38
3 Crystals
Fig. 3.4 The 14
three-dimensional Bravais
lattices: cubic (sc: simple
cubic, bcc: body-centered
cubic, fcc: face-centered
cubic), tetragonal (simple
and body-centered),
orthorhombic (simple,
centered, body-centered
and face-centered),
monoclinic (simple and
centered), triclinic,
rhombohedral and
hexagonal
fcc
bcc
tetragonal
monoclinic
rhombohedral
orthorhombic
triclinic
hexagonal
In the fcc lattice, one lattice point sits in the center of each of the six faces of the usual cubic unit
cell. The vectors spanning the primitive unit cell are
a 1 =
a
2
(1, 1, 0) , a 2 =
a
2
(0, 1, 1) , a 3 =
a
2
(1, 0, 1) .
(3.2)
In the bcc lattice, one extra lattice point sits at the intersection of the three body diagonals at (a 1 +
a 2 + a 3 )/2. The vectors spanning the primitive unit cell are
a 1 =
a
2
(1, 1, −1) , a 2 =
a
2
(−1, 1, 1) , a 3 =
a
2
(1, −1, 1) .
(3.3)