The different typesof order

What is order?

‘’An infinite set of points is geometrically ordered, if it is generated by adeterminist algorithm of finite complexity’’

D. Gratias et al., Annu. Rev. Mat. Res. (2003)

« Disposition organisée, structurée selon certains principes, chaque élément ayant la place qui lui convient »

Larousse

‘’The arrangement or disposition of people or things in relation to each other according to a particular sequence, pattern, or method’’

Oxford dictionnary

Order

O

𝒓 , 𝑡𝑑3𝒓

Average atomic volume

Correlation functions

𝑡=0

Time-dependant pair correlation function

Temporal, statistical, volume means

Pair distribution function (pdf) instantaneous

: Space and time Fourier transform by Neutron scattering

X-ray scattering: Fourier Transform of

Density-density correlation function:

⟨𝑑𝑛(𝒓 , 𝑡)⟩=𝐺(𝒓 , 𝑡)𝑑3𝒓

⟨𝑑𝑛(𝒓 , 𝑡=0) ⟩=𝛿 (𝒓 )𝑑3𝒓+𝑔(𝒓)𝑣𝑎

𝑑3𝒓

𝐺 (𝒓 ,𝑡 ) ¿

0

1

Pair distribution function

Peaks: First neighbour

Second neighbouretc.

Peak width:Distance fluctuation

Peak integral:Number of neighbours

𝑑𝑛 (𝒓 )=𝛿 (𝑟 )𝑑3𝒓+𝑔 (𝒓 )𝜌𝒂𝑑3𝒓

𝑔 (𝒓 )

𝑟

Orientation correlation

𝑔 (𝒓 )

Here, only depends on It is not the general case!

Orientationnal correlation function : 𝜃𝜓6 (𝒓 )=𝑒𝑖6 𝜃 (𝒓)

01

𝑔 (𝒓 )

𝑟

• Short-Range Order (SRO)

• : correlation length• Ex: glass, liquids

• Maximum order in 1D

• Large distance behaviour of defines three types of order :

𝑑𝑛 (𝒓 )=𝛿 (𝑟 )𝑑3𝒓+𝑔 (𝒓 )𝜌𝒂𝑑3𝒓

01

Three types of order

𝑔 (𝒓 )

𝑟

• Long-Range Order (LRO)

• has no limit• Ex: Crystals• Bragg peaks

• Quasi Long-Range Order (QLRO)

• No length scale• Ex: Smectics, 2D crystals• Maximum order in 2D

Experimental evidence of order

Long-Range Order: diffraction

Otherwise: diffuse scattering

X ray Electrons Neutrons

Existence of Bragg peakswidth are resolution limited

Crystal of C60 Quasi-crystal

Water

Continuous scattering

Smectic liquid crystal

• Order in 1D

• Liquids, amorphous, glass

Short-Range Order (SRO)

• Amorphous state (disordered, non-crystalline)

• Amorphous recrystallize when heated.Metals, silicon, water.

• Glass becomes liquid through a vitreous transition.Silicon, Sulfur, Glycerol, Se (+As), obsidian, diatoms

• Liquid : same pdf, but dynamics.

𝑎+𝛿𝑎

𝑛𝑎+𝑛𝛿𝑎LRO is lost when thus

LRO

LRO

Melting in 3D

Solid Liquid

Melting and Quasi Long-Range Order

1er orderPhase transition

• Melting in 2D

Unlike classical melting,2D crystals melt through an intermediate phase:the hexatic phase

2D crystal Hexatic Liquid

LRO

Evidence in liquid crystals Brock, PRL57, 98 (1986), Colloids (Petukhov, 2006)

1st order phase transition?

2nd order phase transition?

Melting and Quasi Long-Range Order

Chou, Science 1998Hexatic phase in

Liq Xtal films

Kosterlitz-ThoulessTransition

• 2D crystals (Orientionnal LRO)• Order is lost very gradually

Quasi Long-Range Order

• Vortices in type II superconductors

• Between Hc1 and Hc2 Abrikosov phase• Bragg Glass (Giamarchi et al. 1994)

h

106 µm, 37003 vortices

Vortices decoratedby Fe clusters,

observed by SEM (Kim et al., PRB60, R12589)

Map of vortices displacementswith respect to perfect lattice

impu.

Bragg glass aredislocation free

BEC

BCS

Fermions Li :Cooper pairs

Bosons (Li2 molecules) :Bose-Einstein condensate

BEC

BCS

Li-LiInteractions

High T

SRO

Low T

QLRO

Murthy et al. PRL 115, 010401 (2015)

Kosterlitz-ThoulessTransition

Order can be studied by Measure of in 2D confined 6Li ultracold gas

Suprafluids, Supraconductors (BCS) and Bose-Einstein condensates (BEC)

are described by a macroscopic wave function:

QLRO and macroscopic quantum systems

Fractal structures

• Self-similarity• Scale invariance

Sierpiński carpet

von Koch snowflake

Menger sponge

Regular fractalsdo not exist un nature...

• Hausdorff dimension of fractal (1918):

n(k)=kD

D=log(3)/log(2)= 1,5849...

D=log(4)/log(3) = 1,261...

D=log(20)/log(3) = 2,7268...

Irregular fractals

• Fractal dimension• Minkowski-Bouligant

Gold nanoparticle clusters

Structure of a 2D lattice of Ising spins at its critical temperature

Brownian motion boundary (W. Werner)

Lichtenberg figures

Broccoli

*=

• A crystal is a basis associated to a lattice

Nucleosom

Macromolecule

C60

Molecule

Basis Crystal

NaCl

Atoms

Na

Atom

Periodical crystals

• Incommensurate modulated crys. • Local property (ex: polarisation) has a

periodicity , incommensurate with lattice period .• Ex: Charge density wave, NaNO2

• Incommensurate composite crystals• Compounds with at least two subsystems with lattices parameters

mutually irrational.• Ex: Rb, Ba, Cs under pressure, Hg3-dAsF6

irrational number

• Quasicristals• Systems with long-range order

and forbidden symmetry (5, 8, 10...)

