Introduction
A family of structures can be derived from the cubic closest packed(ccp) and the hexagonal closest packed (hcp) structures by combinations of interpenetrating ccp lattices and of interpenetrating hcp lattices. It will be shown how the simple cubic, cubic diamond, fluorites/antifluorites, and body centered cubic structures are derived from ccp lattices and how the hexagonal diamond (wurtzite) and the nickel arsenide structures are derived from hcp lattices. The interpenetrated structures are no longer closest packed, their densities will vary with the types of structural combinations and the size of the atoms. In this project we will be limiting our concerns to structures of "infinite" 3-dimensional complexes.
This project originates from my explorations in modeling of the isotropic vector matrix (ivm) in Synergetics (Ref. 1). This work would not have been possible without Gerald de Jong's unique CAD program, Struck, which was used in building all the structures shown in this project. The structures to follow are shown as 3D stereo illustrations. The images are shown in the cross-eye (convergent) stereo format. If you are not able to see in cross-eye stereo format please go to stereo viewing.
The ivm (Fig. 1) is topologically the same as the cubic closest packed (ccp) lattice or the face centered cubic (fcc) lattice (Fig. 2a and 2b). We obtain this lattice by connecting the centers of adjacent spheres or atoms in the ccp configuration. This ccp lattice can be viewed as composed of tetrahedra and octahedra with twice as many tetrahedra as octahedra, and with the tetrahedron always face bonded to an octahedron and vice versa.
We will be using nomenclature and points of view familiar to chemists whenever possible.
The method of interpenetrating lattices shows where the atoms are in space in relation to each other just the same as traditional representations of chemical structures. Although it is a more complex system, the interpenetrating lattices method simplifies the description and building of more complex structures and at times it could elucidate some puzzling aspects of a structure. It would also eliminate the need for describing unit cells. The traditional structural view is usually the net or lattice interconnecting the interpenetrating lattices, i.e. the diamond net is the 4-connected net interconnecting two fcc lattices (Notes 1). The combination of the two views might be more ideal for visualizing crystal structures, and it is how the illustrated structures are built with Struck, although they will be shown separate at times for comparison.The lattices shown below (Fig. 1, 2a, and 2b), the ivm, ccp, and fcc lattices are all the same.
Fig. 1. The isotropic vector matrix (ivm), 2 frequency.
Fig. 2a. The ccp or fcc lattice, 8 unit cells.
Fig. 2b. This is the same structure as (Fig. 2a), from the view of (Fig. 1).
Fig. 3a. Traditional unit cells of ccp, face centered cubic (fcc).
Fig. 3b. Alternative unit cells of ccp, cuboctahedrally coordinated as shown on page 130 (Ref 2).
Interpenetrating ccp lattices
First we will identify the four lattices and their relationships, then we will show the interpenetrating combinations.
The four lattices are the ccp lattice itself and 3 other interstitial lattices which are all topologically identical ccp lattices. This is the filling of voids or filling of interstitial holes view.
The synergetics view comes from the observation of the concentric polyhedral growth within the ivm or ccp structure suggesting the existence of the 4 ivms, which was observed in Synergetica in 1985, by Russell Chu (Notes 9).
The origin of the four lattices can be seen in the concentric hierarchy in (Fig. 4a. 4b), this is their relationship to each other in space. The four lattices are topologically identical but they occupy different space when they are interpenetrated. The spatial relationship between the 4 lattices are always the same. They are as shown in the concentric hierarchy. The 13 vertexial axes of rotational symmetry shown in (Fig. 4a) are the same as the 4-fold, 3-fold, and 2-fold rotational symmetry of the cube shown on pg. 21 (Ref. 2).
The 13 vertexial axes:
3 axes of 4-fold symmetry, corresponding to the XYZ coordinate axes, rotational axes through the faces of the cube, or the vertexial axes of the octahedron.
4 axes of 3-fold symmetry, corresponding to the corners of the cube, or the vertexial axes of the dual tetrahedra.
6 axes of 2-fold symmetry, corresponding to the edges of the cube or the vertexial axes of the cuboctahedron.
Fig. 4a. The concentric hierarchy.
Fig. 4b. The concentric hierarchy showing the origin of the 4 ccp lattices: A, B, C, and D.
Fig. 4c. The concentric hierarchy showing the relationships with other polyhedra occurring in chemical structures. From inside out we have: tet+, tet-, cube in blue, octahedron, rhombic dodecahedron in red, truncated octahedron, and the cuboctahedron. The icosahedron (five-fold symmetry) has been left out from this presentation. More about concentric hierarchy from Kirby Urner.
The four ccp lattices:
1. The ccp lattice obtained by connecting the centers of adjacent spheres in the cubic closest packing configuration. This is the cuboctahedrally coordinated ccp lattice "D", where the cuboctahedron is at the center of the lattice (Fig. 3b, 5d)
2. The ccp lattice obtained by connecting all octahedra centers, which are the octahedral interstices or octahedral voids. This is the octahedrally coordinated ccp lattice "C" (note that this is the familiar view of the fcc unit cell), (Fig. 3a, 5c)
3. The ccp lattice obtained by connecting all negative tetrahedra ( Tet-) centers. This is the Tet- coordinated fcc lattice "B" (Fig. 5b), (negative and positive tetrahedra are arbitrary designations to differentiate the two kinds of tetrahedra that are dual of each other)
4. The ccp lattice obtained by connecting all positive tetrahedra ( Tet+) centers. This is the Tet+ coordinated fcc lattice "A" (Fig. 5a)
Organized in order of hierarchy we have as shown in (Fig. 4b):
- A as Tet+ coordinated ccp, shown as red tetrahedron,
- B as Tet- coordinated ccp, shown as blue tetrahedron,
- C as Octahedrally coordinated ccp , and
- D as Cuboctahedrally coordinated ccp.
Although A, B, C, and D lattices are topologically identical they occupy different location in space relative to each other, as described above and shown in the concentric hierarchy.
See below for a view of the expanded A, B, C, and D lattices:
Fig. 5a. The A ccp lattice, with center tetrahedron in red. The blue cube was added to show the cubical relationship.
This is a 3 frequency Tet+. Tetrahedron of 3 unit edge.
Fig. 5b. The B ccp lattice, with center tetrahedron in blue.
Fig. 5c. The C ccp lattice, with center octahedron in red.
Fig. 5d. The D ccp lattice, with center cuboctahedron in red.
This is a 4 frequency (4f) ccp lattice, or 8 ccp unit cells.
Fig. 5e. All four lattices A, B, C, and D together.
This is the body centered cubic (bcc) configuration, or two interpenetrating simple cubic (sc) structures.
The outer simple cubic "cage" are C+D lattices, each of 4f or 8 fcc unit cells, same as (Fig. 5c and 5d).
Together they make up a simple cubic lattice of 64 sc unit cells or they can be identified as simple cubic of 4 frequency.
The inner simple cubic lattice are the A+B lattices, each of 3f. The A+B sc lattice would be the inner shell or the "caged" lattice at 3f, or 27 sc unit cells.
In Struck the structure in (Fig. 5e) is built with all 4 ccp lattices together with each lattice in a separate group. Figures 5a, 5b, 5c and 5d all come from the same structure (Fig. 5e) by highlighting one group at a time.
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