Some Questions Concerning the Intensities of Spectral Lines

S. E. Frisch

1. Introduction

The intensities of spectral lines play a major role in a number of questions of both theoretical and practical character. Measurement of the intensities of spectral lines makes it possible to determine such an important characteristic of intra-atomic processes as the probability of transitions between different energy states of an atom. From the intensities of spectral lines one may also judge the processes occurring in sources of light, both terrestrial and celestial (stellar atmospheres, nebulae, etc.). The methods of quantitative spectral analysis are based on comparisons of the intensities of spectral lines. The allowance for intensities also plays an essential role in the construction of modern gas-discharge lamps and various laboratory light sources.

Questions concerning the intensities of spectral lines are very diverse and complex, and in the present article we shall consider only a few of them, moreover the simplest. Such a consideration nevertheless seems to us not superfluous, since in the available literature ambiguities are often encountered in the definitions and in the interpretation of various phenomena connected with the emission of spectral lines.

2. Photometric Relations

The measurement of the intensities of spectral lines is a photometric problem. Therefore the first question that arises is what photometric quantity is meant by “intensity.” What exactly is measured under particular concrete experimental conditions?

In photometry, as is well known, the basic quantity considered is the flux of radiant energy*), representing

*) We shall consider only “energetic” photometric quantities.

INTENSITY OF SPECTRAL LINES

the amount of energy carried per unit time through a given surface. The flux is measured in units of power, i.e., in erg/sec, watts, etc. In the case of nonmonochromatic radiation we shall consider the flux \(d\Phi_\nu\) corresponding to a very small frequency interval, enclosed between the given frequencies \(\nu\) and \(\nu + d\nu\). If one singles out a beam of rays propagating within the solid angle \(d\Omega\) (Fig. 1), then the flux \(d\Phi_\nu\) can be represented in the form:

\[ d\Phi_\nu = k_\nu\, d\nu \cos \vartheta\, dS\, d\Omega = k_\nu\, d\nu\, dS_n\, d\Omega, \tag{1} \]

where \(\vartheta\) is the angle between the normal to the area \(dS\) and the axis of the beam, \(dS_n = dS \cos \vartheta\). The quantity \(k_\nu\) is the brightness of the beam, referred to a unit frequency interval. In what follows, for brevity, we shall simply call \(k_\nu\) the brightness. The quantity \(k_\nu\) is a function of the frequency \(\nu\) and, generally speaking, has different values for different directions.

Fig. 1.      Fig. 2.

Let us single out within the luminous gas a volume \(d\tau\) (Fig. 2). Let the energy corresponding to the frequency interval \((\nu, \nu + d\nu)\) and emitted by the volume \(d\tau\) per unit time in all directions be equal to \(w_\nu\, d\nu\, d\tau\). Surround the volume \(d\tau\) by a sphere of arbitrary radius \(R\). Then the flux \(\Delta \Phi_\nu\) through the entire surface of the sphere is equal to

\[ \Delta \Phi_\nu = w_\nu\, d\nu\, d\tau. \tag{2} \]

If the volume \(d\tau\) radiates uniformly in all directions, then the flux \(d\Phi_\nu\) within the solid angle \(d\Omega\) will be equal to

\[ d\Phi_\nu = \frac{1}{4\pi}\, w_\nu\, d\nu\, d\tau\, d\Omega. \tag{3} \]

On the basis of equalities (1) and (3) we establish the following relation between the brightness \(k_\nu\) and the radiative power \(w_\nu\):

\[ k_\nu = \frac{1}{4\pi}\cdot \frac{w_\nu\, d\tau}{\cos \vartheta \cdot dS} = \frac{1}{4\pi}\cdot \frac{w_\nu\, d\tau}{dS_n}. \tag{4} \]

In the presence of an absorbing medium, the flux is attenuated as it propagates in the medium. The decrease of the flux \(d(d\Phi_\nu)\) over a length \(dl\) is equal to

\[ d(d\Phi_\nu)=-\chi_\nu\,d\Phi_\nu\,dl, \tag{5} \]

where \(\chi_\nu\) is the coefficient of absorption of radiation of the given frequency. Integrating expression (5), we obtain (in the case of a homogeneous medium):

\[ d\Phi_\nu=(d\Phi_\nu)_0 e^{-\chi_\nu l}. \tag{6} \]

The quantity \(\chi_\nu l\) is called the optical thickness of the medium. In the case of an inhomogeneous medium, the optical thickness is expressed by the integral \(\int \chi_\nu\,dl\).

Fig. 3.

Finally, let us introduce into consideration the energy density of radiation \(\rho(\nu)\,d\nu\). To do this, consider an infinitely short segment of a beam \(dl\). Its volume is equal to \(d\tau=dS_n\,dl\) (Fig. 3). The amount of energy \(dE_\nu\) contained in this volume is equal to \(d\Phi_\nu\,dt\), where \(dt\) is the time during which the light propagates over the segment \(dl\). Taking the speed of propagation of light to be equal to \(c\), we obtain

\[ dt=\frac{dl}{c}, \]

whence we find

\[ dE_\nu=d\Phi_\nu\,\frac{dl}{c}, \]

which gives us, for the desired energy density \(\rho(\nu)\,d\nu\), the following expression:

\[ \rho(\nu)d\nu=\frac{dE_\nu}{d\tau}=\frac{1}{c}\cdot\frac{d\Phi_\nu}{dS_n}. \tag{7} \]

Using equality (1), we express \(\rho(\nu)\) through the brightness of the beam:

\[ \rho(\nu)=\frac{1}{c}\,k_\nu\,d\Omega. \tag{8} \]

The quantity \(\rho(\nu)\) represents the distribution function of the energy density over frequencies.

Let us now turn to the case of a spectral line of finite width. For such a line the quantities \(k_\nu\) and \(w_\nu\) are definite functions of the frequency \(\nu\). The form of these functions characterizes the contour of the spectral line. Let us introduce, for a line of finite width, the integral flux:

\[ \Delta\Phi=d\sigma\int_0^\infty w_\nu\,d\nu. \tag{9} \]

Here the limits of integration have been taken from \(0\) to \(\infty\), although for each spectral line the intensity of radiation differs noticeably from zero only in a narrow interval of frequencies \(\nu_2-\nu_1\). Since, however, for \(\nu<\nu_1\) and \(\nu>\nu_2\) the integrand is practically equal to zero, the limits of integration may be taken as \(0\) and \(\infty\).

Along with the integral flux, we shall also introduce into consideration the integral brightness of the line \(K\) and the integral power of radiation \(W\), defining them by the equalities:

\[ K=\int_{0}^{\infty} k_\nu\,d\nu; \tag{10} \]

\[ W=\int_{0}^{\infty} w_\nu\,d\nu. \tag{11} \]

As for the absorption coefficient \(\chi_\nu\), for it one cannot introduce the concept of an integral value analogous to the integral values \(K\) or \(W\), since for all media for which \(\chi_\nu\) depends on \(\nu\), the decrease of flux according to the exponential law (6) takes place only for monochromatic radiation.

