REPORT 
 
 ON THE 
 
 QUANTUM THEORY 
 OF SPECTRA 
 
 BY 
 
 L. SILBERSTEIN, PH.D. 
 
 PUBLISHED BY 
 
 ADAM HILGER, LTD. 
 
 75A CAMDEN ROAD, LONDON, N.W. i. 
 IQ20 
 
s 
 
 COPYRIGHT 
 
PREFACE 
 
 THE present report was written originally for the private use of 
 Messrs. Adam Hilger, and was then, at their instance, amplified 
 and prepared for publication. But even with this extension it 
 will be found to contain a brief account of only the most important 
 contributions to the Quantum Theory of Spectra, a new field of 
 inquiry opened only six or seven years ago by Niels Bohr, but 
 already very vast and rapidly growing and ramifying in multiple 
 directions. It will, no doubt, soon call for a supplement. As it 
 is, however, it is hoped to be a helpful guide for those desiring to 
 enter upon this new and strangely fascinating line of thought and 
 investigation. 
 
 I take the opportunity of expressing my best thanks to Messrs. 
 Hilger for bringing me into closer contact with this refreshing new 
 line of thought of the modern spectroscopists. My thanks are also 
 due to my beloved teacher, Prof. Max Planck, and to my friend, 
 Prof. A. W. Porter, for reading the proof slips, and to Messrs. 
 MacLehose for the indefatigable care they have bestowed upon 
 this book. L. S. 
 
 RESEARCH DEPT. ADAM HILGER, LTD., 
 LONDON, February 17, 1920. 
 
 439996 
 
CONTENTS 
 
 PAGE 
 
 1. BIBLIOGRAPHY - I 
 
 2. INTRODUCTORY REMARKS. BOHR'S ASSUMPTIONS AND 
 
 FIRST RESULTS - 2 
 
 3-4. RYDBERG'S CONSTANT AND FINITE MASS OF NUCLEUS 
 
 5. QUANTIZING PRINCIPLES. ELLIPTIC ORBITS AND 
 
 POTENTIAL MULTIPLICITY OF SPECTRUM LINES 9 
 
 6. RESTRICTING INEQUALITIES. QUANTIZATION OF ORBITS 
 
 IN SPACE - 13 
 
 7. SOMMERFELD'S THEORY OF FINE-STRUCTURE. HY- 
 
 DROGEN DOUBLETS AND HELIUM TRIPLETS - 16 
 
 8. THEORY OF ROENTGEN SPECTRA 27 
 
 9. THEORY OF THE STARK-EFFECT - 28 
 
 10. ZEEMAN-EFFECT. PHOTOELECTRIC EFFECT. PLANCK'S 
 THEORY OF ROTATION SPECTRA. MISCELLANEOUS 
 NOTES ON RECENT WORK - 33 
 
 INDEX 41 
 
REPORT ON THE QUANTUM THEORY OF SPECTRA. 
 
 1. The present report, which I shall attempt to bring up to date, 
 is based upon the reading and scrutiny of the following original 
 papers : 
 
 1. N. Bohr, Phil. Mag., xxvi, pp. 1-25, 1913. 
 
 2. xxvi, 476-502, 1913. 
 3- ,, xxvi, 857-875, 1913. 
 
 4. ' xxvii, 506-524, 1913. 
 4*. ,, M xxix, 332-335, 1915. 
 
 5. Th. Wereide, Annalen der Physik, xlix, 966-1000, 1916. 
 $a. ,, ibid., Hi, 276-290, 1917 (three notes). 
 
 6. M. Planck, ibid., 1, 385-418, 1916. 
 
 7. P. S. Epstein, ibid., 489-520, 1916. 
 
 8. ,, 815-840, 1916. 
 
 9. F. Paschen, ibid., 901-940, 1916. 
 
 10. A. Sommerfeld, Ann. d. Physik, li, 1-94, 1916. 
 
 11. 125-167, 1916. 
 na. K. Glitscher, ibid., Hi, 608-630, 1917. 
 
 12. P. S. Epstein, ibid., li, 168-188, 1916. 
 
 13. K. F. Herzfeld, ibid., 261-284, 1916. 
 
 14. P. Ehrenfest, ibid., 327-352, 1916. 
 
 15. N. Bohr, Danish Acad. Sc., IV, I, Parts I and II, pp. 
 
 i-ioo, 1918. 
 
 1 6. M. Planck, Annalen der Physik, Hi, pp. 491-505, and liii, 
 
 pp. 241-256, 1917. 
 
 17. J. M. Burgers, ibid., pp. 195-202, 1917. 
 
 18. H. A. Kramers, Memoires Acad. Sc., Copenhagen, 8th ser., 
 
 iii, No. 3, pp. 287-384, 1919- 
 
 19. L. Silberstein, Phil. Mag., xxxix, January 1920. 
 
 These papers will, whenever the need occurs, be referred to by 
 the attached numbers [in square brackets]. In addition to this 
 
2 THE^ptJANTUM; THEORY OF SPECTRA 
 
 first-hand material may be mentioned Mr. J. H. Jeans' Report on 
 Radiation and the Quantum Theory, Physical Society of London, 
 1914 (pp. iv-f 90), in which a few pages (pp. 50-57) are dedicated 
 to Bohr's elementary theory of line spectra. Jeans' Report 
 may be recommended for a rapid and easy initiation into Planck's 
 theory of quanta in connection with black-body radiation, the 
 knowledge of which I here presuppose, at least as far as its rudi- 
 ments are concerned. With regard to the nucleus structure of 
 the atoms (upon which all these spectrum theories are based) and 
 to the experimental evidence for such a structure, it is advisable 
 to consult Rutherford's papers, which are easily accessible (Phil. 
 Mag.*) and as easily read. There is no need for incorporating 
 here a description of Rutherford's work. 
 
 2. The strong side of the quantum theory of spectra, as first 
 proposed by Bohr [i] and further developed by Sommerfeld and 
 others, consists in its very remarkable agreement with experiment, 
 which in certain directions can be traced even to minute details 
 of observations, such as the fine-structure of the ' lines ' or groups 
 of Fowler's helium series ; the weak side of the theory consists in 
 the heavy sacrifices it requires at the very outset, e.g. the 
 abandonment of otherwise well-established principles of mechanics 
 and of electromagnetism, in addition to the radical innovation 
 inherent in the discontinuity of the very concept of Planck's 
 quantum (quantum of 'action,' that is, ' Wirkungsquantum '), 
 which is one of the most essential parts of the spectrum theory 
 under consideration. The agreement, however, is of such a 
 startling nature that the theory deserves, in spite of all these heavy 
 sacrifices, and notwithstanding its somewhat magically arithmetical 
 character, the greatest active interest of the modern physicist. 
 Moreover, some of the recent predictions of the new theory require, 
 for their experimental verification, a further refinement of spectro- 
 scopic apparatus and methods of observation, which soon may find 
 an echo in the optician's workshop. 
 
 The best way to become acquainted with the fundamental 
 assumptions and with the chief results of the quantum theory 
 of spectra will be to consider, in some detail, the simplest atomic 
 
 * The chief of these papers is published in vol. xxi of Phil. Mag- (1911), 
 p. 669. 
 
BOHR'S ASSUMPTIONS 3 
 
 system, to wit, the hydrogen atom, which according to Ruther- 
 ford and his school consists of a single electron (of mass m 
 and charge -e) circling round a single positive nucleus, i.e. an 
 electric positive charge (+e) contained in such a small volume that 
 it represents almost entirely the mass of the hydrogen atom.* 
 
 In Bohr's first attempt at a spectrum theory [i, also 2, 3], the 
 electrodynamic as well as the relativistic complications of the 
 motion are disregarded, and only the purely electrostatic attraction 
 between the nucleus and the electron is taken into account. In 
 short, the motion is treated as ordinary Keplerian planetary motion 
 (inverse square law), well-known since the beginnings of Celestial 
 Mechanics. Moreover, to begin with, the mass of the nucleus is 
 treated as infinite, in comparison with that of the planet or the 
 electron, so that the vector equation of motion of the electron 
 is simply 
 
 e being the charge in ordinary or so-called ' irrational,' electro- 
 static units, and r = m, the vector drawn from the nucleus to the 
 electron. It is scarcely necessary to say that writing down (i) as 
 the equation of motion, the force-component corresponding to 
 electromagnetic radiation is disregarded, as already stated. 
 But it is of importance .to note that radiation is thus ' disregarded ' 
 not simply as a means of getting a first approximation, but it is 
 disregarded radically, with purpose ; it is suspended, in virtue of a 
 postulate of the theory, precisely for the whole duration of validity 
 of this smooth, Newtonian motion. In other words, apart from 
 the small corrections due to relativity and due to the finite mass 
 of the nucleus, the above equation (i) is assumed to hold rigorously, 
 and the electron is assumed not to radiate at all while it describes 
 round the nucleus any of the Keplerian orbits contained in (i). 
 
 And herein lies the chief of the sacrifices : the validity of Max- 
 wellian principles is denied to these electronic orbits ; notwith- 
 standing that the motion of the electron is not uniform, it does not 
 
 * It will be kept in mind that ' mass ' stands here for electromagnetic 
 mass, and this is inversely proportional to the radius of the, say, spherical 
 charge. Thus, according to Rutherford, the linear dimensions of the 
 hydrogen nucleus would be about 1840 times smaller than those of an 
 electron, and, therefore, of the order of io~ 16 cm. 
 
4 THE QUANTUM THEORY OF SPECTRA 
 
 radiate any electromagnetic energy while describing any of these 
 orbits.* 
 
 Excluding hyperbolic and parabolic orbits (since we are not in- 
 terested here in electrons which are abandoning their atoms), 
 the orbits corresponding to (i) are ellipses described by the electron 
 according to the ordinary laws of Kepler. Thus, if 2a be the 
 major axis of such an ellipse, and T the period of revolution, we 
 have the well-known relations [with v= velocity at any instant] 
 
 . 
 
 whence, denoting the negatived total energy of the system, 
 
 e 2 i 
 
 ~ mv 2 , by W, 
 
 e 2 I A/ 2 \V^ 
 2a = u/; J-- ' ^77= ( 2 ) 
 
 giving the major axis and the frequency of revolution =, in terms 
 
 of the negatived total energy W, which, of course, remains constant 
 during the Keplerian motion in question. 
 
 Now comes Bohr's further assumption, which is not less revolu- 
 tionary than the suspension of radiation already mentioned. It is 
 most important to make this assumption as plain as is possible : 
 
 If the nucleus were our Sun and the electron a planet, say 
 our Earth, the latter could under given ' initial ' conditions 
 describe any one of the multiply-infinite class of Keplerian orbits 
 given above, and each of these would be stationary and, as such, 
 possible for the planetary system for ever. 
 
 Now, according, to Bohr's assumption, only certain of these 
 Keplerian orbits and motions are possible as ' stationary ' states of 
 motion of the atomic system (hydrogen atom = nucleus and electron), 
 viz. in each plane through the nucleus a certain discrete series of 
 Keplerian motions, corresponding to a, discontinuous series of values 
 of the angular momentum and, therefore, of the energy constant 
 W, appearing in (2). 
 
