materials chapter 3

crystal

a solid where atoms are arranged in a highly ordered, repeating pattern

crystalline material

a material composed of crystals (grains) with ordered atomic structure

amorphous (non crystalline)

atoms are arranged randomly, with no long-range order

semi-crystalline

a mix of ordered (crystalling) and disordered (amorphous) regions

lattice

a repeating array of points in space

basis

the group of atoms attached to each lattice point

crystal structure

lattice + basis

unit cell

the smallest repeating volume that represents the entire crystal

how to choose a unit cell

must tile spaces without gaps; should reflect symmetry; usually the simplest repeating shape

number of crystal systems

7

number of bravais lattices

14 total

what defines a crystal system

unit cell edge lengths (a, b, c); and angles (alpha, beta, gamma)

APF

volumes of atoms in cell / volumes of unit cell; fraction of volume occupied by atoms in a unit cell

point coordinates

positions inside the unit cell expressed as fractions of cell dimensions

crystallographic direction

a vector that shows direction between two points in a crystal; identify two points; subtract coordinates; reduce to smallest integers;

close packed structure

atoms arranged to maximize packing efficiency; FCC and HCP;

difference between FCC and HCP

same apf; both close packed; FCC has ABCABC stacking; HCP has ABAB stacking

density equation

nA/V_cN_A; n = atoms/unit cell; A = atomic weight; V_c = unit cell volume; N_a = avogadro’s number

miller indices

a set of integers that describe the orientation of a plane in a crystal

steps to find miller indices

find intercepts; take reciprocals; multiply fractions out; large intercept means small index; parallel to axis means index = 0;

draw a plane from (hkl)

take recipricols of hkl; plot intercepts; connect points;

atomic arrangement

how atoms are positioned along a line or within a plane, used to determine density

linear density

number of atoms per unit length along a direction

planar density

number of atoms per unit area on a plane

polycrystalline materials

a material made of many small crystals (grains) with different orientations

anisotropy

properties that depend on direction in the material; in single crystals or materials with preferred grain orientation

x-ray diffraction

used bc their wavelength is similar to atomic spacing, allowing diffraction; provides crystal structure, lattice spacing, phase identification

components of an XRD experiment

xray source; specimen; detector; geometry (angles/mechanics)

bragg’s law

nw = 2dsin(x); n = order of diffraction; w = wavelength; d = spacing between planes; x = angle of incidence

diffraction conditions

when reflected waves constuctively interfere (path difference matches wavelength)

relationship between d, hkl, and cubic

d = a / sqrt(h²+k²+l²); a = unit cell length; (hkl) = plane