Basics symmetry in condensed matter physics

Sylvain Ravy Laboratoire de physique des solides CNRS, Université Paris-Saclay • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX: only mirror symmetry Definitions

• Transformation, Group • Évariste Galois 1811, 1832.

Symmetry:

Property of invariance of an objet under a space transformation Transformation

• Bijection which maps a geometric set in itself

M f(M)=M’

• Affine transformation maps two points P and P’ such that:

f(M) = P’ + O(PM)

P’ P P f : positions O : vectors Isometries f(M) = P’ + O(PM) • Isometry ||O(u)||=||u|| distance-preserving map • Two types of isometry:

• Affine isometry: f(M) • Transforms points – space groups • Microscopic properties of crystals (electronic structure)

• Translation • Helix of pitch P • Rotations • Reflections (a, Pa /2p)

• Linear isometry O(PM) • Transforms vectors (directions) – point groups • Macroscopic properties of crystals (response functions) 60° • Rotations E ? • Reflections Linear isometry- 2D

||O(u)|| = ||u|| • In the plane (2D)

• Rotations • Reflections (reflections through an axis)

q q/2

• Determinant +1 • Determinant -1 • Eigenvalues eiq, e-iq • Eigenvalues -1, 1 Linear isometry - 3D

• ||O(u)|| = |l| ||u|| Eigenvalues |l | = 1 • In space (3D) : • l : 3rd degree equation (real coefficients) ±1, eiq, e-iq (det. = ± 1) • det. = 1 • det. = -1 • Direct symmetry • Indirect symmetry ퟏ ퟎ ퟎ −ퟏ ퟎ ퟎ ퟎ cos 휽 − sin 휽 ퟎ cos 휽 − sin 휽 ퟎ sin 휽 cos 휽 ퟎ sin 휽 cos 휽

Rotations Rotoreflections

a) Rotation by angle q q q b) Roto-reflection q Improper rotation

c) Inversion (p) q d) Roto-inversion (p+q ) c) Reflection (0) Stereographic projection

• To represent directions preserves angles on the sphere

NN

Direction OM

M

O P’ P P’ M’ P

P, projection of OM : Intersection of SM and equator S

• Conform transformation (preserves angles locally) but not affine Main symmetry operations

• Conventionally

• Direct • Rotations (A ) p n • n-fold rotation An (2 /n) • Reflections (M) • Represented by a polygon of same symmetry. • Inversion (C)_ • Rotoinversion (An) ......

A2 vertical A2 horizontal A3 A4 A5

• Indirect ~ • Rotoreflections (An) • Symmetry element • Reflection (M) • Locus of invariant points • Inversion (C) _ • Rotoinversions (An) ......

M vertical M horizontal M Inversion A4 Composition of symmetries

• Two reflections with angle a = rotation 2a

M M’M=A 2a M’ a • Euler construction

A AN3 N2 AN1

p/N2 p/N1 Composition of two rotations = rotation

AN2AN1=AN3

• No relation between N1, N2 et N3 Point group: definition

• The set of symmetries of an object forms a group G : point group

• A and B  G, AB  G (closure) • Associativity (AB)C=A(BC) 1 2 • Identity element E (1-fold rotation) • Invertibility A, A-1  • No commutativity in general (rotation 3D) 2 1 • Example: point group of a rectangular table (2mm)

Mx

*E Mx My A2 My EE Mx My A2 Mx Mx E A2 My A2 My My A2 E Mx A2 A2 My Mx E

• Multiplicity: number of elements 2mm Composition of rotations Constraints

AN2 AN1 AN3

p/N2 p/N1

234

Spherical triangle, angles verifies:

AN

A2 22N (N>2), 233, 234, 235 A2 Dihedral groups Multiaxial groups groups Points groups ... Monoclinic Triclinic Cubic Trigonal Orthorhombic Hexagonal Tetragonal Curie’s

A n • Sorted by 1 2 3 4 6  Symmetry degree AnA2 • Curie‘s limiting groups 222 32 422 622 2 _ An • Chiral, propers _ _ _ _ _ 1 2=m 3 4 6=3/m • Impropers

An/M  /m • Centrosymmetric 2/m 4/m 6/m

An M

2mm 3m 4mm 6mm  m _ An M

_ _ _ _ _ 3m 42m (4m2) 62m (6m2)

 An /MM’ /mm

mmm 4/mmm 6/mmm

An An’

23 432   _ An An’ _ _ _ m3 43m m3m /m /m 23 432 532

_ _ _ __ m3 43m m3m 53m

Tetrahedron Octahedron Icosahedron

Cube Dodecahedron

Multiaxial groups Platonic solid Points group: Notations • Hermann-Mauguin (International notation - 1935)

• Generators (not minimum) • Symmetry directions • Reflection ( - ): defined by the normal to the plane

Primary Direction: higher-order symmetry

Secondary directions : lower-order

4 2 2 Notation 4 mm m m m réduite m

Tertiary directions : lowest-order

• Schönflies : Cn, Dn,Dnh (D4h) The 7 limiting point groups (Curie’s Groups)

Rotating cone Axial + polar vector (SO(2))



Twisted cylinder Axial tensor (optical gyration)

 2

Rotating cylinder Axial vector (H)

 /m

Cone Polar vector (E, F) (O(2))

 m

Cylinder Polar tensor (Compressive stress)

 /mm

Rotating sphere Axial scalar (chirality) (SO(3))

 

Sphere Polar scalar (pressure, mass) (O(3))

 /m  /m Symmetry of position: periodic order • Lattice : • Set of points (nodes):

