Multiple Scattering Theory for Spectroscopies pp 375-380 | Cite as

# Cumulant Approach for Inelastic Losses in X-ray Spectra

## Abstract

The theory of inelastic losses in x-ray spectra has long been of interest. These losses include plasmons, particle-hole pairs, Debye–Waller factors, and other many-body effects that damp and shift the spectra. The losses give rise to additional features known as satellites that are not present in independent electron or quasi-particle (QP) approaches.

## 27.1 Introduction

Two classes of inelastic losses have been identified: (i) *intrinsic* losses (the static part of \(S_0^2\)) from excitations due to the sudden creation of a core hole, such as shake-up, shake-off excitations; and (ii) *extrinsic* losses (including the mean-free path term \(\lambda _k\)), which arise from similar excitations during the propagation of the photo-electron. Interference between these losses is also important. The extrinsic losses are often approximated in terms of an inelastic mean free path \(\lambda _k\), which is related to the imaginary part of the photoelectron self-energy [3, 4].

Recently a new approach has been developed to treat these effects based on a particle-hole cumulant expansion and the quasi-boson approximation. Remarkably the method can also account for effects of vibrations and other neutral (bosonic) excitations such as charge-transfer satellites. We refer to the full paper for more details [5]. Here we briefly summarize the main results and explain how this approach can be used to calculate the many-body amplitude reduction factor \(S_0^2\).

Briefly, the cumulant expansion for the one particle Green’s function *g*(*t*) is an alternative to the Dyson equation, and is based on an exponential representation in the time-domain \(g(t)=g^0(t)\mathrm{e}^{C(t)}\), where \(g^0(t)\) is the non-interacting Green’s function and *C*(*t*) is the cumulant. This expansion was treated extensively by Hedin and collaborators [6, 7], and a new derivation based on a functional differential equation has recently been developed [8, 9]. However, the single-particle Green’s function alone is inadequate to describe x-ray spectra, which involves the simultaneous creation of both a particle and a hole. Consequently a generalization to an analogous exponential representation for the *particle-hole Green’s function* is needed, \(G_K(t) = G_K^0(t) \mathrm{e}^{{\tilde{C}}_K(t)}\), where the particle-hole cumulant \({\tilde{C}}_K(t)\) is calculated to second order in the couplings to the excitations in the system. The structure of \(G_K\) is related to the effective Green’s function for x-ray spectra introduced by Campbell et al. [10], transformed to the time-domain. Here \(K=(c,k)\) labels the transition from a given core-level \(|c\rangle \) to a photoelectron state \(|k\rangle \). A related cumulant model for the 2-particle Green’s function has recently been introduced by Zhou et al. [9].

## 27.2 Particle-Hole Spectral Function

*c*), extrinsic (

*k*), and interference terms (

*kc*), respectively, i.e.,

## 27.3 EXAFS Reduction Factor \(S_0^2\)

*R*is an interatomic distance and \(k(\omega )=\sqrt{2\omega }\) is the photoelectron wave vector. Neglecting the smoothly varying vectors, the result of the convolution can be written in terms of a complex-valued amplitude factor \(S^{2}(\omega ,R)=|S(\omega ,R)|^2\exp (\mathrm{i}\psi (\omega ,R))\), which is given by an energy dependent

*phasor sum*over the effective normalized spectral function,

*many-body phase factor*\(\psi _k(\omega )\) is usually small but can be important. An additional factor from the core-core overlap factor and from edge-singularity enhancement may also be needed in some cases, but this factor is usually near unity and neglected in this summary. The qualitative behavior of \(S^{2}(\omega ,R)\) can be understood as follows: At low energies compared with the excitation energy \(\omega _{p}\), the satellite terms strongly cancel so \(A(\omega ,\omega ^{\prime })\approx \delta (\omega -\omega ^{\prime })\) and hence, \(S_0^{2}(R)\rightarrow 1\). At high energies the sudden approximation prevails, and \(A\approx A_\mathrm{qp}+A_\mathrm{intr}\), which has a strong satellite structure. However, the phase difference \(2[k(\omega -\omega ^{\prime })-k(\omega )]\) between the primary channel and satellite becomes small at high energies (\(\omega \gg \omega _{p}\)) so that also \(S^{2}(\omega ,R)\rightarrow 1\), with a correction of order \((\omega _{p}R/\sqrt{\omega })\) similar to an additional mean-free path term. At intermediate energies comparable to \(\omega _p\), the value of \(S^{2}(\omega ,R)\) has a minimum.

As an example, we show the experimental XAS for fcc Al metal compared to the calculated results using the FEFF9 code including the cumulant convolution, and those of the single particle calculation (Fig. 27.1 top). Both calculations agree fairly well with experiment, although the single particle spectrum does not contain enough broadening at about 1590 eV, where the dip is slightly too large. The figure (bottom panel) also shows the many-body amplitude reduction factor including extrinsic losses, \(|\tilde{S}_{0}^{2}| \mathrm{e}^{-2R/\lambda _k}\), and the associated phase shift as a function of EXAFS wavenumber *k*. Note the limiting behavior \(S_{0}^{2} \rightarrow 1\) at both low and high *k* as explained above. In addition, we see appreciable reduction in the amplitude at the minimum \(k \approx 3 (\AA ^{-1})\) where \(S_{0}^{2} \approx 0.65\).

## 27.4 Summary

The particle-hole cumulant expansion with a partition of the cumulant into extrinsic, intrinsic and interference contributions. provides practical approach for calculating inelastic losses due to intrinsic, extrinsic and interference effects in x-ray spectra. These losses are included in the spectra in terms of a convolution with a particle-hole spectral function that accounts for their energy dependence. The cumulant approach simplifies the formalism and facilitates practical calculations. The theory elucidates both their behavior and the differences between the spectral functions for XAS and XPS which may be important to their interpretation. The cumulant approach can also account for edge singularities in the spectrum. Physically, the treatment of inelastic losses here is analogous to an *excitonic polaron*, i.e., the interaction of the particle-hole created in photoexcitation with the density fluctuations produced by the particle-hole system. This is in contrast to the *electronic polaron* described by the *GW* approximation [6], where the single-particle excitations arise from the much stronger density fluctuations due to a core-hole.

## Notes

### Acknowledgements

We thank G. Bertsch, C. Draxl, C.R. Natoli, L. Reining, E. Shirley, and T. Devereaux for comments and suggestions. This work was supported by DOE Grant DE-FG03-97ER45623 (JJR and JJK).

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