# EXAFS spectroscopy: a powerful tool for the study of local vibrational dynamics

*Microstructures*2021;1:2021004.

## Abstract

Extended X-ray absorption fine structure (EXAFS) spectroscopy is an ideal technique for studying the local vibrational dynamics of materials due to its sensitivity to short-range order, correlation of atomic motion and anharmonicity. However, despite this, EXAFS is widely employed to investigate the local structure but its use in the study of local dynamics is far more limited. In this brief review, the potential of EXAFS as a vibrational probe is presented with the aim of promoting its application in the study of the local dynamics of solid-state materials.

## Keywords

*,*local dynamics

*,*thermal disorder

## INTRODUCTION

Knowledge of vibrational dynamics is fundamental for understanding the physical properties of materials. Various experimental and theoretical approaches are used to study vibrational dynamics in materials, ranging from Bragg diffraction to vibrational spectroscopy^{[1-6]}, or from *ab initio* calculations to molecular dynamics^{[7-9]}. From an experimental perspective, X-ray and neutron diffraction can give precise parameters that describe the dynamic (but also static) displacements of atoms in crystals, such as mean-square atomic displacements^{[10, 11]}. Inelastic neutron and X-ray scattering can be used to measure phonon dispersion curves ^{[12-14]}. Nuclear inelastic scattering, although not appropriate as with the latter two techniques in the measurement of dispersion curves, provides a direct measurement of the density of phonon states in shorter times^{[15]}. Vibrational spectroscopy, such as infrared or Raman spectroscopy, allows for the determination of the vibrational frequency of selected vibrational modes at the center of the Brillouin zone^{[5, 6, 16, 17]}.

Alternatively, extended X-ray absorption fine structure (EXAFS) spectroscopy^{[18]} provides both unique and complementary information compared to that obtained from the techniques described above. In fact, in addition to being sensitive to short-range order and atomic sites, EXAFS is particularly sensitive to the correlation of vibrational motion, both parallel and perpendicular to the bond direction, and anharmonicity^{[19-22]}. However, although these properties make EXAFS unique, it is largely employed as a structural probe rather than as a dynamic one.

In this brief review, the capabilities of EXAFS spectroscopy for the study of local dynamics are reported. The intention is to present, for non-specialists, an introduction to the use of EXAFS in studying the vibrational dynamics of materials. For readers who wish to deepen their knowledge of EXAFS, other more exhaustive references are available^{[23-25]}.

The review is organized as follows. Section 2 to control sequence contains a synthetic introduction to the theory of EXAFS and the cumulants method. Section 3 shows a comparison between EXAFS spectroscopy and Bragg diffraction. The relations connecting EXAFS parameters to atomic thermal displacements and the sensitivity of EXAFS to atomic correlation are also presented here. Section 4 shows the phenomenological relationship of EXAFS to anharmonicity, emphasizing the differences between the one-dimensional potential probed by EXAFS and the three-dimensional crystal potential. Section 5 provides some recommendations regarding the data analysis procedure to obtain more reliable results. Section 6 provides some examples of studies that show the potential of EXAFS in the study of local dynamics. Section 7 presents the conclusions of this review.

## FUNDAMENTALS OF EXAFS

The aim of an EXAFS experiment is to measure the X-ray absorption coefficient of a selected atomic species as a function of energy [Figure 1A]. After the absorption edge, the absorption coefficient ^{[25, 26]}. The region between the absorption edge and ^{[27]}. XANES is not considered here because it is beyond the scope of this review.

Figure 1. (A) X-ray absorption coefficient (measured at the Cu K-edge of metallic copper at room temperature) plotted as a function of the incident beam energy. (B) Schematic view of EXAFS mechanism.

