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

1. The internal order of minerals: Lattices, Unit Cell & Bravais Lattices Geol 3055 Klein (22nd ed), pages 213-221 & 229-234

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 γbetween 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 are @ 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. 5.62 & 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”)

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)