It is a 2 & times; 2 Hadamard matrix, and its rows form the basis of a diagonal square lattice consisting of the integer points whose coordinates both have the same body-centered cubic lattice.
32.
For the square lattice, this is the group of the square, or the dihedral group of order 8; for the three-dimensional cubic lattice, we get the group of the cube, or octahedral group, of order 48.
33.
This arrangement is known as the simple cubic lattice, and it has a packing efficiency of only 52 percent, that is, the gaps among the spheres are almost as big as the space occupied by the spheres.
34.
Assuming one atom per lattice point, in a primitive cubic lattice with cube side length " a ", the sphere radius would be and the atomic packing factor turns out to be about 0.524 ( which is quite low ).
35.
The IVM scaffolding or skeletal framework was defined by cubic closest packed spheres ( CCP ), alternatively known as the FCC or face-centered cubic lattice, or as the octet truss in architecture ( on which Fuller held a patent ).
36.
For example, for a spherical particle with a cubic lattice, the factor of 5.56 simply becomes 6.96, when the size is the diameter " D, " i . e ., the diameter of a spherical nanocrystal is related to the peak FWHM by
37.
Evolution of ?-PtCl 2 structure : Start with cubic lattice, remove corner and centered lattice points, inscribe octahedron ( red lines ), label corners as X ( twelve Cl " centers ) and face-centered atoms as M ( six Pt ( II ) centers ).
38.
Evolution of " ? "-PdCl 2 structure : Start with cubic lattice, remove corner and centered lattice points, inscribe octahedron ( red lines ), label corners as X ( twelve Cl " centers ) and face-centered atoms as M ( six Pd ( II ) centers ).
39.
These include the cubic lattice with a density of | 6 } } C " 0.5236, the hexagonal lattice with a density of | 3 } } C " 0.6046, and the tetrahedral lattice with a density of | 16 } } C " 0.3401, and loosest possible at a density of 0.0555.
40.
The same ideas apply for other types of lattices where the coordination number is not six, but these are not as easy to visualize and work with in theory, so the remainder of the discussion will focus on simple cubic lattices . "'Table 2 "'indicates the number of neighboring atoms for a bulk atom in some other crystal lattices.
