Cumulant Approach for Inelastic Losses in X-ray Spectra

Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 204)

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

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. These features lead to a path- and energy-dependent many-body amplitude reduction factor \(S_0^2(k,R)\) in the extended x-ray absorption fine structure (EXAFS) [1],
$$\begin{aligned} \chi (k) = \sum _R \frac{|f_\mathrm{eff}(k)|}{kR^2} \sin (2kR + \varPhi _k) |S_0(k,R)|^2 \mathrm{e}^{-2R/\lambda _k} \mathrm{e}^{-2\sigma ^2 k^2} \; . \end{aligned}$$
(27.1)
Typically \(|S_0^2| \approx 0.8-0.9\). Formally such effects can be treated with many-body techniques such as CI, DMFT, or the multi-channel multiple-scattering formalism of Natoli et al. [2], but these methods are all computationally intensive. As a result, these effects are usually neglected on the belief that the error is small or only contributes a smooth background, and hence conventional theories of x-ray spectra are typically only semi-quantitative.

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].

The real-time representation of \(G_K(t)\) considerably simplifies the theory. The formalism leads directly to an expression for the many-body XAS \(\mu (\omega )\) at photon energy \(\hbar \omega \) as a convolution of the spectrum calculated in the presence of a static core hole with an effective particle-hole spectral function \(A_K(\omega )\)
$$\begin{aligned} \mu (\omega ) = \int \mathrm{d}\omega '\, A_{K'}(\omega ') \mu ^{0}(\omega -\omega ') \; . \end{aligned}$$
(27.2)
Here \(\mu ^{0}(\omega )\) is the independent-particle XAS calculated in the presence of a core-hole, \(K' = (c,k(\omega -\omega '))\), and \(A_{K'}(\omega )=-(1/\pi ) \mathfrak {I}\left[ G_{K'}(\omega ) \right] \). We now discuss how a similar convolution [(49) in [10]] over the XAS fine structure \(\chi _K(\omega )\) yields \(S_0^2(R)\). Effects of thermal vibrations and disorder can be included implicitly by averaging over the structural variations. Convolutions similar to that in (27.2) have also been used to incorporate inelastic losses in the XPS photocurrent \(J_k(\omega )\) [6, 11, 12]. Inelastic losses beyond the independent-particle approximation are embedded in the cumulant \({\tilde{C}}_K(t)\). Partitioning the cumulant into intrinsic, extrinsic, and interference terms then facilitates practical calculations. This factorization of the particle-hole Green’s function \(G_K\) is analogous to that in the classic treatment of the x-ray edge singularities by Nozières and de Dominicis [13]. Similarly, our generalized treatment also accounts for edge-singularities from low-energy particle-hole excitations in metals [5].

27.2 Particle-Hole Spectral Function

The generalized cumulant \(\tilde{C}_K(t)\) is obtained by transforming equation (32) of Campbell et al. to the time-domain, and matching the leading terms in powers of the quasi-boson coupling constants,
$$\begin{aligned} \tilde{C}_K(t) = \int d\omega \, \gamma _K(\omega ) (\mathrm{e}^{\mathrm{i}\omega t} - \mathrm{i}\omega t -1) \; . \end{aligned}$$
(27.3)
Practical calculations of the kernel \(\gamma _K\) are based on a partition of \(C_K(t)\) into intrinsic (c), extrinsic (k), and interference terms (kc), respectively, i.e.,
$$\begin{aligned} \tilde{\gamma }_K(\omega )= & {} \gamma _{c}(\omega ) + \gamma _{k}(\omega ) + \gamma _{ck}(\omega ), \end{aligned}$$
(27.4)
$$\begin{aligned} \tilde{C}_K(t)= & {} C_{c}(t) + C_{k}(t) + C_{ck} \; . \end{aligned}$$
(27.5)
The Landau representation [14] of (27.3) ensures that the particle-hole spectral function
$$\begin{aligned} \tilde{A}_K(\omega ) = -\frac{1}{\pi } \mathfrak {I}\left[ \int \mathrm{d}t\, \mathrm{e}^{\mathrm{i}\omega t}\, {\tilde{G}}_{K}^0(t)\mathrm{e}^{{\tilde{C}}_K(t)} \right] \; , \end{aligned}$$
(27.6)
remains normalized with an invariant centroid. Thus the effect of the bosonic excitations is a transfer of spectral weight from the main peak to the satellites while the overall strength is conserved. Note that lifetime broadening due to the photoelectron interactions is included naturally, while the core-hole lifetime is included by adding a damping term, \(-\Gamma _c |t|,\) to the cumulant. This representation is similar to that in the treatment of inelastic losses in XPS [6, 11, 15].