The EXAFS signal is defined as:

where ^{[24, 28]}. The EXAFS signal is conveniently expressed as a function of the photoelectron wavenumber

Figure 2. k

The starting point of EXAFS theory^{[24, 25, 28]} is Fermi's golden rule within the dipole approximation, according to which the absorption coefficient can be written as:

where

where

However, thermal and structural disorder, including anharmonicity, can be better considered by removing the Gaussian approximation and by introducing a general one-dimensional distribution of distances between the absorbing and scattering atoms, ^{[25]}:

Note that, due to the spherical-wave nature of the EXAFS photoelectron and its limited mean free path

The integral in Equation (5) represents the structural part of the EXAFS formula and its logarithm can be developed in a Maclaurin series to obtain the final EXAFS formula, which is particularly suitable for the study of dynamics^{[25, 29, 30]}:

where the parameters

As stated above, the EXAFS cumulants

while the higher order cumulants,

The exact relationships between ^{[30]}. As detailed below, the interatomic distance

## EXAFS VS. DIFFRACTION FOR ASSESSING LOCAL DYNAMICS

EXAFS spectroscopy and X-ray (or neutron) diffraction are complementary techniques.

● Owing to the limited mean free path of the EXAFS photoelectron, EXAFS is sensitive to short-range order, typically up to

● Because the mean lifetime of the EXAFS photoelectron (*true* bond distance, while diffraction measures the difference between the average atomic positions, *apparent* bond distance.

● Finally, EXAFS measures the relative atomic vibrations, including the correlation of atomic motion, while diffraction only measures uncorrelated atomic vibrations from the atomic thermal factors included in the Rietveld analysis.

Let us now consider a pair of atoms, where the absorber atom of EXAFS and one of its neighbors are labeled as 0 and 1, respectively. Let

Figure 3. Instantaneous atomic displacements

and considering the projections of the relative displacement

the scalar distance

and thus its average value results in:

For harmonic displacements ^{[31]}:

The final term in Equation (14) is proportional to the atomic mean square relative displacement (MSRD) perpendicular to the bond direction, ^{[21, 32, 33]}. Quantitative examples are provided in the next sections.

The second cumulant of EXAFS, ^{[19, 30]}:

which can be developed as:

The first two terms on the right-hand side of Equation (16) are the atomic mean square displacements (MSDs) of the absorber and backscatterer atoms along the bond direction, respectively, which can be obtained from Bragg diffraction. The third and final term in Equation (16) is known as the displacement correlation function (DCF) and is a measure of the degree of correlation of vibrational motion along the bond direction. As a result, the stronger the in-phase vibrational motion, the smaller the parallel MSRD ^{[34, 35]}.

Finally, once the parallel and perpendicular MSRDs are known, we can determine the anisotropy of the relative thermal vibrations through the ratio:

which depends on the particular dynamic properties of the system. In the case of parallel-perpendicular isotropy, ^{[36]}. A higher value of ^{[37, 38]}. The different anisotropy between germanium and copper, which can be attributed to the contribution of transverse optical modes, cannot be detected by Bragg diffraction. Indeed, owing to the crystal symmetry of copper and germanium, the atomic MSDs measured by diffraction have to be isotropic in both cases.

In summary, EXAFS provides unique information on the local vibrational dynamics. This is because EXAFS measures the true bond distance, including the correlation of the atomic thermal vibrations, while Bragg diffraction measures the apparent bond distance, where the correlation is not included. The relationship between EXAFS and diffraction distances, given by Equation (14), allows us to determine the perpendicular MSRD of a given pair of atoms. The second cumulant of EXAFS, instead, allows us to determine the parallel MSRD by Equation (15) and, ultimately, the anisotropy of the relative thermal vibrations. The practical procedure for obtaining this information is presented in Section 5.

## ANHARMONICITY

The distribution ^{[39]}:

where *k* is the Boltzmann constant.

