#
Anomalous Behavior in the Atomic Structure of Nb_{3}Sn under High Pressure

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{*}

## Abstract

**:**

_{3}Sn superconductor sample has been probed by X-ray absorption fine structure (XAFS) as a function of hydrostatic pressure (from ambient up to 26 GPa) using a diamond anvil cell set-up. The analysis of the Nb-K edge extended X-ray absorption fine structure (EXAFS) data was carried out combining standard multi shell structural refinement and reverse Monte Carlo method to provide detailed in situ characterization of the pressure-induced evolution of the Nb local structure in Nb

_{3}Sn. The results highlight a complex evolution of Nb chains at the local atomic scale, with a peculiar correlated displacement of Nb–Nb and Nb–Nb–Nb configurations. Such a local effect appears related to anomalies evidenced by X-ray diffraction in other superconductors belonging to the same A15 crystallographic structure.

## 1. Introduction

_{3}Sn is a brittle intermetallic material belonging to the class of the A15 compounds (space group $Pm$-$3n$) that in 1954 reached a superconducting transition temperature of T

_{c}≈ 18 K [1]. Even more important, is its capability to carry high current densities ${J}_{c}$ > 10

^{3}A/mm

^{2}[2], allowing to make compact magnets reaching high critical fields ${B}_{c2}$ up to 30 T [3,4], essential for high field superconductor applications. These properties contributed to make Nb

_{3}Sn the most widely used high field superconductor in top science projects (CERN High luminosity LHC project [5], ITER [6] project) and industrial applications (NMR instruments, compact cyclotrons), with a record production in the period 2009–2014 of 150 Tons/Year for the ITER toroidal field magnets only [7]. The exceptional requirements of these magnet-based projects revamped the interest in this material and considerable efforts are now undertaken to further improve their critical performances and efficiencies during the applications [8,9,10]. In particular, the effects of strains (axial, transverse, hydrostatic) on ${J}_{c}$, ${T}_{c}$ and the electrical resistivity, which may be caused by thermal contractions and strong Lorentz forces due to the high currents, were extensively explored [4,9,11,12,13,14,15]. However, less is known about the structural modifications induced by pressure, especially on the crystallographic and atomic scale [15,16,17]. These informations are crucial for achieving accurate model of the Nb

_{3}Sn properties. As a matter of fact, density functional theory (DFT) calculations have shown that squeezing the structure actually affects the phonon spectra and electronic density of states $N\left({E}_{f}\right)$ of Nb

_{3}Sn [15]. However, accurate models require deep knowledge about the atomic structure at the crystallographic and atomic (local) scale.

_{3}Sn wires, showed an anomaly in the unit cell volume compression (around 5 GPa) not associated with any structural phase transition. A similar effect, ascribed to isostructural transitions, was observed around 15 GPa for other A15 systems, e.g., Nb

_{3}Ga [18] and Nb

_{3}Al [19]. Whereas, low temperature (T = 10 K) HP-XRD measurements, performed on a slightly non stoichiometric Nb

_{3}Sn

_{1−x}single crystal, have evidenced structural instabilities around 3 GPa, suggesting a dimerization of Nb–Nb chains, providing alternating shorter/longer Nb–Nb pairs [17]. Recent simulations on Nb

_{3}Al suggested the $Pm$-$3n$ phase to be energetically not favoured with respect to the C

_{2}/C [20] phase, where the lower symmetry allows for distortions of the Nb chains.

_{3}Sn are quite rare, even at ambient pressure/temperature conditions [22,23].

_{3}Sn between ambient pressure (AP) and 26 GPa. XAFS data analysis was carried out combining well established multi-shell data refinement [24] and Reverse Monte Carlo (RMC) approach [25]. In this way, it was possible to build a 3D model of Nb

_{3}Sn structural details around the Nb sites, and to extract further information on the pressure-dependence of the relative arrangements of Nb neighbors (many-body distribution functions) in the structure.

## 2. Materials and Methods

^{2}using the micro-XAS facility the data of which were collected in transmission mode using two gas filled ionization chambers to measure both the incident (${I}_{o}$) and transmitted (${I}_{t}$) X-ray intensities. For sake of comparison, additional Nb K-edge XAS spectra were measured at the P65 beamline (Petra-III, DESY synchrotron radiation facility in Hamburg, Germany) [27] on powder of the same sample, in standard transmission geometry at ambient conditions.

_{3}Sn samples were obtained from a polycrystalline bulk piece sintered by Hot Isostatic Pressure (HIP) technique (2 kbar Argon pressure at 1250 °C for 24 h) at the University of Geneva [30]. A grain size of about 20 μm and a composition very close to stoichiometry (24.8 at.% Sn) were determined from SEM/EDS analysis. A sharp superconducting transition was observed at 17.9 K by AC susceptibility. Finally, a Rietveld refinement yielded a lattice constant at ambient conditions of 5.291 Å and a Bragg–Williams [31] long-range order parameter S of 0.98. Further details on the synthesis procedure and preliminary characterization can be found in Ref. [32]. Samples from the same batch were characterized by means of microwave measurements [33], yielding a normal state resistivity ${\rho}_{n}\simeq $ 14.8 $\mathsf{\mu}\Omega $·$\mathrm{cm}$, a critical temperature T

_{c}= 17.8 K and an extrapolated upper critical field ${H}_{c2}$, giving $\mu {H}_{c2}\left(0\right)$ of ≃ 27 T [34], in full agreement with the accepted values [4] for pure Nb

_{3}Sn. The sample for the XAS measurements was prepared by grinding a piece of Nb

_{3}Sn bulk in an agate mortar to obtain a fine powder. The latter was then squeezed between two diamond anvils in order to obtain a thin and homogeneous pellet which was cut and loaded in the DAC’s high pressure chamber. A ruby chip was placed a few μm away from the sample and used as pressure gauge. Once the good quality of the sample loading was checked by means of X-rays (providing an absorption jump of 0.3 at the Nb K-edge), the high pressure chamber of the DAC was filled with Ne gas to ensure good hydrostatic conditions for the investigated pressure range [35]. During the experiment, the pressure inside the DAC was measured, before and after each energy scan, using the ruby fluorescence method [36], following the calibration of Dorogokupets and Oganov [37]. High pressure measurements (HP) were carried out at room temperature at seven pressure values in the range between 0.3 GPa and 26 GPa. In order to get a good statistics, the data at each pressure point was obtained by collecting at least four energy spectra, which were checked for energy scale alignment and then averaged.

