![cross.png](/logos/buttonlogos/cross.png)
![cross.png](/logos/buttonlogos/cross.png)
research papers
Multiple-electron excitation in X-ray absorption: a screened model of the core-hole–photoelectron potential
aDepartment of Physics and Astronomy, University of Leicester, University Road, Leicester, England
*Correspondence e-mail: mr6@le.ac.uk
The probability of secondary electron shake-off in X-ray absorption is calculated using a model form for the time- and energy-dependent core-hole–photoelectron potential, screened by the single plasmon pole dielectric function of the surrounding material. The resultant excitation probabilities are related to the energy-dependent intrinsic loss function in
data analysis and compared with experiment. Reasonable agreement is obtained close to the although the calculation is less accurate at higher photon energies. The theory described allows the losses to be calculated with little computational effort, making the method suitable for routine data analysis.1. Introduction
X-ray absorption theory is usually given in purely one-electron terms. This gives very good results for the X-ray ) and reasonable results for the extended X-ray absorption fine structure (EXAFS) except for the amplitude. In a single-electron formalism, Coulomb interactions between the created photoelectron and the bystander electrons are neglected. These interactions can give rise to multiple electron excitations which remove photoelectrons from the elastic-scattering channel, hence reducing the primary-channel (Rehr et al., 1978
). In data analysis, excitations at the absorbing atom (intrinsic losses) are usually accounted for by multiplying the single-electron result by a constant loss factor, so2, while the losses at the scattering atoms (extrinsic losses) are included by introducing an imaginary part to the scattering potential. A constant value for the intrinsic loss factor is, however, a poor approximation, especially close to the X-ray In this paper we investigate the energy dependence of so2.
In a previous paper (Roy et al., 2001) we extended the time-dependent perturbation theory of Thomas (1984
) to describe the energy dependence of the intrinsic losses for all atoms using a simple generic model. In this formalism, however, the strength of the core-hole–photoelectron potential was not given explicitly, but had to be included by normalizing the results to experiment or theory (Roy & Gurman, 1997
) in the high-energy limit. In the present paper we attempt to rectify this problem by using a model form for this potential, screened by the energy- and frequency-dependent dielectric function. We know that the previous form of the theory gives a good description of the losses. In this paper we attempt to obtain an equally good description without the inclusion of any `fiddle factors'. The aim is to describe the inherently difficult problem of multiple electron excitations in terms of a physically intuitive theory, following such authors as Gadzuk & Sunjic (1972
), Noguera & Spanjaard (1981
) and Vatai (1988
). The form of the theory described is simple enough to be easily programmable and can be evaluated sufficiently quickly to be of use in routine data analysis. It is found to give reasonable agreement with experiment for rare-gas atoms.
In the present paper (as in the previous work) we calculate only shake-off transition probabilities. The shake-up, or bound-to-bound, transitions are much less probable than excitations into the continuum (Carlson et al., 1968; Hasnain, 1990
) and are usually ignored for data-analysis purposes (Hayes & Boyce, 1982
). Also, shake-up transitions typically involve only the weakly bound initial states. The energies of these bound-to-bound transitions will therefore tend to be small, and the photoelectron energy will not differ from that of the elastically scattered primary photoelectron by more than a few eV. Thus, experimentally, the two contributions to the signal tend not to be resolved and, in practice, the shake-up channels cause no diminution to the measured signal.
The calculation of the e.g. Chang & Poe, 1975; Carter & Kelly, 1977
) and are therefore unsuited for data-analysis purposes. Recently, work has been done on a full description of losses including the intrinsic and extrinsic losses and the interference (Hedin, 1989
) between these terms (e.g. Fujikawa et al., 2000
; Natoli, 1995
; Newville et al., 1993
; Tyson et al., 1992
); however, again we are unaware of any of these methods currently being used for routine data analysis. At present most data-analysis programs use the Hedin–Lundqvist exchange and correlation potential to model inelastic effects. Empirically, this potential includes all the losses to the amplitude (Roy & Gurman, 2001
) although it was designed to model only the extrinsic losses. The absolute accuracy of the Hedin–Lundqvist potential is therefore uncertain (Tyson, 1991
).
