文档介绍：Chapter 2: Crystal Structures and Symmetry
Laue, Bravais
December 28, 2001
Contents
1 Lattice Types and Symmetry 3
Two-Dimensional Lattices . . . . . . . . . . . . . . . . . 3
Three-Dimensional Lattices . . . . . . . . . . . . . . . . 5
2 Point-Group Symmetry 6
Reduction of plexity . . . . . . . . . . . . 6
Symmetry in Lattice Summations . . . . . . . . . . . . . 7
Group designations . . . . . . . . . . . . . . . . . . . . . 11
3 Simple Crystal Structures 13
FCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
HCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
BCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1
A theory of the physical properties of solids would be practically
impossible if the most stable elements were not regular crystal lattices.
The N-body problem is reduced to manageable proportions by the ex-
istence of translational symmetry. This means that there exist a set
of basis vectors (a,b,c) such that the atomic structure remains invari-
ant under translations through any vector which is the sum of integral
multiples of these vectors. As shown in Fig. 1 this means that one may
go from any location in the lattice to an identical location by following
posed of integral multiples of these vectors.
a α
b
Figure 1: One may go from any location in the lattice to an identical location by
following posed of integral multiples of the vectors a and b.
Thus, one may label the locations of the ”atoms”1. pose
the lattice with
rn = n1a + n2b + n3c (1)
1we will see that the basic building blocks of periodic structures can be plicated than
a single atom. For example in NaCl, the basic building block posed of one Na and one Cl ion
which is repeated in a cubic pattern to make the NaCl structure
2
where n1, n2, n3 are integers. In this way we may construct any periodic
structure.
1 Lattice Types and Symmetry
Two-Dimensional Lattices
Thes