{kj} = N_j \cdot A_{kj}\frac{g_k}{g_j}\frac{mc^2}{8\pi^2 e^2 \nu_{kj}^2}\left(1-\frac{N_k}{N_j}\frac{g_j}{g_k}\right) = N_j \cdot f_{kj}(1-Q_{kj}). \tag{1} \]

In this equation \(j\) refers to the lower, \(k\) to the upper level of the spectral line, \(N_j\) and \(N_k\) are the numbers of atoms in \(1\ \mathrm{cm}^3\) at these levels, \(g_j\) and \(g_k\) are the statistical weights, \(\nu_{kj}\) is the frequency of this line, and \(A_{kj}\) is the transition probability. Hence, if the fraction

\[ Q_{kj}=\frac{N_k g_j}{N_j g_k} \tag{2} \]

is small in comparison with unity, and \(N_j\) is known, as is usually the case for a gas or vapor in the normal state at known temperature and pressure, then from measurements of anomalous dispersion one can calculate the so-called “number of dispersion electrons”

\[ f_{kj}=A_{kj}\cdot \left(\frac{g_k}{g_j}\right)\frac{mc^3}{8\pi^2 e^2 \nu_{kj}^2}. \tag{3} \]

* Reviews of Modern Phys. 5, No. 4, 243, 1933; translated by S. M. Levi.
** See, for example, the review article by Korff and Breit on dispersion.

and the transition probability \(A\) of this line. Measurements near the resonance lines of Na and Hg have confirmed\({}^{2,3}\) these relations, since \(\frac{1}{A}\) coincides with the lifetime \(T\) of the resonance level, determined directly, and quantum-mechanical calculations of the transition probability for the \(D\)-lines of sodium\({}^{4}\) confirm (within the limits of experimental error) the measured value of \(A\).

In excited gases or vapors, where the lower level of the spectral lines is always excited and where the number of atoms \(N_j\) in this state is unknown, determination of the absolute value of \(A_{jk}\) by measuring anomalous dispersion alone is impossible. But it is possible to determine the relative values \(A_{kj}\) for different lines having one and the same lower level (and therefore the same number \(N_j\)) and different upper levels. Experiments were carried out with the Balmer lines of hydrogen\({}^{5}\), and the ratio \(A\) thus determined for \(H_\alpha\) and \(H_\beta\) is in agreement with that calculated quantum-mechanically. Further experiments of the same kind were made with He, Ne, and Hg in the positive column of these gases\({}^{6}\). In particular, in the case of neon excited by a constant current, the relative values of \(f\) and \(A\) were thus determined for a large number of lines.

Such determinations are possible, as we saw in the discussion of equation (1), only when \(Q\) [equation (2)] is small in comparison with unity. This condition can be tested by means of the “reversal temperature” method, usually used for determining flame temperature. Hedwig Kohn\({}^{7}\) showed that this method can be applied to measuring the relative number of excited atoms in a luminous gas. Light from a source giving a continuous spectrum—a black body, a carbon arc, or an incandescent lamp—passes through the luminous gas and is analyzed by means of a spectrograph of large dispersion. In the resulting spectrum the spectral lines of the gas will be dark or bright depending on whether the black temperature of the source is below or above the critical temperature, called the reversal temperature, at which the lines just disappear against the continuous background. If there is statistical equilibrium between the excited states of the gas corresponding to the ordinary temperature, all lines are reversed at one and the same temperature, but usually this is not so. Each line has its own characteristic reversal temperature. The slit width, the amount of absorption, and the resolving power of the spectrograph affect the accuracy of the determination, but not the value of \(T_r\).

The only assumption is that the layer of luminous gas through which the light passes is homogeneous, and that the ratio \(\frac{N_k}{N_j}\) of the number of atoms in both states has one and the same value along the entire light path. This ratio, or

the ratio \(Q\) from (2) is expressed, as Hedwig Kohn\(^7\) has shown, by an equation similar to the Boltzmann equation:

\[ Q=e^{-\frac{\varepsilon_k-\varepsilon_j}{kT_r}}, \tag{4} \]

because the disappearance of a line against the background of a continuous spectrum means that an arbitrary volume of gas absorbs exactly as much radiant energy as it itself emits; this is expressed by the equation:

\[ N_k A_{kj}h\nu_{kj}=(N_jB_{jk}-N_kB_{kj})u_{kj}h\nu_{kj}, \]

where \(u_{kj}\) is the radiation density of a light beam of frequency \(\nu_{kj}\), and \(A\) and \(B\) are Einstein coefficients of the probabilities of the emission and absorption transition\(^8\). If \(T_r\) is the black temperature of the luminous flux, then \(u_{kj}\) is determined from Planck’s formula:

\[ u_{kj}=\frac{8\pi h\nu^3}{c^3}\frac{1}{e^{\frac{h\nu}{kT_r}}-1}. \tag{5} \]

Hence equation (4) follows directly.

