Distribution of Energy in Spectral Series

L. Vegard. Lichterzeugung in Glimmlicht und Kanal-Strahlen. Ann. d’Phys. 39, p. 111, 1912.

J. Holstmark. Versuche über die Lichterregung durch Kathodenstrahlen in Wasserstoff. Phys. ZS. 15, p. 605, 1914.

R. Beatty. Energy Distribution in Spectra. Phil. Mag. Vol. 33, p. 49, 1917.

P. Foote and W. Meggers. Atomic Theorie and Low Voltage Arcs in Caesium Vapour. Phil. Mag. Vol. 40, p. 80, 1920.

At the present time, in view of the continuing development of Bohr’s theory, the study of the distribution of energy in spectral series is necessary both for confirming the theory and for clarifying the physical meaning of its details.

The question of the distribution of energy in line spectra has long been of interest to physicists. The first investigators of this question were Lockyer, Vogel, Angström, Langenbach, Pflüger, and others. From their work it followed that the law of the distribution of energy in the spectrum of an absolutely black body is applicable, to a certain extent, also to line spectra. One need only compare the lines of one and the same series with the corresponding places in the spectrum of a black body at a known temperature. The principal cause of line spectra was considered to be the high temperature of the gas.

As, over time, the role of chemical ionic processes in the radiation of a gas became increasingly clear, the question of energy distribution became more complicated. Before speaking of the distribution, it was necessary carefully to check all the conditions under which the radiation takes place. First of all, it was necessary to determine the influence of the electric field strength and of the pressure in the gas, for the value of the kinetic energy of the electron at the moment of the ionizing collision depends on these two quantities. From Vegard’s work on luminescence in canal rays and Holstmark’s work on luminescence in cathode rays it became clear that the magnitude of the electric field strength in the discharge tube does not affect the distribution of energy, while with a change in pressure the distribution of energy changes somewhat. When the pressure was decreased, the intensity of the short wavelengths increased by 10% more than the intensity of the long wavelengths. Vegard and Holstmark did not give explanations of these facts.

In 1917 there appeared a more extensive work on this question by Beatty, who investigated the distribution of energy in the lines of the Balmer series, \(H_{\alpha}\), \(H_{\beta}\), \(H_{\gamma}\), excited in a Geissler tube at different current strength and different pressure. Beatty gave absolute values of the energy of the given lines, using a photocell connected with an electrometer. Preliminary calibration of the readings was carried out by illuminating the photocell with a source of known energy distribution.

The most important results of Beatty’s work are as follows.

The maximum energy falls on the line \(H_{\alpha}\). The ratio of the intensities for the lines \(H_{\gamma}\), \(H_{\beta}\), \(H_{\alpha}\),

\[ \frac{H_{\beta}}{H_{\alpha}}, \quad \frac{H_{\gamma}}{H_{\alpha}}, \quad \frac{H_{\gamma}}{H_{\beta}} \]

remains constant at pressures above 3 mm. If, however, the pressure is below 3 mm, then the ratios

\[ \frac{H_{\beta}}{H_{\alpha}}, \quad \frac{H_{\gamma}}{H_{\alpha}}, \quad \frac{H_{\gamma}}{H_{\beta}} \]

increase up to twofold and more. In view of the fact that the independence of the distribution of energy from the intensity of the electric field had previously been established by the experiments of Vegard and Holstmark, Beatty explains the change he observed in the intensities of the spectral lines as follows.

According to Bohr’s theory, the lines \(H\alpha\), \(H\beta\), \(H\gamma\), are emitted in the transitions of an electron to the inner ring from various outer orbits. The radii of these orbits increase as the wavelength of the emitted spectral line decreases. Therefore an atom emitting the lines \(H\alpha\), \(H\beta\), \(H\gamma\) has a different diameter and a different mean free path between two collisions. Since interference of light is observed when the path difference of the rays is up to \(1{,}000{,}000\) and more waves, it follows that the quantum is not emitted instantaneously, and, in Beatty’s opinion, if the interval of time between two collisions is less than the interval during which the quantum can be emitted, then only part of the quantum \(h\nu\) is emitted, while the remaining part is converted into energy of another frequency. If the quantum is denoted by \(S_1\), the free path of the emitting atom by \(\lambda_1\), and the path over which the quantum would have time to be emitted completely by \(f_1\), then, according to Beatty, the amount of energy emitted by the atom between two collisions may approximately be taken as equal to \(S_1 \dfrac{\lambda_1}{f_1}\); for another frequency, \(S_2 \dfrac{\lambda_2}{f_2}\); the ratio of intensities is

\[ \frac{H\alpha}{H\beta}=S_1\frac{\lambda_1}{f_1}:S_1\frac{\lambda_2}{f_2}. \]

Since the quantities \(S_1, S_2, f_1, f_2\) are constant, the change of \(\dfrac{H\alpha}{H\beta}\) depends only on \(\dfrac{\lambda_1}{\lambda_2}\). In view of the fact that \(\lambda_1\) and \(\lambda_2\) vary inversely as the pressure, their ratio \(\dfrac{\lambda_1}{\lambda_2}\) does not depend on the pressure; consequently \(\dfrac{H\alpha}{H\beta}\) also does not depend on the pressure, as was observed at pressures above \(3\) mm.