^{[40, 41]}:

where, indicating with

Note that, to the first order and in the classical approximation, the first and second cumulants grow linearly with temperature [Equations (19) and (20)], while the third and fourth cumulants grow, respectively, with the square and cube of the temperature [Equations (21) and (22)]. As an example, Figure 4 shows the first four cumulants of the first Se-Cd coordination shell of CdSe measured by EXAFS and calculated by classic molecular dynamics simulations^{[42, 43]}. Analogous results were found in other compounds, such as copper or gold [Figure 5]^{[36, 44]}. Equation (20) also shows the second-order anharmonic correction to ^{[22]}. In the case that anharmonic effects are negligible and

Figure 4. First four cumulants of the first Se-Cd coordination shell of CdSe measured by EXAFS (solid circles) and calculated by classic molecular dynamics simulations (open circles)^{[43]}. The dashed lines show the linear behavior for the first and second cumulants, in the classical limit, and the quadratic and cubic behavior for the third and fourth cumulants, respectively. © IOP Publishing. Reproduced with permission. All rights reserved.

Figure 5. Third (upper panel) and fourth (lower panel) cumulants measured by EXAFS for the first Cu-Cu coordination shell of copper. The continuous lines are the best fitting theoretical model with ^{[36]}. © APS Publishing. Reproduced with permission. All rights reserved.

However, in the treatment presented above, the effective pair potential ^{[30]}:

Therefore, even in a harmonic crystal, the one-dimensional real distribution

and the position of the maximum distribution thus obtaining:

Equations (26) and (27) correspond to Equations (14) and (15), respectively, because in the case of isotropy,

to include the shift with temperature of the effective pair potential. The bond thermal expansion ^{[38, 43]} and it seems to work satisfactorily [Figure 6]. However, its connection with Equation (14), rewritten as:

Figure 6. Comparison between bond thermal expansions ^{[38]}. © APS Publishing. Reproduced with permission. All rights reserved.

remains not entirely clear. Indeed, if the shift of the effective pair potential ^{[37]}, while for its outer shells and for other compounds^{[32, 36]}, the agreement completely fails, as shown in Figure 6. This is likely due to the fact that the shape of the effective pair potential ^{[45, 46]}, and in CdSe^{[42, 43]}. In addition, more refined treatments with a quantum approach do not lead to better results^{[47, 48]}. Although many works have been devoted to the study of the relationship between the effective pair potential and crystal potential^{[49-53]}, this problem remains open and is not simple to solve.

Finally, as a last observation, according to Equation (28), the transverse vibrations also produce an intrinsic asymmetry (here with ^{[30]}. Consequently, it can be concluded that, in general, the third cumulant of EXAFS is mostly related to the real anharmonicity of the crystal potential.

## DATA ANALYSIS STRATEGY

The aim of this section is to provide information regarding the data analysis procedure to obtain accurate EXAFS results for studying the local dynamics. It is assumed that the reader has knowledge of the standard EXAFS analysis procedure.

Equation (7) is employed to best fit the EXAFS signal and to determine the cumulants *vs*. temperature of a given coordination shell. In this equation, the scattering-path amplitude and phase shift, *ab initio* methods, typically using the FEFF code^{[54, 55]}. This step can be avoided by utilizing the "ratio method" to analyze the data of the first shell where multiple scattering can be neglected^{[29]}. The photoelectron mean free path

The latter two terms have to be handled with careful attention. Indeed, ^{[56, 57]}, proving the significant potential of EXAFS in the study of relative differences.

Firstly, all the experimental spectra have to be aligned in energy before the extraction of the EXAFS signals. This can be achieved through the reference spectra collected during the experiment, or, in the absence of these, through the spectrum at the lowest temperature, which can be used as reference. Obviously, in this second case, we must ensure that there is no shift in the edge position due to valence changes with temperature^{[27]}. The error in the spectral alignment should not exceed 0.1 eV. The EXAFS signals can now be extracted by taking care to adopt the same process for all spectra.