II. TRANSITION PROBABILITY AND LIFETIME OF THE ENERGY LEVELS OF NEON

In what follows we shall deal mainly with the red-yellow lines of neon, corresponding to \(s—p\) transitions. As is seen from Fig. 1, the first excited levels of neon are four \(s\)-terms: \(s_5\), \(s_4\), \(s_3\), and \(s_2\) (\(^3P_2\), \(^3P_1\), \(^3P_0\), and \(^1P_1\)), which are located close to one another; of these, \(s_5\) and \(s_3\) are metastable, whereas \(s_4\) and \(s_2\) combine with the normal level \(p_0\) and give the ultraviolet resonance lines 743 and 763 Å. Combinations of these four \(s\)-terms with the ten terms \(p_{10}\ldots p_1\) (\(^1S_0\), \(^3S_1\), \(^1P_0\), \(^3P_{012}\), \(^1D_{23}\), \(^3D_{123}\)) form about thirty spectral lines giving the red-yellow light of neon advertising lamps. The anomalous dispersion near these lines was thoroughly studied by Ladenburg and his collaborators. They excited the gas in tubes 50 and 80 cm long and 8 and 10 mm in diameter by means of a direct-current machine with a strength of 0.1–700 mA. The electrodes were large nickel cylinders placed in

Fig. 1. Scheme of the terms of neon.

side tubes. The long positive column of luminous gas (“plasma”) appears quite homogeneous, but if it is examined with the aid of a rotating mirror, it reveals so-called moving striations (“laufende Schichten”). To obtain reliable results the tubes and electrodes must be very carefully degassed, and the gas under investigation (helium, neon, argon, etc.) must be very pure, in particular free from hydrogen. The light of an arc lamp was divided by one of the plates of a Jamin interferometer^10 into two beams; one of them passed through the long luminous column, the other through an evacuated tube of the same length. The two coherent beams were combined into one by means of the second plate of the interferometer; the interference fringes thus formed were focused on the slit of a spectrograph with a large plane Michelson grating, giving in the third order a dispersion of about 3 Å per 1 mm. The continuous spectrum, on which the bright lines of the luminous gas were superposed, was traversed by horizontal interference fringes (Fig. 2), or by inclined ones if a plane-parallel plate was introduced into the beam passing through the evacuated tube (“compensating plate”).

Fig. 2. Photographs of anomalous dispersion near some lines of neon.

On both sides of the spectral line the horizontal interference fringes were bent in accordance with the rapid change of the refractive index (Puccianti method), while the inclined fringes revealed “hooks” (Rozhdestvenskii method). The distance between two hooks, in wavelengths, is in a simple dependence on the value \(F\) of the spectral line [equation (1)]. Near some neon lines this phenomenon is already noticeable starting from 1 mA, because the number of atoms in the metastable states \(s_5\) and \(s_3\) is quite large even at weak currents. In Fig. 2 several photographs are given of anomalous dispersion and of “hooks” near the lines 6402 \((s_5 p_9)\), 6383 \((s_4 p_7)\), and 6334 \((s_5 p_8)\) at a current of about 50 mA. In order for the phenomenon of anomalous dispersion to be detected, a considerable number of atoms must be present on the lower term of the spectral line. This is the reason why this phenomenon is noticeable only in comparatively few gases. The circumstance that,

that, as is evident from Fig. 2, the line \(s_4 p_7\), which has the non-metastable lower term \(s_4\), gives anomalous dispersion, can be explained by energy exchange between atoms in the states \(s_5\), \(s_4\), and \(s_3\), because the average current energy at room temperature \((0.04\ \mathrm{V})\) is of the same order as the energy difference of these terms. Then the number of atoms at the level \(s_4\), although it decreases owing to the radiation of the line 743, is nevertheless constantly replenished by strong reabsorption of this radiation \(^{11}\). Therefore it is of the same order as the number of atoms at the metastable levels \(s_5\) and \(s_3\), and for this reason the term \(s_4\) is called “quasi-metastable.” The energy of the term \(s_2\) is \(0.23\ \mathrm{V}\) higher than that of \(s_5\), and it behaves somewhat differently (see Fig. 6, part III). However, at high current intensities \(^{12}\) anomalous dispersion can also be measured for lines terminating in this term.

Simultaneous measurements of the reversal temperature of these lines with a carbon arc as the light source show \(^{13}\) that the ratio \(Q\) in (1) is small in comparison with unity so long as the current does not exceed \(100\ \mathrm{mA}\). The statistical weight for the \(s\)- and \(p\)-terms is known, and the relative values \(F_{kj}\) of different neon lines *, terminating in the same \(s\)-term and having different \(p\)-terms, give the relative \(f_{sp}\) and, consequently, the relative \(A_{sp}\)-values [equation (3)].

Table 1 therefore contains the relative values \(A\), determined in this way, for various \(sp\)-lines **, arranged according to their common lower term \(s_5 \ldots s_2\). The table also contains the values \(f\) (see below).

Combining these experiments on anomalous dispersion with measurements of the intensity ratio of lines arising from one and the same upper, but from different lower, terms, one can also calculate the relative \(A\)- and \(f\)-values of these lines with different lower terms, since the intensity of the spectral line \(k—j\) of an infinitely thin layer of gas (its “true” intensity) is:

\[ H_{kj} = N_k \cdot A_{kj} \cdot h\nu_{jk}. \tag{6} \]

In general, the intensity of the light of a luminous gas of finite layer thickness \(l\) will not be \(l\) times greater, owing to the influence of self-absorption of light in the gas; this is especially the case for \(s—p\) lines in excited neon, where the number of atoms at the \(s\)-levels (and the resulting self-absorption) is large. Indeed, the absorption of these lines is so great that it is almost impossible to avoid it completely. Dorgelo \(^{14}\) attempted to determine the “true” intensity ratios of gas lines by using small amounts of hydrogen in a neon discharge \(^{15}\). Meissner and Dorgelo showed that hydrogen destroys the metastable neon atoms, and therefore also the absorption and anomalous dispersion. But

* Measurements for different lines were, of course, made at the same current.

