Sommerfeld formula

This decrease of work function W is relatively small at reasonable values of electric field E. The Schottky effect can result, however, in essential change of the thermionic current because of its strong exponential dependence on the work function in accordance with the Sommerfeld formula (2-100). Dependence of the Schottky W decrease and thermionic emission current density on electric field are illustrated in Table 2-12 together with corresponding data on field and thermionic field emission (Raizer, 1997). The 4x change of electric field results in 800 x increase of the thermionic current density. 

Sommerfeld Formula for Thermionic Emission. Using the Sommerfeld formula (2-100), calculate the saturation current densities of thermionic emission for a tungsten cathode at 2500 K. Compare the calculated value of cathode current density with a typical value for the hot thermionic cathodes presented in Table 2-12. 

Fig. 6.3. H+H2. a-c) Three different discretisations for the angles 0, 45 and 90" are shown (r, R = [0,8] ao). The discretisation is based on the combined use of the Bohr-Sommerfeld formula in one dimension for r and R with Emax = 0.13 Eh d) A smaller range for triangulation is shown for 0° and with Emax = 0.08 Eh (collinear arrangement). (Reprinted, by permission, from Jaquet, R., Kumpf, A., Heinen, M. J. Chem. Soc., Farad. Trans. 93 (1997) 1027. Copyright 1997 by Royal Society of Chemistry.)...

The energy levels formed by single electron removal are similar to those of a hydrogen-like atom, obtained by replacing Z by (Z-a) and (Z-s) in the Sommerfeld formula, where a and s are screening constants. The energy of a level is given by the modified Sommerfeld formula as... 

From the first term of the Sommerfeld formula, one obtains the screening doublet law, which states that the difference between the square roots of the term values of a given doublet is constant, i.e. independent of Z. This term also gives the irregular doublet law which states that the difference between term values of an irregular (screening) doublet is a linear function ofZ. 

The second term of the Sommerfeld formula gives the separation in energy for a spin-relativity doublet as proportional to the fourth power of the screened atomic number, i.e. (Z-s). This is referred to as the regular doublet law. For example, for the LjpLjjj doublet (same a), Av(cm ) = (Rayi6) Z-3.5)". ... 

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,... 

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. 

It is interesting to note that straightforward Bohr-Sommerfeld quantization of the action (6.1.11) yields the exact result (6.1.25) for the bound state energies. In our units the Bohr-Sommerfeld condition results in / = n, n = 1,2,. Inserting this result into (6.1.13) indeed reproduces (6.1.25) exactly. This is the same happy accident which allowed Bohr (1913) to obtain the Balmer formula from a simple solar system model of a one-electron atom. 

Wilson in 1915 and Sommerfeld in 1916 generalized Bohr s formula for the allowed orbits to... 

The case of hydrogen is peculiar in one respect. Experiment gives distinctly fewer terms than are specified in the term scheme of fig. 8 for = 2 only two terms are found, for n = 3 only three, and so on. The theoretical calculation shows that here (by a mathematical coincidence, so to speak) two terms sometimes coincide, the reason being that the relativity and spin corrections partly compensate each other. It is found that terms with the same inner quantum number j but different azimuthal quantum numbers I always strictly coincide, for instance, the ns and the np, term, the p. , and the d, term, and so on such pairs of terms are drawn close together in fig. 8. For the value of the terms a formula was given by Sommerfeld (1916), even before the introduction of wave mechanics the same formula is also obtained when the hydrogen atom is calculated by Dirac s relativistic (E908) 11... 

It should be particularly emphasized that the X-ray terms also are well represented for all elements by Sommerfeld s formula. 

Sommerfeld has given a relativistic generalisation of the Rydberg formula. This has the form ... 

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. 

Sommerfeld s formula for the energy of hydrogen-like atoms is derived under the assumptions of relativistic motion under an inverse square attractive force, subject to the above quantum postulates. It exhibits only two quantum numbers, n and kt The third, m, appears explicitly if the atom is situated in a... 

The evidence in support of Sommerfeld s formula was too strong to be set aside its essential correctness could hardly be doubted. Yet in order to interpret the observations, both on hydrogen and on ionized helium, it was necessary to violate the selection rule Ak = J-l, a rule inescapable on the Correspondence Principle, which itself was so strongly supported both by its appeal to classical physics and by its success in predicting the Stark effect in hydrogen. 

The resolution of the dilemma is now well known and will be treated later the property of electron spin brings about a re-labelling of the energy levels, but does not displace them. Sommerfeld s formula remains valid, but not the assumptions on which it was derived. The Correspondence Principle is not violated, but in its application it must now be realized that the electron possesses an extra degree of freedom. 

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