At lower pressures, however, \(\lambda_1\) is greater than or equal to \(f_1\), i.e. the quantum has time to be emitted between two collisions; therefore \(\dfrac{H\alpha}{H\beta}\) now depends only on \(\lambda_2\). With a further lowering of the pressure the energy \(H\alpha\) remains unchanged, while the energy \(H\beta\) increases, \(\dfrac{H\beta}{H\alpha}\) increases, until, finally, \(\lambda_2\) becomes equal to \(f_2\). When such a pressure has been reached, \(\dfrac{H\beta}{H\alpha}\) attains a maximum, and with a further lowering of the pressure \(\dfrac{H\beta}{H\alpha}\) no longer changes. In confirmation of this Beatty refers to the experiments of Vegard, who worked at very low pressures, where the change in the ratio of the energies \(\dfrac{H\beta}{H\alpha}\) was very small. Calculating how many complete oscillations of \(H\alpha\) an atom can emit at [[unclear: pressure value]] mm in the interval between two collisions, Beatty obtains the number \(8\cdot 10^5\), close to the data from experiments on the interference of light.

It is interesting to note that Beatty, proceeding from the foundations of Bohr’s theory, does not obtain a quantum of radiation at high pressures in the gas. It should also be noted that in Beatty’s treatment the electron emits the quantum on the initial orbit, and only afterward jumps to the corresponding inner one. Beatty does not at all touch upon the question of what the probability is of finding the electron on one or another initial orbit, and the related probability of one or another jump.

Foot and Meggers in 1920 came considerably closer to this question. Their work is entitled “Atomic Theory and the Arc at Low Potential in Caesium Vapors.” Foot and Meggers distinguish atoms excited and not excited to radiation. In an unexcited atom the electron is located on the innermost Bohr orbit. If the principal spectral series is denoted by

\(1.5S—pm\), then this inner orbit may be denoted by \(1.5S\). Under the influence of an external impulse the electron is thrown from this orbit onto some external one. The atom enters an excited state, for in the reverse transition of the electron it emits energy. If the kinetic energy of the electron striking the atom corresponds to the ionization potential, then the electron in the atom is ejected from the inner orbit to infinity. If, however, the kinetic energy corresponds to the resonance potential, then the electron from the inner stable orbit is thrown onto the outer orbit nearest to it. In the reverse transition of the electron the so-called resonance radiation is obtained. If the kinetic energy of the striking electron assumes some intermediate value between the above-mentioned values, then upon impact the electron is ejected from the inner orbit not onto the nearest resonance orbit, but onto some other one, lying farther from the center. Then the reverse transition of the electron to the inner orbit takes place, with the corresponding frequency being emitted. But now, instead of a single reverse transition, several successive transitions may occur; these occur because the electron does not at once jump to the inner orbit, but passes successively from one of the possible orbits to another, until it reaches the final inner orbit. Under these conditions the atom is capable of emitting several frequencies. The amount of energy falling on a line of a given frequency depends on the probability, under the given conditions, of the corresponding transition. If the given conditions are such that the maximum energy of the striking electron is equal to the resonance potential, then only one resonance line can appear in the spectrum (a one-line spectrum). If, however, the kinetic energy of the colliding electron is increased, then new spectral lines must appear in the spectrum in a definite order, which can be determined by knowing the number and arrangement of the quantum orbits between the inner orbit and the outer orbit corresponding to the given potential.

For their investigation Foot and Meggers take caesium, since in it the principal and subordinate spectral series lie in such a region of the spectrum that it can easily be photographed on specially sensitized plates. The stream of electrons emerged from a heated cathode. Between the cathode and the anode an electric field of known voltage was created. On their way to the anode the electrons, striking caesium atoms in collision, excited them to luminosity. At the low potential of \(1.5\ \mathrm{V}\) (the resonance potential), Foot and Meggers expected to obtain a spectrum consisting of a single resonance line, in the present case the first doublet of the principal series \(8541, 8943\). This was confirmed experimentally. With a constant approach from a somewhat higher potential to the resonance potential, Foot and Meggers found that the ratio of the intensity of the above-mentioned doublet of the principal series to the intensity of the other lines gradually approaches infinity. Thus, at the limit, at \(1.5\ \mathrm{V}\) there exists only a one-line spectrum. At the higher potential, however, new spectral lines appear, while the intensity of the first doublet weakens because some electrons which at relatively low velocity gave only the beginning of the series \(1.5S—2p\), now produce a more complex serial spectrum, and those lines of the \(1.5mp\) series for which \(m>2\) are excited at the expense of a decrease in the intensity of the line \(1.5S—2p\). Above a known voltage the intensity of the lines, referred to a definite number of electrons reaching the anode, receives a certain definite value, in agreement with the quantum theory, which requires that the number of emitted quanta be proportional to the number of collisions and, consequently, to the number of electrons present.

† N. Metelik.