phase transitions in aperiodic crystals and tilings landau

Phase Transitions in Aperiodic Crystalsand Tilings

Ted JanssenTheoretical Physics

University of NijmegenNijmegen, The Netherlands

Landau-Ginzburg TheorySuperspace DescriptionModulated PhasesIncommensurate CompositesQuasicrystalsMagnetic Symmetry Changes

‘Numeration’Lorentz CenterLeiden 2010

Phase transitions in solids

Quasiperiodicity and embedding

Incommensurate modulated phases

Incommensurate composites

Quasicrystals and quasiperiodic tilings

Aperiodic Crystal: diffraction pattern with delta peaks

Rank n > Dimension DExamples: Incommensurate modulated phases Aperiodic composites Quasicrystals

Phase transitions: From periodic crystal to incommensurate modulated phase, between incommensurate phases,between various phases of aperiodic composites,in quasicrystals and between quasicrystals and approximants

hi integers

Landau theory

Order parameter η describes change in structure / symmetry

Landau free energy

Free energy is invariant under high-symmetry group G0 ; order parameter

belongs to irreducible representation of this group; order parameter is invariant under low-symmetry group G.In general, higher-order order parameter if dimension of the irreduciblerepresentation is higher than 1.

4

+ ...

Transition from periodic to aperiodic crystal:characterised by irrep of the 3D space grouplabelled by star of k vectors and irrep of the little group of k

Example: crystal with space group Pcma toward modulated phase with wave vector

Point group of the little group: 2mm

5

Embedding into superspacewith dimension equal to the rank

High symmetry: without modulation

Low symmetry: with modulation

H subgroup of G : n -dimensional space group: superspace group

Incommensurate modulated phasesPositions 1D system: n + rj + fj (q.n )

Transition form unmodulated to incommensurate modulated.

Modulation functions f and g may be smooth or discontinuous:The type changes at the discommensuration transition: effects on the dynamics

Usually the wave vector of the modulation changes with temperature,but the superspace group symmetry remains the same.

Exceptions: - when higher order terms appear in the modulation; - when more wave vectors are involved ; e.g. 1q -> 2q

High temperature:Unmodulated crystal.Positions: n + rj

VE

VI

Transition from 1D periodic to 1D aperiodic = 2D periodicSymmetry from G0×E1 -> G

Phys. Space

Red Blue

9

Rank is 2 Diffraction h1 a* + h2 q/2

VI

VE

Transition from 2D periodic to 2D periodic

Rank remains 2 Diffraction h1 a* + h2 q/2

10

Semi-microscopic models

Discrete frustrated φ4 model:

Phases: para-phase un =0 commensurate superstructure incommensurate modulated phase

11

Studied with J.A. Tjon, A. Rubtsovand V. Savkin

Phase Diagram of DIFFOUR model(discrete frustrated φ4 model)

!1

!5 !4 3 2 1 0 1 2 3 4 5 B/D

0

1

2

c,c,c,c,c,c,c,c

0,0,0,0,0,0,0,0

3

4

5

6

A/Da = !A/D, d = B/D

P

01

29

18

38

16

14

15

310

13

c,-c,c,-c,c,-c

c,C,c,-c,-C,-cc,c,-c,-c c,c,-C

12

17

19

110

12

T

p or c

Lifshitz point

Example of a transition from a 2D periodic to a 2D aperiodic (3D periodic) system:

soft mode at the zone boundary at (α, 1/2)

13

Typical phase diagram: high T : space group symmetric below critical T: incommensurate phase plane wave limit (continuous modulation) discommensuration limit (discontinuous) below lock-in T: commensurate

A/D in the model corresponds to T in mean field phase diagram

Phase transition: rank 2 (periodic) to rank 3 (aperiodic)

(0,0) (π,0)

(0,π)A

A =(6/17,1/2)

A

C

B

T2 < T < T1 T < T2

Free energy on the Brillouin Zone

T > T1 : Unmodulated : AT1 : Phase transition to 1q state -> rank 4 :BT2 : Phase transition to 2q state -> rank 5 :C

Phase transition 1q-2q

15

Biphenyl and BaMnF4

Cf. K. Parlinski (19920

Composites

Embedding of the composite

Composites

18

If sum reflections are present: each subsystemis modulated by the other one.

Diffraction peaks

Embedding of composite with continuous modulation.

Four types of reflections: common to the subsystems, belonging to oneof the subsystems, combinations of reflections of both .19

Discommensuration transition: symmetry the same

Embedding of composite with discontinuous modulation.

Example of a symmetry preserving phase transition with weakanomaly in the spec. heat, and consequences for dynamics.

20

Phase transition between phases of the same rank

Example: rank = 3

21

In superspace doubling of the unit cell.Changes in the diffraction affect main reflections of one system only.

Change of the rank in the transition

Model example

22

In superspace additional dimension.Changes in the diffraction affect sum peaks.Example: urea-nonadecane (Toudic)

New diffraction peaks

Simple model: soft optic mode

Parameters with A, B, C and D are temperature dependent

Quasicrystals

AperiodicLocal structure with clustersPossibly, but not necessarily, non-crystallographic symmetry

As model: tiling.

Example: Ammann-Beenker tiling: a1*

a2*a3*

a4*

Quasicrystals in nD superspaces

(n-3)D atomic surfaces in nD unit cell

Phase transitions:-Phason strain, lower rank-Commensurate modulation, same symmetry-Commensurate modulation, lower symmetry-Incommensurate modulation, higher rank

Quasicrystals1. Phason strain

Embedding gives n-dimensionallattice periodic structure, with lattice characterised by the metric tensor g

Approximant: periodic structure, locally similar tothe quasicrystal. The embedding of the former is obtainedfrom the latter by a ‘phason strain’.In the transition the point group changes from 532 to 23.

