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

2. What is a Unit Cell?
Unit cell for ZnS, Sphalerite
Smallest divisible unit of a mineral that possesses the symmetry and chemical properties of the mineral
Atoms are arranged in a "box" with parallel sides - an atomic scale (5-15 Å) parallelepiped
The “box” contains a small group of atoms proportional to mineral formula
Atoms have a fixed geometry relative to one another
Atoms may be at the corners, on the edges, on the faces, or wholly enclosed in the “box”
Each unit cell in the crystal is identical
Repetition of the “box” in 3 dimensions makes up the crystal

3. The Unit Cell
Unites chemical properties (formula) and structure (symmetry elements) of a mineral
Chemical properties:
Contains a whole number multiple of chemical formula units (“Z” number)
Same composition throughout
Structural elements:
14 possible geometries (Bravias Lattices); six crystal systems (Isometric, Tetragonal, Hexagonal, etc.)
Defined by lengths of axes (a, b, c) and volume (V)
Defined by angles between axes (α, β, γ)
Symmetry of unit cell is at least as great as final crystal form

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 (coordination polyhedron)
Halite (NaCl) unit cell
Galena (PbS) unit cell

5. The Unit Cell and “Z” Number
Determined through density-geometry calculations
Determined in unit cell models, by “fractional ion contribution” (ion charge balance) calculations
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
1/8x charge contribution
Halite (NaCl) unit cell; Z = 4

6. The Unit Cell and “Z” Number
Galena and Halite

7. The Unit Cell and “Z” Number
Fluorite, CaF2
Cassiterite, SnO2

8. Unit Cell Dimensions
Size and shape of unit cells are determined on the basis of crystal structural analysis (using x-ray diffraction)
Lengths of sides (volume)
Angles between faces
nλ=2dsinθ

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

10. Unit Cell Geometry Arrangement of atoms determines unit cell geometry:
Primitive = atoms only at corners
Body-centered = atoms at corners and center
Face-centered = atoms at corners and 2 (or more) faces
Lengths and angles of axes determine six unit cell classes
Same as crystal classes

11. Coordination Polyhedron and Unit Cells
They are not the same!
BUT, coordination polyhedron is contained within a unit cell
Relationship between the unit cell and crystallography
Crystal systems and reference, axial coordinate system
Halite (NaCl) unit cell; Z = 4
Cl CN = 6; octahedral

12. Unit Cells and Crystals
The unit cell is often used in mineral classification at the subclass or group level
Unit cell = building block of crystals
Lattice = infinite, repeating arrangement of unit cells to make the crystal
Relative proportions of elements in the unit cell are indicated by the chemical formula (Z number)
Sphalerite, (Zn,Fe)S, Z=4

13. Unit Cells and Crystals
Crystals 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
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

14. Crystal Systems
Unit cell has at least as much symmetry as crystal itself
Unit cell defines the crystal system
Geometry of 3D polyhedral solids
Defined by axis length and angle
Applies to both megascopic crystals and unit cells
Results in geometric arrangements of
Faces (planes)
Edges (lines)
Corners (points)

15. Crystal Systems: 6 (or 7, including 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 360° 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) Tetragonal High symmetry: 4A3
a = b = c
All angles 90°
Tetragonal
1A4
a = b ≠ c

17. The 6 Crystal Systems Hexagonal (trigonal) Orthorhombic 1A6 or 1A3
a = b ≠ c
α = β = 90°; γ = 120°
Orthorhombic
3A2 (mutually perpendicular)
a ≠ b ≠ c
All angles 90°

18. The 6 Crystal Systems Monoclinic Triclinic 1A2 a ≠ b ≠ c
α = γ = 90°; β ≠ 90°
Triclinic
1A1
α ≠ β ≠ γ ≠ 90°

19. Collaborative Activity:
In groups answer the following:
Calculate the density of fluorite (CaF2). The Z-number for fluorite is 4, and unit cell axis length is 5.46 Å.
A. Find V, the unit cell volume (5.46 Å)3 and convert this value to cm3 (1 Å3 = cm3)
B. Find M, the gram atomic weight of fluorite (Ca + 2F)
C. Calculate G (density) using: G = (Z x M)/(A x V)
Using the chemical analysis of pyrite (FeS2), calculate the Z-number. The density (G) of pyrite is 5.02 g/cm3 and the unit cell axial length is 5.42 Å.
A. Find V, the unit cell volume: (5.42 Å)3 – Note: you don’t need to convert to cm3 in this case because the final formula uses Å3.
B. In the table, calculate the atomic proportions of each element (P/N = wt% / atomic weight)
C. Calculate Z for each element using: (P/N)(VG/166.02) [Note: this formula is a slightly reorganized version of the one in the homework]
D. Sum the metals and use the chemical formula to determine the Z-number