research papers
Predicting
scattering path cumulants and spectra for metals (Cu, Ni, Fe, Ti, Au) using simulations^{a}Department of Chemistry, University of Alberta, Saskatchewan Drive, Edmonton, AB, Canada T6G 2G2, ^{b}Pacific Northwest Consortium Synchrotron Radiation Facility, APS Sector 20, 9700 South Cass Avenue, Bldg 435E, Argonne, IL 60439, USA, ^{c}Australian Synchrotron, Clayton, VIC 3168, Australia, and ^{d}Department of Electronic Materials Engineering, Research School of Physics and Engineering, Australian National University, ACT 0200, Australia
^{*}Correspondence email: karolewski@alum.mit.edu
The ability of ΔR, σ^{2}, C_{3}) for scattering paths are calculated for the metals Cu, Ni, Fe, Ti and Au at 300 K using 28 interatomic potentials of the embeddedatom method type. The MD cumulant predictions were evaluated within a cumulant expansion fitting model, using global (pathindependent) scaling factors. Direct simulations of the corresponding spectra, χ(R), are also performed using MD configurational data in combination with the FEFF ab initio code. The cumulant scaling parameters compensate for differences between the real and effective scattering path distributions, and for any errors that might exist in the MD predictions and in the experimental data. The fitted value of ΔR is susceptible to experimental errors and inadvertent lattice in the simulation crystallites. The unadjusted predictions of σ^{2} vary in accuracy, but do not show a consistent bias for any metal except Au, for which all potentials overestimate σ^{2}. The unadjusted C_{3} predictions produced by different potentials display only orderofmagnitude consistency. The accuracy of direct simulations of χ(R) for a given metal varies among the different potentials. For each of the metals Cu, Ni, Fe and Ti, one or more of the tested potentials was found to provide a reasonable simulation of χ(R). However, none of the potentials tested for Au was sufficiently accurate for this purpose.
(MD) simulations to support the analysis of Xray absorption finestructure (XAFS) data for metals is evaluated. The loworder cumulants (Keywords: XAFS; metals; molecular dynamics; potentials; cumulants.
1. Introduction
Xray absorption finestructure (XAFS) spectroscopy is a versatile technique for determination of local atomic structure (Bunker, 2010). Atomistic [Monte Carlo and (MD)] simulations are increasingly used to calculate the input data (cumulants, synthetic spectra) required for structural analysis of solids (Di Cicco et al., 2002; a Beccara et al., 2003; Okamoto, 2004; Witkowska et al., 2006; Kuzmin & Evarestov, 2009; Kalinko et al., 2009; Higginbotham et al., 2009; Roscioni et al., 2011; Price et al., 2012). The primary motivation for using MD in analysis is often to generate trial structures that can be tested or refined during the course of a fitting procedure. Conversely, the theoretical basis of is sufficiently well established that can be used to test the structural and dynamical predictions of atomistic simulations (Hayes & Boyce, 1980; Mousseau & Thorpe, 1992; Newville, 1995; Edwards et al., 1997; Binsted et al., 2005). Further integration between and atomistic simulation techniques is desirable for both purposes. In particular, emerging materials technologies (Marletta et al., 2011; Hellborg et al., 2010) have led to an increasing need to characterize, through both experiment and simulation, the structures of nanoscale materials that are fabricated by modification of the nearsurface region of solids (e.g. by ion bombardment or overlayer deposition) (MoberlyChan et al., 2007; Krasheninnikov & Nordlund, 2010; Baglin & Ila, 2011). This study benchmarks the accuracy of MD simulations for metallic solids against data, using a simulation methodology that would be appropriate for nanoscale materials applications.
The physics of a MD simulation is largely bound up in the properties of the interatomic potential. MD simulations of surfaceterminated crystallites of the metals Cu, Ni, Fe, Ti and Au at 300 K are reported here for a representative selection of manybody interatomic potentials of the embedded atom method (EAM) type (Daw et al., 1993) that differ in terms of their functional forms and ranges. These metals include examples of the facecentered cubic (f.c.c.), bodycentred cubic (b.c.c.) and hexagonal close packing (h.c.p.) crystal types. The structural properties used for fitting the EAM potentials are predominantly those of ideal crystals, whereas most nanoscale applications of (clusters, films, radiation effects) involve materials with defects or reduced dimensionality. The MD simulations were performed using surfaceterminated crystallites, in order to better resemble typical experimental substrates and conditions. The presence of a surface is a necessary feature in any simulation that seeks to model materials modification by particle bombardment or deposition. The chosen boundary conditions provide a robust test of the ability of the EAM potentials to reproduce bulk structural properties in simulation crystallites that lack the rigid volume and shape constraints imposed by threedimensional periodicity.
The MD configurational data that arise from the various metal/potential combinations are employed for calculation of (i) the cumulants of
scattering path distributions, and (ii) synthetic spectra. The ability of MD to support the analysis of experimental data for these metals is evaluated for each potential by comparing the MD predictions of scattering path cumulants and synthetic spectra, respectively, to the corresponding data derived from experimental measurements. The potentials for a given metal are found to vary significantly in their ability to model data at the level of precision required for structural analysis.2. Experimental
Transmission Kedge) and Au (L_{3}edge) foil standards were recorded at the Pacific Northwest Consortium Collaborative Access Team (PNCCAT) bendingmagnet beamline (sector 20BM) at the Advanced Photon Source (APS). Transmission Kedge data were also collected for Cu and Ti foil standards at the Australian National Beamline Facility bendingmagnet beamline (BL20B) at the Photon Factory (PF), and for a Cu foil standard at the wiggler beamline at the Australian Synchrotron (AS). All data were recorded at 300 K, utilizing a conventional ion chamber configuration, and with harmonic rejection accomplished by detuning the monochromator. The χ(k) extraction and fitting procedures were performed using the Ifeffit package, with Artemis version 0.8.014, and Athena version 0.8.061 (Ravel & Newville, 2005). The Fourier transform operations required for the χ(k) to χ(R) conversion were performed using a Hanning window function (with Δk = 1.0 Å^{−1}). Further data transformation details are given in the footnotes to Tables 4 to 8.
data for Cu, Ni, Fe (3. Computational methods
3.1. MD simulations
MD simulations of metallic crystallites at 300 K were performed with the Kalypso package, version 3.1 (Karolewski, 2005). The arrangements and orientations of the atomic layers used to construct the simulation crystallites are summarized in Table 1. The lattice constants for Cu (Kroeger & Swenson, 1977), Ni (Collins & Gehlen, 1971), Fe (Basinski et al., 1955), Ti (Collins & Gehlen, 1971) and Au (Martienssen, 2005) refer to 300 K. Periodic boundary conditions were applied to the crystallite along the x and y directions (side faces). The crystallites were terminated by free surfaces in the z direction (top and bottom faces). Crystallite sizes much larger than the information range of (∼10 Å, determined by electron mean free paths of 20 Å or less) were employed, both to avoid coarsesampling of which could lead to an underestimation of thermal vibrational amplitudes (Winkler & Dove, 1992), and to avoid surface relaxation effects in the configurational data. Temperatures were maintained by means of a Berendsen thermostat (Berendsen et al., 1984). MD simulations were carried out using a 1 fs timestep. After equilibration for 10 ps, atomic configurational data were stored at 5 ps intervals until termination at 60 ps.

