Concluding Remarks.

In the deductions of this paper the following general assumptions are used:—

1. That an elementary system containing rotating electrons will not emit energy radiation in the continuous way assumed in ordinary electrodynamics, but that radiation is only emitted during the passing of the system between a certain number of stationary states.

2. That the dynamical equilibrium of the system in the stationarystates is governed by the ordinary laws of mechanics, while these laws do not hold for the passing of the system between the different stationary states.

3. That the radiation emitted during the passing of the system between two stationary states is homogeneous; and that only in the region of slow vibrations does the frequency approach that to be expected on ordinary electrodynamics, while in general the frequencyis determined by the relation,whereis the total amount of energy emitted andPlanck’s constant.

It has been attempted to show that, applying these assumptions to Rutherford’s theory of the structure of atoms, it seems possible to obtain an explanation of the laws of line-spectra discovered by Balmer, Rydberg, and Ritz.

It has further been attempted to show that it seems possible to account for some of the general features of the effect of magnetic and electric fields on spectral lines discovered by Zeeman and Stark. In the case of an electric field it is assumed that no alterations in the above assumptions take place. In the case of a magnetic field, however, it is found necessary to modify the third assumption in order to retain the connexion with ordinary electrodynamics in the region of slow vibrations.

FOOTNOTES:[1]Communicated by Sir E. Rutherford, F.R.S.[2]Phil. Mag. vol. xxvi. pp. 1, 476, 857 (1913).[3]Sitzungsb. d. Kgl. Akad. d. Wiss. Berlin, 1913, p. 932.[4]While this paper was in course of preparation, a theoretical paper dealing with the same subject was published by E. Warburg (Verh. d. deutsch. Phys. Ges. xv. p. 1259 (1913)). The later finds that the effect of electric and magnetic fields to be expected on my theory of the hydrogen spectrum is of the same order of magnitude as determined by experiment. However, contrary to the conclusions of the present paper, Warburg concludes that it does not seem possible on the theory to account in detail for the experimental results. In his opinion the theory leads to a broadening of the hydrogen lines in an electric field, instead of the appearance of the homogeneous components observed by Stark. He also calculates that the Zeeman effect should vary from line to line in a manner inconsistent with experiment.[5]In Planck’s original theory certain other assumptions about the properties of the vibrating systems were used. However, Debye (Ann. d. Phys. xxxiii. p. 1427 (1910)) has shown that it is possible to deduce Planck’s formula of radiation without using any assumption about the vibrators, if it be supposed that energy can be transferred between them and the radiation, only in finite quanta.It may further be remarked that Poincaré (Journ. d. Physique, ii. p. 5 (1912)) has deduced the necessity of assuming that the transference of energy takes place in quantain order to explain the experimental laws of black radiation.[6]A series of lines, first observed by Pickering in stellar spectra and recently by Fowler in vacuum-tubes containing a mixture of hydrogen and helium, is generally also ascribed to hydrogen. These lines, however, can be accounted for on the present theory, if we ascribe them to helium. (Phil. Mag.loc. cit. p. 10: comp. also ‘Nature’ xcii. p. 231 (1913).)[7]Comp. A. v. d. Broek,Phys. Zeitschr. xiv. p. 32 (1913), comp. also several recent contributions to ‘Nature.’[8]Phil. Mag.loc. cit. p. 487.[9]Comp. J. W. Nicholson, Month. Not. Roy. Astr. Soc. lxxii. p. 679 (1912).[10]Note added during the proof.—In thePhys. Zeitschr.of Feb. 1, E. Gehrcke has attempted to represent the theory of the hydrogen spectrum in a way somewhat different from that in my former paper. Like the procedure in my paper, Gehrcke does not attempt to give a mechanical explanation of Planck’s relation between the frequency of the radiation and the amount of energy emitted; but he does also not try to give a mechanical interpretation of the dynamical equilibrium of the atom in its possible stationary states, or to obtain a connexion to ordinary mechanics in the region of slow vibrations.[11]Note added during the proof—In thePhys. Zeitsehr. of Feb. 1, A. Garbasso and E. Gehrcke (c. f. note, p. 6) have deduced expressions forwhich differ from (22) only by a numerical factor of 2 andrespectively. The arguments of Garbasso are stated very briefly, but seem of a type similar to those of the present paper. The line of arguments of Gehrcke differs essentially from that used here.[12]Note added during the proof.—InVerh. d. Deutsch. Phys. Ges. 1914, p. 20, K. Schwarzschild has discussed the problem of the effect of the field on the motion of the electron in some detail. In contrast to the above considerations he attempts to apply the results on the explanation of the Stark effect without leaving ordinary electrodynamics.[13]Since the value ofdiffers widely from unity for all series of lines in the spectra of the heavier elements, it is to be expected that the electric effect should be very small, or undetectable for such elements.[14]See G. Runge,Phys. Zeitschr. viii. p. 232 (1907).[15]Ann. d. Phys. xxxix. p. 897 (1912), xl. p. 900 (1913).[16]Ann. d. Phys. xl. p. 368, xli. p. 403, xlii. p. 210 (1913).[17]Ann. d. Phys. xl. p. 748 (1913).[18]‘Æther and Matter,’ Cambridge, 1900, p. 341.[19]Ann. de Chim. et de Phys. v. p. 70 (1905). In this connexion it may be remarked that on the present theory the rotation will give rise to diamagnetism only, since the kinetic energy of the electrons in the stationary states cannot be transferred into heat motion such as is supposed by Langevin in his theory of magnetism. This conclusion seems consistent with experiments which show that the monatomic gases helium and argon are diamagnetic (see P. Tanzler,Ann. d. Phys. xxiv. p. 931 (1907)), although the structure of these atoms, proposed in my former paper, was of a type which on Langevin’s theory should show paramagnetism.[20]See Fr. Croze,Journ. de Phys. iii. p. 882 (1913).[21]Note added during the proof.—InPhys. Zeitschr. of Feb. 15, K. Herzfeld has discussed in detail the different possibilities of the effect of a magnetic field which might be expected on the theory of the hydrogen spectrum proposed by the writer. His conclusions are equivalent with those obtained above. In addition he considers the effect of terms proportional to the square of the magnetic force and shows that in a strong magnetic field these terms may be expected to have an appreciable influence on the magnetic resolution of the hydrogen lines corresponding to high numbers in the Balmer series. This is a consequence of the large orbits of the electron in the stationary states corresponding to high values of.[22]The lines of the ordinary hydrogen spectrum from a vacuum-tube also appear as close doublets with high dispersion. Considering, however, the want of sharpness of the lines and the discrepancies between the distance of components found by different observers, it seems probable that the lines are not true doublets, but are due to an effect of the electric field in the discharge. This is also indicated by the fact that the distance between the components observed increases with the number of the line, contrary to the behaviour of ordinary double lines. The distance between the components observed by Paschen and Back (loc. cit.) wascm. andcm. forandrespectively. According to Stark’s experiments onthis corresponds to a resolution produced by an electric force of about 900 volt per cm. The ratio between an electric resolution ofandshould, according to the calculations of section 2, be 0.76; the ratio between the components observed isor 0.83.