Secondly, since there is no reason that

Figure 7. EXAFS best-fit results obtained for the first Sc-F shell of scandium fluoride[65]. Panels A and B show, respectively, the temperature behavior of

Thirdly, EXAFS cumulants higher than the second order must be considered in the analysis when possible. Indeed, the third and fourth cumulants have to be used to include anharmonicity and to thus obtain accurate values of the first and second cumulants, respectively^{[44, 46, 58]}. This is particularly true for the interatomic distance expansion (^{[59]}. Neglection of the third cumulant leads to an underestimation of ^{[38, 43, 60, 61]}. This can be seen, for example, in Figure 6, where the anharmonic contribution

Figure 8. Bond thermal expansion

Finally, once the relative values ^{[62]}:

where

Equation (32) describes the thermal motion including the zero-point energy at 0 K and it can be used as the starting point to derive phenomenological models to account for the temperature dependence of the parallel MSRD. The first one is the correlated Debye model where only acoustic branches with linear dispersion are considered and the Brillouin zone is substituted by a Debye sphere^{[63, 64]}. An alternative model that is easier to use is the correlated Einstein model, which considers three-dimensional quantum harmonic oscillators with the same frequency ^{[49, 64]}. With this simple assumption, Equation (32) becomes:

where

The Einstein model Equation (33) can be utilized to recover the absolute values of the parallel MSRD, i.e., ^{[65]}.

Figure 9. Temperature dependence of

Similarly, the Einstein model can also be used to determine the absolute values of the perpendicular MSRD by fitting the temperature dependence of ^{[64]}:

Figure 10A shows an example comparison between the true bond expansion ^{[66]}.

Figure 10. (A) Comparison between true bond expansion

Figure 11. Resulting anisotropy of the relative thermal vibrations,

Although the correlated Einstein model is only a simple phenomenological model, it has the ability to reproduce satisfactorily the temperature behavior of ^{[67]}. This makes the Einstein model the most used model in this type of EXAFS analysis and is preferable to other models, such as the Debye model, due to its simplicity.

## APPLICATIONS

In this section, examples of studies are provided that show the capability of EXAFS in the study of local dynamics. Obviously, good quality EXAFS data are required to obtain reliable information on the local vibrational dynamics. This is particularly true for the perpendicular MSRD, which is far more complicated to obtain than the parallel MSRD.

### EXAFS as a benchmark for theoretical calculations

It is obvious that the local dynamics information obtained by EXAFS represents an excellent benchmark for theoretical calculations, because the EXAFS MSRDs depend not only on the frequency of the vibrational modes, but also on their eigenvectors. This is in addition to the EXAFS cumulants depending on the effective pair potential that is related to the real three-dimensional crystal potential. For example, EXAFS was recently used by Kuzmin and co-workers^{[68-70]} to test the accuracy of several force-field models employed in molecular dynamics simulations. Force-field models that give lattice parameters, bulk moduli and elastic constants in good agreement with experimental data can still fail to reproduce the temperature dependence of EXAFS data. Thus, EXAFS provides additional information that enables discrimination between different models.

Various theoretical calculations have been carried out to predict EXAFS parameters and to validate interatomic potential models. The temperature dependence of the EXAFS cumulants up to the third or fourth order was calculated by Yokoyama *et al*.^{[71]} for a Br*et al*.^{[72]} for fcc copper and the sensitivity to the choice of the interaction potential was tested, indicating the need to adopt a multi-body potential to satisfactorily reproduce the experimental EXAFS third cumulant [Figure 13]. Vila *et al*.^{[52]} introduced an *ab initio* equation of motion method to calculate the temperature dependence of the MSRDs of EXAFS and successfully it applied to germanium, Zn^{[73-78]}, nanoparticles^{[79-81]}, glasses^{[82-84]} and liquids^{[85-87]}.

Figure 13. Third EXAFS cumulant for the first shell of fcc copper: from EXAFS data (full circles) and from path-integral Monte Carlo calculations with a many-body potential (open circles) and a two-body potential (crossed squares). The dashed line is the best-fitting model of experimental data. Figure taken from Ref.^{[72]}. © APS Publishing. Reproduced with permission. All rights reserved.

