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

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.

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.

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°

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)

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)

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)

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.

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

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.

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

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 %

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

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.

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.

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 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.

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

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

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

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.

GRAPHITE STRUCTURE

BUCKMINSTERFULLERENE

Structure of the buckminsterfullerene (C60) molecule. Thispolymorphic form of carbon was discovered in 1985. Asingle molecule is often referred to as a "buckyball".

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.

CARBON NANOTUBE

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

SINGLE CRYSTAL

\Single crystal of garnet (a silicate) from Tongbei, FujianProvince, 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

completion of solidifica- appearance of grains un-tion der microscope

SILICON DIOXIDE

crystalline SiO2 non-crystalline SiO2

Two-dimensional analogs of crystalline and non-crystallinesilicon dioxide.

SILICATE GLASS

Schematic representation of sodium ion positions in asodium-silicate glass.

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.