CRYSTALLOGRAPHIC POINT AND SPACE...

28
CRYSTALLOGRAPHIC POINT AND SPACE GROUPS Andy Elvin June 10, 2013

Transcript of CRYSTALLOGRAPHIC POINT AND SPACE...

Page 1: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

CRYSTALLOGRAPHICPOINT AND SPACE

GROUPSAndy Elvin

June 10, 2013

Page 2: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

Contents

Point and Space Groups

Wallpaper Groups

X-Ray Diffraction

Electron Wavefunction

Neumann’s Principle

Magnetic Point Groups

Page 3: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

Point Groups

Subgroups of O(3)

32 Crystallographic Point Groups

E, i, Cn, σh, σv, σd, Sn

relevant point groups will be referred to as group P

Page 4: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.
Page 5: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

D3 symmetries

reflection axes

C3 symmetries

no reflection axes

Page 6: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

Cube and Octahedron are dual

Symmetries under Oh

Page 7: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

Space GroupsSubgroups of E(3)

Point Group + Translation

{ R | 0 }{ E | t }a = { R | t }a = Ra + t

230 Space Groups

73 symmorphic space groups

relevant space groups will be referred to as group S

Page 8: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

Bravais Lattices

14 Lattice Types

Defines translations of space group

t=n1a1+n2a2+n3a3

Page 9: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

Point Group is C3

Left: Space Group is P32

Right: Space Group is P31

Molecule is Chiral

Page 10: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

Wallpaper Groups

Wallpaper Groups

Frieze Groups

Spherical and Hyperbolic Symmetry Groups

Page 11: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

The 17 Wallpaper Groups

Page 12: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

p1

p2

Page 13: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

p4g

p6m

Page 14: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

Reciprocal Lattice

b1 = (a2 × a3)(a1 ⋅ [a2 × a3])-1

cyclically permute indices

Fourier Transformed spatial functions

Momentum Space

k vector

Page 15: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

X-Ray Diffraction

Bragg’s Law/Bragg Scattering

nλ = 2dsin(θ)

X-ray diffraction patterns

Fourier Transform

Page 16: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

Bragg Scattering

X-ray plane waves with propagation vector k

electron number density n(r)

n(r + t) = n(r)

translation in reciprocal lattice space: g

Page 17: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

n(r) = ∑g ng exp[ig ⋅ r]

n(r + t) = ∑g ng exp[ig ⋅ (r + t)]

n(r + t) = ∑g ng exp[ig ⋅ r] exp[ig ⋅ t] = n(r)

n(r) = ∑g ng exp[ig ⋅ r] exp[ig ⋅ t] = ∑g ng exp[ig ⋅ r]exp[ig ⋅ t] = 1 => g ⋅ t = 2πn

and finally

n(r) = ∑g ng exp[ig ⋅ r] = n(r + t)for all t ∈ S

Page 18: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

scattering amplitude of incident plane waves, F(∆k) is given by

F(∆k) = ∫n(r)exp[ir ⋅ (k - k’)]dr and ∆k ≡ k’ - k

where k’ is the direction of propagation of the scattered wave and the integral is taken over all lattice sitesand noting that n(r) = ∑g ng exp[ig ⋅ r],

F(∆k) = ∑g∫ngexp[ir ⋅ (g - ∆k)]dr

which allows us to conclude that the scattering amplitude is at a maximum when g = ∆k and using the definition of ∆k…

k’ = k + g

Page 19: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

then the elastic scattering of the x-rays implies(k + g)2 = k’2 = k2

completing the square yields the two bragg conditions:g2 + 2(k ⋅ g) = 0

2(k ⋅ g) = g2

after noting that if g is a reciprocal lattice basis vector, so is -g

k ⋅ g = |k||g|cos(ß)ß = θ - π/2 => k ⋅ g = |k||g|sin(θ), now…

2|k|sin(θ) = |g|2|t|sin(θ) = |g||t| / |k|

after noting |t| = d and |k| = 2π/λ then |g||t| / |k| = nλ so finally2dsin(θ) = nλ