#### 2.1. Standard EXAFS Data Analysis

^{−1}k-range to the model function ${k}^{2}{\chi}_{th}$. The non-linear least-square data refinement and error analysis has been implemented using the MINUIT routine package [40]. The theoretical EXAFS function is given by a sum of partial contributions (coordination shells) ${\chi}_{th}={\sum}_{i}{\chi}_{i}$, each ${\chi}_{i}$ being calculated from the standard EXAFS formula [38]:

_{3}Sn structure within the $Pm$-$3n$ space group [43]. The squeezing of the lattice parameters in the investigated pressure range [15] is less than 5% and it is expected to have a weak effect on the ${A}_{i}$, ${\varphi}_{i}$ and ${\lambda}_{i}$; therefore they were not recalculated as a function of pressure.

_{3}Sn. The adopted unit cell schematized in Figure 2 was cubic (edge length a). The Sn atoms were located at the cube corners and center, forming a body-centred cubic ($bcc$) lattice, while the Nb atoms were arranged in pairs along chains parallel to the x, y, and z axis, at the center of each face of the cube.

_{0}absorber in the A15 structure (Table 1), namely the two Nb

_{I}along the Nb chain ($a/2$ away from Nb

_{0}), the four Sn

_{I}at the close cube corners and centers of adjacent cubes ($a\sqrt{5}/4$ away) and the eight Nb

_{II}on the close faces ($a\sqrt{6}/4$ away). The features in the FT in the 3–6 Å region derived from single (SS) and multiple scattering (MS) contributions in the more distant region. For the analysis the SS and MS were selected on the basis of their amplitude and statistical significance in the fitting. In particular, the intense MS paths Nb

_{0}-Nb

_{I}-Nb

_{IV}and Nb-Sn

_{I}-Nb

_{V}, being enhanced by collinear arrangements (forward and double forward scattering) were considered in the fitting model. For sake of clarity, the neighboring shells assignment used for the EXAFS data analysis is resumed in Table 1, giving the atomic position relative to the generic Nb

_{0}absorber at (1/4, 0, 1/2) in the unit cell.

_{I}) and the second (Sn

_{I}) shells.

#### 2.2. RMC Data Analysis

^{−1}and R between 0–6 Å. The Nb

_{3}Sn crystallographic structure from Ref. [43] was used as the starting configuration for a population of 32 supercells, each one consisting of 3 × 3 × 3 cells with periodic boundary conditions, containing a total of 216 atoms (162 Nb + 54 Sn) per supercell. The theoretical ${\chi}_{CA}\left(k\right)$ EXAFS spectra were calculated considering MS paths with up to 4 scattering legs with 6 Å maximum length. At each RMC iteration, new supercell configurations were generated by randomly displacing all the atoms of the simulation box by a maximum displacement of 0.2 Å. The simulated annealing approach was used to efficiently reach the global minimum. The acceptance ratio of the new atomic configuration was not fixed but it decreased slowly following a cooling scheme. The length of the cooling scheme, set by the number of iterations after which only the atomic displacements improving the agreement between experiment and theory were accepted, was set to 1500. Figure 3c,d show an example of the ${k}^{2}$-weight multi-shell data fit and RMC best fit in the WT space, obtained for the HP data at 23 GPa. The ${k}^{2}$-weighted configuration averaged curve ${k}^{2}{\chi}_{CA}\left(k\right)$ and its Fourier transform modulus, as obtained from the EvAX output for the HP data at at 23 GPa, are shown in the Supporting Material.

## 3. Results and Discussion

_{3}Sn wires [15]. Noticeably, the Nb–Nb nearest neighbor distance ${R}_{N{b}_{I}}$ is significantly larger than $a/2$ at low pressures but converges to $a/2$ with increasing pressure. Interestingly, the values of $a/2$ and ${R}_{N{b}_{I}}$ found at 26 GPa coincide with the half of the lattice parameter obtained by HP-XRD at the same pressure, on Nb

_{3}Sn powders from technological wire sample [15]. Such a behavior suggests a general anticorrelated displacement of the Nb–Nb neighbors perpendicularly to the average Nb chain directions at ambient conditions (as schematized in the inset of Figure 4a). The application of an external pressure acts against this anticorrelation, by increasing the alignment of the Nb–Nb bonds with the cell axis. The average tilting angle of the Nb–Nb bond with respect to the Nb chain, calculated assuming a perpendicular anticorrelated displacement of the Nb–Nb pairs, decreases from 6.5(5)° at 0.3 GPa to 3(1)° at 26 GPa.

_{I}shell [41]. However, we found the correlations between the first cumulant ${C}_{1}$ (distance) and the third one ${C}_{3}$ (skewness), taking into account the asymmetry of the distribution, to be higher than 90%. The same is true for the correlations between ${C}_{2}$, being the MSRD, and ${C}_{4}$ (Kurtosis) describing the tailedness of the distribution. This gave large uncertainties on the parameters making the results less reliable. Therefore, we did not use the cumulant expansion in the standard analysis but exploited the RMC analysis to obtain deeper structural details.

_{I}(panel a) and Sn

_{I}(panel b) pair distribution functions as obtained from the RMC models. The Nb

_{0}and Sn

_{I}distributions are broad and asymmetric at low pressures. Raising the pressure makes the distributions narrower and shortens the interatomic distances, visually showing the “squeezing” and the overall ordering of the structure. To obtain quantitative information about the local atomic structure around Nb atoms, we calculated the parameters characterizing the neighboring distributions (mean values, variances, higher moments of the distributions) as a function of pressure directly from the RMC structural models (see Figure 5c). The results obtained by RMC refinement of the EXAFS data measured at 0.3 GPa in DAC were fully consistent with those independently measured at AP on a standard set-up. This reinforces the confidence on the reproducibility of the data and the reliability of RMC analysis on independent data sets. However, from this data it is evident that at low pressures, the Nb

_{I}distances, $a/2$ and the MSRD parameters calculated from the RMC atomic models differed from those obtained using the standard EXAFS analysis (Figure 5c, Distances and ${\sigma}^{2}$).

