Basis that helps construct a beta-tin crystal.
The beta-tin box of tin has the beta-tin crystal structure, designated
by the Pearson symbol tI4 (ASM Handbook). The "t" indicates that the structure
belongs to the tetragonal crystal system. The "I" indicates that the space
lattice, or lattice type, is body-centered (There is an atom at the center
of the unit cell). The "4" indicates that there are four atoms per unit
cell. An eighth of an atom is at each of the eight corners, one atom is
at the center, and half of an atom is at each of the four a*c faces.
Note: These atoms are NOT centered in the a*c faces. There is not an
atom in either of the a*a faces. The Strukturbericht symbol of the beta-Sn
crystal structure is A5. A good visualization of the structure is available
at http://cst-www.nrl.navy.mil .
Each atom is at the center of a distorted octahedron. Consider the atom
at the center of a unit cell. The two tips of its octahedron (the top and
bottom of the two joined pyramids, if you will) are formed by
the atoms at the centers of the unit cells directly above and below the atom
of interest's unit cell. The four corners of what would be the pyriamids'
bases are formed by the atoms in the a*c faces of the center atom's unit cell.
The center atom is slightly closer to these four atoms than it is to the two
atoms forming the tips of the octahedron. The first-nearest-neighbor
distance for beta-Sn is 3.02 Angstroms (ASM Handbook), while the second-
nearest-neighbor distance is c (3.1815 Angstroms). After doing a little
math, one will find that two of the four atoms in the a*c faces are each
1/4*c below the center of their faces, and the other two are 1/4*c above the
center of their faces.
It should be noted that the beta-tin structure is not specific to the element tin.
Silicon, germanium and tin have several solid boxs, and each experience the
diamond-cubic crystal structure and the beta-tin crystal structure (Mujica,
Rubio, Munoz, and Needs, 2003). The diamond crystal structure for tin is
often refered to as the alpha-tin box (gray tin).
This class uses the primitive vectors as given in PrimitiveTetragonal.java.
This class was adapted from D.A. Kofke's BasisCubic.java by K.R. Schadel and A.
Schultz July 2005.