Anharmonic atomic vibrations in the relaxor ferroelectric Pb(Mg${}_{1/3}$Nb${}_{2/3}$)O${}_3$ under pressure

Structural analyses of the relaxor ferroelectric material Pb(Mg${}_{1/3}$Nb${}_{2/3}$)O${}_3$ (PMN) with single-crystal x-ray diffraction under pressure in a diamond anvil cell indicate static atomic displacement and chemical disorder. A difference Fourier analysis within the framework of a harmonic oscillator model for the atomic vibrations reveals residual electron density on both the Pb and Nb(Mg) sites. Pb atoms in the $A$ site of the $AB$O${}_3$ perovskite structure exhibit a greater displacement than the Nb(Mg) atoms in the $B$ site, despite the fact that Pb is a much heavier atom. The displacement is interpreted in terms of an anharmonic statistical atomic motion. At pressures above 2.5 GPa the displacement disappears, consistent with previous observations. The difference Fourier maps reveal no residual electron densities greater than 1 $e$/Å${}^3$ at any atomic position, and these appear to be induced by the violation of local electrical neutrality arising from both Mg${}^{2+}$ and Nb${}^{5+}$ ions located at the octahedral site. Similar electron densities are observed at all experimental pressures and in refinements based on both harmonic and anharmonic models. The anharmonic parameters taken into account are the higher-order tensors of atomic elastic motion. At high pressure, where the relaxor transforms to a paraelectric phase, the residual electron densities disappear.

Figure 1

Unit cell volume of PMN as a function of pressure. Error bars on the data points are smaller than the symbol size. Unit cell volume data from the powder diffraction experiments reported by Chaabane et al. (Ref. 10) are also presented.

Figure 2

Difference Fourier maps of the projection on (001) for refinements based on the harmonic and anharmonic oscillator models. The residual electron density around the Pb atom is shown with a contour interval of 0.02 $e$/Å${}^3$ in the section of $z$ $=$ 0.0, $x=-0.5$ to 0.5, and $y=-0.5$ to 0.5.

Figure 3

Difference Fourier maps of the projection on (001) for refinements based on the harmonic and anharmonic oscillator models. The residual electron density around the Nb(Mg) atom is shown with a contour interval of 0.02 $e$/Å${}^3$ in the section of $z$ $=$ 0.5, $x$ $=$ 0.0 to 1.0, and $y$ $=$ 0.0 to 1.0.

Figure 4

Three-dimensional difference Fourier map of PMN at 1 atm for the harmonic oscillator model with a contour interval of 0.02 $e$/Å${}^3$, showing the residual electron density distribution.