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Some Relations between the Optical Spectra of Different Atoms of the same Electronic Structure. II. Aluminium-like and Copper-like Atoms

Published online by Cambridge University Press:  24 October 2008

D. R. Hartree
D. R. Hartree
Affiliation:
Fellow of St John's College
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Extract

The relevant results from the writer's previous paper on the relation between spectra of atoms of different atomic structure are summarised.

For non-penetrating orbits no new theoretical results are obtained, and there are few known spectra (other than those of lithium-like or sodium-like atoms already treated in the first paper) on which to test the relations previously obtained. Values of the polarisability for the A1+ and Si++ ions are calculated from terms of Al I and Si II respectively corresponding to non-penetrating orbits, and are shown to be very much greater than the values of the polarisability of the neon-like ions Al+++ and Si++++.

The main new results are those for penetrating orbits. Assuming a central field of force, it is shown that the quantum defect q for such an orbit can be expressed as the sum of contributions from the electrons in groups of core orbits of different principal quantum number n, and further that if for a given atom in different states of ionisation corresponding orbits of the series electron are compared, the contribution to q from a set of core orbits of given n is very nearly independent of the degree of ionisation so long as the number of electrons in core orbits in the group remains the same. It follows that, if q is the quantum defect for a term of the spectrum of an atom core charge C, the core of which contains SM orbits of principal quantum number M and none of higher quantum number, and q″ is the quantum defect for the corresponding term in the spectrum of the atom of the same element with core charge C + sM, which differs from the atom of core charge C only in lacking the core orbits of principal quantum number M, then q −; q″ is approximately the contribution from the core orbits of principal quantum number M to the quantum defect for the term of the atom core charge C.

Further, it is shown that if corresponding terms of different atoms of the same electronic structure are compared, then for large values of C the contribution to q from any group of core orbits should tend asymptotically to be proportional to the average time mean radius of these orbits, and its reciprocal should tend asymptotically to be linear in C.

Somewhat similar relations are obtained for the quantity Q = dq/d (ν/R) which measures the variation of quantum defect within a sequence.

These theoretical results, and in particular the result that l/(q − q″) should tend asymptotically to be proportional to C, are compared with the values of q deduced from the terms of such observed spectra of aluminium-like and copper-like atoms as are available, and it is found that though the theoretical relations are only established as asymptotically true for large C, there is a considerable measure of agreement with spectra for small values of C, which are the only ones which can be observed.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 23 , Issue 3 , July 1926 , pp. 304 - 326
Copyright
Copyright © Cambridge Philosophical Society 1926

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References

* Hartree, D. R., Proc. Roy. Soc. vol. cvi, p. 552 (1924).CrossRefGoogle Scholar

† For sodium-like atoms better data are now available for P V and S VI from Millikan's recent work (Phys. Rev. vol. xxv, p. 295 (1925)), and they confirm the agreement.Google Scholar

‡ Fowler, A., Phil. Trans. Roy. Soc. vol. ccxxv, p. 1 (1925) (Spectra Si II and III)Google Scholar; Millikanand, R. A. and Bowen, I. S., Phys. Rev. vol. xxv, pp. 591, 600 (1925) (Spectra P III and IV, S IV and V)Google Scholar; Carroll, J. A., Phil. Trans. Roy. Soc. vol. ccxxv, p. 357 (1926) (Spectra Ge III, Ge IV, In III, Hg II, Tl III, Pb IV)CrossRefGoogle Scholar; Salis, G. v., Ann. der Phys. vol. LXXVI, p. 145 (1924) (Spectra Zn II and Cd II)Google Scholar; Saltmarsh, M. O., Proc. Boy. Soc. vol. cviii, p. 332 (1925) (Spectra P III and IV).CrossRefGoogle Scholar

§ Formulae quoted from the previous paper will be numbered with a Roman I prefixed to their number in that paper.

* Bohr, N. and Coster, D., Zeit. f. Phys. voL xii, p. 361 (1923).Google Scholar

* The agreement is best if integer values of k are adopted. In the writer's previous paper this agreement was advanced as evidence for integral values of k, but it seems probable now that the appropriate values of k to take are the half-integral values ½, , …for s, p, d…terms. The consequent divergence of the experimental values from the theoretical relation (2·5) may be due to the presence of further terms in the formula for the added potential, or to a variation of the coefficient β in (I, 3·4) from one orbit to another. This coefficient β is proportional to the polarisability of the core, and it was shown in (I, § 4) that the observations indicate a considerable variation of β with the k of the orbit of the series electron, whether integer or half-integer values of k are adopted.

* Fues, E., Ann. der Phys. vol. LXXVI, p. 299 (1924).Google Scholar

† Hartree, D. R., Proc. Camb. Phil. Soc. vol. xxii, p. 464 (1924).CrossRefGoogle Scholar

* Fowler, A., Report on Series in Line Spectra (A1 I), Phil. Trans, (loc. cit.) (Si II).Google Scholar

† As for sodium-like atoms, the agreement with the values calculated for an inverse fourth power added potential is beat if integer values of k are assumed. Cf. footnote, p. 307.Google Scholar

‡ The value k = 4 has been used in determining the values of the polarisability tabulated; if the value k = 3½ were used, all the values tabulated would be reduced by a factor of 2·18.Google Scholar

* If is not constant but tends to y as N→∞, bn contains a term depending on i.e. on how tends to y as N→∞Google Scholar

* The values of the quantum defect tabulated for P III and S IV have been found from the term values given by Millikan and Bowen (loc. cit.). For P III Miss Saltmarsh (loc. cit.) calculates term values from the observed lines by assuming that the quantum defects for the first two s terms are the same, and obtains 2s = 127004; the present writer has attempted to obtain a more accurate value by estimating the difference of quantum defect between the first two a terms from the value of the corresponding difference in Al I and Si II, and obtained 2s = 125560 with an estimated limit of error of 150. Millikan and Bowen's value is 2s = 125498.Google Scholar

† If v is the wave number of, the term in question, it is theoretically more accurate to replace q″ in these formulae by q′, the quantum defect for the orbit of the sodium-like atom, with the same k, which would give a term of wave number (cf. formulae 6·3—6·6).Google Scholar

* These results hold equally well on the whole but not appreciably better when the theoretically more accurate q′ is substituted for the practically more convenient q″; as it happens the effect of the difference between q′ and q″ is not large and, for a given term, it varies little from atom to atom.

* It is unlikely that the variation of p3/pm is large enough to counteract entirely the decrease of (q−g″)/pm.Google Scholar

* The values of q have been calculated from the term values given by Fowler, A. for Cu I (Report on Series of Line. Spectra) by Salis, G. V. for Zn II (loc. cit.) and by J. A. Carroll for Ga III and Ge IV (loc. cit.).Google Scholar

† If the first d term corresponds to a penetrating orbit.

* The third decimal place in the values of q″ given in the table is probably reliable; the method of extrapolation used is such that the figures are smooth to the four places given, though the fourth figure is probably not correct.

* Loc. cit. § 3.Google Scholar

† Loc. cit. Tables VI and VII.Google Scholar