THE ROLE OF EFFECTIVE CROSS SECTIONS OF ATOMS IN THE EXCITATION OF SPECTRA

S. E. Frisch

1. INTRODUCTION

As is well known, the luminescence of an atom occurs when it passes from a higher energy level to one of the lower levels. The excitation of an atom, i.e., its transition from a lower to a higher energy level, can occur by one of the following processes: 1) collisions of the first kind with electrons, atoms, or ions; 2) collisions of the second kind with atoms or ions; 3) absorption of photons. The first two processes can be described by means of the corresponding effective cross sections \(Q_{ik}\).

Let us specify the concept of the effective cross section of an atom with respect to a given type of collision. To this end we shall first introduce the concept of the “total” cross section of an atom. Suppose that a narrow beam of particles, moving along parallel trajectories with identical velocities, passes through a gas in which there are \(N_0\) atoms per unit volume. Let \(n_0\) particles pass at the beginning of the beam path through a unit of its transverse cross section per unit time. The number of collisions \(dn\) which the particles undergo over a path length \(dx\) of the beam may be taken equal to

\[ dn = N_0 Q_0 n\, dx . \tag{1} \]

If we assume that every particle which has undergone a collision leaves the beam (because it has been deflected*), or because it has lost velocity), then we find that the decrease in the number of particles in the beam over the path \(dx\) is \(dn' = -dn\). Hence, using formula (1) and carrying out the integration, we find that the number of particles continuing to move in the beam after it has traversed a gas thickness \(x\) is equal to

\[ n_x = n_0 \cdot e^{-N_0 Q_0 x}. \tag{2} \]

The quantity \(Q_0\) is called the “total” cross section of the atom. From formula (2) it is seen that \(Q_0\) has the dimensions of area and, consequently, may be measured in \(\mathrm{cm}^2\),

*) Obviously, in every real experiment the loss due to deflections is taken into account only for those particles which have been deflected through an angle \(\varphi\) not smaller than some definite angle \(\varphi_0\). The value of the “total” cross section found will depend on this angle \(\varphi_0\). For an unambiguous definition of the “total” cross section, some additional criterion must be introduced. We shall not dwell on this, referring the reader to L. A. Sena’s monograph Collisions of Electrons and Ions with Gas Atoms, Gostekhizdat, 1948, in which this question is considered in sufficient detail.

If the particles constituting the beam and the atoms of the gas under consideration were hard spheres of radii \(r_1\) and \(r_2\), respectively, then the “total” cross section \(Q_0\) would be equal to \(\pi(r_1+r_2)^2\).

Let us now suppose that the collisions are partly elastic and partly inelastic in character, and that the latter may lead to various changes in the atom (transition to a definite energy level, ionization, etc.). Let the probability of an elastic collision be \(P_{\varphi}\), and let the probabilities of inelastic collisions be, respectively, \(P_{ik}\). Then

\[ Q_0=P_{\varphi}Q_0+\sum_{i,k} P_{ik}Q_0 . \]

The quantity \(Q_{ik}=P_{ik}Q_0\) is called the effective cross section of the atom with respect to the given type of collision. The effective cross section depends not only on the kind of colliding particles, but also on the velocity with which the particles move relative to one another.

Experiments show that the effective cross sections characterizing inelastic collisions of the first kind of atoms with neutral atoms or ions are appreciable only at large energies of the colliding particles; the effective cross sections of atoms with respect to inelastic collisions of the first kind with electrons, however, assume noticeable values beginning with energies only slightly greater than those required to produce the given process (excitation, ionization). This leads to the fact that, in the glow of a gas under the influence of an electric current passing through it, or as a result of heating at low pressure, the principal role is played by excitation of atoms due to inelastic collisions with electrons. Therefore, in what follows we shall consider mainly such collisions.

Fig. 1. Transitions between the energy levels of an atom.

Let us consider a process which reduces to the fact that an atom, as a result of collision with electrons of a definite velocity \(v\), is transferred from a lower \(i\)-th state to a higher \(k\)-th state (Fig. 1). We shall denote the effective cross section corresponding to this process by \(Q_{ik}(v)\). The question arises: by what experimental methods can \(Q_{ik}(v)\) be determined? There may be two such methods. One of them reduces to determining the number of electrons knocked out of the beam as a result of collisions of the given type. The second method is based on determining the number of excitations of the atom from the \(i\)-th to the \(k\)-th state from the intensities of spectral lines emitted in transitions from the \(k\)-th level. We shall discuss only the second method.

If a gas of sufficiently low pressure is penetrated by a beam of electrons, then, in accordance with formula (1), one may assume that the number of excitations of atoms from the \(i\)-th to the \(k\)-th state due to impacts with electrons whose velocities lie in the given interval \((v, v+dv)\) is equal to:

\[ dN_{ik}=N_i Q_{ik}(v)n_e(v)\,dv, \tag{3} \]

where \(N_i\) is the number of atoms in the \(i\)-th state per unit volume, \(n_e(v)\,dv\) is the number of electrons crossing a unit cross section of the beam per unit time whose velocities lie between \(v\) and \(v+dv\). If electrons of different velocities are present in the beam, the number of excitations from the \(i\)-th to the \(k\)-th level \(\Delta N_{ik}\)

will be found by integrating expression (3):

\[ \Delta N_{ik}=N_i\int_{v_{ik}}^\infty Q_{ik}(v)n_e(v)\,dv, \tag{4} \]

where \(v_{ik}\) is the smallest velocity of the electrons in the beam, beginning with which the given transition \(i\to k\) starts to occur. In formulas (3) and (4), the quantities \(dN_{ik}\) and \(\Delta N_{ik}\) denote the number of transitions per unit time in a unit volume of the gas being excited.

Formulas (3) and (4) correspond to the case of excitation of a gas by an electron beam passing through it. This can be realized only in a specially arranged experiment. Usually, however, excitation of a gas occurs owing to free electrons present in this gas and moving in a random manner. Then the number of collisions in a unit volume of gas per unit time will be proportional to the electron velocity \(v\). Therefore, for the number of excitations from the \(i\)-th to the \(k\)-th level we now obtain:

\[ \Delta N_{ik}=N_i\int_{v_{ik}}^\infty Q_{ik}(v)N_e(v)v\,dv, \tag{5} \]

where \(N_e(v)\,dv\) is the number of electrons in a unit volume of the gas whose velocities lie in the interval \((v,\ v+dv)\).

The effective cross section \(Q_{ik}(v)\), as is evident, determines the number of excitations of an atom due to collisions with electrons. Sometimes, to characterize the excitation process, another quantity is introduced, called the excitation function. We shall define the excitation function \(f_{ik}(v)\) by the following equality:

\[ Q_{ik}(v)=Q_{ik\max}\cdot f_{ik}(v), \]

where \(Q_{ik\max}\) is the value of \(Q_{ik}(v)\) at the maximum (at the largest of the maxima, if there are several). The quantity \(Q_{ik}(v)\) may be assigned the dimension of area; then \(f_{ik}(v)\) is dimensionless. Obviously, at the maximum \(f_{ik}(v)=1\); the values of \(f_{ik}(v)\) for different transitions characterize the relative values of the effective cross sections.

In a manner analogous to that described above, one may introduce into consideration effective cross sections for other processes, for example, for transitions of atoms from one energy level to another under the influence of impacts of the second kind, or for ionization of atoms by electron impact.

To find the effective cross sections of collisions leading to excitation of atoms from the intensities of spectral lines, it is necessary to determine what set of processes determines, under the given experimental conditions, the intensity of the spectral line.

2. PROCESSES DETERMINING THE INTENSITY OF SPECTRAL LINES

The intensity of a spectral line emitted in a transition from the \(k\)-th to the lower \(i\)-th level is equal to:

\[ I_{ki}=N_kA_{ki}h\nu_{ki}+N_kB_{ki}\rho(\nu_{ki})h\nu_{ki}, \tag{6} \]

where \(\nu_{ki}\) is the frequency of the line, \(h\) is Planck’s constant, and \(A_{ki}\) and \(B_{ki}\) are coefficients

Einstein coefficients, which determine the probability of spontaneous and induced transitions from the \(k\)-th to the \(i\)-th level. \(\rho(\nu_{ki})\) is the radiation density of frequency \(\nu_{ki}\). The role of the second term in expression (6), which pertains to induced transitions, is noticeable only at considerable radiation densities \(\rho(\nu_{ki})\). If \(\rho(\nu_{ki})\) is small, then induced transitions may be neglected, and then

\[ I_{ki}=N_k A_{ki}h\nu_{ki}. \]

Let us emphasize that by the “intensity” of a line we mean a quantity proportional to the power of radiation of the given frequency \(\nu_{ki}\). In this connection it must be borne in mind that the power of the radiation emerging beyond the boundaries of the light source depends on the absorption of light within the light source itself (reabsorption), whose role may sometimes be substantial\(^2\). In the present paper, however, for simplicity we shall assume that reabsorption is absent.

As is evident, formula (6), which determines the intensity of a line, does not contain explicitly the number of excitations \(\Delta N_{ik}\), which according to formula (5) is connected with the effective cross section \(Q_{ik}(v)\). In order to establish the connection between the intensity of a line and the effective cross section \(Q_{ik}(v)\), it is also necessary to use the condition of stationarity.

Let us consider a monatomic gas in a state of sufficiently high ionization, and let us assume that the numbers of positively and negatively charged particles per unit volume are equal, so that the gas is quasineutral in all its parts. As is known, a gas in such a state is called a plasma. In order to write the stationarity condition for a plasma, let us consider in more detail the set of processes leading to the population and depopulation of a given energy level of the atom (level \(k\), Fig. 1).

I. Processes leading to population of the level

1) Collisions of the first kind (with electrons, atoms, ions):
a) with atoms in the normal state (“direct” excitation);
b) with atoms in an excited state (“stepwise” excitation).

2) Collisions of the second kind (with atoms, ions).

3) Spontaneous and induced transitions from higher levels (“cascade” transitions).

4) Absorption of photons.

5) Recombination of ions.

