Physics 927
E.Y.Tsymbal
1
Section 1: Crystal Structure
A solid is said to be a crystal if atoms are arranged in such a way that their positions are
exactly
periodic
. This concept is illustrated in Fig.1 using a twodimensional (2D) structure.
Fig.1
A perfect crystal maintains this periodicity in both the
x
and
y
directions from 
∞
to +
∞
. As follows
from this periodicity, the atoms A, B, C, etc. are
equivalent
. In other words, for an observer located at
any of these atomic sites, the crystal appears exactly the same.
The same idea can be expressed by saying that a crystal possesses a
translational symmetry
. The
translational symmetry means that if the crystal is translated by any vector joining two atoms, say
T
in
Fig.1, the crystal appears exactly the same as it did before the translation. In other words the crystal
remains
invariant
under any such translation.
The structure of all crystals can be described in terms of a
lattice
, with a group of atoms attached to
every lattice point. For example, in the case of structure shown in Fig.1, if we replace each atom by a
geometrical point located at the equilibrium position of that atom, we obtain a crystal lattice. The
crystal lattice has the same geometrical properties as the crystal, but it is devoid of any physical
contents.
There are two classes of lattices: the
Bravais
and the
nonBravais
. In a Bravais lattice all lattice points
are equivalent and hence by necessity all atoms in the crystal are of the same kind. On the other hand,
in a nonBravais lattice, some of the lattice points are nonequivalent.
Fig.2
In Fig.2 the lattice sites A, B, C are equivalent to each other. Also the sites A
1
, B
1
, C
1
, are equivalent
among themselves. However, sites A
and A
1
are not equivalent: the lattice is not invariant under
translation AA
1
.
A
B
C
A
1
B
1
C
1
y
x
A
B
C
a
1
T
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E.Y.Tsymbal
2
NonBravais lattices are often referred to as a
lattice with a basis
. The basis is a set of atoms which is
located near each site of a Bravais lattice. Thus, in Fig.2 the basis is represented by the two atoms A
and A
1
. In a general case crystal structure can be considered as
crystal structure = lattice + basis.
The lattice is defined by fundamental translation vectors. For example, the position vector of any
lattice site of the two dimensional lattice in Fig.3 can be written as
T
=n
1
a
1
+n
2
a
2
,
(1.1)
where
a
1
and
a
2
are the two vectors shown in Fig.3, and n
1
,n
2
is a pair of integers whose values depend
on the lattice site.
Fig.3
So, the two noncollinear vectors
a
1
and
a
2
can be used to obtain the positions of all lattice points
which are expressed by Eq.(1). The set of all vectors
T
expressed by this equation is called the
lattice
vectors
. Therefore, the lattice has a translational symmetry under displacements specified by the lattice
vectors
T
. In this sense the vectors
a
1
and
a
2
can be called
the primitive translation vectors
.
The choice of the primitive translations vectors is not unique. One could equally well take the vectors
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 Fall '11
 STAFF
 Physics, Crystallography, Crystal system

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