[1]Communicated by Sir E. Rutherford, F.R.S.

[2]Phil. Mag. vol. xxvi. pp. 1, 476, 857 (1913).

[3]Sitzungsb. d. Kgl. Akad. d. Wiss. Berlin, 1913, p. 932.

[4]While this paper was in course of preparation, a theoretical paper dealing with the same subject was published by E. Warburg (Verh. d. deutsch. Phys. Ges. xv. p. 1259 (1913)). The later finds that the effect of electric and magnetic fields to be expected on my theory of the hydrogen spectrum is of the same order of magnitude as determined by experiment. However, contrary to the conclusions of the present paper, Warburg concludes that it does not seem possible on the theory to account in detail for the experimental results. In his opinion the theory leads to a broadening of the hydrogen lines in an electric field, instead of the appearance of the homogeneous components observed by Stark. He also calculates that the Zeeman effect should vary from line to line in a manner inconsistent with experiment.

[5]In Planck’s original theory certain other assumptions about the properties of the vibrating systems were used. However, Debye (Ann. d. Phys. xxxiii. p. 1427 (1910)) has shown that it is possible to deduce Planck’s formula of radiation without using any assumption about the vibrators, if it be supposed that energy can be transferred between them and the radiation, only in finite quanta.It may further be remarked that Poincaré (Journ. d. Physique, ii. p. 5 (1912)) has deduced the necessity of assuming that the transference of energy takes place in quantain order to explain the experimental laws of black radiation.