II. Processes leading to depopulation of the level

1) Spontaneous radiation (to all lower-lying levels).

2) Induced radiation.

3) Collisions of the first kind, leading to transition to higher levels and to ionization.

4) Collisions of the second kind:
a) with atoms and ions;
b) with electrons.

5) Depopulating collisions with the walls of the vessel.

Of all these processes we shall neglect depopulating collisions with the walls of the vessel. This may be done if the mean free path of the particles is small in comparison with the dimensions of the vessel containing the gas. Then the stationarity condition for the \(k\)-th level may be written in the fol-

in the following form:

\[ \sum_m \sum_{r=0}^{k-1} \Delta N_{rk}(N_r, N_m) + \sum_m \sum_{l=0}^{\infty} \Delta N_{lk}(N_l, N_m) + \sum_{l=k+1}^{\infty} \Delta N_{lk}(N_l) + \]

\[ + \sum_{l=k+1}^{\infty} \Delta N_{lk}\bigl[N_l, \rho(\nu_{lk})\bigr] + \sum_{r=0}^{k-1} \Delta N_{r,k}\bigl[N_r, \rho(\nu_{rk})\bigr] + \Delta N_{\infty k}(N_j, N_e) = \]

\[ = \sum_{r=0}^{k-1} \Delta N'_{kr}(N_k) + \sum_{r=0}^{k-1} \Delta N'_{kr}\bigl[N_k, \rho(\nu_{kr})\bigr] + \sum_m \sum_{l=k}^{\infty} \Delta N'_{kl}(N_k, N_m) + \]

\[ + \sum_m \sum_{l=0}^{\infty} \Delta N'_{kl}(N_k, N_m). \tag{7} \]

To this one must add the condition

\[ \sum_{k=0}^{\infty} N_k + N_j = N, \]

where \(N\) is the total number of atoms of the given kind per unit volume.

In equality (7), \(\Delta N\) and \(\Delta N'\) denote, respectively, the numbers of acts of excitation or depletion per unit time per unit volume. In parentheses after each \(\Delta N\) or \(\Delta N'\) are written the arguments on which they depend. Here \(N_r\), \(N_k\), and \(N_l\) denote the concentrations of excited atoms in the \(r\)-th, \(k\)-th, and \(l\)-th states, respectively; \(N_m\) is the concentration of incident particles (for \(m=e\) this is an electron, for \(m=j\) an ion); \(\rho(\nu_{lk})\) is the radiation density of frequency \(\nu_{lk}\). The separate terms in equality (7) are written in the order in which the processes leading to the population and depletion of the level under consideration were listed above. Thus, the first two terms on the left-hand side of equality (7) correspond to the population of the \(k\)-th level due to collisions of the first and second kind with all types of particles (electrons, atoms, ions), while the last term of the same part of the equality corresponds to the population of the \(k\)-th level due to recombination of ions with electrons. On the right-hand side of the equality, the last two terms correspond to the depletion of the excited level due to collisions of the first and second kind. All these processes can be expressed explicitly through the concentrations of the colliding particles and the corresponding effective cross sections \(Q_{ik}\).

The sums

\[ \sum_{l=k+1}^{\infty} \Delta N_{lk}(N_l) \quad \text{and} \quad \sum_{r=0}^{k-1} \Delta N'_{kr}(N_k) \]

represent spontaneous transitions \(l \to k\) and \(k \to r\), while all sums containing among their arguments \(\rho(\nu)\) represent transitions caused by absorption of light, or induced transitions. All of them can be expressed explicitly through the concentrations of atoms in the corresponding states and the Einstein coefficients \(A_{kr}\), \(B_{rk}\), and \(B_{kr}\).

From equality (7) it is seen that the concentration of atoms in the \(k\)-th state, \(N_k\), is connected with the concentration of atoms in all the other states. Therefore equations (7) must be written for all \(k\) from 1 to \(\infty\), i.e. an infinite set. The problem of finding \(N_k\) is reduced to solving a system of equations with an infinite number of unknowns. The solution for \(N_k\) will be

contain, among other parameters, also the effective cross sections \(Q_{ik}\). On the other hand, by (6) \(N_k\) is connected with the intensity of the line of frequency \(\nu_{ki}\). Thus, the effective cross sections \(Q_{ik}\) turn out to be connected with the intensity of the line \(I_{ki}\).

Obviously, the system of equations (7) cannot in general be solved, and one must look for conditions under which a number of processes could be knowingly neglected.

As was indicated, the effective cross sections for collisions of the first kind with neutral atoms and ions are appreciable only at high energies (of the order of \(10^3\) eV); therefore in equality (7) they can usually be disregarded, and it may be assumed that exciting collisions of the first kind occur only with electrons. In addition, in many cases one may neglect collisions of the second kind with atoms and ions, since they occur with sufficient probability only when the excitation energies of the colliding particles are close to one another. However, these simplifications are insufficient, and it is necessary to proceed to the consideration of special cases, when a number of other terms can also be neglected.

3. DETERMINATION OF EFFECTIVE CROSS SECTIONS BY THE ELECTRON-IMPACT METHOD

Work on determining the effective cross sections of atoms with respect to inelastic collisions of the first kind leading to excitation was first undertaken more than 25 years ago by Hanle, Schaffernicht, Larché, and a number of other authors[^3]. In their experiments a beam of electrons of definite velocity passed through a monatomic gas at low pressure. The glow of the gas was observed. The intensity of lines was measured at various velocities of the exciting electrons. As a result of such measurements, the dependence of the line intensity on the electron velocity, \(I_{ki}(v)\), was obtained—the so-called optical excitation function of the line. In plotting excitation functions, often instead of the electron velocity \(v\) one takes the value of the accelerating potential \(V\) or of the electron energy expressed in electron-volts.

The results of the measurements, as is known, proved to be as follows: the optical excitation function rises steeply, beginning at the potential corresponding to the excitation energy (the “critical potential”), then reaches a more or less sharp maximum, after which, in the majority of cases, it decreases monotonically. For helium, the alkaline-earth elements, mercury, cadmium, and zinc, a markedly different course of the excitation functions was found for singlet and triplet lines: the excitation functions of triplet lines have a sharp maximum lying at potentials 2–3 volts above the critical potential; the excitation functions of singlet lines have a flatter maximum lying farther from the critical potential. In a small number of cases (some lines of mercury, zinc, cadmium) a second maximum is observed after the first; with the exception of the line \(\lambda\,2537\) Hg, this is a flat maximum situated approximately at an accelerating potential 20–30 V above the critical potential.

The presence, as a rule, of one maximum in optical excitation functions and their smooth course were interpreted as indicating that, under the conditions of the experiments carried out, cascade transitions were absent. Indeed, excitation of some higher level \((l>k,\ \text{Fig. }1)\) and spontaneous transition from it to the \(k\)-th level would have had to lead to an increase in the intensity of the line \(I_{ki}\), for which the \(k\)-th level is the initial one. As a result, on the curve representing the optical excitation function of the line \(I_{ki}(V)\), an additional maximum should have appeared, or at least—

to some extent, by a break. From the absence of such secondary maxima and breaks it was concluded that the form of the optical excitation function of a spectral line directly reproduces the form of the excitation function of the atomic energy level that is the initial one in the emission of the given spectral line. In other words, it was believed that, from measurements of the intensities of a spectral line excited by an electron beam, one could directly determine the relative values of the effective cross section of collisions leading to excitation of the initial level of the given line.

In the case of lines whose optical excitation functions have two maxima, some authors assumed that cascade transitions occur here. Attempts were made to decompose the curve having two maxima into two curves, one of which would take into account excitation of the level that is the initial one for the given line, and the other—excitation of higher levels. Such an attempt, for example, was made for the line \(\lambda 2537\) Hg in the book by Burhop and Massey on atomic collisions\(^4\). However, decompositions of this kind were, first, not unique, and, second, in general it was unclear why cascade transitions play a role only in the emission of some lines, while in the emission of the majority of spectral lines they do not appear.

All this made it necessary to analyze more rigorously the conditions of excitation of spectral lines under the conditions of the experiments of Hanle, Schaffernicht, and other authors, and to carry out new, more accurate measurements.

The analysis of the conditions occurring in the excitation of spectral lines by an electron beam can be made starting from equality (7). This equality was written by us for the case of excitation of spectral lines in a plasma; however, it is also suitable for the case of excitation by an electron beam. The fact that the atoms are excited by a beam of electrons will require only the use, for the relation between the number of excitation acts and the effective cross section, of formula (4) instead of (5).

In the experiments of Hanle and other authors, a low-pressure monatomic gas at a not very high temperature was penetrated by an electron beam, and the number of electrons passing through a unit cross section per unit time was not very large. In this case, of all the processes listed above that lead to excitation of the \(k\)-th level, there remain: 1) excitation by electron impact from the normal level; 2) spontaneous transitions from higher levels (cascade transitions). Of all the processes leading to depopulation of the \(k\)-th level, only spontaneous transitions to lower levels remain. Then the stationarity condition (7) takes the form

\[ \Delta N_{0k}(N_0,\ n_e) + \sum_{l=k+1}^{\infty} \Delta N_{lk}(N_l) = \sum_{r=0}^{k-1} \Delta N'_{kr}(N_k). \tag{7a} \]

To this we add the expression for the intensity of a line, in which we shall not take induced transitions into account:

\[ I_{ki}=N_k A_{ki} h\nu_{ki}. \tag{6a} \]

In equality (7a) \(n_e\) denotes the number of electrons passing through a unit cross section of the beam per unit time. Let us suppose that the beam consists of electrons sufficiently homogeneous in velocities and that, at an accelerating potential \(V\), the number of passing electrons is equal to \(n_e(V)\). Then the number of excitation acts \(\Delta N_{0k}(N_0,\ n_e)\) will be written in the following

in the form:

\[ \Delta N_{0k}(N_0,\ n_e)=N_0 n_e(V)\cdot Q_{0k}(V), \]

where \(Q_{0k}(V)\) is the effective collision cross section leading to the transfer of an atom from the normal state to the \(k\)-th state.