 * Th. Wereide's attempt to replace the usual radiation formula by a 
 slightly different one which would give no radiation for uniformly described 
 circular orbits [50, first note] does not change very mu.ch the state of things. 
 Nor is Prof. Wereide's reasoning, leading to the new radiation formula, 
 free from serious objections. 
 
BOHR'S ASSUMPTIONS 5 
 
 The procedure of picking out or ' quantizing ' [Sommerfeld says : 
 ' quanteln '] these privileged values and the corresponding 
 Keplerian motions is based upon Planck's universal constant h, 
 the ' Wirkungsquantum,' which appears in his famous formula for 
 the black-body radiation.* The principle of such quantizing has 
 undergone of late several subtle modifications (or, in part, generali- 
 zations), of which we will speak later on. To begin with, however, 
 it will be best to give the principle as it appears in Bohr's first 
 paper [i]. Bohr limits himself here to the consideration of circular 
 orbits, and then the principle in question is equivalent to the 
 following assumption : 
 
 Stationary states of motion are those and only those for which the 
 angular momentum of the electron is equal to an exact multiple of 
 an universal constant, to wit, of A/2?r, i.e. 
 
 2?r x angular momentum =nh, (3) 
 
 where n is an integer. 
 
 This is, in Bohr's case under consideration, equivalent to putting, 
 for the ' stationary ' states, 
 
 W = W n = ^. ( 3 a) 
 
 By (30) and (2) we have 
 
 _ I 
 
 VV n ~ z i 
 
 and 
 
 h 2 . n 2 
 
 At this stage it may be well to note, parenthetically, that if q 
 be the angle between the variable radius vector and a fixed radius 
 vector, the kinetic energy for circular orbits is 
 
 and the moment p (in Hamilton's sense of the word) corresponding 
 to the configurational coordinate q is 
 
 p = ~- = mr 2 q = angular momentum. 
 oq 
 
 * It will be recalled that, according to Planck's theory, radiation from 
 an atomic system takes place in quanta (separate emissions), the energy 
 radiated out by an oscillator of frequency v in each single emission being 
 equal to n . hv, where n is an integer. Planck's constant equals about 6-5 . io- 2? 
 C.G.S. Dimensions : [h] = [energy x time] = \ml-t- r \ = [angular momentum]. 
 
 A2 
 
6 THE QUANTUM THEORY OF SPECTRA 
 
 Thus the quantizing principle (3) can be written 
 
 and since p is constant (Kepler's second law), this is the same as 
 
 or, in words : p dq integrated over a full period of the motion is 
 equal to an exact multiple of h. It is precisely this form which has 
 turned out more recently to be adaptable to cases more general 
 than that treated by Bohr in his first essay. 
 
 Thus, out of all circular orbits only such whose radii are given 
 by (5) are * stationary '.* Imagine, now, a series of such stationary 
 orbits, No. I, 2, 3, etc., to which correspond the energy constants 
 W lt W 2 , W s , etc., given by (4), with n = i, 2, 3, etc. As long as 
 the atom is left to itself, the electron will move along one of these 
 (and these only) without ever emitting any radiation. [Max- 
 wellianism suspended, ordinary mechanics fully valid.] But, if 
 through some foreign agency whose mechanism cannot yet be 
 treated mathematically the electron is thrown out from its 
 stationary orbit (n) to another stationary orbit (n f ), it suddenly 
 radiates out the difference of energies in the form of monochromatic 
 waves, of a frequency v, say, and Bohr's new assumption is 
 precisely this : 
 
 E n = - W n being the total energy on the initial, and E n , = - W n -, 
 that on the final stationary orbit, the emitted frequency v is suck 
 
 W..-W n -*. (6) 
 
 Needless to say the founder of the new theory and his followers 
 do not attempt to describe the mechanism of such an extraordinary 
 performance, one, that is, that enables the atomic system to hit 
 precisely upon the frequency v required by the assumption (6). 
 [But future generations may be able to reduce this assumption to 
 something more plausible and more familiar, when it will cease 
 to be a mere assumption. After all, (6), is not more and not less 
 extravagant than what Planck requires for his black-body radiation.] 
 
 /i- 
 * Notice that the smallest of these quantized radii is, by (5), a, = -. , 
 
 ^.TT W< 
 
 which is about 0-6 . io~ 8 cm., i.e. of the atomic order. 
 
BALMER SERIES AND RYDBERG CONSTANT 7 
 
 The last assumption, as well as the preceding ones, being 
 granted, introduce into (6) the values of W n , W n , from (4). Then 
 the result will bef the corresponding spectrum series, 
 
 (/') 
 
 where n', n are integers. This is one of Bohr's most elegant results. 
 Putting n r = const, and letting n assume a series of integral values 
 (n > n'), we have in (7') a series of spectrum lines, converging 
 towards v J)r=M as the * head ' of the series. 
 
 In (7') v is the reciprocal of the period of the emitted waves ; 
 in experimental work it is usual to quote the reciprocal i/A. of the 
 wave-length in vacuo. Since this is=i>/c, we have, ultimately, 
 
 where 
 
 c being the light velocity in vacuo. * If we take n' const. =2, we 
 have in (7) the well-known Balmer series of hydrogen ( = 3, 4, 5, ...) ; 
 n' = const. =3 gives the infra-red series predicted by Ritz and 
 observed by Paschen. 
 
 A further, very remarkable result of Bohr's elementary theory 
 is that the factor N in (7), as defined by (8), is not only of the 
 order of Rydberg's well-known constant, ^ = 109675 cm." 1 , but 
 coincides with it numerically pretty well when the values of the 
 electronic constants e, m and that of h (derived from experiments 
 on black-body radiation, and from other sources) are substituted. 
 In fact, with 
 
 = 47- io- 10 ; - = 5'3- io 17 (z>. w=o-9 . io~ 27 ) ; 
 
 h = 6-$ . io~ 27 ; c = 3 . io 10 
 (everything in C.G.S.), formula (8) gives 
 
 N = 1-005 i 5 , 
 
 differing from Rydberg's constant only by 9 per cent., which is well 
 within the limits of error of m, e*, h*. This coincidence seems to 
 be a very strong support of the theory. 
 
 Notice that (7), (8) do not contain any new empirical constants. 
 
8 THE QUANTUM THEORY OF SPECTRA 
 
 % 
 3. For spectroscopic purposes, of course, the 5 or 6-figure value 
 
 of N will not be calculated from the very uncertain values of e, 
 m, h, but from the observed lines of the spectrum in question 
 (here that of hydrogen). It is well known that JV = 109675 repre- 
 sented excellently (with n' 2\ = 3, 4, 5, etc.) Balmer's series 
 or the so-called diffuse series of hydrogen. For other elements 
 (notably for helium) the factor N is slightly different. But, and 
 this is again a very remarkable result, even these slight differences 
 can be accounted for by taking into consideration the finite mass 
 (say, M) of the nucleus in question. In fact, this converts m in 
 (8) into 
 
 Jf+m' 
 
 so that the quantum value of the Rydberg factor becomes 
 slightly smaller, viz., instead of N, 
 
 N'=j^N. (8') 
 
 M+m 
 
 A very elegant application of this correction-factor has been made 
 by Fowler, and later by Paschen. It will be mentioned presently- 
 
 4. The details of the quantum theory of spectra for atoms whose 
 nuclei carry the charge 2e (as helium) or $e, etc., are up to the 
 present not quite satisfactory. But even in the present crude state 
 the theory as applied to helium has yielded some very interesting 
 results. 
 
 According to Rutherford's well-supported opinion the atom of 
 helium consists of a nucleus whose charge is 2e and, therefore, in 
 its neutral state, there must be two electrons circling round the 
 nucleus. Thus e 2 = e . e in the equation of motion (i), p. 3, is now 
 to be replaced by 
 
 e . 2e = 2e 2 . 
 
 If the perturbations of the motion of one of the electrons due to 
 the other electron are simply neglected,* or better, if the helium 
 
 * As in Bohr's first paper [i, p. 10]. The neglect of such perturbations 
 does not seem satisfactory. Equally unsatisfactory is Bohr's later assump- 
 tion [2, p. 477], that the electrons of more complicated atomic systems 
 * are arranged at equal angular intervals in coaxial rings,' and that a whole 
 ring of electrons jumps simultaneously from one stationary state of motion 
 to another. It is scarcely necessary to point out the unlikelihood of such 
 configurations and processes. The assumptions of Bohr's [2] lead him 
 again to the same result (9) ; cf. [2, p. 488]. 
 
FINITE MASS OF NUCLEUS 9 
 
 atom in question is ionized and thus retains only one of its electrons f 
 everything will be as before, except that ** in (8) will be replaced by 
 
 i.e., still disregarding the finite mass of the nucleus, N will be 
 replaced by ^N, so that, instead of (7) we shall have, 
 
 - rX.1 (0) 
 
 '/2)2 (n/2)*J 
 
 Now, if we put here w' = 3=const., and w=4, 6, 8, etc., orn = 5, 7, 9, 
 etc., we get Fowler's ' first ' and ' second principal ' series of helium, 
 respectively (cf. M. N. of R. A. S., vol. 73, Dec. 1912, p. 62 et seq.), 
 which properly constitute but one series. [Notice in passing that 
 n' = const. =4 gives the series observed by Pickering in the spectrum 
 of the star.f Puppis.] 
 
 Now, Fowler's measurements are tolerably well represented 
 when we take in (9), 
 
 (a) |A^f 
 while I of 109675 (hydrogen) is 
 
 (b) t N H 
 According to (8') we should have 
 
 , JV He _ i+m/M 
 
 N H i+frn/M' 
 
 where m/M is the ratio of the mass of the electron to that of the 
 hydrogen atom (remember that the atomic weight of He is four 
 times that of H). Now, on substituting in the right-hand member 
 of (c) the numbers (a), (b), we get 
 
 m 
 
 which is in excellent agreement with other determinations of the 
 mass ratio of the electron and the hydrogen atom. This is the 
 remarkable result hinted at before : a refinement of the theory 
 due to taking into account the finite mass of the central body of 
 the atomic system. 
 
 5. A further refinement, which is due to Sommerfeld [10], 
 consisted in taking account of the relativistic complications of the 
 original Keplerian motion of the electron. This refinement, which 
 yielded the fine-structure of spectrum lines, will now occupy our 
 
io THE QUANTUM THEORY OF SPECTRA 
 
 attention, together with the generalization of Bohr's original 
 quantizing principle, made by Sommerfeld himself, on the lines 
 laid down by Planck, Ehrenfest, and others. 
 
 Since bi- and poly-electronic atomic systems are not yet worked 
 out satisfactorily, it will be best to speak henceforth exclusively 
 of the single-e nucleus with its single electron, as in the hydrogen 
 atom.* Such, in fact, is the case of Sommerfeld, paper [io], and 
 also of Epstein's investigations concerning the Stark effect, to be 
 treated in a subsequent section. 
 