MD simulations were performed using six EAM (or EAMlike) potentials for Cu, Ni, Fe and Au, and four EAMlike potentials for Ti. Table 2 lists the abbreviations used in this paper for each potential, and indicates the functional type of each potential. Most of the potentials have analytic forms. The TB potentials are distinguished in terms of whether they extend to the second (TB2) or fifth (TB5) neighbour distance. The Voter (Cu, Ni, Au), GRS (Au) and ZM (Ti) potentials were implemented using numerical tables provided by their originators.
‡Note that coefficients f_{6} and V_{5} are reported with incorrect signs therein. 
Minor adjustments were made to some potentials. An interpolation function was used to truncate all abruptly terminating potentials smoothly in the region beyond the potential cutoff distance. The length scale parameter for the original RTS potential (Cu) was adjusted from 3.6100 to 3.6149 Å in order to reproduce the lattice constant of Cu at 300 K. The length scale of the OLS (Fe) potential was adjusted by a factor of 0.99875 in order to reproduce the lattice constant of Fe at 300 K (2.8665 Å). Other potentials were used without adjustment.
3.2. prediction and fitting methods
3.2.1. Cumulant predictions
Cumulants of ) were located in bulklike sites that were situated at more than twice the potential cutoff distance from the free and periodic surfaces of the simulation crystallites. The loworder cumulants (C_{n}) and moments (μ_{n}) of the scattering path distribution, ρ(r), are related through the following expressions, which depend on the variable r (the instantaneous halflength of the scattering path, with mean value R),
scattering path distributions were extracted from the MD configurational data using a nested neighbour search procedure. All atoms selected for use in cumulant predictions and simulations (§3.2.2Thus, R represents the mean scattering path halflength, while σ^{2} represents the mean square variation of the scattering path halflength (Fornasini, 2001). For singlescattering paths, the MD cumulants reflect the distribution of the instantaneous relative displacements of two atomic centres. For singlescattering paths, R is the mean distance between the absorber and scattering atoms, while σ^{2} is their mean square relative displacement. The first cumulant is conventionally described using a related quantity, ΔR,
where R_{0} represents the scattering path halflength when the absorber and scattering atoms are located at their ideal lattice site positions. In this work, R_{0} is always identical to the experimental crystallographic bond length (for singlescattering paths), or the sum of such bond lengths (for multiplescattering paths). MD predictions of ΔR, σ^{2} and C_{3} will be presented later in this paper. Scattering paths are distinguished by the notation used by Binsted et al. (2005). For example, `020' designates a (twoleg) singlescattering path that involves the absorber atom (`0') and an atom in the second coordination shell (`2').
The cumulants arising from ρ(r), the cumulants are the moments of a weighted or `effective' distribution, P(r),
fitting procedures have a different significance from those predicted by MD. Whereas the MD cumulants represent the true moments of the scattering path distribution (the `real' distribution),that arises from the λ is the photoelectron mean free path) (Bunker, 1983; Fornasini et al., 2004). The distinction between the real and effective distributions is mainly of importance for predictions of the first cumulant.
equation (3.2.2. simulations
Synthetic i.e. the modulation functions χ(k), were calculated from the MD configurational data using the ab initio FEFF code, version 8.2 (Rehr & Albers, 2000). The calculation procedure for χ(k) entails the production and execution of a FEFF input file for each absorber site and simulation time. The resulting χ(k) includes contributions from all scattering paths with one to four legs up to a maximum half path length R_{max} = 6 Å. The total number of absorber atoms sampled for the χ(k) calculations ranged from 35000 to 80000 for the various metals. The averaged χ(k) for simulation times of 10–30 ps did not differ significantly from those for times of 35–60 ps. The site and timeaveraged χ(k) functions were adjusted to take account of the energy origin shift (E_{0}) and amplitude reduction factor (S_{0}^{ 2}) deduced from experimental data, as discussed in §4.2. A normalization (McMaster) correction (Rehr et al., 1991) was also applied to χ(k). The conversion of the adjusted χ(k) function to the synthetic spectrum in Rspace, χ(R), was performed with Athena (Ravel & Newville, 2005), using a Hanning window function (with Δk = 1.0 Å^{−1}).
spectra,4. Results
4.1. MD cumulant predictions
MD simulations can support two distinct approaches to the analysis of et al., 1997; Bunker, 2010), or synthetic spectra can be predicted from MD configurational data (Binsted et al., 2005). In this section the MD cumulants predicted by different potentials for each metal are first compared.
data: the MD cumulants can be extracted for use in an cumulant analysis procedure (EdwardsFigs. 1 to 5 plot the first and second MD cumulants (ΔR and σ^{2}, respectively) predicted for single and multiplescattering paths of Cu, Ni, Fe, Ti and Au, that have scattering path halflengths (R_{0}) up to 5 Å. The third MD cumulants (C_{3}), predicted for the 010 scattering paths only, are listed in Table 3, while Fig. 6 displays C_{3} predictions for a range of scattering paths in Cu and Au.
‡Mean values from fits to APS, AS and PF data. 
Among different potentials, the predictions of ΔR for the 010 scattering paths (i.e. the paths with lowest R_{0} in Figs. 1 to 5) are dispersed over a typical range of 0.01 Å or less. However, the ΔR predictions for the 010 paths based on the RTS potential for Cu, and the ZLL potential for Ni, deviate significantly from the respective group mean values. The ΔR predictions for different potentials tend to maintain their relative size order, but diverge from each other as R_{0} increases.
