Crystal Chemistry

Crystal System

long-range orderly arrangements of atoms or ions

Crystal System: Triclinic

a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90

V = abc(1-cos²α-cos²β-cos²γ+2cosαcosβcosγ)^1/2

Crystal System: Monoclinic

a ≠ b ≠ c, α = γ = 90, β ≠ 90

V=abcsinβ

simple, end-centered

Crystal System: Orthorhombic

a ≠ b ≠ c, α = β = γ = 90

V=abc

simple, body-centered, face-centered, end-centered

Crystal System: Tetragonal

a = b ≠ c, α = β = γ = 90

V=a²c

simple, body-centered

Crystal System: Hexagonal

a = b ≠ c, α = β = 90, γ = 120

V=a²c(3/4)^1/2

Crystal System: Trigonal/Rhombohedral

a = b = c, α = β = γ ≠ 90

V=a³(1-3cos²α+2cps³α)^1/2

Crystal System: Cubic

a = b = c, α = β = γ = 90

V=a³

simple, body-centered, face-centered

Density

how crystal chemistry can be used to predict the magnitude of physical property

ρ = [(Z)(M.W.)]/(V No) (Z=#M/Unit cell, M.W.=Molecular

Weight, No.=Avogadro's constant, V=Volume of unit cell)

Planes

described by miller indices

Miller indices

a set of numbers in parentheses (hkl)

reciprocals of the intercepts of the plane with the crystal axis

Direction

a vector moving from the origin to ua along x, vb along y, and wc along z

Stereographic Projection

easy way of presenting three dimensional objects in two dimensions

Point Groups: Triclinic

1, bar1

Point Groups: Monoclinic

2, m, 2/m

Point Groups: Orthorhombic

222, mm2, mmm

Point Groups: Tetragonal

4, bar4, 4/m, 422, 4mm, bar42m, 4/mmm

Point Groups: Trigonal

3, bar3, 32, 3m, bar3m

Point Groups: Hexagonal

6, bar6, 6/m, 622, 6mm, bar6m2, 6/mmm

Point Groups: Cubic

23, m3, 432, bar43m, m3m

Magnetization Unchanged

about inversion center

magnetization parallel to rotation axis

magnetization perpendicular to mirror plane

Magnetization Changed

magnetization perpendicular to rotation axis

magnetization parallel to mirror plane

Time reversal

symmetry operation which always changes the direction of magnetizationm

m’

mirror accompanied by time reversal

Pyroelectricity

developing an electric polarization with change in temperature

Most abundant element in earth’s crust

oxygen

Hydrogen

Atomic #: 1

Symbol: H

Orbital: single electron in S orbital of K shell

Configuration: 1s¹

Valence: 1

Oxide: H₂O

Helium

Atomic #: 2

Symbol: He

Orbital: two electrons in S orbital

Configuration: 1s²

Lithium

Atomic #: 3

Symbol: Li

Configuration: 1s²2s¹

Valence: 1

Oxide: Li₂O

Beryllium

Atomic #: 4

Symbol: Be

Configuration: 1s²2s²

Valence: 2

Oxide: BeO

High melting point, low density, high thermal conductivity, low dielectric constant, high dielectric strength

Boron

Atomic #: 5

Symbol: B

Configuration: 1s²2s²2p¹

Valence: 3

Oxide: B₂O₃

constituent of glass compositions

Hund’s Rule of Maximum multiplicity

as many of the orbitals as possible are occupied by a single electron before pairing

Carbon

Atomic #: 6

Symbol: C

Configuration: 1s²2s²2p²

Valence: 4

Oxide: CO₂

two crystalline states, diamond, graphite

Oxygen

Atomic #: 8

Symbol: O

Configuration: 1s²2s²2p⁴

Valence: 2

Oxide: O₂

most ceramics have this element, abundance of O in Earth’s Crust: 46.6 %

Sodium

Atomic #: 11

Symbol: Na

Configuration: 1s²2s²2p⁶3s¹

Valence: 1

Oxide: Na₂O

Magnesium

Atomic #: 12

Symbol: Mg

Configuration: 1s²2s²2p⁶3s²

Valence: 2

Oxide: MgO

Aluminum

Atomic #: 13

Symbol: Al

Configuration: 1s²2s²2p⁶3s²3p¹

Valence: 3

Oxide: Al₂O₃

Alumina: strength, hardness, high melting point

Silicon

Atomic #: 14

Symbol: Si

Configuration: 1s²2s²2p⁶3s²3p²

Valence: 4

Oxide: SiO₂

mineral quartz, second to oxygen in earth’s crust, glass forming element, semiconductors

