Treatment of disorder effects in X-ray absorption spectra beyond the conventional approach
TTreatment of disorder e ff ects in X-ray absorption spectrabeyond the conventional approach Alexei Kuzmin a,b, ∗ , Janis Timoshenko c , Aleksandr Kalinko d , Inga Jonane a , AndrisAnspoks a a Institute of Solid State Physics, University of Latvia, Kengaraga street 8, LV-1063 Riga, Latvia b International Research Organization for Advanced Science and Technology (IROAST), KumamotoUniversity, 2-39-1 Kurokami, Chuo-ku, Kumamoto 860-8555, Japan c Department of Materials Science and Chemical Engineering, Stony Brook University, Stony Brook, NY11794, USA d Universit¨at Paderborn, Naturwissenschaftliche Fakult¨at, Department Chemie, Warburger Strasse 100,33098 Paderborn, Germany
Abstract
The contribution of static and thermal disorder is one of the largest challenges for theaccurate determination of the atomic structure from the extended X-ray absorption finestructure (EXAFS). Although there are a number of generally accepted approaches tosolve this problem, which are widely used in the EXAFS data analysis, they oftenprovide less accurate results when applied to outer coordination shells around the ab-sorbing atom. In this case, the advanced techniques based on the molecular dynamicsand reverse Monte Carlo simulations are known to be more appropriate: their strengthsand weaknesses are reviewed here.
Keywords:
X-ray absorption spectrocopy, Extended X-ray absorption fine structure(EXAFS), Molecular dynamics, Reverse Monte Carlo, Static and thermal disorder
1. Introduction
X-ray absorption spectroscopy (XAS) is an excellent tool to probe the local en-vironment in crystalline, nanocrystalline and disordered solids, liquids and gases in awide range of in situ and in operando conditions (van Oversteeg et al. (2017); Mino ∗ Corresponding author
Email address: [email protected] (Alexei Kuzmin)
URL: (Alexei Kuzmin)
Preprint submitted to Radiation Physics and Chemistry (XAFS-2018) February 25, 2020 a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b t al. (2018)). With the increased availability of synchrotron radiation sources and thetremendous improvement in their parameters, the popularity of the technique has in-creased, and the quality of the experimental X-ray absorption spectra has improvedsignificantly. As a result, more accurate and reliable structural information can be ex-tracted from the extended X-ray absorption fine structure (EXAFS) located above theabsorption edge of an element.The quantitative analysis of EXAFS became possible due to significant advance-ments in the theory (Rehr and Albers (2000); Natoli et al. (2003); Rehr et al. (2009)),however, accurate treatment of disorder e ff ects is still the biggest di ffi culty. The prob-lem becomes especially acute when it comes to the outer coordination shells aroundthe absorbing atom, where the overlap of the shells and the e ff ect of the disorder aremixed with the multiple-scattering (MS) contributions.This paper reviews the existing approaches commonly used to solve the problemof disorder in EXAFS and discusses the strengths and weaknesses of two advancedtechniques based on the molecular dynamics and reverse Monte Carlo methods.
2. Conventional approach to disorder in EXAFS
In this section, we will briefly summarize di ff erent conventional approaches to thetreatment of disorder in EXAFS.The X-ray absorption coe ffi cient µ ( E ) in the one-electron approximation is propor-tional to the transition rate between the initial core-state i and the final excited-state f of an electron, which is given by the Fermis Golden rule µ ( E ) ∝ (cid:88) f (cid:12)(cid:12)(cid:12) (cid:104) f | ˆ H | i (cid:105) (cid:12)(cid:12)(cid:12) δ ( E f − E i − E ) (1)where E = (cid:126) ω is the X-ray photon energy, and the transition operator ˆ H = ˆ (cid:15) · (cid:126) r in thedipole approximation. Note that the final state of the electron is the relaxed excitedstate in the presence of the core-hole screened by other electrons.The characteristic time of the photoabsorption process (1) is about 10 − –10 − sand is determined by several processes: the transition time between initial ( i ) and fi-nal ( f ) states, the core-hole lifetime, the excited photoelectron relaxation time and the2ifetime of the photoelectron out of atom related