Sommerfeld’s theory

Further development of Sommerfeld s theory of metals would extend well outside the intended scope of this textbook. The interested reader may refer to any of several books for this (e.g. Seitz, 1940). Rather, this book will discuss the band approximation based upon the Bloch scheme. In the Bloch scheme, Sommerfeld s model corresponds to an empty lattice, in which the electronic Hamiltonian contains only the electron kinetic-energy term. The lattice potential is assumed constant, and taken to be zero, without any loss of generality. The solutions of the time-independent Schrodinger equation in this case can be written as simple plane waves, = exp[/A r]. As the wave function does not change if one adds an arbitrary reciprocal-lattice vector, G, to the wave vector, k, BZ symmetry may be superimposed on the plane waves to reduce the number of wave vectors that must be considered ... 

Like Bohr s model of the hydrogen atom, Sommerfeld s theory flowered only briefly. The creation of quantum mechanics and the discovery of electron spin, both in 1925, followed by Paul Dirac s theory in 1928, provided a solid theory-based underpinning for... 

According to Sommerfeld s theory of metallic conduction, Wi, the kinetic energy of the conducting electrons, is connected with 0, the internal potential, and A, the work of escape or work ijpnction measured thermionically or photo-electrically, by the simple equation... 

On Sommerfeld s theory, however, the coefficients turn out to A. Sommerfeld, Zeitschr.f. Physik, 47, 1 (1928). 

As might be expected, Sommerfeld s theory also makes the longitudinal effect vanish. Thus there are two contradictions between theory and experiment. A third difficulty arose when Kapitza and Meissner and Scheffers f made experiments with strong magnetic fields at low temperatures and showed that even at intensities which are quite easily attained the effect ceases to be proportional to the square of the intensity. In the strongest fields hitherto obtained the change of resistance is a linear function of the intensity. 

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... 

A simple calculation in the spirit of Sommerfeld s theory of metals for the two-dimensional case leads to the equation... 

According to Sommerfeld s theory the difference between two L absorption frequences is due to the difference in shape of a circular and an elliptic orbit. His formula contains an undetermined constant. Professor Patterson and I have shoro that if we assume four electrons in the L orbit the undetermined constant is done away with, and that Sommerfeld s formula represents roughly the difference between the Li and L% absorption frequencies. It may be that a formula calculated on the basis of three dimensional orbits would give more accurate results. 

Stanton s breadth of outlook is demonstrated by the fact that in 1923 he published a book entitled Friction in which he included the fundamentals of aerodynamic friction as well as fluid lubrication. In 1927 he reported to Section G of the British Association the results of a series of experiments in which the relative displacement of a bearing and journal were measured by electrical inductance. Up to an eccentricity ratio of about 0.5 the agreement between observation and theory was reasonably good but a further increase produced a deviation of the point of nearest approach towards the crown of the bearing instead of in the lateral direction as predicted by Sommerfeld s theory. This led to considerable controversy until it was accepted that the negative pressures postulated by this theory could not be withstood by lubricant. 

Fowler proposed a theory in 1931 which showed that the photoelectric current variation with light frequency could be accounted for by the effect of temperature on the number of electrons available for emission, in accordance with the distribution law of Sommerfeld s theory of metals. Sommerfeld s theory (1928) had resolved some of the problems surrounding the original models for electrons in metals. In classical Drude theory, a metal had been envisaged as a three-dimensional potential well (or box) containing a gas of freely mobile electrons. This adequately explained their high electrical and thermal conductivities. However, because experimentally it is found that metallic electrons do not show a gaslike heat capacity, the Boltzman distribution law is inappropriate. A Fermi-Dirac distribution function is required, consistent with the need that the electrons obey the Pauli exclusion principle, and this distribution function has the form... 

I should also mention Sommerfeld, who extended Bohr s theory to try and account for the extra quantum numbers observed experimentally. Sommerfeld allowed the electrons to have an elliptic orbit rather than a circular one. 

Examples of other work on research schools M. Eckert, "Sommerfeld s School and the Electron Theory of Metals," HSPS 17 (1987) 191234 Gerald Geison, Michael Foster and the Cambridge School of Physiology The Scientific Enterprise in Late Victorian Society (Princeton Princeton University Press, 1978) L. J. Klosterman, "A Research School of Chemistry in the Nineteenth Century Jean Baptiste Dumas and His Research Students," Annals of Science 43 (1985) 180 H. A. M. Snelders, "J. H. van t Hoffs Research School in Amsterdam (18771895)," Janus 71 (1984) 130 F. L. Holmes, "The Formation of the Munich School of Metabolism," in William Coleman and F. L. Holmes, eds., The Investigative Enterprise Studies on Nineteenth-Century Physiology and Medicine (Berkeley, Los Angeles, and London University of California Press, 1988). 

Born s essays making up The Constitution of Matter were intended to be an aid to those who did not have "time to study the larger work of Sommerfeld," namely, Atomic Structure and Spectral Lines (1915), which laid out Bohr s first quantum theory and Sommerfeld s emendations.6 Bom and Sommerfeld were to have enormous influence on the development of quantum mechanics and its application to atoms and molecules. Not only did many European, British, and American scientists come to study at their physics institutes but they each gave influential series of lectures at foreign institutions, nowhere with more influence than in the United States. 

Following Sommerfeld s proposal of elliptical electron orbits in 1915, Bohr amended his original theory, which had included only circular orbits. 14 A 1922 paper in Zeitschriftfur Physik outlined the "Aufbauprinzip" by which electrons are fed into atomic subshells. There was a neat correlation between periodic groups containing 2, 8, 8, 18,... 

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. 

Sommerfeld s work was based on Bohr s model of the hydrogen atom. In this work, he brought relativity theory and the quantum idea together and was able to account for the fine details of the hydrogen spectrum. After Sommerfeld s paper on this work was published in 1916, he received a letter from Niels Bohr. In it Bohr wrote, I do not believe ever to have read anything with more joy than your beautiful work. ... 

We may obtain the order of magnitude as follows in Sommer-feld s theory it was the first order effect that vanished, and we had to include higher terms in kTjEQ, Here even the first order effect will differ from zero. Thus here this effect is caused solely by the deviation from the motion of free electrons, i.e. it is proportional to a quantity k giving the relative deviation. In all, then, it will differ from Sommerfeld s result by a factor k E jkTY, To obtain the order of magnitude observed, we must assume that K lies between OT and 0 01, an assumption which seems entirely reasonable. 

It is well known that Bohr s theory of radiation explains very simply the above relation between emission and absorption wave-numbers. The relation, however, should not be absolutely exact, if, as Sommerfeld supposes, some of the atoms contain M orbits that are elliptic, and other atoms of the same chemical element contain M orbits that are circular. In this case the L absorption drops should have a complex structure. We have not been able to observe such a structure in the drops on our curves, and the effect must be exceedingly small, if it exits at all. 

At the present time, by far the most useful non-empirical alternatives to Cl are the methods based on density functional theory (DFT) . The development of DFT can be traced from its pre-quantum-mechanical roots in Drude s treatment of the electron gas" in metals and Sommerfeld s quantum-statistical version of this, through the Thomas-Fermi-Dirac model of the atom. Slater s Xa method, the laying of the formal foundations by... 

Fano U (1941) The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld s waves). J Opt Soc Am A 31 213-222... 

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