2. Theory
In the standard is absorbed by an atom. The photon excites an electron from a given initial state,
, into a continuum state of energy
. Following the the resulting core-hole–photoelectron system acts as a perturbation on the other, passive, electrons in the absorbing atom. We approximate this time-dependent perturbation with a screened model potential and use first-order time-dependent perturbation theory to calculate the probability of exciting each of the secondary electrons.
The temporal variation of the potential arises because the photoelectron takes a finite time to leave the atom, the time depending on the size of the atom and the speed of the photoelectron. In the same way as in the previous paper (Roy et al., 2001), following Gadzuk & Sunjic (1975
) or Thomas (1984
), we approximate the unscreened time- and position-dependent perturbation V(r,t) as a product of time-dependent and position-dependent parts,
where we assume
With this form of the potential the passive electrons see the full core-hole potential, V(r), as when the photoelectron has left the atom. In equation (2)
the characteristic time, to = Rc/v, where Rc is a characteristic distance of the atom and represents the size of the orbital of passive electrons. v is the speed of the photoelectron, which, following Thomas (1984
), we have assumed to be time independent. Finally,
is the Heaviside step function which ensures that the perturbation is switched on at t = 0 when the photoelectron is created.
Thomas (1984) treats Rc as an adjustable parameter. We, however, take Rc to be the mean radius of each arguing that the majority of the interaction between the passive electron and the photoelectron will take place within this radius. We set Rc from the binding energy of each orbital, using the relation for hydrogenic wavefunctions, Rc = no/(2Eo)1/2, where Eo is the binding energy and no is the principle quantum number of each orbital.
The unscreened core hole can be described by a simple coulomb 1/r potential. However, using this potential for the core hole gives values for the probabilities which are much too large. The 1/r perturbation is too strong: the majority of passive electrons will not see a bare core hole because of the screening effects of the other passive electrons. This screening will, in general, be energy dependent. We can model the screening using the single plasmon pole dielectric function of Hedin & Lundqvist (1969). This energy-dependent dielectric function will reduce the magnitude of the hole potential seen by the passive electrons. It will also introduce some many-body effects into our single-electron theory.
To screen the time-dependent potential we must first write it in Fourier transform,
The screened time-dependent perturbation is then given by
where is the causal single plasmon pole dielectric function,
is the appropriate plasma frequency and
is the q-dependent plasmon excitation energy.
Using the above results we can calculate the probability of a multiple-electron excitation following the creation of the core-hole–photoelectron system. As discussed in the previous work (Roy et al., 2001), time-dependent perturbation theory gives the probability amplitude of a passive electron being excited from an
, into an
, as
where the convergence factor corresponds physically to the finite core-hole lifetime. Substituting result (4)
for V(r,t) and integrating over time we obtain the probability amplitude as
The integral above may be performed using contour integration by closing contours in the lower half plane. All the poles in
and
must lie in the upper half plane so that V(r,t) is zero for
. We therefore simply pick up a contribution from the pole at
=
to obtain
We define a slightly simpler form for to that normally used (Hedin & Lundqvist, 1969
) so that we may evaluate the q-integrals analytically. We choose
which exhibits the correct high and low q limits and should therefore give a good approximation to the true excitation frequencies. As we are dealing with atoms and not a free-electron gas we make use of the local density approximation to write the plasma frequency in terms of the radially varying atomic charge density.
Then, using this approximate form for and taking the infinitesimals in
to zero we can integrate equation (8)
directly using a result from Gradsteyn & Ryzhik (1965
),
where
and
There are weak divergences in the r-integrand for both and
of the form
. However, the integrals are easily evaluated numerically using a standard Romberg integration routine (Press et al., 1992
).
At this point we concentrate on the secondary electron shake-off and ignore the possibility of bound-to-bound transitions. We can then calculate the probability of exciting any of the et al. (2001). We find
where the angular parts of the spatial integral have been evaluated to give .
is a bound while Rlo(k,r) is the photoelectron final state. In numerical work we take the initial states from tables by Clementi & Roetti (1974
) while the photoelectron final state may be easily calculated from the Schrödinger equation in the Hartree approximation using a standard Runge–Kutta routine.
