Almost all solids have regular molecule, ion, or atom arrays {crystal}|.
types
Crystals {cubic crystal} can have eight atoms around small atom, three perpendicular four-fold same-length axes, and five crystal classes.
Crystals {rhomboid crystal} can have twelve atoms around similar size atom with every third layer directly above another.
Crystals {rhombohedral crystal} can have one three-fold axis, with one axis perpendicular to the other two but with different length, and two axes with same length at 120-degree angle to perpendicular axis, and make five crystal classes.
Crystals {hexagonal crystal} {hexagon crystal} can have twelve atoms around similar size atom with alternate layers directly above each other.
Crystals {tetrahedral crystal} {tetrahedron crystal} can have four large ions around each small ion.
Crystals {octahedral crystal} {octahedron crystal} can have six large ions around each small ion.
Crystals {triangular crystal} can have three large ions around each small ion.
Crystals {planar crystal} can have two large ions around each small ion.
symmetry
Only one, two, three, four, or six rotational symmetries can fill all space with no gaps or overlaps, so only seven crystal types are common.
packing
The more similar in size atoms or ions are, the more atoms or ions can surround one atom or ion. Number of atoms or ions surrounding one atom also depends on ion charge or covalent-bond number.
lattice
Crystals have lattice structure. Including crystal type and lattice type, 32 crystal classes exist. 14 unit cell and 32 crystal class translations and transformations make 234 possible crystal shapes.
crystal growth
Crystals grow at dislocations, because binding molecules can contact two surface atoms. Impurities, long bond lengths during fast growth, and screw dislocations can cause dislocations and irregularities.
Small crystal faces grow fastest by deposition. Large crystal faces can adsorb other materials.
Crystal surfaces are never flat but are lumpy. Perfect crystals cannot grow.
Diagrams {Miller index} can show crystal planes, using three perpendicular axes. Crystal vertices are one unit length or less apart. Coordinate reciprocals indicate planes through vertices. Putting origin at one vertex and using coordinates for other vertices indicates edges. If plane is parallel to axis, coordinate reciprocal is zero.
Crystal growth can be along one axis {dendritic growth}, because heat leaves best at tips, so deposition is easiest there.
Unit cells can undergo translation and reflection {glide plane}.
Unit cells can undergo translation and turn around axis {screw axis} 0, 180, 120, 90, 72, or 60 degrees.
Crystals {clathrate} can be so open that they can hold small molecules inside, without bonding. Examples are very-cold-water forms and very-cold methane and hydrogen gas mixtures.
Metal crystals {metal crystal} can have hexagonal close packing, face-centered cubic close packing, or body-centered cubic close packing.
Substances {polymorphic solid} can have more than one crystal form.
Crystals {liquid crystal}| {dynamic scattering liquid crystal} (LCD) can have regularity in only one or two dimensions, allowing unit cells to slide past each other in third dimension. Liquid crystals are anisotropic, flow in sheets or steps, and are asymmetric molecules. Numbers can display electronically using reflected or transmitted light, as electric field makes crystal tinted, and no electric field makes crystal transparent.
Liquid crystals {nematic crystal} can have linear crystals with regularity in only one dimension, are like threads with no planes, orient, and are not periodic.
Liquid crystals {smectic crystal} can have planar crystals with regularity in two dimensions, orient, and are not periodic.
Missing atoms, extra atoms, or different atom types {crystal defect} alter regular crystal structure.
Crystal defects {dislocation, crystal} can displace unit cells from usual positions. Inserted atoms wedged into lattice can cause dislocations {edge dislocation}. Dislocations {screw dislocation} can be around axis to make helical unit-cell arrangements. Dislocations in crystals affect brittleness, ductility, and other mechanical crystal properties. Alloys lack dislocations and so do not slide, because odd atoms move to lowest free-energy positions.
Crystal ions can move to interstitial places and so leave vacancies {Frenkel defect}.
Different molecules or atoms {impurity}| can be in crystals.
Extra ions {interstitial ion} can be in ionic crystals.
In ionic crystal, cations and anions can be missing {Schottky defect}.
Ions can be missing from ionic crystals {vacancy}|.
Crystals have repeating atom groups {unit cell}. Crystals have lattice structure.
Only 14 possible arrangements {Bravais lattice} of identical spheres can make unit cells {space lattice}. Three are cubic, two monoclinic, four orthorhombic, two tetragonal, one triclinic, one hexagonal, and one rhombohedral.
