AS LIV. The Helium-Hydrogen Series Constants. S is well known, Bohr proposed to modify the Balmer formula for these series by a term depending on the relativity-effect on the moying electron t. The formula for Helium thus becomes where P and mare integers, m>p, Nнe=RMнe/(MHe+μ), R being Rydberg's constant 22euch, MHe the mass of the helium atom and that of the electron, while a=2me2/he. The Helium lines, however, display a fine-structure under high resolving power, and Sommerfeld in explanation of this developed a theory of adjacent (elliptical) orbits with eccentricities depending on quantum considerations. He gives a physical meaning as the ratio of angular momenta, and proposes to measure it from fine-structure observations. We know from other considerations that a2 has approximately the value 5'3 x 10-5. Paschen § tested this theory by an exhaustive set of measurements in the helium spectrum and substantial verification was apparently obtained. His value for a from the line 4686 is 5.315 x 10-5. According to Sommerfeld's theory the series (1) when p = 3, m = 4, 5, ..., which is the one most easily resolved, consists of a set of triplets having components I, II, III, in diminishing order of wave-length, their wavenumber differences being constant. III-II=1·73 cm. ̄1, II-I='58 cm.-1. Further, each component is bordered towards the red by a set of fainter lines rapidly closing in on them as higher terms of the series are reached. Formula (1) is to apply only to I, and can therefore be accurately tested. Paschen of course did this, implicitly, but the new table from the Bureau of Standards for the refractive index of air slightly alters his derived constants. His measurements are given in Table I., along with the new additive correction to reduce to vacuum, the figures in italics referring to the series p=4, m=-, 6, 7, ..., the others to p=3, m=4, 5, .... * Communicated by the Author. + Phil. Mag. Feb. 1915. Ann. d. Physik, li. No. 17 (1916). $ Ann. d. Physik, 1. No. 16 (1916). Bureau of Standards, Washington, Scientific Papers,' No. 327. (1918). Phil. Mag. S. 6. Vol. 40. No. 238. Oct. 1920 2 K we see that if the formula is to hold the second or relativity term must be just large enough at each value to reduce the corresponding first one to a constant. Both terms are given in the table, and their difference is under NHe. For the series p=3 a constant value is reached, average 109722-31; but such is not the case for the unresolved series p=4. Now this series is according to the theory one of quadruplets I, II, III, IV, each bordered towards the red by fainter lines as before; but I, II, III were not resolved, and it is most likely that Paschen's readings apply to some kind of a mean among them, and consequently the wavenumbers are all too high for I to which the formula applies. Theoretically, IV-III=730 cm.-1, III-II=243 cm-1, II-I=123 cm.-1. If we assume the wave-numbers to be all too high by dv, then by (2) we need to apply a correction: It is rather surprising that this should so exactly, as far as present accuracy in measurements goes, be the relativity correction. To equal it we require dv = Na24/p1·091 cm. ̄Î ̧ However, it is readily seen that we cannot correct these terms by subtracting a suitable multiple of the relativity correction, for although a multiple 4 or 5 would do away with the diminishing trend the resultant NHe would be much lower than that from the other series. It will be seen from the following that this difficulty is due to systematic error among unresolved lines. The components III of the triplet series and IV of the quadruplets stand farthest out, differing from I by 2.31 cm.-1 and 109 cm. respectively. Their corresponding N values have therefore to be corrected by these differences multiplied -1 *If the differences in wave-number among the standard iron lines are all counted correctly (by interference methods; but the reference Cadmium line is in error by dv, we should have a spurious relativity effect oλ =λ throughout the spectrum. This error 09 cm. would, however, represent a miscount by about one part in 200,000 for a red line. An accuracy of one part in ten million is claimed. by p2m2 in each case. Paschen was able to record 4(m2-p2) these lines in some cases, as shown in Table II. It will be seen that the tendency to lower values in the higher terms has disappeared in the series p=4. The results are not very consistent among themselves, but this is readily explained if we assume that in some cases only the bordering satellites were observed. The errors are all on the same side of the determined NHe. Paschen's estimates of possible error of observation are also given, reduced to our scale. Turning now to the Hydrogen (Balmer) series, we have the formula a NH being RM/(M+μ), M mass of hydrogen nucleus. The careful measurements of Curtis* and Paschen (loc. cit.) are given in Table III. Unfortunately they exhibit a small systematic drift-Curtis's values increasing relatively to Paschen's roughly linearly in terms of wave - lengths. The first line H. is easily resolved into a doublet, e. g. by Paschen in the third-order spectrum dλ=·124 A.Ŭ. Sv=289 cm.-1. The second line has also been resolved. In Sommerfeld's theory the series is a set of doublets I, II, of constant wave-number difference II-I=365 cm.-1, the stronger component I being towards the red, but both components being bordered on the long-wave side by fainter lines rapidly converging in the higher terms. Paschen's measurements are for "centres of gravity," while Curtis endeavoured to record the centres of the diffuse lines (they were not resolved). Now for the fainter lines component II would probably vanish first, and the diffuse mean of the photographic plate would tend towards I-that is, longer wave-lengths. This may partly explain the minute drift. Again, Curtis used Burns's secondary iron lines, while Paschen used a variety-Fabry and Buisson, Neon lines by Meissner, &c. The values of 4m2v/(m2-4) for the two observers are given under C. and P., and in each case we find that a minimum seems to have been reached for Paschen at Hy, Curtis at He. Now the stock corrections to the Balmer formula all have the property of steadily decreasing, i. e. no maximum or minimum, as for example the relativity. correction given in the table. Such a correction could * Proc. Roy. Soc. vol. xc. (1914) and vol. xcvi. (1919). |