3. WHAT QUANTITIES ARE MEASURED BY SPECTROPHOTOMETRIC METHODS?

Let us now see what is measured by means of spectrophotometric methods. A spectral instrument, generally speaking, may be reduced to the scheme shown in Fig. 4. The entrance slit \(S_1\) is located in the principal focal plane of the objective of the collimator \(L_1\). Next are placed the dispersing system \(G\) (prism, grating), the objective of the telescope \(L_2\), and the exit slit \(S_2\). Let the focal lengths of the objectives \(L_1\) and \(L_2\) be equal to \(F_1\) and \(F_2\). For simplicity, we shall consider the diameters of the objectives \(L_1\) and \(L_2\) to be equal to each other and equal to \(d\).

Fig. 4.

Let the area of the open part of the entrance slit be equal to \(\Delta S_1\). The beam passing through the entrance slit and falling on the objective of the collimator, by (1), is equal to:

\[ \Delta\Phi_\nu = k_\nu\,d\nu\,\Delta S_1 \int \cos\vartheta\,d\Omega. \]

Integration must be extended over the entire solid angle under which the objective \(L_1\) is seen from the position of the entrance slit. Carrying out the integration, we obtain:

\[ \Delta\Phi_\nu=\pi k_\nu\,d\nu\,\Delta S_1\sin^2 u, \tag{12} \]

where \(u\) is the angle under which the radius of the objective \(L_1\) is seen from the position of the slit. Taking approximately \(\sin u \simeq \dfrac{d/2}{F_1}\), we rewrite equality (12) in the form:

\[ \Delta\Phi_\nu=\frac{\pi}{4}k_\nu d\nu\,\Delta S_1\left(\frac{d}{F_1}\right)^2. \tag{12a} \]

Through the exit objective \(L_2\) there will pass a beam attenuated as a result of light losses in the instrument. Denoting it by \(\Delta\Phi'_\nu\), we may write:

\[ \Delta\Phi'_\nu=a\Delta\Phi_\nu=\frac{a\pi}{4}k_\nu d\nu\,\Delta S_1\left(\frac{d}{F_1}\right)^2, \tag{13} \]

where the quantity \(a\) indicates the light losses in the instrument; obviously, always \(a<1\).

For a line of finite width, the integral flux \(\Delta\Phi'\) will be equal to

\[ \Delta\Phi'=\frac{a\pi}{4}\Delta S_1\left(\frac{d}{F_1}\right)^2\int_0^\infty k_\nu\,d\nu \]

or, by (10),

\[ \Delta\Phi'=\frac{a\pi}{4}\Delta S_1\left(\frac{d}{F_1}\right)^2 K. \tag{14} \]

In doing this we assumed that, for the frequency interval within which the brightness of the line is different from zero, the quantities \(a\) and \(F_1\) are constant.

Let us now consider various methods of recording a line. In objective methods of recording with the aid of a photocell or a photomultiplier, the beam is emitted from the spectral instrument through the exit slit and falls on the photosensitive layer of the recording apparatus. If the exit slit is widened so much that it transmits the flux \(\Delta\Phi'\) completely, then the reading of the recording apparatus will be proportional to the quantity \(b\Delta\Phi'\), where \(b\) denotes the sensitivity of the recording apparatus to radiation of the given frequency \(\nu\). Consequently, in the case under consideration the integral flux \(\Delta\Phi'\) is measured. Since the right-hand side of expression (14) contains \(K\), by measuring the flux \(\Delta\Phi'\) we can thereby determine also the integral brightness of the line \(K\).

When comparing the brightnesses of two different lines, one should take into account the dependence on the frequency $\nu$ of the coefficients $a$ and $b$, and also the possible dependence on $\nu$ and on the focal length $F_1$ (incomplete achromatization of the objective).

Working with an instrument with sufficiently large dispersion and resolving power, and using narrow slits, one can, with the indicated registration scheme, also find the line contour, i.e., determine $k_\nu$ as a function of $\nu$. However, this problem is much more complicated, and we shall not dwell on it here.

Let us now turn to the widely used method of photographic photometry. In this method the intensities of lines are compared by the blackenings which they produce on the photographic plate.

To photograph the spectrum, the plate is placed in the principal focal plane of the exit objective $L_2$. Let us suppose first that we are dealing with a flux $d\Phi'_\nu$ belonging to an infinitely narrow frequency interval $\nu, \nu + d\nu$. Then in the focal plane of the objective $L_2$ an image of the slit in light of frequency $\nu$ will be obtained, whose area $\Delta S_2$ will satisfy the relation (neglecting diffraction phenomena):

\[ \frac{\Delta S_1}{\Delta S_2}=\left(\frac{F_1}{F_2}\right)^2 . \tag{15} \]

The photographic action is determined by the illuminance within the image of the slit

\[ dA=\frac{d\Phi'_\nu}{\Delta S_2}. \]

Using relations (13) and (15), we obtain the following expression for the illuminance:

\[ dA=\frac{d\pi}{4}\,k_\nu\,d\nu\left(\frac{d}{F_2}\right)^2 . \tag{16} \]

In the case of a line of finite width, the image of the slit will be broadened. In addition, because of the finite width of the entrance slit (and also because of diffraction, which leads to a finite value of the resolving power of the instrument), light not of one frequency, but of different frequencies $\nu$, will fall on each given point of the photographic plate. In other words, on the photographic plate the contour of the line will turn out to be distorted. Finding from it the true contour $k_\nu=f(\nu)$ is, generally speaking, a difficult problem, on which we likewise shall not dwell (see, for example, the article by V. M. Chulanovskii and A. V. Timoreva$^{1}$).

Let us turn, however, to another, simpler case. Let the entrance slit be so broadened that the width of its image is much greater than the width of that line which would arise in the focal plane of the objective $L_2$ with an infinitely narrow entrance slit solely due to the true contour of the spectral line and the role of diffraction.

and dispersion in the instrument. In that case the distribution of illumination in the focal plane of the objective \(L_2\) will take the form shown in Fig. 5. The flat part of the graph corresponds to the illumination \(A\) at those places on the photographic plate onto which light of all those frequencies has fallen, within whose limits the intensity of the spectral line

Fig. 5.

Fig. 6.

is appreciably different from zero. Assuming that these frequencies belong to the interval \((\nu_1,\nu_2)\), we obtain from (16):

\[ A=\frac{a\pi}{4}\left(\frac{d}{F_2}\right)^2\int_{\nu_1}^{\nu_2} k_\nu\,d\nu . \]

The limits of integration may be replaced by \(0\) and \(\infty\), since for \(\nu<\nu_1\) and \(\nu>\nu_2\) the illumination is practically equal to zero. Then we obtain:

\[ A=\frac{a\pi}{4}\left(\frac{d}{F_2}\right)^2\int_0^\infty k_\nu\,d\nu =\frac{a\pi}{4}\left(\frac{d}{F_2}\right)^2K. \tag{17} \]

Thus, in the indicated case of a wide entrance slit, the illumination \(A\), and consequently also the degree of blackening of the photographic plate, are determined by the integral brightness \(K\).