_{I}shell. During the fit, two ${R}_{N{b}_{I}}(a,b)$ distances were refined with the same multiplicity numbers ($N(a,b)=1$) and MSRD (${\sigma}_{N{b}_{I}}^{2}\left(a\right)={\sigma}_{N{b}_{I}}^{2}\left(b\right)$). The obtained fit is shown in Figure 7 and the corresponding results are reported in Table 3 along with the square residual function ${R}_{w}^{2}$, a statistical indicator representing the best fit quality [38]. The ${R}_{N{b}_{I}}(a,b)$ distances (Table 3) matched well those in Figure 5a) and ${\sigma}_{N{b}_{I}}^{2}(a,b)$ were half of the MSRD of the single shell model. This demonstrated the consistency between RMC and the standard EXAFS analysis and reinforces the reliability of the analysis procedures. It is important to notice that, as the double shells model slightly improved the fitting by about 5%, the number of free parameters increased by one. To evaluate the statistical significance of the best fit improvement we evaluated the associated Fisher F function [57]:

_{I}shell, ${R}_{{w}_{1}}^{2}$ and ${R}_{{w}_{2}}^{2}$ the corresponding square residual functions and ${N}_{i}\simeq 51$ is the number of independent experimental points in the fit [57]. The experimental $F\simeq 1.7$ corresponded to a p-value $\simeq 0.2$, which established that the double shell model was not statistically justified in the standard analysis despite the improvement of the residual function.

_{I}distribution were calculated directly from the 3D atomic models (Figure 5c—Skewness and Kurtosis). At ambient pressure, the Nb

_{I}pair distribution is strongly asymmetric (Skewness $\approx -1$), with a broad tail at low R. Under applied pressures (up to 20 GPa), the asymmetry slightly increased (Skewness absolute value), then it suddenly decreased and vanished at 26 GPa. The pressure-induced evolution of the Kurtosis showed an initial negative value at AP and 0.3 GPa. When raising the pressure, its value increased systematically up to 23 GPa then suddenly dropped down once 26 GPa are reached. These findings established a complex pressure-induced evolution of the Nb–Nb pair distribution functions and pointed out an anomaly above 23 GPa. Noticeably, no structural changes were found by XRD in the same pressure range, underlining the local nature of these changes.

_{3}Sn (technological wires) have been reported [15].

_{c}-Nb isosceles triangles (${R}_{1}\approx {R}_{2}$) with Nb

_{c}randomly displaced, preferentially in the direction perpendicular to the Nb chains (Figure 8c, top scheme). ii. The other fraction (highlighted in green), was associated to the asymmetric arrangement of Nb

_{c}(with ${R}_{1}<{R}_{2}$ or ${R}_{2}<{R}_{1}$) where the Nb

_{c}was randomly displaced, preferentially in the direction parallel to the Nb chains, thus providing a local dimerization of the Nb–Nb bonds (Figure 8c, bottom scheme). The results of such a simulation are shown for the 14 GPa analysis in Figure 8c (center panel) for sake of qualitative comparison. Under compression, the fraction of anticorrelated configurations decreased also reducing $\Delta R=|{R}_{2}-{R}_{1}|$. The fraction of dimerized configurations disappeared at 23 GPa. Upon further compression (up to 26 GPa) the R

_{2}vs. R

_{1}distribution results were more squeezed. In particular, the value of the average distribution of ${R}_{N{b}_{I}}$ became very close to $a/2$ (Figure 4 and Figure 5). Interestingly, while a fraction of the configurations was localized at the points ${R}_{1}\sim {R}_{2}$, at such a pressure a novel ${R}_{2}\text{}vs\text{}{R}_{1}$ correlation mode was established, where shorter (around 2.5 Å) and sharper R

_{2}vs. R

_{1}) distances were associated to longer (around 2.6 Å) and wider ${R}_{2}$ (${R}_{1}$) distances, corresponding to the bimodal ${R}_{N{b}_{I}}$ distribution highlighted in Figure 5.

_{3}Sn as a function of pressure. In particular, the EXAFS data analysis showed an anticorrelated displacement of Nb atoms perpendicularly to the Nb chains which decreased when raising the applied pressure. Our results also suggested the loss of the cubic ($Pm$-$3n$) symmetry at the local scale. Noticeably, recent ab initio simulations [58] showed that lower symmetry structures were energetically favoured with respect to the $Pm$-$3n$ in other A15 systems such as Nb

_{3}Al. In particular, the ${C}_{2}/c$ symmetry allowing anticorrelated Nb atom displacements perpendicularly to the Nb chains, was energetically favored with respect to the $Pm$-$3n$ one at ambient conditions and in a wide range of pressures. This work suggested that a similar behavior is observed in Nb

_{3}Sn at the local scale.

_{3}Sn structure thanks to the direct access to the 3D atomistic models. The analysis of the Nb

_{I}nearest neighbors distributions pointed out non-Gaussian contributions (Figure 5), in particular, the negative Skewness ($\sim -1$) being associated to a fractions of closer Nb–Nb pairs. The Kurtosis parameter increased with raising pressure. Raising the pressure above 23 GPa sudden reduced the non-Gaussian contributions. This behavior is better understood looking at the parameters defining the three body distribution function ${g}^{\left(3\right)}$ of the the Nb–Nb–Nb chains (Figure 8). The RMC atomistic model suggested a bimodal behavior for the Nb–Nb–Nb configurations along the Nb chains: some of them depicted the symmetric displacement of the central Nb perpendicularly to the chain (isosceles triangles) coherently with the anticorrelated Nb–Nb displacement discussed above. Nevertheless a fraction of the Nb–Nb–Nb triangles had R

_{1}≠ R

_{2}, with $\Delta R/R\sim 9\%$. These scalene triangles produced the partial dimerization of the Nb chains. It is noticeable that recent single crystal HP-XRD measurements performed at low temperatures (10 K) [17] on a slightly non-stoichiometric Nb

_{3}Sn

_{1−x}demonstrated a pressure induced symmetry lowering to the $P{4}_{2}/mmc$ space group at around 3 GPa where some of the Nb–Nb chains were dimerized in agreement with the present RMC EXAFS analysis. Our results coherently recognized that the anticorrelated displacement along the Nb–Nb–Nb chains was locally stabilized in that low pressure room temperature region. Finally, our EXAFS analysis showed that raising the pressure up to 26 GPa suppressed most of the local disorder.