to its mean-free path (MFP). Notethat this time is significantly shorter than the characteristic time ( ∼ − –10 − s) ofthermal vibrations. Therefore, atoms can be considered as frozen at their instanta-neous positions during the excitation process, and the experimental X-ray absorptionspectrum corresponds to the average over all atomic configurations during the time ofexperiment.The oscillating part of the absorption coe ffi cient χ l ( E ) located above the absorptionedge of orbital type l is defined as χ l ( k ) = ( µ ( E ) − µ ( E ) − µ b ( E )) /µ ( E )) (2)where µ b ( E ) is the background absorption, and µ ( E ) is the atomic-like absorption dueto an isolated absorbing atom (Lee et al. (1981)). The wave number k of the excitedphotoelectron is related to its kinetic energy ( E − E ) by k = (cid:112) (2 m e / (cid:126) )( E − E ),where m e is the electron mass, (cid:126) is the Plank’s constant, and E is the threshold energy,i.e., the energy of a free electron with zero momentum.Within the framework of MS theory, EXAFS χ l ( k ) is described using a series χ l ( k ) = ∞ (cid:88) n = χ ln ( k ) ,χ ln ( k ) = (cid:88) j A ln ( k , R j ) sin[2 kR j + φ ln ( k , R j )] (3)which includes contributions χ ln ( k ) from the ( n − k values due to the finite lifetime of the excitation, thescattering path lengths, interference cancellation e ff ects and path disorder. In practice,the MS contributions up to the 8th-order can be calculated using ab initio FEFF code(Ankudinov et al. (1998); Rehr et al. (2010)).An alternative description of the EXAFS χ l ( k ) in terms of the n -order distributionfunctions g n ( R ) is also known χ l ( k ) = (cid:90) π R ρ g ( R )[ χ oio ( k ) + + . . . ] dR (cid:90) (cid:90) (cid:90) π R R sin( θ ) ρ g ( R , R , θ ) × [2 χ oi jo ( k ) + χ oio jo ( k ) + . . . ] dR dR d θ + . . . (4)where ρ is the average density of a system and χ m ( k ) are the MS EXAFS signals ofthe ( m −
1) order generated within a group of atoms (o, i, j, . . . ) described by g n (Fil-ipponi et al. (1995); Filipponi and Di Cicco (1995)). This approach was realized in theGNXAS code (Di Cicco (1995); Filipponi and Di Cicco (2000)), which is able to ac-count for the two-body ( g ), three-body ( g ) and four-body ( g ) distribution functions.The analytical expression for EXAFS can be greatly simplified when one needs toextract information only on the first coordination shell of the absorbing atom.The contribution of the first coordination shell to the total EXAFS spectrum can beusually isolated by Fourier filtering procedure and analysed within the single-scatteringapproximation, since the length of all MS paths is longer than the first coordinationshell radius. Thus, only the first term of the series given by Eq. (3) remains. In the caseof a Gaussian distribution (or in the harmonic approximation), the EXAFS expressiontakes a simple form χ l ( k ) = S (cid:88) i N i | f l e ff ( k , R i ) | kR i exp (cid:34) − R i λ ( k ) (cid:35) × sin[2 kR i + φ l ( k , R i )] exp( − σ i k ) (5)where S is a scaling factor; N i is the coordination number; R i is the interatomic dis-tance; λ ( k ) is the photoelectron MFP; f l e ff ( k , R ) and φ l ( k , R ) are the photoelectron ef-fective scattering amplitude and phase shift functions (Sayers et al. (1971); Lee andPendry (1975)). The sum in Eq. (5) is taken over groups of atoms located at di ff erentdistances from the absorber.For moderate disorder, when distribution of interatomic distances becomes asym-metric, the EXAFS equation can be expressed using the cumulant decomposition (Bunker(1983); Dalba et al. (1993)). The cumulant model is often useful for the analysis of an-harmonic and thermal expansion e ff ects (Tranquada and Ingalls (1983); Fornasini et al.