 Sommerf eld's investigations, more especially [io], deserve careful 
 attention, and will, therefore, be treated here at some length. 
 
 It will be remembered that Bohr, in his first papers (1913-14), 
 limited himself to the consideration of circular orbits. But, as Som- 
 merfeld justly remarks, the electron cannot be prevented from 
 choosing an elliptic orbit, and he shows that contrary to Bohr's 
 original statement this is by no means irrelevant for the final result. 
 In fact, if the angular momentum alone is quantized, the result 
 would be a series of diffuse bands, not sharp lines (as will be shown 
 presently). In short, in Bohr's treatment the atomic system 
 (hydrogen) appeared as a system with one degree of freedom, while 
 admitting ellipses of every eccentricity, and still confining himself 
 to a plane Sommerfeld treats it more completely, that is, as a 
 system of two degrees of freedom. 
 
 Let q 1} q 2 be the two configurational coordinates, say the polar 
 coordinates </>, r of the electron, T the kinetic energy as function of 
 
 the q and q, pi = ^, ^2 = 3 > tne corresponding moments. More 
 
 especially let q 1} q 2 be canonical coordinates [such are precisely the 
 said r, <], that is, in which, the equations of motion assume the 
 well-known Hamiltonian canonical form. Then, basing himself in 
 part on Planck's investigations [6] and in part upon his own 
 modifications of Planck's reasoning (cf. also [12] and [14]), Som- 
 merfeld adopts as a quantizing principle, for each of the q's, and 
 for the case of two or of any number of degrees of freedom, 
 
 dft-rtA (10) 
 
 * The reasoning, with N replaced by +N, will apply also to ionized (not 
 to neutral) He-atoms, such, that is, as have been deprived of one of their 
 two electrons and retain but a single electron. 
 
QUANTIZING PRINCIPLES 11 
 
 where n t are positive integers, independently chosen for each degree 
 of freedom, and h, as before, Planck's constant. He states himself 
 explicitly that he has no support (' Berechtigung ') for this * Quan- 
 tenansatz ' or for the elementary, differential quantum cells which 
 are equivalent to (10). As to the integration limits to be adopted 
 in each of the equations (10) nothing very definite can be said in 
 general. But when the motion, as in the present case [without 
 relativity], is strictly periodic, he integrates p dq over a full period, 
 i.e. over q 1 = ^ from O to 27r, and over q<^ = r from r min . (perihelion) 
 to r max (aphelion) and back again, or, which is the same thing, 
 
 (writing dr = ~- d<f>] over </> again from O to 2ir. When the relati- 
 
 vistic complications are taken into account the motion is no longer 
 strictly periodic. Yet for moderate velocities (v/c = small fraction) 
 the electron can be considered as describing a Keplerian ellipse 
 with slowly moving perihelion [ =peri-nucleon], and Sommerfeld 
 says, therefore [10, p. 8], that ' the statistical manifold of the 
 orbits is also here determined by a direction varying from O to 27r, 
 as for instance that of the initial line of apses, so that here also the 
 integration domain in </> is 2:r, and does not depend on the shape of 
 the orbit.' In short, without being able to support the procedure 
 adopted, he applies it both without and with the relativistic compli- 
 cations. Properly speaking, its only support is a post facto one, 
 to wit, the admirable agreement of the final results, especially those 
 concerning the fine-structure of helium lines, with the experimental 
 observations. 
 
 Sommerfeld treats first the atomic system without the relativistic 
 terms. The motion is then strictly Keplerian. One first integral 
 is the well-known p 1 = mr 2 <j> = const., and the equation of the ellip- 
 tical orbit is 
 
 I me 2 , 
 
 ^ = -^-2( I + COS <t>}> ( JI ) 
 
 r p 1 
 
 where e is the eccentricity. The total energy (E = - W) is easily 
 found to be ^* 
 
 __, //Z-C> / AX. t \ 
 
 -^i ('-*> < 12 ) 
 
 Now, if we were to quantize with respect to q l only, i.e. to 
 make, by (10), 
 
12 THE QUANTUM THEORY OF SPECTRA 
 
 "! 
 
 J 
 
 or t> t **-, 
 9* 'Mo 27r' 
 
 we should have, by (12), and by (6) which Sommerfeld takes over 
 from Bohr's theory without modification, the spectrum series 
 
 (13) 
 
 where the eccentricities e n , e n , can assume all possible values. Thus 
 we should have no discrete series of sharp lines but ' ein verwaschenes 
 Band,' to use Sommerfeld's own expression. To remedy this, 
 Sommerfeld quantizes also with respect to the second variable 
 (r), and through this he quantizes the eccentricity, which converts 
 (13) into a series of sharp lines. 
 
 In fact, the moment p 2 or p r is = mf, and thus, by (10), 
 
 i L ; f-,7 F- dr j' 
 n^fi \pzdr = m\r dr = m\ r j- a<p, 
 
 J J Jo ^9 
 
 where n z is another positive integer, independent of n v Now, 
 
 r = - sin 9 and dr/d^^l- Sm * 
 
 in virtue of (ll), so that 
 
 whence, with 27rp 1 = n l h (as before), 
 
 the required quantizing of eccentricity. Not only the angular 
 momentum, but also the eccentricity of the ' stationary ' orbits 
 can have only certain values, to wit, those prescribed by (14), with 
 
 ^ = l, 2, 3, .... This sounds strange; but by no means more 
 n 2 
 
 strange than 27rp l = n 1 h. 
 
 Now, substituting (14) into (13), the terms n^ cancel out,* and the 
 
 * Here Sommerfeld becomes mystical ; for in the circumstance that ' nf 
 hat sich gerade herausgehoben,' so that n 1 + w a only remains, he sees 
 something particularly profound, and exclaims [10, p. 20] : ' Es scheint 
 ausgeschlossen, dass ein so prazises und folgenreiches Ergebnis einem alge- 
 braischen Zufall (!) zuzuschreiben sein konnte.' 
 
ELLIPTIC ORBITS 13 
 
 result is, with N as in (8), and with n^, n 2 characterizing the final, 
 and n 1} n 2 the initial orbit of the electron, 
 
 J _A7/ I 
 
 i.e. again the Balmer series of hydrogen, consisting of sharp lines, 
 depending on four integers, * die sich aber beim Wasserstoff 
 sozusagen zufallig auf zwei ganze Zahlen [n l + n 2j n-^ + n^] redu- 
 zieren.' In short, the Balmer series of lines reappears completely, 
 but ' in ausserordentlich vervielfachter Mannigfaltigkeit ihrer 
 Erzeugungsmoglichkeiten.' For now the total integer n l + n 2 can 
 be obtained in a variety of ways. In other words, to each possible 
 value of the energy, which is given by 
 
 W =- = -, cNk . = W n (16) 
 
 K+>* 2 ) 2 
 
 belong several elliptic orbits. Sommerfeld discusses these at some 
 length [10, 5, 6] in connection with the intensity question of the 
 emitted lines. This interesting subject deserves some attention. 
 
 9 6. Notice that, by (2), p. 4, all the ellipses belonging to the same 
 *W have equal major axes, 2a = e 2 /W ; and, remembering that the 
 
 p 2 i 
 semi-minor axis is b = -^ 2 / > we nave > by (16) and (14), 
 
 i + n 2) 2 * b = -^-. (17) 
 
 a 
 
 The Balmer series of hydrogen is given by (15), when we put 
 
 const. =2, 
 
 3, 4, 5, -00. 
 
 The lines are emitted when the electron passes from one of the 
 latter, initial, to one of the former, final, orbits. Both n^ and 
 n 2 are positive integers (such being the integrals J^^, \p 2 dr). As 
 to zero, we can well have n 2 = O, which, by (14), means e = o, i.e. a 
 circular orbit; but n x equal zero would mean pi = o, i.e. a rectilinear 
 orbit passing through the nucleus, and this cannot be treated (even 
 approximately) without taking into account the relativistic terms, 
 
 * i.e., with the value (8) for N, a t> (n-. + n.^ 2 . 
 
 47r-mtf 2 
 
 A3 
 
14 THE QUANTUM THEORY OF SPECTRA 
 
 for the velocity would increase beyond limit. Therefore, in the 
 present treatment, we have to assume 
 
 n 1 >o, n^Q. (18) 
 
 Thus the final orbits (n 1 / + w 2 ' = 2) are but two in number, viz. 
 fw 2 '=0, < = 2; /. b = a, by 
 jn 2 ' = l, < = i ; /. b = ", 
 
 The initial orbits are, always by (17) and (18), 
 
 , 
 
 line H a of [-3 orbits, to wit - = I, -, -; 
 spectrum series,; 
 
 forw 1 + w 2 = 4, ) , . . b 311 
 
 line//* 'J4 orbits, to wit-- 1, |, -, -, 
 
 and so on. 
 
 Thus the line H a can be generated by a passage of the electron 
 from three different initial to either of the two final orbits, i.e. in 
 all in 
 
 2 .3=6 ways; 
 
 similarly, H$ in 2.4=8 ways ; 
 
 H y in 2.5=10 ways, 
 
 and so on, in general, in (n^ + n 2 ')(n 1 + n 2 ) ways. 
 
 This number of possibilities, which thus far is only ' arithmeti- 
 cally ' determined, Sommerfeld first reduces, on the ground of some 
 quantist guesses (* quantentheoretische Vermutungen '), and then 
 increases on the ground of some * generalized assumptions,' in 6 
 and 7, respectively, of his paper [10]. These guesses are partly 
 supported by spectroscopic experience. Of course, the quantists 
 have still to seek, groping, their way through the marvellously 
 complicated labyrinth of spectroscopic phenomena. 
 
 We have obviously, for every actual passage associated with 
 radiation, 
 
 n 1 + n 2 > n t f + n 2 f - 
 
 Now, Sommerfeld assumes, in the way of a guess, that only suck 
 passages are ' possible ' for which, separately, 
 
 n ^ w/ and w 2 ^ n z '. ' (19) 
 
 These are Sommerfeld's ' Quantenungleichungen,' which he 
 attempts to bring into relation with intensity questions. These 
 
RESTRICTING CONDITIONS 15 
 
 considerations assume a more definite aspect when the hitherto 
 single spectrum lines are resolved into several separate com- 
 ponents ; for, it will appear later that every one of the possible 
 1 ways of generation ' gives rise to a distinct line. This is due 
 to the variability of the mass of the electron, and it will be better, 
 therefore, to postpone details concerning this subject until we come 
 to consider the relativistic complications. [The actual resolution 
 becomes even more conspicuous through the agency of an electric 
 field ; Stark-effect. Cf. infra.] 
 
 Here it will be enough to state that Sommerfeld finds the rules 
 (19), which instead of n . n' ways [I write n^ + n^ n t n^ +n 2 '=n'] 
 give only 
 
 n'(n-n' + i) (190) 
 
 ways of generation, in general well corroborated by the observations. 
 Yet he adds that the first of (19) ' nur im groben richtig ist,' while 
 the second alone, i.e. 
 