We shall express the number of spontaneous transitions from the \(k\)-th level to lower-lying levels in terms of the Einstein coefficients \(A_{kr}\):

\[ \sum_{r=0}^{k-1}\Delta N'_{kr}(N_k)=N_k\sum_{r=0}^{k-1}A_{kr}. \]

Since all \(A_{kr}\) are constants, from (6a) and (7a) we obtain:

\[ I_{ki}(V)\sim N_0 n_e(V)Q_{0k}(V)+\sum_{l=k+1}^{\infty}N_l A_{lk}. \tag{8} \]

It follows from equality (8) that the form of the optical excitation function \(I_{ki}(V)\) coincides with the form of the excitation function \(Q_{0k}(V)\) only in the case where one may neglect the term

\[ \sum_{l=k+1}^{\infty}N_l A_{lk}, \]

which takes into account transitions to the given level from higher levels (cascade transitions). The magnitude of this term under the conditions considered is determined by the number of excitation events \(\Delta N_{0l}\) of each of the upper levels. Since, at sufficiently high energies of the exciting electrons \(V\), the number of excitations \(\Delta N_{0l}\) \((l>k)\) is, generally speaking, of the same order as the number of direct excitations of the \(k\)-th level, stepwise transitions cannot be neglected. Consequently, we come to the conclusion that the earlier accepted assumption that the form of the optical excitation function of a spectral line directly reproduces the form of the excitation function of the atomic energy level that is the initial one for the given line is incorrect.

The absence of secondary maxima and breaks in the optical excitation functions should be explained not by the absence of cascade transitions, but by the insufficiency of experiments, which yielded crude, smoothed curves.

4. EXPERIMENTAL OBSERVATION OF CASCADE TRANSITIONS

In order to detect the role of cascade transitions, in the work of S. E. Frisch and I. P. Zapesochny\(^5\) the intensities of mercury lines were determined as a function of the energy of the exciting electrons by means of a photoelectric technique; special attention was paid to the velocity homogeneity of the exciting electron beam. For accelerating the electrons, an apparatus with an equipotential cathode and a system of guard cylinders was used. The electron receiver was equipped with a series of grids in order to avoid the appearance of secondary and scattered electrons. Lines from the mercury spectrum were isolated by a monochromator; their intensities were measured with the aid of a photoelectric multiplier with subsequent current amplification. The photoelectric method of measuring intensities had substantial advantages over the photographic

method: it had greater sensitivity and greater accuracy (error \(\sim 2\%\)) and ensured rapid measurements. Measurements were made both with a pointer galvanometer and with a device giving automatic recording of the readings in the form of a curve. In the first case the curves were constructed from many tens of points (up to 70 for one curve), which were taken, where necessary, every 0.1 volt.

A number of control experiments were carried out. The absence of reabsorption, the absence of excitations from intermediate levels, and of quenching collisions were checked from the linearity of the dependence of the line intensity \(I_{kl}\) on the pressure of the excited mercury vapor and on the current in the electron beam. The distribution of electrons by energies was determined by the retarding-field method. Fig. 2 shows the dependence of the current \(I\) reaching the collector on the retarding potential \(V_{\mathrm{ret}}\). The same figure gives the differential curve, from which it follows that 90% of the electrons had energies in an interval about \(1.0\ \mathrm{eV}\) wide (at a mean electron energy of \(12.7\ \mathrm{eV}\)). On decreasing the current density and the vapor pressure this interval could be reduced to \(0.5\ \mathrm{eV}\).

Fig. 2. Distribution of electrons by energies.

Eight lines of the visible part of the mercury spectrum were investigated; their wavelengths and series assignments are given in Table I, and the scheme of the corresponding transitions is given in Fig. 3. The results of the measurements showed the following: 1) The optical excitation functions of the mercury lines \(\lambda\lambda\) 5461, 4358, 4047, 4078,

Table I

Wavelength \(\lambda\)	Series symbol	Excitation potential	Number of observed maxima	Position of the observed maxima
5461	\(6^3P_2 — 7^3S_1\)	7.69	6	8.2; 8.9; 9.6; 10.5; 11.0; 12.4
4358	\(6^3P_1 — 7^3S_1\)	7.69	6	The same
4047	\(6^3P_0 — 7^3S_1\)	7.69	6	The same
4078	\(6^3P_1 — 7^1S_0\)	7.90	\(4\ (+1)^*)\)	8.7; 10.0; 11.1; 13.1; (\(\sim 30\))
4916	\(6^1P_1 — 8^1S_0\)	9.18	\(2\ (+1)\)	10.2; 11.1; (\(\sim 35\))
4108	\(6^1P_1 — 9^1S_0\)	9.67	\(2\ (+1)\)	10.4; 11.1; (\(\sim 35\))
5791	\(6^1P_1 — 6^1D_2\)	8.80	\(1\ (?)\)	18
4347	\(6^1P_1 — 7^1D_2\)	9.51	\(1\ (?)\)	20

*) In parentheses are indicated flat maxima lying in the region of energies of the exciting electrons \(30—35\ \mathrm{eV}\).

4916, 4108 have several well-pronounced maxima located at potentials somewhat greater (by \(1—4\ \mathrm{V}\)) than the excitation potential. The number and positions of these maxima are indicated in Table 1. 2) The optical excitation functions of the lines \(\lambda\lambda\) 4078; 4916 and 4108 have, besides the maxima located at the beginning of the curves, also a flat maximum at energies of the exciting electrons \(30—35\ \mathrm{eV}\). This circumstance was

as was already noted earlier by a number of authors^3. 3) The optical excitation functions of the lines \(\lambda 5791\) and \(\lambda 4347\) do not have pronounced maxima. 4) The optical excitation functions of the lines \(\lambda\lambda 5461, 4358, 4047\), which have the common upper level \(7^3S_1\), coincide completely with one another.

When the homogeneity of the electron beam was artificially worsened, which was achieved by applying an additional alternating potential \(V_g\), the maxima,

Fig. 3. Transition scheme in the spectrum of mercury.

located near the excitation potential, were smoothed out, and when the electron energies were distributed over an interval about \(3.5\) eV wide, they merged into one (Fig. 4). In this case the curves agreed, within the errors of observation, with the curves obtained earlier for the corresponding lines by Schaffernicht^3.

Thus it was shown directly that the absence of secondary maxima on the curves obtained in earlier work was the result of insufficient homogeneity in the velocities of the electrons in the exciting beam.

The data presented here on the presence of secondary maxima in the optical excitation functions of mercury lines have recently been confirmed by measurements of the Dutch physicists Smit and Jongerius^6.

The presence of several maxima in the optical excitation functions near the excitation potential is explained by the role, indicated above, of cascade transitions. Each experimental curve with several maxima

can be decomposed into curves representing the excitation functions of individual energy levels of the mercury atom. Since for each level its excitation potential is known, and since the observed maxima

Fig. 4. Change in the form of the optical excitation function of the mercury line \(\lambda 5461\) with deterioration of the homogeneity of the electron beam.

are sufficiently sharp, such a decomposition near the excitation potential is practically unambiguous. Ambiguity occurs only at relatively high excitation energies, where the experimental curves run smoothly. Figure 5 gives the observed optical excitation function of the line \(\lambda 5461\) (it coincides with the optical excitation functions of the lines \(\lambda 4358\) and \(\lambda 4047\)). The dotted lines show the excitation functions of the energy levels into which the experimental curve is decomposed. The first small maximum on the experimental curve at \(8.2\ \mathrm{eV}\) corresponds to excitation of the level \(7^3S_1\); the next three maxima correspond to excitation of the levels \(7^3P_J\), \(8^3P_J\), and \(9^3P_J\), having, respectively, excitation potentials \(8.6\), \(9.4\), and \(9.8\ \mathrm{eV}\) (see Fig. 3). The fifth maximum corresponds to excitation of the unresolved group of levels \(n^3P_J\) for \(n > 10\). The broadened maximum near \(12.4\ \mathrm{eV}\), in all probability, corresponds to the process of recombination of mercury ions.

The lines \(\lambda\lambda\ 4078,\ 4916\) and \(4108\) have as their initial levels, respectively, the levels \(7^1S_0,\ 8^1S_0\), and \(9^1S_0\). The observed optical excitation function of the line \(\lambda 4078\) (Fig. 6) has three well-defined maxima, which are associated with the excitation of the levels \(7^1S_0,\ 7^1P_1\), and of the unresolved group of levels \(n^1P_1\) for \(n \gg 8\).

Fig. 5. Decomposition of the observed excitation function of the mercury line \(\lambda 5461\) into the excitation functions of individual levels.

The maxima belonging to this group of levels \(n^1P_1\) are also present on the curves depicting the excitation functions for the lines \(\lambda 4916\) and \(\lambda 4108\) (Fig. 6). The first maxima on these two curves correspond, respectively, to excitation of the levels \(8^1S_0\) and \(9^1S_0\). Thus, the form of the curves for these three lines is in full agreement with one another. What remains unexplained are the blurred maximum in the excitation function of the line \(\lambda 4078\), lying near 13 eV, and the flat maxima on the curves of all three excitation functions, lying in the region 30–35 eV. The first of these, in all probability, is associated with the process of recombination of the mercury ion; the flat maxima in the region 30–35 eV are evidently inherent in the excitation functions themselves of the individual levels, possibly the levels \(n^1S_0\).

The absence of noticeable maxima in the optical excitation functions of the lines \(\lambda 5791\) and \(\lambda 4347\) is explained by the fact that above their initial levels there is a considerable number of levels close to one another, from which stepwise transitions are possible. Under the conditions of the experiments described, these levels remained unresolved.

From the above analysis of the curves it follows that the excitation functions of mercury accepted up to now were obtained as the result of the merging of curves corresponding to excitation of individual energy levels. The true excitation functions of energy levels have sharp maxima located near the excitation potentials. For mercury no noticeable difference is found in the position of these maxima for the excitation functions of triplet and singlet levels. Whether this conclusion also applies to other elements having triplet and singlet levels remains, for the time being, unclear.

