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The Unit Cell

Crystallography, Crystal Symmetry, and Crystal Systems

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What is a Unit Cell?– 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

Unit cell for ZnS, Sphalerite

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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

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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

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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 unitcell

• 1/4x charge contribution

– Ions on the corners of the unitcell

• 1/8x charge contributionHalite (NaCl) unit cell; Z = 4

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The Unit Cell and “Z” Number

Galena and Halite

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Cassiterite, SnO2

Fluorite, CaF2

The Unit Cell and “Z” Number

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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

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X-ray DiffractogramsX

-ray

ene

rgy

Reflection angle

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• 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

Unit Cell Geometry

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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

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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

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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

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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)

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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)

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The 6 Crystal Systems• Cubic (isometric)

– High symmetry: 4A3

– a = b = c– All angles 90°

• Tetragonal– 1A4

– a = b ≠ c– All angles 90°

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The 6 Crystal Systems• Hexagonal (trigonal)

– 1A6 or 1A3

– a = b ≠ c– α = β = 90°; γ = 120°

• Orthorhombic– 3A2 (mutually perpendicular)– a ≠ b ≠ c– All angles 90°

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The 6 Crystal Systems• Monoclinic

– 1A2

– a ≠ b ≠ c

– α = γ = 90°; β ≠ 90°

• Triclinic– 1A1

– a ≠ b ≠ c

– α ≠ β ≠ γ ≠ 90°

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Collaborative Activity:In groups answer the following:1. 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 = 10-24 cm3)B. Find M, the gram atomic weight of fluorite (Ca + 2F)C. Calculate G (density) using: G = (Z x M)/(A x V)

2. 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