Page 20: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

X-ray diffraction pattern for NaCl Structure of NaCl

Page 21: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

Electron Wavefunctions

Bloch’s Theorem

Probability Density

Energy invariance

Degeneracy of electron energy levels

Page 22: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

Bloch’s Theorems ∈ S, then

s(exp[ik ⋅ r]) ≡ exp[isk ⋅ r]and sf(r) = f(s-1r) implies

s(exp[ik ⋅ r]) = exp[ik ⋅ s-1r]

now considering the crystal system described by the space group S, then the potential has periodicity defined by t such that

V(r) = V(r + t)

Bloch’s Theorem States:Ψk(r) = uk(r)exp[ik ⋅ r]where uk(r) = uk(r + t)

Page 23: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

Consider the electrons in the crystal defined by SHΨ = EΨ

[H - E]uk(r)exp[ik ⋅ r] = 0[H - E]uk(r) = 0

uk(r) is then a solution of Schrodinger’s equation for the crystal

Probability density|Ψk(r + t)|2 = |uk(r + t)|2exp[i(k ⋅ r) - i(k ⋅ r)] = |uk(r + t)|2

|Ψk(r + t)| = |uk(r + t)| = |uk(r)| = |Ψk(r)|

EnergyΨk(r + t) = exp[ik ⋅ t + ik ⋅ r]uk(r + t) = exp[ik ⋅ t]Ψk(r)H(Ψk(r + t)) = H(exp[ik ⋅ t])Ψk(r) = E(exp[ik ⋅ t])Ψk(r)E(k) = E(exp[ik ⋅ t]) = E(exp[i(k + g) ⋅ t]) = E(k + g)

Page 24: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

DegeneracyO E 8C3 3C2 6C2 6C4

Γ1 1 1 1 1 1

Γ2 1 1 1 -1 -1

Γ3 2 -1 2 0 0

Γ4 3 0 -1 -1 1

Γ5 3 0 -1 1 -1

O E 8C3 3C2 6C2 6C4

D0 1 1 1 1 1D1 3 0 -1 -1 1D2 5 -1 1 1 -1D3 7 1 -1 -1 -1D4 9 0 1 1 1

Character Table of theIrreducible Representations

Character Table of theReducible Representations, DL, corresponding to the spherical

harmonics, YLM

D0 = Γ1

D1 = Γ4

D2 = Γ3 + Γ5

D3 = Γ2 + Γ4 + Γ5

D4 = Γ1 + Γ3 + Γ4 + Γ5

Page 25: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

Neumann’s Principle

Any physical property of a crystal possesses the symmetry of its point group, P

Tensors representing a property are invariant under P

Susceptibility, Stress, Polarizability, Inertial, etc...

Page 26: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

Polarizability TensorP = C2h = {E, C2, i, σh}

Tij = Tji

E C2 i σh

x2 x2 x2 x2 x2

y2 y2 y2 y2 y2

z2 z2 z2 z2 z2

xy xy xy xy xy

xz xz -xz xz xz

yz yz -yz yz yz

C2 operation implies Txy = -Txy = 0Tyz = -Tyz = 0

This leaves the polarizability tensor with at most four

components

Page 27: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

Magnetic Point GroupsMagnetic Crystals display different symmetries

M is the antisymmetry operator (“black and white”, “time-reversal”, “current-reversal”)

P = N ∪ (P - N)

N is an invariant subgroup of P

P’ = N ∪ M(P - N)

P’ is the Magnetic Point Group (“black and white point group”)

There are 58 Magnetic Point Groups

Page 28: CRYSTALLOGRAPHIC POINT AND SPACE GROUPSscipp.ucsc.edu/~haber/archives/physics251_13/ace13_crystals.pdf · Primer for Point and Space Groups. Springer, 2004 Bradley and Cracknell.

ReferencesMichael Tinkham. Group Theory and Quantum Mechanics. Dover Publications, 1992.

Richard L. Liboff. Primer for Point and Space Groups. Springer, 2004

Bradley and Cracknell. The Mathematical Theory of Symmetry in Solids. Oxford, 1972.

Burns and Glazer. Space Groups for Solid State Scientists. Academic Press, 1990

Bhagavantarn and Venkatarayudu. Theory of Groups and its Application to Physical Problems. Academic Press, 1969.