Multiple-electron excitation in X-ray absorption: a simple generic model

One of the authors was omitted in the published version of the paper by Strange et al. [J. Synchrotron Rad. (2002). 10, 71±75]. The full author list is given above. Since the acceptance of our paper, an atomic-resolution (1.16 AÊ ) structure of MoFe-protein has emerged [Einsle et al. (2002), Science, 297, 1696±1700]. We take this opportunity to update Table 1 of the paper, demonstrating improved agreement of the three-dimensional XAFS re®nement with atomicresolution metrical information.

Constrained re®nements of the Mo K-edge EXAFS using a three-dimensional re®nement approach.
In the ®rst re®nement, distances are kept at crystallographic values and Debye±Waller (' 2 ) values are re®ned; in the second re®nement, the distances are also allowed to vary. A higher value than 0.03 A Ê 2 signi®es that the atom is incorrectly placed and that little contribution to the EXAFS signal is made. ÁR is the difference in MoÐligand distance from the crystallographic value averaged over the two independent units. The ®nal column includes information from a 1.16 A Ê structure (Einsle et al., 2002).

Introduction
The theory of X-ray absorption is usually described in one-electron terms. This theory gives very good results for the X-ray absorption coef®cient (Gurman, 1983). A one-electron theory also gives good results for the extended X-ray absorption ®ne structure (EXAFS), except for the amplitude. However, it is known from studies on the noble gases (e.g. Bartlett et al., 1992) that 20±30% of X-ray absorption processes give rise to multiple-electron excitation. This value is in line with the empirical amplitude reduction factor applied in early EXAFS analyses. In order to describe such processes, and so obtain a correct value for the EXAFS amplitude, we need to use a many-body theory. Such theories are complicated and computationally intensive. We describe here a simple model which correctly gives the multipleexcitation probability (and hence the EXAFS amplitude reduction factor) as a function of photon energy for all atoms with very little computational effort. We also show why the X-ray absorption coef-®cient is only very slightly altered by the inclusion of these manybody effects, despite the high probability of multiple-electron excitation.
In EXAFS studies, multiple-electron excitation on X-ray absorption (intrinsic losses) is not the only loss process which needs to be considered. There are also the extrinsic losses suffered by the photoelectron during its passage through the material. These are usually described in terms of a mean-free-path factor. We do not consider such losses here. Strictly, the intrinsic and extrinsic losses cannot be separated in this simple way, since they may interfere with one another. This interference is signi®cant at low photoelectron energies (Hedin, 1989). Since it treats only the intrinsic losses, our theory completely neglects these interference processes.
We ®rst consider multiple-electron excitation in the limit of high photon energy, where we use the`sudden approximation' (Schiff, 1968). We need a form for the one-electron wavefunctions of all atomic states and use the Slater form [the single zeta functions of Clementi & Roetti (1974), which are themselves ®tted to Hartree± Fock wavefunctions]. The screening parameters which appear in these wavefunctions are obtained using a modi®ed form of Slater's rules. Thus we obtain an accurate representation of the atomic wavefunctions without the need for large amounts of input data. The results of the sudden approximation are compared with the values of the EXAFS amplitude reduction factor S 2 0 derived from many EXAFS analyses.
We go on to consider the energy dependence of the multipleelectron excitation probability, using a development of the model proposed by Thomas (1984). We ®nd that an analytic expression for the energy-dependent probability can be obtained which is in good agreement with experimental data from rare gases. This form provides an energy-dependent form for S 2 0 which could be used in EXAFS analysis to avoid the need for calculations using a complex potential.
Finally, we calculate the X-ray absorption coef®cient itself, including multiple-electron excitations, and show why these do not greatly alter the results obtained, again in agreement with experiment.
We therefore obtain a model description of multiple-electron processes in X-ray absorption which involves very little computing or input data and which provides an accurate description of the observed phenomena.
2. The high-energy limit