## 4. Conclusions

_{3}Sn has been characterized in situ between ambient pressure and 26 GPa by XAS. The Nb K-edge EXAFS data were analysed using a standard multi-shell method, based on small Gaussian disorder model and a RMC method, providing a 3D atomic structural model. The results obtained from analysis carried out independently on the same sample, using different experimental set-up (AP and 0.3 GPa) are completely consistent. This demonstrates the reproducibility of the adopted procedures. The results of AP multishell EXAFS data analysis are in good agreement with previous literature [23]. The results obtained with RMC and the standard EXAFS analysis differ at low pressure, likely due to the different sensitivity of the two data analysis techniques to the structural disorder, and the presence of alarge non-Gaussian contributions. In fact, they become fully consistent at high pressures, where the sharper radial distributions are better described by the small Gaussian disorder model at the basis of the standard EXAFS formula. This strengthens the consistency between standard and RMC EXAFS analysis.

_{3}Sn [59,60,61]. The present results represent a new knowledge, relevant for providing accurate physical models in view of a better understanding of the superconductive properties, thus helping to individuate a route towards improving their critical properties.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AP | Ambient Pressure |

DAC | Diamond Anvil Cell |

DFT | Density Functional Theory |

DOS | Density of State |

EA | Evolutionary Algorithm |

EXAFS | Extended X-ray Absorption Fine Structures |

FT | Fourier Transform |

HP | High Pressure |

MS | Multiple Scattering |

MSRD | Mean square relative displacement |

RMC | Reverse Monte Carlo |

SS | Single Scattering |

WT | Morlet wavelet transform space |

XAFS | X-ray Absorption Fine Structures |

XAS | X-ray Absorption Spectroscopy |

XRD | X-ray Diffraction |

## References

- Matthias, B.T.; Geballe, T.H.; Geller, S.; Corenzwit, E. Superconductivity of Nb
_{3}Sn. Phys. Rev.**1954**, 95, 1435. [Google Scholar] [CrossRef] - Kunzler, J.E.; Buehler, E.; Hsu, F.S.L.; Wernick, J.H. Superconductivity in Nb
_{3}Sn at High Current Density in a Magnetic Field of 88 kgauss. Phys. Rev. Lett.**1961**, 6, 89–91. [Google Scholar] [CrossRef] - Parrell, J.A.; Field, M.B.; Zhang, Y.; Hong, S. Advances in Nb
_{3}Sn strand for fusion and particle accelerator applications. IEEE Trans. Appl. Supercond.**2005**, 15, 1200–1204. [Google Scholar] [CrossRef] - Godeke, A. A review of the properties of Nb
_{3}Sn and their variation with A15 composition, morphology and strain rate. Supercond. Sci. Technol.**2006**, 19, R68. [Google Scholar] [CrossRef][Green Version] - Bottura, L.; Rijk, G.; Rossi, L.; Todesco, E. Advanced accelerator magnets for upgrading the LHC. IEEE Trans. Appl. Supercond.
**2012**, 22, 4002008. [Google Scholar] [CrossRef][Green Version] - Vostner, A.; Salpietro, E. Enhanced critical current densities in Nb
_{3}Sn superconductors for large magnets. Supercond. Sci. Technol.**2006**, 19, S90. [Google Scholar] [CrossRef] - ITER Home Page. Available online: http://https://www.iter.org/factsfigures (accessed on 1 February 2021).
- Zhang, R.; Gao, P.; Wang, X.; Zhou, Y. First-principle study on elastic and superconducting properties of Nb
_{3}Sn and Nb_{3}Al under hydrostatic pressure. AIP Adv.**2015**, 5, 107233. [Google Scholar] [CrossRef][Green Version] - Ren, Z.; Gamperle, L.; Fete, A.; Senatore, C.; Jaccard, D. Evolution of T
^{2}resistivity and superconductivity in Nb_{3}Sn under pressure. Phys. Rev. B**2017**, 95, 184503. [Google Scholar] [CrossRef] - Quiao, L.; He, Y.; Wang, H.; Shi, Z.; Li, Z.; Xiao, G.; Yang, L. Effect on grain boundary deformation on the critical temperature degradation of superconducting Nb
_{3}Sn under hydrostatic pressure. J. Alloys Compd.**2021**, 864, 158116. [Google Scholar] [CrossRef] - Nishijima, G.; Watanabe, K.; Araya, T.; Katagiri, K.; Kasaba, K.; Miyoshi, K. Effect of transverse compressive stress on internal reinforced Nb
_{3}Sn superconducting wires and coils. Cryogenics**2005**, 45, 653–658. [Google Scholar] [CrossRef] - Lu, J.; Han, K.; Walsh, R.P.; Miller, J.R. I
_{C}Axial Strain Dependence of High Current Density Nb_{3}Sn Conductors. IEEE Trans. Appl. Supercond.**2007**, 17, 2639–2642. [Google Scholar] [CrossRef] - Nijhuis, A.; van Meerdervoort, R.P.P.; Krooshoop, H.J.G.; Wessel, W.A.J.; Zhou, C.; Rolando, G.; Sanabria, C.; Lee, P.J.; Larbalestier, D.C.; Devred, A.; et al. The effect of axial and transverse loading on the transport properties of ITER Nb
_{3}Sn strands. Supercond. Sci. Technol.**2013**, 26, 084004. [Google Scholar] [CrossRef] - Zhang, W.J.; Liu, Z.Y.; Liu, Z.L.; Cai, L.C. Melting curves and entropy of melting of iron under Earth’s core conditions. Phys. Earth Planet. Inter.
**2015**, 244, 69–77. [Google Scholar] [CrossRef] - Loria, R.; Marzi, G.D.; Anzellini, S.; Muzzi, L.; Pompeo, N.; Gala, F.; Silva, E. The Effect of hydrostatic pressure on the superconducting and structural properties of Nb
_{3}Sn: Ab-initio modeling ans SR-XRD investigation. IEEE Trans. Appl. Supercond.**2017**, 27, 8400305. [Google Scholar] [CrossRef] - Chu, C.W. Pressure-Enhanced Lattice Transformation in Nb
