A crystal is a periodic array of atoms. Many elements and quite a few compounds are crystalline at low enough temperatures, and many of the solid materials in our everyday life (like wood, plastics and glasses) are not crystalline. Nevertheless, typical solids state physics texts start with the discussion of crystals for a good reason: The treatment of a large number of atoms is immensely simplified if they are arranged into a periodic order.
Figure 1.1:
Examples of crystals in two dimension. Dots, curved lines or shaded areas
represent various molecules or atomic arrangements.
Figure 1.1 shows a few two-dimensional ``crystals''.
All crystals have discrete translational symmetry: If displaced by a
properly selected lattice vector 
, every atom moves to the
position of an identical atom in the crystal.
Due to this translational symmetry, a crystal can be constructed
by repeating the basis at every Bravais lattice point.
The basis is the ``building block" of the crystal.
It may be simple, a spherical atom, or as complex as a DNA molecule.
Sometimes we have to use a basis made up of two (or more) atoms, even if there
is only one type of atom in the crystal (see the example in Figure
1.2).
The Bravais lattice, or space lattice, is an infinite array of
points, determined by the lattice vectors , where
such that every 
is an integer. The 
's are the three
primitive vectors of the Bravais lattice; in three dimensions they must
have a nonzero 
product.
There are an infinite number of different choices for the primitive
vectors of a given lattice. For example, 
; 
; and 
will describe the same lattice.
The lattice spacings
are the lengths of the shortest possible set of primitive vectors.
All three crystals in Figure 1.1 have the same Bravais lattice. Note that not all symmetric arrays of points are Bravais lattices! For example, Figure 1.2 shows a honeycomb lattice and a choice for its Bravais lattice and basis.
Figure 1.2: Example of a regular array of points that is not a Bravais lattice
(honeycomb lattice).
In addition to the translational symmetry, most crystals also have other symmetries, including reflection, rotation, or inversion symmetry, or more complicated symmetry operations, like the combination of rotation and translation by a fraction of the lattice vector. In Figure 1.1 the three-fold rotational symmetry around the point P is common to all three crystals. The honeycomb lattice (Figure 1.1c) also has a ``mirror line'' m, while the other two crystals do not have this symmetry. A less trivial symmetry operation is mirroring the honeycomb lattice with respect of the line m', and then shifting it parallel to the line, until it overlaps with itself.
The collection of symmetry operations forms a symmetry group.
The important property that defines a symmetry group is the relationship
between the symmetry elements - i.e., what happens if two symmetry
operations are applied subsequently. In the language of group theory,
this relationship is described by the multiplication (direct product)
table.
The symmetry group can be represented in many ways (collections of matrices,
symmetry operations of a simple geometric object, and so on). As long as the
multiplication table is the same, we are dealing with the same group.
The crystals in Figures 1.1a and 1.1b have equivalent
symmetry groups, while some of the symmetries of the honeycomb lattice are
different.
When all possible symmetry operations are taken into account we talk about
crystallographic space groups. Any given three-dimensional
crystal belongs to one of the 230 possible crystallographic space groups.
(Two-dimensional crystals are much simpler; there are only 17 inequivalent
``crystallographic plane groups''.)
The symmetries are often identified by the name of a representative material,
like ``sodium chloride structure'', ``diamond structure'', ``wurzite (or zincblende,
zinc sulfide) structure'', and so on. More sophisticated group theoretical
notations are used by crystallographers.
For a complex structure the identification of the symmetry group may be
a rather nontrivial task.
A subset of symmetry operations that leaves at least one point invariant makes up the crystallographic point group. There are 32 different crystallographic point groups in three dimensions, and 10 in two dimensions. Considering the examples in Figure 1.1, the rotations around P and the mirror line m are point group symmetries, but the combination of mirroring around m' and the subsequent shift is not a point group operation.
Sorting out the symmetries of the Bravais lattices is much simpler. There are 14 different space groups for three-dimensional Bravais lattices, including the simple cubic (sc), face centered cubic (fcc), body centered cubic (bcc), simple tetragonal, body centered tetragonal, and others. Figure 1.3 shows all possible Bravais lattices in two dimensions. It is important to emphasize that the symmetries of the Bravais lattice are intimately related to the symmetries of the original lattice. For example, the three-fold rotational symmetry of the honeycomb lattice results in the requirement that its Bravais lattice must have three-fold rotational invariance (which leaves the hexagonal lattice as the only choice, see Figure 1.2).
Figure 1.3: Two-dimensional Bravais lattices.
Finally, when the point group symmetries of the Bravais lattices are considered, the choices are further limited, and in three dimensions only seven distinct groups are left. These define the seven crystal systems: Cubic, tetragonal, orthorhombic, monoclinic, triclinic, trigonal, and hexagonal. (In two dimensions there are four crystal systems. The rectangular and centered rectangular Bravais lattices shown in Figure 1.3 make up the ``rectangular'' system.)
The primitive unit cell or primitive cell is a volume which will fill space completely, without overlap, if shifted by each of the lattice vectors. The primitive unit cell contains exactly one Bravais lattice point and the atoms in it can be used as the basis to construct the crystal. The volume made up by the primitive vectors is a possible primitive unit cell, but there are many other possibilities. More often than not, the primitive unit cell is less symmetric than the Bravais lattice.
The unit cell is a volume that fills up space with an integer multiplicity, if shifted by each of the lattice vectors. It contains an integer number of lattice points. Sometimes it is more convenient than the primitive unit cell (as shown in Figure 1.4).
