# Crystallography

This is a single crystal XRD unit. Credit: Anjali Merin.

Crystallography is the science that examines crystals, which can be found everywhere in nature—from salt to snowflakes to gemstones.

It uses the properties and inner structures of crystals to determine the arrangement of atoms and generate knowledge.

## Crystallography and Mineral Evolution

Within the past century, crystallography has been a primary force in driving major advances in the detailed understanding of materials, synthetic chemistry, the understanding of basic principles of biological processes, genetics, and has contributed to major advances in the development of drugs for numerous diseases. It is one of the most important tool to study solids, since most of the materials in solid state exhibits crystalline nature. Crystalline solids are known to show different structural forms depending on different conditions of temperature, pressure etc. Since minerals are naturally occurring inorganic crystalline, the phase transitions involving minerals will be interesting. Phase transition studies of minerals deserve special attention as they can provide clues to mechanism of mineral evolution on earth crust. They also opens up the possibility of utilizing the naturally abundant minerals for generating novel functional materials processing properties such as ionic conductivity, ferroelecticity, ferromagnetism etc .[1][1]

A mineral is an element or chemical compound that is normally crystalline and that has been formed as a result of geological process. It has highly ordered atomic structure and specific physical properties . Mimicking the mineral evolution process in laboratory by phase transition studies can throw lights on our understanding of mineral evolution. A large number of minerals occur in hydrated form, especially the bimetallic sulfate minerals are widely interested in its varying non stoichiometric crystal structure with the levels of hydration. Bimetallic sulfates are more interested due to their phase transition with temperature, eg; langbenites, krohnkite. The non-stoichiometric structures getting trapped in small kinetically stabilized energy wells, which are intermediates between individual monosulfates and bimetallic sulfate minerals, may have valuable structural hints about the process of origin of such minerals. The phase transition studies reveal that those structures are precursors of the original mineral and the ubiquitous role of water in the formation of minerals in the earth crust .[2][2]

In particular, Langebenite minerals are distinctive geological minerals found in only a few locations in the world. These deposits were formed millions of years ago when a variety of salts were left behind after the evaporation of ancient ocean beds. These type crystals have general chemical formulae A2B2(SO4)3 where A denotes a monovalent cation such as K, NH4… and B a divalent cation such as Mg, Mn, Ni… At high temperatures, they crystallizes isomorphously in the cubic space group P213. They are having a wide variety of applications due to its ferroelectric, ferroelastic, spectroscopic and magnetic properties [2]. Their best-known applications are in dosimetry of ionizing radiation, CTV screen phosphors, projection T V phosphors, scintillators, fluorescent lamps, full color displays, X-ray storage and screens intensifying phosphors, and laser materials .[3][3]

## Theoretical crystallography

Def. an "experimental science of determining the arrangement of atoms in solids"[4], or the "study of crystals"[4] is called crystallography.

## Formula units

This is a model of how NaCl formula units could form a cube. Credit: BruceBlaus.
Illustration shows the close-packing of equal spheres in both hcp (left) and fcc (right) lattices. Credit: Twisp.

Flame emission spectroscopy of a test mineral described in the above section suggests some conclusions about the mineral.

The halide present was chlorine. The mineral is most likely halite. The formula unit is NaCl. The mineral grains appear to be cuboidal. At the left is a model of how NaCl formula units could form a cube.

When NaCl is dissolved in water, it has the formula [Na(H2O)8]+. The chloride ion is surrounded by an average of 6 molecules of water.[5] As the water evaporates, the cations of sodium and the anions of chlorine should be drawn back together.

If the sodium and chlorine ions can be represented by equal-sized hard balls, they would be expected to form close-packed solids.

An examination of the two types of close-packed structures shows a problem that may disqualify such structures for representing NaCl. Each sphere is the same distance from every other sphere. An effort to use some as sodiums and the others as chlorines always results in at least two sodiums contacting each other and the same thing happens with the chlorines.