27.3 EXAFS Reduction Factor \(S_0^2\)

In the usual MS theory, [16] the XAFS spectrum \(\chi ^{(0)}(\omega )\) is a rapidly varying energy dependent factor in the one-particle expression for the x-ray absorption,
$$\begin{aligned} \mu ^{(0)}(\omega )=\mu _{0}^{(0)}(\omega )[1+\chi ^{(0)}(\omega )] \;, \end{aligned}$$
(27.7)
where \(\mu _{0}^{(0)}\) is the generally smooth absorption from the central embedded atom alone, in the absence of MS. The many body XAFS function \(\chi (\omega ) = (\mu -\mu _{0})/\mu _{0}\) then becomes
$$\begin{aligned} \chi (\omega )\approx \!\int \! \mathrm{d}\omega ^{\prime }{\tilde{A}}(\omega ,\omega ^{\prime })\chi ^{(0)} (\omega -\omega ^{\prime }) \; , \end{aligned}$$
(27.8)
where \({\tilde{A}}(\omega ,\omega ^{\prime })= A_K(\omega ^{\prime })\) with \(\omega = (1/2)k^2-E_c\). If interference is neglected, the particle-hole Green’s function would simply be a product of the core-hole Green’s function \(g_c(t) = g_c^0(t) \mathrm{e}^{C_{c}(t)}\), and the damped final state Green’s function in the presence of a core hole \({\tilde{g}}_k(t) = {\tilde{g}}^{0}_k(t) \mathrm{e}^{C_{k}(t)}\). This approximation would imply that the intrinsic and extrinsic losses are independent and additive, but that yields XAS satellite strengths that are generally too large. Consequently the interference terms are essential. They provide an energy dependence which tends to cancel the extrinsic and intrinsic losses near threshold, due to the opposite signs of the hole and photoelectron charges, while at very high energies only the intrinsic losses remain. This difference is characterized as an adiabatic to sudden transition, and can be used to justify the adiabatic approximation and the usual neglect of inelastic losses near threshold, i.e., well below the characteristic excitation energy \(\omega _p\) of order 10–15 eV.
Fig. 27.1

(top) Theoretical Al K-edge XAS spectrum compared to the quasi-particle theory in this work and experimental data [17]; and (bottom) the many-body amplitude factor for the first shell R including extrinsic losses \(|\tilde{S}_0^2| = |S_0^2(R)| \mathrm{e}^{-2R/\lambda _k}\) as well as the associated phase \(\psi _k\) in radians.

The top figure was adapted from that in [5]

The full many body XAS \(\mu (\omega )\) can then be expressed as a convolution of an independent particle XAS with a spectral function as in (27.2), where \(\mu ^{0}(\omega )\) is the independent particle XAS calculated in the presence of a core hole. The net effect of the convolution over a particle-hole spectral amplitude \(\tilde{A}(\omega ,\omega ^{\prime })\) in (27.8) is clearly a decreased XAFS amplitude and a phase shifted oscillatory signal compared to the one-particle XAFS \(\chi ^{(0)}\). In the single scattering approximation the oscillatory energy dependence of \(\chi _\mathrm{qp}(\omega )\) enters primarily through the complex exponential \( \mathfrak {I}\{\exp [\mathrm{i}2k(\omega )R]\}\), where 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,
$$\begin{aligned} S_0^{2}(R)=\int _{0}^{\omega } \mathrm{d}\omega ^{\prime }\tilde{A}(\omega ,\omega ^{\prime })\mathrm{e}^{\mathrm{i}2[k(\omega -\omega ^{\prime })-k(\omega )]R} \; . \end{aligned}$$
(27.9)
The 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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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of WashingtonSeattleUSA

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