X-ray diffraction by a crystal in a permanent external electric field: general considerations

Abstract

The variations of X-ray diffraction intensities from a crystal in the presence of a permanent external electric field is modeled analytically using a first-order stationary perturbation theory. The change in a crystal, induced by an external electric field, is separated into two contributions. The first one is related to a pure polarization of an electron subsystem, while the second contribution can be reduced to the displacements of the rigid pseudoatoms from their equilibrium positions. It is shown that a change of the X-ray diffraction intensities mainly originates from the second contribution, while the influence of the pure polarization of a crystal electron subsystem is negligibly small. The quantities restored from an X-ray diffraction experiment in the presence of an external electric field were analyzed in detail in terms of a rigid pseudoatomic model of electron density and harmonic approximation for the atomic thermal motion. Explicit relationships are derived that link the properties of phonon spectra with E-field-induced variations of a structure factor, pseudoatomic displacements and piezoelectric strains. The displacements can be numerically estimated using a model of independent atomic motion if the Debye-Waller factors and pseudoatomic charges are known either from a previous single-crystal X-ray diffraction study or from density functional theory calculations. The above estimations can be used to develop an optimum strategy for a data collection that avoids the measurements of reflections insensitive to the electric-field-induced variations.

Electric-field-induced local and mesoscale structural changes in polycrystalline dielectrics and ferroelectrics

“…A formalism for analysis of diffraction data collected on electrically biased single crystals has been developed previously1 and used to determine field-induced changes of bond lengths in several piezoelectrics, including α-GaPO 4 2, LiNbO 3 3 and LiSO 4 ·H 2 O2. However, for many useful systems, large single crystals are unavailable and applications rely on more easily produced polycrystalline materials.…”

Electric‐field‐induced internal deformation in piezoelectric BiB3O6 crystals

“…According to our previous studies [7,16], the Bragg intensities are mainly changed due to the electric fieldinduced displacements of pseudoatoms, where a pseudoatom is defined as a rigid fragment of the electron density attached to a single nucleus [20]. Note that there is a certain ambiguity on how the above partitioning of the electron density into pseudoatoms is done.…”

“…The displacement tensor, ˆ, a with the components a i j (µ) and its dependence on the microscopic parameters of the crystal, such as phonon spectra and electron density distribution, were introduced in [7]. Note that ˆ*) µ a( of an atom μ* which is related to an atom µ by the symmetry operation { } S, d (Ŝ is a rotation matrix and d is a translation vector) is given by:…”

“…Considering the reasons mentioned above the total number of the Bragg reflections that showed a measurable change of diffraction intensities was not more then 48, for 10 reflections the measurements were conducted at two different wavelengths (see Table A1 in Appendix for more details). Ιn the range of the applied high voltage (U < 3 kV) ΔI/I is supposed to be linear with the external electric field [7], that was generally well confirmed in the past experiments [5,6,9,14]. For the further refinements of atomic displacements we averaged the relative changes in the intensity of each reflection measured at a positive, ΔI/I(E + ), and negative, ΔI/I(E -), electric field direction to the quantity …”

“…Both of these tiny structural effects can be simultaneously investigated by one and the same X-ray diffraction experiment at a single crystalline sample [6]. In particular, the internal strains are evaluated from the variations of Bragg intensities [7], whereas the change of lattice parameters is extracted from the angular shifts of Bragg peak positions [8,10]. Nowadays such kind of experiments is of a special interest, because the converse piezoelectric effect, the dielectric polarization and finally their interdependency are still poorly understood on the atomic scale [6].…”