(2017)), nanoparticles (Clausen and Nørskov (2000); Sun et al. (2017)) and disorderedmaterials (Dalba et al. (1995); Okamoto et al. (2002)).4ometimes, the first coordination shell around the photoabsorber is so stronglydistorted that the cumulant series does not converge. In this case, the EXAFS formulaexpressed in terms of the radial distribution function (RDF) G ( R ) χ l ( k ) = S (cid:90) R max R min G ( R ) | f l e ff ( k , R ) | kR × sin[2 kR + φ l ( k , R i )] exp (cid:34) − R i λ ( k ) (cid:35) dR (6)should be used instead (Stern et al. (1975); Lee et al. (1981)). The RDF G ( R ) definesthe probability of finding an atom in a spherical shell dR at the distance R from thephotoabsorber. The number N of atoms located in the range between R min and R max is given by the integral N = (cid:82) R max R min G ( R ) dR . To determine RDF G ( R ) from Eq. (6),the regularization technique (Babanov et al. (1981); Ershov et al. (1981); Kuzmin andPurans (2000)) can be used to solve this integral equation as an ill-posed problem with-out any preliminary assumption on the shape of the RDF. This approach was recentlyused to reconstruct the local structure in several tungstates MWO (M = Ni, Cu, Zn andSn) (Kalinko and Kuzmin (2011); Anspoks et al. (2014); Kuzmin et al. (2015)) and inmolybdate CuMoO (Jonane et al. (2018b)), where the Jahn-Teller e ff ect is responsiblefor a strong distortion of structural units. It was demonstrated recently that the RDF G ( R ) of atoms can be reliably extracted up to distant coordination shells using neuralnetwork approach (Timoshenko et al. (2018)).In crystalline and nanocrystalline materials, the experimental EXAFS spectrum of-ten contains a significant amount of structural information on outer coordination shells,which is challenging to extract. It is possible to estimate the region of a structure aroundthe absorber, which can potentially contribute into EXAFS, from the photoelectronMFP. Examples for bulk and nanocrystalline nickel oxide (Anspoks et al. (2012)) andbody-centred-cubic (bcc) tungsten (Jonane et al. (2018a)) are shown in Fig. 1. A halfof the MFP λ ( k ) gives an estimate of how far the excited photoelectron can propagateto be able to return back to the absorbing atom. The MFP λ ( k ) depends strongly onthe photoelectron wavenumber k and increases at large k -values. It is equal to about10–20 Å for NiO or bcc W at k ≈ − . This means that when high-qualityexperimental EXAFS data are available in large k -space range, one can expect to see5tructural contributions from atoms located in distant coordination shells. For exam-ple, the structural peaks in Fourier transforms of EXAFS can be recognized up to about11 Å in Fig. 1 for bulk and nanosized NiO at T =
10 K and for bcc W at T =
300 K.The possibility to analyse contributions from distant coordination shells is usefulsince it provides access to additional structural information. However, such analysisbased on the conventional approaches faces a number of problems even for crystallinematerials with a known structure, in which at least the mean-square relative displace-ment (MSRD) factors are variable model parameters.The main problem is related to the number of model parameters, which increasesexponentially when more coordination shells are included to the model (Kuzmin andChaboy (2014)). For example, in the case of bulk NiO with a rock-salt structure, thetotal number of scattering paths, the number of unique paths due to the cubic symmetryand the maximum number of fitting parameters, which can be used in the EXAFSmodel according to the Nyquist criterion ( N par = ∆ k ∆ R /π ) evaluated for relativelylong EXAFS signal with ∆ k =
20 Å − , are shown in Fig. 2 as a function of the clusterradius R around the photoabsorbing nickel atom. Note that the Nyquist criterion is notsatisfied above R ∼ R ∼ ff erent Einstein or Debye temperatures arerequired for each MS path. Besides, these models of lattice dynamics ignore anisotropyof the phonon spectra.Another approach is to calculate MSRD parameters from the phonon projecteddensity of states using the Debye integral σ R ( T ) = (cid:125) µ R (cid:90) ∞ ω coth (cid:32) (cid:125) ω k B T (cid:33) ρ R ( ω ) d ω (7)where µ is the reduced mass associated with the MS path, and k B is the Boltzmann’sconstant. The vibrational density of states ρ R ( ω ) projected on R can be obtained fromfirst-principles calculations of the dynamical matrix of force constants (Vila et al.(2007); Rehr et al. (2009, 2010)). However, this approach uses (quasi-)harmonic ap-6roximation, requires a priori knowledge of structure and can be computationally ex-pensive.An alternative solution which allows one to account simultaneously for the MScontributions and disorder e ff ects is to rely on atomistic simulations such as the molec-ular dynamics (MD) and reverse Monte-Carlo (RMC) methods combined with ab initioMS calculations.