** The largest value of \(A\) for a given level is taken as unity.

In the experiments described, the gas was purified of hydrogen. Ladenburg[^16] gave another method for calculating the “true” intensities, taking absorption into account and introducing the corresponding corrections into the measured brightnesses. This is possible, of course, only when the “intensity distribution” of the coefficient of absorption and emission is known. In the positive column of a neon discharge this intensity distribution is determined by the Doppler effect[^16] (cf. Fig. 3,

Fig. 3. Distribution of intensity in the neon line 6304; measurement by means of a Perot–Fabry standard; xxx—measurements in a layer of length 80 cm with a standard 15 mm thick; ...—measurements in a layer of 0.8 cm with a standard 15 mm; 000—measurements in a layer of 0.8 with a standard 30 mm. The dashed and solid curves are calculated for layers of 80 and 0.8 cm by the formula:

\[ H = 1 - e^{-Ce^{-\beta^2(v_0-v)^2}}, \]

where

\[ C=\frac{2e^2Fl}{mv_0q}, \qquad \beta^2=\frac{4c^2}{\pi v_0^2 q^2} \]

and

\[ F=0.6\cdot 10'', \qquad q=5.05\cdot 10^4 \frac{\text{cm}}{\text{sec}}. \]

taken from the work of Ladenburg–Levy) and the authors showed that the intensity of spectral lines in a luminous gas of finite layer thickness \(l\) is equal to the intensity in an infinitely thin layer, multiplied by the expression \(l\cdot S\), where

\[ S = 1 - \frac{C}{2\sqrt{2}} + \frac{C}{2\cdot 3\cdot \sqrt{3}} - \cdots (-1)^n \frac{C^n}{(n+1)!\sqrt{n+1}} + \cdots \]

and

\[ C=\frac{2Fe^2l}{v_0mq}, \]

where \(q\) denotes the mean velocity of the atoms. Values of \(S\) are given in the table in the work of Ladenburg–Levy (see 9, VI) for values of \(C\) between 0.1 and 1000. The values of \(F\) for neon lines.

TABLE 1

Relative values of \(A\) and \(f\) for the \(s\)-\(p\) lines of neon

were known from measurements of anomalous dispersion. By means of this method the ratios of the intensities of the lines 6678 \((s_2p_4)\) to 5944 \((s_5p_4)\) and 6598 \((s_2p_2)\) to 5882 \((s_5p_2)\) were determined, and then also the ratios \(A\). In this way the values of \(A\) for the \(s_2\) lines relative to the \(s_5\) lines were calculated.^16 In a similar manner the values of \(A\) for the \(s_4\) and \(s_3\) lines relative to the \(s_5\) lines were calculated, using the intensity ratios from Dorgelo’s measurements, given in Table 2.

TABLE 2

The results of these calculations are given in Tables 1 and 2. Table 1 contains the values of \(f\) for different lines, referred to the value of \(f\) for the line 6402, which is taken equal to 0,5. In Table 3^36 one can find the values of \(A\) of the neon lines, referred to the \(A\)-line 6402 \((s_5p_9)\), corresponding to \(f = 0,5\). The method for calculating the absolute values of \(f\) and \(A\) is given in Section III. The abbreviation (o. m.) in tab-

only means that the value \(A\) of this line is so small that the anomalous dispersion of this line could not be measured; (?) means that the value was not measured and is unknown; (0) means that this combination does not exist, i.e. that \(A\) is practically equal to zero. The largest \(A\)’s are possessed by the resonance line and the “quasi-resonance” lines \(s_4p_3\) and \(s_2p_1\). Their upper levels each combine with one more lower level, but the corresponding transition probabilities are so small that they cannot be measured. In addition, the table gives the sums \(\Sigma A\) of all lines originating from one and the same upper level. The reciprocal of this sum is the lifetime \(T\) of the upper state \(p_1, p_2 \ldots p_{10}\). It is seen from the table that \(T\) for different \(p\)-levels varies within the limits from \(0.8\cdot10^{-8}\) to \(2\cdot10^{-8}\). The lifetime increases systematically as the energy of the level increases. Small deviations from this lie within the limits of the errors of measurement (about 10–15%).

TABLE 3

Values of the quantities \(A\) for neon lines, referred to the absolute value \(\mathring{A}\) for \(\lambda\ 6402\) \((s_5p_g)\), corresponding to \(f=0.5\)

III. The number of atoms in excited states in the positive column, statistical equilibrium between different levels, and the relation to the electron temperature

The same measurements of the anomalous dispersion and of the “true” intensities of the neon lines make it possible to calculate the relative number of atoms at the various excited levels of neon. This follows from the equations:

\[ F_{kj}=N_j i_{kj}=N_j A_{kj}\,\frac{g_k}{g_j}\cdot \frac{mc^3}{8\pi e^2\nu_{kj}^{\,2}} \tag{1a} \]

and

\[ H_{kj}=N_k\cdot A_{kj}\,h\nu_{kj}. \tag{6} \]

The first of these equations contains the assumption that the number of atoms in the upper state is so small that the ratio \(Q=\dfrac{N_k g_j}{N_j g_k}\) may be neglected (see Section I). If we compare two lines with the same upper level (let it be \(p_2\)) and different lower levels (for example \(s_5\) and \(s_3\)), then from the ratio of the intensities of these lines we obtain the ratio of their \(A\)-values and, from measurements of their anomalous dispersion, the ratio of their \(F\)’s; this gives us the ratio \(N_{s_5}:N_{s_3}\). Whereas the ratio of the “true” intensities of two lines with the same upper level does not depend on the current strength, the values of \(F\) and their ratios for lines with different lower levels depend strongly on the current strength in the positive column. Fig. 4 gives the variation of \(F\)* for certain neon lines for currents between 5 and 50 mA.

