THE LONDON, EDINBURGH, AND DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [SIXTH SERIES.] APRIL 1918. XXXV. The X-Ray Spectra and the Constitution of the Atom. By L. VEGARD, Dr. phil. University Christiania *. § 1. CCORDING to Rutherford † the atom consists of a positive nucleus and electrons circulating round it. A number of different methods have led to the result that the number of electrons in the neutral atom is equal to the atomic number N, and that consequently the charge of the nucleus is + Ne. The stability of the system is not secured by the usual electrodynamic forces, but Bohr by his ingenious explanation of the series spectra has been able to fix orbits of stability by means of the energy quanta introduced by Planck. The fundamental assumptions underlying Bohr's theory of atoms and their light emission are the following I. In the normal state of the atom the electrons are arranged in groups (rings) in such a way that for each electron the angular momentum is equal to or h 27 where m is the mass, w the angular velocity of the electron, a the radius of the circular orbit, and h Planck's constant. * Communicated by the Author. The results of the present investigation were communicated to the Kristiania Vid. Selsk. on October 23 and 30, and read before the Society November 23. + E. Rutherford, Phil. Mag. xxi. p. 669 (1911), and xxviii. p. 488 (1914). N. Bohr, Phil. Mag. xxvi. pp. 1, 476, 857 (1913). Phil. Mag. S. 6. Vol. 35. No. 208. April 1918. Y II. To produce the line of a series it is necessary that an electron should be removed in some way or other from one of the rings. The recombination of an electron towards the broken ring may take place in steps between stability orbits determined by the condition : III. To each stability circle of the recombining electron corresponds a certain energy of the atomic system. When the electron in one step passes from a stability circle 71⁄2 to one 7 the energy must be removed from the system, and Bohr assumes it to be radiated out in one quantum homogeneous radiation, or hv=W2-W1, (2) The where is the frequency and W the total energy. latter relation we may call Bohr's frequency law. The orbits corresponding to the normal state of the atom we shall call primary orbits, and the stability orbits which the electron may take up during its recombination we denote as secondary orbits. By means of these assumptions Bohr was able to deduce the complete series spectrum of hydrogen and a spectrum of helium which was emitted when a single electron recombined to the isolated nucleus. He was also able to give an explanation of the appearance in the series formula of the universal frequency constant (R) of Rydberg, and his frequency law explained in a simple manner the combination principle of Ritz. But up to the present the atomic model based on the assumption I. has not been able to yield any theoretical deduction of the series spectra in general. In the case of hydrogen and those spectra similar to that of hydrogen, the theory of Bohr has proved to be very successful indeed. Through the introduction of noncircular orbits already Bohr* was able to deduce a formula for the Stark-effect which gave the right order of magnitude and the right type of variation. Later on the question of noncircular orbits has been much more completely treated by Sommerfeld †, Schwarzschild ‡, and Epstein §. Sommerfeld gave a generalization of the quant-conditions * N. Bohr, Phil. Mag. xxviii. p. 506 (1914). † A. Sommerfeld, Sitz. Ber. d. Münchener Akad. d. Wiss. 1915; Ann. d. Phys. li. pp. 1 & 125 (1916). K. Schwarzschild, Berliner Akad. d. Wiss. p. 548 (1916). which proved to be of very great importance. He introduces generalized coordinates q; (coordinates of position) and p; (coordinates of momentum), and takes as the general quant-condition for stable orbits : where the integral is to be extended over one period. (3) This general quant-condition involves the new question as to the choice of coordinates. Sommerfeld carries out the calculation in polar coordinates (re) for a single electron moving round a positive charge. The three quant-conditions then take the form So" Pody=n,h, Spd.