Various theoretical calculations have been carried out to predict EXAFS parameters and to validate interatomic potential models. The temperature dependence of the EXAFS cumulants up to the third or fourth order was calculated by Yokoyama *et al*.^{[71]} for a Br*et al*.^{[72]} for fcc copper and the sensitivity to the choice of the interaction potential was tested, indicating the need to adopt a multi-body potential to satisfactorily reproduce the experimental EXAFS third cumulant [Figure 13]. Vila *et al*.^{[52]} introduced an *ab initio* equation of motion method to calculate the temperature dependence of the MSRDs of EXAFS and successfully it applied to germanium, Zn^{[73-78]}, nanoparticles^{[79-81]}, glasses^{[82-84]} and liquids^{[85-87]}.

### Nanoparticles

Detailed EXAFS studies have been conducted to explore the local dynamics of nanoparticle systems. In one of the earliest studies, Rockenberger *et al*.^{[88]} investigated the size dependence of the structural and dynamic properties of CdS nanoparticles in a size range from 1.3 to 12 nm. They observed that these properties are related to the surface-to-volume ratio and to the method of surface stabilization of the nanoparticles. A temperature-independent static contribution to the MSRD of the Cd-S bonds was identified. This static contribution is strongly size dependent. In contrast, the vibrational amplitudes of the Cd-S bonds are slightly reduced with decreasing particle size. The asymmetry of the interatomic Cd-S pair distribution, evaluated through the third EXAFS cumulant, is increased with reducing particle size and shows the possibility of distinguishing the crystal structure of CdS nanoparticles.

A decade later, EXAFS was used by Araujo *et al*.^{[89]} to study the vibrational properties of Ge nanocrystals (NCs). The EXAFS Debye-Waller factor revealed that the Ge-Ge bonds of Ge NCs are stiffer than both bulk crystalline and amorphous Ge, with a static contribution that lies between theirs [Figure 14]. The higher static disorder in Ge NCs with respect to the crystal was attributed to the presence of undercoordinated atoms on the surface and the resulting internal strain. Moreover, the bond thermal expansion obtained for Ge NCs is smaller than for crystalline Ge, in good agreement with existing data for other nanocrystalline semiconductors^{[90, 91]}. Somewhat different behavior was observed by Sprouster *et al*.^{[92]}, who investigated the size-dependent vibrational properties of Co nanoparticles by temperature-dependent EXAFS. Compared to bulk Co, the authors noted a softening of the Co-Co bonds in the smallest nanoparticles, attributed to the strong influence of undercoordinated bonds, as well as reductions in the average bond length and coordination number. Enhanced structural disorder and asymmetrical deviation from a Gaussian-like interatomic distance distribution with decreasing particle size were confirmed.

Figure 14. EXAFS Debye-Waller factor measured as a function of temperature in Ge nanocrystals (circles), bulk crystalline Ge (squares and down-triangles) and amorphous Ge (right-triangles), with the respective correlated Einstein model fits (dashed lines). The solid line is the Ge-Ge dynamic contribution in c-Ge from *ab initio* calculations^{[93]}. The reduced slope of Debye-Waller for Ge NCs means that the Ge-Ge bonds in Ge NCs are stiffer than crystalline and amorphous Ge. Moreover, Ge-Ge static disorder is observed in Ge NCs, lying between that of c-Ge and a-Ge. Figure taken from Ref.^{[89]}. © APS Publishing. Reproduced with permission. All rights reserved.