A photon incident on an atom may excite electrons from any occupied core state, , provided that
. To compare directly with experiment, we have therefore weighted the probability of exciting a secondary electron given that the photoelectron is from state
with the probability that the photon is absorbed by the electron in state
. Thus
is the contribution to the total atomic
, from all electrons in the initial state
.
In term in the equation above may be removed to give
, the contribution to the probability from a given edge. Only those photoelectrons which do not excite passive electrons into the continuum contribute to the signal, thus the amplitude reduction factor is simply the probability that none of the are excited,
3. Results
In this section we calculate the secondary electron excitation probabilities using equation (13) and compare with experimental data for the neon L edge and argon K edge. We also examine the energy-dependent amplitude reduction factor and calculate an example spectrum for copper. We choose these systems primarily because of the availability of experimental data, but also to demonstrate the flexibility of the method: the probabilities may be easily calculated for any of any element.
3.1. Secondary electron excitation probabilities
Fig. 1 compares calculated and measured shake-off probabilities for neon following an L-edge The magnitudes of the calculated results agree fairly well with the experimental data over the whole of the energy range investigated, certainly to the level of scatter in the various experimental results. The calculated result for the neon L edge matches the data well at low energies and appears to exhibit the correct energy dependence up to a photon energy of
. The theoretical result has not, however, reached its sudden limit even by 700 eV above threshold but gives shake-off probabilities which are still rising with energy. This could be due to a fault in our model dielectric function. With the simplified single plasmon pole, LDA, dielectric function [equation (5)
] it is possible that we are under-screening the core hole at high primary photoelectron energies.
![]() | Figure 1 The probability of secondary electron excitation as a function of photon energy following the creation of a photoelectron from the L shell in neon. The solid line shows the result calculated using equation (13) ![]() ![]() ![]() ![]() ![]() ![]() |
Fig. 2 shows the shake-off probabilities against photon energy following an argon K-edge The experimental results (points with error bars) have been measured by Armen et al. (1985
) while the dotted line is reproduced from Roy et al. (2001
). The calculated result, using the screened core hole, gives a reasonable approximation to the measured data although the energy dependence is not as good as for the neon L edge. Again, the calculated excitation probabilities are still rising at high photoelectron energies where we might have expected them to have reached their sudden limit. Over the whole of the energy range investigated the screened core hole result is not as good as the simple Thomas approximation result from the previous paper. However, the screened core hole result appears to better match the energy dependence of the experimental data at low energies, up to about 70 eV above the edge.
![]() | Figure 2 The probability of secondary electron excitation as a function of photon energy for the argon K-edge. The points are experimental data from Armen et al. (1985 ![]() ![]() ![]() |
3.2. Amplitude reduction factor
From equation (14) we can calculate the energy-dependent amplitude reduction factor. To obtain the total loss factor for the amplitude we must also account for extrinsic losses to the photoelectron beam at the scattering atom and in the medium in between the central and scattering atoms. We do this by multiplying by a mean-free-path term, exp[-2(r1-ro)VPI/k], where r1 is the distance to the scattering atom, ro is the muffin tin radius of the central atom and VPI is the magnitude of a constant imaginary potential, usually taken to be 4 eV.
Once we have obtained the total loss factor we can compare ) exchange and correlation potential used in most current data-analysis programs [e.g. EXCURV98 (Binsted, 1998
)]. This comparison is shown in Figs. 3
and 4
for copper.
![]() | Figure 3 EXAFS loss factors for copper. The points are obtained by ratioing the EXAFS peak heights calculated using the Hedin–Lundqvist potential, which empirically includes all the losses, and the ![]() |
![]() | Figure 4 k3-weighted spectra for copper foil. The solid curve shows the spectra obtained using the real ![]() |
The Daresbury program EXCURV98 was used to calculate an spectrum for Cu foil with both the Hedin–Lundqvist (solid line) and the potentials. We used only the first shell of scattering atoms in order to obtain an unambiguous value for the distances used in the mean-free-path term. The refined fit gave a nearest-neighbour distance of r1 = 2.541 ± 0.006 Å, a first-shell of n1 = 11.3 ± 1.5, a of Ef = −13.2 ± 0.8 eV and a Debye–Waller factor of a1 = 0.016 ± 0.002 Å2. These parameters were then used as input to calculate the spectra using the real
potential which includes no losses.
