Multiscale Statistics & Crystals

Magnesium Oxide (MgO) is a ceramic with molecular units distributed in space in an essentially cubic arrangement.  As with many materials, the energy corresponding to a given plane within this cubic arrangement depends on the normal vector to that plane.  For example, a point (x,y,z) on the surface, the z axis represents the energy and (x,y)  represents a particular choice of the normal vector to a given plane within the crystal.  Purely experimental determination of the energy corresponding to a wide variety of planes is tedious.  Consequently, accurate representations are rare. 

One way to circumvent this problem is to experimentally collect (via automation) vast amounts of geometric data from specimens and then employ such data in physically based equations wherein the data acts as coefficients, and the corresponding surface energy as unknowns.  This typically leads to thousands of simultaneous and overdetermined linear equations. Through intelligent multiscale sorting into urns and averaging of these equations, accurate representations of the surface energy can be developed.  The images at left depict computed representations of the surface energy of the MgO at two consecutive levels of urn refinement.

This work was completed in 2000 as part of a larger effort called the Mesoscale Interface Mapping Project (MIMP), an NSF funded research center at Carnegie Mellon University.  Darren E. Mason in the Albion College Department of Mathematics and Computer Science is an affiliated faculty member of this project. For more information about this work and other exciting interdisciplinary projects in materials science, check out the MIMP web page as well as Dr. Mason's web page.