With a few exceptions, the MD predictions of σ^{2} for the 010 scattering paths show a tendency to cluster in a range ≤0.002 Å^{2}. For longer paths, the range of predicted values tends to increase. The σ^{2} estimates for some potentials (Cu: RTS; Fe: EFS; Ti: IKV; Au: AV) consistently deviate from their respective group mean values. The σ^{2} predictions for multiplescattering paths are usually close to those for the singlescattering paths that have comparable path lengths. The σ^{2} predictions for different potentials also tend to maintain their relative size order as R_{0} increases. However, this order is unrelated to the size order observed for the ΔR predictions.
The C_{3} predictions produced by different potentials display only orderofmagnitude consistency (Fig. 6; C_{3} predictions for Ni, Fe and Ti may be found in the supplementary material for this paper, Figs. S1 to S3^{1}). This behaviour is discussed in §5.1. For most metal/potential combinations, C_{3} has its highest value for the first shell (Cu, Ni and Au) or for the first and second shells (Fe and Ti). All potentials for Cu, Ni and Au predict minimum C_{3} values for the second shell (020 path). The ZLL potential for Ni predicts negative C_{3} for all paths, whereas the predictions for other Ni potentials are typically positive for paths other than 020.
4.2. cumulant fitting
The MD cumulants predicted by different potentials were evaluated for use in , 2010). In the fitting procedure, the first (R) and second (σ^{2}) MD cumulants of each path were optimally adjusted using global (pathindependent) scale factors α and β, respectively. The scale factors were included in order to compensate for (i) intrinsic errors in the predicted MD cumulants, (ii) the systematic differences that are known to exist between the MD and cumulants (§3.2.1) and (iii) experimental measurement errors. Specifically, the first and second cumulants for the nth scattering path were fitted by making the following adjustments to the predicted MD cumulants: and (where R_{n} and are the first and second MD cumulants). In this scheme, = . For the 010 paths of the f.c.c. metals (Cu, Ni, Au), the third MD cumulant (C_{3}) was also included in the fit, with a corresponding scale factor γ.
analysis by employing them to fit experimental data for metal foil samples in a cumulant expansion model (Bunker, 1983Since the MD cumulants are scaled during the fitting procedure, the fitting error metric (Rfactor) characterizes the ability of the MD simulations to predict the relative values of the optimum cumulants for a range of scattering paths. The fitted values of α, β and γ represent the correction factors that must be applied to the MD cumulants in order to produce the best fit with the experimental data.
An accurate potential would be expected to produce a small Rfactor, with physically reasonable fitted values of α and β. Since = R_{n}R_{0}, the fitted value of α is expected to be small but nonzero. The fitted value of β represents a scaling factor applied to the predicted second cumulants, and adjusts the latter to optimize the fit with the experimental second cumulant (if β = 1.0, no adjustment is required). It is worth emphasizing that the values of both α and β are also influenced by any measurement errors in the spectra to which they are fitted.
The values of E_{0} and S_{0}^{ 2} used for the MD cumulant fits were first estimated independently of the MD predictions by fitting the experimental data (single and multiplescattering paths) with a simple model. The first cumulant was fitted as = , where α is treated as a pathindependent coefficient. The correlated Debye (CD) model (Sevillano et al., 1979) was used with a fitted Debye temperature, to estimate the second cumulants. The third cumulant was also fitted, for the first shell only. The estimated values of E_{0} and S_{0}^{ 2} were not found to be sensitive to the fitting scheme, and several different approaches yielded similar estimates. These fits intentionally covered a limited range of scattering paths (details in Tables 4 to 8). Since R_{0} ≃ R (to within <0.1%), the α parameters in the MD and CD fitting models have a similar significance, although they are not strictly identical.
‡Typical fitting errors for the MD models are α: ±5 × 10^{−4}; β: ±0.015; γ: ±0.2. §Fitted Debye temperatures (±4 K) are 329 K (AS), 319 K (APS), 323 (PF). 
‡Typical fitting errors for the MD models are α: ±7 × 10^{−4}; β: ±0.01; γ: ±0.08. §Fitted Debye temperature is 193 ± 3 K. 
Tables 4 to 8 summarize the results of fitting the experimental spectra for Cu, Ni, Fe, Ti and Au with the MD cumulant models. The fitting procedures include all scattering paths that influence the spectra within the fitting ranges of ∼1.5 to 5 Å (details in Tables 4 to Table 7). Tables 4, 5, 6 and 8 (for the f.c.c. and b.c.c. metals) also include comparisons with a CD model that fits the spectra over the same range (and includes the third cumulant in the fits for the 010 paths of the f.c.c. metals). With a few exceptions (Ni: ZLL potential; Ti: IKV potential), the fitting capabilities of the optimally scaled MD cumulants for different potentials are found to be comparable for a given metal. The MD and CD models employ an equal number of fitting parameters (three for Cu, Ni, Au; two for Fe, Ti). On the basis of the Rfactor, the scaled MD cumulant models typically fit the experimental spectra as well as, or better than, the CD models. Fig. 7 displays the results of fitting the MD cumulants predicted by different EAM potentials for Cu to the experimental χ(R) data, using the kspace weighting employed in the fits (details in Table 4). Similar graphs showing the fitting results for Ni, Fe, Ti and Au may be found in the supplementary material to this paper (Figs. S4 to S7).
‡Typical fitting errors for MD models are α: ±5 × 10^{−4}; β: ±0.025; γ: ±0.3 (for the ZLL model α: ±9 × 10^{−4}; β: ±0.032; γ: ±4). §Fitted Debye temperature is 407 ± 5 K. 
‡Typical fitting errors for the MD models are α: ±5 × 10^{−4}; β: ±0.04. §Fitted Debye temperature is 426 ± 8 K. 
‡Typical fitting errors for MD models are α: ±1 × 10^{−3}; β: ±0.05. 
In Table 4, the variation of α between different Cu potentials (6 × 10^{−4}) is an order of magnitude smaller than the variation of α between different beamlines (4.8 × 10^{−3}). Both the sign and the magnitude of α vary with synchrotron source, e.g. in the range −0.0044 to 0.0004 for the CD fitting model. A reviewer of this paper has pointed out that discrepancies of this kind might arise from various angletoenergy conversion errors in the measurements that are not necessarily removable by shifting ΔE. The range of values obtained in this work for α is not unusual for recent studies of Cu (at 300 K) reported by other groups. For example, a study of Cu spectra from 11 synchrotron sources fitted α in the range −0.002 to 0.000 (threeshell fits) (Kelly et al., 2009). Firstshell fits to Cu spectra by Newville et al. (2009) fitted α between −0.0073 and −0.0050 for a single beamline using different fitting procedures. Fourshell fits to Cu spectra from three established beamlines fitted α between −0.002 and −0.0004 (Gaur et al., 2013). The elimination of beamlinedependent errors in remains an active field of investigation (Chantler et al., 2012). In this study the variation in α between different beamlines contributes an instrumental error of about ±0.005 Å to the estimate of the Cu first shell distance.
