The internal order of minerals: Lattices, Unit Cell & Bravais

Lattices

Geol 3055Klein (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

• The connection of four nodes in the figure represent a unit cell (smallest building unit). Various unit cells produce a plane lattice.

γ

Unit cell

x

y

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

Rhombus: a1=a2, γ=60o or 120o; Fig 5.50d

Parallelogram: a≠b, γ≠90o

Fig 5.50aRectangles a≠b, γ=90oFigs 5.50a & b

Diamond: a1=a2,γ≠90o,60o,120o; fig 5.50c

Square, a1=a2, γ=90o fig 5.50 e

P= primitive (only nodes that produce the unit cell are @ corners of figureC = 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 = primitiveC = centeredI = body centered node at center of figureF = face centered (node at the center of face(s)