Fine structure: Sommerfeld

This increase should be observed for all screening constants, and in particular for the Sommerfeld fine-structure screening constants. The constancy of s reported for the Z-doublet throughout the periodic system (s = 3.50 Ai 0.08 from Z = 41 to Z = 92, Sommerfeld, Atom-bau pp. 447, 462) seems to contradict this. But actually this constancy proves the point. For s has been calculated with the complete formula,... 

Fine structure. Evidently the set of term values is exactly the same as on the usual theory but the quantum numbers are different, making new transitions possible and changing the intensities of the fine structure. The hydrogen fine structiure is so obscured by the natural breadth of the lines that no information can be obtained from it, and we must turn to the spectrum of ionized helium. For Paschen s data the reader is referred to Sommerfeld, figures 89-92. The only measurements of value for the... 

But no fine structure - yet - until in 1915 Bohr considered the effect of relativistic variation of mass with velocity in elliptical orbits under the inverse square law of binding, and pointed out that the consequential precessional motion of the ellipses would introduce new periodicities into the motion of the electron, whose consequences would be satellite lines in the spectra. The details of the dynamics were worked out independently by SOMMERFELD [38] and WILSON [39] in 1915/16 based on a generalisation of Bohr s quantization, namely, the quantization of action the values of the phase integrals Jf = fpj.d, - of classical mechanics should be constrained to assume only integral multiples of h. 

But experiments to resolve the fine structure of the Balmer lines were difficult as you all know, resolution was impeded by the Doppler broadening of components. So ionized helium comes into the picture, because, as Sommerfeld s formula predicted, fine structure intervals are a function of (aZ)2, so in helium they are of order four times as wide as in hydrogen and one has more chance of resolving the Doppler-broadened lines. So PASCHEN [40], in 1916. undertook an extensive study of the He+ lines and in particular, 4686 A (n = 4->3). Fine structure, indeed, was found and matched against Sommerfeld s formula. The measurements were used to determine a value of a. But the structure did not really match the theory in that the quantum numbers bore no imprint of electron spin, so even the orbital properties - which dominated the intensity rules based on a correspondence with classical radiation theory - were wrongly associated with components, and the value of a derived from this first study was later abandoned. 

Medium fields with ALs < Ac/ Aei It is the relative energies that are important the multiplet splitting may vary from 10 l to 104 cm-1 (see Sommerfeld s (594) fine structure formula). In this case Fei is the first perturbation effect, and the cubic part of Vcf is treated as a perturbation before spin-orbit effects are calculated. For distortions from cubic symmetry, it is necessary to consider carefully the relative magnitudes of Vt (that part of VCf due to departures from cubic symmetry) and Vis S. Weak fields with ACf ls - In this case the ligand fields merely perturb the multiplet structure of the free atom. 

Now we can expand exp(iK r) in a Taylor series, reminding ourselves that for atoms, in any event, the wavelength of a photon of visible light is of the scale of very many atomic diameters, so that K r is of the order of the Sommerfeld fine structure constant a (Problem 3.42.1 below) thus we are justified in keeping the leading term only ... 

Table G Definitions of the Electric Field E, the (Di)electric Polarization P, the Electric Displacement D, the Magnetic Field H, the Magnetization M, the Magnetic induction or flux density B, statement of the Maxwell equations, and of the Lorentz Force Equation in Various Systems of Units rat. = rationalized (no 477-), unrat. = the explicit factor 477- is used in the definition of dielectric polarization and magnetization c = speed of light) (using SI values for e, me, h, c) [J.D. Jackson, Classical Electrodynamics, 3rd edition, Wiley, New York, 1999.]. For Hartree atomic u nits of mag netism, two conventions exist (1) the "Gauss" or wave convention, which requires that E and H have the same magnitude for electromagnetic waves in vacuo (2) the Lorentz convention, which derives the magnetic field from the Lorentz force equation the ratio between these two sets of units is the Sommerfeld fine-structure constant a = 1/137.0359895...

