The Unit Cell

1
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 (a, ß, ?)
• 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
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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 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

n?2dsin?
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X-ray Diffractograms
X-ray energy
Reflection angle
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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
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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, Z4
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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 Systems6 (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

17
The 6 Crystal Systems

• Hexagonal (trigonal)
• 1A6 or 1A3
• a b ? c
• a ß 90 ? 120
• Orthorhombic
• 3A2 (mutually perpendicular)
• a ? b ? c

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The 6 Crystal Systems

• Monoclinic
• 1A2
• a ? b ? c
• a ? 90 ß ? 90
• Triclinic
• 1A1
• a ? b ? c
• a ? ß ? ? ? 90

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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 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)
• 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 dont 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