Anharmonic thermal vibrations and atomic potentials in lead fluoride (β-PbF₂) as a function of temperature

Bragg intensities of β-PbF₂ (with the cubic fluorite structure) have been measured in the step-scan mode up to sin θ/λ = 1.17 Å^-1 at 295, 461 and 625 K by X-ray diffraction: a = 5.925 (2), 5.952 (2), 5.982 (4) Å, respectively; R = 0.013 for 103 reflections, 0.015 for 103 reflections, 0.018 for 106 reflections, respectively. The intensities were used for the refinement of anharmonic temperature factors up to third order for F^- and up to sixth order for Pb²⁺. The temperature-factor formalism was based on the Gram-Charlier expansion. The probability density function (p.d.f.) of the F^- and Pb²⁺ ions were calculated from the coefficients of the anharmonic temperature factors. The p.d.f. maps gave clear evidence that the Pb²⁺ ions do not carry out isotropic thermal motion. The atomic potentials of the Pb²⁺ ions (derived from the p.d.f. maps) indicate that the thermal motion of the excited Pb²⁺ ions is strongly influenced by the repulsion terms along the Pb²⁺-F^- bonds. The F^- ions carry out their largest thermal vibrations along 〈111〉 towards the centre of the elementary cell.