_{3}Sn Single Crystal. Phys. Rev. Lett.**1974**, 33, 1283–1286. [Google Scholar] [CrossRef] - Svitlyk, V.; Mezouar, M. Pressure-Induced Symmetry Lowering in Nb
_{3}Sn_{1−x}Superconductor. 2020. Available online: http://xxx.lanl.gov/abs/2011.14982 (accessed on 23 March 2021). - Mkrtcheyan, V.; Kumar, R.; Baker, J.; Connolly, A.; Antonio, D.; Cornelius, A.; Zhao, Y. High pressure transport and structural studies on Nb
_{3}Ga superconductor. Physica B**2015**, 459, 21–23. [Google Scholar] [CrossRef][Green Version] - Yu, Z.; Li, C.; Liu, H. Compressibility anomaly in the superconducting material Nb
_{3}Al under high pressure. Physica B**2012**, 407, 3635–3638. [Google Scholar] [CrossRef] - Mao, J.; Chen, Y. Ground-state crystal structures of superconducting Nb
_{3}Al and the phase transformation under high pressures. J. Appl. Phys.**2018**, 124, 173902. [Google Scholar] [CrossRef] - Bunker, G. Introduction to XAFS. A Practical Guide to X-ray Absorption Fine Structure Spectroscopy; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Sakashita, H.; Kamon, K.; Terauchi, H.; Kamijo, N.; Maeda, H.; Toyota, N.; Fukase, T. EXAFS Study on Premartensitic Phase in Nb
_{3}Sn. J. Phys. Soc. Jpn.**1987**, 56, 4183–4187. [Google Scholar] [CrossRef] - Heald, S.; Tarantini, C.; Lee, P.; Brown, M.; Sung, Z.; Ghosh, A.; Larbalestier, D. Evidence from EXAFS for different Ta/Ti site occupancy in high critical current density Nb
_{3}Sn superconductor wires. Sci. Rep.**2018**, 8, 4798. [Google Scholar] [CrossRef][Green Version] - Battocchio, C.; Meneghini, C.; Fratoddi, I.; Venditti, I.; Russo, M.V.; Aquilanti, G.; Maurizio, C.; Bondino, F.; Matassa, R.; Rossi, M.; et al. Silver Nanoparticles Stabilized with Thiols: A Close Look at the Local Chemistry and Chemical Structure. J. Phys. Chem. C
**2012**, 116, 19571–19578. [Google Scholar] [CrossRef] - Timoshenko, J.; Kuzmin, A.; Purans, J. EXAFS study of hydrogen intercalation into ReO
_{3}using the evolutionary algorithm. J. Phys. Condens. Matter**2014**, 26, 055401. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mathon, O.; Beteva, A.; Borrel, J.; Bugnazet, D.; Gatla, A.; Hino, R.; Kantor, I.; Mairs, T.; Munoz, M.; Pasternak, S.; et al. The Time-resolved and Extreme-conditions XAS (TEXAS) facility at the European Synchrotron Radiation Facility: The energy-dispersive X-ray absorption spectroscopy beamline ID24. J. Synchrotron Radiat.
**2015**, 22, 1548–1554. [Google Scholar] [CrossRef] [PubMed] - Welter, E.; Chernikov, R.; Herrmann, M.; Nemausat, R. A beamline for bulk sample x-ray absorption spectroscopy at the high brilliance storage ring PETRA III. AIP Conf. Proc.
**2019**, 2054, 040002. [Google Scholar] [CrossRef] - Ishimatsu, N.; Matsumoto, K.; Maruyama, H.; Kawamura, N.; Mizumaki, M.; Sumiya, H.; Irifune, T. Glitch-free X-ray absorption spectrum under high pressure obtained using nano-polycrystalline diamond anvils. J. Synchrotron Radiat.
**2012**, 19, 768–772. [Google Scholar] [CrossRef] - Irifune, T.; Kurio, A.; Sakamoto, S.; Inoue, T.; Sumiya, H. Ultrahard polycrystalline diamond from graphite. Nature
**2003**, 421, 599–600. [Google Scholar] [CrossRef] - Spina, T. Proton Irradiation Effects on Nb
_{3}Sn Wires and Thin Platelets in View of High Luminosity LHC Upgrade. Ph.D. Thesis, Deparement de Physique de la Matiere Quantique (DQMP), Universite de Geneve, Geneva, Switzerland, 2015. [Google Scholar] - Bragg, W.L.; Williams, E.J. The effect of thermal agitation on atomic arrangement in alloys. Proc. R. Soc. A
**1934**, 145, 699–730. [Google Scholar] - Flükiger, R.; Spina, T.; Cerutti, F.; Ballarino, A.; Scheuerlein, C.; Bottura, L.; Zubavichus, Y.; Ryazanov, A.; Svetogovov, R.D.; Shavkin, S.; et al. Variation ofTc, lattice parameter and atomic ordering in Nb
_{3}Sn platelets irradiated with 12 MeV protons: Correlation with the number of induced Frenkel defects. Supercond. Sci. Technol.**2017**, 30, 054003. [Google Scholar] [CrossRef] - Alimenti, A.; Pompeo, N.; Torokhtii, K.; Spina, T.; Flükiger, R.; Muzzi, L.; Silva, E. Surface Impedance Measurements on Nb
_{3}Sn in High Magnetic Fields. IEEE Trans. Appl. Supercond.**2019**, 29, 3500104. [Google Scholar] [CrossRef] - Alimenti, A.; Pompeo, N.; Torokhtii, K.; Spina, T.; Flükiger, R.; Muzzi, L.; Silva, E. Microwave measurements of the high magnetic field vortex motion pinning parameters in Nb
_{3}Sn. Supercond. Sci. Technol.**2021**, 34, 014003. [Google Scholar] [CrossRef] - Klotz, S.; Chervin, J.C.; Munsch, P.; Le Marchand, G. Hydrostatic limits of 11 pressure. J. Phys. D Appl. Phys.