To summarize, we may say: although the methods of registration by means of a photoelement (or photomultiplier) and by means of a photographic plate differ essentially, since one of them measures flux and the other illumination, nevertheless they both make it possible to determine one and the same photometric quantity—the integral brightness \(K\). The determination precisely of the integral brightness \(K\) is, in this case, the simplest problem. Measurement of \(k_\nu\) as a function of \(\nu\) (also possible by both methods) proves to be a more complicated problem.

Finally, let us note that, having found \(K\) or \(k_\nu\), one can from them also determine the radiation powers \(W\) or \(w_\nu\).

Up to now we have spoken only about emission lines. Let us now say a few words about absorption lines. To determine the “intensity” of an absorption line, one must measure its contour against the background

continuous emission spectrum. Then, if \(k_{\nu 0}\) gives the distribution of brightness over frequencies in the continuous spectrum in the absence of absorption (dashed line in Fig. 6), and \(k_\nu\) in the presence of absorption (solid line), then the “intensity” of the absorption line is determined by the integral

\[ \int_{0}^{\infty} (k_{\nu 0} - k_\nu)\, d\nu . \]

4. INTENSITIES OF EMISSION LINES WITHOUT TAKING SECONDARY PROCESSES INTO ACCOUNT

We now turn to the question of what intra-atomic constants and external physical parameters determine the intensity of spectral lines. For the time being we shall consider only primary emission processes, completely neglecting all kinds of secondary processes (reabsorption, etc.), which, as we shall see below, play an essential role in most real sources.

When a gas glows, regardless of whether the glow occurs under the influence of an electric discharge or of thermal heating, the gas is ionized to a considerable extent. If, moreover, there are no space charges in the gas, it constitutes a quasi-neutral ionized medium—the so-called “plasma.” As is known, a plasma may be in a peculiar nonisothermal state*, when the distribution of electron velocities follows Maxwell’s law corresponding to a much higher temperature \(T_e\) than the temperature \(T\) at which the gas itself is found. The temperature \(T_e\) is called the electron temperature; it can be measured by the method of probe characteristics. Nonisothermal plasma occurs in an electric discharge at low pressures. In an electric discharge at high pressures, and also under thermal heating, the plasma is in an isothermal state \((T_e = T)\). The character of the excitation of spectral lines is determined to a considerable degree by whether the plasma is isothermal or nonisothermal.

Let us consider the simplest scheme of two energy levels \(i\) and \(k\) with energies \(E_i\) and \(E_k\) (Fig. 7); let \(E_i > E_k\). The number of atoms in these two states will be denoted, respectively, by \(\delta N_i\) and \(\delta N_k\). In a transition between the levels, emission of a photon with energy \(h\nu_{ik} = E_i - E_k\) occurs.

In reality, spectral lines are never ideally monochromatic. Therefore we shall assume that the dis—

* The words “nonisothermal” and “isothermal” are used here in a sense different from that in thermodynamics, where one speaks of processes that are isothermal or nonisothermal depending on whether they proceed at constant temperature or not.

the group of atoms under consideration emits light lying in a very narrow interval \(d\nu\) near the given \(\nu_{ik}\). We shall denote the power of this radiation, as before, by \(w_\nu\,d\nu\).

In a nonisothermal plasma the excitation of atoms occurs practically only at the expense of electron impacts. In the absence of secondary processes (collisions of the second kind, cascade transitions, reabsorption), the number of acts of photon emission is equal to the number of acts of excitation. Therefore the radiation power \(w_\nu\,d\nu\) is equal to:

Fig. 7.

\[ w_\nu\,d\nu=\delta n_i\cdot h\nu_{ik}, \tag{18} \]

where \(\delta n_i\) is the number of acts of excitation of level \(i\) per unit time per unit volume. The number of acts of excitation (from state \(k\) to state \(i\)) may be represented in the form:

\[ \delta n_i=\delta N_k\cdot N_e \int_{V_i}^{\infty} Q_{ki}(V)\cdot \sqrt{V}F(V)\,dV, \tag{19} \]

where \(N_e\) is the number of free electrons per unit volume, \(Q_{ki}(V)\) is the effective cross section of the atoms with respect to exciting electron impacts, \(F(V)\) is the distribution function of the electrons by energies (velocities), and \(V_i\) is the excitation energy of the \(i\)-th level. Thus, the intensity of a spectral line is determined by the concentration of atoms in the normal state \(\delta N_k\), by their effective cross section \(Q_{ki}\), and also by the concentration and energy (velocity) distribution of the electrons.

The indicated two-level scheme corresponds to the case of excitation of resonance lines. V. A. Fabrikant\(^2\) applied formulas (19) and (18) to the calculation of the intensities of resonance lines of sodium and mercury when the vapors of these elements glow at low pressures, and obtained good agreement with the experimental data. The inverse problem may also be posed, namely, from measured intensities of spectral lines to determine the effective cross sections \(Q_{ki}\). Determinations of this kind were also carried out by V. A. Fabrikant and a group of his collaborators\(^3\). In the case of nonresonance lines it is necessary to take into account the possibility of stepwise excitations and cascade transitions from higher levels (see also the work of Yu. M. Kagan and V. M. Zakharova\(^4\)).

The radiation power \(w_\nu\,d\nu\) can be represented not only through the number of acts of excitation, as was done with the aid of formulas

(18). Let us introduce the transition probability \(A_{ik}\) from state \(i\) to state \(k\). Then the number of emission events per unit time is equal to \(A_{ik}\delta N_i\), and the radiant power is

\[ w_\nu\, d\nu = A_{ik}\delta N_i h\nu_{ik}. \tag{20} \]

Expression (20) is more general than (18), since it is also applicable to the case when several transitions from level \(i\) to a number of lower levels are possible.