Fig. 3 shows a direct comparison between loss factors calculated using the screened core hole and those given by the Hedin–Lundqvist potential. To obtain the data points we have ratioed the peak heights of the spectra calculated with the Hedin–Lundqvist potential to the equivalent peak heights of the spectra calculated with the
potential. The solid line shows the result for the loss factor calculated using the screened core hole and multiplied by the appropriate mean-free-path term to describe the losses outside of the central muffin tin.
Within the range of an level required by calculations. The energy dependence of the total calculated loss factor is unphysical at small k because of the constant value for VPI used in the mean-free-path term. This completely kills the as
. To model the mean-free-path term more accurately we need an energy-dependent VPI, the simplest form being one which simply cuts off at the binding energy of the most weakly bound state. Below this energy, no is possible and thus we obtain no losses to the EXAFS.
In Fig. 4 we compare calculated spectra for copper metal. To obtain the solid curve we multiply the
spectra with the calculated energy-dependent loss factor and by a mean-free-path term, with the values of the parameters as given above. The dashed line is the best-fit curve to the experimental data (dotted line) fitted using EXCURV98 with the Hedin–Lundqvist potential and including only a single shell of scattering atoms. The two methods of calculation appear to agree fairly well, although the agreement with experiment is poor because we have fitted using only a single shell of scattering atoms to provide an unambiguous description of the mean-free-path term. The loss factor calculated using a screened core hole is certainly more accurate than the constant so2 and VPI loss factors used historically.
4. Conclusion
In this paper we have used time-dependent perturbation theory and a model core-hole–photoelectron potential, V(r), to obtain results for the secondary electron shake-off probabilities as a function of photon energy above the X-ray absorption edge.
We found the results to agree reasonably well with experiment. The energy dependence of the shake-off probabilities is, however, not quite correct. The dielectric function used appears to underestimate the screening at high photoelectron energies, leading to secondary electron excitation probabilities which are too large in this energy region.
The magnitudes obtained are, however, generally within the experimental error over the region of energy of interest in et al., 1987). Using this real scattering potential and an energy-dependent loss factor such as that described here we would expect to obtain good spectra without the complications of the complex phase-shift approach.
The calculation using a screened core hole is not as accurate as the results obtained with the time-dependent model discussed in the previous paper (Roy et al., 2001). The current method, however, has the advantage that it does not include `fiddle factors'. It may point the way to a more accurate but still physically intuitive description of the loss factors, possibly using a more accurate form of the dielectric function.