Using half-unit circles as atoms, in a 3 x 4 surface area make two-dimensional figures: squares; triangles, hexagons, and rhombuses; centered rectangles; rectangles, and oblique figures.
From most symmetric to least symmetric two-dimensional space lattices (see Figure 1).
Square: Cell is a unit-length-side square, as in the first figure above. The two axes have equal length, both axes are mirror planes, and both axes have 90-degree rotation symmetry. Atoms are at corners. Cell has 90-degree rotation symmetry and two planes with mirror symmetry. Atoms have 4 atoms 1 unit away and 4 atoms SQR(2) away. Density is 12/12 = 1.
Hexagonal: Cell is a unit-length-side triangle with angles 60 degrees, a rhombus with angles 60 and 120 degrees, and a hexagon, as in the second figure above. The two axes have equal length, both axes are mirror planes, and both axes have 60-degree rotation symmetry. Atoms are at corners. Rhombus has 180-degree rotation symmetry and two planes with mirror symmetry. Triangle has 60-degree rotation symmetry and no planes with mirror symmetry. Atoms have 6 atoms 1 unit away. Density is ~13/12 > 1.
Centered rectangular: Cell is a parallelogram with angles not 30, 45, 60, 90, or 120 degrees, short diagonal unit length, and a rectangle with two sides greater than unit length and two side greater than that, as in the third figure above. The two axes have unequal length, one axis is a mirror plane, and both axes have 180-degree rotation symmetry. Atoms are at corners. Cell has 180-degree rotation symmetry and two planes with mirror symmetry. Corner atoms have 4 atoms 1 unit away, 2 atoms farther away, and 2 atoms even farther away, and central atoms have 4 atoms 1 unit away and 4 atoms farther away. Density is ~10/12 < 1.
Rectangular: Cell is a rectangle, as in the fourth figure above. The two axes have unequal length, both axes are mirror planes, and both axes have 180-degree rotation symmetry. Atoms are at corners. Cell has 180-degree rotation symmetry and two planes with mirror symmetry. Atoms have 2 atoms 1 unit away, 2 atoms farther away, and 4 atoms even farther away. Density is 6/12 = 0.5.
Oblique: Cell is a parallelogram with angles not 30, 45, 60, 90, or 120 degrees, as in fifth figure above. The two axes have unequal length, no axes are mirror planes, and both axes have 180-degree rotation symmetry. Atoms are at corners. Cell has 180-degree rotation symmetry and no planes with mirror symmetry. Each atom has 2 atoms 1 unit away, 2 atoms farther away, 2 atoms even farther away, and 2 atoms much farther away. Density is <6/12 < 0.5.
From most symmetric to least symmetric three-dimensional space lattices:
Isometric crystal: Cubic cell base is a square with angles 90 degrees, and height is perpendicular to base. All faces are squares. All axes have equal length. Atoms are at corners, can be in body center, and can be in face centers (three Bravais lattices). Point-group symmetry has three rotations by 90 degrees. For corners only, each atom has 4 atoms around in a plane and 2 atoms (above and below) along the perpendicular, total 6. For face-centered, corner atoms have 8 atoms around in a plane and 2 atoms (above and below) along the perpendicular, and centered atoms have 4 atoms around in a plane and 2 atoms (above and below) along the perpendicular. For body-centered, corner atoms have 4 atoms around in a plane, 2 atoms (above and below) along the perpendicular, and 4 atoms along diagonals, and centered atoms have 8 atoms along diagonals.
Hexagonal crystal: Cell base is a parallelogram with angles 120 and 60 degrees, and height is perpendicular to base. Two faces are parallelograms, and four faces are rectangles. Parallelogram axes have equal length, and height has any length. Atoms are at corners (one Bravais lattice). Point-group symmetry has one rotation by 60 degrees. Each atom has 6 atoms around in a plane and 2 atoms (above and below) along the perpendicular, total 8.
Tetragonal crystal: Cell base is a square, and height is perpendicular to base. Four faces are rectangles, and two faces are squares. Square axes have equal lengths, and height is not equal to square-side length. Atoms are at corners, and can be in body center (two Bravais lattices). Point-group symmetry has one rotation by 90 degrees. For corners only, each atom has 4 atoms around in a plane and 2 atoms (above and below) along the perpendicular, total 6. For body-centered, corner atoms have 4 atoms around in a plane, 2 atoms (above and below) along the perpendicular, and 4 atoms along diagonals, and centered atoms have 8 atoms along diagonals.