Let us now proceed to take into account the integral radiant power belonging to a line of finite width. To this end, we shall first consider the causes of spectral-line broadening. We point to the following causes of broadening: a) the Doppler effect due to the thermal motion of atoms; b) the natural damping of radiation; c) interaction between the particles making up the luminous gas. We shall discard the last cause, since at low pressures of the luminous gas (and in what follows we shall consider only the case of low pressures) it does not play a noticeable role. The natural width, the same for all lines \((\Delta\lambda = 1.18\cdot 10^{-4}\ \text{\AA})\), is so small that under ordinary conditions of spectral-line emission it has no appreciable effect on the appearance of their contour. Thus, we shall restrict ourselves solely to allowing for the Doppler contour of the lines. In this case, in formula (20), \(\delta N_i\) must be understood as the number of atoms moving relative to the observer with velocities lying in a specified interval of velocities (in magnitude and direction). Since the transition probabilities \(A_{ik}\) do not depend on the motion of the atoms relative to the observer, for the integral radiant power we obtain:

\[ W_{ik} = A_{ik}h\int \nu_{ik}\delta N_i = N_i A_{ik}h\nu_{ik}, \tag{21} \]

where \(N_i\) is the total number of atoms in state \(i\), and \(\nu_{ik}\) now denotes the mean frequency of the line, i.e. the frequency corresponding to its maximum. It follows from formula (21) that the intensity of a spectral line is determined by two factors: a) the intra-atomic constant—the transition probability \(A_{ik}\); b) the number of excited atoms \(N_i\), which depends on the conditions of excitation of the atoms.

5. PARTICULAR CASES OF EMISSION OF SPECTRAL LINES

Let us now turn to the consideration of several particular cases of emission of spectral lines. First of all, let us consider the case in which two spectral lines with frequencies \(\nu_{ik}\) and \(\nu_{il}\) arise as a result of transitions from a common excited level \(i\) to two lower levels \(l\) and \(k\) (Fig. 8). Since two different transitions are now possible from the upper excited level to

lower, then the probabilities of both these transitions \(A_{ik}\) and \(A_{il}\) must be taken into account. The radiation powers for both lines, by (21), are equal to:

\[ \begin{aligned} W_{ik} &= N_i A_{ik} h\nu_{ik},\\ W_{il} &= N_i A_{il} h\nu_{il}, \end{aligned} \tag{22} \]

and their ratio is

\[ \frac{W_{ik}}{W_{il}}=\frac{A_{ik}}{A_{il}}\,\frac{\nu_{ik}}{\nu_{il}} . \tag{23} \]

As is seen from formula (23), the ratio of the radiation powers is determined only by the atomic constants and by the frequencies of the lines themselves, and does not depend on external conditions (since secondary processes are neglected). Knowing \(\nu_{ik}\) and \(\nu_{il}\), one can, by measuring the ratio of the line intensities, find from (23) the ratio of the transition probabilities \(\dfrac{A_{ik}}{A_{il}}\).

Fig. 8.

Fig. 9.

We shall obtain different conclusions if two spectral lines with frequencies \(\nu_{ik}\) and \(\nu_{lk}\) arise as a result of transitions from two different levels \(i\) and \(l\) to a common lower level \(k\) (Fig. 9). Now the radiation powers of the two lines are equal to:

\[ \begin{aligned} W_{ik} &= N_i A_{ik} h\nu_{ik},\\ W_{lk} &= N_l A_{lk} h\nu_{lk}, \end{aligned} \tag{24} \]

and their ratio is

\[ \frac{W_{ik}}{W_{lk}} = \frac{N_i}{N_l}\, \frac{A_{ik}}{A_{lk}}\, \frac{\nu_{ik}}{\nu_{lk}} . \tag{25} \]

The ratio of the radiation powers, as is evident, in this case depends not only on the frequencies of the lines and the atomic constants, but also on the ratio of the numbers of excited atoms \(\dfrac{N_i}{N_l}\). The numbers of excited atoms \(N_i\) and \(N_l\) depend on the processes occurring in the light source. In a nonisothermal plasma, as we indicated,

excitation occurs through collisions of atoms with electrons. It may be assumed that in most other light sources as well, atoms are excited predominantly by impacts of electrons, since the effective cross sections of atoms with respect to exciting impacts from other neutral atoms or ions are appreciable only at high collision energies, of the order of thousands of electron-volts. The results of collisions with electrons, however, can be quite different depending on the general nature of the processes occurring in the light source. Let us consider separately the following three cases:

1) Low gas pressure (of the order of 0.01 mm Hg and below) and low electron concentration; the distribution of electrons by energy is not Maxwellian. In this case the number of excited atoms is determined by the number of exciting impacts of electrons, which, according to (19), depends on the effective cross section \(Q(V)\) and the electron energy distribution function \(F(V)\). Depending on the form of the function \(F(V)\), the excitation of levels may be very different. As an example, let us cite the excitation of atoms by a beam of electrons uniform in velocity. If in this case the levels \(i\) and \(l\) are sufficiently far apart from one another and the effective cross sections of both levels (“excitation functions”) \(Q_{ki}(V)\) and \(Q_{kl}(V)\) have sufficiently sharp maxima, then the ratio \(\frac{N_i}{N_l}\) may, generally speaking, take any value.

2) Intermediate pressure (greater than 0.01 mm Hg) and intermediate electron concentration; the plasma is nonisothermal; the electron velocities are distributed according to the Maxwell law corresponding to the electron temperature \(T_e\). In this case an equilibrium is possible between the distribution of electrons by energy and that of atoms among their energy levels. In the presence of such an equilibrium, atoms are distributed among the energy levels according to the Boltzmann formula corresponding to the electron temperature \(T_e\):

\[ \begin{aligned} N_i &= N_k \frac{g_i}{g_k} e^{-\frac{E_i-E_k}{kT_e}},\\ N_l &= N_k \frac{g_l}{g_k} e^{-\frac{E_l-E_k}{kT_e}}, \end{aligned} \tag{26} \]

whence for the ratio \(\frac{N_i}{N_l}\) we obtain:

\[ \frac{N_i}{N_l}=\frac{g_i}{g_l}e^{-\frac{E_i-E_l}{kT_e}}. \tag{27} \]

In formulas (26) and (27) the quantities \(g_i\), \(g_l\), and \(g_k\) are the statistical weights of the corresponding levels.

It is essential to note that, for a Boltzmann distribution of atoms over levels, an equilibrium distribution of plasma electrons over energies alone is not sufficient. At low pressures and small electron concentrations, the distribution of atoms over excited levels may differ considerably from the corresponding equilibrium distribution at a temperature equal to the electron temperature \(T_e\). This was shown by direct measurements carried out by N. P. Penkin and A. M. Shukhtin by the method of reversal of spectral lines\(^5\), and by Yu. M. Kagan and N. P. Penkin by the method of anomalous dispersion\(^6\). In the first of these works, cesium vapors were studied, and it was shown that the population of the levels CsI \(7p^2P_{1/2}\) and \(7p^2P_{3/2}\) at a cesium vapor pressure \(p = 0.012\) mm Hg is much lower than the equilibrium one. Only at a cesium vapor pressure \(p = 0.07\) mm Hg and discharge-current strengths above one ampere did the population of the levels \(7p^2P_{1/2}\) and \(7p^2P_{3/2}\) approach the Boltzmann population corresponding to the electron temperature (the latter under these conditions was low—about \(2000^\circ\)K). In the second of the works mentioned, the population of mercury levels was determined. It turned out that the population of the levels \(6p^3P_2\) and \(6p^2P_1\) at low pressures is much less than the equilibrium one (corresponding to \(T_e\)) and only at a mercury vapor pressure of \(3 \cdot 10^{-2}\) mm Hg and higher does it approach the equilibrium value. The conditions under which equilibrium should set in were investigated by A. D. Sakharov\(^7\).

