The internal order of minerals:
Lattices, Unit Cell & Bravais
Lattices
Geol 3055
Klein (22nd ed), pages 213-221 &
229-234
Definition of a mineral
• Naturally occuring
• Homogenous solid
• Definite (but not fixed) chemical
composition
• Defined physical properties
• Highly ordered atomic arrangement
• Usually formed by inorganic processes
…
• Ordered atoms distinguished crystals (solids)
from liquids, gases and glasses
• Ordered…periodic repetition of atoms of atom or
ion througout an infinite atomic array.
• An atom is surrounded by an identical
arrangement of neighboring atoms, which are n
quantity of unit cells
• Unit cells dimensions: 5-20 angstroms
(1A=10-8cm)
Translation
• Example of translation (vectors):
,
,
Translation in y-axis
,
,
,
,
Translation in x-axis
,
,
Translation symbols are: t1 for the y axis translation and
t2 for the x-axis translation for 2-D figures. 3-D figures have a
t3
One-dimensional order (rows)
• Motifs, nodes or objects in a row
• In a row the magnitude of one translation
determines spacing (distance)
Two dimensional order (plane lattices)
• Regular translation in two different directions
y
γ
x
• The connection of four nodes in the figure represent a
unit cell (smallest building unit). Various unit cells
produce a plane lattice.
Unit cell
Lattices
• When motifs (commas) are substitute by points (nodes)
the pattern is called a lattice. The nodes represent
atoms or ions.
• Lattice is an imaginary pattern of points (or nodes) in
which every point has an environment that is identical to
that of any other point (node) in a pattern. A lattice has
no specific origin, as it can be shifted parallel to itself
α
Plane lattices
• The are ONLY 5 possible and distinct
plane lattices or nets (see figure 5.50)
– Result by the repetition of a row (translation
along y)
– Depend on the angle γ between x and y, and
the size of the b translation along y
• See Fig 5.50
Unit cell’s produce by arrays of
nodes
Parallelogram: a≠b, γ≠90o
Fig 5.50a
Diamond: a1=a2,
γ≠90o,60o,120o; fig 5.50c
Rectangles a≠b, γ=90o
Figs 5.50a & b
Rhombus: a1=a2, γ=60o
or 120o; Fig 5.50d
Square, a1=a2,
γ=90o fig 5.50 e
P= primitive (only nodes that produce the unit cell are @ corners of figure
C = centered (node at center of unit cell, is called non primitive
Three-dimensional order
• Three vectors (a, b, c) instead of two (a & b)
• The stacking in the c-axis, of the five planar nets
discussed in 2-dimensional figures (fig. 5.50),
will produce 14 different lattice types known as
the Bravais Lattices (see figs. 5.62 & 5.63)
– ONLY possible ways which points can be
arranged periodically in 3 dimensions
– Coincide with the 32 crystal classes studied in
class!
– (see CD-ROM: ”Three dimensional order:
Generation of the Bravais Lattices”)
Three-dimensional order & unit
cells
• Since a lot of unit cells are possible in 3-d
figures, crystallographer drawn some rules
to minimize the number:
– Edges of unit cells should coincide, if
possible, with symmetry axes of the lattice
– Edges should be related to each other by the
symmetry of the lattice
– The smallest possible cell should be chosen
in accordance with first two rules.
14 Bravais Lattices
P = primitive
C = centered
I = body centered
node at center
of figure
F = face centered
(node at the
center of
face(s)