3. Atomistic simulations of EXAFS
MD (Alder and Wainwright (1957)) and RMC (McGreevy and Pusztai (1988))methods are known for a long time, however their application in the field of X-rayabsorption spectroscopy is still scarce. The use of both methods requires significantcomputing resources, so their development has been directly related to the advances incomputer technologies.The first use of MD simulations to reproduce the experimental EXAFS is datedback to the middle of nineties, when the method was applied to study the hydration ofions in aqueous solutions (D’Angelo et al. (1994, 1996); Palmer et al. (1996); Kuzminet al. (1997)). The advantages of the RMC method were realized even earlier at thebeginning of nineties, when it was used to interpret EXAFS of amorphous Si and crys-talline AgBr (Gurman and McGreevy (1990)), liquid KPb alloys (Bras et al. (1994))and superionic glasses (Wicks et al. (1995)).There are several common features for the MD and RMC methods. The simulationresult is represented as one or more atomic configurations (“snapshots”), suitable togenerate the configuration-averaged (CA) EXAFS, which includes static and dynamicdisorder and can be directly compared to experimentally measured EXAFS. The staticdisorder is due to a number of di ff erent atomic dispositions, corresponding to minima ofthe potential energy surface. Examples of systems with the static disorder include non-crystalline materials such as glasses, amorphous solids and liquids, nanocrystals andthin films with atomic structure relaxed due to the size or thickness reduction e ff ect,and materials with structural defects (e.g., vacancies or grain boundaries). Dynamicdisorder arises from temperature-dependent fluctuations in the atomic positions from7he equilibrium structure.The CA EXAFS spectra for di ff erent absorption edges can be calculated from thesame set of atomic coordinates and used in the analysis, thus improving the reliabilityof the structural model (Timoshenko et al. (2014a)). During a simulation, the atoms areplaced in a cell of the required size and shape, often with periodic boundary conditions(PBC) in order to avoid e ff ects associated with the surface. Note that using PBC limitsthe maximum cluster radius, for which EXAFS calculations can be safely performedto avoid artificial correlation e ff ects, to half the minimum cell size. There are also twonon-structural parameters, ∆ E and S , which are required for comparison with theexperimental EXAFS. They can be determined from the analysis of reference materialsor obtained by best matching the experimental and calculated EXAFS spectra.The scheme of the MD and RMC methods is shown in Fig. 3. The structural modelof a material is constructed first in both cases, and the ab initio MS code, such asFEFF (Ankudinov et al. (1998)) or GNXAS (Filipponi and Di Cicco (2000)), is usedto calculate EXAFS for each atomic configuration during the simulation.The principal di ff erence between two methods is that no fitting of experimental EX-AFS is performed in the MD-EXAFS approach, and the structure obtained in the MDsimulation is used “as-is” for the calculation of the CA EXAFS. Note that the numberof required atomic configurations and the time step between them should be carefullyestimated for each particular case to obtain the proper CA signal. On the contrary, thestructural model is modified at each RMC iteration to minimize the di ff erence betweenthe experimental and CA EXAFS in the RMC-EXAFS approach.To perform MD simulations, a model of interactions between atoms is required. Inclassical MD (CMD), the empirical interatomic potential is employed, that significantlyreduces the requirements for computing resources. Besides, the MD-EXAFS approachis suitable for a validation of interatomic potential along with other conventionally em-ployed properties of a material (Di Cicco et al. (2002); Kuzmin and Evarestov (2009);Kuzmin et al. (2016); Bocharov et al. (2017)). Ab initio
MD (AIMD) based on den-sity functional theory (DFT) formalism is also accessible nowadays but is extremelycomputationally expensive. It is important that in the MD simulation, initial model ofthe atomic structure is evolving in time within one of the canonical (NVT), isothermal-8sobaric (NpT) or microcanonical (NVE) ensembles following to classical Newtonianlaws of motion both in CMD and AIMD. Therefore, such simulations cannot be usedto model the motion of atoms at low temperatures, where the zero-point oscillations ofatoms play an important role (Yang and Kawazoe (2012)). In this case, instead, morecomplex methods should be used, such as, for example, the path-integral MD (Marxand Parrinello (1996)).Note that recent developments of X-ray free-electron laser (X-FEL) facilities opennew possibilities to probe the ultrafast excited state dynamics using Xray absorptionspectroscopy (Lemke et al. (2017)). Such experiments provide information on thefemtosecond nuclear wavepacket dynamics, which can be described by first-principlesquantum dynamics simulations (Capano et al. (2015)).The MD simulations can be performed, for example, either by one of the CMDcodes as LAMMPS (Plimpton (1995)), GULP (Gale and Rohl (2003)) or DL POLY(Todorov et al. (2006)), or using AIMD codes as CP2K (VandeVondele et al. (2005)),VASP (Kresse and Furthm¨uller (1996)) or SIESTA (Soler et al. (2002)). After accu-mulating the required number of atomic configurations, one can employ, for example,the EDACA code (Kuzmin and Evarestov (2009); Kuzmin et al. (2016)) to generate theCA EXAFS spectrum.In RMC simulation, the position of atoms in the configuration is usually randomlymodified at each iteration, and the CA EXAFS signal is calculated. The decision toaccept or reject the new atomic configuration is made based on the Metropolis algo-rithm (Metropolis et al. (1953)), taking into account the di ff erence (residual) betweenthe experimental and simulated data in either k or R space, or