## Models

This is a body-centered cubic unit cell. Credit: Baszoetekouw.
Crystal structure of NaCl shows coordination polyhedra. Credit: Solid State.
Relative radii of atoms and ions for neutral atoms colored gray, cations red, and anions blue. Credit: Popnose.
Common dodecahedra
Ih, order 120
Regular- Small stellated- Great- Great stellated-
Th, order 24 T, order 12 Oh, order 48 Td, order 24
Pyritohedron Tetartoid Rhombic- Trapezoidal-
D4h, order 16 D3h, order 12
Rhombo-hexagonal- Rhombo-square- Trapezo-rhombic- Rhombo-triangular-

A slightly more open structure is the body-centered cubic shown in the image on the left. Here, one sodium ion could be surrounded by six chlorines in an octahedron, and one chlorine anion could be surrounded by six sodium cations. These are shown in the second image down on the left.

Using the approximate ionic radii from the third image down on the right for Na+ as 116 pm and 167 pm for Cl- to calculate a radius ratio yields 0.695. Such a size ratio falls in the octahedron range of ≥ 0.414 and < 0.732.

The ball and stick model on the left shows what's inside the cube.

## Habits

The image shows that halite can occur in a massive habit. Credit: Helix84.

Def. form "of growth or general appearance of a variety or species of plant or crystal"[6] is called a habit.

The image on the left shows that halite can occur in a massive habit but is apparently always crystalline.

Habit[7][8][9] Image Description Common Example(s)
Acicular Needle-like, slender and/or tapered Natrolite, Rutile[10]
Amygdule (Amygdaloidal) Almond-shaped Heulandite, subhedral Zircon
Botryoidal or globular Grape-like, hemispherical masses Hematite, Pyrite, Malachite, Smithsonite, Hemimorphite, Adamit, Variscite
Columnar Similar to fibrous: Long, slender prisms often with parallel growth Calcite, Gypsum/Selenite
Coxcomb Aggregated flaky or tabular crystals closely spaced. Barite, Marcasite
Cubic Cube shape Pyrite, Galena, Halite
Dendritic or arborescent Tree-like, branching in one or more direction from central point Romanechite and other Manganese (Mn)-oxide minerals, magnesite, native copper
Dodecahedral Rhombic dodecahedron, 12-sided Garnet
Drusy or encrustation Aggregate of minute crystals coating a surface or cavity Uvarovite, Malachite, Azurite, Quartz
Enantiomorphic Mirror-image habit (i.e. crystal twinning) and optical characteristics; right- and left-handed crystals Quartz, Plagioclase, Staurolite
Equant, stout Length, width, and breadth roughly equal Olivine, Garnet
Fibrous Extremely slender prisms Serpentine group, Tremolite (i.e. Asbestos)
Filiform or capillary Hair-like or thread-like, extremely fine many Zeolites
Foliated or micaceous or lamellar (layered) Layered structure, parting into thin sheets Mica (Muscovite, Biotite, etc.)
Granular Aggregates of anhedral crystals in matrix Bornite, Scheelite
Hemimorphic Doubly terminated crystal with two differently shaped ends Hemimorphite, Elbaite
Hexagonal Hexagon shape, six-sided Quartz, Hanksite
Hopper crystals Like cubic, but outer portions of cubes grow faster than inner portions, creating a concavity Halite, Calcite, synthetic Bismuth
Mammillary Breast-like: surface formed by intersecting partial spherical shapes, larger version of botryoidal, also concentric layered aggregates Malachite, Hematite
Massive or compact Shapeless, no distinctive external crystal shape Limonite, Turquoise, Cinnabar, Realgar
Nodular or tuberose Deposit of roughly spherical form with irregular protuberances Chalcedony, various Geodes
Octahedral Octahedron, eight-sided (two pyramids base to base) Diamond, Magnetite
Platy Flat, tablet-shaped, prominent pinnacoid Wulfenite
Plumose Fine, feather-like scales Aurichalcite, Boulangerite, Mottramite
Prismatic Elongate, prism-like: well-developed crystal faces parallel to the vertical axis Tourmaline, Beryl
Pseudo-hexagonal Hexagonal appearance due to cyclic twinning Aragonite, Chrysoberyl
Radiating or divergent Radiating outward from a central point Wavellite, Pyrite suns
Reniform or colloform Similar to botryoidal/mamillary: intersecting kidney-shaped masses Hematite, Pyrolusite, Greenockite
Reticulated Crystals forming net-like intergrowths Cerussite
Rosette or lenticular (lens shaped crystals) Platy, radiating rose-like aggregate Gypsum, Barite (i.e. Desert rose)
Sphenoid Wedge-shaped Sphene
Stalactitic Forming as stalactites or stalagmites; cylindrical or cone-shaped Calcite, Goethite, Malachite
Striated Not a habit per se, but a condition of lines that can grow on certain crystal faces on certain minerals Tourmaline, Pyrite, Quartz, Feldspar, Sphalerite
Stubby or blocky or tabular More elongated than equant, slightly longer than wide, flat tablet shaped Feldspar, Topaz
Tetrahedral Tetrahedra-shaped crystals Tetrahedrite, Spinel, Magnetite
Wheat sheaf Aggregates resembling hand-reaped wheat sheaves Stilbite

