Kronig structure

R. de L Kronig, Band Spectra and Molecular Structure, Cambridge University Press, New York, 1930, p. 6. 

Quantitative data on local structure can be obtained via an analysis of the decaying slope next to the absorption edge. The absorption of an X-ray photon boosts a core electron up into an unoccupied band of the material which, in a metal, is the conduction band above the Fermi level. Electrons in such a band behave as if nearly free and no fine structure would be expected on the absorption tail . However, fine structure is observed up to 500 to 1000eV above the edge (see Figure 2.73(b)). The ripples are known as the Kronig fine structure or extended X-ray absorption fine structure (EX AFS). 

Absorption of X-ray radiation of energy well above the threshold for an X-ray transition will result in the ejection of a photoelectron since the initial unoccupied band stale to which the transition takes place will be above the vacuum level. The Kronig fine structure is due to oscillations induced in the absorption cross-section of the absorbing atom as a result of interference... 

Figure 2.74 Schematic representation of the electron wave interference effects giving rise to the Kronig fine structure on X-ray absorption edges (sec text).

Band Theory of Metals, Three approaches predict the electronic band structure of metals. The first approach (Kronig-Penney), the periodic potential method, starts with free electrons and then considers nearly bound electrons. The second (Ziman) takes into account Bragg reflection as a strong disturbance in the propagation of electrons. The third approach (Feynman) starts with completely bound electrons to atoms and then considers a linear combination of atomic orbitals (LCAOs). 

The Kronig theory has been used to explain the extended fine structure, occurring in the energy range 100 to 500 ev. above the principal edge. Closer to the edge, difficulties arise, perhaps related to the interactions of the slow photoelectron with the lattice. 

A new approach to the explanation of fine structure has been offered by Hayasi (S). Like Kossel, he assumes bound states, but of short lifetime and of positive energy. These states, which he refers to as quasi-stationary, are stabihzed by Bragg reflections from important planes in the crystal. In the theory of Kronig the Bragg reflections of the photoelectron lead to minima in the absorption coefficient, in that of Hayasi to maxima. 

The spectrum of titanium metal foil is in Fig. 23. The foil, originally of 12/t thickness, was reduced by etching in HF to about 4/. A corresponding spectrum of titanium metal powder (not shown here) contained the same peaks but with only about half the amplitude, and with greater scatter of data points. This is an illustration of the advantages of having uniformity of thickness, which is especially difficult to achieve with metal powders. The titanium metal spectrum appears to be Kronig type fine structure. 

The Kronig-Penney band structure can range from free-electron-like behaviour to free-atom-like behaviour by changing the strength of the barrier, . It follows from eqn (5.26) that as i -+ 0, cos Ka - cos ka, and hence... 

The experimental 5s XPS spectrum in Xe7,8 (Fig. 32a) looks quite similar to the 4p spectrum in Ba, suggesting that the 5 s level lies just below the 5 1 level structure, forming a well-defined 5 s1/2 quasi-particle excitation and giving considerable strength to a prominent satellite spectrum, mainly 5 p25 d94, 95. The basic process is again giant Coster-Kronig fluctuation of the core hole... 

Finally, Figs. 32 b, c demonstrate that in La and Ce metal, the shift of the 5 s level is at least as large as in Xe, indicating that the giant Coster-Kronig fluctuation process (Eq. (81)) is very strong and atomic-like. However, since the 5 d/6 s electrons now form rather broad bands, one can understand that the satellite structure appears to be washed out in the metal. These types of problems will be further discussed in Section 8. 

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