 H 2 > H 2 ', (20) 
 
 is, * under normal conditions,' satisfied without exception. The 
 corresponding number of ways of generation is 
 
 '(*" ~T^)' (2<>) 
 
 \ / 
 
 As the measure of intensity of a line, due to the passage from 
 an orbit n^ n 2 to n^, n 2 ', Sommerfeld proposes to consider, 
 provisionally, 
 
 inasmuch as none of the ' Quantenungleichungen ' is violated ; 
 in the latter case the intensity is to be * small ' or zero. 
 
 The above rules reduce the arithmetically computed number 
 of ways (nn'}. On the other hand, an increase of this number 
 is brought about by taking into account the orientation of the orbit 
 in space. In fact, proceeding to quantize also the inclination a 
 of the plane of the orbit with respect to an arbitrarily fixed 
 reference plane, he finds that cos a can assume only rational 
 values, viz. 
 
 ^ 
 
 where w 3 is a new integer, corresponding to I p^d\// = n 3 h, where \f/ 
 
 Jo 
 
16 THE QUANTUM THEORY OF SPECTRA 
 
 is an angular coordinate (fixing, together with r and another angle, 
 the position in space), and p^ the corresponding moment. But, in 
 the absence of any physically given direction (say, an actually given 
 axis of symmetry) such an * orientation of the orbit plane in space ' 
 is obviously meaningless. Sommerfeld is well aware of this fact, 
 and he discusses this matter only in view of considerations con- 
 cerning such phenomena as the Stark effect, in which the external 
 (homogeneous) electric field offers an axis of reference. He 
 mentions, in fact, in this connection Epstein's investigations on 
 the Stark effect, without, however, entering deeply into them. We 
 shall come to the quantum theory of the Stark effect, as worked 
 out by Epstein and Bohr himself, a little later. Meanwhile let us 
 go on with the review of Sommerfeld's own researches. 
 
 7. The most important and, at the same time, the most beautiful 
 part of Sommerfeld's investigation is his explanation of the fine- 
 structure of the hydrogen and similar series-lines by means 
 of the relativistic treatment of the motion of the electron round 
 the nucleus. 
 
 Sommerfeld does not omit to mention that the importance of 
 the theory of relativity for the completion of his atom model was, 
 on several occasions, pointed out by Bohr himself, who [40] proposed 
 also to view the hydrogen doublets as a relativistic effect, of the 
 
 r fV 
 
 order of ( - 
 
 But in taking up Bohr's suggestion, Sommerfeld modifies 
 essentially the standpoint. While Bohr confined his attention 
 to ellipses of evanescent eccentricity, Sommerfeld in accordance 
 with the first part of his investigations explained above seeks 
 the origin of the doublets and the more complicated ' lines ' in the 
 finitely different, discrete values of eccentricity of his quantized 
 elliptic orbits. [Cf. especially formula (14), p. 12, of the present 
 report.] 
 
 To simplify matters Sommerfeld assumes the nucleus to be 
 
 fixed [which, provided that m is ultimately replaced by - '-**, is 
 
 sufficiently correct for the purpose in hand]. Then the force 
 upon the electron is, also according to the theory of relativity, 
 
SOMMERFELD'S THEORY OF FINE-STRUCTURE 17 
 
 rigorously given by e*/r 2 , and directed towards M, but the 
 electron's mass is no longer constant ; it becomes 
 
 and our previous equation of motion, (i), becomes 
 
 (*)-- (*-*) (23) 
 
 This gives a plane orbit, as before, and the integral of areas is 
 again [we shall now write p instead of p ] 
 
 p = mr z (j> = const., (24) 
 
 with the difference, however, that m -is now a function of the 
 
 P= ~' 
 
 m being the so-called rest mass of the electron. The total energy 
 E, which is another first integral, is easily found to be 
 
 . (26) 
 
 [For a simple deduction of this from (23) see, for instance, my 
 ' Theory of Relativity.'] 
 
 The equation of the orbit is easily found to assume the form 
 [with <j>=O as initial perihelion] 
 
 - =C(i +e . cos y<), (27) 
 
 where 
 
 (28) 
 
 and = const. If y differs but slightly from I, (27) represents an 
 ellipse with slowly moving perihelion, to wit, the motion of the 
 perihelion is progressive (as the astronomers say), i.e. in the sense 
 of revolution, and amounts, angularly, per period of revolution, to 
 
 Put - =p [more generally, /> = - x charge of nucleus] ; then 
 c c 
 
 (28*) 
 
18 THE QUANTUM THEORY OF SPECTRA 
 
 Thus for small ^ the orbit is almost an ellipse with slowly moving 
 
 perihelion. But when p approaches p , the orbit ceases to exhibit 
 any resemblance to an ellipse and can, under certain circumstances, 
 assume the form of a spiral. The manifestly all-important p , 
 as defined above, will hereafter be referred to as the critical value 
 of p = mr 2 ij>. 
 
 Using (24) and (26), and eliminating /3 = v/c, the energy belonging 
 to a relativistic elliptic orbit will be seen to be 
 
 The equation of the orbit, (27), (28), now becomes 
 
 where the signs correspond to initial F ^K^^T ta ken as </> = < 
 Notice that the quantized moment p is 
 
 m *f* 
 and for n = i, 
 
 so that the ratio of the critical or, as Sommerfeld calls it also, the 
 1 universal ' moment p to p l is a small fraction, to wit, 
 
 for hydrogen (nucleus e), and for a nucleus of charge ne, 
 
 ^ = K. 7 . io- 3 . 
 
 Pi ' 
 
 If p approaches p and, at the same time, e-> i, we have, under 
 some further specifications which need not detain us here, spiral 
 orbits ; but since in this case (velocity tending to c and therefore 
 mass m growing without limit) the assumption of a fixed nucleus is 
 illegitimate, Sommerfeld is not able to utilize these orbits for the 
 spectrum theory. We can, therefore, omit here his short dis- 
 cussion of spiral orbits altogether, and confine ourselves to quasi- 
 Keplerian ellipses. 
 
SOMMERFELD'S THEORY OF FINE-STRUCTURE 19 
 
 Passing to the quantizing process, with respect to 9 and r which 
 continue to be canonical variables, we have, by (10), for the 
 ^-coordinate, 
 
 p-?, (33) 
 
 while \p r dr = n 2 h gives (since, owing to the motion of the peri- 
 helion we have now to integrate from 
 
 d /i\ f 2 ^ d /i\ dr , 
 
 T-, [-} . dr= p\ T7I I ! T: 09, 
 
 d<b\rj ^JQ d<f>\r/ d<p 
 
 and this becomes, by (27), 
 
 1 -^= (34) 
 
 f 
 
 n h = -p 
 r } 
 
 which quantizes again the eccentricity of- the stationary orbits. 
 
 In virtue of (33) and (34) the expression (30) for the energy E, 
 
 belonging to any stationary orbit, becomes, for a nucleus containing 
 
 the charge KC, 
 
 I 
 
 E= 
 
 where a is the fraction (32), i.e. 
 
 a^^y.io- 3 ; (36) 
 
 n lt n 2 are independent positive integers, as before. Let us shortly 
 denote the energy value following from (35) 'for a pair of integers 
 n ij H 2 by E(ni, n 2 ) ; then, if n 1? n 2 characterize the initial and 
 n-^, n 2 the final orbit, the spectrum series corresponding to (35) 
 will be, by Bohr's assumption (6), p. 6 : 
 
 l = s{ Ki) -,%')} (37) 
 
 We see that, while according to Newtonian mechanics n t and n 2 
 entered only through their sum n 1 + n 2 (giving a Balmer series of 
 single lines), the integers n^ n 2 enter into the relativistic energy (35) 
 separately, so that the energies of the n- L -\-n 2 different orbits, corre- 
 sponding to the same value of n^+n^ are no longer equal to one 
 another. It is this which accounts for the complexity or fine- 
 
20 THE QUANTUM THEORY OF SPECTRA 
 
 structure of the spectrum lines, each ' line ' now consisting of two 
 or more separate components. 
 
 Since a 2 , appearing in (35), is a small fraction, of the order 
 5 . I0~ 5 , Sommerfeld develops the energy expression into a power 
 series of a 2 . For the visible, and also the usual ultra-violet, 
 region of the spectrum the series is needed only up to a 2 . [The 
 higher terms are of interest when the Rontgen 'spectra are aimed 
 at, for we have then large K ; cf. infra.} Thus, rejecting a 4 and 
 higher terms, (35) gives, with the previous meaning of N, 
 
 g-g(...^-- ( ^;^(i +- ( ^F 5 i(J+r 1 )}- (38/) 
 
 For hydrogen we have K = I ; for helium, K = 2, etc. The term inde- 
 pendent of a, i.e. the factor of {} , is the previous, Newtonian value 
 of E (corresponding to Newtonian mechanics) ; taken by itself, 
 it would give the previous Balmer series ; the a 2 term, due to relati- 
 vistic complications (variability of mass), appears as responsible 
 for the fine structure. To bring this into evidence, let us put 
 n 1 + n 2 = n, and let us denote the said * Newtonian ' energy by E n * 
 Then the last formula will become 
 
 -2 /T W-\\ 
 
 (38) 
 
 This, together with (37), shows us at a glance the modifications due 
 to the relativistic variability of the electron's mass. There is, 
 firstly, a general increase of E (and therefore of the corresponding 
 frequencies) whose relative amount is 
 
 This Sommerfeld calls the relativistic correction of energy for circular 
 orbits (since n z =o means : circular orbit, = o). Secondly, we have 
 an increase of E of the several ellipses, whose relative amount is 
 
 which is thus seen to be different for different ellipses (all cor- 
 responding to the same n = n 1 + n z ), and increasing with their 
 eccentricity. To these different increases corresponds a splitting of 
 
 -I.e. let 
 
SOMMERFELD'S THEORY OF FINE-STRUCTURE 21 
 
 the ' lines,' which now become groups of two and more components ; 
 Sommerfeld calls, therefore, the relative energy increase (B) the 
 splitting of energy (' Aufspaltung der Energie '). By what was 
 said before, the approximate value of the pure number a 2 , funda- 
 mental for these shifts and splittings, is a 2 1=5 . icr 5 . 
 
 Sommerfeld, with a view to Rontgen spectra, proceeds to develop- 
 the energy up to a 6 inclusively ; but this matter need not detain 
 us here. 
 
 What is here of importance is that the theoretical rules laid down 
 in (A), (B) or in (38), with (37), have turned out to be admirably 
 obeyed by the observed spectroscopic facts. 
 
 It would greatly exceed the limits of this report to quote all the 
 detailed instances of the marvellous agreement between theory 
 and experiment. We shall, therefore, confine ourselves to a 
 few instances. 
 