3) Isothermal plasma. \(T_e = T\). The number of excited atoms is again expressed by the Boltzmann formulas (26) and (27), but with the gas temperature \(T\) in the exponent. This case of excitation may be called thermal. The author, together with N. P. Penkin and A. M. Shukhtin\(^8\), showed by the method of reversal of spectral lines that an isothermal low-pressure plasma is realized in a vacuum high-temperature furnace. In this case, for nonresonance lines the secondary processes play an insignificant role, and therefore the observed radiation power of any line is expressed by (24) and (26) by the formula

\[ W_{ik} = N_k \frac{g_i}{g_k} A_{ik} h\nu_{ik} e^{-\frac{E_i - E_k}{kT}} . \tag{28} \]

Using the relation \(E_i - E_k = h\nu_{ik}\), we rewrite the last formula in the form:

\[ W_{ik} = N_k \frac{g_i}{g_k} A_{ik} h\nu_{ik} e^{-\frac{h\nu_{ik}}{kT}} \tag{28a} \]

Measuring the radiation power \(W_{ik}\), one can determine from (28) the transition probability \(A_{ik}\).

Isothermal plasma is also realized in electric arcs and in sparks at high pressures (of the order of atmospheric pressure); however, in these sources secondary processes play such a substantial role that use of formula (28) is impossible.

6. ABSORPTION LINES

Let us now consider absorption lines. Again let us single out the number of atoms \(\delta N_k\) which absorb radiation in a narrow frequency interval \(d\nu\) near the given frequency \(\nu_{ik}\). If the atom has two energy levels (Fig. 7), the power of the absorbed energy \(w'_\nu d\nu\) may be represented in the form:

\[ w'_\nu d\nu=\delta N_k\cdot B_{ki}\rho(\nu_{ik})h\nu_{ik}, \tag{29} \]

where \(\rho(\nu_{ik})\) is the distribution function of the radiation density at the place where absorption occurs. The coefficient \(B_{ki}\) is related to the transition probability \(A_{ik}\) by the relation

\[ B_{ki}=\frac{c^3}{8\pi h}\cdot\frac{g_i}{g_k}\cdot\frac{1}{\nu_{ik}^3}A_{ik}. \tag{30} \]

In order to find the integral absorption power \(W'_{ki}\) for lines of finite width, it is necessary to integrate expression (29) over all those frequencies within whose limits the line gives appreciable absorption.

Considering emission lines, we neglected the natural width and took into account only the Doppler broadening. For absorption lines such a neglect cannot always be made. The point is that the contour caused by the natural width (dashed line in Fig. 10) falls off at considerable distances from the center of the line more slowly than the Doppler contour (solid line in Fig. 10). These distant regions, the so-called “wings” of natural broadening, remain unnoticeable in emission lines; however, in absorption they can produce a considerable effect when the total absorption is strong, which occurs for large thicknesses of the absorbing medium. In what follows we shall restrict ourselves only to cases of insignificant total absorp-

Fig. 10.

tion and shall again assume that the main role is played by Doppler broadening. Then, integrating formula (29), we obtain:

\[ W'_{ki}=N_k B_{ki}\cdot \rho(\nu_{ik})\,h\nu_{ik}, \tag{29a} \]

where now \(\nu_{ik}\) is to be understood as the frequency corresponding to the center of the absorption line.

Along with absorption it is also necessary to consider the so-called induced radiation, whose power is likewise proportional to \(\rho(\nu_{ik})\). Induced radiation has the peculiarity that the photon corresponding to it is emitted in the direction of the primary photon that caused the induced transition in the atom between the corresponding energy levels. This circumstance makes it possible to regard induced radiation as a kind of “negative absorption” and to consider that, in fact, the energy loss per unit volume is equal to:

\[ W'_{ki}=N_k B_{ki}\cdot \rho(\nu_{ik})\,h\nu_{ik}-N_i B_{ik}\cdot \rho(\nu_{ik})\,h\nu_{ik}, \tag{31} \]

where \(B_{ik}\) is the coefficient determining the probability of an induced transition. According to Einstein

\[ B_{ik}=\frac{g_k}{g_i}B_{ki}, \tag{32} \]

which makes it possible to rewrite expression (30) in the form:

\[ W'_{ki}=N_k B_{ki}\cdot \rho(\nu_{ik})\,h\nu_{ik} \left[1-\frac{g_k}{g_i}\cdot\frac{N_i}{N_k}\right]. \tag{31a} \]

For a Boltzmann distribution of atoms among the levels, expression (31a) takes the form:

\[ W'_{ki}=N_k B_{ki}\cdot \rho(\nu_{ik})\,h\nu_{ik} \left[1-e^{-\frac{h\nu_{ik}}{kT}}\right]. \tag{33} \]

If the distribution function of the radiation density \(\rho(\nu_{ik})\) corresponds to an absolutely black body, then, according to Planck:

\[ \rho(\nu_{ik})=\frac{8\pi h}{c^3}\nu_{ik}^{3}\, \frac{1}{e^{\frac{h\nu_{ik}}{kT}}-1}. \]

Substituting this value of \(\rho(\nu_{ik})\) into (33) and assuming that the absolutely black body is at the same temperature as corresponds to the distribution of atoms among the levels, we find:

\[ W'_{ki}=\frac{8\pi h^2}{c^3}N_k B_{ki}\nu_{ik}^{4}\cdot e^{-\frac{h\nu_{ik}}{kT}}. \]

If one uses relation (30) between the coefficients \(B_{ki}\) and \(A_{ik}\) and Boltzmann’s formula for the distribution of atoms

…to levels, the latter expression will be equal to:

\[ W'_{ki}=N'_i A_{ik}h\nu_{ik}. \]

Comparing this result with formula (21), we obtain \(W'_{ki}=W_{ik}\)—the absorption power is equal to the emission power. This equality is the basis of the above-mentioned method of reversal of spectral lines. A beam of rays from an absolutely black body, whose temperature \(T_a\) can be varied, is passed through a luminous gas.