The presence of cascade transitions can be verified by direct observations. Excitation of the level \(7^3P_j\) should be accompanied by emission of the infrared triplet \(\lambda\lambda\ 11282,\ 13673\), and \(13950\ \text{Å}\); excitation of the level \(8^3P_j\), by emission of the triplet \(\lambda\lambda\ 6907,\ 7082\), and \(7092\ \text{Å}\). The photomultiplier used in the work

Fig. 6. Decomposition of the observed excitation functions of the mercury lines \(\lambda\lambda\ 4078,\ 4916\), and \(4108\) into excitation functions of individual levels.

did not have the necessary sensitivity in the infrared region for it to be possible to record the triplet \(\lambda\lambda\ 11282,\ 13673,\ 13950\ \text{Å}\). For the triplet \(\lambda\lambda\ 6907,\ 7082\), and \(7092\ \text{Å}\), however, the optical excitation function was measured. The value of the excitation function of the line \(\lambda 6907\) at the maximum was compared with the position of the corresponding maximum on the curve representing the excitation function of the line \(\lambda 5461\ \text{Å}\). Within the observational errors, coincidence of the two maxima was obtained.

It should be noted that there exists only a very small number of atoms for which one can expect to detect experimentally, in the excitation functions, secondary maxima and breaks caused by cascade transitions. The point is that it is difficult to obtain an electron beam with a narrower distribution of electron velocities than within approximately 0.5 and 0.3 ev. However, such a distribution is not sufficient to reveal the role of individual energy levels, since in most atoms the levels lying above the second are closely spaced. For example, in sodium the level \(4s^2S_{1/2}\) lies approximately 1 ev above the level \(3p^2P\), and then come levels separated from one another by tenths of an ev. Under these conditions the measurements give, to one degree or another, a smoothed curve, from which it is difficult to determine the form of the excitation functions for individual energy levels.

5. THE ROLE OF EFFECTIVE CROSS SECTIONS IN THE EXCITATION OF SPECTRAL LINES IN A GAS-DISCHARGE PLASMA

As is known, in a gas-discharge plasma the electron velocities may be distributed according to Maxwell’s law, corresponding to a temperature \(T_e\) higher than the temperature of the atomic gas. This temperature, called the electron temperature, may reach many tens of thousands of degrees. Excitation of atoms in such a plasma occurs mainly through collisions with electrons. The number of exciting collisions, in accordance with formula (5), may be written in the form

\[ \Delta N_{ik}=N_iN_e\int_{v_{ik}}^{\infty} Q_{ik}(v)F(v)v\,dv, \tag{9} \]

where \(F(v)\) is the Maxwellian distribution function of electrons with respect to velocities.

The stationarity condition (7), when collisions of the second kind with atoms and ions, recombination phenomena, as well as photon absorption and induced transitions are neglected, takes the form

\[ \sum_{r=0}^{k-1}\Delta N_{r,k}(N_r,N_e)+\sum_{l=k+1}^{\infty}\Delta N_{lk}(N_l)= \]

\[ =\sum_{r=0}^{k-1}\Delta N'_{kr}(N_k)+\sum_{l=k+1}^{\infty}\Delta N'_{kl}(N_k,N_e)+\sum_{r=0}^{k-l}\Delta N'_{kr}(N_k,N_e). \tag{10} \]

Here in the left-hand side the first term corresponds to the population of the \(k\)-th level due to excitation of the atom by electron impacts, and the second to spontaneous transitions from higher levels (cascade transitions). In the right-hand side of equality (10) the separate terms correspond to spontaneous transitions to lower levels, transitions to higher levels due to electron impacts, and transitions to lower levels due to collisions of the second kind with electrons.

If the role of collisions of the first and second kind with electrons is large in comparison with the role of spontaneous transitions, then the atoms will be distributed over the energy levels according to Boltzmann’s law, corresponding to the electron temperature \(T_e\). Consequently, the number of atoms in some \(k\)-th state will depend only on the number of atoms in the normal state \(N_0\), on the statistical weights \(g_0\) and \(g_k\), and on the temperature \(T_e\); it will not depend on the values of the effective cross sections \(Q_{ik}\). Under these conditions the intensity of the line

\[ I_{ki}=N_kA_{ki}h\nu_{ki} \]

will be determined only by the temperature, by the statistical weights \(g_0\) and \(g_k\), by the transition probability \(A_{ki}\), and by the frequency \(\nu_{ki}\). This is the case in which, for multiplets, the “intensity rules,” into which, as is known, the values of the effective cross sections do not enter, are fulfilled.

Along with this let us consider the case in which the role of collisions of the second kind with electrons is so small that it may be neglected. In addition, for simplicity, let us assume that excitation by collisions of the first kind occurs practically only from the normal level. Then the stationarity condition (10) is further simplified and takes the form

\[ \Delta N_{0k}(N_0,N_e)+\sum_{l=k+1}^{\infty}\Delta N_{lk}(N_l) = \sum_{r=0}^{k-1}\Delta N'_{kr}(N_k). \tag{10a} \]

Here \(\Delta N_{0k}(N_0,N_e)\) is expressed by formula (9), while the other two terms are respectively in terms of the transition probabilities \(A_{lk}\) and \(A_{kr}\):

\[ \Delta N_{lk}(N_l)=N_l A_{lk}, \]

\[ \Delta N'_{kr}(N_k)=N_k A_{kr}. \]

Using these values, we obtain from (10a):

\[ N_k= \frac{ N_0N_e\displaystyle\int_{v_{0k}}^{\infty} Q_{0k}(v)F(v)v\,dv }{ \displaystyle\sum_{r=0}^{k-1} A_{kr} } + \frac{ \displaystyle\sum_{l=k+1}^{\infty} N_l A_{lk} }{ \displaystyle\sum_{r=0}^{k-1} A_{kr} } \]

and, consequently, by (6a), for the intensity of a line:

\[ I_{ki}= \frac{ N_0N_e\displaystyle\int_{v_{0k}}^{\infty} Q_{0k}(v)F(v)v\,dv }{ \displaystyle\sum_{r=0}^{k-1} A_{kr} } A_{ki}h\nu_{ki} + \frac{ \displaystyle\sum_{l=k+1}^{\infty} N_l A_{lk} }{ \displaystyle\sum_{r=0}^{k-1} A_{kr} } A_{ki}h\nu_{ki}. \tag{11} \]

The conditions under which formula (11) is applicable can be realized with a sufficient degree of approximation. They are realized in the positive column of a glow discharge in a monatomic gas, at low gas pressure and low discharge-current density. As is seen, under these conditions the intensity of a line is determined by the sum of two terms, of which the first, depending on the effective cross section \(Q_{0k}\), takes into account the role of direct excitations by electron impacts, and the second—the role of cascade transitions. The latter, in turn, are determined by the effective cross sections \(Q_{0l}\) \((l=k+1, k+2,\ldots,\infty)\). Thus, in the case under consideration, the intensity of a line depends on the effective cross sections corresponding to transitions from the normal level to the \(k\)-th level and to all higher-lying levels.

The role of cascade transitions, depending on the electron temperature \(T_e\), and also depending on the values of \(Q_{0l}\) and \(A_{lk}\), can be different. We have seen that it is significant even in the excitation of atoms by an electron beam, possibly homogeneous in velocities. In a plasma, however, with the distribution of electrons by velocities according to Maxwell’s law, there are electrons

different velocities. As is known, when Maxwell’s law is satisfied, about 30% of the total number of particles have velocities exceeding the most probable velocity by more than one and a half times.

However, in order to reveal more clearly the role of the effective cross sections corresponding to direct excitation, let us neglect cascade transitions. Then the last term in formula (11) drops out and we obtain:

\[ I_{ki}= \frac{ N_0N_e\displaystyle\int_{v_{0k}}^{\infty} Q_{0k}(v)F(v)v\,dv }{ \displaystyle\sum_{r=0}^{k-1} A_{kr} } \,A_{ki}h\nu_{ki}. \tag{11a} \]

Let us consider, for this case, the ratio of the intensities of two close lines with different upper levels and a common lower level. Suppose that the excitation functions of the upper levels are identical, so that the corresponding effective cross sections \(Q_{0k}\) and \(Q_{0l}\) differ only in their maximum values. Then, since we consider \(v_{0k}\) and \(v_{0l}\) to differ little from one another,

\[ \frac{ \displaystyle\int_{v_{0k}}^{\infty} Q_{0k}(v)F(v)v\,dv }{ \displaystyle\int_{v_{0l}}^{\infty} Q_{0l}(v)F(v)v\,dv } \simeq \frac{Q_{0k\,\max}}{Q_{0l\,\max}} \]

and for the ratio of the line intensities we obtain:

\[ \frac{I_{ki}}{I_{li}} = \frac{\nu_{ki}}{\nu_{li}}\, \frac{A_{ki}}{A_{li}}\, \frac{ \displaystyle\sum_{r=0}^{l-1} A_{lr} }{ \displaystyle\sum_{r=0}^{k-1} A_{kr} } \cdot \frac{Q_{0k\,\max}}{Q_{0l\,\max}}. \tag{12} \]

As is evident, the ratio of the intensities of two lines now depends explicitly on the values of the effective cross sections at their maxima. If these lines are components of a spectral multiplet, then the ratio of their intensities does not obey the “intensity rules.”

A case of such a deviation from the “intensity rules” was observed for doublets of the diffuse series of thallium by I. P. Bogdanova\(^{7}\). By photographic photometry, the intensity ratio of the Tl I lines, \(6^2P_{3/2}-6^2D_{3/2}\), \(\lambda 3529\) Å and \(6^2P_{3/2}-6^2D_{5/2}\), \(\lambda 3519\) Å, was measured. The measurements were made for the positive column in a discharge in pure thallium vapor and in thallium vapor with an admixture of argon. Reabsorption was taken into account. The intensity ratios (corrected for reabsorption) at different argon pressures \(P_{\mathrm{Ar}}\) are given in Table II.

