1. The internal order of minerals: Lattices, Unit Cell & Bravais Lattices
Geol 3055
Klein (22nd ed), pages &

2. 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

3. …
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)

4. 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

5. One-dimensional order (rows)
Motifs, nodes or objects in a row
In a row the magnitude of one translation determines spacing (distance)

6. Two dimensional order (plane lattices)
Regular translation in two different directions
The connection of four nodes in the figure represent a unit cell (smallest building unit). Various unit cells produce a plane lattice. y γ x
Unit cell

7. 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

8. 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

9. Unit cell’s produce by arrays of nodes
Parallelogram: a≠b, γ≠90o
Fig 5.50a
Rectangles a≠b, γ=90o
Figs 5.50a & b
Square, a1=a2, γ=90o fig 5.50 e
Diamond: a1=a2,
γ≠90o,60o,120o; fig 5.50c
Rhombus: a1=a2, γ=60o
or 120o; Fig 5.50d
P= primitive (only nodes that produce the unit cell corners of figure
C = centered (node at center of unit cell, is called non primitive

10. 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 & 5.63)
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”)

11. 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.

12. 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)