Section 4

The effect of an external electric field on the hydrogen lines

by Niels Bohr

A detailed theory of the characteristic effect of an external homogeneous electric field consideration obtains a striking confirmation by the observation of the appearance of new series of lines in the ordinary series spectra of helium and other elements, when the atoms are exposed to an intense external electric field.

As it will be discussed more closely in Part III, it is possible in this way to account in detail for the manifold results, regarding the appearance of such series in the helium spectrum, which have been published quite recently by J. Stark (Ann. d. Phys. LVI, p. 577 (1918)) and by G. Liebert (ibid. LVI, p. 589 and p. 610 (1918)).

on the hydrogen spectrum, discovered by Stark, has been given by Epstein and Schwarzschild on the basis of the general theory of conditionally periodic systems which allow of separation of variables. Before we enter on the discussion of the results of the calculations of these authors, we shall first, however, show how the problem may be treated in a simple way by means of the considerations about perturbed periodic systems, developed in § 2. Consider an electron of mass m and charge −e, rotating round a positive nucleus of infinite mass and of charge Ne, and subject to a homogeneous electric field of intensity F, and let us for the present neglect the small effect of the relativity modifications. Using rectangular coordinates, and taking the nucleus as origin and the z-axis parallel to the external field, we get for the potential of the system relative to the external field, omitting an arbitrary constant, Ω = eF z. Calculating now the mean value of Ω over a period σ of the undisturbed motion, we see at once, from considerations of symmetry, that this mean value ψ will depend only on the component of the external electric force in the direction of the major axis of the orbit. We have therefore

ψ = eF cos ϕ 1 σ Z σ 0 r cos ϑ dt,

where ϕ is the angle between the z-axis and the major axis, taken in the direction from the nucleus to the aphelium, and 138 where r is the length of the radius-vector from the nucleus to the electron, and ϑ the angle between this radius-vector and the major axis. By means of the well known equations for a Keplerian motion r cos ϑ = α(cos u + ε), dt σ = (1 + ε cos u) du 2π , where 2α is the major axis, ε the eccentricity and u the so called eccentric anomaly, this gives ψ = eF cos ϕ 1 2π Z 2π 0 α(cos u + ε)(1 + ε cos u) du