## Cleavages

Image shows a mineral specimen of Halite. Credit: Vassia Atanassova - Spiritia.

The image on the right shows a more familiar crystal habit of halite.

Halite might be expected to break leaving behind a cube-like face.

Def. the "tendency of a crystal to split along specific planes"[11] is called cleavage.

The cleavage described for halite is "{001} perfect Fracture conchoidal. Brittle."[12]

Miller indices {h k ℓ} describe potential cleavage planes. Credit: DeepKling.

Cleavage forms parallel to crystallographic planes:[13]

• Basal or pinacoidal cleavage occurs when there is only one cleavage plane. Graphite has basal cleavage. Mica (like muscovite or biotite) also has basal cleavage; this is why mica can be peeled into thin sheets.
Meroxene biotite
• Cubic cleavage occurs on when there are three cleavage planes intersecting at 90 degrees. Halite (or salt) has cubic cleavage, and therefore, when halite crystals are broken, they will form more cubes.
Halite
• Octahedral cleavage occurs when there are four cleavage planes in a crystal. Fluorite exhibits perfect octahedral cleavage. Octahedral cleavage is common for semiconductors. Diamond also has octahedral cleavage.
Fluorite
• Rhombohedral cleavage occurs when there are three cleavage planes intersecting at angles that are not 90 degrees. Calcite has rhombohedral cleavage.
Calcite
• Prismatic cleavage occurs when there are two cleavage planes in a crystal. Spodumene exhibits prismatic cleavage.
Spodumene
• Dodecahedral cleavage occurs when there are six cleavage planes in a crystal. Sphalerite has dodecahedral cleavage.
Pyrite

## Unit cells

This is a possible unit cell for crystalline NaCl. Credit: Ed Caruthers.

Def. the "smallest repeating structure (parallelepiped) of atoms within a crystal, from which the structure of the complete crystal can be inferred"[14] is called a unit cell.

An examination of the model unit cell for crystalline NaCl shows a repeat pattern along the left front edge starting with a Cl atom at the corner. Moving along the line of atoms at this lowest edge, next is a smaller Na atom, then another Cl atom. This second Cl atom is a repeat of the first.

Going back to the corner Cl atom and moving straight up above it is a Na atom. Above that Na atom is again another Cl that is another repeat of the first.

Looking at the eight corners of this perspective view of the model there is a Cl atom at each corner. Starting again at the lower left corner Cl, directly behind the left rear face of this unit cell is another such unit cell not shown. Further left of these two unit cells are two more. One is along the base diagonal though the corner Cl and the second is in front sharing the face of the model unit cell shown. Counting the model shown there are four unit cells in this layer that share the lower left corner Cl atom.

Right below this layer of four unit cells are four identical unit cells all cornered to this same Cl atom in the lower left corner of the model. Summarizing, the corner of a unit cell is shared by eight unit cells total.

From visualizing nearer unit cells next to the one drawn, an edge of this cell is shared by four unit cells, and each face by two unit cells.

Looking at the numbers of Cls and Nas:

1. eight corner Cls are each shared by eight unit cells so one corner Cl per unit cell,
2. along each edge between each corner Cl is a Na, each of 12 edges is shared by four unit cells, so 3 Nas per unit cell,
3. each face shares a centered Cl with two unit cells, 6 such Cls, each shared by two unit cells, is 3 Cls,
4. inside the unit cell pictured there is one and only one Na, shared with no other unit cell, so one Na,

totalling the anions, there are 4 Cls, totalling the cations there are 4 Nas. In a unit cell for this model of halite, there are four formula units. The formula content of a unit cell is often denoted as Z. For this model, Z = 4.

