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An ideal crystal is a repetition of
identical structural units in three dimensional space. The periodicity is
described by a mathematical lattice (which are
mathematical points at specific coordinates in space), the identical structural
units (or base of the crystal) are the
atoms in some specific arrangement which are unambiguously placed at every
lattice point. Note that a lattice is not a
crystal, even so the two words are often used synonymously in
colloquial language, especially in the case of elemental crystals where the
base consists of one atom only. |
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All possible lattices can be described by a set of three linearly
independent vectors a1,
a2, and a3, the
unit vectors of the lattice. Each lattice point than can be reached by a
translation vector T of the lattice given by |
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(u · a1, v ·
a1, w · a3) |
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With u,v,w = integers. |
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It is convenient, to classify
lattices according to some basic symmetry groups. This yields the 14
Bravais
lattices, which are commonly used to
describe lattice types. Their basic features are shown below (For sake of
clarity, the lattice points are shown as little spheres and occasionally only
"visible" lattice points are shown. These are not atoms, however!) |
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Name of crystal system
Length of Base vectors |
Angles
between axes |
Bravais Lattices |
Cubic
a1= a2 = a3 |
a =
b = g =
900 |
cubic primitive |
cubic body centered (bcc) |
cubic face centers (fcc) |
Tetragonal
a1= a2 not a3 |
a =
b = g =
900 |
Tetragonal primitive |
Tetragonal body centered |
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Hexagonal
a1= a2 not a3 |
a =
b = 900,
g = 1200 |
Hexagonal (elementary cell continued to show hex. symmetry) |
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Rhombohedral
a1= a2 = a3 |
a =
b = g
¹ 900 |
Rhombohedral |
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Orthorhombic
a1 ¹ a2
¹ a3 |
a =
b = g
¹ 900 |
Orthorhombic primitive |
Orthorhombic body centered |
Orthorhombic base face centered
Orthorhombic face centered |
Monocline
a1 ¹ a2¹ a3 |
a =
b = 900, g
¹ 900 |
Monocline primitive |
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Monocline base face centered |
Tricline
a1 ¹ a2
¹ a3 |
a
¹ b
¹ g
¹ 900 |
Tricline |
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A
crystal now is obtained by taking a
Bravais lattice and adding a
base! |
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The base can just be one atom (as in the case of
many elemental crystals, most noteworthy the metals), two identical atoms (e.g.
Si, Ge, C(diamond)), two different atoms (NaCl,
GaAs, ...) three atoms, ... up to huge complex molecules as in the case
of protein crystals. |
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An arbitrary example is shown below |
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For certain applications, a Bravais
lattice may not be the best choice. Whereas, for example, it shows best the
cubic symmetry of the cubic lattices, its elementary cell is not a
primitive unit cell of the lattice,
i.e. there are unit cells with a smaller volume (but without the cubic
symmetry). For other cases (especially if working in reciprocal lattices) the
choice of a Wigner-Seitz cell may be appropriate, which is
obtained by intersecting all lines from one lattice point to neighboring points
at half the distance with planes at right angles to the lines |
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This is shown schematically below: The blue lines
connect lattice points, the red lines denote the intersection at right angles.
The resulting Wigner-Seitz cell and its use in constructing the lattice are
shown in yellow. |
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In practical work, one oftens refers
to crystal types instead of lattices by using the name of prominent
crystals, crystallographers or minerals etc.; e.g. "diamond type,
Perovskites, "Zinkblende" structure and so on. A
few examples are given in the
link. |
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2.1.1 Simple Vacancies and Interstitials
