MATSE 512 - Symmetry

Point groups, space groups, symmetry operations for crystal chemistry

Triclinic

a≠b≠c

α≠β≠γ

Min. Symmetry: None

Monoclinic

a≠b≠c

α=γ=90°

β≠90°

Min. Symmetry: 2-fold along [010]

Orthorhombic

a≠b≠c

α=β=γ=90°

Min. Symmetry: 3 parallel 2-fold along [100],[010],[001]

Trigonal

a=b≠c

α=β=90°

γ=120°

Min. Symmetry: 3-fold along [001]

Rhombohedral

a=b=c

α=β=γ≠90°

Min. Symmetry: 3-fold along [111]

Tetragonal

a=b≠c

α=β=γ=90°

Min. Symmetry: 4-fold along [001]

Hexagonal

a=b≠c

α=β=90°

γ=120°

Min. Symmetry: 6-fold along [001]

Cubic

a=b=c

α=β=γ=90°

Min. Symmetry: 4 3-fold along ⟨111⟩

Unit cell volume

V = abc(1+2cos(α)cos(β)cos(γ)-cos²(α)-cos²(β)cos²(γ))^1/2

Density

ρ = MZ / (N_AV)

M = molar mass, Z = formula units per U.C., N_A = Avogadro, V = U.C. volume

Bond Length

d² = a²(Δx)² + b²(Δy)² + c²(Δz)² + 2ab(Δx)(Δy)cos(γ) + 2ac(Δx)(Δz)cos(β) + 2bc(Δy)(Δz)cos(α)

Bond Angle

cosθ = (R₁₂²+R₂₃²-R₁₃²) / (2R₁₂R₂₃)

Point group: 1

No symmetry elements

Point group: 1bar

only inversion (1bar) symmetry

Point group: 2

1 2-fold axis

Point group: m

1 mirror

Point group: 2/m

1 2-fold, 1 inversion, 1 mirror

Point group: 222

3 2-fold rotation, 1 inversion, 3 mirrors

Point group: mm2

1 2-fold, 2 mirrors

Point group: mmm

3 2-fold, inversion, 3 mirrors

Point group: 3

1 3-fold axis

Point group: 3bar

1 3-fold, 1 inversion, 1 3bar

Point group: 32

3 2-fold, 1 3-fold

Point group: 3m

1 3-fold, 3 mirrors

Point group: 3bar m

3 2-fold, 1 3-fold, inversion, 3 mirrors, 1 3bar

Point group: 4

1 2-fold, 1 4-fold

Point group: 4bar

1 2-fold, 1 4bar

Point group: 4/m

1 2-fold, 1 4-fold, inversion, 1 mirror, 1 4bar

Point group: 422

5 2-fold, 1 mirror

Point group: 4mm

1 2-fold, 1 4-fold, 4 mirrors

Point group: 4bar 2m

3 2-fold, 2 mirrors, 1 4bar

Point group: 4/mmm

5 2-fold, 1 4-fold, inversion, 5 mirrors, 1 4bar

Point group: 6

1 2-fold, 1 3-fold, 1 6-fold

Point group: 6bar

1 3-fold, 1 mirror, 1 6bar

Point group: 6/m

1 2-fold, 1 3-fold, 1 6-fold, inversion, 1 mirror, 1 3bar, 1 6bar

Point group: 622

7 2-fold, 1 3-fold, 1 6-fold

Point group: 6mm

1 2-fold, 1 3-fold, 1 6-fold, 6 mirrors

Point group: 6bar m2

3 2-fold, 1 3-fold, 4 mirrors, 1 6bar

Point group: 6/mmm

7 2-fold, 1 3-fold, 1 6-fold, inversion 7 mirrors, 1 3bar, 1 6bar

Point group: 23

3 2-fold, 4 3-fold

Point group: m3

3 2-fold, 4 3-fold, inversion, 3 mirrors, 4 3bar

Point group: 432

9 2-fold, 4 3-fold, 3 4-fold

Point group: 4bar 3m

3 2-fold, 4 3-fold, 6 mirrors, 3 4bar

Point group: m3m

9 2-fold, 4 3-fold, 3 4-fold, inversion, 9 mirrors, 4 3bar, 3 4bar

Unit cell basis naming conventions

F = face-centered

P = primitive

I = body-centered

A,B,C = base-centered

Glide plane (a,b,c)

Reflection across the plane containing the axis, then translation by ½ in the direction of the glide plane

A diamond glide (denoted by “d”) has a translation of ¼ instead

A net plane (denoted by “n”) has translation in two directions

These are all space group elements, NOT found in point groups! a/b/c/n/d are all mirrors when you remove the translation aspect to convert to a point group

axis/plane notation

The subsequent plane is perpendicular to the preceding axis

Group-subgroup phase transitions

Often associated with a 2^nd order phase transition (anomaly in the 2^nd order derivative of free energy function) — could be 1^st or 2^nd order transition though

No group-subgroup relationship in the phase transition

Must be a 1^st order phase transition (anomaly in the 1^st derivative of the free energy function)