For metals other than Au, the fitted values of β (the scale factor for the second MD cumulant) are scattered around a mean value near 1.0. For Au, the fitted values of β are always <1.0, with a mean value of 0.65. In Table 4, the variation of β among different Cu potentials (±19% of the mean value) is much larger than the variation of β between different beamlines (±3% of the mean value). The relative values of β predicted by different Cu potentials are beamlineindependent (e.g. the TB2:TB5 ratio of β values is 0.8622 ± 0.0005 for all beamlines). This suggests that the small variation in β between different beamlines is due to experimental error rather than structural differences (e.g. static disorder) in the unannealed foil standards.
The fitted values of γ (the scale factor fitted for the third MD cumulant of Cu, Ni and Au only) span a relatively broad range that reflects the scatter in the corresponding MD cumulants (§4.1). The large relative errors in the estimates of γ (typically ±20%) are due to the high correlation (r > 0.7) that exists between γ and α. In Table 4, the variation of γ among different Cu potentials (±65% of the mean value) is somewhat larger than the variation of γ between different beamlines (±20% of the mean value). To a good approximation, the relative values of γ predicted by different Cu potentials are beamlineindependent (e.g. the TB2:TB5 ratio of γ values is 0.71 ± 0.02 for all beamlines). This suggests that the scatter in the γ values between different Cu potentials is a real effect, rather than an artefact of the fitting procedure. For Ni with the ZLL potential, α and γ are anticorrelated (r = −0.76), in contrast to other f.c.c. models. For this potential, both fitted parameters display anomalous values that are suggestive either of a systematic error in the pair correlation function predicted by the ZLL potential, or fitting errors caused by interactions between α and γ in the fitting model.
For the metals other than Cu, the fitted values of α, β and γ found in Tables 5–8 refer to a single beamline and thus provide no information about any beamlinedependent error in the experimental data. To address this, Table 9 compares R and σ^{2} values for one (Cu, Ni, Au) or two (Fe, Ti) coordination shells obtained in this work using the CD model whose fitted data are summarized in Tables 4 to 8, with representative fitted data derived from recent studies (only limited data for Ti are available at 300 K). Although the CD model was constrained to use only two (b.c.c., h.c.p.) or three (f.c.c.) fitting parameters over multiple coordination shells, its results are in reasonable agreement with the previous studies. The maximum discrepancy in R of 0.015 Å is observed for the Au data, while the σ^{2} fits agree to within the fitting errors with one exception (Fe first shell).

4.3. Synthetic spectra
The relative accuracy of synthetic FEFF calculations automatically assign the correct weights to each scattering path contribution, so the distinction between the real and effective distributions [equation (5)] does not have to be considered when evaluating synthetic spectra. The first cumulants (ΔR in Figs. 1 to 5) determine the positions of peaks in χ(R). The distribution of ΔR values for any given metal is relatively narrow for the majority of potentials (in a range ±0.005 Å for most 010 paths). The MD cumulant predictions are more clearly differentiated in terms of their ability to predict the second cumulants. The fitting parameter β (Tables 4 to 8) optimally scales the MD second cumulants (σ^{2}) that determine the in χ(R). Values of β near 1.0 imply a greater similarity between the predicted MD second cumulants and the second cumulants. If β > 1, the predicted second cumulants will be too small, so the peaks in χ(R) will be too narrow, and vice versa. For the Au potentials, the fitted values of β range from 0.57 to 0.78 (Table 8), and thus the in synthetic spectra will be too high for all Au potentials. For the other metals (Cu, Ni, Fe and Ti) there is at least one potential for which β lies in the range 1.00 ± 0.05.
spectra can be expected to correlate, to a large extent, with the accuracy of the corresponding MD cumulant predictions. TheThe most realistic synthetic χ(R) functions for Cu, Ni, Fe and Ti (as identified on the basis of the fitted β parameters) are compared with experimental χ(R) data in Fig. 8 (for Cu, the AS experimental data are selected). The synthetic χ(R) functions employ the same values of E_{0} and S_{0}^{ 2} as the fits to the cumulant expansion models (details in Tables 4 to 8). The unadjusted synthetic χ(R) are inferior in quality to the fits provided by the cumulant expansion model based on scaled MD cumulants (§4.2). However, for Fe (FSA potential), the synthetic and experimental χ(R) functions are in fair agreement up to 5 Å. For Cu (GTL potential), Ni (EFS potential) and Ti (LREP potential), the synthetic χ(R) data tend to display inaccuracies in peak positions and/or shapes above 3 Å. Binsted et al. (2005) performed direct simulations for bulk Cu based on use of the TB5 potential, but found limited agreement of peak heights, except for the 010 path. Higginbotham et al. (2009) did not report χ(R) predictions, but observed that the form of EAMtype potentials could significantly modify the χ(k) predictions for Fe. Taking into account the beamlinedependence of the fitted β values for Cu, it is not straightforward to decide which Cu potential produces the most accurate predictions of in χ(R), as judged by the proximity of β to 1.0. The GTL potential is most accurate for the AS and PF data, while the TB2 potential is most accurate for the APS data.
5. Discussion
5.1. Reliability of MD cumulant predictions
5.1.1. First cumulants
MD simulations predict cumulants of the real path length distribution (§3.2.1). The first MD cumulant, R, differs significantly from the site–site distance, R_{0}, due to the effects of atomic thermal motion. The instantaneous relative thermal displacements, = , of atoms at two sites (labelled 1 and 2) can be resolved into components and that are oriented parallel and perpendicular, respectively, to the site–site vector (Fornasini et al., 2004). The deviation of the first MD cumulant from R_{0} is approximately related to as follows,
If required, and can be predicted by projecting the current internuclear vector on the site–site vector (Sanson, 2010) (however, angular momentum is not conserved under periodic boundary conditions, so the prediction of requires a simulation method that explicitly eliminates errors due to rotational drift).