In order to use the results of measurement to determine the number of atoms in the excited terms, we must clarify how the anomalous dispersion and the quantity \(F\) depend on the current strength. The increase of \(F\) with current strength might, according to (1a), be caused either by an increase in the number of atoms in the lower level of this line or by an increase in the quantity \(A\), the transition probability. This quantity is a property of the atom, and variation of this quantity with current seems highly improbable. Theoretically, very strong electric fields can affect the quantity \(A\), but in the experiments under discussion here such a phenomenon could not occur, because if all the \(F\)’s for the different lines of Fig. 4 that have one and the same lower level \(s_5\) are reduced to one scale, the reduced curves coincide (Fig. 5); consequently, with current it is the number \(N_{s_5}\) that changes (common to all these lines). We know that, as the current strength increases, the number of electrons increases, and that these electrons, if they are sufficiently fast, excite the gas atoms by collisions of the “first kind,” either directly or through \(p\)-levels, which are excited much more easily than the \(s\)-levels \(^{18}\). On the other hand, collisions of an electron with an excited atom can destroy the excitation through impacts of the “second kind.” An excited atom may also be destroyed by collision with the wall of the vessel or with other atoms, or else as a result of spontaneous emission; but, on the other hand, as has already been said, reabsorption of radiation can raise atoms from the normal level to an excited one, provided only that it is not metastable. In the state of equilibrium the number of atoms rising to a given level must be equal to the number of excited—

* In German works, instead of our \(F\), the German letter \(\mathfrak{R}\) is used as the ordinate in Figs. 4, 5.

atoms at this level which during the same time are destroyed. As a first approximation it may be assumed that the number of atoms excited and destroyed owing to collisions with electrons is proportional to the current. But at large currents one may also take into account higher powers of the current, in view of the fact that, as a result of electron impact, interaction arises between atoms. Therefore, in the state of equilibrium, the number of atoms at a certain level \(j\) may be approximately represented by the expression 19:

\[ N_j=\frac{AI+BI^2}{CI+DI^2+E}, \tag{7} \]

where \(I\) is the current strength, and \(A\), \(B\), \(C\), \(D\), and \(E\) are positive constants, independent of the current, characterizing the probability of excitation

Fig. 4. Change in the value of \(F\) for different \(s_5\)-lines with increasing current.

Fig. 5. Values of \(F\) for different \(s_5\)-lines as a function of current. The curves are reduced to one scale.

and destruction of the excited state of atoms \(j\) in the various processes mentioned above. The term \(E\), independent of the current, determines the destruction of excitation as a result of spontaneous radiation and of collision with the walls of the tube or with other atoms.

At small currents the quadratic terms may be neglected, and we obtain the simple equality:

\[ N_j=\frac{\alpha I}{\beta I}+1, \tag{8} \]

where \(\alpha=\dfrac{A}{E}\) and \(\beta=\dfrac{C}{E}\), so that \(\dfrac{1}{N_j}\) is a linear function of \(\dfrac{1}{I}\). This dependence was checked experimentally on various neon lines (9, 11), and it was found that it is approximately valid for currents between 10 and 60 mA*.

* Deviations from linear dependence at very small currents are still not sufficiently satisfactorily explained.

Fig. 6 gives the increase in the number of atoms at the different \(s\)-levels at a current of 60 mA and a pressure of 1.3 mm. The curves are obtained from the quantity \(F\) for different lines having the same lower level and capable of being expressed by means of one and the same curve, i.e. by the same function \(I\).

We see that the curves for \(s_5\), \(s_4\), and \(s_3\) are very similar to one another in form, but that the curve for \(s_2\) behaves quite differently, in accordance with our expectations (Section II); the curves \(s_5\), \(s_4\), and \(s_3\) at first rise rapidly with the current, then more slowly, and at 50 mA approach “saturation,” when the number of atoms does not increase with increasing current. This phenomenon is more clearly expressed at a higher gas pressure (9, II Fig. 6) and indicates that in (8) the term \(\beta I\) becomes large in comparison with unity, i.e. the number

Fig. 6. Number of atoms at the \(s\)-levels as a function of current at a pressure of 1.3 mm.

of collisions with electrons that destroy the excitation is large in comparison with the number of collisions with atoms and with the walls of the vessel, and also in relation to the number of atoms passing to the lower level as a result of spontaneous radiation. This is undoubtedly something like a statistical equilibrium between the colliding electrons and the excited atoms.

This approach to statistical equilibrium is also seen in the aggregate of the different \(s\)-states. At a current of about 50 mA the ratio of the numbers of atoms in the states \(s_5\), \(s_4\), and \(s_3\) is \(100 : 52.5 : 17.5\) (Fig. 6). The ratio of their statistical weights is \(5 : 3 : 1\); if Boltzmann’s law is applied at a high “specific temperature,” for example at \(10\,000^\circ\), the ratio obtained is \(100 : 56 : 18\). The number of atoms in the state \(s_2\) at the currents used is far from the equilibrium state (15 instead of 46), and, as is seen from Fig. 6, is still far from saturation. The reason for this is that the \(s_2\)-level

lies 0.2 V above the three other levels, and the mean energy of the atoms at room temperature is insufficient for energy exchange; moreover, the spontaneous destruction of the level \(s\) owing to radiation is very large.*

Since the \(s\) levels lie too close to one another, measurements of the number of atoms are not sufficiently accurate for testing Boltzmann’s law and for computing in this way the “specific temperature.” Under certain conditions, however, general theoretical considerations reveal the existence of statistical equilibrium between excited atoms and electrons and the validity of Boltzmann’s law as applied to excited atoms. For this it is necessary only to assume a Maxwellian distribution of electron velocities in the plasma \(^{20,21}\), and also that excitation of a state, as well as its destruction, occurs predominantly as a result of electron impact, i.e., that electron impacts outweigh other processes of excitation and annihilation of excitation, in particular the action of spontaneous radiation and collisions with the walls of the vessel or with other atoms.

