Quantitative understanding of negative thermal expansion in scandium trifluoride from neutron total scattering measurements

Open Access

Quantitative understanding of negative thermal expansion in scandium trifluoride from neutron total scattering measurements

Martin T. Dove, Juan Du, Zhongsheng Wei, David A. Keen, Matthew G. Tucker, and Anthony E. Phillips

Phys. Rev. B 102, 094105 – Published 16 September 2020

Abstract

Negative thermal expansion (NTE)—the phenomenon where some materials shrink rather than expand when heated—is both intriguing and useful but remains poorly understood. Current understanding hinges on the role of specific vibrational modes, but in fact thermal expansion is a weighted sum of contributions from every possible mode. Here we overcome this difficulty by deriving a real-space model of atomic motion in the prototypical NTE material scandium trifluoride, $Sc F_{3}$ , from total neutron scattering data. We show that NTE in this material depends not only on rigid unit modes—the vibrations in which the scandium coordination octahedra remain undistorted—but also on modes that distort these octahedra. Furthermore, in contrast with previous predictions, we show that the quasiharmonic approximation coupled with renormalization through anharmonic interactions describes this behavior well. Our results point the way towards a new understanding of how NTE is manifested in real materials.

Received 11 March 2020
Revised 22 June 2020
Accepted 27 July 2020

DOI:https://doi.org/10.1103/PhysRevB.102.094105

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Lattice dynamics Thermal expansion

Monte Carlo methods Neutron pair-distribution function analysis

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Martin T. Dove ^1,2,3,*, Juan Du ³, Zhongsheng Wei ³, David A. Keen ⁴, Matthew G. Tucker ⁵, and Anthony E. Phillips ³

¹Schools of Computer Science and Physical Science & Technology, Sichuan University, Chengdu 610065, People's Republic of China
²Department of Physics, School of Sciences, Wuhan University of Technology, 205 Luoshi Road, Hongshan district, Wuhan, Hubei 430070, People's Republic of China
³School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom
⁴ISIS Facility, Rutherford Appleton Laboratory, Harwell Campus, Didcot, Oxfordshire OX11 0QX, United Kingdom
⁵Oak Ridge National Laboratory, Neutron Scattering Division, 1 Bethel Valley Road, Oak Ridge, Tennessee 37831, USA

^*martin.dove@icloud.com

Article Text

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Supplemental Material

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References

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Issue

Vol. 102, Iss. 9 — 1 September 2020

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Images

Figure 1
Representation of a linear arrangement of corner-linked structural octahedra showing rotations. For clarity the upper atoms are not shown.
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Figure 2
Crystal structure of $Sc F_{3}$ at a temperature of 1200 K obtained by Rietveld refinement of neutron powder diffraction data reported in this paper. It has a primitive cubic structure (Strukturbericht symbol $D 0_{9}$ , space group $P m \bar{3} m$ ) with one formula unit per unit cell). The ellipsoids (Sc pink, F green) represent the thermal motion along different directions, with the volume enclosing 50% of the total distribution of atom positions.
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Figure 3
Comparison of the temperature dependence of half the lattice parameter, $a / 2$ (black squares, filled squares representing data from longer measurements and open squares representing data from shorter measurements), half of the average instantaneous nearest-neighbor Sc–Sc distance obtained from analysis of the RMC configurations (gray filled circles), the average instantaneous nearest-neighbor F–F distance obtained from analysis of the RMC configurations scaled by $1 / \sqrt{2}$ (open black circles), and the average instantaneous nearest-neighbor Sc–F distance also obtained from analysis of the RMC configurations (black filled circles. In each plot statistical error bars are smaller than the sizes of the data symbols. The scaling means that each data set should converge to a value of $a / 2$ at low temperature. The lines are guides to the eye; the guides for the F–F and Sc–F distances were obtained by fitting functions of the form $d = d_{0} + α coth (θ / T)$ , and the guides for the lattice parameter and Sc–Sc distances were obtained by fitting functions of the form $d = d_{0} - γ T + α coth (θ / T)$ , where the parameters $d_{0}$ , $α$ , $γ$ , and $θ$ were variables in the fitting process.
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Figure 4
(a) Comparison of the variances of three angles associated with local motions taken from the RMC configurations. The black points show the departure of the Sc–F–Sc angle from its nominal value of $180^{\circ}$ , the green points show the variance of the F–Sc–F angle as it fluctuates from its nominal value of $90^{\circ}$ , and the red points show the mean-square angle of rotation of the $Sc F_{6}$ octahedra as a whole body. (b) Breakdown of the total motion (excluding overall polyhedral displacements) of the atoms in the $Sc F_{3}$ crystal from the GASP analysis of the RMC configurations, where the red, green, and blue points and guides to the eye represent the fraction of the motion that is associated with whole-body rotations of the $Sc F_{6}$ octahedra, deformations of the F–Sc–F right angles, and stretching of the Sc–F bonds, respectively. In both plots statistical error bars are smaller than the sizes of the data symbols, and the lines/curves are given as guides to the eye.
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Figure 5
(a) and (b) show the variation of volume of one unit cell of $Sc F_{3}$ with the interoctahedral Sc–F–Sc and intraoctahedral F–Sc–F angle force constants, respectively, $A$ and $k$ (see Supplemental Material [39] for the equations). (c) and (d) show the corresponding coefficients of volumetric thermal expansivity for the cases $k = 1.2$ eV and $A = 0.025$ eV respectively, corresponding to the values that give best match to the DFT dispersion curves [16] as discussed in the Supplemental Material [39].
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Figure 6
(a) and (b) show histograms of the lateral displacements of the F atoms at the different temperatures (circles). At all temperatures the distributions are well described by Gaussian functions (thin lines). (c) shows the variance of the Gaussian fits as a function of temperature. The data follow a linear dependence on temperature throughout the range studied, as indicated by the fitted straight line, demonstrating that a renormalized phonon model is sufficient to describe the anharmonic effects in ${ScF}_{3}$ .
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