CRYSTAL STRUCTURE TERMS

CRYSTAL STRUCTURE TERMS crystalline material - a material in which atoms, ions, or molecules are situated in a periodic 3-dimensional array over large atomic distances (all metals, many ceramic materials, and certain polymers are crystalline under normal conditions) crystal structure - the manner in which atoms, ions, or molecules are arrayed spatially in a crystalline material; it is defined by (1) the unit cell geometry, and (2) the positions of atoms, ions, or molecules within the unit cell coordination number (in the hard sphere representation of the unit cell) - the number of nearest (touching) neighbors that any atom or ion in the crystal has atomic packing factor (APF) (in the hard sphere representation of the unit cell) - the ratio of solid sphere volume to unit cell volume for a crystal structure

UNIT CELL lattice points unit cell The unit cell is the basic structural unit of a crystal structure which, when tiled in 3-dimensions, would reproduce the crystal. All atomic, ionic, or molecular positions in a crystal may be generated by translating the unit cell integral distances along each of its edges. Lattice points are those points representing the positions of atoms, ions, or molecules in a crystal structure.

UNIT CELL LATTICE PARAMETERS z c β α a γ b y x A unit cell with x, y, and z coordinate axes (not necessarily mutually orthogonal), showing the axial lengths (a, b, c) of the unit cell and the interaxial angles (α, β, γ) of the unit cell. These parameters (a, b, c; α, β, γ) are the lattice parameters of the unit cell.

CRYSTAL SYSTEMS cubic tetragonal orthorhombic a = b = c a = b = c a = b = c α = β = γ = 90 α = β = γ = 90 α = β = γ = 90 rhombohedral hexagonal monoclinic a = b = c a = b = c a = b = c α, β, γ = 90 α = β = 90 γ = 120 α = γ = 90 = β triclinic a = b = c α, β, γ = 90

FACE-CENTERED CUBIC (FCC) a a hard sphere model unit cell reduced sphere model unit cell hard sphere model radius: R =.354a unit cell volume: 1.000a 3 number of atoms in unit cell: 4 atomic packing factor: 0.740 coordination number: 12 examples: aluminum (a =.1431 nm) copper (a =.1278 nm) gold (a =.1442 nm) nickel (a =.1246 nm)

BODY-CENTERED CUBIC (BCC) a a hard sphere model unit cell reduced sphere model unit cell hard sphere model radius: R =.433a unit cell volume: 1.000a 3 number of atoms in unit cell: 2 atomic packing factor: 0.680 coordination number: 8 examples: chromium (a =.1249 nm) niobium (a =.1430 nm) tungsten (a =.1371 nm) iron (a =.1241 nm)

HEXAGONAL CLOSE-PACKED (HCP) reduced sphere model unit cell hard sphere model unit cell hard sphere model radius: R =.500a unit cell volume: 2.121a 3 number of atoms in unit cell: 6 atomic packing factor: 0.740 coordination number: 12 examples: cadmium (a =.1490 nm) beryllium (a =.1140 nm) titanium (a =.1445 nm) zinc (a =.1332 nm)

ESTIMATING METAL DENSITIES The density of a metal can be estimated using: ρ = na/v C N A where n = number of atoms in a unit cell A = atomic weight of metal V C = volume of the unit cell N A = 6.02x10 23 atoms/mole FCC BCC HCP a 2.828R 2.309R 2.000R V C 1.000a 3 1.000a 3 4.243a 3 example: Copper has an atomic radius of.128 nm, an atomic weight of 63.5 g/mole, and a FCC crystal structure. Compute its theoretical density using this information, and find the percent error between this value and the accepted value of 8.94 g/cm 3.

TABLE OF IONIC RADII cation ionic radius anion ionic radius Al 3+.053 nm Br -.196 nm Ba 2+.136 nm Cl -.181 nm Ca 2+.100 nm F -.133 nm Cs +.170 nm I -.220 nm Fe 2+.077 nm O 2-.140 nm Fe 3+.069 nm S 2-.184 nm K +.138 nm Mg 2+.072 nm Mn 2+.067 nm Na +.102 nm Ni 2+.069 nm Si 4+.040 nm.061 nm Ti 4+

ANION-CATION STABILITY stable stable unstable A cation-anion combination is unstable if the cation cannot coordinate with any of the anions which create the interstitial position that the cation occupies.