This essential result follows directly \(^{22}\), if one turns to the arguments of Klein and Rosseland \(^{23}\). On the basis of the Maxwellian distribution, the number of electrons with energy between \(E\) and \(E+dE\) is equal to:

\[ n(E)\,dE = C\cdot \sqrt{E}\, e^{-\frac{E}{kT_e}}\cdot dE, \]

where \(T_e\) denotes the “electron temperature.” If \(E_j\) and \(E_k\) are the energies of two states of the atom, then the level with the greater energy \(k\) is excited only by electrons with energy \(E''>E_k-E_j\); the corresponding probability will be \(s_{jk}(E'')\). The energy of the electron after excitation is \(E'=E''-(E_k-E_j)\). In a collision of the second kind such an electron destroys the excitation with probability \(s_{kj}(E')\), taking away the energy \(E_k-E_j\). According to Klein and Rosseland \(^{23}\), there exists between the two probabilities the well-known relation:

\[ g_j\sqrt{E''}\,s_{jk}(E'') = g_k\sqrt{E'}\,s_{kj}(E'); \tag{9} \]

\(g_k\) and \(g_j\) denote, as before, the statistical weights of the two atomic levels. Klein and Rosseland derive this relation under the assumption of temperature equilibrium between atoms and electrons. But the probabilities \(s\), just like the weights \(g\), are properties of atoms and electrons and do not depend on the existence of statistical equilibrium and do not change with external conditions. The same is true for the transition probabilities. Therefore we may apply relation (9) to processes in a plasma without the special assumption of statistical equilibrium between atoms and electro-

* The probability of transition from it is approximately 13 times greater than from \(s_4\), according to the most recent unpublished calculations of G. Chartley.

by us. But we must make the essential assumption that the current density in the plasma is very large, i.e., that the number of excited atoms destroyed per unit time by electron impact of the second kind is equal to the number of excitations by electron impact of the first kind in the same time. Of course, the excitation is also destroyed by other, already mentioned, means, but we shall assume the number of these processes to be very small compared with the number of collisions of the second kind with electrons. Our assumption will therefore be valid only for very strong currents. Equating to one another the number of impacts of the first and second kind, we obtain:

\[ N_j \int_{E_k-E_j}^{\infty} s_{jk}(E'') e^{-\frac{E''}{kT_e}} \cdot \sqrt{E''}\, dE'' = \]

\[ = N_k \int_{0}^{\infty} s_{kj}(E') e^{-\frac{E'}{kT_e}} \cdot \sqrt{E'}\, dE' . \]

Substituting on the left, instead of the variable \(E''\): \(E' = E'' - (E_k - E_j)\), and replacing the probability \(s_{jk}(E'')\) by \(s_{kj}(E')\) according to (9), we obtain:

\[ N_j \int_{0}^{\infty} s_{kj}(E') e^{-\frac{E'}{kT_e}} \cdot e^{-\frac{E_k-E_j}{kT_e}} \cdot \frac{g_k}{g_j}\sqrt{E'}\, dE' = \]

\[ = N_k \int_{0}^{\infty} s_{kj}(E') e^{-\frac{E'}{kT_e}} \sqrt{E'}\, dE' \]

and hence

\[ \frac{N_k}{N_j}=\frac{g_k}{g_j}\, e^{-\frac{E_k-E_j}{kT_e}}, \tag{10} \]

i.e., at strong currents in the plasma the numbers of atoms in two atomic states correspond to statistical equilibrium at the electron temperature. This result can be checked experimentally, because from experiments with anomalous dispersion one can calculate the ratio \(\frac{N_k}{N_j}\).

As was indicated above, the quantity \(F\), determined from anomalous dispersion, gives only the product \(N_j f_{kj}\). The absolute value of \(f\), however, can be approximately calculated from the Thomas–Reiche–Kuhn \(f\)-sum rule \(^{25}\):

\[ \sum_a f_a - \sum_e f_e = Z, \tag{11} \]

where \(Z\) denotes the number of electrons in the atom.

