Wave Propagation in Solids: Wave Surface "Cross-Overs"

Wave surfaces intersect when there exists a eigenvalue degeneracy in the Christoffel's equations. At these intersections the eigenvalues (wave vibration directions, 0-90 degrees shown as color: purple-0 (longitudinal) and red-90(transverse)) remain associated with the same surface and appear to "jump" or "hop" to an adjoining surface when viewed in three-dimensions (3D) as shown below. We refer to these intersections as wave surface "cross-overs". A closer examination reveals that these cross-overs are the basis for new types of wave surface geometries that can be used for crystal classification.

For example wave surface cross-overs in a Tetragonal symmetry crystal called Tellurium Dioxide (TeO2) are shown below in 3D.

To view larger images select: Top View / Rotated View / Cross-Over Close-up

To view these wave surfaces in three dimensions, two different VRML file formats are provided here that can be viewed with a VRML file web browser plug-in (e.g. google: cortona, cosmo, FreeWRL): VRML1 velocity surfaces, VRML2 velocity surfaces

When all of these wave surface cross-overs are considered the Christoffel's equations
predicted three seperate wave surfaces as expected and is shown in the isometric
figure shown below.

However for an orthorhombic symmetric crystal called Calcium Formate (HCOO2) wave surface cross-overs results in a fully connected "single-surface" geometry.

First construct three wave surfaces using Christoffel's equation. The figure below illustrates how the fourth order stiffness tensor is related to the nano-structure of a unit cell.

Consider three wave surfaces intersecting three orthogonal planes in two-dimensions (2D): x1-x2, x1-x3, and x2-x3.

Now construct these three 2D images into a 3D isometric view and note that there is
only one connected surface, because of four wave surface cross-overs at points a,b,c, and d.
To view these wave surfaces in three dimensions, two different VRML file formats
are provided here that can be viewed with a VRML file
web browser plug-in (e.g. google: cortona, cosmo, FreeWRL):
VRML1 velocity surfaces,
VRML2 velocity surfaces

In fact for orthorhombic symmetry there are four unique types of geometries that can result from wave surface cross-overs.

**Engineering Science and Mechanics: ESM5344 Spring 2007**

Any comments and suggestions about the material presented above,

please contact Ron Kriz at rkriz@vt.edu

bio

Virginia Tech

College of Engineering

http://www.sv.vt.edu/classes/ESM5344/ESM5344_kriz_NoteBook/TeO2_CrossOvers/TeO2_CrossOvers.html