1.2 The Structure of Solids 5
and the Penrose tilings. The latter allows a local fivefold symmetry, which
is prohibited in crystalline solids. In contrast, this lo ng-range order is missing
completely in amorphous solids which are characterized by disorder in the
spatial configuration (structural disorder). The X-ray diffraction patterns of
quasi-crystals show sharp p eaks owing to the long-range order, while those of
amorphous solids are diffuse. Liquid crystals have long-range order but not
in all spatial directions. The building blocks – usually large rod-like or cyclic
molecules – are arranged such that a long-range order exists with respect to
the orientation of these molecules in at least one direction, whereas in other
directions, a liqui d structure prevails. Because of their order–disor der phase
transitions at roo m temperature which can be triggered by applied voltages,
liquid crystals have seen widespread use in displays and large scale television
screens. Finally, we refer to soft matter, a class of materials that comprises
foams, polymer melts, biological membranes, and colloid systems. Their par-
ticular material properties result from structuring on a mesoscopic scale on
which normal liquids and solids are homogeneous.
Let us return to the periodic structures. Their systematic description can
be found at the beginning of most Solid State Physics books and shall be
repeated here only briefly (see Problem 1.1). An infinite periodic structure
can be characterized by a point lattice, which in three dimensions is defined
by the set of lattice vectors
R
n
= n
1
a
1
+ n
2
a
2
+ n
3
a
3
(1.1)
with linearly independent vectors a
i
, (i =1, 2, 3), the primitive lattice vec-
tors, and integers n
i
, (i =1, 2, 3) combined to n =(n
1
,n
2
,n
3
). Point lattices
in one and two dimensions are defined analogously. While in one dimension
there is only one point lattice, there are 5 in two (Problem 1.3) and 14 in
three dimensions [
29].
The point lattice is used to define the crystal unit cell or its particu-
lar choice, the Wigner–Seitz
2
cel l, which by repetition fills the whole space.
Clearly, the lattice structure is mapped onto itself under a translation by a
lattice vector. Mathematically, these operations form the translation group
of the point lattice (see Appendix). Lattice translations commute with the
system Hamiltonian and allow one to characterize the quantum states of the
solid by a wave vector k. It corresponds to the linear momentum. However,
as the crystalline solid is only invarian t under the discrete lattice (and not
under infinitesimal) translations, the meaning of this mo mentum is modified
as will be explained below. Therefo re, it is called crystal momentum.
A crystal structure is obtained by assigning an atom or a group of atoms
to each lattice point. The former case corresponds to the Bravais
3
lattices.
For the latter case, called lattice with basis, the position of the atoms can be
2
Eugene Paul Wigner 1902–1995, Nobel prize in physics 1963; Frederick Seitz
1911–2008.
3
Auguste Bravais 1811–1863.