If \(T_a<T\), where \(T\) is the temperature corresponding to the distribution of atoms over levels, then, when observed in a spectral instrument, the gas emission lines appear brighter than the continuous spectrum of the absolutely black body. For \(T_a>T\) the lines become less bright, i.e., they appear against the background of the continuous spectrum as absorption lines. With a Boltzmann distribution of atoms over levels, all lines must disappear against the background of the continuous spectrum simultaneously when the condition \(T_a=T\) is fulfilled. Deviations in the distribution of atoms over levels from the Boltzmann distribution will manifest themselves in the fact that the emission lines will not disappear simultaneously against the background of the continuous spectrum of an absolutely black body. In the aforementioned work of N. P. Penkin and A. M. Shukhtin, not only was a discrepancy observed between the reversal temperature \(T_a\) and the electron temperature \(T_e\), but also a non-simultaneous disappearance of the emission lines against the background of the continuous spectrum. This indicated that the distribution of cesium atoms over levels at low pressures is nonequilibrium. On the contrary, in the work of N. P. Penkin, A. M. Shukhtin, and the author, when vapors were introduced into a high-temperature vacuum furnace, the lines disappeared simultaneously and at a temperature coinciding, within the limits of observational errors, with the temperature of the furnace walls. In this case complete thermodynamic equilibrium was present.

7. ABSORPTION COEFFICIENT

Let us now consider the absorption coefficient \(\chi_\nu\), defined by equality (5):

\[ d(d\Phi_\nu)=-\chi_\nu\,d\Phi_\nu\,dl. \tag{5} \]

To calculate \(\chi_\nu\), let us note that the attenuation of the energy flux \(d(d\Phi_\nu)\) over the length of the beam \(dl\) (Fig. 3) is equal to the amount of energy absorbed per unit time in the volume \(d\tau=dS_n\,dl\). Hence, taking account of “negative absorption,” we obtain:

\[ d(d\Phi_\nu)=\delta N_k B_{ki}h\nu_{ik}\rho(\nu_{ik}) \left[1-\frac{g_k}{g_i}\frac{\delta N_i}{\delta N_k}\right]dS_n dl. \]

The flux \(d\Phi_\nu\), according to (7), is equal to \(d\Phi_\nu=c\rho(\nu)\,d\nu\,dS_n\). Substituting the obtained values of \(d(d\Phi_\nu)\) and \(d\Phi_\nu\) into (5), we write:

\[ \delta N_k B_{ki}h\nu_{ik}\rho(\nu_{ik}) \left[1-\frac{g_k}{g_i}\cdot\frac{\delta N_i}{\delta N_k}\right]dS_n\,dl = \chi_\nu c\rho(\nu_{ik})\,d\nu\,dS_n\,dl, \]

whence we find the following expression for the product \(\chi_\nu\,d\nu\):

\[ \chi_\nu\,d\nu= \frac{1}{c}\,\delta N_k B_{ki}h\nu_{ik} \left[1-\frac{g_k}{g_i}\cdot\frac{\delta N_i}{\delta N_k}\right]. \tag{34} \]

Integrating (34) over all frequencies for which the absorption within the line differs from zero, we obtain:

\[ \int_0^\infty \chi_0\,d\nu= \frac{1}{c}\,N_k B_{ki}h\nu_{ik} \left[1-\frac{g_k}{g_i}\cdot\frac{N_i}{N_k}\right]. \tag{35} \]

If the number of excited atoms \(N_i\) is small in comparison with the number of atoms in the normal state \(N_k\), then approximately:

\[ \int_0^\infty \chi_0\,d\nu= \frac{1}{c}\,N_k B_{ki}h\nu_{ik}. \tag{35a} \]

From formula (35a) it is seen that from the magnitude of \(\int_0^\infty \chi_\nu\,d\nu\) one can find the product of the number of atoms \(N_k\) and the coefficient \(B_{ki}\). The quantity \(\int_0^\infty \chi_\nu\,d\nu\) remains constant when the width of the spectral line changes due to changes of external parameters (with \(N_k\) constant).

Let us introduce another quantity \(a_\nu\), called the absorptive capacity. We shall define this quantity as the ratio of the energy \(w'_\nu d\nu\,d\tau\), absorbed per unit time in the given volume \(d\tau\), to the quantity of light energy \(\rho(\nu)d\nu\,d\tau\) contained in the same volume.

Then we have:

\[ a_\nu= \frac{w'_\nu\,d\nu\,d\tau}{\rho(\nu)\,d\nu\,d\tau} = \frac{\delta N_k B_{ki}h\nu_{ik} \left[1-\frac{g_k}{g_i}\cdot\frac{\delta N_i}{\delta N_k}\right]}{d\nu}. \]

Let us form the ratio of the power \(w_\nu d\nu\,d\tau\), emitted by the volume \(d\tau\), to the absorptive capacity \(a_\nu\). Using expression (20) for \(w_\nu d\nu\), we find:

\[ \frac{w_\nu\,d\nu\,d\tau}{a_\nu} = \frac{A_{ik}\delta N_i\,d\nu\,d\tau} {B_{ki}\delta N_k \left[1-\frac{g_k}{g_i}\cdot\frac{\delta N_i}{\delta N_k}\right]}. \]

or, by (31),

$$ \frac{w_\nu\, d\nu\, d\tau}{a_\nu} = \frac{8\pi h}{c^3}\nu_{ik}^3\, \frac{g_k}{g_i}\cdot \frac{\delta N_i\, d\nu\, d\tau} {\delta N_k\left[1-\dfrac{g_k}{g_i}\dfrac{\delta N_i}{\delta N_k}\right]} . \tag{36} $$

In the equilibrium case the atoms are distributed over the levels according to Boltzmann’s law (27), and then formula (36) takes the form:

$$ \frac{w_\nu\, d\nu\, d\tau}{a_\nu} = \frac{8\pi h}{c^3}\nu_{ik}^3 \frac{1}{e^{h\nu_{ik}/kT}-1}\,d\nu\,d\tau . \tag{36a} $$

On the right here an expression has been obtained which coincides with Planck’s formula. Thus, equality (36a) expresses Kirchhoff’s law: the ratio of the radiation power to the absorptive capacity is equal to the radiation power of an absolutely black body. This result, indicating that for Kirchhoff’s law to hold an equilibrium distribution of atoms over the levels (according to Boltzmann’s law) is necessary, is essential for understanding many processes in light sources.

8. CONDITIONS FOR THE FULFILLMENT OF THE “INTENSITY RULES”

The transition probabilities $A_{ik}$ can be determined experimentally by methods entirely independent of the measurement of the intensities of spectral lines. The theory of anomalous dispersion shows that the behavior of the refractive index $n$ near a sharp absorption line $\nu_{ik}$ is determined by the formula

$$ n-1=\frac{e^2}{2\pi m}\cdot\frac{N_k f_{ki}}{\nu_{ik}^2-\nu^2}, $$

where $N_k$ is the number of atoms in the lower energy level (Fig. 7), and $f_{ki}$ is an atomic constant related to the transition probability $A_{ik}$ by the relation

$$ f_{ki}=\frac{g_i}{g_k}\cdot\frac{mc^3}{8\pi^2 e^2\nu_{ik}^2}\,A_{ik}. \tag{37} $$

Thus, by measuring the anomalous dispersion one can find the product $N_k f_{ki}$. If the number of atoms in state $k$ is known, then the transition probability $A_{ik}$ is also found from this.

