Anharmonicity and scissoring modes in the negative thermal expansion materials ${\mathrm{ScF}}_3$ and ${\mathrm{CaZrF}}_6$

We use a symmetry-motivated approach to analyzing x-ray pair distribution functions to study the mechanism of negative thermal expansion in two ${\mathrm{ReO}}_3$-like compounds: ${\mathrm{ScF}}_3$ and ${\mathrm{CaZrF}}_6$. Both average and local structures suggest that it is the flexibility of $M$-F-$M$ linkages $(M$ = Ca, Zr, Sc) due to dynamic rigid and semirigid “scissoring” modes that facilitates the observed negative thermal expansion (NTE) behavior. The amplitudes of these dynamic distortions are greater for ${\mathrm{CaZrF}}_6$ than for ${\mathrm{ScF}}_3$, which corresponds well with the larger magnitude of the thermal expansion reported in the literature for the former. We show that this flexibility is enhanced in ${\mathrm{CaZrF}}_6$ due to the rocksalt ordering mixing the characters of two of these scissoring modes. Additionally, we show that in ${\mathrm{ScF}}_3$ anharmonic coupling between the modes responsible for the structural flexibility and the rigid unit modes contributes to the unusually high NTE behavior in this material.

Figure 1

Pair distribution functions for ${\mathrm{ScF}}_3$ (top) and ${\mathrm{CaZrF}}_6$ (bottom) at 400 K (black circles). A small box fit with the modes belonging to $X_5^{+}$ refined is shown for both compounds (blue lines), with the $R_2^{-}$ mode additionally refined for ${\mathrm{CaZrF}}_6$. Labeled peaks correspond to Sc-F (1a), F-F (2a, 3b), Sc-Sc (3a), Zr-F (1b), Ca-F (2b), and Ca-Zr (4b).

Figure 2

Transverse atomic displacement parameters from Rietveld refinement (top), the best weighted-phase $R$ factor for each irrep at each temperature (middle), and the Boltzmann weighted mode amplitude (bottom). Results for ${\mathrm{ScF}}_3$ are displayed on the left; those for ${\mathrm{CaZrF}}_6$ are on the right.

Figure 3

Representations showing the effect of (a) $X_5^{+}$, (b) $X_5^{-}$, (c) $M_5^{+}$, and (d) $M_5^{-}$ on the crystal structure of ${\mathrm{CaZrF}6}_6$. The distortions are taken from the refinements at 400 K with the lowest $R_{wp}$ and plotted using the vesta software [33].

Figure 4

Comparison of weighted $R$ factors for restricted irreps $X_5^{+},X_5^{-},M_5^{+}$, and $M_5^{-}$, unrestricted irreps $M_2^{+}$ and $R_5^{-}$, and coupled $X_5^{+}\oplus M_5^{-}$ and $X_5^{-}\oplus M_5^{+}$.

Figure 5

Comparison of fits to ${\mathrm{ScF}}_3$ (left) and ${\mathrm{CaZrF}}_6$ (right) PDF data at 400 K using restricted $X_5^{+},X_5^{-},M_5^{+}$, and $M_5^{-}$; unrestricted $R_5^{-}$; and restricted $X_5^{+}\oplus M_5^{-}$.

Figure 6

Comparison of coupled and two-phase fits to PDF data as a function of temperature for ${\mathrm{ScF}}_3$ (top) and ${\mathrm{CaZrF}}_6$ (bottom), as described in the text.

Figure 7

Comparison of fits for $X_5^{+}\oplus R_5^{-}$ using a two-phase model (blue) and a coupled model (red) and $X_5^{+}(a,b;0,0;0,0)\oplus M_5^{-}(0,0;c,d;0,0)\oplus R_5^{-}(e,f,g)$ (yellow) for ${\mathrm{ScF}}_3$.