Case of Cubic Structures
For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length (usually denoted a); similar to the reciprocal lattice. Thus, in this common case, the Miller indices (hkℓ) and both simply denote normals/directions in Cartesian coordinates.
For cubic crystals with lattice constant a, the spacing d between adjacent (hkℓ) lattice planes is (from above):
- .
Because of the symmetry of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes:
- Coordinates in angle brackets such as ⟨100⟩ denote a family of directions which are equivalent due to symmetry operations, such as, or the negative of any of those directions.
- Coordinates in curly brackets or braces such as {100} denote a family of plane normals which are equivalent due to symmetry operations, much the way angle brackets denote a family of directions.
For face-centered cubic and body-centered cubic lattices, the primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic supercell and hence are again simply the Cartesian directions.
Read more about this topic: Miller Index
Famous quotes containing the words case, cubic and/or structures:
“Consumer wants can have bizarre, frivolous, or even immoral origins, and an admirable case can still be made for a society that seeks to satisfy them. But the case cannot stand if it is the process of satisfying wants that creates the wants.”
—John Kenneth Galbraith (b. 1908)
“Mining today is an affair of mathematics, of finance, of the latest in engineering skill. Cautious men behind polished desks in San Francisco figure out in advance the amount of metal to a cubic yard, the number of yards washed a day, the cost of each operation. They have no need of grubstakes.”
—Merle Colby, U.S. public relief program (1935-1943)
“The American who has been confined, in his own country, to the sight of buildings designed after foreign models, is surprised on entering York Minster or St. Peter’s at Rome, by the feeling that these structures are imitations also,—faint copies of an invisible archetype.”
—Ralph Waldo Emerson (1803–1882)