 As an immediate consequence of the above formulae, the Rydberg 
 factor in the Balmer series of hydrogen, instead of being equal, 
 to N say, for all lines, should become 
 
 where n = 3, 4, 5, etc., for the lines H a , Hp, H y , etc. Now, Paschen's 
 measurements of 
 for //-//, gave 
 
 AA 7 "' 
 measurements of =-* which were most exact and most reliable 
 
 N 
 
 while (39) gives, for H a - Hp, 
 
 *****. 
 
 which is certainly a satisfactory agreement, giving the same order 
 of magnitude, and hardly more than this could be expected in 
 view of the difficulty of such refined observations and of the 
 uncertainty of a 2 , and the same sign of AAT'. [N.B. A certain 
 theoretical modification due to Planck would give the wrong sign 
 
 * Paschen quotes (Ann. Phys. 50, p. 935) for H a Hp the ' corrected ' 
 wave-lengths 6564-6592, 4862-7116 and, while he tries to represent the 
 
 series by A 7 "'. (--:aj ne finds N' H =109678-205, N' H =109678-164. For 
 details cf. Paschen, I. cit. 
 
22 THE QUANTUM THEORY OF SPECTRA 
 
 and the wrong order of magnitude, to wit, -0-48. I0 3 , instead of 
 Sommerfeld's + 0-61 ; this speaks in favour of Sommerfeld's as 
 against Planck's proposed ' Quantenansatz.' For details see [10], 
 
 -PP- 57-59-] 
 
 But much more remarkable are the many instances of agreement 
 of theory and observation relating to the fine- structure (splitting) 
 of the lines of hydrogen and of ionized helium, not only with regard 
 to the configuration and distance between the components or 
 satellites, but very often also with regard to their intensity. 
 Again, in several cases, Paschen [in assiduous, reciprocal, communi- 
 cation with Sommerfeld] was able to discover a new component or 
 satellite only after its existence was insistently predicted to him 
 by his theoretical colleague. 
 
 Let us quote at least a few examples of such fine-structure 
 agreement. 
 
 For n = n 1 + n 2 = 2 there are, as already mentioned, two possi- 
 bilities, 2 +oand I + I. Thus the constant term, i/2 2 , of the Balmer 
 series of hydrogen (corresponding to two final orbits) should 
 give rise to a doublet,* whose components should show a. frequency 
 difference, which to a first approximation is equal to 
 
 No. 2 
 
 ^- (40) 
 
 (where N = 1-097 . io 5 , a 2 = 53 . IO~ 5 ), i.e. constant f for all lines 
 (H a , ///B, etc.) and amounting to 
 
 A (V =0-363 cm.- 1 (Theory) 
 
 Now, the best observations on H a , which are due to Fabry and 
 
 Buisson, gave 
 
 (Obs.) 
 
 (For H y , Michelson finds 0-42, but it is the last quoted value 
 which is considered to be the most reliable.) The agreement is 
 good. Moreover, Sommerfeld shows that if the distance is 
 measured not from the chief line of the first component, but from 
 the mid-point between la and Ib (cf. Sommerfeld's paper, Fig. 2) 
 
 * Each component of these doublets will further consist of several sub- 
 components, owing to the multiplicity of the initial orbits. 
 t To a first approximation. 
 
FINE-STRUCTURE OF HELIUM LINES 
 
 to the mid-point of lib and lie of the second component, Fabry 
 and Buisson's value is increased by twenty-five per cent., thus 
 giving 0-382 cm." 1 , which is still nearer the theoretical value 0*363. 
 
 Again, according to Sommerf eld's theory, [10], pp. 62-63, tne 
 stronger of the two components of the doublet should be the one 
 on the red side, and such is also the observed fact. 
 
 Owing to the multiplicity of the initial orbits (variable term 
 of Balmer series) each of the above components of the doublet 
 should, according to the relativistically refined theory, consist of 
 several distinct lines, and the intensity of these lines should 
 decrease towards the red end of the spectrum (just the reverse of 
 the components of the doublet, each taken as a whole). Each 
 of the H a components, for instance, should be a triplet, each of 
 those of Hp and of H y a quartet and a quintet, respectively, and so 
 on. Some of these fainter lines or * satellites ' are theoretically 
 expected to be but very weak, some others should (in virtue of the 
 ' Quantum-ungleichungen ') be absent. 
 
 T I 
 
 c ba 
 
 He 4686. 
 
 Now, many of these details, even those concerning the intensities, 
 but more especially those concerning the multiplicity of a * line ' 
 or, properly, group, and the relative distances between its com- 
 ponents and satellites, are in most remarkable agreement with 
 Paschen's careful spectroscopic observations, especially of helium. 
 
 To see this, a glance at Sommerfeld's figures 4, 5, 6 (pp. 73, 75 
 [10]) will suffice. These three figures, which, with Sommerfeld's 
 kind permission, are here reproduced, contain Sommerfeld's theo- 
 
24 THE QUANTUM THEORY OF SPECTRA 
 
 retical configurations of components with Paschen's observations 
 juxtaposed.* The coincidence is striking. 
 
 I IF TIT 
 
 Hi 
 
 He 3203. 
 
 Further, some slight 'defects in the constancy of the frequency 
 difference between the components of the doublets are accounted 
 for. According to Sommerfeld's theory this difference should tend 
 
 i 
 
 \ 
 
 He 2733. 
 
 to constancy only for the higher members of the series, while in the 
 first members there should be deviations from constancy, which 
 again seems to bring better harmony into the relation between 
 the observational material and theory. This subject, intimately 
 connected with the fine-structure of the components themselves, 
 need not detain us any further. [For details, cf. [10], pp. 76 et seq.} 
 
 * The Roman numerals correspond to the final orbits as determined by 
 W 1 / +w 2 ' = 3, thus 
 
 3+0 I 
 
 2 + 1 II 
 
 1+2 III. 
 
 The last row in the diagram of He 4686 corresponds to a spark spectrum, 
 the remaining shaded rows in all three diagrams correspond to the usual 
 method of excitation. 
 
HELIUM TRIPLETS 25 
 
 Some of the observational material alluded to in the above lines 
 and pages concerns hydrogen. But an important part of the 
 verification of the theory was furnished by the observed spectra 
 of helium, to which also the diagrams here given refer. Since 
 the nucleus of the helium atom contains twice the charge (2e) 
 belonging to the hydrogen nucleus, the frequency differences of the 
 components of each line are, for helium, 2 4 = i6 times greater than 
 for hydrogen's analogous infra-red series. We have here in mind 
 the principal series of helium, observed by Fowler, which apart 
 from the relativistic refinements is represented by 
 
 X -*(?-;?) " = 4,5,6,7,-, (41) 
 
 and it will be enough to mention here some details concerning 
 this series only. The constant term being n 1 + n 2 = ^ 1 each of the 
 lines, coarsely represented by (41), should be a triplet, and the 
 frequency differences between the 1st and 2nd, 2nd and 3rd 
 component should bear to one another the ratio 1:3; to wit, 
 
 Na 2 
 if j- = 0-363, mentioned previously, is shortly denoted by A^, 
 
 the said two frequency differences are, theoretically, 
 
 128 128 . v 
 
 A.-jj-'A* **-T 7 -*'- (42) 
 
 Now, Paschen's careful observations of the helium series (41), 
 stimulated by Sommerfeld's predictions, have not only revealed, 
 for the first time, the triplet-nature of each ' line,' but also the fine 
 structure of each of the three components, especially for the first 
 member of the series, n=4, but also for n = $, n = 6. Several of 
 the satellites, even the fainter ones, predicted by the theory, were 
 detected, for the first time, on Paschen's spectrophotographic 
 plates. 
 
 The Figures 4, 5, 6, of Sommerfeld, [10], given above con- 
 cern precisely the ' lines ' n=4, n = 5, and n = 6 of the series (41) 
 with its relativistic correction and refinement terms. 
 
 The observations on helium, ionized helium that is,* provided, 
 in fact, according to Sommerfeld's own opinion, the best confir- 
 mation of his theory, extending to almost all of its details. The 
 
 * For such is, according to general opinion, the ' helium ' treated above ; 
 it has lost one of its electrons and, like hydrogen, retains but one. 
 
26 THE QUANTUM THEORY OF SPECTRA 
 
 details being all due to the relativistic refinement of Bohr's original 
 theory, these experimental confirmations of the fine-structure 
 constitute a new triumph of Einstein's * old ' theory of relativity 
 (1905). It is well to bear this in mind, because of late many 
 conservative physicists have thought that Relativity, after 
 having accounted for Fresnel's dragging coefficient and for the 
 variability of mass of the /2-particles, had become barren. 
 
 The reader will have noticed that Sommerfeld's investigation is 
 based on the Lorentz-Einstein electron, the mass of which is 
 
 It may be interesting to note that K. Glitscher [iifl], on W. Lenz's 
 and Sommerfeld's suggestion, took up the question whether 
 Abraham's rigid electron, for which the momentum is 
 
 . . . f 
 
 log - '' mStead f 
 
 would show any observable differences with respect to the fine- 
 structure of spectrum lines. With this purpose in view, Glitscher 
 develops the general formulae, putting mv equal to any function 
 of /?, and applies them then to Abraham's electron, breaking off 
 the series development of the energy with the a 2 term. The result 
 is that, whereas Lorentz's electron, as in Sommerfeld's theory 
 just reported, yielded a 2 = 5-30. icr 5 , harmonizing well with the 
 best observed values ^=476 . icr 10 and h = 6'$i . icr 27 , the rigid 
 electron of Abraham would require at least a 2 = 6-37 . icr 5 , while 
 (according to Glitscher) e, h cannot be pushed beyond 4-8 . icr 10 and 
 6-3 . icr 27 , giving only 5-86 . icr 5 for the spectroscopic constant a 2 . 
 In fine, Glitscher's pronouncement is decidedly against the rigid, and 
 for the Lorentz electron. One cannot help pointing out a certain 
 exaggeration and superfluity in this evidence. The superiority 
 of the Lorentz electron was demonstrated some years ago by 
 safer and more convincing methods, and there is no need to 
 return to Abraham's electron. 
 
 We can omit here Sommerfeld's short discussion of the spectra 
 of neutral helium (retaining both electrons) and of lithium, as well 
 as his remarks on ' universal ' spectroscopic units, such as his a 
 used above, and shall pass on to give a short account of his in- 
 vestigations on the theory of Rontgen spectra, as laid down in [11], 
 This will occupy our attention in the next section. 
 
ROENTGEN SPECTRA 27- 
 
 8. Rontgen spectra. Let Z be the ordinal number of a chemical 
 element in the periodic system (commonly denoted by N), and 
 let KC be the charge of the nucleus of the atom of this element ; put 
 l = Z-K. Then* according to Sommerf eld's theory, the ' lines ' of 
 the so-called K- and L-series should be doublets whose components 
 are separated by a frequency distance 
 A = (Z-/) 4 . A,, 
 
 where A /7 is as in (40). Thus, the value of ,-~ ^ should be- 
 
 (Z, I) 
 
 constant, and equal to ^ ff = o-^6. 
 
