Crystals Structure- Exam 1

Long-range order (LRO)

a defining property of crystals showing translational symmetry

Intrinsic properties

material properties that are microstructure-insensitive

Extrinsic properties

material properties that are microstructure-sensitive

Intrinsic property examples (4)

coefficient of thermal expansion, modulus of elasticity, refractive index, saturation magnetization

Extrinsic property examples (4)

ductility/toughness/brittleness, electrical conductivity, magnetic coercivity, yield strength

Formula unit

a repeating unit of atoms (building blocks)

Motif/basis

a repeating structure made of formula units

crystalline density __ amorphous density

>

crystalline thermal experimental coefficient __ amorphous thermal experimental coefficient

>>

crystalline index of refraction __ amorphous index of refraction

>

requirements for crystalline state

low temperature, has LRO, k_BT << bonding energy

Lattice parameters (2D)

{a,b,y}

Symmetry class

the number of mirror lines in a motif or basis

Point/space lattice

an infinite array of points that fill space

Coordinate format (abc)

(a,b,c)

Direction format (abc)

[abc]

Plane format (abc)

(abc)

Crystallographically equivalent direction format (abc)

<abc>

Equivalent plane format (abc)

{abc}

Coordinate in reciprocal space format (abc)

abc

Primitive unit cell

a unit cell containing one lattice point

Planar lattices

hexagonal, oblique, rectangular, square

Square lattice parameters

{a,a,90}

Rectangular lattice parameters

{a,b,90}

Oblique lattice parameters

{a,b,y}

Hexagonal lattice parameters

{a,a,120}

Lattice parameters (3D)

{a,b,c,a,B,y}

Crystal systems

cubic (c), hexagonal (h), monoclinic (m), orthorhombic (o), rhombohedral/trigonal (R), tetragonal (t), triclinic/anorthic (a)

Types of Bravais symmetry

C (centered on two faces), F (face symmetry), I (centered in body), P (primitive)

Cubic lattice parameters

{a,a,a,90,90,90}

Triclinic/anorthic lattice parameters

{a,b,c,a,B,y}

Monoclinic lattice parameters

{a,b,c,90,B,90}

Hexagonal lattice parameters

{a,a,c,90,90,120}

Rhombohedral/trigonal lattice parameters

{a,a,a,a,a,a}

Orthorhombic lattice parameters

{a,b,c,90,90,90}

Tetragonal lattice parameters

{a,a,c,90,90,90}

Bravais lattices

aP, cF, cI, cP, hP, mC, mP, oC, oF, oI, oP, R, tI, tP

Cubic lattices

3- cF, cI, cP

Tetragonal lattices

2- tI, tP

Orthorhombic lattices

4- oC, oF, oI, oP

Monoclinic lattices

2- mC, mP

Zone law/zonal equation

vector calculus

Metric tensor

|a² a·b a·c|

|a·b b² b·c|

|a·c a·b c²|

Distance equation between points P and Q

D^2 = [Q-P](row) x [g] x [Q-P](col)

Magnitude equation of a vector [v]

|v| = sqrt([v(row)] x [g] x [v(col)])

Reciprocal basis equations

a* = (bxc)/(a·(bxc))

b* = (cxa)/(a·(bxc))

c* = (axb)/(a·(bxc))

Properties of reciprocal base vectors

a* orthogonal to b and c, a*·a=1

Reciprocal metric tensor calculations

metric tensor equation with reciprocal vectors or inverse matrix of real metric tensor

Volume of unit cell

V^2 = det|g|

Distance between planes (hkl)

1/d^2 = [hkl](row) x [g*] x [hkl](col)