**2009**, 42, 075413. [Google Scholar] [CrossRef] - Syassen, K. Ruby under pressure. High Press. Res.
**2008**, 28, 75–126. [Google Scholar] [CrossRef] - Dorogokupets, P.I.; Oganov, A.R. Ruby, metals, and MgO as alternative pressure scales: A semiempirical description of shock-wave, ultrasonic, x-ray, and thermochemical data at high temperatures and pressures. Phys. Rev. B
**2007**, 75, 024115. [Google Scholar] [CrossRef] - Meneghini, C.; Bardelli, F.; Mobilio, S. Estra-Fitexa: A Software Package for Exafs Data Analysis. Nucl Instrum. Methods Phys. Res. Sect. B Beam Interact. Mater. Atoms
**2012**, 285, 153–157. [Google Scholar] [CrossRef] - Meneghini, C.; Matteo, S.D.; Monesi, C.; Neisius, T.; Paolasini, L.; Mobilio, S.; Natoli, C.R.; Metcalf, P.A.; Honig, J.M. Antiferromagnetic–paramagnetic insulating transition in Cr-doped V
_{2}O_{3}investigated by EXAFS analysis. J. Phys. Condens. Mat.**2009**, 21, 355401. [Google Scholar] [CrossRef] [PubMed] - James, F. MINUIT: Function Minimization and Error Analysis Reference Manual Version 94.1. CERN Program Library D506. 1994. Available online: http://cdssls.cern.ch/record/2296388/files/minuit.pdf (accessed on 23 March 2021).
- Fornasini, P.; Monti, F.; Sanson, A. On the cumulant analysis of EXAFS in crystalline solids research papers. J. Synchrotron Radiat.
**2001**, 8, 1214–1220. [Google Scholar] [CrossRef] [PubMed][Green Version] - Rehr, J.; Kas, J.; Vila, F.; Prange, M.; Jorissen, K. Parameter-free calculations of X-ray spectra with FEFF9. Phys. Chem. Chem. Phys.
**2010**, 12, 5503–5513. [Google Scholar] [CrossRef] - Shirane, G.; Axe, J.D. Neutron Scattering Study of the Lattice-Dynamical Phase Transition in Nb
_{3}Sn. Phys. Rev. B**1971**, 4, 2957–2963. [Google Scholar] [CrossRef] - Kuzmin, A.; Timoshenko, J.; Kalinko, A.; Jonane, I.; Anspoks, A. Treatment of disorder effects in X-ray absorption spectra beyond the conventional approach. Radiat. Phys. Chem.
**2020**, 175. [Google Scholar] [CrossRef][Green Version] - Timoshenko, J.; Kuzmin, A.; Purans, J. Reverse monte carlo modeling of thermal disorder in crystalline materials from EXAFS spectra. Comput. Phys. Commun.
**2012**, 183, 1237–1245. [Google Scholar] [CrossRef] - Jonane, I.; Cintis, A.; Kalinko, A.; Chernikov, R.; Kuzmin, A. Low temperature X-ray absorption spectroscopy study of CuMoO
_{4}and CuMo_{0}.90W_{0}.10O_{4}using reverse monte-carlo method. Radiat. Phys. Chem.**2020**, 175, 108411. [Google Scholar] [CrossRef] - Rehr, J.; Albers, R.C. Theoretical approaches to x-ray absorption fine structure. Rev. Mod. Phys.
**2000**, 72, 621. [Google Scholar] [CrossRef] - Timoshenko, J.; Kuzmin, A. Wavelet data analysis of EXAFS spectra. Comput. Phys. Commun.
**2009**, 180, 920–925. [Google Scholar] [CrossRef] - Filipponi, A.; Di Cicco, A.; Natoli, C.R. X-ray-absorption spectroscopy and n-body distribution functions in condensed matter. I. Theory. Phys. Rev. B
**1995**, 52, 15122–15134. [Google Scholar] [CrossRef] - Harris, C.R.; Millman, K.J.; van der Walt, S.J.; Gommers, R.; Virtanen, P.; Cournapeau, D.; Wieser, E.; Taylor, J.; Berg, S.; Smith, N.J.; et al. Array programming with NumPy. Nature
**2020**, 585, 357–362. [Google Scholar] [CrossRef] - Meneghini, C.; Ray, S.; Liscio, F.; Bardelli, F.; Mobilio, S.; Sarma, D.D. Nature of “Disorder” in the Ordered Double Perovskite Sr
_{2}FeMoO_{6}. Phys. Rev. Lett.**2009**, 103, 046403. [Google Scholar] [CrossRef] - Timoshenko, J.; Anspoks, A.; Cintins, A.; Kuzmin, A.; Purans, J.; Frenkel, A.I. Neural Network Approach for Characterizing Structural Transformations by X-Ray Absorption Fine Structure Spectroscopy. Phys. Rev. Lett.
**2018**, 120, 225502. [Google Scholar] [CrossRef][Green Version] - Egami, T.; Billinge, S. (Eds.) Underneath the Bragg Peaks, 2nd ed.; Pergamon Materials Series; Elsevier: Pergamon, Turkey, 2012; Volume 16. [Google Scholar]
- Provost, K.; Beret, E.; Muller, D.; Marcos, E.S.; Michalowicz, A. Impact of the number of fitted Debye-Waller factors on EXAFS fitting. J. Phys. Conf. Ser.
**2013**, 430, 012015. [Google Scholar] [CrossRef][Green Version] - Filipponi, A.; Di Cicco, A. X-ray-absorption spectroscopy and n -body distribution functions in condensed matter. II. Data analysis and applications. Phys. Rev. B
**1995**, 52, 15135–15149. [Google Scholar] [CrossRef] [PubMed] - Tucker, M.G.; Keen, D.A.; Dove, M.T.; Goodwin, A.L.; Hui, Q. RMCProfile: Reverse Monte Carlo for polycrystalline materials. J. Phys. Condens. Matter
**2007**, 19, 335218. [Google Scholar] [CrossRef] - Michalowicz, A.; Provost, K.; Laruelle, S.; Mimouni, A.; Vlaic, G. F-test in EXAFS fitting of structural models. J. Synchr. Radiat.
**1999**, 6, 233–235. [Google Scholar] [CrossRef][Green Version] - Mao, H.K.; Chen, X.J.; Ding, Y.; Li, B.; Wang, L. Solids, liquids, and gases under high pressure. Rev. Mod. Phys.
**2018**, 90, 015007. [Google Scholar] [CrossRef][Green Version] - Markiewicz, W. Elastic stiffness model for the critical temperature Tc of Nb
_{3}Sn including strain dependence. Cryogenics**2004**, 44, 767–782. [Google Scholar] [CrossRef] - Valentinis, D.F.; Berthod, C.; Bordini, B.; Rossi, L. A theory of the strain-dependent critical field in Nb
_{3}Sn, based on anharmonic phonon generation. Supercond. Sci. Technol.**2013**, 27, 025008. [Google Scholar] [CrossRef][Green Version] - Godeke, A.; Hellman, F.; ten Kate, H.H.J.; Mentink, M.G.T. Fundamental origin of the large impact of strain on superconducting Nb
_{3}Sn. Supercond. Sci. Technol.**2018**, 31, 105011. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**(