The first sum (with index \(a\)) represents all possible transitions with absorption from the state under consideration; the second sum (with index \(e\)) refers to all possible transitions from the given state associated with emission. For the metastable state of neon \(s_5\) the second sum is zero. Multiplying (11) by \(N_{s_5}\), we obtain:

\[ \sum_a N_{s_5} f_{s_5 a}=\sum_k F_{s_5 k}=Z\cdot N_{s_5}. \]

It should be taken into account that only one of the outer electrons in the neon atom participates in the absorption of lines from the \(s_5\)-state, but that the \(f\)-sum of the inner electrons may be less than their number \(^{26}\), so that the \(f\)-sum of the single outer electron may be greater than unity, and may be equal to two. Further uncertainty arises from the fact that we know the value of \(f\) only for combinations of \(s_5\) with \(p\)-levels, but not for higher transitions, and also not for the adjoining continuum \(^*\). Taking this uncertainty into account \(^{28}\), we obtain for \(N_{s_5}\), at a current of 100 mA and at 1 mm pressure in a tube of diameter 0.8 cm, the limits \(2.6\) and \(13.8\cdot 10^{12}\). The absolute value \(f_{6420}\) lies between 0.85 and 0.21 (mean value \(0.5 \pm 0.3\), see Table 1 \({}^{**}\)), and from the known number of neon atoms in the normal state we obtain the ratio

\[ Q=\frac{N_{s_5}g_{p_0}}{N_{p_0}g_{s_5}} \]

within the limits \(3\cdot 10^{-5}\) and \(16\cdot 10^{-5}\).

The corresponding “specific temperature,” determined by equation (10), is found to be \(20000^\circ \pm 10\%\); for \(s_4\)- and \(s_3\)-atoms this temperature is \(300^\circ\) lower, i.e. the same within the limits of experimental error. At stronger currents (see the following section) the specific temperature is approximately \(400^\circ\) lower. We shall see that at stronger currents statistical equilibrium is also attained for higher levels, with correspondingly similar temperature.

The electron temperature was determined by Seligер and Girchert \(^{21}\) in the positive column of a low-voltage arc in neon. They used tubes of diameter 40 and 20 mm, without obtaining any noticeable difference.

Since the measurements of anomalous dispersion were made only with a tube 8 mm in diameter, the comparison is not entirely reliable. The data of Seligер and Girchert are given in Table 4 \({}^{***}\) (accuracy 10%). With

\(^*\) We know only that the value of \(f\) for the second line of the series \(s_5-p_9\) (3473) is very small, certainly less than \(\frac{1}{7}\), than \(f\) for 6402 (see Agathe Carst, Ztschr. Physik 48, 59, 1928).

\({}^{**}\) The sum of the values of \(f\) for the four \(s\)-levels in Table 1 is incomplete, since there exist transitions with emission from \(s_4\) and \(s_2\) to the normal level, for which \(f\) cannot be considered equal to zero.

\({}^{***}\) The authors give quantities in volts and assume that 1 V corresponds to \(7750^\circ\) \(\left(eV=\frac{3kT}{2}\right)\).

TABLE 4

Electron temperature in neon as a function of pressure and current strength, according to the measurements of Selig and Gierert

At a pressure of 1 mm the electron temperature at 100 mA is equal to 26,600°, and at 300 mA about 22,000°. These values are in satisfactory agreement with the electron temperature \(20\,200 \pm 10\%\), calculated from measurements of the number of excited atoms at a pressure of 1 mm.*

Further, with increasing pressure the specific temperature of the excited atoms decreases similarly to the electron temperature and falls at 9 mm to 15,000°, whereas the electron temperature (according to Table 4) falls to approximately 16,000°. Experiments show that the statistical equilibrium of excited atoms at high pressures is attained at lower current strengths. This is not difficult to understand, since with increasing pressure the number of collisions of excited atoms with the walls, which destroy the excitation, decreases, because their diffusion decreases; on the other hand, the number of slow electrons responsible for collisions of the second kind with excited atoms increases with increasing pressure at constant current. Both these phenomena act in one and the same direction and produce the observed effect.

Moler \(^{30}\) investigated the positive column of a cesium discharge; he determined the electron temperature and, at the same time, the number of excited atoms. In agreement with our results in a neon discharge he found that, with increasing current, the number of excited atoms approaches an equilibrium value corresponding to the Boltzmann distribution at the electron temperature. He then derives an analogous result for the positive column of mercury on the basis of Killian’s experiments \(^{31}\). It therefore seems probable that the relation between the electron temperature and the specific temperature, and the approach to statistical equilibrium between electrons and excited atoms as the current strength increases, is in fact a general property of monatomic gases, as is evident from the general theoretical considerations given above.

* M. Druyvesteyn \(^{29}\) gives values considerably larger than those of Selig and Gierert; this discrepancy has not yet been explained.

IV. The influence of strong currents on the dispersion of excited gases and the phenomenon of negative dispersion

If the current exciting and destroying the excitation of neon atoms reaches 100 mA, then the simple equation (8) no longer holds, since the quadratic terms (7) become appreciable and, in addition, the number of atoms in the excited levels \(p_1 \ldots p_{10}\) increases, so that the expression

\[ 1-Q_{kj}=1-\frac{N_k g_j}{N_j g_k} \]

in Eq. (1) for \(F\) cannot be omitted. Experiments with very large current densities, as was shown by Kopfermann and

Fig. 7. Change of \(F\) for the \(s_5\) lines at strong currents.

Ladenburg \(^{32}\), gave the first experimental proof of the existence of negative terms in the dispersion expression, i.e. of “negative dispersion.”

The results of these experiments are given in Fig. 7, which gives the values of \(F\) for different neon lines having the common lower level \(s_5\), at currents up to 700 mA.

For such strong currents, a narrow tube (0.8 cm in diameter) is best made of quartz and cooled with running water. We see (Fig. 7) that \(F\) for the lines 6402 (\(s_5p_9\)), 6334 (\(s_5p_8\)), 6143 (\(s_5p_6\)), 5945 (\(s_5p_4\)) increases appreciably only up to approximately 60 mA, between 60 and 100 mA increases only slowly, and at stronger currents decreases.