3 2 εαeF cos ϕ. (74) We see thus that ψ is equal to the potential energy relative to the external field, which the system would possess, if the electron was placed at a point, situated on the major axis of the ellipse and dividing the distance 2εa between the foci in the ratio 3 : 1. This point may be denoted as the “electrical centre” of the orbit. From the approximate constancy of ψ during the motion, proved in § 2, it follows therefore in the first place that, with neglect of small quantities of the same order of magnitude as the ratio between the external force and the attraction from the nucleus, the electrical centre will during the perturbations of the orbit remain in a fixed plane perpendicular to the direction of the external force. From the considerations in § 2 it follows further, that the total energy in the stationary states of the system 139 in the presence of the field, with neglect of small quantities proportional to F 2 , will be equal to En + ψ, where En is the energy of the hydrogen atom in its undisturbed stationary state. Since both ε and cos ϕ are numerically smaller than one, we obtain therefore at once from (74) a lower and an upper limit for the possible variations of the energy in the stationary states, due to the field. Introducing from (41) the values of En and αn, and neglecting, here as well as in the following calculations in this section, the small correction due to the finite mass of the nucleus—not only in the expression for the additional energy but, for the sake of brevity, also in the main term—we get for these limits E = − 2π 2N2 e 4m h 2n2 ± 3h 2n 2 8π 2Nem F, (75) which formula coincides with the expression previously deduced by the writer by applying the condition I = nh to the two (physically not realisable) limiting cases, corresponding to z = 1 and cos ϕ = ±1, in which the orbit remains periodic in the presence of the field.1 ) In order to obtain further information as to the values of the energy in the stationary states in the presence of the 1 ) See N. Bohr, Phil. Mag. XXVII, p. 506 (1914) and XXX, p. 394 (1915). Compare also E. Warburg, Verh. d. D. Phys. Ges. XV, p. 1259 (1913), where it was pointed out, for the first time, that the effect of an electric field on the hydrogen lines to be expected on the quantum theory was of the same order of magnitude as the effect observed by Stark. 140 field, it is necessary to consider more closely the variation of the orbit during the perturbations. Since the external forces possess axial symmetry, the problem of the stationary states might be treated by means of the procedure indicated in § 2 on page 107. In the present special case, however, the stationary states of the atom may be very simply determined, due to the fact that the secular perturbations are simply periodic independent of the initial shape and position of the orbit, so that we are concerned with a degenerate case of a perturbed periodic system. This property of the perturbations follows already from some calculations given by Schwarzschild1 ) in a previous attempt to explain the Stark effect of the hydrogen lines, without the help of the quantum theory, by means of a direct consideration of the harmonic vibrations into which the motion may be resolved, according to the analytical theory of conditionally periodic systems. Starting from the above result, that the electrical centre moves in a plane perpendicular to the direction of the external field, the periodicity of the perturbations may also be proved in the following way, by means of a simple consideration of the variation of the angular momentum of the electron round the nucleus, due to the effect of the external electric force. Using again rectangular coordinates with the nucleus at the origin and the z-axis parallel to the direction of the electric force, and calling the coordinates of the electrical centre ξ, η, ζ, we have 1 ) K. Schwarzschild, Verh. d. D. Phys. Ges. XVI, p. 20 (1914). 141 according to formula (74) ξ 2 + η 2 + ζ 2 =  3 2 εα2 , ζ = const. (1∗ ) Denoting the components parallel to the x y and z-axis of the angular momentum of the electron round the nucleus, considered as a vector, by Px, Py and Pz, we have next P 2 x + P 2 y + P 2 z = (1 − ε 2 )(2πmα2ω) 2 , Pz = const. (2∗ ) Since the angular momentum is perpendicular to the plane of the orbit, we have further ξPx + ηPy + ζPz = 0. (3∗ ) Now we have for the mean values of the rates of variation of Px and Py with the time DPx Dt = eF η, DPy Dt = −eF ξ. (4∗ ) From this we get, differentiating (1∗ ) and (2∗ ) with respect to the time, and remembering that α and ω remain constant during the perturbations, ξ Dξ Dt

• η Dη Dt = −K2  Px DPx Dt
• Py DPy Dt  = −eFK2 (ηPx − ξPy), (5∗ ) where K = 3 4πmαω . (6∗ ) 142 On the other hand we have, differentiating (3∗ ) and introducing (4∗ ), Px DPx Dt
• Py DPy Dt = 0, which together with (5∗ ) gives Dξ Dt = eFK2Py, Dη Dt = −eFK2Px, from which we get, by means of (4∗ ), D2 ξ Dt2 = −e 2F 2K2 ξ, D2η Dt2 = −e 2F 2K2 η, the solution of which is ξ = A cos 2π(vt + a), η = B cos 2π(vt + b), (7∗ ) where A, a, B and b are constants, and where, introducing (6∗ ), we have v = eFK 2π = 3eF 8π 2mαω . (8∗ ) During the perturbations the electrical centre will thus perform slow harmonic vibrations perpendicular to the direction of the electric force, with a frequency which is proportional to the intensity of the electric field, but, for a given value of F, quite independent of the initial shape of the orbit and its position relative to the direction of the field. For the value of this frequency in the multitude of states of the perturbed system, for which the mean value of the inner energy is equal to the energy En in a stationary state of the 143 undisturbed system corresponding to a given value of n, we get from the above calculation, introducing for α and ω the values of αn and ωn given by (41), vF = 3hn 8π 2Nem F. (76) Now from the periodic motion of the electrical centre we may conclude that, in the presence of the field, the system will be able to emit or absorb a radiation of frequency vF , and that accordingly the possible values of the additional energy of the system in the presence of the field will be given directly by Planck’s fundamental formula (9), holding for the possible values of the total energy of a linear harmonic vibrator, if in this formula ω is replaced by the above frequency vF . Since further a circular orbit, perpendicular to the direction of the electric force, will not undergo secular perturbations during a slow establishment of the field, and therefore must be included among the stationary states of the perturbed system, we get for the total energy of the atom in the presence of the field E = En + nvF h = − 2π 2N2 e 4m n2h 2

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