**a**) Experimental (dots) and best fit (lines) Nb K-edge ${k}^{2}$-weighted extended X-ray absorption fine structure (EXAFS) spectra as a function of pressure. (

**b**) Moduli of the FT of ${k}^{2}$-weighted EXAFS spectra (dots) and best fit curves (full lines). In each frame the corresponding pressures are marked and the curves are vertically shifted for sake of clarity.

**Figure 2.**Representation of the Nb

_{3}Sn structure, where Nb and Sn atoms are reported in light red and purple, respectively. For sake of clarity some of the atoms have been labelled following the nomenclature of Table 1, in order to identify the various paths used for the EXAFS analysis. The dimension of the Nb

_{0}atom has been enhanced to help the reader finding the absorbing atom.

**Figure 3.**Example of the best fit obtained at 23 GPa for the EXAFS signal of Nb

_{3}Sn at the Nb K-edge using the standard (left panels) and Reverse Monte Carlo (RMC) (right panels) analysis. In particular, (

**a**,

**b**) panels report the ${k}^{2}$-weighted EXAFS signal and the corresponding Fourier transform (FT) moduli |FT|, respectively. In both the panels, the experimental data are reported as green dots, whereas the best fit curves as black lines. The partial contribution used in the analysis are shown (yellow lines), vertically shifted for sake of clarity. Right panels (

**c**,

**d**) show the ${k}^{2}$-weighted Morlet wavelet transform for the experimental and CA-model from the RMC refinement. The Fourier transform (FT) moduli |FT| and the Back Fourier filtered (FF) curves obtained from the RMC refinement are shown in Figure S1 of the Supporting Information.

**Figure 4.**(

**a**) ${R}_{N{b}_{I}}$ distance (orange symbols) and the $a/2$ parameters (light blue symbols) are shown. The ${R}_{N{b}_{I}}$ is systematically larger than $a/2$, signifying an anticorrelated Nb–Nb neighbor displacement perpendicularly to the Nb chain, as schematized in the inset. (

**b**) The mean square relative displacement (MSRD) of the first (${\sigma}_{N{b}_{I}}^{2}$), second (${\sigma}_{S{n}_{I}}^{2}$) and third (${\sigma}_{N{b}_{I}I}^{2}$) neighbors are shown as a function of pressure.

**Figure 5.**Synthesis of the main RMC results. (

**a**) ${R}_{N{b}_{I}}$ and (

**b**) ${R}_{S{n}_{I}}$ distributions histograms as a function of pressure. (

**c**) Top panel (Distance): pressure dependence of ${R}_{N{b}_{I}}$ and $a/2$ compared to the one obtained from the standard analysis. (

**c**) Second panel from the top (MSRD): pressure-induced evolution of the variance (${\sigma}^{2}$) of the first three shells as obtained from the RMC simulations. (

**c**) Third panel from the top (Skewness): pressure-induced evolution of the asymmetry of the distribution of the Nb–Nb nearest neighbor pair distribution function. (

**c**) Bottom panel (Kurtosis): pressure-induced evolution of the Kurtosis of the distribution of the first shell

**Figure 6.**Best fit of the AP data comparing the ${k}^{2}{\chi}_{exp}^{\left(AP\right)}$ (

**top**figures: (

**a**,

**b**)) and ${k}^{2}{\chi}_{CA}^{\left(AP\right)}$ (

**bottom**figures: (

**c**,

**d**)). In particular, (

**a**,

**c**) panels report the ${k}^{2}$-weighted EXAFS signal and (

**b**,

**d**) panels their corresponding Fourier Trasform (FT) moduli |FT|. In both the panels, the experimental data are reported as green dots, the best fit curves as black lines and the residuals are reported as grey dots at the bottom of the figures.

**Figure 7.**Best fit of the ${k}^{2}\chi \left(k\right)$ data at 26 GPa. In panel (

**a**) the best fit obtained with a single shell model is reported while panel (

**b**) shows the fit obtained with a bimodal distribution model of the Nb

_{I}shell. In both the panels, the experimental data are reported as green dots and the best fit curves as a black line. The contributions used to fit the Nb

_{I}shell (S

_{1}for the single shell model, S

_{1}and S

_{2}for the bimodal distribution model) are reported by orange lines vertically shifted for clarity. The contributions from the other coordination shells are not shown for sake of clarity. The best fit residuals (${k}^{2}({\chi}_{exp}-{\chi}_{fit})$) are reported as grey dots at the bottom of the panels.

**Figure 8.**Pressure induced evolution of the ${R}_{1}-{R}_{2}$ distances (

**a**) and corresponding angles (

**b**). Explanatory model (

**c**) of the correlated disorder for ${R}_{1}-{R}_{2}$ distribution compared with the experimental data obtained from RMC model at 14 GPa (background). The points are calculated by randomly displacing Nb

_{c}. For the red points the average distances are the same ${\overline{R}}_{1}={\overline{R}}_{2}$ and the random displacement is preferentially perpendicular to the Nb chain (as schematized in the top scheme). For the green points the average distances are different ${\overline{R}}_{1}>(<){\overline{R}}_{2}$ and the random displacement is preferentially parallel to the Nb chain (as schematized in the bottom scheme).

**Table 1.**Definitions of the seven neighboring shells used for the EXAFS data analysis calculated assuming the generic Nb

_{0}absorber located at $(\frac{1}{4},0,\frac{1}{2})$. The atom labels are those shown in Figure 2. For each shell, the labels of the neighboring atoms used in the text and the half path length R(a) as a function of the cube edge (lattice parameter) a are reported. For each atomic configuration contributing to the shell the multiplicity (N), the scattering model (single SS, or multiple MS, scattering) and the neighboring relative position respect to Nb

_{0}(in units of the lattice parameter a) are shown. The scattering paths are also reported for sake of completeness (path) along with the relative intensity respect to the Nb

_{I}shell as given by FEFF, the MS terms include the three and four leg paths.