If the values of \(F\) for different lines are again reduced to one and the same scale so that they coincide at small currents (approximately 10 mA) (Fig. 8), then the reduced values \(F\) coincide, as already mentioned, within the experimental errors up to 60 mA. But at stronger currents, especially above 100 mA, the lines behave differently; the reduced values of \(F\) no longer coinci-

give, but differ noticeably from one another. \(F\) falls faster for large wavelengths, and more slowly for short wavelengths, i.e., the smaller the difference between the energies of the common lower level \(s_5\) and the various upper states \(p_k\), the more strongly \(F\) falls. This is precisely what should be expected as a result of negative dispersion and the influence of the expression \(Q_{kj}\) in formula (1) (\(j\) corresponds to the term \(s_5\), and \(k\)—to \(p_4\), \(p_6\), \(p_8\), and \(p_9\)): the stronger the current, the greater the number of atoms on the upper level \(p_k\); at the same time the number of atoms on the lower level \(s_5\) does not increase, but rather decreases at currents somewhat above 100 mA, as is shown by investigation (8) (see the following section). Therefore, with increasing current the ratio \(\frac{N_k}{N_j}\) increases. Further, the number of \(N_k\)-atoms on the various upper levels (i.e., \(p_{10}\ldots p_2\)) is the greater, the lower the energy

Fig. 8. \(F\) from Fig. 7, reduced to one scale.

of this level and the greater its statistical weight, which is the result of statistical equilibrium, as we shall see in Section V.

The difference in the decrease of \(\bar F\) for different lines is essential. This difference cannot be explained by the decrease in the number of atoms on the level \(s_5\), since it is common to all the lines. Likewise, the change of the magnitude \(A\) with current, different for different lines, is not the cause of this difference; the following observations\(^{33}\) show beyond doubt that the number of atoms on the upper levels \(p_4\), \(p_6\), \(p_8\), and \(p_9\) increases with current to such an extent that the expression \(Q\) in (1) for \(F\) can no longer be omitted.

First, measurements of the absorption of the line \(p_{10}—s_1\) (7059) showed that noticeable absorption begins at approximately about 50 mA and increases with current. Second, the “reversal temperature” of the neon lines was measured, i.e., the black temperature of the light beam at which the neon lines just disappear against the background of the continuous spectrum; the measurements were made using a carbon arc as the light source. This reversal temperature was below 4000° for currents of 50 mA and increased strongly with increasing current. According to Hedwig Kohn this means that the value of \(Q\) for our

lines is less than 0.01 for 50 mA, but increases with increasing current. Thirdly, the increase in the intensity of the \(k—j\) lines at currents above 50 mA shows that the number of \(N_k\)-atoms continues to grow.

All these diverse experiments leave no doubt that the number of atoms at the \(p\) levels increases with the current, and does so so rapidly that the ratio \(Q\) at currents above 100 mA reaches a considerable value. Further, the experiments show (Fig. 8) the influence of the negative term of the dispersion formula. This “negative dispersion” corresponds to negative absorption in the theory of radiation—to the term \(-1\) in the denominator of Planck’s formula for black-body radiation (5), as is easily seen in Einstein’s derivation of this formula. In radiation measurements this term manifests itself only at high temperatures or at long wavelengths, as was shown by the classical measurements of Lummer-Pringsheim and Rubens-Kurlbaum ^34. At high temperatures and long wavelengths the expression \(\frac{h\nu}{kT}\) in Planck’s formula becomes so small that \(-1\) cannot be omitted and, in the limit, Planck’s formula passes into Rayleigh’s formula. In an analogous way, here we find the influence of negative dispersion only at large current intensities, and this influence is the greater the smaller the difference of the energies of the two levels, i.e. the greater the wavelength of the line whose anomalous dispersion is being investigated.

V. Statistical Equilibrium between the \(s\) and \(p\) States of Neon

We are now in a position to calculate from the experiments mentioned the relative number of atoms in the various excited \(s\) and \(p\) states of neon. According to (1)

\[ Q \equiv \frac{N_k g_j}{N_j g_k}=1-\frac{F_{kj}}{N_j f_{kj}}. \]

The quantity \(F\) is determined directly by experiment, but the quantity \(N_j\) and its variation with current are known only up to 60 mA, as long as the linear dependence (8) is valid and negative dispersion has not yet made itself felt. At high currents an extrapolation is necessary for the dependence of \(N_j\) on the current. When using the quadratic expression (7), which was derived on the basis of theoretical considerations concerning the processes that excite and destroy excitation, this expression may be written in a somewhat simplified form:

\[ N_j=\frac{\alpha I+\beta I^2}{\gamma I+\delta I^2+1}. \tag{7a} \]

For constant experiments ^35 it admits only the values:

\[ \alpha=7\cdot 10^{10},\quad \beta=1\cdot 10^8,\quad \gamma=5\cdot 10^{-2},\quad \delta=1.5\cdot 10^{-4}. \]

The values of \(N_{s_5}\), calculated in this way, are seen from Fig. 9.

Below \(100\ \mathrm{mA}\) the number of atoms at the level \(p_k\) is very small in relation to \(N_{s_5}\); as the current is increased \(N_p\) grows, but soon reaches saturation and statistical equilibrium. At \(700\ \mathrm{mA}\)

Fig. 9. Change in the number of atoms of the \(s_5\) level and of various \(p\)-levels with current (neon at a pressure of \(1\ \mathrm{mm}\)).

the atoms \(p_k\) are distributed approximately according to their energies and their statistical weights. If we put

\[ Q_{kj}=e^{-\frac{E_k-E_j}{k\Theta}}, \]

\[ \Theta=\frac{0.621}{\lambda_{kj}}\left(-\log Q_{kj}\right), \]

where \(j\) corresponds to \(s_5\), and \(k\) to the various values of \(p\), and where

\[ E_k-E_j=\frac{hc}{\lambda_{kj}}, \qquad \frac{hc}{\log e}=0.621, \]

then \(\Theta\) is the specific temperature characterizing the ratio \(\dfrac{N_k}{N_j}\).

