24
3 Broken Symmetry
by the structure of a unit cell, infinitely repeated by translation in three dimensions. There are 32 possible crystal classes. Johann Hessel (1831) derived them,
without the benefit of mathematical group theory, by studying actual crystal forms.
This classification, published in an obscure Physics Dictionary, went unnoticed until reprinted posthumously in 1897. Auguste Bravais (1850) proved the existence of
14 Bravais lattices in three dimensions, based on 14 types of unit cells that can be
repeated by translating in all three dimensions to build up all crystalline structures.
Additional crystal classes can be obtained by rotating unit cells.
Fig. 3.1 Ionic lattices with the coordination numbers 6 (left) and 8 (center) and a metal lattice with
the coordination number 12 (right)
It is far more difficult to predict the
shape of a unit cell and hence the type
of lattice formed by a particular chemical species: this depends on the arrangement of atoms and a preferred orientation of bonds that would optimize
their interactions and reduce the energy
of the crystal. Even apparently similar
crystals with a simple composition, like
pure metals composed of identical atoms held together by delocalized electrons or
ionic crystals containing ions of opposite sign may settle into different unit cells. In
both cases, the energy minimum can be attained in different compounds with different coordination numbers, i.e., different numbers of adjacent metal atoms or ions
of the opposite sign, which interact attractively. For ions of different sizes, like Na +
and Cl − in common table salt (NaCl), the optimal three-dimensional ionic arrangement excluding contacts between mutually repulsive ions of the same sign has the
coordination number six – along the three mutually perpendicular directions in a cubic lattice (Fig. 3.1, left). Ions of about the same size can be packed more efficiently
with the coordination number eight, with each ion sitting in a body-centered cubic
cell surrounded by the corresponding counter-ions (Fig. 3.1, center). For identical
metal atoms, the maximum coordination number with twelve neighbors is possible
(Fig. 3.1, right).
Structures held by covalent bonds are more varied, since such bonds have a certain direction. Carbon atoms commonly have four covalent bonds directed to the
vertices of a tetrahedron. This is the basis of the diamond structure (Fig. 3.2, left).
δ−
δ−
δ−
δ−
δ+
δ+
δ+
δ+
H
H
O
1
Fig. 3.2 The structure of diamond (left) and graphite (center). Right: Hydrogen bonds (shown by
dotted lines) in ice
3 Broken Symmetry
by the structure of a unit cell, infinitely repeated by translation in three dimensions. There are 32 possible crystal classes. Johann Hessel (1831) derived them,
without the benefit of mathematical group theory, by studying actual crystal forms.
This classification, published in an obscure Physics Dictionary, went unnoticed until reprinted posthumously in 1897. Auguste Bravais (1850) proved the existence of
14 Bravais lattices in three dimensions, based on 14 types of unit cells that can be
repeated by translating in all three dimensions to build up all crystalline structures.
Additional crystal classes can be obtained by rotating unit cells.
Fig. 3.1 Ionic lattices with the coordination numbers 6 (left) and 8 (center) and a metal lattice with
the coordination number 12 (right)
It is far more difficult to predict the
shape of a unit cell and hence the type
of lattice formed by a particular chemical species: this depends on the arrangement of atoms and a preferred orientation of bonds that would optimize
their interactions and reduce the energy
of the crystal. Even apparently similar
crystals with a simple composition, like
pure metals composed of identical atoms held together by delocalized electrons or
ionic crystals containing ions of opposite sign may settle into different unit cells. In
both cases, the energy minimum can be attained in different compounds with different coordination numbers, i.e., different numbers of adjacent metal atoms or ions
of the opposite sign, which interact attractively. For ions of different sizes, like Na +
and Cl − in common table salt (NaCl), the optimal three-dimensional ionic arrangement excluding contacts between mutually repulsive ions of the same sign has the
coordination number six – along the three mutually perpendicular directions in a cubic lattice (Fig. 3.1, left). Ions of about the same size can be packed more efficiently
with the coordination number eight, with each ion sitting in a body-centered cubic
cell surrounded by the corresponding counter-ions (Fig. 3.1, center). For identical
metal atoms, the maximum coordination number with twelve neighbors is possible
(Fig. 3.1, right).
Structures held by covalent bonds are more varied, since such bonds have a certain direction. Carbon atoms commonly have four covalent bonds directed to the
vertices of a tetrahedron. This is the basis of the diamond structure (Fig. 3.2, left).
δ−
δ−
δ−
δ−
δ+
δ+
δ+
δ+
H
H
O
1
Fig. 3.2 The structure of diamond (left) and graphite (center). Right: Hydrogen bonds (shown by
dotted lines) in ice