The sudden approximation
When an atom absorbs an X-ray photon, a photoelectron and a hole in a deep core state are produced. The core-hole/photoelectron system corresponds to a time-dependent change in potential that is, in general, extremely complex. This more general case is considered in the next section. We ®rst consider the limit of extremely high photon energies. In this limit, the photoelectron has a very high kinetic energy and leaves the atom very rapidly. The other electrons, which we shall refer to as passive electrons, then relax under the in¯uence of the core-hole potential alone. Within this approximation, the passive electrons experience an abrupt change in Hamiltonian so that there is a possibility that they too may be excited into the continuum (we neglect the very weak bound±bound transitions) giving rise to a multiple-electron excitation. The effect of the core-hole potential on these electrons can be described in terms of the sudden approximation (Schiff, 1968).
We describe the wavefunction of the electrons in the atom in Hartree form, as a single product of one-electron wavefunctions. Within the sudden approximation, the boundary condition is that the wavefunction shall be continuous across the abrupt change in Hamiltonian. The probability that a given passive electron is then not excited is given by where the unprimed wavefunctions relate to the unperturbed atom and the primed wavefunctions to the atom with a core hole present. The probability that no passive electrons are excited is then the product of such terms over all passive electrons. This is the probability that only one photoelectron is produced. The theory of EXAFS (Stern, 1988) shows that this is equal to S 2 0 , the amplitude reduction factor for EXAFS. Thus we have and 1 À S 2 0 is equal to the probability of a multiple-electron excitation occurring. This expression for S 2 0 was ®rst obtained by Rehr et al. (1978). The proportion of multiple-electron events can be measured from the charge distribution of the ions produced by photoionization (e.g. Carlson et al., 1968). The EXAFS amplitude reduction factor for very many atoms can be obtained from the many early EXAFS analyses, where this parameter had to be ®tted to data obtained from a standard sample with known coordination number (usually a metal foil).

Wavefunctions
In order to be able to calculate the multiple-electron probability we need to know the form of the atomic wavefunctions. Accurate wavefunctions are available, derived from self-consistent Hartree± Fock calculations (Clementi & Roetti, 1974). In order to obtain a generic form, we use wavefunctions of the Slater type, ®tted to these accurate wavefunctions. These are the single zeta functions of Clementi & Roetti (1974). Thus we have The quantity which appears in these wavefunctions can be written in terms of a screening factor ', where we use atomic units, such that r is given in terms of the Bohr radius. The values of ', obtained from the tables of Clementi & Roetti (1974), can be ®tted by a modi®ed version of Slater's rules. In this way we obtain a generic form for the wavefunctions of all electrons in all atoms. Our modi®cation of Slater's rules is described in Appendix A: they give a good ®t to the tabulated zeta values and so give good wavefunctions for the calculation of multiple-electron excitation probabilities in the high-energy limit. The Slater wavefunctions give, for the probability of a given passive electron remaining unexcited, the expression where, once again, unprimed values relate to the unperturbed atom and primed values to the atom with a core hole present. Since the perturbation is due to a deep core hole, Slater's rules give unless the passive electron state is either below, the same as or next above the core-hole state. For hard X-ray excitation, none of these conditions apply for passive electrons which have an appreciable probability of excitation. Thus equation (6) corresponds to the commonly used Z + 1 approximation for the relaxed ®nal states used when calculating atomic scattering factors in EXAFS. is fairly large for all atomic states. Thus we can approximate the result given in (5) to give Since the energy E i of the state is approximately given by the square of (in atomic units) we see that the probability of excitation of a passive electron falls off roughly as E À1 i . Thus only weakly bound passive electrons are likely to be excited, as we would expect. We actually use the full form, equation (5), in our calculations.
We note that our calculation is appropriate to an isolated atom in that we take the atomic states to be discrete levels. We can extend the calculation to include bonding effects in the tight-binding approximation (Roy, 1999). For full bands this extended calculation gives the same results as the atomic calculation, while for partially ®lled bands the correction is always less than 10% (Roy et al., 1997). Thus we use the simpler atomic calculations in our comparison with EXAFS data.