Experimental firstshell values estimated for δ at 300 K are 0.0039 Å for Cu (Fornasini et al., 2004) and 0.0011 Å for Au (Comaschi et al., 2009). These values will increase at higher temperatures. An accurate MD simulation should provide an estimate of the first real cumulant such that R ≃ , and thus ≃ δ. However, the first MD cumulant predictions (Figs. 1 to 5) are typically larger, such that ΔR = 0.01–0.02 Å for many 010 paths (e.g. 0.008 to 0.024 Å for Cu, and 0.017 to 0.026 Å for Au). These larger shifts are artefacts that originate from the practice of fitting interatomic potentials to static lattice properties, without regard for the lattice expansion that takes place after thermalization (Sheng et al., 2011). Specifically, simulation crystallites that are constrained to be periodic in two dimensions will relieve thermal stress via an expansion in the third, unconstrained, dimension. As a result, the MD prediction of ΔR is artificially high for any scattering path that has legs with a vector component lying normal to the free surface. These small structural distortions, due to anisotropic cannot easily be avoided if the surface–vacuum interface is to be retained in the MD simulation. Scattering paths that lie entirely in the periodically constrained plane do not show this dispersion, but such paths are relatively rare, e.g. the 040 path for Ti (R_{0} = 4.683 Å in Fig. 4), which lies parallel to the c axis.
5.1.2. Second cumulants
The accuracy of predicted thermal displacements is necessarily related to the ability of MD to model both (i) phonon occupation statistics, and (ii) phonon dispersion behaviour, at the temperature of interest. These are distinct issues.
MD ensembles reflect the properties of particle systems that obey classical (Boltzmann) statistics. However, in real metals, phonon occupation numbers, and thus properties that depend on phonons, are determined by quantum (Bose–Einstein) statistics (Cahill et al., 2003; Turney et al., 2009). The classical and quantum occupation number schemes begin to converge around the Debye temperature, θ_{D}, which is indicative of the thermal energy required to excite the highfrequency modes in the phonon For Au, θ_{D} = 170 K, while for other metals examined in this study θ_{D} takes values of 343 K (Cu) and 420–467 K (Ni, Ti, Fe). On this basis, the lattice dynamics of Au can be described by a classical model at 300 K, whereas Cu is a borderline case. At 300 K, the phonon occupation numbers for the highfrequency modes of Ni, Ti and Fe are significantly influenced by quantum statistics, although it should be noted that their phonon contributions to the specific heat are all within 10% of the classical value (i.e. 3k_{B}). Below θ_{D}, MD simulations always underestimate the thermal displacements. At 0 K, classical particles become stationary at their lattice sites, whereas quantum particles retain their energies and associated vibrational displacements.
For metals with cubic structures (e.g. Cu, Ni, Fe and Au), EAM potentials can often reproduce the experimental elastic constants (C_{11}, C_{12}, C_{44}) with a precision of 1–2%. This implies that these potentials are able to reproduce phonon dispersion curves near the Brillouinzone origin, because the elastic constants involve phonons near the zone origin. However, EAM potentials often show poor reproduction of the phonon dispersion curves near the Brillouinzone boundaries. This is particularly true for potentials with a small number of fitting parameters (Bian et al., 2008). Phonon dispersion curves have been predicted for only a few of the potentials used in this study. The TB5 potentials overestimate the phonon cutoff frequencies for Cu and Ni by 5% and 16%, respectively, but underestimate that for Au by 30% (Cleri & Rosato, 1993). This has the effect of narrowing the phonon in Au, leading to excessive thermal displacements (Kallinteris et al., 1997), and is a probable explanation for the consistently low values of β fitted to the MD second cumulants for Au in Table 8. The tendency for EAMtype potentials to overestimate thermal displacements in MD simulations of bulk Au (Kallinteris et al., 1997; Chamati & Papanicolaou, 2004) and Au nanoparticles (Roscioni et al., 2011) has been noted previously. The underlying cause is probably the influence of noncentral manybody forces in Au that cannot be modelled using simple EAM functional forms (Cleri & Rosato, 1993; Bian et al., 2008). The phonon cutoff frequency for Cu is also substantially (20%) underestimated by the RTS potential (Kimura et al., 1998), which accounts for the unusually low value of β fitted for this potential.
The choice of potential cutoff distance in the fitting procedure has a minor influence on the predicted bulk material properties, but significantly affects the forces that act between neighbouring atoms, and hence the corresponding vibrational dynamics. Cutoff distances are arbitrarily employed in MD simulations for computational efficiency, and there are presently no clear criteria for fitting them optimally (Baskes et al., 2001). Similar remarks apply to the interpolation functions that are necessary to truncate some analytic potentials smoothly at the cutoff distance. The TB5 and TB2 potentials (Cleri & Rosato, 1993; Karolewski, 2001) used in this work for Cu, Ni and Au are fitted to similar material properties, and have identical (exponential) functional forms, but differ in terms of their cutoff distances, which lie above the second and fifth nearest neighbour distances, respectively. This difference is sufficient to produce a 10–30% variation in the predicted second cumulants.
Thermal lattice distortions also influence the MD predictions of σ^{2}, to the extent of increasing mean values by typically 5% (this can be established from detailed analysis of the directional behaviour of the σ^{2} predictions). For the first shell of Cu, the MD σ^{2} predictions obtained in this work using the TB5 potential differ by <1% from those obtained for bulk Cu in NVT Monte Carlo simulations by a Beccara et al. (2003), but are 7% higher than those reported by Binsted et al. (2005) in NVE MD simulations. Fitting errors in the experimental determination of σ^{2} by for the first Cu shell are typically 5% (Newville et al., 2009).
5.1.3. Third cumulants
For most metals at 300 K, the third cumulants are small and ) argue that an EAMtype potential should include oscillatory terms, and be fitted to at least three coordination shells, for accurate modelling of thirdorder elastic constants. Few currently used EAMtype potentials meet these criteria. artefacts (see above) might also be expected to unphysically skew the distribution of atomic positions, and thus artificially inflate any predicted values of C_{3}. However, no evidence was found for this. The C_{3} predictions obtained with the TB5 potential for the first eight scattering paths of Cu (Fig. 6) are in close agreement with those reported by Binsted et al. (2005) for bulk Cu using the same potential. Some reasonable predictions of C_{3} (in the sense that γ ≃ 1) do arise, probably fortuitously, from several of the potentials used for Cu, Ni and Au (Tables 4, 5 and 8).
data can be fitted reasonably well without using them. This is fortunate, because the prediction of third cumulants represents a severe test for MD simulations. The values of the third cumulants are associated with anharmonic thermal vibrations in lattices, which ultimately determine the thirdorder elastic constants. However, none of the potentials used in this study was explicitly fitted to thirdorder elastic constants. Chantasiriwan & Milstein (19965.2. Prospects for analysis
After the energy origin shift (E_{0}) and amplitude reduction factors (S_{0}^{ 2}) have been established for the sample of interest (by fitting data for experimental standards), it is relatively straightforward to incorporate MD results in data analysis, either via a cumulant expansion model (§4.2) or by direct simulation of spectra (§4.3).
