Christopher J. Kimmer, Ph.D.
IU Southeast Informatics
iSci
the informatics of scientific computing
The Physical Space Structure of the Surface
So far, several tools to describe the real space structure of the surface
have been assembled. The real space data points used to compute the
discrete free energy functional provide the most accurate data.
It is reasonable to assign a value
of the order parameter to a point in parallel space (at
x
⊥
= 0
)
based on which real space grid point it is closest to, but this is in
general a cumbersome procedure for the high resolution data.
The function of the two-Fourier modes,
(
x
)
, and the Voronoi construction provide
a means to map the surface in physical space, and the existence of these
distinct tools provides self-consistency checks between the Voronoi
construction and
(
x
)
.
The value of
(
x
)
may be quickly calculated for any point in
physical space, or the zero surface, and hence an approximation to the
minimal surface, in physical space
can be computed. It is
also a fast operation to compute which extremum's Voronoi
cell a given point belongs to and associate a sign of the order parameter
field with it.
Near the origin in physical space, all three ways of generating the isosurface give similar results and agree on the topology of the surface. Further from a lattice point where the 32 Voronoi cells of the extrema are centered, as the boundaries of the Voronoi cells are approached, the approximations become less accurate, and care must be taken. Elsewhere in physical space, points further away are modelled correctly corresponding to the core of the Voronoi construction that is expected to be modified little if at all from the present structure as the t = 0 limit is reached. Nonetheless, all these methods offer at best educated guesswork towards determining the real space topology of the structure, and it is hard to be too confident about any predictions made far away from the origin.
The Voronoi construction provides a means to relate the minimal surface in physical space with a quasilattice; a sampling of points in a region of physical space can be used to obtain the set of projected lattice points in that region whose extrema's Voronoi cells are intersected by the cut-plane. The local labyrinth structure of the minimal surface is determined by the neighborhoods of the quasilattice points and is dominated by four motifs. Most notably, surface patches with local icosahedral symmetry such as the one about the origin have 12 labyrinths issuing forth along the 5-fold symmetry directions and correspond to a quasilattice point whose nearest neighbors along all 12 5-fold and 20 3-fold directions are icosahedrally symmetric. Additional local surface patches involve two-, three-, or five-fold saddle-shaped regions. Since this classification of the local geometries of the surface in physical space is based primarily on visual inspection, it is not necessarily exhaustive. Further details about the local environments in physical space are deferred until Chapter 4.
Chris Kimmer 2011-06-01