3.4 Miller Indices and Crystal Forms

Crystal structures are the building blocks of minerals. Miller indices provide a standardized way to describe planes and directions within these structures, making it easier to analyze and communicate about crystal geometry.

Understanding Miller indices is crucial for interpreting crystal forms and symmetry. This knowledge helps geologists predict mineral properties, analyze X-ray diffraction patterns, and unravel the complex relationships between a crystal's internal structure and external appearance.

Miller indices in crystallography

Fundamentals of Miller indices

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• Miller indices consist of three integers (h, k, l) describing planes and directions in crystal lattices
• Determine Miller indices by reducing reciprocals of plane intercepts with crystallographic axes to smallest integers
• Provide standardized notation for crystal planes, faces, and directions across all crystal systems
• Denote negative Miller indices with a bar over the number (h̄, k̄, l̄)
• Use {hkl} notation for symmetrically equivalent plane families and (hkl) for specific planes
• Employ Miller-Bravais indices (hkil) in hexagonal crystal systems to maintain symmetry relationships

Applications of Miller indices

• Calculate interplanar spacing in crystal structures
• Determine crystal symmetry and predict X-ray diffraction patterns
• Analyze crystal growth and morphology
• Identify cleavage planes and slip systems in minerals
• Describe orientation relationships between different crystal phases
• Facilitate communication of crystal structures in scientific literature

Determining Miller indices

Finding Miller indices for crystal planes

• Identify plane intersections with crystallographic axes
• Take reciprocals of these intersections
• Reduce reciprocals to smallest set of integers (multiply or divide by common factor)
• Assign zero index for planes parallel to an axis
• Example: Plane intersecting at (2, 1, 1) becomes (1/2, 1, 1), reduced to Miller indices (211)
• Example: Plane parallel to y-axis intersecting at (1, ∞, 2) becomes (1, 0, 1/2), reduced to (201)

Determining Miller indices for crystal directions

• Write direction indices as [uvw], representing vector components along unit cell edges
• Identify vector parallel to desired direction
• Reduce vector components to smallest set of integers
• Use notation for symmetrically equivalent direction families
• In cubic systems, (hkl) plane indices match [hkl] direction perpendicular to that plane
• Example: Direction vector [2, 2, 4] reduces to [110]
• Example: In body-centered cubic iron, close-packed directions are <111>

Crystal forms and Miller indices

Common crystal forms

• Crystal forms comprise symmetrically related crystal faces
• Include pinacoids, prisms, pyramids, dipyramids, and pedions
• General form {hkl} encompasses all symmetrically equivalent faces with permutations and sign changes of h, k, and l
• Special forms occur when some indices are zero or equal (cubic system examples: {100}, {110}, {111})
• Number of faces in a form depends on crystal system and specific Miller indices
• Higher symmetry systems (cubic) have fewer general forms due to increased face equivalence
• Example: Octahedron in cubic system represented by {111}, with 8 equivalent faces
• Example: Hexagonal prism in hexagonal system represented by {101̄0}, with 6 equivalent faces

Relationship between crystal forms and Miller indices

• Predict possible growth faces and cleavage planes in crystals
• Determine form multiplicity and its impact on crystal habit
• Analyze crystal morphology and symmetry
• Identify twin planes and intergrowth relationships
• Relate external crystal morphology to internal atomic structure
• Example: Calcite {101̄4} cleavage rhombohedron reflects its internal structure
• Example: Pyrite cubes {100} and pyritohedra {210} demonstrate different growth conditions

Applications of Miller indices

Geometric calculations in crystallography

• Calculate angle between crystal planes using dot product of normal vectors expressed in Miller indices
• Determine zone axis of crystal plane set by taking cross product of their Miller indices
• Apply Weiss zone law to find additional planes in a zone or verify plane belonging to specific zone
• Calculate interplanar spacing using appropriate formula for each crystal system
• Example: In cubic systems, angle θ between planes (h₁k₁l₁) and (h₂k₂l₂) given by $cos θ = \frac{h₁h₂ + k₁k₂ + l₁l₂}{\sqrt{(h₁² + k₁² + l₁²)(h₂² + k₂² + l₂²)}}$
• Example: Interplanar spacing d in cubic systems calculated as $d = \frac{a}{\sqrt{h² + k² + l²}}$ where a is the lattice parameter

Miller indices in diffraction and crystal analysis

• Predict relative intensities of X-ray diffraction peaks based on structure factor calculations
• Employ Miller indices in stereographic projections for 2D representation of 3D crystal symmetry
• Analyze texture and preferred orientation in polycrystalline materials
• Interpret electron backscatter diffraction (EBSD) patterns
• Determine crystal orientations in transmission electron microscopy (TEM)
• Example: Structure factor F for a body-centered cubic crystal given by $F = f(1 + e^{πi(h+k+l)})$ where f is the atomic scattering factor
• Example: Pole figure analysis using Miller indices to represent crystal orientations in rolled metal sheets