As is known, D. S. Rozhdestvenskii created a very reliable and accurate method for measuring anomalous dispersion near sharp absorption lines—the so-called “hook” method. By this method, D. S. Rozhdestvenskii himself and a number of his collaborators and successors (A. N. Filippov, V. K. Prokof’ev, G. S. Kvater, N. P. Penkin) measured the constants $f_{ki}$ for a large number of transitions in various atoms.

Using relation (37), let us express the ratio of the radiation powers of two spectral lines in terms of the constants \(f_{ki}\). For the case of two lines having a common upper level, from formula (23) we obtain:

\[ \frac{W_{ik}}{W_{il}}= \frac{g_k}{g_l}\cdot \left(\frac{\nu_{ik}}{\nu_{il}}\right)^3\cdot \frac{f_{ki}}{f_{li}}, \tag{38} \]

and for the case of two lines with a common lower level—from (25):

\[ \frac{W_{ik}}{W_{lk}}= \frac{g_l}{g_k}\cdot \frac{N_k}{N_l}\cdot \left(\frac{\nu_{ik}}{\nu_{lk}}\right)^3 \frac{f_{ki}}{f_{kl}}. \tag{39} \]

As a result of work carried out by D. S. Rozhdestvenskii in the period 1910–1915, he established that for the principal doublets of the principal series of Na, K, Rb, and Cs the ratio \(\dfrac{f_{ki}}{f_{li}}\) is equal to an integer, namely 2 (the measurement error did not exceed 2.5%). A generalization of subsequent work on anomalous dispersion made it possible to arrive at the following conclusion: for the components of a doublet with a common upper level (subordinate series, Fig. 8), \(\dfrac{f_{ki}}{f_{li}}=1\); for the components of a doublet with a common lower level (principal series, Fig. 9), the ratio \(\dfrac{f_{ki}}{f_{li}}\) is equal to the ratio of the statistical weights of both upper levels, \(\dfrac{g_i}{g_k}\).

In the first of these cases, from (38), for the ratio of the radiation powers of the two components of the doublet, we have:

\[ \frac{W_{ik}}{W_{il}}= \frac{g_k}{g_l} \left(\frac{\nu_{ik}}{\nu_{il}}\right)^3. \tag{38a} \]

As we have already indicated, this ratio does not depend on the conditions of excitation of the spectral lines (in the absence of secondary processes). For a narrow doublet \(\nu_{ik}\simeq \nu_{il}\), and approximately \(\dfrac{W_{ik}}{W_{il}}=\dfrac{g_k}{g_l}\), which gives the well-known “intensity rule”: the intensities of the components of a spectral doublet having a common upper level are related as the statistical weights of the lower levels. The latter are expressed through the quantum numbers \(J\) of the corresponding levels:

\[ g_k=2J_k+1;\quad g_l=2J_l+1. \]

In the second case, from (39):

\[ \frac{W_{ik}}{W_{lk}}= \frac{g_l}{g_k}\cdot \frac{N_k}{N_l}\cdot \left(\frac{\nu_{ik}}{\nu_{lk}}\right)^3 \frac{f_{ki}}{f_{kl}}. \tag{39a} \]

Here, as we noted, any value of the ratio \(\dfrac{W_{ik}}{W_{lk}}\) is possible, depending on the value of the ratio \(\dfrac{N_k}{N_l}\). For a close doublet \((\nu_{ik}\simeq \nu_{lk})\), with a Boltzmann distribution of atoms over the levels,

\[ \frac{N_k}{N_l}=\frac{g_k}{g_l}e^{-\frac{h(\nu_{lk}-\nu_{ik})}{kT}}\simeq \frac{g_k}{g_l}, \]

and from (39a) it approximately follows:

\[ \frac{W_{ik}}{W_{il}}=\frac{f_{ki}}{f_{kl}}=\frac{g_k}{g_l}, \]

i.e. again the “intensity rule” is fulfilled: the intensities of the components of a spectral doublet having a common lower level are related as the statistical weights of the upper levels.

It follows from what has been said that simple integral rules apply only to the ratio of the constants \(f_{ki}\), or of the transition probabilities \(A_{ik}\). The intensities of spectral lines obey the “intensity rules” only approximately, when the conditions indicated above are fulfilled.

These conclusions, as is known, are also generalized to the components of complex spectral multiplets.

In conclusion, let us note that, according to the measurements of D. S. Rozhdestvenskii and his collaborators, strong deviations from the indicated rules are encountered for the ratios of the constants \(f_{ki}\). Thus, for the second doublet of the principal series of cesium,

\[ \frac{f_{ki}}{f_{kl}}=4.07, \]

whereas the ratio of the statistical weights of the upper levels in this case is equal to 2.

9. THE INFLUENCE OF REABSORPTION ON LINE INTENSITIES

The relations obtained by us, giving the power of the radiation of an elementary volume of a luminous gas, generally speaking still do not make it possible to judge the power emitted by a layer of gas of finite thickness. The point is that the radiation emitted by each elementary volume \(d\tau\) will be absorbed to one degree or another before it leaves the bounds of the light source. This phenomenon is called reabsorption of light.

We shall restrict ourselves to the case of a perfectly homogeneous luminous layer of thickness \(l\) (Fig. 11). Let us select an infinitely thin layer \(dx\), located from the origin of coordinates at a distance \(x\). The flux emitted by the volume of this layer \(d\tau=dS_n\,dx\) within the solid angle \(d\Omega\) is equal to:

Fig. 11.

\[ (d\Phi_\nu)_0=\frac{1}{4\pi}w_\nu\,d\nu\,d\tau\,d\Omega . \]

Before emerging from the luminous gas, the flux will traverse a gas thickness \(l-x\), and, consequently, the flux that leaves the gas will be

\[ d\Phi_\nu=(d\Phi_\nu)_0 e^{-\chi_\nu(l-x)} =\frac{1}{4\pi}\,w_\nu\,d\nu\,dS_n\,d\Omega\,e^{-\chi_\nu(l-x)}dx, \tag{40} \]

where \(\chi_\nu\) is the absorption coefficient.