Figure 1.4: Some choices of primitive unit cells and the conventional unit
cell for a centered rectangular lattice.
The Wigner-Seitz cell (WS cell) is a volume made up of space which is closer to a given lattice point than to any other point. There is a practical recipe for the construction of this cell: Select a lattice point, draw the lines connecting it to its neighbors (nearest and next-nearest is usually sufficient), and draw the perpendicular bisecting planes to these lines. The smallest volume enclosed within these planes is the WS cell. The WS cell is a primitive unit cell that preserves the symmetries of the Bravais lattice.
Imagine that a crystal is made of spheres with their diameters being equal to the nearest-neighbor distance. The filling factor is the volume fraction of the spheres relative to the total volume. The coordination number is the number of nearest-neighbors to any sphere.
When waves are scattered from a periodic array the constructive interference
is often described by the Bragg condition:
, where 
is the wavelength,
d is the spacing between subsequent lattice planes (that is, planes containing a high
density of lattice points) and 
is the
angle between the incident beam and the lattice planes. The scattered
beam will have the same angle with respect to the planes as the incident
beam, so the total scattering angle is 
.
Instead of wavelength, the concept of the wavevector is often used
to characterize the a plane wave. The wavevector 
points in the
direction of the propagation of the wave, and the magnitude of the vector is
. The condition for constructive
interference can be expressed in terms of wave vectors as
, where n is an integer,
and
are the incident and scattered wavevectors, and 
is
the vector pointing from one scattering center to another.
The reciprocal lattice is a very useful tool to handle the diffraction of waves; it is generally used to describe all things of ``wavy nature" (like electrons and lattice vibrations). Definitions of the reciprocal lattice are as follows:
vector is a
possible period of the Bravais lattice).
satisfying 
, or 
.
and cyclic permutations of 1 2 3, where 
is
the volume of the unit cell.
The volume of the primitive unit cell in the reciprocal lattice is 
.
The crystal system of the reciprocal lattice is the same as the direct
lattice (for example, cubic remains cubic), but the Bravais lattice may be
different (e.g., fcc becomes bcc).
The Brillouin zone is the WS cell in the reciprocal lattice.
Using the reciprocal lattice, the condition for constructive interference
becomes quite simple: If the difference between the incident
( ) and scattered ( ) wave vectors is equal to a reciprocal
lattice vector, the diffracted intensity may be nonzero. This is the
Laue condition. With 
, this leads to
the simple equation 
, or 
. The Ewald construction is a geometric
representation of these equations.
The Miller indices, h, k, and l, are obtained from the
``coordinates" of a reciprocal lattice
vector 
.
By definition, the Miller indices are integers. For a simple cubic
lattice these numbers are real coordinates in a Cartesian coordinate system.
There is an interesting relationship between Miller indices and lattice
planes. For any plane there is an infinite number of other, parallel lattice
planes, separated by a distance d. It is easy to see that the ratio
is the same for all parallel planes, where
x, y, and z are the intercepts of a given plane with the coordinate
axis defined by the primitive vectors 
, 
, 
.
Sometimes the need arises to classify these planes. There is a convenient
mapping between a given class of lattice planes and a lattice vector
in reciprocal space: For any family of lattice planes separated by a
distance d, there is a reciprocal vector with length
, and this vector is perpendicular to the
lattice planes. One can show (nontrivially) that 
.
The Laue condition is based solely on the Bravais lattice,
so the positions of the diffraction peaks are independent of the atomic basis.
However, the intensities of the peaks are strongly influenced by the basis.
The structure factor, 
, and the form factor,
, tell us how the
intensities of the peaks depend on the atoms making up the crystals.
These quantities are
calculated as a sum (or integral) within the unit cell; therefore they may
be totally different for two different crystals, even if the crystals have
the same Bravais lattice. In the simplest approximation the scattering
depends on the atomic charge distribution 
, and the
intensity is proportional to the absolute value squared of
and
where e is the electron charge, the sum is over the atoms in the unit cell, and the integration is over the volume of an atom. Similar formulae work for electron and neutron scattering, except the form factor integral is different depending on the microscopic interaction at play. Even for X-rays, the calculation of the form factor as an integral over the charge density works only for the simplest cases. For a realistic calculation of scattered radiation intensities one has to include factors representing the directional dependence of the scattering by a single point charge, the absorption of the radiation, and other effects. For powder samples this process is called Rietveld analysis. The expression becomes much more complicated for example, if there is a match between the energy of atomic transitions and the X-ray quanta.
When the atomic positions are time-dependent (for example, if lattice waves
are excited in the crystal), the crystal scatters radiation at a frequency
different from the incident frequency. In this case energy is either
absorbed or emitted by the crystal. The process can be described by the
dynamic structure factor, which
depends also on the frequency difference 
between the incident
and scattered radiation: 
. The general expression
relates the structure factor to
,
the density-density correlation function.
(Here N is the number of primitive unit cells and
is the charge density at time t.)
This formula is
equally useful when dynamics of the system are described by quantum
mechanics (and the 
expectation value is that of
the density
operators) or at finite temperature [when the (classical) atoms have thermal
motion]. For a static array of classical atoms the
quantity 
is identical to the
structure factor defined in Eq. 1.2.
C 
C
Laszlo Mihaly
Thu Oct 31 13:23:11 EST 1996
Cover Page
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