Shell | Atoms | R(a) | N | Scattering | Position | Path |
---|---|---|---|---|---|---|

I | Nb_{I} | $\frac{a\sqrt{4}}{4}$ | 2 | SS | $(\pm \frac{1}{2},0,0)$ | Nb_{0}-Nb_{I} (100% ) |

II | Sn_{I} | $\frac{a\sqrt{5}}{4}$ | 2+ | SS | $(-\frac{1}{4},0,\pm \frac{1}{2})$ | Nb_{0}-Sn_{I} (173% ) |

2 | SS | $(\frac{1}{4},\pm \frac{1}{2},0)$ | ||||

III | Nb_{II} | $\frac{a\sqrt{6}}{4}$ | 4+ | SS | $(-\frac{1}{4},\pm \frac{1}{2},\pm \frac{1}{4})$ | Nb_{0}-Nb_{II} (247% ) |

4 | SS | $(\frac{1}{4},\pm \frac{1}{4},\pm \frac{1}{2})$ | ||||

IV | Sn_{II} | $\frac{a\sqrt{13}}{4}$ | 2+ | SS | $(\frac{3}{4},0,\pm \frac{1}{2})$ | Nb_{0}-Sn_{II} (45% ) |

2 | SS | $(-\frac{3}{4},\pm \frac{1}{2},0)$ | ||||

V | Nb_{III} | $\frac{a\sqrt{14}}{4}$ | 4+ | SS | $(\frac{3}{4},\pm \frac{1}{2},\pm \frac{1}{4})$ | Nb_{0}-Nb_{III} (152% ) |

4+ | SS | $(\frac{1}{4},\pm \frac{3}{4},\pm \frac{1}{2})$ | ||||

4+ | SS | $(-\frac{3}{4},\pm \frac{1}{4},\pm \frac{1}{2})$ | ||||

4 | SS | $(-\frac{1}{4},\pm \frac{1}{2},\pm \frac{3}{4})$ | ||||

VI | Nb_{IV} | $\frac{a\sqrt{16}}{4}$ | 2+ | MS | $(\pm 1,0,0)$ | Nb_{0}-Nb_{1}-Nb_{IV} (163% ) |

2+ | SS | $(0,\pm 1,0)$ | Nb_{0}-Nb_{IV} (46% ) | |||

2 | SS | $(0,0,\pm 1)$ | ||||

VII | Nb_{V} | $\frac{a\sqrt{20}}{4}$ | 2+ | MS | $(\frac{1}{2},\pm 1,0)$ | Nb_{0}-Sn_{I}-Nb_{V} (142% ) |

2+ | MS | $(-\frac{1}{2},0,\pm 1)$ | ||||

2 | SS | $(-\frac{1}{2},\pm 1,0)$ | Nb_{0}-Nb_{V} (42% ) |

**Table 2.**Comparison between the structural parameters obtained from the EXAFS and RMC fit of the experimental AP data ${k}^{2}{\chi}_{exp}^{\left(AP\right)}$ and from the EXAFS fit of the ${\chi}_{CA}^{\left(AP\right)}$ curve.

AP Data | ${\mathit{R}}_{{\mathit{Nb}}_{\mathit{I}}}$ [Å] | a/2 [Å] | ${\mathit{\sigma}}_{{\mathit{Nb}}_{\mathit{I}}}^{2}$ |
---|---|---|---|

EXAFS fit of ${\chi}_{exp}$ | 2.644(4) | 2.638(3) | 0.0064(4) |

RMC fit of ${\chi}_{exp}$ | 2.60(1) | 2.56(1) | 0.020(2) |

EXAFS fit of ${\chi}_{CA}$ | 2.649(4) | 2.644(4) | 0.0064(4) |

**Table 3.**Quantitative parameters obtained from the EXAFS fit of the data at 26 GPa. The results obtained with a single shell model and a bimodal distribution model are compared. Parameters indicated with * were constrained during the analysis. The multiplicity number N was fixed to 2 for the single shell model while for the bimodal distribution model it was constrained to 1 for each shell. In the bimodal distribution model the ${R}_{N{b}_{I}}$ parameters was left free to vary for each shelll and the ${\sigma}_{N{b}_{I}}^{2}$ of the two contributions was constrained to be the same. In the last column of the table the ${R}_{W}^{2}$ parameters indicates the best fit quality.

26 GPa | N | ${\mathit{R}}_{{\mathit{Nb}}_{\mathit{I}}}$ [Å] | ${\mathit{\sigma}}_{{\mathit{Nb}}_{\mathit{I}}}^{2}$ | ${\mathit{R}}_{\mathit{W}}^{2}$ |
---|---|---|---|---|

single shell model | 2 * | 2.545(4) | 0.0044(4) | 0.0615 |

bimodal distribution model | 1 * | 2.498(5) | 0.0022(2) * | 0.0587 |

2.607(4) |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Schiesaro, I.; Anzellini, S.; Loria, R.; Torchio, R.; Spina, T.; Flükiger, R.; Irifune, T.; Silva, E.; Meneghini, C. Anomalous Behavior in the Atomic Structure of Nb_{3}Sn under High Pressure. *Crystals* **2021**, *11*, 331.
https://doi.org/10.3390/cryst11040331

**AMA Style**

Schiesaro I, Anzellini S, Loria R, Torchio R, Spina T, Flükiger R, Irifune T, Silva E, Meneghini C. Anomalous Behavior in the Atomic Structure of Nb_{3}Sn under High Pressure. *Crystals*. 2021; 11(4):331.
https://doi.org/10.3390/cryst11040331

**Chicago/Turabian Style**

Schiesaro, Irene, Simone Anzellini, Rita Loria, Raffaella Torchio, Tiziana Spina, René Flükiger, Tetsuo Irifune, Enrico Silva, and Carlo Meneghini. 2021. "Anomalous Behavior in the Atomic Structure of Nb_{3}Sn under High Pressure" *Crystals* 11, no. 4: 331.
https://doi.org/10.3390/cryst11040331