According to the “intensity rules,” the ratio \(I_{3519}/I_{3529}\) should be equal to 9. As is seen, in a discharge in pure thallium vapor there is a sharp violation of the “intensity rule”: the line \(\lambda 3519\) is approximately two times wea-

line \(\lambda 3529\), instead of being 9 times brighter. This violation of the “intensity rule” is not connected with any anomalies in the values of the transition probabilities \(A_{ki}\). According to measurements by G. S. Kvater\(^8\), carried out by D. S. Rozhdestvenskii’s “hook” method, the ratio of the numbers \(f_{ik}\) for the Tl I lines \(\lambda\lambda 3519;\ 3529\) is equal to 9, which exactly corresponds to the “intensity rules.”

Table II

Tl I; \(t=600^\circ\mathrm{C};\ i=40\ \mu\mathrm{A}\)

\(P_{\mathrm{Ar}}\), mm Hg	0	0.8	1.8	2.1	3.1	4.2	5.8
\(\dfrac{I_{3519}}{I_{3529}}\)	0.45	1.15	1.32	1.54	1.71	1.90	2.00

Obviously, the excessively small value for the intensity ratio of the thallium lines \(\lambda\lambda 3519;\ 3529\) is explained, in accordance with formula (12), by the large difference in the effective cross sections leading to excitation of the levels \(6^2D_{3/2}\) and \(6^2D_{5/2}\). It should be considered that excitation of these levels takes place from the normal level of Tl I \(6^2P_{1/2}\) (Fig. 7), since at the comparatively low temperature (\(600^\circ\mathrm{C}\)) at which the experiment was performed, the equilibrium population of the \(6^2P_{3/2}\) level is small. Thus, we obtain that

\[ Q_{\max}(6^2P_{1/2}\to 6^2D_{3/2}) \gg Q_{\max}(6^2P_{1/2}\to 6^2D_{5/2}). \]

This circumstance is understandable from the theoretical point of view: the transition \(6^2P_{1/2}\to 6^2D_{5/2}\) is associated with a reorientation of the direction of the spin angular momentum of the valence electron in the thallium atom, whereas the transition \(6^2P_{1/2}\to 6^2D_{3/2}\) is not associated with such reorientation. A transition with reorientation is less probable than one without reorientation. The reverse spontaneous (radiative) transition \(6^2D_{5/2}\to 6^2P_{1/2}\) is altogether forbidden by the selection rule for the quantum number \(J\).

Fig. 7. Scheme of thallium levels.

Analogously excessively small intensity ratios have been found for two other components of the doublets of the diffuse series of thallium, namely for the lines:

\[ 6^2P_{3/2} - 7^2D_{3/2,\ 5/2},\ \lambda\lambda 2918;\ 2921\ \text{Å}, \]

\[ 6^2P_{3/2} - 8^2D_{3/2,\ 5/2},\ \lambda\lambda 2709;\ 2710\ \text{Å}. \]

The intensity ratio \(I_{3519}/I_{3529}\), as follows from Table II, increases upon the addition of argon; the line \(\lambda 3519\) becomes brighter than the line \(\lambda 3529\), although the ratio of their intensities still remains much smaller than 9. The same increase in the intensity ratio upon addition of argon is also observed for the lines \(\lambda\lambda 2918;\ 2921\) and \(\lambda\lambda 2709;\ 2710\).

Direct measurements carried out by the anomalous-dispersion method show that the addition of argon does not change the values of the numbers \(f_{ik}\). Thus, the change in the line intensities upon the addition of argon occurs because of a change in the populations of the levels \({}^{2}D_{3/2}\) and \({}^{2}D_{5/2}\). A change in the population of these two levels may occur either through transitions between them caused by collisions of the second kind, or by a change in the character of their excitation. That the first of these causes does not play a substantial role was shown by means of the following experiment, carried out by I. P. Bogdanova. Pure thallium vapors were optically excited by the line \(\mathrm{Tl\ I}\ 6^{2}P_{1/2} — 6^{2}D_{3/2}\), \(\lambda 2768\ \text{Å}\). Then in the fluorescence spectrum, in accordance with the arrangement of the energy levels in the thallium atom (Fig. 7), only two lines were observed: \(\lambda 2768\) and \(\lambda 3529\). The line \(\lambda 3519\) was absent. Upon addition of argon the line 3519 appeared, but was very weak. This meant that, upon addition of argon, transitions \(6^{2}D_{3/2} \to 6^{2}D_{5/2}\) did occur, but rarely. Consequently, they could not produce a substantial change in the populations of the levels \(6^{2}D_{3/2}\) and \(6^{2}D_{5/2}\).

Measurements carried out by the anomalous-dispersion method showed that, upon addition of argon (at unchanged temperature), the population of the metastable levels \(6^{2}P_{3/2}\) of thallium increases in the discharge. At the same time, the concentration of atoms in the normal state \(6^{2}P_{1/2}\) decreases. In some cases the number of thallium atoms in the metastable state reached 60% of the total number of atoms. Such an increase in the population of the metastable level \(6^{2}P_{3/2}\) explains the change in the intensities of the lines \(\lambda\lambda 3529;\ 3519\). Now excitation of the levels \(6^{2}D_{3/2}\) and \(6^{2}D_{5/2}\) occurs not only from the normal level \(6^{2}P_{1/2}\), but also from the metastable level \(6^{2}P_{3/2}\). The transition \(6^{2}P_{3/2} \to 6^{2}D_{5/2}\), however, is not connected with reorientation of the spin moment of the valence electron, and to it there must correspond a large value of the effective cross section \(Q_{ik}\). As a result, the level \(6^{2}D_{5/2}\) is populated more strongly and the intensity of the line \(\lambda 3519\) increases, as is also observed experimentally.

6. EXCITATION OF RESONANCE LINES

As is known, a resonance line is a line arising in a transition from the first excited level to the normal level (Fig. 8).

Fig. 8. Excitation of a resonance line.

When a resonance line is emitted in a low-pressure gas-discharge plasma, at small electric-current densities, all processes may be neglected except direct excitations and spontaneous transitions. Then the intensity of the resonance line is expressed by formula (11), in which the sum \(\sum_{r=0}^{k-1} A_{kr}\) reduces to one term \(A_{10}\); as a result one obtains:

\[ I_{10}=N_{0}N_{e}\int_{v_{01}}^{\infty} Q_{01}(v)F(v)v\,dv\cdot h\nu_{10} +\sum_{l=2}^{\infty}N_{l}A_{l1}\cdot h\nu_{10}. \tag{13} \]

It follows from formula (13) that the intensity of the line \(I_{10}\) does not depend on the probability of the transition \(1\to0\) corresponding to it. This occurs because, under the discharge conditions considered, the number of acts of emission of the line

ROLE OF THE EFFECTIVE CROSS SECTIONS OF ATOMS IN THE EXCITATION OF SPECTRA

is equal to the number of acts of excitation of its initial level. However, it should be borne in mind that in reality the intensity of resonance lines is strongly affected by the phenomenon of reabsorption, which we do not consider here.

The second term in formula (13) expresses the role of cascade transitions; as we have indicated, they cannot be neglected. Nevertheless, in a number of works this term was neglected, and the intensity of the resonance line was calculated by the formula

\[ I_{10}=N_0N_e\int_{v_{01}}^{\infty} Q_{01}(v)F(v)v\,dv\cdot h\nu_{10}. \tag{13a} \]

For example, V. A. Fabrikant\(^9\) calculated by formula (13a) the intensity of the resonance lines of sodium and mercury and obtained good agreement with experimental data. Likewise, V. M. Zakharova and Yu. M. Kagan\(^ {10}\) used relation (13a) to calculate the intensities of lines of the subordinate series of sodium. The question naturally arises why, in these works, the role of cascade transitions did not make itself felt. The point is that the authors used, for \(Q_{ik}\), data in which what was determined was not the true effective cross section with respect to an impact leading to a transition from the normal level to the level initial for the given line, but the optical excitation function\(^*\). Optical excitation functions already take into account the role of cascade transitions. If the calculation of the intensity of the resonance line were carried out according to formula (13a), in which \(Q_{01}(v)\) meant the true effective cross section with respect to an impact leading to the transition \(0\to1\), then a noticeable discrepancy with experiment would have been obtained. The use of “worse,” “smoothed” values for \(Q_{01}(v)\) gives a better result, since in this case cascade transitions are taken into account.

The excitation functions of individual levels of mercury, as follows from the measurements described above by S. E. Frisch and I. P. Zapesochnyi, possess a sharp maximum and have the form shown in Fig. 9, a. In all probability, for the majority of other transitions in various atoms the excitation functions of individual levels have approximately the same form. “Smoothed” excitation functions, which take cascade transitions into account, can be approximately represented by the curve shown in Fig. 9, b.

In order to reveal the dependence of the line intensity on the form of the excitation function, we shall carry out a calculation using a simple algebraic approximation for \(Q_{ik}\). In doing so we shall regard \(Q_{ik}\) not as a function of the electron velocity \(v\), but as a function of the accelerating potential \(V\), related to the velocity \(v\) by the relation

\[ eV=\frac{mv^2}{2}, \]

where \(e\) and \(m\) are the charge and mass of the electron. As an approximation convenient for us, we use the following\(^ {11}\):

\[ \left. \begin{array}{ll} Q(V)=0 & \text{for } V<V_k,\\[6pt] Q(V)=\dfrac{(Q_m-Q_\infty)V_k}{V}+Q_\infty & \text{for } V\gg V_k. \end{array} \right\} \tag{14} \]

\(^*\) In carrying out the calculation, V. A. Fabrikant used a certain algebraic approximation to express the dependence of \(Q_{01}\) on the velocity of the exciting electrons. However, this approximation was chosen so as to represent in the best way the experimental data, which in fact referred to the optical excitation functions.

Here \(V_k\) denotes the critical potential. The form of the curves corresponding to approximation (14) is shown in Figs. 10, \(a, b, c\). Figure 10, \(a\) pertains to the case of certain arbitrary values of \(Q_m\) and \(Q_\infty\), while Figs. 10, \(b\) and 10, \(c\), respectively, pertain to the cases \(Q_\infty=0\) and \(Q_\infty=Q_m\). As can be seen, for \(Q_\infty \ll Q_m\) one obtains a curve with a sharp maximum, similar to

Fig. 9. Form of the excitation functions:
a) excitation function of an individual energy level;
b) excitation function taking cascade transitions into account.