## Structures

Even when the mineral grains are too small to see or are irregularly shaped, the underlying crystal structure is always periodic and can be determined by X-ray diffraction.[15] Minerals are typically described by their symmetry content. Crystals are restricted to 32 point groups, which differ by their symmetry. These groups are classified in turn into more broad categories, the most encompassing of these being the six crystal families.[16] All of these six crystal families when combined with the 32 point groups result in 230 space groups that symmetrically describe all space-filling three-dimensional crystal structures.

These families can be described by the relative lengths of the three crystallographic axes, and the angles between them; these relationships correspond to the symmetry operations that define the narrower point groups. They are summarized below; a, b, and c represent the axes, and α, β, γ represent the angle opposite the respective crystallographic axis (e.g. α is the angle opposite the a-axis, viz. the angle between the b and c axes):[16]

Crystal family Lengths Angles Common mineral examples
Isometric (Cubic crystal system) a=b=c α=β=γ=90° garnet, halite, pyrite
Tetragonal a=b≠c α=β=γ=90° rutile, zircon, andalusite
Orthorhombic a≠b≠c α=β=γ=90° olivine, aragonite, orthopyroxenes
Hexagonal a=b≠c α=β=90°, γ=120° Quartz, calcite, tourmaline
Monoclinic a≠b≠c α=γ=90°, β≠90° clinopyroxenes, orthoclase, gypsum
Triclinic a≠b≠c α≠β≠γ≠90° Anorthite, albite, kyanite
Crystal Family Lattice System Schönflies notation 14 Bravais Lattices
Primitive (P) Base-centered (C) Body-centered (I) Face-centered (F)
Triclinic Ci
Monoclinic C2h
Orthorhombic D2h
Tetragonal D4h
Hexagonal Rhombohedral D3d
Hexagonal D6h
Cubic Oh

The unit cells are specified according to the relative lengths of the cell edges (a, b, c) and the angles between them (α, β, γ). The volume of the unit cell can be calculated by evaluating the triple product a · (b × c), where a, b, and c are the lattice vectors. The properties of the lattice systems are given below:

Crystal family Lattice system Volume Axial distances (edge lengths)[17] Axial angles[17] Corresponding examples
Triclinic ${\displaystyle abc{\sqrt {1-\cos ^{2}\alpha -\cos ^{2}\beta -\cos ^{2}\gamma +2\cos \alpha \cos \beta \cos \gamma }}}$ (All remaining cases) potassium dichromate (K2Cr2O7), copper(II) sulfate (CuSO4·5H2O), boric acid (H3BO3)
Monoclinic ${\displaystyle abc\,\sin \beta }$ ac α = γ = 90°, β ≠ 90° monoclinic sulphur, sodium sulfate (Na2SO4·10H2O), lead(II) chromate (PbCrO3)
Orthorhombic ${\displaystyle abc}$ abc α = β = γ = 90° rhombic sulphur, potassium nitrate (KNO3), barium sulfate (BaSO4)
Tetragonal ${\displaystyle a^{2}c}$ a = bc α = β = γ = 90° white tin, tin dioxide (SnO2), titanium dioxide (TiO2), calcium sulfate (CaSO4)
Hexagonal Rhombohedral ${\displaystyle a^{3}{\sqrt {1-3\cos ^{2}\alpha +2\cos ^{3}\alpha }}}$ a = b = c α = β = γ ≠ 90° calcite (CaCO3), cinnabar (HgS)
Hexagonal ${\displaystyle {\frac {\sqrt {3}}{2}}\,a^{2}c}$ a = b α = β = 90°, γ = 120° graphite, zinc oxide (ZnO), cadmium sulphide (CdS)
Cubic ${\displaystyle a^{3}}$ a = b = c α = β = γ = 90° sodium chloride (NaCl), zinc blende, copper metal, potassium chloride (KCl), diamond, silver metal

## Visual crystallography

The image shows an example of Mitscherlich's goniometer. Credit: Samuel Orgelbrand's Universal Encyclopedia.

## References

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13. Hurlbut, Cornelius S.; Klein, Cornelis, 1985, Manual of Mineralogy, 20th ed., Wiley, isbn=0-471-80580-7
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17. Hahn, Theo, ed. (2002). International Tables for Crystallography, Volume A: Space Group Symmetry. International Tables for Crystallography. A (5th ed.). Berlin, New York: Springer-Verlag. doi:10.1107/97809553602060000100. ISBN 978-0-7923-6590-7.