For five lines of \(s_5\) we give the values in Table 5. The difference in \(\Theta\) for different values of \(p\) at a current of \(700\ \mathrm{mA}\) does not exceed the observational errors; therefore one may speak of statistical equilibrium among the \(p\)-states; their common mean temperature is about \(23000^\circ\mathrm{K}\). This is in sufficiently good agreement with the temperature of the \(s\)-states relative to the normal state calculated in Section III, and also with the electron temperature measured by Seeliger and Kircher \(^{21}\).

We may summarize these results as follows. In the positive column (the so-called “plasma”) of pure neon, and in general in monatomic gases, when the electron velocities have a Maxwellian distribution, statistical equilibrium is attained between the electrons and the excited atoms. The specific temperature determining the relative number of atoms in the exc...

TABLE 5

Values of \(Q\left(=\dfrac{N_k g_j}{N_j g_k}\right)\) and \(\Theta\) for different currents in neon

of the excited levels approaches the electron temperature if the destruction of excitation is caused only by collisions of the second kind with electrons, i.e., when these collisions predominate over other factors, such as, for example, collisions with other atoms and with the walls of the vessel, as well as spontaneous radiation. This “equilibrium current” is the higher, the higher the excitation energy of the level. The kinetic energy of normal atoms does not take part in this equilibrium. The actual average temperature is of an entirely different order of magnitude.

LITERATURE

1. S. A. Korff and G. Breit, Rev. Mod. Phys. 4, 471, 1932; see also R. Ladenburg, Z. Physik 48, 15, 1928.
2. R. Ladenburg and R. Minkowski, Z. Physik 6, 153, 1927.
3. R. Ladenburg and G. Wolfsohn, Z. Physik 63, 616, 1930.
4. W. Prokofjew, Z. Physik 58, 255, 1929; R. Ladenburg and E. Thiele, Z. Physik. Chem. (B) 7, 161, 1930.
5. Agathe Carst and R. Ladenburg, Z. Physik 48, 192, 1928.
6. R. Ladenburg, H. Kopfermann and Agathe Carst, Sitzungsber. d. Preuss. Akad. d. Wissensch. p. 256, 1926.
7. H. Kuhn, Physik. Z. 29, 49, 1928; 33, 957, 1932.
8. A. Einstein, Physik. Z. 18, 121, 1917.
9. See the works of R. Ladenburg, Z. Physik 48, 15, 1928 (I); R. Ladenburg and H. Kopfermann, Z. Physik 48, 26 and 51, 1928 (II and III); Z. Physik 65, 167, 1930 (V); R. Ladenburg and S. Levy, Z. Physik 65, 189, 1930 (VI); subsequently these works are cited as 9, I…VI.
10. See Korff and Breit’s report, p. 482 et seq., Fig. 5.
11. “ 9, II, p. 42.
12. “ 9, VI, p. 198.
13. “ 9, II, p. 38.
14. H. B. Dorgelo, Physica 5, 90, 1925; H. B. Dorgelo and W. de Groot, Z. Physik 36, 897, 1926.
15. See 9, III, p. 55.
16. “ 9, VI, p. 200.
17. R. Ladenburg, Z. Physik 4, 455, 1921.

1. M. I. Druyvesteyn, Z. Physik. 64, 787, 1930.

2. See 9, VI p. 178.

20, 21. As was shown for Hg vapor by I. Langmuir and Mott-Smith, Phys. Rev. 28, 727, 1925; see K. K. Darrow, Discharges in Gases, 1932, and for Ne—R. Seeliger and R. Hirchert, Ann. Physik. 11, 817, 1931.

1. H. Kopfermann and R. Ladenburg, Naturwiss. 19, 512, 1931.

2. O. Klein and S. Rosseland, Z. Physik. 4, 46, 1921.

3. Tolman, Statistical Mechanics, 165, 1927.

4. W. Thomas, Naturwiss. 19, 627, 1925; W. Kuhn, Z. Physik. 33, 408, 1925; F. Reiche and W. Thomas, Z. Physik. 34, 510, 1925.

5. R. de L. Kronig and H. A. Kramers, Z. Physik. 48, 174, 1928.

6. Agathe Carst, Z. Physik. 48, 59, 1928.

7. For details of the calculation see Z. Physik. 65, 185, 1930.

8. Z. Physik. 81, 571, 1933.

9. P. L. Mohler, Bur. Stand. Research Pap. 485, 9, 1932.

10. Killian, Phys. Rev. 35, 1238, 1930.

11. See 9, VI; see also H. Kopfermann and R. Ladenburg, Z. physik. Chem. (A) Haberband, 378, 1928.

12. See 9, V and VI.

13. O. Lummer and E. Pringsheim, Verh. d. dtsch. physik. Ges. 2, 163, 1900; H. Rubens and F. Kurlbaum, Berliner Akad. Ber. 929, 1900; Ann. Phys. 4, 649, 1901.

14. See 9, V p. 180.

15. R. Ladenburg and S. Levy, Z. Physik. 88, 461, 1934.