The EXAFS amplitude reduction factor
We may compare our results with experiment in the form of the EXAFS amplitude reduction factor S 2 0 , which is assumed to be energy independent, or the high-energy limit of ion charge data in photoionization.
First of all we note that the amplitude reduction factor will be very similar for K and L edges of the same element, since it only differs in the two cases by the excitation probability of the 1s state, which is very small for X-ray photons.
Using the modi®ed Slater's rules to obtain and H we can easily calculate P for any orbital and S 2 0 for any atom, using equations (2) and (5). The values of S 2 0 obtained in this way are shown in Fig. 1. The calculated amplitude reduction factor has a characteristic dependence on atomic number, essentially following the ionization potential as we argued it would following equation (7). For Z > 10, the region of interest for EXAFS, it lies between 0.65 and 0.80. Our calculated values for S 2 0 are very similar to those calculated (Roy et al., 1997) using the full tabulated wavefunctions of Clementi & Roetti (1974). Thus it appears that the Slater wavefunctions, ®tted to the single zeta function approximations of these wavefunctions, give a suf®ciently accurate form for the wavefunction. Fig. 1 also shows values of the amplitude reduction factor obtained from many EXAFS data analyses (Roy et al., 1997) which used the real X scattering potential (and hence required empirical amplitude factors) in early versions of the data-analysis program EXCURV. Also shown on Fig. 1 are the results obtained by Carlson (1967) using Hartree±Fock wavefunctions and some data from ion charge measurements (Carlson et al., 1968;Holland et al., 1979;Armen et al., 1985;Bartlett et al., 1992). We note that our results are in fairly good agreement with the full calculation and essentially always agree with experiment to within the experimental uncertainties. The poorest agreement occurs for the 3d and 4d elements: we believe this is due to the lower accuracy of the single zeta form for these orbitals, which have two large components in their full form (Clementi & Roetti,   Carlson et al. (1968). Points with error bars: experimental data from K-edge EXAFS ®ts (S 2 0 ). Large points: experimental data from ion charge measurements. 1974). There is also a problem when the hole is located in the outermost orbital (e.g. Ne L edge). This is due to electron correlation and occurs in all calculations. It is of no signi®cance for X-ray studies, but is rather a problem for UV photoemission.
Thus we conclude that we can successfully model the high-energy limit of the multiple-electron excitation probability in a way which requires little input data or computing.