The fine-structure constant is another fundamental constant, which first appeared in Sommerfeld s work on the hydrogen atom. Its value is a = 0.00729735308. More often, this constant is written as follows ... 

This is called the fine structure of the spectral lines. Its theory was given by Sommerfeld for the case of atoms of the hydrogen type (H, He+, Li++), and was tested by Fowler and Paschen on the spectrum of singly ionized helium (He+), which was found in complete agreement with the theory. The test is easier wdth He+ than with H for this reason, that the eiu. rgy terms of He+ are four times as far apart, on account of the nuclear charge number Z being doubled, w hereas the... 

An important generalization of the quantum theory by Sommerfeld [125] and independently by Wilson [142] allowed a detailed study of the non-radiating non-circular orbits, and led to Sommerfeld s celebrated fine structure formula which represents the energy levels of hydrogen-like atoms to a precision which was substantiated by the most refined experiments over the twenty years following its derivation. Comparison with experiment, however, implies a consideration, not only of energy levels, but also of the relative intensities of spectral lines. We shall see that on this point the theory failed. 

Although Sommerfeld made certain proposals concerning the relative intensities of trie fine structure components, they were rather arbitrary and were not supported by experiment, in particular, by observations on the line 4686 A of He+. The development of the Correspondence Principle by Bohr and Kramers was more successful, and more convincing in its logical basis. 

These were produced by Pasehen in DC and in condensed spark discharges, and resolved by a powerful grating. More components appeared in the condensed spark than in the DC discharges, but in neither pattern was the structure completely resolved. The condensed spark spectrograms showed the closest agreement with the predicted patterns, and measurements of these allowed an experimental determination of the fine structure constant a. In this comparison between theory and experiment, Sommerfeld s original intensity rules, later abandoned, were used. 

Fasohen s 1916 paper also records observations on the fine structure of the Balmer lines, which appeared double to him as to earlier observers. Recognizing in these doublets a blend of the components predicted by Sommerfeld s theory, Pasohen... 

While the new theory predicts the same energy levels as that of Sommerfeld, it does not predict the same fine structure since the different labelling of the energy levels permits the appearance of components which on the old theory were forbidden. In particular, the component c observed by Hansen (section 5-.2) now corresponds to an allowed transition (cf. Figs. 1,2). 

The fine structure of the ionized helium lines, a d in particular of the line 4686 A (n = 4—3) is sixteen times as wide as the corresponding structure in hydrogen. A4686 A therefore offers to optical spectroscopists a better opportunity than Ha for the resolution of its components. For this reason, and because of its greater complexity, Paschen s observations provided Sommerfeld with more material than did the Baimer lines for testing his fine structure formula (section 5.1). In so far as 4686 A is a spark line, its excitation requires a relatively violent gas discharge, which favours neither a small Doppler width nor freedom from the influence of electric fields. Its attraction as a test of the radiation theory lies particularly in the possibility of checking the predicted dependence of term. shifts on n and on Z. 

As a measure of the relativistic fine structure we take, following Sommerfeld, that of the limiting term (n=2) of the Balmer series of hydrogen. This has the theoretical value... 

Abstract. Among many other results, Arnold Sommerfeld gave in his article the correct expression for the relativistic bound-state energy levels of the hydrogen atom, well before the development of wave mechanics, clear ideas about the electron spin, and Dirac s relativistic wave equation. He correctly attributed the fine structure of atomic spectra to relativistic effects, and thus published the first paper giving a quantitative perspective on relativistic quantum chemistry. 

Besides making implicit use of these really puzzling properties of the relativistic Kepler problem, the second major impact of Sommerfeld s article lies in several notions introduced which lie at the foundation of relativistic quantum chemistry and have since been instrumental in the field the notion of scalar (kinemat-ical) relativistic effects versus fine-structure effects, the introduction of the fine-structure constant, a = jhc, and the expansion of the relativistic expressions in powers of the square of this constant. The idea that relativistic effects decisively influence the structure of the outer electrons of the atoms is at the root of relativistic quantum chemistry. 

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