In this study, global scaling parameters α, β and γ have been used to incorporate the first, second and third MD cumulants, respectively, into data fitting procedures based on standard data analysis software (Ravel & Newville, 2005) and a cumulant expansion model. The scaling parameters compensate for the inherent differences between the real and effective distributions, and for any errors that might exist in the MD predictions and in the experimental data. In Cu, the fitted value of α is largely determined by beamlinedependent factors (e.g. experimental errors), whereas the beamline influence on the fitted values of β and γ is somewhat weaker. Therefore, for Cu (and possibly for other metals), β and γ are mainly determined by the properties of the potential (i.e. MD prediction errors).
For systems that display high structural disorder (e.g. clusters, irradiated materials), a cumulant expansion fitting model is not suitable (Bunker, 1983, 2010). Under these circumstances, data can in principle be analysed by coupling a heuristic search procedure, such as reverse Monte Carlo simulation (McGreevy & Pusztai, 1988; Di Cicco & Trapananti, 2005) or genetic algorithms (Dimakis & Bunker, 2006), to a multiplescattering code that generates the signal for trial structures. In contrast to the purely numerical optimization performed by heuristic techniques, MD simulations of χ(R) can provide a firstprinciples approach to analysis in cases where it is possible to specify (actual or hypothetical) initial conditions for the system of interest. Unlike predicted MD cumulants, the fitting properties of direct simulations of χ(R) are not influenced by differences between the real and effective distributions, but they remain similarly sensitive to MD simulation errors and experimental errors. The references given in §1 provide examples of MD applications in analysis. In particular, Binsted et al. (2005), Roscioni et al. (2011) and Price et al. (2012) discuss fitting strategies (for nanoscale and bulk materials) in some detail.
6. Conclusions
This study evaluates the ability of MD simulations to support the analysis of ΔR, σ^{2}, C_{3}) for scattering paths were calculated for the metals Cu, Ni, Fe, Ti and Au at 300 K using 28 interatomic potentials of the EAM type. The MD cumulant predictions were evaluated within a cumulant expansion fitting model, using global (pathindependent) scaling factors. Direct simulations of the corresponding spectra, χ(R), were also performed using MD configurational data in combination with the FEFF ab initio code (Rehr & Albers, 2000).
data for metals. The loworder MD cumulants (The scaling parameters that are fitted in the cumulant expansion model compensate for differences between the real and effective scattering path distributions, and for any errors that might exist in the MD predictions and in the experimental data. The fitted value of ΔR is particularly susceptible to errors that arise both from experimental factors and inadvertent lattice in the simulation crystallites. The unadjusted predictions of σ^{2} vary in accuracy, but do not show a consistent bias for any metal except Au, for which all potentials overestimate σ^{2}. The unadjusted C_{3} predictions produced by different potentials display only orderofmagnitude consistency. The suitability of current EAMtype potentials for predictions of C_{3} is questionable, since their fitting databases do not include thirdorder elastic constants.
The accuracy of direct simulations of χ(R) for a given metal varies among the different potentials. For each of the metals Cu, Ni, Fe and Ti, at least one of the tested potentials was found to provide a reasonable simulation of χ(R), in the sense of reproducing the major peak positions and widths in the region R < 5 Å (Fig. 8). However, none of the potentials tested for Au was able to reproduce the with sufficient accuracy to be of value for data analysis.
It is difficult to anticipate how a given interatomic potential will perform when used for ΔR could be improved if the lattice expansion due to thermal motion were included explicitly in the fitting procedure (Sheng et al., 2011). Accurate predictions of σ^{2} require potentials that can reproduce phonon dispersion properties (in addition to elastic constants). Even then, quantum statistical effects may limit the accuracy of σ^{2} predictions below the Debye temperature. The potential cutoff distance may also be a significant fitting parameter for applications of MD.
applications. MD predictions ofThe systematic differences between the cumulants of the real and effective scattering path distributions complicate the fitting of λ(k), which can be obtained from FEFF or other theoretical codes. Theoretical relationships between the real and effective cumulants have been derived that can be used for consistency checks (Bunker, 1983, 2010).
data by MD cumulants. However, it is possible to compute cumulants for the effective distribution. Such calculations require a knowledge of the electron dependence on the wavevector,Supporting information
MD third cumulant predictions for Ni, Fe and Ti; 10.1107/S0909049513010303/hf5227sup1.pdf
data for Ni, Fe, Ti and Au fitted with scaled MD cumulant. DOI:Acknowledgements
PNC/XSD facilities at the Advanced Photon Source, and research at these facilities, are supported by the US Department of Energy (Basic Energy Sciences), a Major Resources Support grant from NSERC, the University of Washington, Simon Fraser University, the Canadian Light Source and the Advanced Photon Source. Use of the Advanced Photon Source, an Office of Science User Facility operated for the US Department of Energy (DOE) Office of Science by Argonne National Laboratory, was supported by the US DOE under Contract No. DEAC0206CH11357. We thank the University of Alberta, the Australian Synchrotron, the Photon Factory and the Australian Research Council for support. We also thank the two anonymous reviewers of this paper for their comments and suggestions.