Example: 6D lattice correspondingto icosahedral standard tiling.

Order parameter is the 6D strain (phason strain).This transforms with an irrep of 532 (icosahedral).The lowest energy structure has symmetry 23 (tetrahedral).

Change of the metric tensor:

Symmetry change:

Order parameter

Through phason strain:transition to lower symmetry

2. Through phason modulation:transition to higher rank

Commensurate modulation, same rank same symmetry

Projection Voronoi cell=fundamental region of the 4Doctagonal latticeon perp space P8mm(8mm)

Fundamental regions of the2D space group pgg

what are aperiodic tilingswith non-symmorphic symmetry(space group not a semidirect product)?

3. Symmetry change in superspace

Superspace group

Extension Zn -> G -> K, with K an nD point group.

Infinite G-orbit of point x

Fundamental region: set of points closer to x thanto any other point of the G-orbit of x

Atomic surface: copy of the projection of the f.r. ontointernal space VI placed in x

Sum of the projections of f.r. in all points of the G-orbitinside the Delone cell of the lattice gives the atomicsurface corresponding to this Delone cell.

Projection fundamental region of space group P8um(8mm)on internal space VI .This may be used as atomic surface to produce a decorated tilingwith P8um(8mm) symmetry.In contrast to the usual projection of the Delone cell, thisatomic surface does not have the point group symmetry of the lattice: 8mm

P8um(8mm)

Projection of the 8 fundamental 4Dregions on VI fill the projection ofthe Delone cell of the lattice

8mm(8mm)P 8mm(8mm)I≈8mm(8mm)P

H=(h,k,l,m)/2h+k+l+m=even

Centering transition

p8um(8mm)

1/4,1/4,3/4,3/4

p8um(8mm)

3/4,3/4,1/4,1/4

Double tiling with nonsymmorphic superspace group

Elementssymmetry group

(a) (b)

(c) (d)

Decomposition into small occupation domains of the four atomic surfacesin the six-dimensional model of Yamamoto. Only the asymmetric unitis shown. The number on each domain refers to the decoration. The level ofgrey indicates an atomic species ( Al, Mn, or Pd).

Compare the atomic surfaces with those of models forrealistic quasicrystals (Yamamoto)

The atomic surface for the substitution A->B, B->C,C->AC isa fractal. The dashed line is the outline of the projection of the 3D unit cell in superspace. The atomic surfaces are located at thenodes of a lattice. Question: are there atomic surfaces such that thenD structure has non-symmorphic space group symmetry?

Generation of the atomic surface

Eigenvalues substitution matrix:1.4656 , -0.2328±0.7926i

Symmetry 1D tiling: scaling by 1.4656

Scale-rotation symmetry 2D atomic surface:scale 0.8261, rotation -73.6o

This atomic surface has a fractal boundary. If the substitution is not invertible, the a.s. itself is fractal.(Lamb, Wen and Wijnands).

In general, tilings obtained by substitutionmay have roto-scaling symmetry.

8

6

4

2

0

–2

–4

–6

–8–8 –6 –4 –2 0 2 4 6 8

Matrix

commutes with the point group,and leaves VE and VI invariant:scale factors, resp.√2+1 and √2-1

This can be used to analysethe set of vertices(cf. Burdik, Frougny, Gazeau, 1998)

Analogously, the dodecagonal tiling withtriangles, squares and hexagons hasscale-rotation symmetryfactor 2+√3, angle 15o

Deformed 6D lattice with basis vectorsTransition with Landau theory

49

So, there is a lattice deformation of phason type which produces from the icosahedral lattice a periodic system in VE with tetrahedral point group symmetry

The transition from the icosahedral to the tetrahedral phase can be considered as a soft phason, as appears also in, e.g., TTF-TCNQ [4]. Cf. Ishii [5].

Free Energy: F = A η2 /2 + B η4 /4 + C η6 /6

First order phase transition

η = η(α a1* )

η

F

50

1-τ

Free Energy: to get the minimum at 1-τ ‘lock-in’ terms are needed.

Energy: Ec compared with Eqc

There are tiling models with decoration and interatomicinteraction, with a stable quasicrystal configuration.The energy might be a polynomial in the n-dimensionalorder parameter, with lock-in terms:

Entropy: Sc compared with Sqc

More microstates may be reached by phason fluctuationsin quasicrystal (projection dense on VI) than for crystal approximant: Sqc > Sc F=E-ST

T

Eqc

Ec

PT

Ground state: depends on interatomicinteractions. It must be determined byrealistic (not phenomenological) calculations: cf. Widom & Mihalkovic. Calculations in the RTM are notsufficient proof.

51

Conclusions

There is a rich variety of phase transitions involvingquasiperiodic structures.

A unified approach to these transitions is by usingLandau theory, where the symmetry groups aresuperspace groups or direct products of a (super)spacegroup en the Euclidean group Em in m dimensions.

Phase transitions may occur between phases of thesame rank, or between phases of different rank. Exampleshave been found in experiments.

For each n-dimensional superspace group G a (decorated)G-invariant tiling may be constructed, with the projectionsof the fundamental regions as atomic surfaces.

Status

Mathematics has played an essential rôle in the understandingof the structure and properties of quasicrystals andother quasi-periodic crystals.

Nowadays ‘realistic’ calculations and structure determinationare possible.

But there remain mathematical questions.

- Which structures give a point spectrum in diffraction?- What is the spectrum of electrons in quasicrystals, what are the types of states, and how are these affected by disorder?- How can we specify the atomic surfaces in a concise way?

phase transitions in aperiodic crystals and tilings landau

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