simultaneously in k and R -spaces using the wavelet transformation (Timoshenko and Kuzmin (2009)). At thispoint, various chemical or geometrical constraints can be easily implemented, by as-signing some penalty to the residual value. For example, one can avoid situations whenthe atoms are getting too close or too far from each other, when non-physical valuesof some bond angle are found (Tucker et al. (2007)), or when the coordination numberfor some atom deviates from the expected one (McGreevy (2001)), etc. The e ffi ciencyof the RMC process can be significantly improved by using an evolutionary algorithm(EA) together with a simulated annealing scheme (Timoshenko et al. (2012, 2014b)).9he RMC method relies on stochastic process, so it will generate di ff erent final setsof atomic coordinates upon restarting simulation several times from di ff erent startingconditions. However, it is expected that the results will be statistically close in termsof the distribution functions. Note that RMC method tends to converge to the mostdisordered solution consistent with the experimental data (Tucker et al. (2007)).Some of the software packages for RMC-EXAFS simulations include RMC-GNXAS(Di Cicco and Trapananti (2005)), RMCProfile (Tucker et al. (2007)), EPSR-RMC(Bowron (2008)), SpecSwap-RMC (Leetmaa et al. (2010)), RMC ++/ RMC POT (Gerebenet al. (2007); Gereben and Pusztai (2012)) and EvAX (Timoshenko et al. (2014b)).Note that in addition to the MD and RMC methods, the average atomic config-uration required to compute CA EXAFS can also be generated from a Monte Carlosimulation based on interatomic potentials (Hansen et al. (1997); Canche-Tello et al.(2014); House et al. (2017)) or atomic displacement parameters obtained from latticedynamics calculations (Duan et al. (2016); Lapp et al. (2018)).
4. Examples of MD / RMC-EXAFS applications
In this section the specific capabilities of the MD-EXAFS and RMC-EXAFS meth-ods will be demonstrated.The first example is concerned with the lattice dynamics in bcc tungsten (Jonaneet al. (2018a)). High-quality experimental W L -edge EXAFS spectrum was recordedat T =
300 K up to k =
18 Å − (Fig. 4 (upper panel)) and includes contributions fromthe coordination shells with a radius of at least up to ∼
11 Å (Fig. 1 (lower panel)).The NVT MD simulations were performed by the GULP code (Gale and Rohl (2003))using a supercell of 7 a × a × a size ( a = .
165 Å) and a time step of 0.5 fs. Theinteractions were described by the second nearest-neighbour modified embedded atommethod (2NN-MEAM) potential (Lee et al. (2001)). After equilibration during 20 ps,the atomic configurations were accumulated during the production run of 20 ps andused to calculate the CA EXAFS. The RMC / EA calculations were performed by theEvAX code (Timoshenko et al. (2014b)) using a supercell of 5 a × a × a size to getbest possible agreement between the Morlet wavelet transforms (WTs) of the experi-10ental and calculated EXAFS spectra. Good agreement with the experimental EXAFSdata was obtained for both MD-EXAFS and RMC-EXAFS approaches (Fig. 4 (upperpanel)). Next, the atomic configurations were used to calculate the RDFs G W − W ( R ) andthe radial dependence of the MSRD factors σ ( R ). At long distances, when correlationin atomic motion becomes negligible, the MSRD σ − W = σ (see the inset in Fig. 4(lower panel)). The obtained mean square displacements (MSD) σ are in agreementwith previously reported experimental and theoretical results (Jonane et al. (2018a)).Thus, the analysis of distant coordination shells allows extracting information on theMSD of atoms, which otherwise requires a di ff raction experiment.In the second example, the use of the MD-EXAFS approach for the validation ofthe interatomic potential model is shown on the example of iron fluoride (FeF ) (Jo-nane et al. (2016)). The crystalline lattice of rhombohedral FeF is composed of FeF octahedra joined by corners with the bond angle Fe–F–Fe between two adjacent octa-hedra equal to ∼ ◦ . The MD simulations were carried out using a simple empiricalpotential, including two-body (Fe–F and F–F) and three-body (Fe–F–Fe) interactions.It was found that di ff erent sets of the optimized potential parameters, correspondingto the iron e ff ective charge q (Fe) in the range of 1.2–3.0, reproduce equally well thestatic crystallographic structure of FeF . This ambiguity was resolved by performingNVT CMD simulations and calculating the CA Fe K-edge EXAFS spectra (Fig. 5).Strong sensitivity of EXAFS to the strength of the Coulomb interactions was found,thus allowing one to select the iron e ff ective charge q (Fe) = N) using the RMC-EXAFS approach (Fig. 6)(Timoshenko et al. (2017)). Cu N has a unique cubic anti-perovskite-type structure(AB X), composed of NCu octahedra joined by the corners with the A sites beingvacant. High symmetry of its lattice is responsible for strong overlap of coordinationshells in the RDF, large MS contributions in EXAFS due to the presence of linear –Cu–N–Cu– chains and an anisotropy of atom vibrations due to tilting motion of NCu octahedra. Since RMC simulation results in a 3D model of the structure, one has anopportunity to analyse separately behaviour of atoms, belonging to di ff erent coordina-11ion shells but located at close distances from the absorber. Temperature dependencesof the MSRD factors for selected Cu–N and Cu–Cu atom pairs were calculated fromatomic configurations obtained by RMC and are shown in Fig. 6 (lower panel). Strongcorrelation in atomic motion was found for atoms (N , Cu a and N a ) located in thechains along the crystallographic axes. Moreover, it is possible to distinguish clearlylarge di ff erence in the MSRD factors of non-equivalent atoms located in the 3rd (Cu a and Cu b ) and 7th (N a and N b ) shells. Strong increase of the MSRD of Cu a , Cu andCu b points to the anisotropic vibration of copper atoms in the direction orthogonal to–N–Cu–N– chains.
