Least-squares thermal expansion tensor of vanadate and arsenate triclinic apatites derived from laboratory X-ray powder diffraction cell data

Cell data for triclinic end-member Ca₁₀(VO₄)₆F₂ and Ca₁₀(AsO₄)₆F₂ apatites were measured in the temperature range from 303 to 773 K. Reversible phase transitions at, respectively, 453 and 583 K are shown and attributed to mobility of contact surfaces for triclinic twins within a mosaic block at about the transition temperature. The simple method developed here is based on 12 separate linear regressions. The first six regressions are on observations for individual lattice parameters a, b, c, α, β or γ. The last six are on linear data sets, each involving a single expansion coefficient α₁₁, α₂₂, α₃₃, α₁₂, α₁₃ or α₂₃. Singular-value decomposition of the least-squares thermal expansion tensors obtained below the transition temperatures shows that both materials actually contract considerably along [24] upon heating. Expansion in the plane perpendicular to this direction differs somewhat for the two materials. In contrast, the expansion above the transition temperature is barely anisotropic in both materials. The ability to measure thermal expansion tensors for triclinic materials with decent accuracy from routine powder data is demonstrated. This possibility extends the applications of the powder method because some samples may not be readily available in single-crystal form.