Fig. 10. Form of the excitation functions corresponding to approximation (14).

the excitation function of an individual energy level of the atom (cf. Fig. 9, \(a\)). For \(Q_\infty=Q_m\) one obtains a constant value of the excitation function for all \(V>V_k\); this case is close to the “smoothed” optical excitation function that takes into account the role of cascade transitions (Fig. 9, \(b\)).

Let us calculate the integral

\[ f(V)=\frac{1}{Q_{\max}}\int_{V_k}^{\infty} Q(V)\,F(V)\sqrt{V}\,dV, \tag{15} \]

using the adopted approximation (14) for \(Q(V)\).

Let us suppose that excitation of the line occurs in a plasma in which the electron velocities are distributed according to the Maxwell law corresponding to the temperature \(T_e\).

Then

\[ F(V)\,dV=\frac{8e^{3/2}}{\sqrt{2\pi}\cdot a^3 m^{3/2}}\sqrt{V}\,e^{-\frac{eV}{kT_e}}\,dV, \tag{16} \]

where

\[ a=\sqrt{\frac{2kT_e}{m}}. \]

Here \(k\) is Boltzmann’s constant.

Let us also introduce the quantity

\[ \tau_e=\frac{V_e}{V_k}, \]

where \(V_e\) is related to the electron temperature \(T_e\) by the relation

\[ eV_e=\frac{3}{2}kT_e . \]

Obviously, \(\tau_e\) has the meaning of an electron temperature expressed in fractions of the critical potential; in this case \(V_e=1\) corresponds to a temperature

Fig. 11. Form of the curves \(f(\tau_e)\) for the cases:
1) \(Q_\infty=0\); 2) \(Q_\infty=0.3Q_m\); 3) \(Q_\infty=0.8Q_m\);
4) \(Q_\infty=Q_m\), in the interval \(\tau_e\) from 0 to 15.

Fig. 12. Form of the curves \(f(\tau_e)\) in the interval \(\tau_e\) from 0 to 2.

7780°K. Substituting \(Q(V)\) and \(F(V)\) from (14) and (16) into formula (15) and carrying out the integration, we obtain (up to a constant factor):

\[ f(\tau_e)=\left\{\tau_e^{-1/2}+\frac{3}{2}\frac{Q_\infty}{Q_m}\tau_e^{1/2}\right\}e^{-\frac{3}{2\tau_e}} . \tag{17} \]

Hence we find that the intensity of the resonance line, when relation (13a) is used, is equal to

\[ I_{10}=aN_0N_eQ_m f(\tau_e)h\nu_{10}, \tag{18} \]

where \(a\) is a constant factor.

Formula (18) gives the intensity of the line \(I_{10}\) as a function of the electron concentration \(N_e\) and the electron temperature \(T_e\). The form of the function \(f(\tau_e)\) for the following four cases: 1) \(Q_\infty=0\); 2) \(Q_\infty=0.3Q_m\); 3) \(Q_\infty=0.8Q_m\) and 4) \(Q_\infty=Q_m\)—is shown in Fig. 11 in the range of values of \(\tau_e\) from 0 to 15, and in Fig. 12 on a larger scale in the range of values of \(\tau_e\) from 0 to 2. Only for the case \(Q_\infty=0\) does the function \(f(\tau_e)\) have a maximum, lying at \(\tau_e=3\). For \(Q_\infty\ne0\), at large \(\tau_e\) the function \(f(\tau_e)\) increases proportionally to \(\tau_e^{1/2}\). At small \(\tau_e\), approximately

\[ f(\tau_e)\sim \tau_e^{-1/2}\,e^{-\frac{3}{2\tau_e}}, \]

i.e., at small \(\tau_e\)

the form of the function \(f(\tau_e)\) does not depend on the value of the ratio \(Q_\infty/Q_m\). Since in most cases the electron temperature of the plasma, expressed in volts, is of the same order as or smaller than the critical potential \(V_k\), one has to deal with small \(\tau_e\). Under these conditions the form of the excitation function has little effect on the dependence of \(I_{10}\) on the electron temperature \(\tau_e\). Substantially different dependences are obtained only at large electron temperatures. Let us also note that at small \(\tau_e\) the function \(f(\tau_e)\) increases rapidly with increasing electron temperature \(T_e\). For example, for the excitation potential \(V_k = 7.5\ \mathrm{V}\) we find that, when the electron temperature is increased from \(25\,000^\circ\mathrm{K}\) to \(35\,000^\circ\mathrm{K}\), the function \(f(\tau_e)\) changes from the value \(2.10 \cdot 10^{-2}\) to \(1.06 \cdot 10^{-1}\), i.e., almost by a factor of 5. This means that, at small \(\tau_e\), the line intensity depends very strongly on the electron temperature \(T_e\).

Let us note that the electron velocity distribution function (16) has a maximum at

\[ V = V_e - \frac{kT_e}{2e} = \frac{1}{3}\,V_e . \]

Consequently, in most cases the maximum of the electron velocity distribution curve lies at potentials smaller than the critical one \((V_e \leq V_k)\), and excitation of the line takes place at the expense of the “tail” of the Maxwellian curve.

7. THE ROLE OF STEPWISE EXCITATIONS

According to formula (18), in direct excitations the intensity of the resonance line is proportional to the electron concentration \(N_e\) and depends on the electron temperature \(T_e\). In gas-discharge plasma in the positive column, as the discharge-current density \(i\) increases, there usually occurs an increase in the electron concentration \(N_e\) and a decrease in the electron temperature \(T_e\). In this case the electron concentration \(N_e\) increases either linearly with \(i\), or somewhat faster than linearly. The decrease in the electron temperature occurs slowly, so that over a certain interval of discharge-current densities \(T_e\) may approximately be regarded as constant. Then the line intensity should increase linearly with the electron concentration:

\[ I_{10} \sim aN_e . \tag{19} \]

Let us consider what the dependence of the line intensity on the electron concentration will be in the presence of stepwise excitations.

In the presence of stepwise excitations*) the stationarity condition (10a) is replaced by the following:

\[ \sum_{r=0}^{k-1} \Delta N_{rk}(N_r,N_e) + \sum_{l=k+1}^{\infty} \Delta N_{lk}(N_l) = \sum_{r=0}^{k-1} \Delta N'_{kr}(N_k) \tag{10b} \]

and for the intensity of the line \(I_{ki}\), instead of expression (11), we obtain the following:

\[ I_{ki} = \frac{ \displaystyle \sum_{r=0}^{k-1} N_r N_e \int_{v_{rk}}^{\infty} Q_{rk}(v)\,F(v)\,v\,dv }{ \displaystyle \sum_{r=0}^{k-1} A_{kr} } \,A_{ki}h\nu_{ki} + \frac{ \displaystyle \sum_{l=k+1}^{\infty} N_l A_{kl} }{ \displaystyle \sum_{r=0}^{k-1} A_{kr} } \,A_{ki}h\nu_{ki}. \tag{11a} \]

*) In all other respects we make the same simplifying assumptions that we used in formulating the stationarity condition (10a).

Let us neglect the role of cascade transitions and assume that excitation of the \(k\)-th level occurs, besides from the normal level, only from one excited level \((r\text{-th})\). Then we obtain:

\[ I_{ki}\sim N_0N_e\int_{v_{0k}}^\infty Q_{0k}(v)F(v)v\,dv +N_rN_e\int_{v_{rk}}^\infty Q_{rk}(v)F(v)v\,dv. \]

Finally, if, as was done above, certain approximations are used for the effective cross sections \(Q_{0k}(v)\) and \(Q_{rk}(v)\), we find:

\[ I_{ki}\sim N_0N_eQ_{0k\max}f_1(\tau_{e_1}) +N_rN_eQ_{rk\max}f_2(\tau_{e_2}). \tag{20} \]

The relative role of the first and second terms in expression (20) depends on a number of factors: on the relative values of \(Q_{0k\max}\) and \(Q_{rk\max}\), on the concentration of excited atoms \(N_r\), and on the electron temperature \(T_e\).

The role of stepwise excitations may be large in the presence of metastable states (relatively large \(N_r\)), and also in the excitation of ionic lines \(^{12}\). Excitation of an ion may occur by the direct route, i.e., as a result of collision of an electron with a normal atom, in which the atom is simultaneously ionized and excited. In addition, excitation of an ion may occur by a stepwise method: first an ion is formed in the normal state, and then it is excited. In the presence of both a direct and a stepwise process, the intensity of the ionic line is expressed by the relation

\[ I_{ki}\sim N_0N_eQ_{0k\max}f_1(\tau_{e_1}) +N_jN_eQ'_{0k\max}f_2(\tau_{e_2}), \tag{21a} \]

where \(N_j\) is the concentration of ions in the normal state, and \(Q_{0k\max}\) and \(Q'_{0k\max}\), respectively, are the effective cross sections at the maximum for collisions leading to direct excitation of the ion and to its excitation from the normal state of the ion.

For a quasineutral plasma \(N_j=N_e\), as a result of which from formula (21) one obtains

\[ I_{ki}\sim \left(\frac{N_e}{N_0}\right)Q_{0k\max}f_1(\tau_{e_1})+ \]

\[ +\left(\frac{N_e}{N_0}\right)^2Q'_{0k\max}f_2(\tau_{e_2}). \tag{21b} \]

Depending on the relation between \(Q_{0k\max}\) and \(Q'_{0k\max}\) and on the discharge conditions, the role of the first and second terms in formula (21b) will be different. In a low-pressure discharge, when the electron temperature is high, the direct excitations should play the main role, and according to formula (21a) the dependence of \(I_{ki}\) on the electron concentration will be close to linear. At higher pressures, when the electron temperature is lower, the second term in formula (21a) will also begin to play a substantial role, and the dependence of \(I_{ki}\) on \(N_e\) will assume a parabolic character.

Fig. 13. Dependence of the intensity of the Hg II line, \(\lambda 3983\), on the discharge-current strength at pressure \(p=3\cdot10^{-4}\) mm Hg.

