crystal structure terms - rdarke - home
TRANSCRIPT
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CRYSTAL STRUCTURE TERMS
crystalline material - a material in which atoms, ions, ormolecules are situated in a periodic 3-dimensional arrayover large atomic distances (all metals, many ceramicmaterials, and certain polymers are crystalline under nor-mal conditions)
crystal structure - the manner in which atoms, ions, ormolecules are arrayed spatially in a crystalline material; itis defined by (1) the unit cell geometry, and (2) the posi-tions of atoms, ions, or molecules within the unit cell
coordination number (in the hard sphere representationof the unit cell) - the number of nearest (touching) neigh-bors that any atom or ion in the crystal has
atomic packing factor (APF) (in the hard sphere repre-sentation of the unit cell) - the ratio of solid sphere volumeto unit cell volume for a crystal structure
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UNIT CELL
lattice points unit cell
The unit cell is the basic structural unit of a crystal struc-ture which, when tiled in 3-dimensions, would reproducethe crystal. All atomic, ionic, or molecular positions in acrystal may be generated by translating the unit cell inte-gral distances along each of its edges.
Lattice points are those points representing the positionsof atoms, ions, or molecules in a crystal structure.
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UNIT CELL LATTICE PARAMETERS
z
c
β α
b yγ
a
x
A unit cell with x, y, and z coordinate axes (not necessarilymutually orthogonal), showing the axial lengths (a, b, c) ofthe unit cell and the interaxial angles (α, β, γ) of the unitcell. These parameters (a, b, c; α, β, γ) are the lattice pa-rameters of the unit cell.
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CRYSTAL SYSTEMS
cubic tetragonal orthorhombic
a = b = c a = b = c a = b = cα = β = γ = 90° α = β = γ = 90° α = β = γ = 90°
rhombohedral hexagonal monoclinic
a = b = c a = b = c a = b = cα, β, γ = 90° α = β = 90° γ = 120° α = γ = 90° = β
triclinic
a = b = cα, β, γ = 90°
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FACE-CENTERED CUBIC (FCC)
a
a
hard sphere reduced spheremodel unit cell model unit cell
hard sphere model radius: R = .354aunit cell volume: 1.000a3
number of atoms in unit cell: 4atomic packing factor: 0.740coordination number: 12
examples: aluminum (a = .1431 nm)copper (a = .1278 nm)gold (a = .1442 nm)nickel (a = .1246 nm)
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BODY-CENTERED CUBIC (BCC)
a
a
hard sphere reduced spheremodel unit cell model unit cell
hard sphere model radius: R = .433aunit cell volume: 1.000a3
number of atoms in unit cell: 2atomic packing factor: 0.680coordination number: 8
examples: chromium (a = .1249 nm)niobium (a = .1430 nm)tungsten (a = .1371 nm)iron (a = .1241 nm)
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HEXAGONAL CLOSE-PACKED (HCP)
reduced sphere hard sphere model unit cell model unit cell
hard sphere model radius: R = .500aunit cell volume: 2.121a3
number of atoms in unit cell: 6atomic packing factor: 0.740coordination number: 12
examples: cadmium (a = .1490 nm)beryllium (a = .1140 nm)titanium (a = .1445 nm)zinc (a = .1332 nm)
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ESTIMATING METAL DENSITIES
The density of a metal can be estimated using:
ρ = nA/VCNA
where n = number of atoms in a unit cellA = atomic weight of metalVC = volume of the unit cellNA = 6.02x1023 atoms/mole
FCC BCC HCP
a 2.828R 2.309R 2.000R
VC 1.000a3 1.000a3 4.243a3
example: Copper has an atomic radius of .128 nm, anatomic weight of 63.5 g/mole, and a FCC crystal structure.Compute its theoretical density using this information, andfind the percent error between this value and the acceptedvalue of 8.94 g/cm3.
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TABLE OF IONIC RADII
cation ionic radius anion ionic radius
Al3+ .053 nm Br- .196 nmBa2+ .136 nm Cl- .181 nmCa2+ .100 nm F- .133 nmCs+ .170 nm I- .220 nmFe2+ .077 nm O2- .140 nmFe3+ .069 nm S2- .184 nmK+ .138 nm
Mg2+ .072 nmMn2+ .067 nmNa+ .102 nmNi2+ .069 nmSi4+ .040 nmTi4+ .061 nm
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ANION-CATION STABILITY
stable stable unstable
A cation-anion combination is unstable if the cation cannotcoordinate with any of the anions which create the inter-stitial position that the cation occupies.