The energy dependence of the multiple-electron excitation probability
In the previous section the sudden approximation was used to calculate the multiple-electron excitation probability in the limit of extremely high photon energies. In this section we consider the photon energy dependence of this probability. The core-hole/photoelectron system is considered using a spatially and temporally varying model potential and analyzed using standard time-dependent perturbation theory. The temporal variation of the perturbation arises because the photoelectron takes a ®nite time to leave the atom, this time depending on the size of the atom and the energy of the photoelectron. This gives rise to an energy dependence of the excitation probability. At very high photoelectron energies, the results of this calculation should tend to those found using the sudden approximation, and we use this to normalize our results, so avoiding much computation.
The problem of multiple-electron excitation in photoionization has been studied by many authors. Most of these calculations are extremely complex (e.g. Chang & Poe, 1975;Carter & Kelly, 1977) and computationally intensive. They also require much input data. Here we seek a simple model which will describe the energy dependence of the multiple electron probability with reasonable accuracy and with little input or computation. Such a model has been described in outline by Thomas (1984) and we use this as the basis of our work.
Following Gadzuk & Sunjic (1975), we ®rst approximate the timeand position-dependent perturbation V(r, t) due to the core-hole/ photoelectron system as a product of time-dependent and positiondependent parts, with a time dependence of the form f t 1 À exp Àtat 0 X 9 t 0 is a characteristic time which we write as where R is a characteristic distance in the atom and represents the size of the orbital of the passive electron; v is the speed of the photoelectron. With this form, the passive electron sees the full corehole potential V(r) as t 3 I, when the photoelectron has left the atom. Time-dependent perturbation theory gives the probability amplitude of a passive electron being excited from an atomic orbital |ii to a propagating state |ki due to the perturbation V(r, t) as Using the form of perturbation given above we can evaluate the integral if we include a convergence factor exp(Àt) in V(r, t). This convergence factor physically represents the ®nite core-hole lifetime. We then ®nd a transition probability In order to obtain the total excitation probability we must sum over all ®nal states which are accessible, i.e. which obey energy conservation. Replacing the sum by an integral and introducing a density of ®nal states we ®nd a total probability with E the energy of the ®nal state |ki and E B the (positive) binding energy of the initial passive electron state |ii. E max is given by energy conservation as E p À E B , where E p is the energy of the primary photoelectron, equal to h -3 À E edge . The factor A includes all the constants.
We use the Slater orbitals as our initial states |ii and propagating spherical waves (free electron approximation) normalized to unit amplitude as the ®nal states |ki. Taking the core-hole potential as a Coulomb potential we then ®nd with B and B H constants. The second form of equation (14) is obtained by replacing k 2 /2 by the ®nal-state energy E (we use atomic units) and 2 /2 by the binding energy of the initial state E B . When we substitute this form into equation (13) the total excitation probability is given by In the high-energy limit, t 0 3 0 since the velocity of the photoelectron v 3 I. The integral can then be performed at once (Gradshteyn & Ryzhik, 1980;3.241.4). Setting the excitation probability to P(I) in this limit gives us P h -3 P I % 2n À 1 3 2 2n n À 1 3 n 1 3 ÃdEX 16 P(I) is known from the results of the sudden approximation. Thus we have only to evaluate the remaining integral. It is this normalization to the high-energy limit which produces much of the simplicity of our ®nal result. Equation (16) is obtained using an atomic calculation, with the initial state of the passive electron an atomic level, not a band. As we noted in the previous section, full bands give the same result as atomic levels in the high-energy limit, with partially ®lled bands changing the result by less than 10%. The integral of (16) remains ®nite as E B goes to zero (as occurs when the passive electron lies at the Fermi level) so, at worst, including bonding effects (which will involve an integral over E B ) will only alter the energy dependence of Ph -3. Such effects are expected to be small (Roy et al., 1997) and so we continue to use the simpler atomic calculation. t 0 is given by equation (10). We ®x R as the radius at which the charge density |2| 2 peaks, which is easily evaluated using the Slater form. We also calculate the speed of the photoelectron from its kinetic energy. Thus we ®nd We substitute this expression into (16)  result for the probability of exciting a passive photoelectron with a photon of energy h -3 is We note that no passive electron can be excited until the energy of the primary photoelectron is greater than its binding energy, as seems reasonable. Thus no secondary electrons are excited until an energy E B above the absorption edge. The integral may be evaluated analytically [by splitting it into partial fractions and using Gradshteyn & Ryzhik (1980); 2.216 and 2.225], giving a result in terms of the photon energy and the binding energies of the core hole and the passive electron. We therefore obtain a general result for the multiple-electron excitation probability which involves very little computation or input data. We can compare the predictions of this model with experimental data obtained for rare gases: we do this below. The result also gives the energydependent EXAFS amplitude reduction factor which could be used in data analysis. We also note that equation (18) gives P as a function of two parameters, the ratio E p /E B , the ratio of the primary photoelectron energy to the binding energy of the passive electron, and n, the principal quantum number of the passive electron orbital. We can therefore plot a canonical form for P as a function of these two variables. This is shown in Fig. 2. We note that P has reached essentially its high-energy limit by a few times E B above the edge, rising somewhat more slowly for higher values of n. Since only the outermost electrons give a signi®cant contribution to the total multiple electron probability, this means that the EXAFS amplitude reduction factor will reach its full value within a few tens of eV above the edge, with a slower rise for heavier atoms. The rapid rise accounts for the success of simple analyses made using a constant value for the reduction factor.
The results shown in Fig. 2 depend on our value for t 0 , i.e. the value we take for R. Values other than the one we use could be tried. However, they all give a very similar form for P. Thus if we choose R as the peak in the radial charge density r 2 |2| 2 , we merely replace n À 1 by n and so can still use Fig. 2. If we choose the expectation value hri we need to replace n À 1 by n + 1/2. Both these models give a slower rise in P for a given n. Comparison with experimental data from rare gases (see below) suggests that our present form gives the best agreement.