Furthermore, the vibrational properties of gold nanoparticles have also been subject to EXAFS studies. Comaschi *et al*.^{[94]} studied the thermal behavior of Au nanoparticles with sizes ranging from 2.4 to 5.0 nm. With respect to the bulk crystal counterpart, the authors observed for the first four shells a progressive reduction of the average bond length and coordination number with decreasing particle size, as well as a simultaneous increase in the static disorder. More interestingly, a reduction of the Au-Au bond thermal expansion was found, with a curious crossover from an initial thermal expansion to a thermal contraction in the smallest particles, tentatively interpreted as being due to the presence of discrete electronic energy levels induced by the finite size particles^{[95]}. Duan *et al*.^{[96]} confirmed the contraction of the average Au-Au bond length, the decrease of coordination number and the presence of static disorder in Au147 nanoparticles compared to the bulk material. However, they were able to observe a bond expansion for the surface-surface bonds, accompanied by a softening of the vibrational modes, while the surface-core and core-core bonds contracted. A recent review by Timoshenko *et al*.^{[97]} reports the latest advancements in the study of the structure and dynamics of metal nanoparticles by combining EXAFS spectroscopy and atomistic simulations.

A noteworthy EXAFS study of the local dynamics of nanoparticles was carried out by Hu *et al*.^{[98]} for scandium fluoride. Static disorder and a decrease in the average bond length were observed for both the Sc-F and Sc-Sc distances with respect to the bulk counterpart. However, the most important result to be noted was the large stiffening of the average Sc-F bond bending, which resulted in a reduction of the Sc-F anisotropy and negative thermal expansion (NTE). The use of EXAFS spectroscopy to investigate NTE materials represents one of the most successful fields of EXAFS applications for studying vibrational dynamics, as described in the next section.

### Negative thermal expansion materials

Thermal expansion represents a problem for many materials and engineering applications and its control is a challenge for materials design. After the discovery of materials that exhibit strong NTE over wide temperature ranges^{[99, 100]}, the interest in NTE has rapidly expanded and the prospect of controlling thermal expansion has become possible. Although it is known that NTE arises from a range of different mechanisms^{[101]}, the general comprehension of NTE (and related phenomena) is still incomplete. In this regard, EXAFS spectroscopy is of great help since it provides unique information on the local vibrational dynamics. Like EXAFS, the neutron pair distribution function (PDF) is sensitive to the atom-atom correlation, but the overlap of neighboring peaks in PDF analysis makes it extremely difficult to investigate the local dynamics of nearest neighboring bonds with sufficient accuracy.

To the best of our knowledge, the first EXAFS studies to investigate the local dynamics of NTE materials were performed on zirconium tungstate^{[102, 103]}, the most popular NTE material, and cuprite structures^{[35, 104, 105]}. For ZrW^{[106]}. As a result, a more flexible model, simply based on rigid nearest-neighbor bonds and a tension effect, has been adopted to explain the NTE of cuprite structures^{[107]}.

Figure 15. Bond thermal expansion (top panels) and parallel MSRDs (bottom panels) of the M-M second-shell distance in Cu^{[35]}). Closed and open symbols refer, respectively, to intra-tetrahedra (edges of M^{[35]}. © APS Publishing. Reproduced with permission. All rights reserved.

More recently, a number of crystals with the diamond-zincblende structure (Ge, CuCl, CdTe, InP and GaAs), affected by NTE at low temperatures, have been investigated by EXAFS^{[21, 32, 33, 37, 108]}. These studies showed that a correlation exists between the strength of NTE, the degree of bond ionicity and some of the parameters determined from EXAFS analysis, such as the anisotropy of the relative vibrations of nearest-neighbor pairs.

Direct experimental evidence of the NTE mechanism in ScF^{[65]}. A positive expansion of the first-shell Sc-F bond distance and a concomitant shrinkage of the second-shell Sc-Sc distance were observed. This shows the existence of large vibrations of fluorine atoms perpendicular to the Sc-F-Sc linkage, indicating that the Sc-F bond is much softer to bend than to stretch, and that the relative motion of the Sc-Sc pair is more pronounced along the bond direction [Figure 16]. Although the latter effect might be too emphasized due to the large error bars in the Sc-Sc perpendicular MSRD, the combination of these findings indicates the presence of a "guitar-string" effect^{[109]} as the local mechanism responsible for NTE in ScF^{[110]}. The EXAFS study of Purans *et al*.^{[111]} on ReO^{[111]}.