The excitation of ionic lines in a discharge in mercury vapor under different conditions was investigated by Yu. M. Kagan and V. M. Zakharova \(^{13}\). Figures 13 and 14 present the dependences they obtained for the intensity of the ionic line Hg II, \(\lambda 3983\) Å, on the discharge-current strength. Figure 13 refers to observations made at a mercury-vapor pressure of \(3.0\cdot10^{-4}\) mm Hg, and Fig. 14 at a pressure of \(3.0\cdot10^{-3}\) mm Hg. As is seen, in the first case the dependence \(I_{3983}\) on the discharge-current strength is approximately linear, while in the second—

parabolic. Since under the given conditions the electron concentration increases approximately linearly with the discharge current, the dependence of \(I_{3983}\) on the electron concentration \(N_e\) retains the same character. It must be borne in mind, however, that in a discharge at a mercury-vapor pressure of \(3\cdot 10^{-3}\) mm Hg, as the current increases the electron temperature drops appreciably, so that in formula (216) \(f_1(\tau_{e_1})\) and \(f_2(\tau_{e_2})\) cannot be regarded as constants.

Comparison of the experimental data with those that may be expected on the basis of formula (216) makes it possible to draw certain conclusions concerning the ratio of the effective cross sections \(Q_{0k\max}\) and \(Q'_{0k\max}\). Let us first examine from this point of view the data relating to Fig. 13. At a mercury-vapor pressure of \(3.0\cdot 10^{-4}\) mm Hg and a discharge current of \(1\) a, probe measurements showed that the electron concentration is \(N_e = 0.72\cdot 10^{11}\ \text{cm}^{-3}\) and the electron temperature is \(T_e = 41100^\circ\) K. The atom concentration is \(N_0 = 1.03\cdot 10^{13}\ \text{cm}^{-3}\). The potential of direct excitation of the Hg II line, \(\lambda 3983\ \text{\AA}\), is \(17.9\) V, while the excitation potential, if reckoned from the normal state of the ion, is \(7.5\) V. This gives \(\tau_{e_1}=0.295\) and \(\tau_{e_2}=0.705\). We shall now determine the values of \(f(\tau_e)\). Since at small \(\tau_e\) the form of the approximation plays no essential role, we use the hyperbolic approximation (formula (14) with \(Q_\infty = 0\)); then we have \(f(\tau_{e_1}) = 1.14\cdot 10^{-2}\) and \(f(\tau_{e_2}) = 9.88\cdot 10^{-2}\). From the calculations presented we obtain that, at a current of \(1\) a, the quantities \(\left(\dfrac{N_0}{N_e}\right) f(\tau_{e_1})\) and \(\left(\dfrac{N_0}{N_e}\right)^2 f(\tau_{e_2})\) are respectively equal to \(7.98\cdot 10^{-5}\) and \(4.84\cdot 10^{-6}\). As is seen, the quadratic term is an order of magnitude smaller than the linear one. Comparing this result with the experimentally obtained linear dependence of \(I_{3983}\) on the discharge current, we come to the conclusion that \(Q'\) is of the same order as, or smaller than, \(Q\). If \(Q'\) were considerably larger than \(Q\), the role of the quadratic term would have shown itself in formula (216).

Fig. 14. Dependence of the intensity of the Hg II, \(\lambda 3983\), line on the discharge-current strength at pressure \(p = 3\cdot 10^{-3}\) mm Hg.

Let us now examine the case to which the curve shown in Fig. 14 refers. This curve was obtained at a mercury-vapor pressure of \(3\cdot 10^{-3}\) mm Hg. At this pressure, for a discharge current of \(1\) a, probe measurements gave: \(T_e = 30500^\circ\) K and \(N_e = 1.50\cdot 10^{12}\ \text{cm}^{-3}\). Hence we find \(\left(\dfrac{N_e}{N_0}\right)=1.46\cdot 10^{-2}\), and \(\left(\dfrac{N_e}{N_0}\right)^2 = 2.13\cdot 10^{-4}\); \(\tau_{e_1}=0.219\) and \(\tau_{e_2}=0.523\). From these data we calculate the values \(f(\tau_{e_1})\) and \(f(\tau_{e_2})\), which turn out to be \(2.27\cdot 10^{-3}\) and \(7.86\cdot 10^{-2}\). Using these results, we find that in the case under consideration

\[ \left(\frac{N_e}{N_0}\right) f(\tau_{e_1}) = 3.31\cdot 10^{-5}, \]

and

\[ \left(\frac{N_e}{N_0}\right)^2 f(\tau_{e_2}) = 1.67\cdot 10^{-5}, \]

i.e., both of these factors are of the same order. This means that, for \(Q' \simeq Q\), the quadratic term in the expression for the intensity \(I_{3983}\) will play an essential role, as is also found experimentally. Thus, we arrive at the conclusion that the effective cross sections for direct excitation of the ion and for excitation from the normal state of the ion are of the same order.

8. DETERMINATION OF EFFECTIVE CROSS SECTIONS FOR COLLISIONS OF THE SECOND KIND

In conclusion, let us consider the possibility of spectroscopic determination of the effective cross section for collisions of the second kind between two atoms. As is known, a collision of the second kind between two atoms is an inelastic collision in which a nonradiative exchange of excitation energies takes place. Let us consider atoms of two kinds, denoting these atoms in their normal states by \(A\) and \(B\), and in excited states by \(A^*\) and \(B^*\). Then a collision of the second kind may be represented as follows:

\[ A + B^* + \Delta W_{\mathrm{k}} \to A^* + B + \Delta W'_{\mathrm{k}}, \]

where \(\Delta W_{\mathrm{k}}\) and \(\Delta W'_{\mathrm{k}}\) are the kinetic energies of the relative motion of the atoms, respectively before and after the collision. The effective cross sections for collisions of the second kind are appreciable only in the case when the excitation energies of atoms \(A\) and \(B\) differ little from one another.

Collisions of the second kind manifest themselves most distinctly in sensitized fluorescence: in a mixture of vapors, atoms of one kind (\(A\)) are optically excited; the fluorescence spectrum contains spectral lines of both atoms. Under these conditions, the lines of atoms of the second kind (\(B\)) are excited as a result of collisions of the second kind with excited atoms \(A^*\).

Although sensitized fluorescence has been the subject of many works, the question of the magnitude of the effective cross sections for collisions of the second kind still remains unresolved. Experimentally, not only have the absolute values of the effective cross sections not been determined, but there are not even reliable data on their relative values. Beutler and Josephi\(^{14}\) attempted to determine, from sensitized fluorescence, the relative values of the effective cross sections for collisions of the second kind between excited mercury atoms and normal sodium atoms. The relative values of the effective cross sections for excitation of the \(S\)- and \(D\)-levels of sodium were given by them in the form of a curve, which was then reproduced in a large number of monographs and textbooks\(^*\). In reality, however, their data are very approximate, since, first, they calculated the relative values of the effective cross sections from the measured line intensities without taking proper account of transition probabilities, and, second, they divided (on the basis of an erroneous generalization of the intensity rules) the intensities of the lines of the diffuse series by 5.

Let us consider in more detail the conditions of excitation of atoms in experiments on sensitized fluorescence. Suppose there is a mixture of mercury and sodium vapors, in which the mercury is excited optically, and the sodium—through collisions of the second kind with excited mercury atoms. Under these conditions it may be assumed that the population of some sodium level occurs as a result of two processes: a) collisions of the second kind with excited mercury atoms; b) cascade transitions from higher levels. The destruction of the same level also occurs as a result of two processes: a) spontaneous radiation; b) quenching collisions with normal mercury atoms.

The number of exciting transitions \(\Delta N(A)\) in atoms \(A\), caused by collisions of the second kind between normal atoms \(A\) and excited atoms \(B^*\),

\(^*\) See, for example, E. V. Shpol’skii, Atomic Physics, vol. II, fig. 286, Gostekhizdat, 1950.

can be written in the following form:

\[ \Delta N(A)=N_0(A)\cdot N_m(B^*)Q_{0k}\cdot \bar v, \tag{22} \]

where \(N_0(A)\) is the number of normal atoms \(A\) per unit volume, \(N_m(B^*)\) is the number of excited atoms \(B^*\) in the \(m\)-th state per unit volume, \(Q_{0k}\) is the effective cross section of the collision under consideration, and \(\bar v\) is the relative velocity of the colliding atoms.

Using expression (22), it is easy to write the stationarity condition. Suppose that only one level of mercury is optically excited (the \(m\)-th level). Then the stationarity condition will have the form:

\[ N_0(\mathrm{Na}^*)\cdot N_m(\mathrm{Hg}^*)Q_{0k}\cdot \bar v+ \sum_{l=k+1}^{\infty} N_l(\mathrm{Na})\cdot A_{lk} = \]

\[ = N_0(\mathrm{Hg})\cdot N_k(\mathrm{Na}^*)\cdot Q_{k0}\cdot \bar v+ N_k(\mathrm{Na})\cdot \sum_{r=k-1}^{0} A_{kr}. \tag{23} \]

Between the effective cross sections \(Q_{0k}\) and \(Q_{k0}\) there is the relation

\[ g_0p^2Q_{0k}=g_kp'^2Q_{k0}, \]

where \(g_0\) and \(g_k\) are statistical weights and \(p,p'\) are the momenta of the atoms.

