ABSTRACT
H. Barium β(II)-Alumina Surfaces
I. Comparison of Surfaces of the Barium Phases
relaxation is similar to that seen in the barium β(I)-alumina and magnetoplumbite systems.
The unrelaxed and relaxed surface structures can be seen in Figures 3.55 and 3.56, respectively. In the unrelaxed surface structure, as found in the barium magnetoplumbite and barium β(I)-alumina {001} surfaces, there is rumpling in the O2- layer. However, in this system, the rumpling is imperceptible by visual examination. Neighboring O2- ions relax in and out of the bulk less than 0.05 A from their unrelaxed positions. The Al3+ ions relax to positions lower in the bulk structure, and the rumpling in the O2- layer immediately below increases to accommodate the relaxed Al3+ ions. This increase in rumpling is also not evident by visual examination. This increases the coordination of the Al3+ ions and reduces the polarization energy of the surface. The change in structure between the bulk and relaxed surface results in a small energy difference and thus a low surface energy value as found in the barium magnetoplumbite and barium β(I)-alumina {001} surfaces. This results in the lowest surface energy structure for the barium β(II)-alumina system.
Figure 3.55. Unrelaxed {001} surface of barium β(II)-alumina.
Ions are color coded as follows: red - O2-, white - Al3+, and blue - Ba2+. Surface image is of four blocks of unit cells of the modeled surface area. The unrelaxed surface shows rumpling in the O2- layer.
Figure 3.56. Relaxed {001} surface of barium β(II)-alumina.
Ions are color coded as follows: red - O2-, white - Al3+, and blue - Ba2+. Surface image is of four blocks of unit cells of the modeled surface area. The O2- layer has a slight
increase in rumpling although it is not evident by visual examination.
Table XX. Surface Energies of the Barium Hexa-Aluminates
Surface Ba-MP (J/m2) Ba β(I) (J/m2) Ba β(II) (J/m2)
{100} 2.09 2.18 2.14
{110} 2.58 2.31 1.97
{120} 2.62 n/a 2.88
{101} n/a 2.62 1.65
{201} n/a 1.65 2.15
{102} n/a 2.22 2.32
{111} n/a 2.47 2.81
{121} n/a n/a 2.82
{112} 2.26 2.41 2.06
{122} 2.40 n/a 3.04
{001} 1.03 1.48 1.25
small enough to lower overall energy to remain stable.
A simple model was used to determine if the surface energy of the barium magnetoplumbite could be stabilized via surface energy effects. The model used the plate- like nature of the crystals, consisting of the {001} and {100} surfaces. The resulting crystal had six {100} and two {001} surfaces for each of the three barium hexa-aluminate phases.
The relative values of the surface energy determined the area of each of the facets for each crystal structure. As the surface energy decreases, the area of the surface increases so that each facet has the same total surface energy for a given size crystal. The total surface energy for each crystal was calculated for a given size crystal. The total surface energy of a crystal plus the lattice energy equals the total energy of the crystal. If stabilization is possible, the total surface energy of the barium magnetoplumbite crystal must be more than 1.79 eV lower than that of the two β-alumina crystals with the same molar concentration of their constituent ions at equilibrium so that the total energy of the barium magnetoplumbite crystal is lower. The size of the barium magnetoplumbite crystal with the same total crystal energy was calculated.
The size of the barium magnetoplumbite crystal for which the reaction would be in equilibrium was 0.405 formula units. Below this size, the barium magnetoplumbite crystal is the stable phase. This result shows that the barium magnetoplumbite cannot be stabilized by its lower surface energy, since the size of 0.405 formula units is smaller than one formula unit of barium magnetoplumbite. This is not physically possible.
Each model used in the surface calculations has several formula units of atoms within the surface block and many times that in the bulk block needed to make these calculations. That is true for each surface calculated. In each direction normal to the surface there must be several blocks of atoms present to make the surface models accurate. Even at eight formula units of barium magnetoplumbite, where all three phases have at least one formula unit, the modeling conducted in this investigation would not create a valid picture of the crystal surfaces.
The overall structures of the {001} surfaces for the three barium hexa-aluminates modeled are very similar. In each of these systems, the Al3+ ions relax into the surface
structure while the O2- layer rumples to accommodate the space needed for the Al3+ ions.
The {001} surface for the barium β(II)-alumina had two termination planes that result in identical calculated surface energies. These corresponded to the O2- and Al3+ layer immediately below the mirror plane. The barium β(I)-alumina had only one lowest energy termination. This occurred on the O2- and Al3+ layer immediately below the mirror plane containing the barium vacancy and the Reidinger defect. Manual surface construction of the {001} plane was necessary to model the barium magnetoplumbite surface.
The more open structure of the barium β(II)-alumina phase resulted in a high number of low dipole moment termination surfaces that were acceptable starting configurations.
Many of these surfaces relaxed to highly polarized structures. Such a degree of polarization results in high surface energies. Excessive polarization and divergent surface energy terminations, where relaxation resulted in an increase in the dipole moment normal to the surface, caused cessation of several calculations. There were some occurrences of this in the barium β(I)-alumina system, but the less open nature of the mirror plane in this phase substantially reduced the number of terminations affected in this way.