The Fine Structure of the Hydrogen lines

An instructive application of the calculations in the last section may be made in connection with the fine structure of the hydrogen lines.

These, according to Sommerfeld’s theory mentioned in Part I on page 31, may be explained by taking into account the small variation of the mass of the electron with its velocity, claimed by the theory of relativity.

All the general considerations in the preceding sections, as regards relations between energy and frequency and as regards the mechanical transformability of the stationary states, hold unaltered if the relativity modifications are taken into account.

This follows from the fact that the Hamiltonian equations (4), which are taken as a basis for all the previous calculations, may be used to describe the motion also in this case.

If, when the relativity modifications are taken into account, the motion of the system is simply periodic independent of the initial conditions, we shall consequently expect that:

• the stationary states are characterised by the condition I = nh only
• the energy and frequency are the same for all states corresponding to a given value of n in this equation.

The stationary states will also in the relativity case be fixed by (22), if the system is conditionally periodic and allows of separation of variables.

The stationary states of a perturbed periodic system, also in the relativity case, will be characterised by the conditions (67), if the secular perturbations are of conditionally periodic type.

When the relativity modifications are taken into account, the motion of the particles in the hydrogen atom will not, as assumed in § 1, be exactly periodic. The orbit of the electron will be of the same type as that, which would appear on ordinary Newtonian mechanics, if the law of attraction between the particles differed slightly from that of the inverse square.

If the mass of the nucleus as infinite, the system will allow of a separation of variables in polar coordinates, and the stationary states may consequently be fixed by the conditions (16).

In this way Sommerfeld obtained an expression for the total energy in the stationary states, which, with neglect of small quantities of higher order than the square of the ratio of the velocity of the electron and the velocity of light c, is given by

…

where, as in the calculations in § 1, the charge and the mass of the electron are denoted by −e and m, and for sake of generality the charge of the nucleus by Ne.

Further n1 and n2 are the integers appearing on the right side of the conditions (16) as factors to Planck’s constant.

While n1 may take the values 0, 1, 2, . . . , it will be seen that n2 can only take the values 1, 2, . . . , because in the present case there will obviously not correspond any stationary state to n2 = 0, since in such a state the electron would collide with the nucleus.

Introducing the experimental values for e, h and c, it is found that e 2/hc is a small quantity of the same order as 10−3 ; and, unless N is large number, the second term within the bracket on the right side of (68) will consequently be very small compared with unity.

Putting n1 + n2 = n, it will further be seen that the factor outside the bracket will coincide with the expression for Wn given by (41) in § 1, if we look apart from the small correction due to the finite mass of the nucleus.

Due to the presence of the second term within the bracket, we thus see that, for any value of n, formula (68) gives a set of values for E which differ slightly from each other and from −Wn. Sommerfeld’s theory leads therefore to a direct explanation of the fact, that the hydrogen lines, when observed by instruments of high dispersive power, are split up in a number of components situated closely to each other.

1. A. Sommerfeld, Ann. d. Phys. LI, p. 53 (1916). Compare also P. Debye, Phys. Zeitschr., XVII, p. 512 (1916).

In the special case of circular orbits (n1 = 0), this expression coincides with an expression previously deduced by the writer (Phil. Mag. XXIX p. 332 (1915)), by a direct application of the condition I = nh to these periodic motions.

By means of formula (68) in connection with relation (1), it was actually found possible, within the limits of experimental errors, to account for the frequencies of the components of this so called fine structure of the hydrogen lines. Moreover the theory was supported in the most striking way by Paschen’s

1. recent investigation of the fine structure of the lines of the analogous helium spectrum, the frequencies of which are represented approximately by formula (35), if in the expression for K, given by (40), we put N = 2.

As it should be expected from (68), the components of these lines were found to show frequency differences several times larger than those of the hydrogen lines, and from his measurements Paschen concluded, that it was possible on Sommerfeld’s theory to account completely for the frequencies of all the components observed.

We shall not enter here on the details of the calculation leading to (68), but shall only show how this formula may be simply interpreted from the point of view of perturbed periodic systems.

1. F. Paschen, Ann. d. Phys. L, p. 901 (1916). See also E. J. Evans and C. Croxson, Nature, XCVII, p. 56 (1916).

Thus, by a simple application of relativistic mechanics, it is found that, if the equation of a Keplerian ellipse in polar coordinates is given by r = f(ϑ), the equation of the orbit of the electron in the case under consideration will be given by r = f(γϑ) where γ is a constant given by

, in which expression p denotes the angular momentum of the electron round the nucleus.

1 In the stationary states the quantity in the bracket, which is of the same order of magnitude as the ratio between the velocity of the electron and the velocity of light, will be very small, unless N is a large number, and it will therefore be seen that the orbit of the electron can be described as a periodic orbit on which a slow uniform rotation is superposed.

Denoting the frequency of revolution in the periodic orbit by ω and the frequency of the superposed rotation by vR, we have, with neglect of small quantities of higher order than the square of the ratio between the velocity of the electron and the velocity of light,

Comparing this formula with equation (62) and remembering that, with the approximation in question, p may be replaced by the quantity denoted in § 2 by α2, we see that the frequency of the secular rotation of the orbit will be the same as that which would appear, if the variation of the mass of the electron was neglected, but if the atom was subject to a small external central force the mean value of the potential

1. See f. inst. A. Sommerfeld, loc. cit. p. 47. of which, taken over a revolution of the electron, was equal to

This is simply shown, however, to be equal to the expression for ψ corresponding to a small attractive force varying as the inverse cube of the distance. In fact, let the potential of such a force be given by Ω = C/r2 , where C is a constant and r the length of the radius vector from the nucleus to the electron. By means of the relation α2 = mr2ϑ, where ϑ is the angular distance of the radius vector from a fixed line in the plane of the orbit, we get then

which expression is seen to coincide with (70), if C = −