The total flux emerging within the solid angle is obtained by integrating expression (40) from \(x=0\) to \(x=l\):

\[ \Delta\Phi_\nu=\frac{1}{4\pi}w_\nu\,d\nu\,dS_n\,d\Omega \int_0^l e^{-\chi_\nu(l-x)}\,dx = \]

\[ =\frac{1}{4\pi\chi_\nu}w_\nu\,d\nu\,dS_n\,d\Omega \left(1-e^{-\chi_\nu l}\right) \tag{41} \]

or

\[ \Delta\Phi_\nu=\frac{(d\Phi_\nu)_0}{\chi_\nu\,dx} \left(1-e^{-\chi_\nu l}\right). \tag{41a} \]

If the optical thickness \(\chi_\nu l \ll 1\), then approximately \(e^{-\chi_\nu l}=1-\chi_\nu l\), and (41a) gives:

\[ \Delta\Phi_\nu=\frac{(d\Phi_\nu)_0}{dx}\cdot l. \tag{41б} \]

Thus, we find that only for a small optical thickness of the luminous layer is the flux emerging from it proportional to its thickness \(l\). In the general case, however, the calculation must be carried out according to formula (41). For a line of finite width, expression (41) must be integrated over all frequencies. Then for the integral flux \(\Delta\Phi\) we find:

\[ \Delta\Phi=\frac{1}{4\pi}dS_n\,d\Omega \int_0^\infty \frac{w_\nu}{\chi_\nu} \left(1-e^{-\chi_\nu l}\right)d\nu. \tag{42} \]

The integrand, as is evident, depends on \(w_\nu\) and on \(\chi_\nu\), i.e., both on the contour of the emission line and on the contour of the absorption line.

Before turning to the general consequences following from formula (42), let us consider two limiting cases:

1) \(\chi_\nu l \ll 1\); it is easy to see that in this case relation (41б) remains valid, from which it follows that the integral brightness of the line increases in proportion to the thickness of the luminous layer \(l\).

2) \(\chi_\nu l \gg 1\); in this case the total absorption becomes large, and consequently the absorptivity of the gas for frequencies lying within the given line becomes close to unity. If, at the same time, the gas is in an equilibrium state (the atoms are distributed among the levels according to Boltzmann’s law), then, according to Kirchhoff’s law, the brightness of the line must approach

to the brightness of an absolutely black body at the temperature to which the distribution of atoms over levels corresponds. Since \(x, l\) for the center of the line are greater than for its edges, this effect is observed above all for the middle of the spectral line. With increasing thickness of the luminous layer, the line contour undergoes changes: it broadens, and its upper part becomes flat. Such a phenomenon was in fact observed by Ladenburg\(^{10}\) on neon lines.

In intermediate cases it is necessary to use formula (42). Ladenburg calculated the value of the integral (42) for the case when the contours of the emission and absorption lines coincide and both are Doppler contours. According to his calculations, the integral flux \(\Delta\Phi\) is equal to:

\[ \Delta\Phi=\frac{(\Delta\Phi)_0}{dx}\,lS, \tag{43} \]

where \(S\) is a function of the product \(x_0l\) (\(x_0\) is the absorption coefficient for the center of the line). The values of the function \(S\) were computed by Ladenburg for various values of \(x_0l\). At \(x_0l=0\) the function \(S\) assumes the maximum value \(S=1\); as \(x_0l\) increases it decreases monotonically.

The general case of the contour of a spectral line was considered by S. L. Mandelstam\(^{11}\) and a number of other authors.

The relations become more complicated when the luminous layer is inhomogeneous\(^{12}\). In this case the spectral line may exhibit self-reversal—a minimum of brightness is obtained at its center.

The phenomena of reabsorption in the light source make it possible to explain a number of long-known facts. The observed intensity ratios of the components of spectral multiplets correspond to the “intensity rules” either when very thin layers of gas are glowing, or only for the components of such multiplets for which absorption is small (small population of the lower levels). As the conditions of thermodynamic equilibrium are approached, the intensities of the components of multiplets are equalized. When observing a series of successive members of one series, the intensities along the series decrease more slowly if the phenomenon is observed along a long luminous tube than if it is observed across the tube. This is explained by the fact that, for the distant members of a series, \(x_\nu\) is small and their intensities increase approximately in proportion to the length of the tube, whereas for the first members of the series they increase much less.

The role of reabsorption phenomena was experimentally investigated on the resonance lines of mercury by L. M. Biberman and I. M. Gurevich\(^{13}\).

At the Physical Institute of Leningrad University, the role of reabsorption has recently been studied in cesium vapor\(^{14}\). The intensities of lines emitted were compared experimentally...

cesium vapors along and across the discharge tube, as well as by means of tubes of different lengths.

Table I gives the intensity ratios for the second member of the principal series of cesium,

\[ \frac{W(\lambda 4555)}{W(\lambda 4593)} \]

and for the doublet of the diffuse series

\[ \frac{W(\lambda 6212)}{W(\lambda 6011)} \]

at different cesium-vapor pressures (for \(\lambda 4555, 4549\) the photographs were taken across the tube, since along the tube the reabsorption was too large).

Table I

For the doublet of the principal series \(\lambda 4555, 4593\), the true ratio \(\dfrac{W_1}{W_2}\) should be equal to 4, in agreement with the indicated measurements of D. S. Rozhdestvenskii; as can be seen, all values of

\[ \frac{W(\lambda 4555)}{W(\lambda 4593)} \]

in Table I are smaller than this ratio, as should be the case in the presence of reabsorption. For the doublet of the diffuse series \(\lambda 6212, \lambda 6011\), the theoretical intensity ratio is equal to 2. At a cesium-vapor pressure \(p=4.3\cdot 10^{-4}\) mm Hg the measurements gave a considerably smaller value—1.2; this is a consequence of the fact that on the \(\lambda 6011\) line reabsorption had not yet appeared, whereas on the \(\lambda 6212\) line it had already begun to appear. With increasing pressure the ratio

\[ \frac{W(\lambda 6212)}{W(\lambda 6011)} \]

approaches the theoretical value; such a result, in a certain sense, is accidental and is due to the fact that at a cesium-vapor pressure \(p=2.5\cdot 10^{-3}\) mm Hg reabsorption affects the \(\lambda 6212\) and \(\lambda 6011\) lines almost equally.

There is also quantitative agreement between the experimental results and theory. From the measured value of \(\dfrac{W_1}{W_2}\) one can find

\[ \frac{S_1}{S_2}, \]

whence, in turn, one can determine the product \(Nf\). The values of \(Nf\) thus found for the lines \(\lambda 6212\) and \(\lambda 6011\)

agreed well with the value of \(Nf\) determined for the same lines by the anomalous-dispersion method. Thus, at a discharge-current strength of \(100\ \mathrm{mA}\), the reabsorption method gave, for \(\lambda 6212\) and \(\lambda 6011\), values of \(Nf\) respectively equal to \(2.8\cdot 10^{10}\) and \(3.1\cdot 10^{10}\); the anomalous-dispersion method gave \(Nf = 3.7\cdot 10^{10}\) and \(3.9\cdot 10^{10}\). Taking into account the approximate nature of the theory, such agreement should be regarded as quite satisfactory. From reabsorption on the line \(\lambda 4593\) and the known value \(Nf = 3\cdot 10^{-3}\), the concentrations of cesium atoms in the normal state \(N\) were found (Table II).

Table II

In practice these concentrations should coincide with the concentration of the total number of cesium atoms at the given vapor pressure. As is seen from Table II, such agreement is indeed present.

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