 Now, this relation was excellently confirmed throughout the whole- 
 domain of the system of elements, from Cr to U (i.e. from Z = 24 
 to Z = 92), for which measurements were available (1916).* To 
 give an idea of the closeness of agreement it will be enough to 
 mention that 
 
 Cr Cu Zn Br Rh Ag (tf -doublets) 
 gave, for A . (Z-/)- 4 , respectively, 
 
 0-43 0-40 0-39 0-47 0-35 044 
 and Ag Cs Pt Hg Th U (L-doublets) 
 whose Z are 47, 55, 78, 80, 90, 92, gave 
 
 0-39 0-39 o-44 5 0-45 0-49 0-50. 
 
 None of the values tabulated in Sommerf eld's paper including 
 many more than those given above exceeds the limits 0-35 and 
 0-50. 
 
 It must be kept in mind that the deviations in the ^-doublets are,, 
 perhaps, mainly due to the fact that their measured A's are un- 
 certain to about 20 per cent. On the other hand, the L-doublets 
 
 show a systematic rising of -r^ j^ from 0-389, for Z = 47, up to 
 
 U* *} 
 
 0-498, for Z = 92, which was to be expected on Sommerfeld's theory^ 
 the value A.(Z 1)~*== & being only a first approximation. 
 Sommerfeld's higher approximation is 
 
 r sa*(z-i)* $ 3 a*(z-in 
 
 * X 1 ~2 2*~ '~8~ 2 4 /' 
 
 * The value of / originally taken by Sommerfeld was i. Later on he 
 took 1 = 3-5, but even with l = i the agreement is very good indeed. We 
 may note that 1= i was adopted by Moseley. 
 
28 THE QUANTUM THEORY OF SPECTRA 
 
 But details of this kind need not detain us here any longer ; nor 
 can we enter, in a short report, into Sommerf eld's further investi- 
 gation and discussion of the detailed structure of the Rontgen 
 series of lines and their satellites, which is partly of a constructive 
 and partly of a critical nature. Suffice it to say that even into 
 this domain of phenomena Sommerfeld's work introduces order 
 and clearness, which are likely to make the enormous observa- 
 tional material more ready for a thorough physico-mathematical 
 treatment. 
 
 Leaving, therefore, the subject of the Rontgen spectra * and of 
 the relativistic refinement of Bohr's theory, we will now pass to 
 another topic, the effect upon spectrum emission of an external 
 electric field, known as the Stark-effect, which offers yet another 
 example of the conspicuous success of the quantum theory of spectra. 
 
 9. Theory of the Stark-effect. This was worked out by Sommer- 
 feld's pupil, Dr. P. S. Epstein [7] and, independently of him, on 
 somewhat different lines, by Schwarzschild, like Moseley and, 
 alas, many others, prematurely lost to science and humanity in 
 what is believed to be the ' last ' war. Schwarzschild's paper 
 (Berl. Ber. p. 548, 1916), the last work of this eminent physicist 
 and mathematician,! being, unfortunately, inaccessible to me for 
 the present, I shall base this part of the report on Epstein's paper 
 alone, which, however, may be found sufficiently illuminating. 
 
 Epstein investigates an atomic system : nucleus of charge ne 
 and a single electron-planet, as in section 2, but placed in an uniform 
 electric field of intensity (. He disregards relativistic complica- 
 tions, so that his differential equation of motion is simply, with 
 the tf-axis along the electric field, 
 
 mr = VU, U=- K -^-ex; r 2 = x* + y*. (44) 
 
 Mathematically this case is a sub-case of the famous problem 
 of Euler-Jacobi (motion about two fixed centres), in which the 
 
 * Glitscher's investigation [na], mentioned in Section 7. also condemns the 
 unfortunate rigid electron in connection with the theory of the Rontgen 
 spectra, inasmuch as it gives for the denominator of the first term of 
 the L-series of doublets the value 1-892 instead of Sommerfeld's 2 ooo, 
 which corresponds to a Lorentz-Einstein electron. 
 
 t Who died not ' in action,' but of an infectious illness contracted 
 at ' the front.' 
 
THEORY OF STARK-EFFECT 29 
 
 method of ' separation of variables ' can be applied. In the 
 present case the two variables , 17 which have this property are 
 those denned by 
 
 *=i(J 2 -^ 2 ), y=V 
 
 The third coordinate, <, of the cylindrical system, with % as axis, 
 has this property automatically. If p^ p^, p$ be the corresponding 
 moments, p^'dT/'dg, etc., we have 
 
 Epstein adopts Sommerfeld's form of the quantizing-principle, and 
 puts, therefore, 
 
 \Psdt~nJi, \pidvj~nji, U*<ty= 3 fc, (45) 
 
 n i, n * n z being three independent integers. The last integral is 
 simply taken between the limits o and 27r, as in the case of periodic 
 orbits. The choice of the integration limits for and ?/, however, 
 requires some special care, since the orbits of the electron are, due to 
 (, no longer periodic. Guided by the analogy of Sommerfeld's inte- 
 grating over r (cf. supra) from r min . to r max . and backwards, Epstein 
 decides to take the first of (45) twice between ^ and ^ 2 , and the 
 second between r/ x and r; 2 , where lt 2 and rj lt rj 2 are those values for 
 which dg/dri=o, dr]/dg = O, respectively.* 
 
 With these limits the first two integrals (45) are evaluated 
 approximately up to (S; 2 , while the third integral offers no difficulty ; 
 the angular momentum, say a\/m, round the #-axis being constant, 
 
 _ pir _ 
 
 the third of (45) or ^/m\ ad<f>=>hn 9 simply gives a<\/m=hn 3 /27r. 
 
 The calculation of the (5; 2 -terms of the first two integrals would 
 offer some technical difficulties, but these higher terms are, for the 
 present, scarcely needed, and Epstein, therefore, does not push 
 his approximations beyond (. The expressions thus obtained 
 
 T| = O has also a third root 77.5, but this is imaginary, and ^- = has a 
 third root 3 , which is real but ( for G small as compared with -5 ) enormous, 
 
 while lf , differ but little from their values for = o. These latter, 
 therefore, are chosen for the purpose in hand. 
 
30 THE QUANTUM THEORY OF SPECTRA 
 
 contain, besides n lt n 2 , w 3 , the energy constant IV, and yield 
 ultimately for the value of W for a stationary orbit characterized 
 by n lt n 2 , 3 , the expression 
 
 TT/ . 
 
 W = - s + o o n (n, - 2 ) , 46) 
 
 2 2 v 1 
 
 where N = p- is, as before, the Bohr value of the Rydberg 
 
 constant, and 
 
 n = n 1 + n 2 + n s . 
 
 The corresponding series of spectrum lines is, by Bohr's principle, 
 <6), P. 6, 
 
 X ' W ^ - n + 8 e {"' - "*'> - " <*' - ) > ' (47) 
 
 The first term is the ordinary series of the Balmer type (for 
 hydrogen K = I, etc.) ; the second term, proportional to the inten- 
 sity ( of the electric field, gives the Stark-effect, i.e. a slight shift 
 and a splitting of the lines given by the first term. For, 
 
 and n' = n- L f +n 2 ' +n 3 ' being fixed, there are still, in general, several 
 possibilities for the addends, and, therefore, for n^ -n 2 and n\ - n 2 
 which appear in the second term. 
 
 Such then is Epstein's final formula of the Stark-effect, for 
 hydrogenic [hydrogen-like] spectrum lines at least. 
 
 The resolution or splitting of lines is here somewhat similar to, 
 but, with the electric fields available, much stronger than the 
 relativistic one discovered by Sommerfeld. The comparison with 
 observations is, therefore, in Epstein's case considerably easier. 
 Paschen had to discover many of the components or satellites 
 predicted by Sommerfeld, whereas owing to the assiduity of Johann 
 Stark and his school, much good material was ready for comparison 
 with the theory in question. 
 
 Now, the agreement of the theoretical effect, which for K = i 
 is, by (47), 
 
 with Stark's rneasurements on hydrogen lines, is even more striking 
 than in the case of the relativistic refinement. 
 
THEORY OF STARK-EFFECT 31 
 
 * According to Stark's measurements the separation is proportional 
 to (, as in (470). [We will, however, not count this agreement as one 
 of the triumphs of the theory ; for the above series development 
 was simply stopped at the ( term ; it would rather seem incumbent 
 upon Epstein to estimate the ( 2 -term and show that it is but a 
 small fraction of the above 1st order term.] . In the second place 
 Stark's observations show that the separation is symmetric with 
 respect to the original position of the line, and (47#) is in full agree- 
 ment with this experimental result (interchanging n-^ with n 2 , which 
 does not change n 1 + n 2 + n 3 , i.e. the original position, we have for 
 every + AX a corresponding AA). But the most remarkable 
 feature is the excellent agreement of the several intervals AA 
 between the electrically separated components, as obtained by 
 Epstein's theory and by observation, to wit, in the case of the 
 lines H a) Hp, H y , H s of the Balmer series of hydrogen. 
 
 If, in (470), we call n, etc., the integers of the constant term and 
 m t etc., those of the variable term, the Balmer series will be given by 
 
 n = 2, m = 3, 4, 5, 6, ... 
 (H a Hp H y HS ...), 
 
 and the last factor in (470) will become 
 
 A [(!- W 2 )]=m(w 1 -m 2 )- 2(n 1 -n 2 )=Z, say, (48) 
 
 where m=m 1 + m 2 + m Z) n 1 + n 2 + n 3 = 2. Epstein adopts, as the 
 basis of selection of possible Z-values, Sommerfeld's inequalities, 
 extended also to the third integer w 3 , i.e. 
 
 l^% n 2^>2, n 3 ^W 8 . (49) 
 
 He thus obtains, for the H line (w = 3), for instance, the following 
 six possible values : 
 
 Z = 5, 4, 3, 2, i, o; (H a ) 
 
 similarly, for m = 4, the seven values 
 
 Z = I2, 10, 8, 6, 4, 2, 0, (H ft ) 
 
 for w = 5 (H y ) as many as twenty different values, and for Hs 
 seventeen different Z-values. 
 
 Now, most of the theoretical components had been actually 
 observed, and in all cases in which they were observed and their 
 (wave-length) shifts measured, the results are in excellent agree- 
 ment with the theory. Thus, for instance : 
 
32 THE QUANTUM THEORY OF SPECTRA 
 
 r (Z = 5 ) (4) (3) (2) (i) (o) 
 
 H a - A/V calculated : 147 117 8-8 5-9 2-9 o A. U. 
 I ,, measured: 11-5 8-8 6-2 2-6 o 
 
 ( (12) (10) (8) (6) (4) (2) (o) 
 
 H J calc. 19-4 16-1 12-9 97 6-5 3-2 o 
 
 I meas. 19-4 16-3 13-2 10 & 97 67 & 6-6 3-3 & 3-4 o. 
 
 V /-comp. j-cotnp. / s p s 
 
 Here p and s stand for ^-component and s-component, the 
 usual notation of Johann Stark and others for light polarized 
 parallel, i.e. with the electric oscillations parallel to the vector (, 
 and perpendicular (s senkrecht) to this vector. 
 
 Now, it is most remarkable that, as Epstein found without 
 exception, 
 
 w 3 -w 3 even gives a ^-component,) , . 
 
 and m 3 n s odd gives an ^-component./ 
 
 This is also a good hint which Nature (observation) gives the 
 groping quantist for the further development of his theory. 
 