Figure 16. (A) Parallel MSRDs, (B) perpendicular MSRDs and (C) anisotropy of the relative thermal vibrations for the Sc-F and Sc-Sc atomic pairs determined by EXAFS in ScF^{[65]}. © ACS Publishing. Reproduced with permission. All rights reserved.

EXAFS was also employed to study the local dynamics of Prussian blue analogues (PBAs), an important family of multifunctional materials with applications in catalysis, energy storage, magnetism, drug delivery and so on^{[112]}. In particular, some PBAs display NTE properties^{[113]}. The chemical composition MM'(CN)^{[114-116]}.

Figure 17. Parallel and perpendicular MSRDs for the Ga-N and Fe-C bonds [panels (A) and (B), respectively] of GaFe(CN)^{[114]}. © ACS Publishing. Reproduced with permission. All rights reserved.

Very interesting is the fact that the insertion of guest ions or molecules in PBA structures can inhibit NTE, thereby opening the possibility of controlling the thermal expansion of materials. For examples, the incorporation of K^{[119]}, while the progressive insertion of Na^{[117]}. In addition, the intercalation of H^{[118]}. EXAFS allows us to understand the local mechanism at the origin of such thermal expansion behavior, i.e., the presence of guest ions or molecules hinders the transverse atomic vibrations, thus inhibiting the NTE. This effect is much more marked for the M-N bonds than for the M'-C bonds. This is clearly evident from Figure 18, where the EXAFS results for both Na: GaFe(CN)

Figure 18. Anisotropy of relative thermal vibrations determined by EXAFS in Na: GaFe(CN)^{[117, 118]}. © ACS Publishing. Reproduced with permission. All rights reserved.

### Superconducting materials

EXAFS was used to study superconducting materials. Lanzara *et al*.^{[120]} studied the local atomic correlations in the HgBa

Figure 19. Temperature dependence of the EXAFS Debye-Waller factors (MSRDs) for the Cu-O and Cu-Ba atomic pairs in HgBa^{[120]}. © APS Publishing. Reproduced with permission. All rights reserved.

More recently, Joseph *et al*.^{[121]} studied the iron-oxypnictide superconductor system as a function of temperature. EXAFS showed a weak temperature and F-substitution dependence of the Fe-As bond length, in agreement with the strong covalent nature of this bond. More importantly, the temperature dependence of the Fe-As MSRDs was found to follow the correlated Einstein model, but with different Einstein frequencies for the superconducting and non-superconducting samples, with a hardening of the Fe-As bond in the former compared to the latter. This finding shows the non-negligible role of the lattice vibrational modes in the superconductivity of pnictide superconductors. This conclusion was supported by Chu *et al*.^{[122]}, who studied the Fe isotopic effect in the (Ba, K)Fe*et al*.^{[123]} investigated the local atomic displacements in the new Li^{[124-126]}.

### Other applications

EXAFS allows for the exploration of a wide range of topics in condensed-matter physics and materials science. Before concluding, some more examples of EXAFS applications in studying local dynamics are presented.