References
a Beccara, S., Dalba, G., Fornasini, P., Grisenti, R., Pederiva, F., Sanson, A., Diop, D. & Rocca, F. (2003). Phys. Rev. B, 68, 140301. Web of Science CrossRef Google Scholar
Ackland, G. J. (1992). Philos. Mag. A, 66, 917–932. CrossRef CAS Google Scholar
Ackland, G. J., Mendelev, M. I., Srolovitz, D. J., Han, S. & Barashev, A. V. (2004). J. Phys. Condens. Matter, 16, S2629–S2642. Web of Science CrossRef CAS Google Scholar
Ackland, G. J., Tichy, G., Vitek, V. & Finnis, M. W. (1987). Philos. Mag. A56, 735–756. (The revised potential parameters for Ni are available online from http://homepages.ed.ac.uk/graeme/moldy/ATVF.txt .) Google Scholar
Ackland, G. & Vitek, V. (1990). Phys. Rev. B, 41, 10324–10333. CrossRef CAS Web of Science Google Scholar
Baglin, J. E. E. & Ila, D. (2011). Mater. Res. Soc. Symp. Proc. 1354, 153–160. CrossRef Google Scholar
Basinski, Z. S., HumeRothery, W. & Sutton, A. L. (1955). Proc. R. Soc. A229, 459–467. CrossRef Google Scholar
Baskes, M. I., Asta, M. & Srinivasan, S. G. (2001). Philos. Mag. A, 81, 991–1008. CrossRef CAS Google Scholar
Berendsen, H. J. C., Postma, J. P. M., van Gunsteren, W. F., DiNola, A. & Haak, J. R. (1984). Comput. Phys. 81, 3684–3690. CAS Google Scholar
Bian, Q., Bose, S. K. & Shukla, R. C. (2008). J. Phys. Chem. Solids, 69, 168–181. Web of Science CrossRef CAS Google Scholar
Binsted, N., Edwards, A. B., Evans, J. & Weller, M. T. (2005). Phys. Scr. T115, 155–158. CrossRef CAS Google Scholar
Bunker, G. (1983). Nucl. Instrum Methods, 207, 437–444. CrossRef CAS Google Scholar
Bunker, G. (2010). Introduction to XAFS: A Practical Guide to Xray Absorption Fine Structure Spectroscopy. Cambridge University Press. Google Scholar
Cahill, D. G., Ford, W. K., Goodson, K. E., Mahan, G. D., Majumdar, A., Maris, H. J., Merlin, R. & Phillpot, S. R. (2003). J. Appl. Phys. 93, 793–818. Web of Science CrossRef CAS Google Scholar
Chamati, H. & Papanicolaou, N. I. (2004). J. Phys. Condens. Matter, 16, 8399–8407. Web of Science CrossRef CAS Google Scholar
Chantasiriwan, S. & Milstein, F. (1996). Phys. Rev. B, 53, 14080–14088. CrossRef CAS Web of Science Google Scholar
Chantler, C. T., Barnea, Z., Tran, C. Q., Rae, N. A. & de Jonge, M. D. (2012). J. Synchrotron Rad. 19, 851–862. Web of Science CrossRef CAS IUCr Journals Google Scholar
Cleri, F. & Rosato, V. (1993). Phys. Rev. B, 48, 22–33. CrossRef CAS Web of Science Google Scholar
Collins, E. W. & Gehlen, P. C. (1971). J. Phys. F, 1, 908–919. CrossRef Web of Science Google Scholar
Comaschi, T., Balerna, A. & Mobilio, S. (2009). J. Phys. Condens. Matter, 21, 325404. Web of Science CrossRef PubMed Google Scholar
Dai, X. D., Kong, Y., Li, J. H. & Liu, B. X. (2006). J. Phys. Condens. Matter, 18, 4527–4542. Web of Science CrossRef CAS Google Scholar
Dai, Y., Li, J. H. & Liu, B. X. (2009). J. Phys. Condens. Matter, 21, 385402. Web of Science CrossRef PubMed Google Scholar
Daw, M. S., Foiles, S. M. & Baskes, M. I. (1993). Mater. Sci. Rep. 9, 251–310. CrossRef CAS Web of Science Google Scholar
Di Cicco, A., Minicucci, M., Principi, E., Witkowska, A., Rybicki, J. & Laskowski, R. (2002). J. Phys. Condens. Matter, 14, 3365–3382. CAS Google Scholar
Di Cicco, A. & Trapananti, A. (2005). J. Phys. Condens. Matter, 17, S135–S144. CAS Google Scholar
Dimakis, N. & Bunker, G. (2006). Biophys. J. 91, L87–L89. Web of Science CrossRef PubMed CAS Google Scholar
Dudarev, S. L. & Derlet, P. M. (2007). J. Phys. Condens. Matter, 19, 239001. Google Scholar
Edwards, A. B., Tildesley, D. J. & Binsted, N. (1997). Mol. Phys. 91, 357–369. CrossRef CAS Google Scholar
Felderhoff, M., Klementiev, K., Grünert, W., Spliethoff, B., Tesche, B., Bellosta von Colbe, J. M., Bogdanović, B., Härtel, M., Pommerin, A., Schüth, F. & Weidenthaler, C. (2004). Phys. Chem. Chem. Phys. 6, 4369–4374. Web of Science CrossRef CAS Google Scholar
Fornasini, P. (2001). J. Phys. Condens. Matter, 13, 7859–7872. Web of Science CrossRef CAS Google Scholar
Fornasini, P., a Beccara, S., Dalba, G., Grisenti, R., Sanson, A., Vaccari, M. & Rocca, F. (2004). Phys. Rev. B, 70, 174301. Google Scholar
Gaur, A., Shrivastava, B. D., Jha, S. N., Bhattacharyya, D. & Poswal, A. (2013). Pramana, 80, 159–171. Web of Science CrossRef CAS Google Scholar
Gong, H. R., Kong, L. T. & Liu, B. X. (2004). Phys. Rev. B, 69, 024202. Web of Science CrossRef Google Scholar
Gordon, R. A. & Crozier, E. D. (2006). Phys. Rev. B, 74, 165405. Web of Science CrossRef Google Scholar
Grochola, G., Russo, S. P. & Snook, I. K. (2005). J. Chem. Phys. 123, 204719. Web of Science CrossRef PubMed Google Scholar
Hayes, T. M. & Boyce, J. B. (1980). J. Phys. C, 13, L731–L737. CrossRef CAS Web of Science Google Scholar
Hellborg, R., Whitlow, H. & Zhang, Y. (2010). Editors. Ion Beams in Nanoscience and Technology. Berlin: Springer. Google Scholar
Higginbotham, H., Albers, R. C., Germann, T. C., Holian, B. L., Kadau, K., Lomdahl, P. S., Murphy, W. J., Nagler, B. & Wark, J. S. (2009). High Energy Density Phys. 5, 44–50. Web of Science CrossRef CAS Google Scholar
Igarishi, M., Khantha, M. & Vitek, V. (1991). Philos. Mag. B, 63, 603–627. Google Scholar