Rhombohedral crystal: Trigonal-point-group cell base is a rhombus, and height is not perpendicular to base. All faces are rhombuses. All axes have equal lengths. Atoms are at corners (one Bravais lattice). Point-group symmetry has one rotation by 120 degrees. Each atom has 4 atoms around in a plane and 2 atoms (above and below) along the perpendicular, total 6.
Orthorhombic crystal: Cell base is a rectangle, and height is perpendicular to base. All faces are rectangles. All axes have unequal lengths. Atoms are at corners, can be in body center, can be in base-face centers, and can be in all face centers (four Bravais lattices). Point-group symmetry has three rotations by 180 degrees and two mirror planes. For corners only, each atom has 4 atoms around in a plane and 2 atoms (above and below) along the perpendicular, total 6. For base-face-centered, corner atoms have 8 atoms around in a plane and 2 atoms (above and below) along the perpendicular, and centered atoms have 4 atoms around in a plane and 2 atoms (above and below) along the perpendicular. For all-face-centered, corner atoms have 8 atoms around in a plane and 2 atoms (above and below) along the perpendicular, and centered atoms have 4 atoms around in a plane, 4 atoms along diagonals, and 2 atoms (above and below) along the perpendicular. For body-centered, corner atoms have 4 atoms around in a plane, 2 atoms (above and below) along the perpendicular, and 4 atoms along diagonals, and centered atoms have 8 atoms along diagonals.
Monoclinic crystal: Cell base is a parallelogram, and height is perpendicular to base. Two faces are parallelograms, and four faces are rectangles. The three axes have unequal length. Atoms are at corners and can be in face centers (two Bravais lattices). Point-group symmetry has one rotation by 180 degrees and one mirror plane. For corners only, each atom has 4 atoms around in a plane and 2 atoms (above and below) along the perpendicular, total 6. For face-centered, corner atoms have 8 atoms around in a plane and 2 atoms (above and below) along the perpendicular, and centered atoms have 4 atoms around in a plane and 2 atoms (above and below) along the perpendicular.
Triclinic crystal: Cell base is a parallelogram, and height is not perpendicular to base. All faces are parallelograms. All axes have unequal lengths. Atoms are at corners (one Bravais lattice). There are no point-group symmetries. Each atom has 4 atoms around in a plane and 2 atoms (above and below) along the perpendicular, total 6.
Lattices {primitive lattice} can have atoms at unit-cell corners.
Lattices {body-centered lattice} can have one or two atoms at unit-cell centers. Lattices {body-centered cubic close packing} can have atoms in cube centers, with identical atoms at cube corners.
Lattices {face-centered lattice} can have one atom in unit-cell face. Unit cells with different atoms {face-centered cubic close packing} can have atoms in cube centers and in cubic-unit-cell face centers.
Unit-cells {cubic close packing} can be cubic. Twelve identical atoms surround each atom, and every third layer is directly above another.
Unit cells {hexagonal close packing} can be hexagonal. Twelve identical atoms surround each atom, and alternate layers are directly above each other.
Unit crystals can have same structure after rotation around axis, reflection across axis, inversion through central point, translation along axis, or any combination {symmetry, crystal}. Nature has six symmetry groups: isometric, hexagonal, tetragonal, orthorhombic, monoclinic, and triclinic.
Symmetry groups {hexagonal} can have rotation by 60 degrees. Hexagonal crystals have one six-fold axis, with one axis perpendicular to the other two axes but with different length, and two axes with same length at 60-degree angle to perpendicular axis, and makes seven crystal classes.
Symmetry groups {cubic symmetry group} {isometric symmetry group} can have rotation by 90 degrees, reflection, and inversion.
Symmetry groups {monoclinic symmetry group} can have rotation by 90 degrees. Crystals {monoclinic crystal}| can have one two-fold axis, two perpendicular same-length axes, and one non-perpendicular different-length axis, to make three crystal classes.
Symmetry groups {orthorhombic symmetry group} can have rotation by 180 degrees and reflection. Crystals {orthorhombic crystal} can have three two-fold axes, which are all perpendicular but have different lengths, to make three crystal classes.
Symmetry groups {tetragonal symmetry group} can have rotation by 90 degrees. Crystals {tetragonal crystal} can have one four-fold axis and three perpendicular axes but only two with same length, to make seven crystal classes.
Symmetry groups {triclinic symmetry group} can have rotation by 120 degrees. Crystals {triclinic crystal}| can have three axes, all not perpendicular but all of same length, to make two crystal classes.
5-Chemistry-Inorganic-Phase-Phases-Solid
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Date Modified: 2022.0225