 The above reproduction of the H a - and //^-tables does not 
 exhibit this remarkable rule (50), but Epstein's full tables (loc. cit. t 
 pp. 512, 513), show it in a striking manner. To quote but a few 
 examples, we have for H s , 
 
 Z = 32 28 24 [w 3 -n 3 even] 
 
 calc. AA = 38-i 33-4 28-6, 
 
 obs. AA = 37-5 33 ^p 2S-6p, 
 
 etc., etc. 
 
 Even the intensities as based on theoretical guesses agree suffi- 
 ciently well with the observations, especially if the last of (49) 
 is replaced by n 3 <m 3 + i, and under certain further clauses. 
 But this aspect of the theory being still in its infancy, we may 
 leave it for the present. 
 
 Epstein is certainly justified in concluding his elegant paper by 
 the expression of the belief " that the communicated results offer 
 a strikingly convincing proof for the ' Richtigkeit ' of Bohr's 
 model of the atom." 
 
 The italicized word would stand for ' correctness.' But I am 
 inclined to think that neither word is the proper one. I should 
 prefer to say of such an atom model that it is a good working 
 hypothesis, that for the time being it has a heuristic value, 
 
ZEEMAN PHENOMENON 33 
 
 and that it is not likely soon to exhaust itself as such. (Of course, 
 it may become barren with time, as almost every theory hitherto 
 constructed by man, but it promises to lead still to some further 
 results agreeing with Nature.) Such, no doubt, was the meaning 
 Dr. Epstein wished to convey by ' richtig.' He justly adds that, 
 in the knowledge of atomic mechanisms and thus far little more 
 than hydrogen is covered we are at the beginning of a new 
 evolution, and that much still, it would seem, is to be expected 
 from the quantum theory for the atomist. 
 
 Accordingly, the quantum theory of spectra should certainly 
 become an attractive field for many investigators. It would 
 seem also that it should stimulate a further refinement in the 
 spectroscopic workshop, for as can be seen from Paschen's and 
 Sommerfeld's requirements a further increase in resolving power 
 and sharpness of images would help very much in testing the theo- 
 retical predictions and even in guiding quantists in elaborating 
 and improving their theory. 
 
 10. The preceding sections describe the birth of the quantum 
 theory of spectra and its definite and, as it seems, well consolr 
 dated progress practically up to the present time. Investigations 
 other than those treated above have, thus far, more or less the 
 character of attempts, in many cases promising, no doubt, but yet 
 immature. Upon these it will perhaps be enough to touch briefly 
 in this concluding section of our report. 
 
 The quantum theory of the effect upon spectrum emission of 
 a magnetic field (Zeeman phenomenon) has been worked out by 
 Sommerfeld, Debye, and Herzfeld, and, more recently, discussed 
 on somewhat different lines by Bohr [15], Part II, and by H. A. 
 Kramers [18], It may be said that these interesting and elegant 
 investigations account for the main features of the Zeeman effect 
 as known from observation ; but many important details are 
 still to be dealt with, and even some questions of a fundamental 
 nature seem to remain unsettled. Moreover, as in the case of the 
 problems previously mentioned, what has been worked out under 
 this head is again the simplest atomic system representing hydrogen 
 and hydrogen-like elements, whereas especially with regard to 
 the Zeeman effect the theory of other, more complex, atomic 
 systems would seem to be most desirable. 
 
34 THE QUANTUM THEORY OF SPECTRA 
 
 A promising attempt at an application of the quantum theory 
 to fat photoelectric effect and to the /3-particles, emitted by radio- 
 active substances was made by Epstein [8]. This is particularly 
 interesting, from the theoretical standpoint, as unlike the 
 previous papers it shows how to quantize the hyperbolic orbits 
 appropriate to the escape of the electron. We may mention here 
 that Epstein, while retaining Bohr's original principle, hv = W'-W, 
 for the. passage of the electron from one closed, i.e. elliptic, orbit 
 to another such orbit, makes for the passage from an elliptic to a 
 hyperbolic orbit the new assumption, 
 
 hv^W^-Wa = h* 9 , (51) 
 
 where v is the frequency of the light illuminating the atom and 
 thus stimulating the passage in question, and v g is the limiting 
 frequency, below which no ' photoelectric ' electron is released. 
 Epstein supports this assumption by some known experimental 
 facts, as those found by Wagner and Kossel. The chief experi- 
 mentally important result of Epstein's theory is the maximum 
 velocity it yields. This latter agrees with observation very closely 
 indeed, at least for Gehrcke and Janicki's experimental conditions 
 (Annalen der Physik, xlvii, p. 679, 1915). With regard to 
 the /?-rays from radio-active substances, Epstein's calculated 
 velocity of the electrons constituting these ' rays ' shows a re- 
 markable agreement with the measurements on radium B and C 
 (Ra B and Ra C] made by Danysz, Rutherford and Robinson, 
 and others. Yet, Epstein himself does not consider his theory 
 in its present form as final, chiefly because he is not satisfied 
 with his own arbitrary artifice by means of which he attempted 
 to avoid the difficulty arising from the quantum-integrals 
 becoming, in general, infinite for hyperbolic orbits. Notwith- 
 standing this, his theory seems promising, especially with regard 
 to the /?-rays. 
 
 Turning to questions of a more general nature, we have to 
 mention an interesting, although not unobjectionable paper 
 by Th. Wereide [5] on the energy exchange * between matter and 
 aether.' Wereide attempts to reduce Planck's quantum hypothesis 
 to things more plausible from the classical point of view. He 
 gives also a very simple deduction of Planck's famous black body- 
 radiation formula, and of the Balmer series, with Bohr's value 
 
PLANCK'S INVESTIGATIONS 35 
 
 of the Rydberg constant. Moreover, the former 
 continuous spectrum appears only as a certain limiting case of 
 line spectra or of what Wereide calls ' atomic radiation.' Un- 
 fortunately all these beautiful and stimulating results are obtained 
 by furnishing the atomic system, and especially its nucleus, 
 with an exact multiple of a certain magnetic quantum, the * Kern- 
 magneton.' Thus, it would seem, the whole difficulty associated 
 with Planck's theory and concepts is only shifted back. Yet, 
 Wereide's theory certainly deserves the attention of specialists. 
 
 A very important contribution to our subject, especially with a 
 view to the hitherto neglected question of absorption, is made by 
 Planck, 1917, in two most elegant and suggestive communications 
 entitled Zur Theorie des Rolationsspektrums [16]. In opposition 
 to Einstein's opinion (Ber. deutsch. phys. Ges., 1916, p. 318), Planck 
 assumes that Bohr's stationary orbits are not the only possible 
 ones, but that they are the only orbits associated with the emission 
 of radiant energy. This assumption has the advantage of being; 
 compatible with the laws of absorption deducible from classical 
 electrodynamics, and Planck's purpose, in comparing the con- 
 sequences of the said assumption with the experimental facts, 
 in perfect harmony with the later form of his quantum theory,* is 
 to save, if possible, these classical laws of absorption. Only if these 
 consequences should turn out to disagree with the facts will it be 
 necessary to abandon the classical theory also for absorption. 
 With this purpose in view, Planck turns his attention to the 
 absorption spectra presumably due to the rotations of rigid 
 electrical dipoles investigated by N. Bjerrum, H. Rubens and 
 others. In the first communication the axes of the dipoles are 
 assumed to be fixed, in order to simplify the investigation in a 
 first attempt. The second communication is devoted to an 
 extension of the results to dipoles with free axes. Let N be the 
 number of dipoles, each of electric moment E, per unit volume of 
 the gas ; of these let NP(&) d& have their angular velocities con- 
 tained between ui and W + dT3. Then Planck's formula for the 
 absorption coefficient a 3 , belonging to an incident radiation of 
 frequency trr, can be written 
 
 : " Concerning only the emission and not the absorption processes. 
 
36 THE QUANTUM THEORY OF SPECTRA 
 
 where / is the moment of inertia of each dipole. The most note- 
 worthy feature of this value of the absorption coefficient is that 
 c vanishes not only for P = 0, but, more generally, for any distri- 
 bution-function P proportional to ST. Thus, if there is no absorption 
 at all in a certain region of the spectrum, such as between two 
 neighbouring absorption lines, 
 
 nh (n + i)h . N 
 
 w n = T-j, and ST M+I = - y-, (S3) 
 
 then, within this region, P(rs) need by no means vanish, but must 
 be proportional to the angular velocity rar itself. In fact, Planck 
 finds also theoretically, by two different methods (a purely 
 statistical, and an electro-dynamical one), that within every 
 elementary domain of the phase-space P is equal to & multiplied by 
 a constant. He then proceeds to evaluate the jump of P(&) at 
 the frontier TS = TS n of the (n- i) th and the w th elementary domain, 
 where TS n is as in (53). This gives for the measure of the intensity 
 of the absorption line at X = X n =4Tr 2 cJ/nh an expression which, 
 apart from a constant factor, is 
 
 *+n 
 
 (54) 
 
 h? 
 where log ^= - o T7 ^, h and k being the two familiar constants 
 
 07T -* 
 
 appearing in Planck's radiation formula, and T the absolute 
 temperature. The coefficient a n vanishes for n=^0 as well as for 
 w = ocT, and attains its maximum for ^ 2n = (2n - i) (2n + l), i.e. for 
 
 ' Thus we should have maximum absorption at the 
 wave-length 
 
 . (55*) 
 
 Again, the intensity of a given absorption line A n , which varies 
 with the temperature, will reach its maximum at the temperature 
 
 T -' tt **J- (55*) 
 
 " iax ~&A n 2 ' 
 
 These theoretical results are compared with the observations 
 of Rubens and Hettner (Berl. Ber., 1916, p. 179) on the rotation 
 spectrum of water vapour, in an avowedly provisional way only, 
 
. PLANCK'S INVESTIGATIONS 37 
 
 since manifestly the f/ 2 (9-molecules cannot be regarded as simple 
 dipoles. To the ordinal number n = i2 of Bjerrum's series corre- 
 sponds the absorption line A M = i4'3fi, whence, by the relation 
 A ?( =47r 2 c//n/z, the moment of inertia 7 = 9-4 . IO~ 41 , which is of the 
 right order. The temperature of the vapour being 7 = 398, this 
 gives ^=0-90, and, by (550), 
 
 which would correspond to the absorption line n = 2 of Bjerrum's 
 series. This agrees well enough with Rubens and Hettner's 
 experimental diagram, which shows a strong increase of absorption 
 with decreasing value of n. Formula (55^) gives for the line n = 12 
 a very high temperature for its maximum intensity, to wit, about 
 2800 C., which is much above the actual temperature (125 C.) 
 at which the observations were made. But for the line n = 5, 
 i.e. A Jl = 33'3/*, we should have only r, nax = 58o, i.e. about '300 C. 
 Thus, if the vapour were heated above 300, this absorption line 
 should be weakened. It seems that these predictions have not 
 up to the present been tested experimentally. Reasons of space 
 prevent us from dwelling any further upon this most attractive 
 subject. We may add only that Planck himself is by no means 
 satisfied with his own beautiful work!* Yet, it can scarcely be 
 doubted that it will stimulate many new and important investi- 
 gations both theoretical and experimental. 
 
