Core electron spectra

Schwarz, K.

doi:10.1107/97809553602060000639

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 2.2, p. 312

Section 2.2.16.3. Core electron spectra

K. Schwarz^a ^*

^a Institut für Materialchemie, Technische Universität Wien, Getreidemarkt 9/165-TC, A-1060 Vienna, Austria
Correspondence e-mail: kschwarz@theochem.tuwein.ac.at

2.2.16.3. Core electron spectra

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In excitations involving core electrons, simplifications are possible that allow an easier interpretation. As one example, (soft) X-ray emission (XES) or absorption (XAS) spectra are briefly discussed. In the one-electron picture, the XES process can be described as sketched in Fig. 2.2.16.4. First a core electron of atom A in state $[n^{\prime}\ell^{\prime}]$ is knocked out (by electrons or photons), and then a transition occurs between the occupied valence states at energy $[\varepsilon]$ and the core hole (the transitions between inner core levels are ignored).

Figure 2.2.16.4 | top | pdf |

Schematic transitions in X-ray emission and absorption spectra.

According to Fermi's golden rule, the intensity of such a transition can be described by $[I_{An^{\prime}\ell^{\prime}}(\nu)=\nu^{3}\textstyle\sum\limits_{\ell}W_{\ell\ell^{\prime} }n_{\ell}^{A}(\varepsilon)M_{A}(\ell,n^{\prime}\ell^{\prime},\varepsilon)^{2}\delta(\varepsilon-E_{n^{\prime}\ell^{\prime}}^{A}-h\nu),\eqno(2.2.16.5)]$ where $[W_{\ell\ell^{\prime}}]$ comes from the integral over the angular components (Table 2.2.16.1) and contains the $[\Delta\ell=\pm1]$ selection rule, $[n_{\ell}^{A}(\varepsilon)]$ is the local (within atomic sphere A) partial ( $[\ell]$ -like) DOS, $[M_{A}(\ell,n^{\prime}\ell^{\prime},\varepsilon)^{2}]$ is the radial transition probability [see (2.2.16.6) below], and the last term takes the energy conservation into account.

Table 2.2.16.1 | top | pdf |
$[W_{\ell\ell^{\prime}}]$ factors for X-ray emission spectra showing the $[\Delta\ell=\pm 1]$ selection rule

$[\ell']$	$[\ell]$
$[\ell']$	0	1	2	3	4
0		1/3
1	1		2/5
2		2/3		3/7
3			3/5		4/11

The $[M_{A}(\ell,n^{\prime}\ell^{\prime},\varepsilon)^{2}]$ are defined as the dipole transition (with the dipole operator r) probability between the valence state at $[\varepsilon]$ and the core state characterized by quantum numbers $[n^{\prime}\ell^{\prime}]$ , $[M_{A}(\ell,n^{\prime}\ell^{\prime},\varepsilon)^{2}={{[\int_{0}^{R_{A}}u_{\ell}^{A}(r,\varepsilon)r^{3}R_{n^{\prime}\ell^{\prime}}^{A\,{\rm core}}(r)\,\,{\rm d}r]^{2}}\over{\int_{0}^{R_{A}}[u_{\ell}^{A}(r,\varepsilon)]^{2}r^{2}\,\,{\rm d}r}}.\eqno(2.2.16.6)]$ In this derivation one makes use of the fact that core states are completely confined inside the atomic sphere. Therefore the integral, which should be taken over the entire space, can be restricted to one atomic sphere (namely A), since the core wavefunction $[R_{n^{\prime}\ell^{\prime}}^{A\,{\rm core}}(r)]$ and thus the integrand vanishes outside this sphere. This is also the reason why XES (or XAS) are related to $[n_{\ell}^{A}(\varepsilon)]$ , the local DOS weighted with the $[\ell]$ -like charge within the atomic sphere A.

The interpretation of XES intensities is as follows. Besides the $[\nu^{3}]$ factor from Fermi's golden rule, the intensity is governed by the $[\Delta \ell=\pm1]$ selection rule and the energy conservation. In addition, it depends on the number of available states at $[\varepsilon]$ which reside inside sphere A and have an $[\ell]$ -like contribution, times the probability for the transition to take place from the valence and to the core hole under energy conservation. For an application, see for example the comparison between theory and experiment for the compounds NbC and NbN (Schwarz, 1977).

Note again that the present description is based on an atomic sphere representation with partial waves inside the spheres, in contrast to an LCAO-like treatment with site-centred basis functions. In the latter, an equivalent formalism can be defined which differs in details, especially for the small components (off-site contributions). If the tails of an orbital enter a neighbouring sphere and are crucial for the interpretation of XES, there is a semantic difference between the two schemes as discussed above in connection with f.c.c. Cu in Section 2.2.16.1. In the present framework, all contributions come exclusively from the sphere where the core hole resides, whereas in an LCAO representation `cross transitions' from the valence states on one atom to the core hole of a neighbouring atom may be important. The latter contributions must be (and are) included in the partial waves within the sphere in schemes such as LAPW. There is no physical difference between the two descriptions.

In XES, spectra are interpreted on the basis of results from ground-state calculations, although there could be relaxations due to the presence of a core hole. As early as 1979, von Barth and Grossmann formulated a `final state rule' for XES in metallic systems (von Barth & Grossmann, 1979). In this case, the initial state is one with a missing core electron (core hole), whereas the final state is close to the ground state, since the hole in the valence bands (after a valence electron has filled the core hole) has a very short lifetime and is very quickly filled by other valence electrons. They applied time-dependent perturbation theory and could show by model calculations that the main XES spectrum can be explained by ground-state results, whereas the satellite spectrum (starting with two core holes and ending with one) requires a treatment of the core-hole relaxation. This example illustrates the importance of the relevant physical process in experiments related to the energy-band structure: it may not always be the just the ground states that are involved and sometimes excited states must be considered.

References

Barth, U. von & Grossmann, G. (1979). The effect of the core hole on X-ray emission spectra in simple metals. Solid State Commun. 32, 645–649.Google Scholar

Schwarz, K. (1977). The electronic structure of NbC and NbN. J. Phys. C Solid State Phys. 10, 195–210.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 2.2, p. 312