5. Conclusions
Atomistic simulation methods such as molecular dynamics and reverse Monte Carloprovide a natural way to include disorder (static and dynamic) into the EXAFS formal-ism taking into account multiple-scattering e ff ects.The two methods have several common points. In both cases, multiple absorptionedges can be easily simulated or fitted, thus improving the reliability of the accessiblestructural information. The analysis of EXAFS contributions from outer coordinationshells of the absorbing atom is feasible, which is rather challenging in conventionalapproach but provides an access to some useful structural and dynamic properties of amaterial as, for example, mean-square displacements.Opposite to conventional analysis, dealing with a set of structural parameters, MD-EXAFS and RMC-EXAFS approaches provide a result in terms of atomic configura-tions, giving information on atom-atom and bond-angle distributions and correlations.Moreover, an access to atomic coordinates makes it possible to distinguish contribu-tions of non-equivalent atom pairs with equal or close path lengths.At the same time, there are also several di ff erences between the two methods.The MD-EXAFS approach does not require any structural fitting parameters, andthe structural model of a material is uniquely defined by the results of the MD sim-ulation. The agreement between the experimental and calculated CA EXAFS spectradepends on the accuracy of interatomic potential model, therefore, EXAFS spectrum12an be used to validate the interatomic potentials.3D structure models obtained by the RMC method from experimental EXAFS canbe directly compared with the results of other atomistic simulations. Moreover, theycan be employed to include disorder e ff ects into first-principles simulations to pre-dict temperature dependent material properties. Note that constraints can be easilyincorporated into the RMC analysis to account for information from other experiments(di ff raction, total scattering, etc) or chemical / geometrical information (bond-lengths,bonding angles, coordination, energetics, etc). Acknowledgements
This work has been partially supported by the Latvian Council of Science projectno. lzp-2018 / References
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Figure 1: Upper panel: Calculated photoelectron mean free path (MFP) λ ( k ) for c-NiO and bcc W. Lowerpanel: Fourier transforms of the experimental W L -edge and Ni K-edge EXAFS spectra χ ( k ) k in bulk andnanosized NiO at T =
10 K and in bcc tungsten at T =
300 K, respectively. Ni O O Ni Ni O Ni O Ni U n i q u e p a t h s N u m b e r o f S ca tt e r i ng P a t h s Distance (Å)