If the experiment is performed at such a low pressure that the time between atomic collisions is greater than the lifetime of the excited states of the sodium atom, then in equality (23) the term \(N_0(\mathrm{Hg}^*)N_k(\mathrm{Na})Q_{k0}\cdot \bar v\) may be neglected. Then equality (23) can be rewritten in the form

\[ N_0(\mathrm{Na})N_m(\mathrm{Hg}^*)Q_{0k}\bar v = N_k(\mathrm{Na})\cdot \sum_{r=k-1}^{0} A_{kr} - \sum_{l=k+1}^{\infty} N_l(\mathrm{Na})A_{lk}. \tag{23a} \]

Hence, using the expression for the intensity of the sodium lines:

\[ I_{ki}=N_k(\mathrm{Na})\cdot A_{ki}\cdot h\nu_{ki}, \]

we obtain:

\[ Q_{0k}\sim \frac{ I_{ki}\displaystyle\sum_{r=k-1}^{0} A_{kr} }{ A_{ki}\cdot h\nu_{ki} } - \sum_{l=k+1}^{\infty} N_l(\mathrm{Na})A_{lk}. \tag{24} \]

As is clear from formula (24), in order to find \(Q_{0k}\) it is necessary not only to measure the intensity of the line \(I_{ki}\) in sensitized fluorescence, but also to know all \(A_{ki}\) and \(A_{lk}\), as well as the concentrations of the atoms \(N_l(\mathrm{Na})\). The Einstein coefficients \(A_{ki}\) and \(A_{lk}\) can, generally speaking, be calculated by the corresponding approximate method. The concentrations \(N_l(\mathrm{Na})\) can be determined only by measuring, in the spectrum of sensitized fluorescence, the intensities of a large number of weak lines. Experimentally this presents considerable difficulties. Therefore we shall make certain simplifying assumptions.

Let us consider the scheme of the arrangement of the energy levels of sodium and mercury. Fig. 15 shows the relative arrangement of the sodium levels and the pred—

compiled transitions leading to the emission of the lines of the subordinate series \(n^{2}S_{1/2}\to 3^{2}P_J\) and \(n^{2}D_J\to 3^{2}P_J\). On the left is marked the position of the mercury level \(6^{3}P_1\). If mercury vapor is excited only by means of the mercury resonance line \(\lambda 2537\ \text{\AA}\), then the excited mercury atoms will be only in this state. As can be seen, the mercury level \(6^{3}P_1\) lies very close to the sodium levels \(7S\), \(8S\), \(9S\) and \(6D\), \(7D\), \(8D\). Excitation of these levels by collisions of the second kind with mercury atoms in the \(6^{3}P_1\) state will be large; excitation

Fig. 15. Scheme of sodium levels.

of the remaining levels by collisions of the second kind will be much smaller. Therefore, in determining the effective cross sections leading to excitation of levels close to the mercury level, the role of cascade transitions may be neglected.

Then formula (24) takes the form

\[ Q_{0k}\sim \frac{ I_{ki}\displaystyle\sum_{r=k-1}^{0} A_{kr} }{ A_{ki}h\nu_{ki} }. \tag{24a} \]

To calculate \(Q_{0k}\) by this formula, one must know not only the coefficient \(A_{ki}\), but also the coefficients \(A_{kr}\) for transitions to all levels lying below the \(k\)-th level under consideration.

For the case, for example, when excitation of the sodium level \(9^{2}S_{1/2}\) is considered, one must know \(A_{kr}\) for the transitions \(9^{2}S_{1/2}\to 8^{2}P_J,\ 7^{2}P_J,\ 6^{2}P_J,\ 5^{2}P_J,\ 4^{2}P_J,\ 3^{2}P_J\). The probabilities of a number of transitions in the sodium atom have been calculated by the approximate

method developed by M. I. Petrashen and I. V. Abarenkov[^15]. The results of these calculations for the transitions \(9^2S_{1/2}\to n^2P_J\) are presented in Fig. 16. As can be seen, the values of the probabilities decrease rapidly as the principal quantum number \(n\) increases. Therefore, as an approximation for a number of high levels, for example \(7^2S_{1/2}\), \(8^2S_{1/2}\), etc., the value of the sum

\[ \sum_{r=k-1}^{0} A_{kr} \]

may be regarded as constant, and it may be assumed that for these levels

\[ Q_{0k}\sim \frac{I_{ki}}{A_{ki}\cdot h\nu_{ki}} . \tag{25} \]

For calculating \(Q_{0k}\) in this simplified approximation it is sufficient to know only one coefficient \(A_{ki}\). This formula was used in the work of S. E. Frisch and E. K. Kraulin1 to calculate the effective cross sections of collisions of the second kind leading to excitation of the \(n^2S_{1/2}\)- and \(n^2D_J\)-levels of sodium, lying close to the mercury level \(6^3P_1\).

Fig. 16. Transition probabilities \(9^2S_{1/2}\to n^2P_J\) in the sodium atom.

The calculations were carried out on the basis of the results of measurements performed by the authors under experimental conditions somewhat improved in comparison with those under which the measurements of Beitler and Josephy were carried out. The sensitized fluorescence of sodium vapor was excited in a tube made of glass resistant to sodium vapor. The tube was illuminated by a cooled mercury lamp. The glass from which the tube was made is transparent in thin layers to the mercury line \(\lambda 2537\,\text{\AA}\), but absorbs all shorter-wavelength radiation. In addition, by means of liquid filters (diphenylbutadiene + cobalt sulfate + nickel sulfate), the longer-wavelength radiation was also absorbed. Thus, excitation of the mercury vapor was produced only by means of the single mercury line \(\lambda 2537\,\text{\AA}\), and collisions of the second kind occurred only between normal sodium atoms and excited mercury atoms in the \(6^3P_1\) state. The pressure of the mercury vapor was \(0.008\) mm Hg, and that of the sodium vapor \(0.01\)—\(0.02\) mm Hg.

By the method of photographic photometry, the relative intensities of 12 lines of the subordinate series of sodium \(n^2S_{1/2}\to 3^2P_J\) and \(n^2D_J\to 3^2P_J\), and of the resonance line of sodium \(3^2P_J\to 3^2S_{1/2}\), were measured in the fluorescence spectrum. The intensities of all lines, by comparison with the continuous spectrum obtained from an incandescent tungsten strip, were referred to the intensity of the line \(\lambda 4983.79\,\text{\AA}\), which was conventionally taken as 100. The photographs were taken with a spectrograph slit so wide that the components of the doublet structure of the lines merged.

Table III gives: the symbols of the transitions and the wavelengths of the lines studied; the energy differences \(\Delta W\) of the sodium levels and the mercury level \(6^3P_1\) (the sign \(+\) means that the sodium level lies above the mercury level \(6^3P_1\)); the measured relative intensities of the lines \(I_\lambda\); the relative effective cross sections \(Q_{0k}\), calculated by formula (25). The effective cross section for excitation of the atom[^15]: M. I. Petrashen and I. V. Abarenkov.

sodium to the level \(5^2D_J\) (the upper level of the line \(\lambda\,4938|79\)) is conventionally set equal to 1.00.

Table III

Transition	\(\lambda\)	\(\Delta W\), eV	\(I_\lambda\)	\(Q_{0k}\)
\(5S \to 3P\)	6161/54	\(-0.76\)	302	2.72
\(6S \to 3P\)	5154/49	\(-0.37\)	34	0.59
\(7S \to 3P\)	4752/48	\(-0.17\)	20	0.57
\(8S \to 3P\)	4545/42	\(-0.05\)	26	1.17
\(9S \to 3P\)	4423/20	\(+0.02\)	144	10.4
\(10S \to 3P\)	4343/40	\(+0.07\)	13	1.31
\(4D \to 3P\)	5688/83	\(-0.60\)	437	2.24
\(5D \to 3P\)	4983/79	\(-0.29\)	100	1.00
\(6D \to 3P\)	4669/65	\(-0.13\)	38	0.74
\(7D \to 3P\)	4498/94	\(-0.03\)	121	3.93
\(8D \to 3P\)	4993/90	\(+0.04\)	122	6.16
\(9D \to 3P\)	4324/21	\(+0.09\)	18	1.25
\(3P \to 3S\)	5896/90	\(-2.76\)	1700	—

In Fig. 17 the relative values of the effective cross sections are presented as a function of the energy difference \(\Delta W\), separately for the \(nS\)-levels and \(nD\)-levels.

From Fig. 17 it is seen that the effective cross sections increase strongly for small cross sections of the energy difference \(\Delta W\). At the same time, the effective cross sections are larger in those cases in which the sodium level lies above the mercury level \({}^3P_1\). This circumstance had already been noted in the work of S. E. Frisch and A. A. Ferkhmin \(^{17}\). The curves in Fig. 17 rise on going to excitation of the levels \(4D\), \(5D\), and \(5S\), which is explained by the role of cascade transitions; consequently, the simplifying assumptions adopted for calculating the effective cross sections of these levels become incorrect. The true values of the effective cross sections of the levels \(4D\) and \(5S\) must be small.

The large role of cascade transitions for levels lying far from the mercury level \(6^2P_1\) becomes especially clear if one pays attention to the sodium \(D\)-lines \(\lambda\,5896|90\) Å. In the spectrum of sensitized fluorescence of sodium vapor these lines are the brightest. At the same time, the effective cross section for collisions of the second kind of the level \(3^2P_J\) cannot be appreciable. Evidently, these lines are excited mainly at the expense of cascade transitions. Then the stationarity condition (23a) takes the form

\[ N_k(\mathrm{Na}) \cdot A_{ki} = \sum_{l=k+1}^{\infty} A_{lk} N_l(\mathrm{Na}), \]

whence for the intensity of the \(D\)-lines one obtains

\[ I_D = \nu_D \cdot \sum_{l=k+1}^{\infty} \frac{I_{lk}}{\nu_{lk}} . \tag{26} \]

Fig. 17. Relative values of the effective cross sections for collisions of the second kind between mercury and sodium atoms.

This expression does not contain the effective cross section determining the excitation of the sodium levels \(3^2P_J\).

The summation sign in expression (26) includes the intensities of all those lines for which the lower level is \(3^2P_J\). If this sum is restricted to the 12 lines of the subordinate series whose intensities were measured, one obtains \(I_D = 1234\). In this case, not only weak transitions from high levels remain unaccounted for, but also the intense transitions \(4^2S_{1/2} \to 3^2P_J\) and \(3^2D_J \to 3^2P_J\). Indeed, the number obtained, 1234, is somewhat smaller than the measured value \(I_D = 1700\). Thus it may be regarded as established that the sodium resonance lines arise in the spectrum of sensitized fluorescence chiefly as a result of cascade transitions.

References

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1. S. E. Frisch and E. K. Kraulin. ↩