Calcium

Atomic #: 20

Symbol: Ca

Configuration: 1s²2s²2p⁶3s²3p⁶4s²

Valence: 2

Oxide: CaO

limestone, rock salt structure, low hardness

Titanium

Atomic #: 22

Symbol: Ti

Configuration: 1s²2s²2p⁶3s²3p⁶3d²4s²

Valence: 4

Oxide: TiO₂

Titania, mineral name rutile

Iron

Atomic #: 26

Symbol: Fe

Configuration: 1s²2s²2p⁶3s²3p⁶3d⁶4s²

Valence: 2 and 3

Oxide: α Fe₂O₃ hematite, bright red color, γ Fe₃O₄ magnetite, Fe₃O₄

steel making

Copper

Atomic #: 29

Symbol: Cu

Configuration: 1s²2s²2p⁶3s²3p⁶3d¹⁰4s¹

Valence: 1 and 2

Oxide: Cu₂O, CuO

chalcopyrite

Zinc

Atomic #: 30

Symbol: Zn

Configuration: 1s²2s²2p⁶3s²3p⁶3d¹⁰4s²

Valence: 2

Oxide: ZnO

Mineral: Zincblende

Chemical Bonds (Weakest —> Strongest)

Van der Waals, Hydrogen, Metallic, Ionic, Covalent

Covalent Bond

shared pair(s) of electrons between atoms, may or may not be equal sharing

strong, high melting point, low thermal expansion, semiconductors, low coordination #

Ionic Bond

complete transfer of electrons

strong, medium/high melting point, low thermal expansion, high coordination #

Metallic Bond

columbic attractive force between nearly free electrons and positive ion cores

plastic deformation, intermediate thermal expansion, high thermal conductivity, high coordination #

Hydrogen Bond

Hydrogen loses an electron and becomes H⁺

low strength, low melting point

Van der Waals Bond

dipole forces between atoms

low strength, very low melting point, insulators, coordination # ~ 12

Covalent Structure

significant electron density between atoms

Ionic Structure

electron density drops to zero

Metals Structure

Electron Belong to all ion cores

Van der Waals Structure

electron clouds belong to all of the ion cores

Hardness

resistance to scratching with the soft material being scratched by those higher in the scale

decreases with increase in ionic size

increases with increase of valency

increases with packing density

decreases with temperature

Carbides, Borides, Nitrides

strong covalent bonds

high hardness

excellent abrasive materials

Distance formula for planes (i think?)

(d_hkl)²=(h/a)²+(k/b)²+(l/c)²

Lattice Planes for the Faces and Diagonal Planes of the Cubic Unit Cell

{100}

Faces: (1 0 0) (0 1 0) (0 0 1) (1̅ 0 0) (0 1̅ 0) (0 0 1̅)

Diagonals: (1 0 1) (1 1 0) (0 1 1) (1 0 1̅) (1 1̅ 0) (0 1 1̅)

Slip Planes in FCC Lattice

{111}

(1 1 1) (1̅ 1 1) (1̅ 1̅ 1) (1 1̅ 1) (1 1 1̅) (1 1̅ 1̅) (1̅ 1̅ 1̅) (1̅ 1 1̅)

Family of directions <hkl>

all possible directions with the same basic coordinates

Zone’s Law

the direction [uvw] is parallel to (hkl) plane if:

hu + kv + lw = 0

Criss-Cross Law

Line formed by the intersection of two planes

U = kl₁-lk₁

V = lh₁-hl₁

W = hk₁-kh₁

Angle Between Two Vectors

cosa = (A . B)/|A||B|

Cleavage

tendency of a mineral to break along planes of weakness in the chemical bonds

breaking along planes to produce parallel sheets, blocks, prism shapes, etc

Inter Atomic Distance

d = r₁-r₂

d²=(x₁-x₂)²a²+(y₁-y₂)²b²+(z₁-z₂)²c²

Ionic Radius

not a fixed property

decreases with increasing positive charge

increases with increasing negative charge

Trends in Radii

positive or higher valence corresponds to stronger bond and shorter distance

Ionic Radii vs. Metallic Radii

ionic radii are smaller than metallic radii

Ionic Radii vs. Covalent Radii

ionic radii are larger than corresponding covalent radii