Ruvw = u a +v b + w c (a, b, c) basis, (u, v, w) integers.

c b a b g a • Unit cell : • Volume with no gaps or overlaps, gal parallelepipedic (a,b,c) • Primitive (one node), multiple (symmetry) : elementary (unit cell)

• Conventionnal unit cells :

P : Primitive F : Face-centred I : Body-centred A,B,C : Base-centred Point symmetry of lattices

• Only 1-, 2-, 3- 4-, 6-fold symmetries are compatible with periodicity

• Every symmetry axe An is normal to a lattice plane

A n A2 A’2

T T

a=p a=p a=2p /n

• Symmetry of this plane An

B B’ • BB’ lattice vector • BB’=T-2Tcosa =mT p cos a a n=2p/a BB' An(T) A-n(-T) -2 -1 p 2 3T cosa =p/2 a -a -1 -0.5 2p/3 3 2T 0 0 p 4 T An T A’n /2 1 0.5 p/3 6 0 2 1 0 1 0 • Tilings • No gaps or overlaps Only symmetry compatible with translation : 1, 2, 3, 4, 6

2 3

5 8 1 4 6

• Kepler (1571-1630) in 1619 : « Harmonices Mundi »

Towards Penrose tilling 2D lattices •In 2D • 4 systems (systems) • 5 latttice modes

Oblic : p Rectangular : p Rectangular : c Square : p Hexagonal : p

•In 3D • Stacking of 2D lattices preserving symmetry (Ex. square)

P I P I F C Bravais _ Triclinic 1 lattices a  b  c a  b  g

b Monoclinic 2/m a  b  c a = g = 90°; b

Orthorhombic 2/mmm a  b  c a = b = g = 90° • In 3D • 7 systems (symmetry) Tetragonal 4/mmm • 14 lattice modes a = b  c a = b = g = 90°

_ Rhomboedric 3m a = b = c a = b = g

Hexagonal 6/mmm a = b  c a=b=90°;g =120°

_ Cubic m3m a = b = c a = b = g =90° 32 crystal Trigonal Monoclinique Triclinique Cubique Hexagonal Tétragonal Orthorhombique Orthorhombique classes

1 2 3 4 6 • Crystallographic point groups

222 32 422 622 • 7 crystal systems _ _ _ _ _ 1 2=m 3 4 6=3/m

2/m 4/m 6/m • Holohedral : with the lattice symmetry 2mm 3m 4mm 6mm Ex : Tétragonal (4/mmm)

_ _ _ _ _ ... hemihedral, tetarto-hedral 3m 42m (4m2) 62m (6m2) Chiral groups (Direct sym) Centrosym groups (Laue class) mmm 4/mmm 6/mmm Improper groups (ind sym.– inv)

23 432

_ _ _ m3 43m m3m Hexagonal Cubic Relations between the 7 systems Tetragonal • Group/subgroup

• Symmetry breaking Trigonal Orthorhombic • Phase transitions

Monoclinic L 4 L 2 Triclinic L L+e L L-e

L 6 L 3 Space groups

• Mauritz Cornelis Escher • Dutch graphic artist (1898-1972)

. Groupe P4 (chiral) New symmetries

Groupe P4gm

Reflections

Glide planes

Glide planes New symmetries 3D • Glide plane (M,t) • After two reflections M, periodicity T • t=T/2 • Combination (O, t) O : Rotation, Reflection T T : translation

• Notation : M a, b, c, n, d, g T/2

• Screw axis (AN, t) • After N translations t periodicity: mc • t = mc/N

• Notation :

Nm

(AN, mc/N) 21 41 42 61 64 Symmetry operations

• Rotations • Reflection • Roto-reflections • Glide plane • Screw axis Space groups

• 230 space groups

• 7 crystalline systems

• Notations • Directions (primary, etc.) • Lattice mode • Generators

Tetragonal Body centered I41/amd

• Point Group • Without translation 4 m m m Symmetry

• Linear Symmetry • Symmetry of position

• Rotations • Translations • Translations • Roto-reflections • T= u a + v b + w c • Rotations _ _ _ • Roto-reflections • Conventionnally • Symmetry allowed + • 1, 2, 3, 4, 6 ( 3, 4, 6) • Screw axis • M, C • Rotations (An) • Glide plane • Reflections (M) • Inversion (C) _ • 14 Bravais lattices • Roto-inversions (An)

Point groups 32 Crystal 230 Space group classes ( 7 systems )

• 7 Curie • 7 crystal systems Phase Transitions Phase I Phase II

G1 Tc G2 T

•Landau theory :

• G1 and G2 have no relation group/sub-group : 1st order transition (sulfur a  sulfur b)

• G1 sub-group of G2 (G1  G2) An order parameter h can be defined, zero in the symmetrical phase h h

• h discontinuous Tc T • h continuous Tc T • 1st order transition • 2nd order transition • Hysteresis, latent heat • Coexistence at critical point • Ferroelectric BaTiO3

• Perovskite ABO3 Order parameter: polarization • T > 120 °C, Cubic Pm3m, paraelectric • 0°C < T < 120 °C, Tetragonal P4mm, ferroelectric P4mm  Pm3m, 1st order transition (domains). • -90°C < T < 0 °C, Orthorhombic Cmm2 Cmm2  P4mm, 1st order transition . • T < -90 °C, Rhombohedral R3m R3m  Cmm2, 1st order transition .

Ba2+, Ti4+, O2-

4 Å O

1er 1er er Ti 1

Rhombohedral Orthorhombic Tetragonal Ba