Parametric Rietveld refinement for the evaluation of powder diffraction patterns collected as a function of pressure

Abstract

Under the assumption that the structural parameters of a crystalline phase change `smoothly' with increasing pressure, the evolution of the parameters can be parameterized as a function of pressure using continuous monotonic functions. Four different approaches to determine the structural evolution of As2O5 with increasing pressure from a set of powder diffraction patterns collected over the pressure range from 2.5 to 19.5 GPa have been investigated. Approach (A) was the common sequential refinement of atomic coordinates with restraints on the geometry and was compared with three parameterization approaches. Approach (B) used direct parameterization by low‐order polynomials of each crystallographically distinct atomic coordinate, (C) described the atoms of the asymmetric unit as a rigid body and allowed the internal degrees of freedom of the rigid body to vary with the change in pressure using rigid unit modes, and (D) described the crystal structure as a distortion of the higher‐symmetry structure of As2O5 (which is here also a high‐temperature phase) by using symmetry‐adapted distortion modes. Approach (D) offers the possibility to directly introduce an order parameter into Rietveld refinement through an empirical power law derived from Landau theory and thus to obtain the value of the critical exponent. In contrast, the rigid‐body approach did not fit the data as well. All parameterizations greatly reduce the number of required parameters.

Revealing phase boundaries by weighted parametric structural refinement

“…composition, temperature or pressure) in order to restrain the Rietveld refinement to fit a three-dimensional space of 2, intensity and a parameter dependent on the external variable (Stinton & Evans, 2007). This has been used in a wide variety of studies to understand material structure with regards to electric field, composition, temperature, magnetic field and pressure (Daniels, 2008;Marlton et al, 2017;Daniels et al, 2010;Olsen et al, 2017;Mabied et al, 2012;Halasz et al, 2010;Scarlett et al, 2009). However, care must be taken when applying the parametric technique to avoid enforcing unphysical restraints on the structural model.…”

Possibilities and limitations of parametric Rietveld refinement on high pressure data: The case study of LaFeO3

“…This linearization is fully valid for cubic materials, whereas it is a reasonable approximation for tetragonal or orthorhombic cases. Considering the main restriction for the use of the Murnaghan EoS, namely, that it is only valid up to a compression of approximately 10% [6,27], a parameterization can be carried out without loss of refinement quality, as this compression level is first reached at the critical pressure of the second order phase transition, which is far beyond the hydrostatic limit of the used pressure medium. This means that below the hydrostatic limit the Murnaghan EoS will be fully applicable in the parameterization, whereas above, the lattice parameters have either to be individually refined or to be fixed to the values determined by the sequential Rietveld refinement.…”

“…The parameterization of the lattice parameters up to the hydrostatic limit can be done by using a linearized inverted Murnaghan EoS [6,9,27]:…”

“…So far parametric Rietveld refinement was successfully applied in for instance the modeling of temperature-dependent structural changes and phase transitions [3], the modeling of kinetics in time-resolved powder diffraction data [4], the modeling of temperature-dependent coupling of strain and order parameters [5] and the modeling of pressure-dependent powder diffractions patterns with different methods [6].…”

“…Approach A with traditional Rietveld refinement of all data, refining fractional atomic coordinates, Approach B where the Rietveld refinement was done by using the method of a rigid body [10,11] with a deformable body model of the FeO 6 octahedron, Approach C where symmetry modes were used [12][13][14][15][16] and Approach D where the newly developed method of the rotational symmetry mode description for a rigid body was used [17]. All four approaches were carried out with sequential as well as parametric Rietveld refinements, the latter allowing to model the evolution of lattice parameters with equations of state (EoS) [18] up to the hydrostatic limit of the used pressure medium [19] while atomic positions can be modeled with power-law equations derived by Landau theory.…”