 A few words must now be said about Bohr's own large memoir 
 on The Quantum Theory of Line Spectra [15], which is to consist 
 of four parts, but of which unfortunately only Parts I and II are 
 yet published. A large portion of these two parts of this admir- 
 ably written memoir is devoted to an account and to the dis- 
 cussion of work done by others since the publication (1913) of 
 Bohr's first paper. Yet, especially in Part I, many new and 
 promising general points of view are expounded, both with regard 
 to the quantizing principles and concerning the probability 
 of ' spontaneous transition ' from a given stationary state to a 
 
 * In the concluding paragraph of his second communication [16] Planck con- 
 siders a certain limiting condition \loc. cit., p. 255, formula (51)] to be the weakest 
 point in his chain of reasoning, while in a recent private letter to the present writer 
 Prof. Planck says that what strikes him " unpleasantly ( ' unbehaglich' ) is that in his 
 theory the frequency of the emitted and the absorbed light is identified with the 
 frequency of rotation, which is in complete contradiction to Bohr's model." 
 
38 THE QUANTUM THEORY OF SPECTRA 
 
 neighbouring one. The latter subject is investigated in connection 
 with the coefficients of the Fourier expansion of the displacement 
 component of the mobile along any given direction. This subject 
 of the intensities of spectral lines, which has since been developed 
 further in a most masterly way by H. A. Kramers, reappears also 
 in connection with attempts at finding criteria for the state of 
 polarization of the emitted light under various external circum- 
 stances. A very valuable investigation on perturbed periodic 
 systems is given in Part II, the purpose of which is to illustrate 
 the General Theory laid down in Part I, using hydrogen as an 
 example. The author happily advocates the use of the method 
 of perturbations borrowed from celestial mechanics, and especially 
 of what in the latter science is known as the method of variation of 
 elements, secular variation, that is, of the elements of Keplerian 
 motion. This facilitates the treatment of even such problems 
 as that of the fine-structure of spectrum lines due to relativistic 
 complications. The Fourier expansions are again taken up, in 
 broad lines, and Kramers's paper, just quoted [18], is announced as 
 an investigation in which it will be shown in detail how such 
 calculations will enable us to account for the relative intensities 
 of the fine-structure components, even as influenced by the experi- 
 mental conditions of spectrum excitation, a subject already 
 pointed out by Sommerfeld in collaboration with Paschen. Both 
 parts of Bohr's new memoir abound in hints which are likely to be 
 very helpful in building up the theory of spectra originated by 
 Bohr himself. Parts III and IV, in preparation, are to contain 
 a discussion of problems arising in connection with ' spectra of 
 other elements ' (other, that is, than hydrogen), and a general 
 discussion of the theory of the constitution of atoms and molecules 
 on the lines of the quantum and the nucleus-atom theory. 
 
 A clear statement of the properties of so-called ' conditionally 
 periodic ' systems (an important concept in these investigations) 
 in connection with the ' adiabatic invariants ' of Ehrenfest [14], is 
 given in a short paper [17] by J. M. Burgers, to which the reader 
 must be referred for further information. Among more recent 
 papers, not included in the list given at the beginning of the Report, 
 we have to mention P. S. Epstein's new theoretical investiga- 
 tion of the Stark-effect in Fowler's helium series (Ann. d. Physik, 
 Iviii, pp. 553-576, April 1919), which proves his assumptions 
 
RECENT PUBLICATIONS 39 
 
 for conditionally periodic motions and Bohr's theory of intensities 
 to be in excellent agreement with Stark's measurements ; further, 
 a paper by Manne Siegbahn on ' Rontgenspektroskopische 
 Prazisionsmessungen ' (ibid., lix, pp. 56-72, June 1919), giving 
 further experimental support to Sommerfeld's theory, and a paper 
 by W. Kossel and A. Sommerfeld (Verh. deutsch. Phys. Ges., xxi, 
 pp. 240-259, April 1919) entitled, ' Auswahlprinzip und Verschie- 
 bungssatz bei Serienspektren,' which, though not devoid of interest, 
 seems thus far to be merely of an orientational and tentative 
 character. It promises at any rate to open new roads to the still 
 unexplored domain of non-hydrogenic spectra and to lay down 
 some sure principles for their rational classification. The concept 
 of atoms being built up of a number of electrons uniformly 
 increasing throughout the periodical system plays a dominant 
 role in Kossel and Sommerfeld's suggestive paper. The selective 
 principle, used by these authors, and the associated polarization 
 rule were deduced by A. Rubinowicz (Phys. Zeitschr., xix, 1918) in 
 a most remarkable way by the aid of the principle of conservation 
 of electromagnetic momentum and moment of momentum. This 
 subject is thoroughly expounded in the sixth chapter of Sommer- 
 feld's admirable book on Atomic Structure and Spectrum Lines 
 (Vieweg, Brunswick, pp. x + 550, end of 1919). This masterly 
 written work, which unfortunately reached me only after all 
 the preceding pages of this Report were finally printed, may be 
 warmly recommended to the care of the reader. Among a wealth 
 of information on all related subjects, Sommerfeld gives also a 
 good account of the chief investigations on the important question 
 of the ionization potential, which for reasons of space could not be 
 incorporated in the present booklet. 
 
 Whereas in all the investigations described in this Report 
 the atomic nucleus is treated as a homogeneous sphere or, which 
 is the same thing, as a point-charge, the present writer attempted 
 in a recent paper [19] to work out the quantum theory of spectra 
 emitted by atomic systems containing a * complex,' i.e. any 
 aspherical nucleus. The resulting series, whose members show 
 a fine-structure altogether different from Sommerfeld's relati- 
 vistic one, deviates also as a whole more or less from the Balmerian 
 type according to the amount of asphericity of the nucleus. Appli- 
 cations of this theory, the chief- results of which are condensed in 
 
40 THE QUANTUM THEORY OF SPECTRA 
 
 formulae (2l)-(2l'3) and (3O)-(30'2), loc. cit., are now in the course 
 of preparation. A short account of the theoretical results was 
 given at the Bournemouth meeting of the British Association 
 (September, 1919). 
 
 The promised Parts III and IV of Bohr's own memoir [15], and, 
 no c^oubt, many new papers due to other investigators will, soon call 
 for a supplement to the present Report which, owing to the rapid 
 progress manifesting itself in this domain of research, will certainly 
 be incomplete and much behind the times before it reaches the 
 reader. 
 
INDEX 
 
 Abraham's electron, 26 
 
 absorption, 35, 37 
 
 action, quantum of, 2 
 
 adiabatic invariants, 38 
 
 aether and matter (Wereide), 34 
 
 angular momentum, 4-5 
 
 aspherical nucleus, 39 
 
 atomic radiation, 35 
 
 axis, major, of elliptic orbit, 4 
 
 Balmer series, 7, et passim 
 Bjerrum, 35, 37 
 Bohr, i, 3-5, 8, 16, 33, 37 
 Burgers, i, 38 
 
 Canonical coordinates, 10, 19 
 conditionally periodic system, 38 
 conservation of electromagnetic 
 momentum (Rubinowicz), 39 
 
 Danysz, 34 
 Debye, 33 
 dipoles, rotating, 35 
 doublets, 22, 27 
 
 Eccentricity quantized, 12, 19 
 
 Ehrenfest, i, 38 
 
 Einstein, 35 
 
 electron, lorentzian versus rigid, 26, 
 
 28 
 
 electronic data, 7 
 elements of Keplerian motion, 38 
 emission and absorption, 35 
 energy, total, 4, 18 
 Epstein, i, 28, 34, 38-39 
 Euler-Jacobi's problem, 28 
 
 Fabry and Buisson, 22 
 fine-structure, 22 
 Fourier expansions, 38 
 Fowler, 9 
 
 Gehrcke, 34 
 Glitscher/i, 26, 28 
 
 Hamiltonian equations, 10 
 head of series, 7 
 helium, principal series, 9 
 helium triplets, 23-25 
 Herzfeld, i, 33 
 Hettner, 36, 37 
 hydrogen doublets, 22 
 hyperbolic orbits, 34 
 
 Inclination, of orbit, quantized, 15 
 
 infra-red series, 7 
 
 intensity of spectrum lines, 13, 15, 
 
 23, 38, 39 
 
 ionization potential, 39. 
 ionized helium, 9, 10, 25 
 
 Janicki, 34 
 Jeans, 2 
 
 K-series, 27 
 Keplerian orbits, 3 
 kernmagneton, 35 
 Kossel, 34, 39 
 Kramers, i, 33, 38 
 
 L-series, 27 
 Lorentz-Einstein electron, 26 
 
 Magnetic quantum, 35 
 mass, of electron, variable, 17 
 Michelson, 22 
 momentum, angular, 4-5 
 
 electromagnetic, 39 
 Moseley, 27 
 
 Nucleus, aspherical, 39 
 
 mass, 8-9 
 
 theory, 2 
 
INDEX 
 
 Orbits, circular, 5 
 elliptic, 11-15 
 
 hyperbolic, 34 
 
 spiral, 1 8 
 
 stationary, 4 
 
 ordinal numbers, 27 
 
 Paschen, i, 7, 8, 21, 22 
 
 perihelion motion, 17 
 
 photoelectric effect, 34 
 
 Pickering, 9 
 
 Planck, i, 35-37 
 
 Planck's constant h, 5, 7 
 
 polarization of spectrum lines, 32, 
 38, 39 
 
 probability of spontaneous tran- 
 sition, 37-38 
 
 Quantizing principles, n, 29, 37 
 Quantum of action, 2 
 magnetic, 35 
 
 Radiation theory, 2, 5 
 
 radium B, and C, 34 
 
 relativistic correction of energy, 20 
 
 relativity, 26 
 
 Ritz, 7 
 
 Robinson, 34 
 
 Roentgen spectra, 20, 27-28, 39 
 
 rotation spectra, 35 
 
 | Rubens, .35, 36, 37 
 Rubinowicz, 39 
 Rutherford, 2, 34 
 Rydberg factor, 7, 2 1 
 
 Schwarzschild, 28 
 
 selective principle, 39 
 
 separation of variables, 29 
 
 sharpness of lines, 12. 
 
 Siegbahn, 39 
 
 Sommerfeld, i, 9, 13, 16, 27, 33, 39 
 
 spiral orbits, 18 
 
 splitting of energy, 2 1 
 
 Stark-effect, 15, 16, 28-32, 38 
 
 stationary orbits, 4 
 
 Temperature, and absorption, 36 
 triplets, 25 
 
 Units, spectroscopic, 26 
 Variation of elements, 38 
 
 Wagner, 34 
 
 water vapour, absorption spectrum,, 
 
 36-37 
 Wereide, i, 4, 34 
 
 Zeeman phenomenon, 33 
 
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