EXAFS is widely used to study the structural and dynamic aspects of magnetic, ferroelectric and multiferroic materials, as well as of glasses and amorphous systems. Yokoyama *et al*.^{[127]} studied the spin-crossover transition of a chain Fe(II) complex by temperature-dependent EXAFS and found an abrupt change in the first-shell Fe-N bond distance and the Debye-Waller factor at the transition temperature. The variation of the effective force constant for the Fe-N stretching vibration revealed a significant weakening of the Fe-N bond in the high-spin state. Later, the same authors investigated the anharmonicity and quantum effects in the Invar alloy Fe^{[128]}. The first nearest-neighbor shells around Fe and Ni both show a suppression of the bond thermal expansion, although the anharmonicity, with important quantum effects at low temperature, is clearly evident. Panchwanee *et al*.^{[129]} explored the local modifications associated with the spin-reorientation transition in DyFeO

Fischer *et al*.^{[130]} detected the antiferrodistortive transition in perovskite SrTiO*et al*.^{[131]} studied the local vibrational dynamics of hematite *et al*.^{[132]} studied the short-range order around Pb in PbTiO*et al*.^{[133]} investigated the first oxygen coordination shell at the manganese site in mixed-valence perovskite manganites as a function of temperature. In the metallic-ferromagnetic phase, manganese is surrounded by regular oxygen octahedra, but above the magnetic transition, the EXAFS Debye-Waller factor shows an increased disorder for the Mn-O distances. Since no structural distortion was detected by the diffraction measurements, this disorder must be dynamic, thus supporting the presence of a large phonon-electronic interaction.

Regarding glass and amorphous systems, in one of the pioneering works, Yang *et al*.^{[134]} used temperature-dependent EXAFS to provide information about the nature and strength of local bonds of arsenic and arsenic-chalcogen compounds. They found that the As-As bonds in c-As are weaker than those in a-As, while the stretching forces of the As-S bonds in crystalline and glass As*et al*.^{[135]} found a softening of the Mo-O bonds in AgI: Ag^{[136]}. Siqueira *et al*.^{[137]} investigated the structural and thermal properties in an amorphous GaSe*et al*.^{[138]} studied amorphous mesoporous titania with EXAFS to clarify the characteristics of the Ti-O chemical bonds. The coexistence of 5- and 6-fold coordinated Ti was demonstrated and the EXAFS Debye-Waller factor suggested that the pores filled with template micelles suppress the thermal vibrations of Ti-O bonds.

The examples given in this section are clearly not exhaustive and other EXAFS studies of local dynamics can be found in the literature.

## CONCLUSION

This paper has presented a brief review on the potential of EXAFS for the study of the local vibrational dynamics in materials. The strength of EXAFS is the selectivity of the atomic species and the sensitivity to short-range order and correlation of atomic motion.

The EXAFS approach, based on the cumulant expansion, gives information on the distribution of distances between the atoms of the absorbing species and their nearest neighbors, including the deviations from a Gaussian shape. In addition to giving information on the distance distribution and the anharmonicity of the interatomic potential, the EXAFS cumulants provide information on the local vibrational dynamics. Specifically, the first cumulant, benefitting from the difference between EXAFS and diffraction, leads to the evaluation of the perpendicular MSRD, while the second cumulant corresponds, in good approximation, to the parallel MSRD. The anisotropy of the relative thermal vibrations can thus be determined by EXAFS.

Some information has been provided regarding the data analysis strategy in order to improve the EXAFS results. In particular, the binding energy

Finally, in order to show the capability of EXAFS to study the local dynamics, several examples of application have been reported, ranging over a wide array of topics. This work encourages the use of EXAFS spectroscopy as a dynamic probe and not only as a structural probe.

## DECLARATIONS

### Authors' contributions

The author contributed solely to the article.

### Availability of data and materials

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### Financial support and sponsorship

None.

### Conflicts of interest

The author declared that there are no conflicts of interest.

### Ethical approval and consent to participate

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### Consent for publication

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### Copyright

© The Author (s) 2021.

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Sanson A. EXAFS spectroscopy: a powerful tool for the study of local vibrational dynamics. *Microstructures* 2021;1:2021004. http://dx.doi.org/10.20517/microstructures.2021.03

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Sanson A. EXAFS spectroscopy: a powerful tool for the study of local vibrational dynamics. *Microstructures*. 2021; 1(1):2021004. http://dx.doi.org/10.20517/microstructures.2021.03

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