References
Armen, G. B., Aberg, T., Karim, Kh. R., Levin, J. C., Crasemann, B., Brown, G. S., Chen, M. H. & Ice, G. E. (1985). Phys. Rev. Lett. 54, 182–185. CrossRef PubMed CAS Web of Science Google Scholar
Bartlett, R. J., Walsh, P. J., He, Z. X., Chung, Y., Lee, E. M. & Samson, J. A. R. (1992). Phys. Rev. A, 46, 5574–5579. CrossRef CAS PubMed Web of Science Google Scholar
Binsted, N. (1998). EXCURV98: Computer Program for EXAFS Data Analysis. CLRC Daresbury Laboratory, Daresbury, Warrington WA4 4AD, UK. (https://srs.dl.ac.uk/xrs/Computing/Programs/excurv97/e61.htm.) Google Scholar
Carlson, T. A., Nestor, C. W., Tucker, T. C. & Malick, F. B. (1968). Phys. Rev. 169, 27–36. CrossRef CAS Web of Science Google Scholar
Carter, S. L. & Kelly, H. P. (1977). Phys. Rev. A, 16, 1525–1534. CrossRef CAS Web of Science Google Scholar
Chang, T. N. & Poe, R. T. (1975). Phys. Rev. A, 12, 1432–1439. CrossRef CAS Web of Science Google Scholar
Chou, S. H., Rehr, J. J., Stern, E. A. & Davidson, E. R. (1987). Phys. Rev. B, 35, 2604–2614. CrossRef CAS Web of Science Google Scholar
Clementi, E. & Roetti, C. (1974). Atom. Data Nucl. Data Tables, 14, 177–478. CrossRef CAS Google Scholar
Fujikawa, T., Hatada, K. & Hedin, L. (2000). Phys. Rev. B, 62, 5387–5398. Web of Science CrossRef CAS Google Scholar
Gadzuk, J. W. & Sunjic, M. (1975). Phys. Rev. B, 12, 524–530. CrossRef Web of Science Google Scholar
Gradsteyn, I. S. & Ryzhik, I. M. (1965). Tables of Integrals, Series and Products. New York/London: Academic Press. Google Scholar
Gurman, S. J. (1983). J. Phys. C, 16, 2987–3000. CrossRef CAS Web of Science Google Scholar
Hasnain, S. S. (1990). Editor. Synchrotron Radiation and Biophysics. Chichester: Ellis Horwood. Google Scholar
Hayes, T. M. & Boyce, J. B. (1982) Solid State Phys. 37, 173–351. CAS Google Scholar
Hedin, L. (1989). Physica B, 158, 344–346. CrossRef CAS Web of Science Google Scholar
Hedin, L. & Lundqvist, S. (1969). Solid State Phys. 23, 2–181. Google Scholar
Holland, D. M. P., Codling, K., West, J. B. & Marr, G. V. (1979). J. Phys. B, 12, 2465–2484. CrossRef CAS Web of Science Google Scholar
Natoli, C. R. (1995). Physica B, 209, 5–10. CrossRef Web of Science Google Scholar
Newville, M., Livins, P. & Yacoby, Y. (1993). Phys. Rev. B, 43, 14126–14131. CrossRef Web of Science Google Scholar
Noguera, C. & Spanjaard, D. (1981). J. Phys. F, 11, 1133–1149. CrossRef CAS Web of Science Google Scholar
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. (1992). Numerical Recipes, The Art of Scientific Computing. Cambridge University Press. Google Scholar
Rehr, J. J., Stern, E. A., Martin, R. L. & Davidson, E. A. (1978). Phys. Rev. B, 17, 560–565. CrossRef CAS Web of Science Google Scholar
Roy, M. & Gurman, S. J. (1997). J. Phys. IV, C2 151–152. Google Scholar
Roy, M. & Gurman, S. J. (2001). J. Synchrotron Rad. 8, 1095–1102. Web of Science CrossRef CAS IUCr Journals Google Scholar
Roy, M., Lindsay, J. D., Louch, S. & Gurman, S. J. (2001). J. Synchrotron Rad. 8, 1103–1108. Web of Science CrossRef CAS IUCr Journals Google Scholar
Samson, J. A. R., Bartlett, R. J. & He, Z. X. (1992). Phys. Rev. A, 46, 7277–7280. CrossRef PubMed Web of Science Google Scholar
Thomas, T. D. (1984). Phys. Rev. Lett. 52, 417–420. CrossRef CAS Web of Science Google Scholar
Tyson, T. A. (1991). PhD thesis, Stanford University, USA. Google Scholar
Tyson, T. A., Hodgson, K. O., Natoli, C. R. & Benfatto, M. (1992). Phys. Rev. B, 42, 5997–6019. CrossRef Web of Science Google Scholar
Vatai, E. (1988). Phys. Rev. A, 38, 3777–3780. CrossRef CAS PubMed Web of Science Google Scholar
Wight, G. R. & Van der Viel, M. J. (1976). J. Phys. B, 9, 1319–1327. CrossRef CAS Web of Science Google Scholar
© International Union of Crystallography. Prior permission is not required to reproduce short quotations, tables and figures from this article, provided the original authors and source are cited. For more information, click here.