On the Broadening of Spectral Lines

(Cf. Holtsmark, Über die Verbreitung von Spektrallinien, Ann. d. Phys. 58, p. 577, 1919).

The causes of the broadening of spectral lines may be very diverse, partly of kinematic origin (translational and rotational motion of molecules), partly physical (damping of oscillations due to radiation or due to collisions of molecules). The discovery of the phenomenon of the splitting of spectral lines in an electric field enabled Stark to point to this effect as one of the factors that substantially influence both the broadening of a line and the distribution of energy within the broadened band. Stark did not give a mathematical formulation of his hypothesis, indicating only a number of facts that qualitatively confirm it: lines that undergo strong splitting in an electric field also possess considerable width; if the components of the split line are situated asymmetrically with respect to the original one, then the distribution of energy within the broadened line is usually also asymmetric. Holtsmark casts Stark’s views into mathematical form. The course of the author’s reasoning is as follows. Suppose there is an emitting atom surrounded by \(N\) atoms of the same type. The electric fields of these atoms cause the splitting of the line emitted by the radiating atom. Owing to the motion of the atoms, the resulting electric field is variable in magnitude and direction; the split line practically merges into a band. Holtsmark first seeks the probability that the electric field \(F^0\) acting on the emitting atom lies within the limits between \(F_0\) and \(F_0 + dF_0\). Using, in part, Markov’s method,\(^1\) Holtsmark solves the problem for certain particular cases that are of the greatest practical importance. In the course of solving the problem it is necessary to compute the potential at some point \(x, y, z\), produced by the charges of a given atom. Denoting the magnitude of the charges by \(e_s\), and by \(R_s\) the distance of a given charge to the point \(x, y, z\), we have for the potential

\[ \Phi = \sum \frac{e_s}{R_s} \ldots \ldots \ldots \ldots \ldots \ldots (1). \]

Let \(r\) be the distance of the point \(x, y, z\) from the geometric center of the atom (the origin of coordinates), and \(x_s, y_s, z_s\) the coordinates of the point charges \(e_s\). Denoting

\[ xx_s + yy_s + zz_s = (WW_s) \]

as the scalar product of two vectors and expanding in a series \(\{R_s^2\}^{-1/2}\), we find from (1)

\[ \Phi = \frac{\sum e_s}{r} - \frac{1}{r^2} \left( \frac{\partial}{r}\cdot \sum e_s \vartheta_s \right) - \frac{1}{2r^3} \left\{ \sum e_s x_s^2 \left(1-\frac{\partial x^2}{r^2}\right) + \sum e_s y_s^2 \left(1-\frac{\partial y^2}{r^2}\right) + \sum e_s z_s^2 \right. \]

\[ \left. \left(1-\frac{\partial z^2}{r^2}\right) + \frac{6\sum e_s x_s y_s xy}{r^2} + \frac{6\sum e_s y_s z_s yz}{r^2} + \frac{6\sum e_s z_s x_s zx}{r^2} \right\} \ldots (2). \]

If \(\sum e_s \ne 0\), then we are dealing with free charges, i.e. ions; if \(\sum e_s = 0\), but \(\sum e_s \vartheta_s \ne 0\), with the case of a dipole with a definite electric moment; the case \(\sum e_s = 0\), \(\sum e_s \vartheta_s = 0\), \(\sum e_s^2 \ne 0\), etc., corresponds to a neutral atom, a quadrupole, as Holtsmark calls it. For three

\(^1\) A. A. Markov, Calculus of Probabilities. St. Petersburg, 1908, p. 48.

the indicated cases, and the probability of the field \(F_0\), \(W(F_0)\), is calculated. The result of the calculation is as follows:

Ions

\[ W(F_0)dF_0=\frac{4}{3\pi}\,\beta^2 d\beta \left[1-0.4629\beta^2+0.1227\beta^4-0.02325\beta^6-\ldots\right]\ldots \tag{3} \]

where \(\beta=\dfrac{F_0}{[[unclear: denominator]]}\), and \(n\) is the number of molecules in \(1\ \mathrm{cm}^3\), \(\varepsilon\) is the charge of the ion.

Dipoles

\[ W(F_0)dF_0=\frac{4}{\pi}\,\frac{\beta^2\,d\beta}{(1+\beta^2)^2}\ldots \tag{4} \]

where \(\beta=\dfrac{F_0}{4.54\,m\,n}\), and \(m\) is the dipole moment.

Quadrupoles

\[ W(F_0)dF_0=\frac{4}{3}\cdot\frac{4}{\pi}\,\beta^2 d\beta \left[1-2.44\beta^2+11.25\beta^4-72\beta^6+\ldots\right]. \tag{5} \]

where \(\beta^3=\dfrac{F_0}{(11.49\,n\,A^{[[unclear: exponent]]})^{4/3}}\), and \(A=\sum e_i x_i^2-\sum e_i y_i^2\) (the atom is assumed to be symmetric with respect to one axis). In the presence of the field \(F_0\), a line with frequency \(\nu_0\) is split in such a way that the intensity is distributed according to some law

\[ J(F,\nu)\,d\nu\ldots \tag{6} \]

Hence, for the intensity distribution in the broadened line, generally speaking, we have

\[ J\,d\nu=d\nu\int_0^\infty J(F,\nu)\cdot W(F)\,dF\ldots \tag{7} \]

Replacing approximately the law (6) by its mean value, i.e., assuming that under the influence of the electric field there occurs only a uniform broadening of the spectral line, while the total emitted energy remains unchanged, Holtsmark writes

\[ \left. \begin{aligned} J(F,\nu)&=\frac{f}{2\nu_m}\quad \text{inside }2\nu_m,\\ J(F,\nu)&=0\quad \text{outside }2\nu_m \end{aligned} \right\}\ldots \tag{8} \]

where \(2\nu_m\) is the “width” of the split line, and \(f\) is the total energy. On the basis of (3), (4), (5), (7), (8), one can find, for all three cases under consideration, the intensity distribution within the broadened line, as well as the half-width of the band, i.e., the difference between the two frequencies corresponding to half the maximum intensity of the line. Holtsmark obtains the following values for this quantity:

\[ \left. \begin{array}{ll} \text{Ion} & 3.25\,c\,n^{2/3}\,\varepsilon\\ \text{Dipole} & 4.54\,c\,n\,m\\ \text{Quadrupole} & 17.2\,c\,n^{4/3}\,A \end{array} \right\}\ldots \tag{9} \]

where \(c\) is the constant of the Stark effect. Thus, only in the case of dipoles is the half-width of the line directly proportional to the concentration; in the other two cases the dependence is more complex. Holtsmark’s theory connects the broadening of spectral lines with very interesting physical constants, in the special case of the quadrupole with those characterizing the structure of the atom or molecule (the constant \(A\)). The author applies his theory to existing exper-

… data, using the models of molecular structure of Bohr, Debye, and Sommerfeld. Unfortunately, the experimental data concerning the Stark effect and the broadening of spectral lines are still extremely scanty and, in most cases, inaccurate. In those cases where the experimental conditions are such that the kinematic causes of line broadening are not of predominant importance, the author succeeds in demonstrating a satisfactory agreement between theory and experiment, at least with respect to the order of magnitude of the corresponding quantities.

S. Vavilov.