[6]A series of lines, first observed by Pickering in stellar spectra and recently by Fowler in vacuum-tubes containing a mixture of hydrogen and helium, is generally also ascribed to hydrogen. These lines, however, can be accounted for on the present theory, if we ascribe them to helium. (Phil. Mag.loc. cit. p. 10: comp. also ‘Nature’ xcii. p. 231 (1913).)

[7]Comp. A. v. d. Broek,Phys. Zeitschr. xiv. p. 32 (1913), comp. also several recent contributions to ‘Nature.’

[8]Phil. Mag.loc. cit. p. 487.

[9]Comp. J. W. Nicholson, Month. Not. Roy. Astr. Soc. lxxii. p. 679 (1912).

[10]Note added during the proof.—In thePhys. Zeitschr.of Feb. 1, E. Gehrcke has attempted to represent the theory of the hydrogen spectrum in a way somewhat different from that in my former paper. Like the procedure in my paper, Gehrcke does not attempt to give a mechanical explanation of Planck’s relation between the frequency of the radiation and the amount of energy emitted; but he does also not try to give a mechanical interpretation of the dynamical equilibrium of the atom in its possible stationary states, or to obtain a connexion to ordinary mechanics in the region of slow vibrations.

[11]Note added during the proof—In thePhys. Zeitsehr. of Feb. 1, A. Garbasso and E. Gehrcke (c. f. note, p. 6) have deduced expressions forwhich differ from (22) only by a numerical factor of 2 andrespectively. The arguments of Garbasso are stated very briefly, but seem of a type similar to those of the present paper. The line of arguments of Gehrcke differs essentially from that used here.

[12]Note added during the proof.—InVerh. d. Deutsch. Phys. Ges. 1914, p. 20, K. Schwarzschild has discussed the problem of the effect of the field on the motion of the electron in some detail. In contrast to the above considerations he attempts to apply the results on the explanation of the Stark effect without leaving ordinary electrodynamics.

[13]Since the value ofdiffers widely from unity for all series of lines in the spectra of the heavier elements, it is to be expected that the electric effect should be very small, or undetectable for such elements.

[14]See G. Runge,Phys. Zeitschr. viii. p. 232 (1907).

[15]Ann. d. Phys. xxxix. p. 897 (1912), xl. p. 900 (1913).

[16]Ann. d. Phys. xl. p. 368, xli. p. 403, xlii. p. 210 (1913).

[17]Ann. d. Phys. xl. p. 748 (1913).

[18]‘Æther and Matter,’ Cambridge, 1900, p. 341.

[19]Ann. de Chim. et de Phys. v. p. 70 (1905). In this connexion it may be remarked that on the present theory the rotation will give rise to diamagnetism only, since the kinetic energy of the electrons in the stationary states cannot be transferred into heat motion such as is supposed by Langevin in his theory of magnetism. This conclusion seems consistent with experiments which show that the monatomic gases helium and argon are diamagnetic (see P. Tanzler,Ann. d. Phys. xxiv. p. 931 (1907)), although the structure of these atoms, proposed in my former paper, was of a type which on Langevin’s theory should show paramagnetism.

[20]See Fr. Croze,Journ. de Phys. iii. p. 882 (1913).

[21]Note added during the proof.—InPhys. Zeitschr. of Feb. 15, K. Herzfeld has discussed in detail the different possibilities of the effect of a magnetic field which might be expected on the theory of the hydrogen spectrum proposed by the writer. His conclusions are equivalent with those obtained above. In addition he considers the effect of terms proportional to the square of the magnetic force and shows that in a strong magnetic field these terms may be expected to have an appreciable influence on the magnetic resolution of the hydrogen lines corresponding to high numbers in the Balmer series. This is a consequence of the large orbits of the electron in the stationary states corresponding to high values of.

[22]The lines of the ordinary hydrogen spectrum from a vacuum-tube also appear as close doublets with high dispersion. Considering, however, the want of sharpness of the lines and the discrepancies between the distance of components found by different observers, it seems probable that the lines are not true doublets, but are due to an effect of the electric field in the discharge. This is also indicated by the fact that the distance between the components observed increases with the number of the line, contrary to the behaviour of ordinary double lines. The distance between the components observed by Paschen and Back (loc. cit.) wascm. andcm. forandrespectively. According to Stark’s experiments onthis corresponds to a resolution produced by an electric force of about 900 volt per cm. The ratio between an electric resolution ofandshould, according to the calculations of section 2, be 0.76; the ratio between the components observed isor 0.83.

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