The Unit Cell;

1. The Unit Cell; Crystallography, Crystal Symmetry, and Crystal Systems

2. The Unit Cell • Atomic scale (5-15 D) parallelepipeds. • Contains 1 or an integral multiple of chemical formula units (“Z” number). • Most minerals do not have “molecules”, per say, but rather almost infinite three dimensional repetition of the unit cell to form crystals. Sphalerite; (Zn,Fe)S (Z = 4)

3. The Unit Cell • Crystals can occur in any size and may (or may not!) express the internal order of constituent atoms with external crystal faces. • Euhedral, subhedral, anhedral. • The unit cell is often used in classification at the subclass or group classification level. • Relates chemistry and structure through coordination number and the geometric relationships of elemental components.

4. The Unit Cell • The unit cell has electrical neutrality through charge sharing with adjacent unit cells • The unit cell geometry reflects the coordination principle

5. The Unit Cell and “Z” Number • On the basis of crystal structural analysis (using x-ray diffraction) the size and shape of orderly, repeating structural units in a mineral are determined nλ=2dsinθ

6. X-ray Diffractograms X-ray energy Reflection angle

7. The Unit Cell and “Z” Number • Determined through density-geometry calculations (see take home exercise #3) • Determined, practically, in unit cell models, by “fractional ion contribution” (ion charge balance) calculations

8. The Unit Cell and “Z” Number Galena and Halite: “Z” = 4

9. Charge Balances and “Z” in Unit Cell Models • Ions entirely within the unit cell • 1x charge contribution • Ions on faces of the unit cell • 1/2x charge contribution • Ions on the edges of the unit cell • 1/4x charge contribution • Ions on the corners of the unit cell • 1/8 charge contribution

10. Charge Balances and “Z” in Unit Cell Models Fluorite, CaF2 Z =4 Cassiterite, SnO2 Z = 2

11. Coordination Polyhedrons and Unit Cells • They are not the same!! • Relationship between the unit cell and crystallography! • Crystal systems and reference, axial coordinate system

12. Unit Cells and Crystals • All crystals have basic building blocks called unit cells • Unit cells are arranged in infinite patterns called lattices • The relative proportions of elements in the unit cell are indicated by the chemical formula • The integral number of formula units in the unit cell is called “Z” the repeating “building blocks” of crystals and a fundamental physical property of all minerals Sphalerite, (Zn,Fe)S, Z=4

13. Unit Cells and Crystals • Crystals (composed of infinite arrays of unit cells) belong to one of six crystal systems • Unit cells of distinct shape and symmetry characterize each crystal system • Total crystal symmetry depends on unit cell and lattice symmetry

14. Crystal systems reference, axial coordinate system • Defines the geometry of 3D polyhedral solids (axis length and angle) • Both megascopic crystals and unit cell building blocks of crystals otbo • Geometric arrangements of • Faces (planes, 2d geometric features) • Edges (lines, 1d geometric features) • Corners (points, dimension-less geometric features)

15. Crystal Systems:6 (or 7including Trigonal) • Defined by symmetry (physical manipulation resulting in repetition) • Symmetry elements • Center of symmetry = center of gravity; Every face, edge and corner repeated by an inversion (2 rotations about perpendicular axis) • Axis of symmetry = line about which serial rotation produces repetition. The number of serial rotations in 3600 rotation determines “foldedness” 1 (A1), 2 (A2), 3 (A3), 4 (A4), 6 (A6) • Plane of symmetry = plane of repetition (mirror plane)

16. The 6 Crystal Systems • Cubic (isometric) • High symmetry: 4A3 (equant xtls) • Tetragonal • 1A4

17. The 6 Crystal Systems • Hexagonal (trigonal) • 1A6 or 1A3 • Orthorhombic • 3A2 (mutually perpendicular)

18. The 6 Crystal Systems • Monoclinic • 1A2 • Triclinic • 1A1