Multiple electron probabilities in rare gases
Experimental data on the photon energy dependence of the multiple-electron excitation probability is available for the rare gases. In Fig. 3 we show a comparison between our results, obtained using equation (18), and the value of P(I) obtained as in x2, and the experimental data of Armen et al. (1985) for the K edge of argon. The argon K edge lies at 3203 eV and we have taken the binding energy of the n = 3 state (the passive electron) as 15 eV, the ®rst ionization potential of argon. This is probably a little low, since the core hole will lower the 3sp energy somewhat (a relaxation shift). In the energy range of the data, only the 3s and 3p passive electrons can be excited. We note that the sudden approximation gives a good estimate of the high-energy end of the data, as is clear from Fig. 1 also. The shape of the energy variation is also well reproduced, although the rise is a little too rapid in the calculated values. The data for the K edge of neon (Carlson & Krause, 1965) is similarly well described (Roy, 1999). In the case of the neon L edge (Bartlett et al., 1992) and the argon M edge (Holland et al., 1979) the energy dependence is accurately reproduced (Lindsay, 2000) by equation (18) but the limit P(I) is poorly calculated by the sudden approximation owing to problems with electron correlation noted in x2.
The use of the experimental binding energy rather than that given by the screening coef®cient calls for some comment. We ®nd much better agreement with experiment, for all rare gases, using the former. It appears that, although the single Slater orbital gives a good representation of the overall wavefunction [and hence good values for P(I)], it does not give accurate values for the binding energies, 2 /2 normally being considerably larger than E B . Nor does it give a good representation of the energy dependence of the matrix elements (Roy, 1999). The use of experimental energies, which are easily accessible for all levels as the X-ray edge energies, appears to solve both these problems. In Fig. 4 we plot the K-edge EXAFS amplitude reduction factor S 2 0 = 1 À Ph -3 for silicon, copper and silver, calculated using this model. The energy dependence agrees with the commonly held view Canonical form for the energy-dependent multiple-electron probability P(E p ). E p is the primary photoelectron energy, E p = h -3 À E edge , E B the binding energy, and n the principal quantum number, of the initial state of the passive electron. Solid line: n = 2; dotted line: n = 3; short dashed line: n = 4; long dashed line: n = 5.

Figure 3
Multiple-electron excitation probability for the K edge of argon. Line: calculation; points: data of Armen et al. (1985). that S 2 0 reaches almost its full magnitude within at most a couple of hundred volts of the edge. The model results also agree closely with those calculated using the local density approximation (LDA) (Roy, 1999;. The LDA calculation is known to give good results for the EXAFS amplitude reduction factor and so we may say that our model will also. We may also note that the integrand of (16) gives the probability that a passive electron will be excited to an energy E [the primary photoelectron will then have an energy E p À (E + E B )]. Thus the integrand can be used to predict the secondary electron spectrum in X-ray photoemission. We (Roy, 1999) ®nd reasonable agreement with the experimental data (Carlson, 1967) for this spectrum.