=n'h, Spyd±=ngh.. (4) If we merely consider orbits in a plane, only the first two conditions will be wanted. Carrying out the calculation in the case of hydrogen he gets : my, m' being the value of n1, n' for the orbit from which the electrons recombine. In order to give the ordinary Balmer series it is necessary that N=n2+n'=2 and that Mm+m' is an integer number > 2. Further, the number of combinations is limited by the conditions m1 n1, m' — n'. This new form of the frequency formula is so far identical with that of Bohr that it gives the same position of the lines, but it is different with regard to the way in which we may imagine the lines to be produced. Thus the H-line may be produced in four different ways, as, e. g., by recombination from an elliptic orbit (m=2, m'=1). The case of circular orbits treated by Bohr is only a special case of many, and corresponds to a recombination from (m1=3, m'=0) to (n=2, n'=0). The H-lines which are produced in these different ways are only with certainty identical provided we may treat the system as consisting of two constant masses attracting each other by a force inversely proportional to the square of the distance. When other forces come into play, or if the masses are not to be regarded as constant, the H-lines produced in the different possible ways will no longer be identical. In fact, when the mass of the electron is supposed to vary with the velocity according to the law given by Lorentz or the principle of relativity, Sommerfeld finds that the lines are split up, and carrying out the calculation he has been able to explain even quantitatively the splitting up of the hydrogen lines and to give a general theory of the formation of multiple lines. Further, we see that the frequency formula of Sommerfeld gives us a possibility of explaining the Stark-effect, for when a uniform electrostatic field is introduced into the system the various ways in which a certain line may be produced will no longer give the same frequency. A complete determination of the Stark-effect was finally given by Epstein*, by an ingenious method of selecting the generalized coordinates by means of the equations of Hamilton-Jacobi. Application of Bohr's Conceptions to the High Frequency Spectra of the Elements. § 2. The law connecting the high-frequency spectra of the elements which was brought out through the beautiful experiments of Moseley † was simply explained by the atomic model of Rutherford ‡ and showed that the atomic number played a fundamental part in the constitution of atoms. In fact, we may say that the evidence gathered from various sources leaves no doubt as to the correctness of Rutherford's conception of the atom. I think we may safely take it as a fact that the normal atom has a positive nucleus of charge Ne, surrounded by N electrons. The atomic problem is then resolved into the following two questions:— 1. The arrangement of the electrons which surround the nucleus and the laws that govern this arrangement. 2. The constitution of the nucleus. The nucleus is the seat of gravitation and the radioactive transformations, and a possible theory of the nucleus would have to gather evidence from these two phenomena. The • P. Epstein, l. c. † H. G. J. Moseley, Phil. Mag. xxvi. p. 1024 (1913), and xxvii. p. 703 (1914). Loc. cit. outer electronic system is responsible for the light emission, for the homogeneous X-rays, and for the chemical properties of the atom. Bohr* was able to show that the K-spectrum could be approximately explained by assuming it to be produced by the removal and recombination of an electron next to the nucleus. Moseley found a better agreement by assuming four electrons in the system next to the nucleus; but the way in which he deduced his formula was open to criticism. His formula may therefore be regarded as empirical, although Nicholson points out that it can be deduced by a proper modification of Bohr's frequency law to systems of electrons. Kossel § was the first to point out some very interesting relations between the lines of the K- and L-series. Denoting the frequency by v, the following relation very nearly holds true: According to Bohr's frequency law (III.) the frequency is proportional to the differences of energy of the electron in the initial and final state, and, as pointed out by Bohr |, the above relations would naturally convey the following conception with regard to the formation of the high-frequency spectra. The electrons may be supposed to be arranged in rings round the nucleus. When an electron is removed from the ring nearest the nucleus, an electron from the next ring may replace it and give rise to the emission of K. If the electron is taken from the third ring, we get Kg. When an electron of the second ring is removed and replaced by one from the third, we might get La, and in this way Kossel's frequency relations should be explained. Sommerfeld, following up this line of thought, has been able to express a number of lines of the X-ray spectra by introducing a number of "terms" peculiar to the various X-ray series. Thus, e. g., he introduces a K-term (N −1·6)2 (N-3.5)2 and a L-term L= and finds 12 22 VK=K-L. K= * N. Bohr, Phil. Mag. (6) xxvi. p. 408 (1913). † Loc. cit. J. W. Nicholson, Phil. Mag. (6) xxviii. p. 562 (1914), || N. Bohr, Phil. Mag. xxx. p. 394 (1915). TA. Sommerfeld, Ann. d. Phys. li. p. 125 (1916). |