T o t a l p a t h s Nyquist criterion for D k=20 Å -1 c-NiO O Figure 2: The dependence of the number of scattering paths on the cluster size around the absorbing nickelatom in NiO. (cid:121)(cid:882)(cid:396)(cid:258)(cid:455) (cid:4)(cid:271)(cid:400)(cid:381)(cid:396)(cid:393)(cid:410)(cid:349)(cid:381)(cid:374) (cid:121)(cid:4)(cid:69)(cid:28)(cid:94)(cid:3)(cid:3)(cid:3) (cid:40)(cid:59)(cid:36)(cid:41)(cid:54) (cid:36) (cid:69) (cid:86) (cid:82) (cid:85) (cid:83) (cid:87) (cid:76) (cid:82)(cid:81) (cid:59)(cid:16)(cid:85)(cid:68)(cid:92)(cid:3)(cid:40)(cid:81)(cid:72)(cid:85)(cid:74)(cid:92) (cid:12)(cid:11)(cid:12)(cid:11) (cid:70)(cid:68) kk lil (cid:70)(cid:70) (cid:32) (cid:94)(cid:258)(cid:373)(cid:393)(cid:367)(cid:286)(cid:94)(cid:455)(cid:374)(cid:272)(cid:346)(cid:396)(cid:381)(cid:410)(cid:396)(cid:381)(cid:374) (cid:38)(cid:258)(cid:272)(cid:349)(cid:367)(cid:349)(cid:410)(cid:455) (cid:349) (cid:1089)(cid:3)(cid:1005)(cid:3)(cid:3) (cid:1006)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:1007)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) (cid:374)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) (cid:857) (cid:94)(cid:410)(cid:396)(cid:437)(cid:272)(cid:410)(cid:437)(cid:396)(cid:258)(cid:367)(cid:3)(cid:68)(cid:381)(cid:282)(cid:286)(cid:367) (cid:894)(cid:400)(cid:349)(cid:460)(cid:286)(cid:853)(cid:3)(cid:400)(cid:346)(cid:258)(cid:393)(cid:286)(cid:853)(cid:3)(cid:282)(cid:286)(cid:296)(cid:286)(cid:272)(cid:410)(cid:400)(cid:853)(cid:857)(cid:895) (cid:68)(cid:437)(cid:367)(cid:410)(cid:349)(cid:393)(cid:367)(cid:286)(cid:882)(cid:400)(cid:272)(cid:258)(cid:410)(cid:410)(cid:286)(cid:396)(cid:349)(cid:374)(cid:336)(cid:410)(cid:346)(cid:286)(cid:381)(cid:396)(cid:455) (cid:47)(cid:374)(cid:410)(cid:286)(cid:396)(cid:258)(cid:272)(cid:410)(cid:349)(cid:381)(cid:374) (cid:68)(cid:381)(cid:282)(cid:286)(cid:367)(cid:47)(cid:374)(cid:410)(cid:286)(cid:396)(cid:258)(cid:272)(cid:410)(cid:349)(cid:381)(cid:374) (cid:68)(cid:381)(cid:282)(cid:286)(cid:367)(cid:47)(cid:374)(cid:410)(cid:286)(cid:396)(cid:258)(cid:272)(cid:410)(cid:349)(cid:381)(cid:374) (cid:68)(cid:381)(cid:282)(cid:286)(cid:367)(cid:90)(cid:286)(cid:448)(cid:286)(cid:396)(cid:400)(cid:286)(cid:3)(cid:68)(cid:381)(cid:374)(cid:410)(cid:286)(cid:3)(cid:18)(cid:258)(cid:396)(cid:367)(cid:381)(cid:3)(cid:94)(cid:349)(cid:373)(cid:437)(cid:367)(cid:258)(cid:410)(cid:349)(cid:381)(cid:374) (cid:28)(cid:454)(cid:393)(cid:286)(cid:396)(cid:349)(cid:373)(cid:286)(cid:374)(cid:410)(cid:94)(cid:349)(cid:373)(cid:437)(cid:367)(cid:258)(cid:410)(cid:349)(cid:381)(cid:374)(cid:400) (cid:857) (cid:1847) (cid:3404) (cid:3533)(cid:1847) (cid:4666)(cid:2869)(cid:4667) (cid:1870) (cid:3036) (cid:3397) (cid:3036) (cid:3533)(cid:1847) (cid:4666)(cid:2870)(cid:4667) (cid:1870) (cid:3036) (cid:481)(cid:1870) (cid:3037) (cid:3397) (cid:1710) (cid:3036)(cid:481)(cid:3037) (cid:68)(cid:286)(cid:410)(cid:396)(cid:381)(cid:393)(cid:381)(cid:367)(cid:349)(cid:400)(cid:3)(cid:258)(cid:367)(cid:336)(cid:381)(cid:396)(cid:349)(cid:410)(cid:346)(cid:373) (cid:2239) (cid:3404) (cid:3533) (cid:2257) (cid:2191)(cid:1805)(cid:1824)(cid:1816) (cid:3398) (cid:2257) (cid:2191)(cid:1803)(cid:1801)(cid:1812)(cid:1803) (cid:2779)(cid:2191) (cid:72) (cid:1005) (cid:72) (cid:1006) (cid:72) (cid:1007)(cid:47)(cid:296)(cid:3) (cid:72) (cid:374)(cid:286)(cid:449) (cid:1092)(cid:3) (cid:72) (cid:381)(cid:367)(cid:282) (cid:410)(