COORDINATION STABILITY CN = 2 r c /r a <.155 CN = 3 CN = 4.155 < r c /r a <.225.255 < r c /r a <.414 CN = 6 CN = 8.414 < r c /r a <.732.732 < r c /r a < 1.000

SOME CERAMIC CRYSTAL STRUCTURES rock salt cesium chloride zinc blende structure structure structure type: AX type: AX type: AX example: NaCl example: CsCl example: ZnS = Na + = Cl - = Cs + = Cl - = Zn +2 = S -2 r C /r A =.564 r C /r A =.939 r C /r A =.402 CN = 6 CN = 8 CN = 4 PIC = 66.8 % PIC = 73.4 % PIC = 18.3 %

SOME CERAMIC CRYSTAL STRUCTURES fluorite structure perovskite structure type: AX 2 type: ABX 3 example: CaF 2 example: BaTiO 3 = Ca + = F - =Ti 4+ = Ba 2+ = O 2- r C /r A =.752 CN = 4 for Ba-O CN = 8 CN = 8 for Ti-O

ESTIMATING CERAMIC DENSITIES The density of a ceramic can be estimated using: ρ = n'a F /V C N A where n' = number of formula units in a unit cell A F = molar weight of one formula unit V C = volume of the unit cell N A = Avogadro's number = 6.02x10 23 atoms/mole example: Cesium chloride is an AX-type ceramic with the cesium chloride crystal structure. Cesium has an atomic weight of 132.9 g/mole, and the cesium cation has an ionic radius of.170 nm. Chlorine has an atomic weight of 35.5 g/mole, and the chlorine anion has an ionic radius of.181 nm. Compute the theoretical density of cesium chloride using this information.

CRYSTAL DIRECTION INDICES (1) obtain components: obtain the components of the vector along the three coordinate axes in terms of the lattice parameters a, b, and c (if vector is not in standard position, subtract the coordinates of the tail of vector from the coordinates of the tip). (2) normalize components: normalize this triple of indices by dividing each index by its corresponding lattice parameter. (3) obtain integers: multiply these three indices by a common factor so that all indices become integers (smallest possible). (4) display results: enclose the three indices (not separated by commas) in square brackets; use a bar over an index to indicate a negative value.

CRYSTAL DIRECTION EXAMPLE z c b y a x

CRYSTAL DIRECTION EXAMPLE z c b y a x

CRYSTAL DIRECTION EXAMPLE z c b y a x

OBTAINING MILLER INDICES (1) obtain components: obtain the distances along the three crystallographic axes where the plane in question intersects those axes; if the plane intersects any axis at its zero value, translate the plane one lattice parameter along that axis and redraw the plane. (2) normalize components: normalize this triple of indices by dividing each index by its corresponding lattice parameter. (3) invert: compute the reciprocals of the three indices above. (4) obtain integers: multiply these three indices by a common factor so that all indices become integers (smallest possible). (5) display results: enclose the three indices (not separated by commas) in parentheses; use a bar over an index to indicate a negative value.

MILLER INDICES EXAMPLE z c b y a x

MILLER INDICES EXAMPLE z c b y a x

MILLER INDICES EXAMPLE z c b y a x

LINEAR AND PLANAR DENSITIES linear density LD number of atoms center- ed on a direction vector divided by the length of the direction vector crystallographic plane (hkl) planar density PD number of atoms centered on a plane divided by the area of the plane crystallographic direction [h'k'l']

LINEAR AND PLANAR DENSITY PROBLEM Indium has a simple tetragonal crystal structure for which the lattice parameters a and c are 0.459 nm and 0.495 nm respectively. Find the cation densities LD 111 and PD 110 for crystalline indium. c a a

DIAMOND STRUCTURE Each carbon atom is single-bonded to three others. This is the same structure as the zinc blende structure, but with carbon atoms occupying all positions. Silicon, germanium, and gray tin (Group IVA elements in the periodic table) have the same structure.

GRAPHITE STRUCTURE

BUCKMINSTERFULLERENE Structure of the buckminsterfullerene (C 60 ) molecule. This polymorphic form of carbon was discovered in 1985. A single molecule is often referred to as a "buckyball".

CARBON NANOTUBE Tube diameters are typically less than 100 nm. It is one of the strongest known materials (based on tensile strength). Illustration by Aaron Cox / American Scientist.

CARBON NANOTUBE Anatomically resolved scanning tunneling microscope (STM) image of a carbon nanotube. Coutesy of Vladimir Nevolin, Moscow Institute of Electronic Engineering.

SINGLE CRYSTAL \ Single crystal of garnet (a silicate) from Tongbei, Fujian Province, China.

PHASE TRANSITION IN TIN White (β) tin (body-centered tetragonal crystal structure) transforms as the temperature drops below 13.2 C to gray (α) tin (diamond cubic crystal structure).

PHASE TRANSITION IN TIN White (β) tin (lower cylinder) and gray (α) tin (upper)

POLYCRYSTALLINITY small crystallite nuclei growth of crystallites appearance of grains un- der microscope completion of solidification

SILICON DIOXIDE crystalline SiO 2 non-crystalline SiO 2 Two-dimensional analogs of crystalline and non-crystalline silicon dioxide.

SILICATE GLASS Schematic representation of sodium ion positions in a sodium-silicate glass.

MARTENSITE A metastable phase in the Fe-Fe 3 C system occuring when the temperature of austenite (FCC) drops rapidly from above the eutectoid temperature (727 o C) to temperatures around ambient. The transformation involves an essentially diffusionless rearrangement of carbon atoms and produces a BCT phase.