The X-ray absorption coef®cient
We have noted that a one-electron calculation gives good results for the X-ray absorption coef®cient (Gurman, 1983). We have also noted that 20±30% of X-ray absorption events give rise to multiple-electron excitation. We now try to show how these two points may be resolved.
If we take Thomas' (1984) model at face value then there is no problem. In this model the photoelectron is emitted before it interacts with the passive electrons, so the absorption process is described solely in one-electron terms. However, this is physically untenable: one-electron and multiple-electron processes form different channels for the excitation of the atom and must be considered as separate processes.
If we consider separate channels, then the measured X-ray absorption coef®cient is a sum of one-electron and multiple-electron processes. In the latter, only the sum of the ®nal energies of the two excited electrons is ®xed. We can therefore write the total absorption coef®cient "(3) as The ®rst term is the one-electron channel. In the second term we have the probability of exciting a passive electron to a ®nal state |ki multiplying the absorption coef®cient appropriate to a photoelectron of diminished ®nal energy, summed over all available ®nal states. Equation (19) looks physically reasonable and can also be derived using a many-body theory of photon absorption. We now expand the one-electron absorption coef®cient which appears in the second term about the photon energy h -3. We then ®nd so that the absorption coef®cient is equal to the one-electron value plus a correction term. The one-electron absorption coef®cient may be fairly accurately described by a power law form with n about 3 or 4 (this is a slightly simpli®ed version of the standard Victoreen form) and h -3 e the energy of the edge. When we substitute this form into equation (20) we ®nd Now, |a ik | 2 is only large for small values of E ik : according to equations (12) and (13) it is effectively the slope of the plot of P(h -3). Thus the factor E ik /h -3 is very small, of the order of the ratio of the binding energy of the passive electron to that of the core electron. For the argon K-edge data used above, this ratio is 15/3200. Also, the sum rises from zero in a manner very similar to that shown by P(h -3).
[The sum is actually given by equation (16) with the power n + 2 replaced by n + 1 in the denominator: Fig. 2 shows that this change has little effect.] We therefore expect to see very little change in the X-ray absorption coef®cient when multiple-electron excitations become energetically allowed: just a very weak kink as is actually observed (e.g. Deutsch & Hart, 1986;Filipponi, 1995). Such small changes may be expected, since both the one-electron and the many-body calculations of the X-ray absorption coef®cient obey the same sum rule, the Thomas±Reiche±Kuhn sum rule.

Conclusions
We have described a simple generic model for many-body effects in X-ray absorption. Our major purpose has been to provide a theory which gives the probability of multiple-electron excitation, and hence the EXAFS amplitude reduction factor, in a form which requires little computing or input data. We achieved this by the use of Slater orbitals, with the screening constants ®tted to modern wavefunction results (Clementi & Roetti, 1974) and so calculated using a modi®ed form of Slater's rules. For the energy dependence we have used a development of the model originally proposed by Thomas (1984) which gives an analytic result for the energy-dependent probability, normalized to the high-energy limit calculated in the sudden approximation.
The results obtained using this model agree well with experimental data, both for the standard EXAFS amplitude reduction factor S 2 0 (which is assumed constant, so is given by the high-energy limit) and for the energy-dependent probability as measured for rare gases.
One possible application of our results is in EXAFS data analysis. At present the major data-analysis codes use complex potentials derived from the local density approximation (the Hedin±Lundqvist potential). This gives good results for the amplitude reduction in the energy region of interest for EXAFS (photoelectron energies up to about 1 keV), although this is largely fortuitous (Roy & Gurman, 1999. However, use of this potential involves calculating the photoelectron scattering using a complex potential. It may be more convenient to calculate the EXAFS spectrum using a real potential, such as the energy-dependent Hara potential which is known to give good results (Woolfson et al., 1982) with the intrinsic losses described by S 2 0 included using this model, which requires very little input data. The extrinsic losses also need to be included, but this can be performed in terms of the standard mean-free-path term. research papers Figure 4 Calculated K-edge EXAFS amplitude reduction factor S 2 0 = 1 À P(E p ). Dashed line: silicon; solid line: copper; dotted line: silver.