Kalinko, A., Evarestov, R. A., Kuzmin, A. & Purans, J. (2009). J. Phys Conf. Ser. 190, 012080. CrossRef Google Scholar
Kallinteris, G. C., Papanicolaou, N. I., Evangelakis, G. A. & Papaconstantopoulos, D. A. (1997). Phys. Rev. B, 55, 2150–2156. CrossRef CAS Web of Science Google Scholar
Karolewski, M. A. (2001). Radiat. Eff. Defects Solids, 153, 239–255. Web of Science CrossRef CAS Google Scholar
Karolewski, M. A. (2005). Nucl. Instrum. Methods Phys. Res. B, 230, 402–405. Web of Science CrossRef CAS Google Scholar
Kelly, S. D., Bare, S. R., Greenlay, N., Azevedo, G., Balasubramanian, M., Barton, D., Chattopadhyay, S., Fakra, S., Johannessen, B., Newville, M., Pena, J., Pokrovski, G. S., Proux, O., Priolkar, K., Ravel, B. & Webb, S. M. (2009). J. Phys. Conf. Ser. 190, 012032. CrossRef Google Scholar
Kimura, Y., Qi, Y., Cagin, T. & Goddard, W. (1998). The Quantum Sutton–Chen ManyBody Potential for Properties of FCC Metals, Caltech ASCI Technical Report 003. Pasadena, CA, USA. Google Scholar
Krasheninnikov, A. V. & Nordlund, K. (2010). J. Appl. Phys. 107, 071301. Web of Science CrossRef Google Scholar
Kroeger, F. R. & Swenson, C. A. (1977). J. Appl. Phys. 48, 853–864. CrossRef CAS Web of Science Google Scholar
Kuzmin, A. & Evarestov, R. A. (2009). J. Phys. Conf. Ser. 190, 012024. CrossRef Google Scholar
Krayzman, V., Levin, I., Woicik, J. C., Proffen, T., Vanderah, T. A. & Tucker, M. G. (2009). J. Appl. Cryst. 42, 867–877. Web of Science CrossRef CAS IUCr Journals Google Scholar
Lau, T. T., Först, C. J., Lin, X., Gale, J. D., Yip, S. & Van Vliet, K. J. (2007). Phys. Rev. Lett. 98, 215501. Web of Science CrossRef PubMed Google Scholar
McGreevy, R. L. & Pusztai, L. (1988). Mol. Simul. 1, 359–367. Web of Science CrossRef Google Scholar
Marchese, M., Jacucci, G. & Flynn, C. P. (1988). Philos. Mag. Lett. 57, 25–30. CrossRef CAS Web of Science Google Scholar
Marletta, G., Oztarhan, A., Baglin, J. & Ila, D. (2011). Editors. MRS Symposium Proceedings, Vol. 1354, Ion Beams: New Applications from Mesoscale to Nanoscale. Warrendale: Materials Research Society. Google Scholar
Martienssen, W. (2005). Springer Handbook of Condensed Matter and Materials Data, edited by W. Martienssen & H. Warlimont, pp. 45–160. Berlin: Springer. Google Scholar
MoberlyChan, W. J., Adams, D. P., Aziz, M. J., Hobler, G. & Schenkel, T. (2007). MRS Bull. 32, 424–432. Web of Science CrossRef CAS Google Scholar
Mousseau, N. & Thorpe, M. F. (1992). Phys. Rev. B, 45, 2015–2022. CrossRef CAS Web of Science Google Scholar
Newville, M. (1995). PhD dissertation, University of Washington, USA. Google Scholar
Newville, M., Kas, J. J. & Rehr, J. J. (2009). J. Phys. Conf. Ser. 190, 012023. CrossRef Google Scholar
Okamoto, Y. (2004). Nucl. Instrum. Methods Phys. Res. A, 526, 572–583. Web of Science CrossRef CAS Google Scholar
Olsson, P. A. T. (2009). Comput. Mater. Sci. 47, 135–145. Web of Science CrossRef CAS Google Scholar
Price, S. W. T., Zonias, N., Skylaris, C.K., Hyde, T. I., Ravel, B. & Russell, A. E. (2012). Phys. Rev. B, 85, 075439. Web of Science CrossRef Google Scholar
RafiiTabar, H. & Sutton, A. P. (1991). Philos. Mag. Lett. 63, 217–224. CAS Google Scholar
Ravel, B. & Newville, M. (2005). J. Synchrotron Rad. 12, 537–541. Web of Science CrossRef CAS IUCr Journals Google Scholar
Rehr, J. J. & Albers, R. C. (2000). Rev. Mod. Phys. 72, 621–654. Web of Science CrossRef CAS Google Scholar
Rehr, J. J., Mustre de Leon, J., Zabinsky, S. I. & Albers, R. C. (1991). J. Am. Chem. Soc. 113, 5135–5140. CrossRef CAS Web of Science Google Scholar
Roscioni, O. M., Zonias, N., Price, S. W. T., Russell, A. E., Comaschi, T. & Skylaris, C.K. (2011). Phys. Rev. B, 83, 115409. Web of Science CrossRef Google Scholar
Sanson, A. (2010). Phys. Rev. B, 81, 012304. Web of Science CrossRef Google Scholar
Sevillano, E., Meuth, H. & Rehr, J. J. (1979). Phys. Rev. B, 20, 4908–4911. CrossRef CAS Web of Science Google Scholar
Sheng, H. W., Kramer, M. J., Cadien, A., Fujita, T. & Chen, M. W. (2011). Phys. Rev. B, 83, 134118. Web of Science CrossRef Google Scholar
Turney, J. E., McGaughey, A. J. H. & Amon, C. H. (2009). Phys. Rev. B, 79, 224305. Web of Science CrossRef Google Scholar
Voter, A. F. (1993). Embedded Atom Method Potentials for Seven FCC Metals: Ni, Pd, Pt, Cu, Ag, Au and Al. Unclassified Technical Report LAUR93–3901. Los Alamos National Laboratory, Los Alamos, NM, USA. Google Scholar
Voter, A. F. (1998). Phys. Rev. B, 57, 13985–13988. CrossRef Google Scholar
Voter, A. F. & Chen, S. P. (1987). Mater. Res. Soc. Symp. Proc. 82, 175–180. CrossRef CAS Google Scholar
Winkler, B. & Dove, M. T. (1992). Phys. Chem. Miner. 18, 407–415. CrossRef CAS Google Scholar
Witkowska, A., Rybicki, J., De Panfilis, S. & Di Cicco, A. (2006). J. NonCryst. Solids, 352, 4351–4355. Web of Science CrossRef CAS Google Scholar
Zhang, Q., Lai, W. S. & Liu, B. X. (1998). Europhys. Lett. 43, 416–421. Web of Science CrossRef Google Scholar
Zope, R. R. & Mishin, Y. (2003). Phys. Rev. B, 68, 024102. Web of Science CrossRef Google Scholar
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