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COORDINATION STABILITY
CN = 2rc/ra < .155 CN = 3 CN = 4
.155 < rc/ra < .225 .255 < rc/ra < .414
CN = 6 CN = 8.414 < rc/ra < .732 .732 < rc/ra < 1.000
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SOME CERAMIC CRYSTAL STRUCTURES
rock salt cesium chloride zinc blendestructure structure structure
type: AX type: AX type: AXexample: NaCl example: CsCl example: ZnS = Na+ = Cl- = Cs+ = Cl- = Zn+2 = S-2
rC/rA = .564 rC/rA = .939 rC/rA = .402CN = 6 CN = 8 CN = 4
PIC = 66.8 % PIC = 73.4 % PIC = 18.3 %
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SOME CERAMIC CRYSTAL STRUCTURES
fluorite perovskitestructure structure
type: AX2 type: ABX3example: CaF2 example: BaTiO3 = Ca+ = F- =Ti4+ = Ba2+ = O2-
rC/rA = .752 CN = 4 for Ba-OCN = 8 CN = 8 for Ti-O
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ESTIMATING CERAMIC DENSITIES
The density of a ceramic can be estimated using:
ρ = n'AF/VCNA
where n' = number of formula units in a unit cellAF = molar weight of one formula unitVC = volume of the unit cellNA = Avogadro's number
= 6.02x1023 atoms/mole
example: Cesium chloride is an AX-type ceramic with thecesium chloride crystal structure. Cesium has an atomicweight of 132.9 g/mole, and the cesium cation has anionic radius of .170 nm. Chlorine has an atomic weight of35.5 g/mole, and the chlorine anion has an ionic radius of.181 nm. Compute the theoretical density of cesium chlo-ride using this information.
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CRYSTAL DIRECTION INDICES
(1) obtain components: obtain the components of thevector along the three coordinate axes in terms of thelattice parameters a, b, and c (if vector is not in standardposition, subtract the coordinates of the tail of vector fromthe coordinates of the tip).
(2) normalize components: normalize this triple of indi-ces by dividing each index by its corresponding latticeparameter.
(3) obtain integers: multiply these three indices by acommon factor so that all indices become integers (small-est possible).
(4) display results: enclose the three indices (not sepa-rated by commas) in square brackets; use a bar over anindex to indicate a negative value.
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CRYSTAL DIRECTION EXAMPLE
z
c
b y
a
x
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CRYSTAL DIRECTION EXAMPLE
z
c
b y
a
x
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CRYSTAL DIRECTION EXAMPLE
z
c
b y
a
x
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OBTAINING MILLER INDICES
(1) obtain components: obtain the distances along thethree crystallographic axes where the plane in questionintersects those axes; if the plane intersects any axis at itszero value, translate the plane one lattice parameter alongthat axis and redraw the plane.
(2) normalize components: normalize this triple of indi-ces by dividing each index by its corresponding latticeparameter.
(3) invert: compute the reciprocals of the three indicesabove.
(4) obtain integers: multiply these three indices by acommon factor so that all indices become integers (small-est possible).
(5) display results: enclose the three indices (not sepa-rated by commas) in parentheses; use a bar over an indexto indicate a negative value.
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MILLER INDICES EXAMPLE
z
c
b y
a
x
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MILLER INDICES EXAMPLE
z
c
b
y
a
x
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MILLER INDICES EXAMPLE
z
c
b y
a
x
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LINEAR AND PLANAR DENSITIES
crystallographic crystallographic plane (hkl) direction [h'k'l']
planar density PD linear density LDnumber of atoms center- number of atoms center-ed on a plane divided by ed on a direction vectorthe area of the plane divided by the length of
the direction vector
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LINEAR AND PLANAR DENSITY PROBLEM
Indium has a simple tetrago-nal crystal structure for whichthe lattice parameters a andc are 0.459 nm and 0.495nm respectively. Find thecation densities LD111 andPD110 for crystalline indium. c
a
a
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DIAMOND STRUCTURE
Each carbon atom is single-bonded to three others. Thisis the same structure as the zinc blende structure, butwith carbon atoms occupying all positions. Silicon, germa-nium, and gray tin (Group IVA elements in the periodictable) have the same structure.
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GRAPHITE STRUCTURE
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BUCKMINSTERFULLERENE
Structure of the buckminsterfullerene (C60) molecule. Thispolymorphic form of carbon was discovered in 1985. Asingle molecule is often referred to as a "buckyball".
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CARBON NANOTUBE
Tube diameters are typically less than 100 nm. It is one ofthe strongest known materials (based on tensile strength).Illustration by Aaron Cox / American Scientist.
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CARBON NANOTUBE
Anatomically resolved scanning tunneling microscope(STM) image of a carbon nanotube. Coutesy of VladimirNevolin, Moscow Institute of Electronic Engineering.
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SINGLE CRYSTAL
\Single crystal of garnet (a silicate) from Tongbei, FujianProvince, China.
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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).
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PHASE TRANSITION IN TIN
White (β) tin (lower cylinder) and gray (α) tin (upper)
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POLYCRYSTALLINITY
small crystallite nuclei growth of crystallites
completion of solidifica- appearance of grains un-tion der microscope
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SILICON DIOXIDE
crystalline SiO2 non-crystalline SiO2
Two-dimensional analogs of crystalline and non-crystallinesilicon dioxide.
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SILICATE GLASS
Schematic representation of sodium ion positions in asodium-silicate glass.
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MARTENSITE
A metastable phase in the Fe-Fe3C system occuring whenthe temperature of austenite (FCC) drops rapidly fromabove the eutectoid temperature (727
oC) to temperaturesaround ambient. The transformation involves an essen-tially diffusionless rearrangement of carbon atoms andproduces a BCT phase.
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