cid:346)(cid:286)(cid:374)(cid:3)(cid:258)(cid:272)(cid:272)(cid:286)(cid:393)(cid:410)(cid:856)(cid:3)(cid:47)(cid:296)(cid:3) (cid:72) (cid:374)(cid:286)(cid:449) (cid:1093)(cid:3) (cid:72) (cid:381)(cid:367)(cid:282) (cid:410)(cid:346)(cid:286)(cid:374)(cid:3)(cid:258)(cid:272)(cid:272)(cid:286)(cid:393)(cid:410)(cid:3)(cid:449)(cid:349)(cid:410)(cid:346)(cid:3)(cid:410)(cid:346)(cid:286)(cid:3)(cid:393)(cid:396)(cid:381)(cid:271)(cid:258)(cid:271)(cid:349)(cid:367)(cid:349)(cid:410)(cid:455)(cid:3)(cid:286)(cid:454)(cid:393)(cid:894)(cid:882) (cid:39)(cid:72) (cid:876) (cid:52) (cid:895)(cid:856)(cid:3) (cid:52) (cid:349)(cid:400)(cid:3)(cid:410)(cid:346)(cid:286)(cid:3)(cid:400)(cid:272)(cid:258)(cid:367)(cid:349)(cid:374)(cid:336)(cid:3)(cid:296)(cid:258)(cid:272)(cid:410)(cid:381)(cid:396)(cid:856) (cid:18)(cid:381)(cid:374)(cid:296)(cid:349)(cid:336)(cid:437)(cid:396)(cid:258)(cid:410)(cid:349)(cid:381)(cid:374)(cid:882)(cid:258)(cid:448)(cid:286)(cid:396)(cid:258)(cid:336)(cid:286)(cid:282)(cid:3)(cid:28)(cid:121)(cid:4)(cid:38)(cid:94)(cid:68)(cid:381)(cid:367)(cid:286)(cid:272)(cid:437)(cid:367)(cid:258)(cid:396)(cid:3)(cid:24)(cid:455)(cid:374)(cid:258)(cid:373)(cid:349)(cid:272)(cid:400)(cid:3)(cid:94)(cid:349)(cid:373)(cid:437)(cid:367)(cid:258)(cid:410)(cid:349)(cid:381)(cid:374) (cid:894)(cid:69)(cid:115)(cid:100)(cid:853)(cid:3)(cid:69)(cid:393)(cid:100)(cid:853)(cid:3)(cid:69)(cid:115)(cid:28)(cid:895) Figure 3: Scheme of EXAFS analysis using reverse Monte Carlo and molecular dynamics methods. RMC MD bcc W T=300 K RD F G W - W ( R ) ( a t o m s / Å ) Distance R (Å)
RMC M S RD s ( Å ) Distance R (Å) MD Exper.
RMC-EXAFS
MD-EXAFSW L -edge in bcc tungstenT=300 K EX A F S c ( k ) k ( Å - ) Wavenumber k (Å -1 ) Figure 4: Upper panel: Comparison of the experimental and calculated by the RMC and MD methodsW L -edge EXAFS χ ( k ) k of bcc W at T =
300 K. Lower panel: Radial distribution functions (RDF) G W − W ( R ) obtained by RMC and MD simulations. Inset: Dependence of the MSRD σ − W ( R ) on distance.Two horizontal lines correspond to a sum of two MSDs of tungsten. Experiment q(Fe)=3.0 q(Fe)=1.71 q(Fe)=1.20 EX A F S c ( k ) k ( Å - ) Wavenumber k (Å -1 )Fe K-edge in FeF T=300 K
Experiment q(Fe)=3.0 q(Fe)=1.71 q(Fe)=1.20
Fe K-edge in FeF T=300 K FT c ( k ) k ( Å - ) Distance R (Å)
Figure 5: Comparison of the experimental and calculated Fe K-edge MD-EXAFS χ ( k ) k spectra and theirFourier transforms (FTs) (modulus and imaginary parts are shown) in FeF at T =
300 K. Only few spectracalculated for di ff erent e ff ective iron charges are shown for clarity. EX A F S c ( k ) k ( Å - ) Wavenumber k (Å -1 ) Cu Cu N N N N Cu Figure 6: Upper panel: Comparison of the experimental and calculated CA Cu K-edge EXAFS spectra ofCu N at T =
10 K. Dashed line shows the total MS contribution. Lower panel: Temperature dependenciesof the MSRD factors for selected Cu–N and Cu–Cu atom pairs in Cu N. Inset: The fragment of the Cu Nstructure. Coordination shells around the absorber Cu are labelled.are labelled.