Penrose tilling

• Long-range order

• No periodicity

a

un

Aperiodic crystals

a

b

𝑛𝒂+𝒖𝑛=𝑛𝒂+𝒖0 sin2𝜋𝜆𝑛𝑎

𝑎𝑏

Atomic force microscopy:Average lattice

Scanning tunneling microscopy:Charge density wave

Incommensurate modulated crystals

• Tantalum dichalcogenide 1T-TaSe2: Charge density wave• Modulation of the electron density at 2kF (twice the kF Fermi vector)

E. Meyer et al. J. Vac. Sci. Technol. 8, 495 (1990)

1313~

• Alkane/Urea• Inclusion of alkane in urea channels

B.Toudic et al., Science 319, 69 (2008)

• Ba under 12 GPa (120000 atm.)• Ba in Ba channels! ( irrational number)

Composite crystals

R.J. Nelmes et al. Phys. Rev. Lett. 83, 4081 (1999)

Entanglement of periodic crystals with incommensurate

lattice parameters

Quasicrystals

Sharp diffraction peaks

Long-range orderAND

5-fold symmetry(not consistent with periodicity)

1

2

3

47

8

9

10

Decagonal Al-Ni-Co :10-fold symmetry

www.cbed.rism.tohoku.ac.jp/saitoh/saitoh.html

Electron diffraction of an Al-Mn alloy

(From D. Shechtman et al. Phys. Rev. Lett. 53, 1951 (1984)) Quasicristals discovered by chance by Schechtman (1982-Nobel 2011)

while he studied rapidly cooled Al alloys.

56

Penrose tiling

• Two types of ‘‘tiles’’• Matching rules

2D quasicristals can be modelled by a Penrose tiling

Al-Fe-Cu alloy(Marc Audier)

36° 72°

Penrose tiling

• Non periodic tilings• Long-range order WITHOUT periodicity

• N-fold symmetry for any N

• Quasiperiodic tilingsbefore Penrose…

12-fold symmety

Darb-i Imam templeIsfahan, Iran, XVe

1 2 3 4 5 6 7 8

-10

-5

0

5

10

Van der Waals Ionique, Covalent, Metallic

En

ergi

e (e

V)

r(Å)

• Interaction potentials

• Interaction potential : minimum around 1,5-2 Å and 3-4 Å

• Ex: In water vapour, mean distance of 30 Å (ideal gas) In liquid water: 3 Å (liquid order)

• Shape of potential determines properties:

• Equilibrium distance given by : structure.• Rigidity given by : elasticity, dynamics (phonon dispersion),

Thermal conductivity, specific heat.• Anharmonicity : thermal dilatation.

Origin of order

• Ionic bonding (heteropolar)• Coulombic interaction between ions.• Strong bonding (eV), nonsaturable and nondirectional.• Ex: NaCl, LiF

• Covalent bond (homopolar)• Electrons shared by two atoms.• Strong bonding (1.5 eV O-O, 3.6 eV C-C ), saturable and directional.• Ex: Diamond

• Metallic bonding • Delocalized electrons.• Intermediate bonding (0.5 eV Cu), nonsaturable and nondirectional.• Ex: All metals (Na, Cu, U), organic conductors.

• van der Waals bonding• Dipole (induced) – dipole interaction.• Weak bonding (10 meV), nonsaturable and nondirectional.• Ex: Noble gas (Ar, Xe), molecular crystals.

• Hydrogen bond• Ionic bonding between H and electronegative atom.• Weak bonding (100 meV) directional.• Ex: Ice (O-H---O 0.26 eV), organic and biologic crystals.

300 K (kBT) 25.8 meV

6.25 THz208.5 cm-1

48 µm

Five types of bondings

• Difficult to predict structure ab initio

• Simplest model: close-packed structures

• In 2D, close-packing: hexagonal infinite lattice • In 3D, close-packing of hexagonal layers: face centred cubic (FCC) and

hexagonal close-packed (HCP) are the more compact (Kepler 1611; Th. Hales 1998); compacity=0,74 Not always periodical (stacking faults)

• Noble gas ~ 2/3 f metals (fcc ou hcc)• But alcaline metals (cc), Fea (cc) Feg(fcc).

Growing a crystal

atom by atom…

From interaction to order-1

B

A

B

C

A

B

3

6

3

1

5

5

1

Icosahedral order HCP CFC

CuboctahedraIcosahedra

a

b

c

Structure of the elements

cfc

hc

cc

From R.K Vainshtein, Structure of Crystals

• 3D close-packing of 4 atoms: tetrahedra

• Impossible to fill Euclidian space by perfect tetrahedra (dihedral angle = 70,528°)

But LOCALLY, tiling of distorded

tetrahedra is possible Icosahedrea

• Impossible to fill Euclidian space with distorted tetrahedra, so that a constant

number of tetrahedra sharing a common edge

Topological frustration

Interactions favor icosahedral local ordernot consistent with infinite system.

Frustration produces defects (liquids, glass)

From interactions to order-2

7.36°

From interactions to order-3

• Small clusters of icosahedral symmetrymore stable

Electron diffraction on Cu, Ni, CO2, N2, Ar Transition icosahedra-fcc observed when size increases (1000 Ar, 30 CO2)

Disorder 1-Effect of temperature• Thermal motion

• At a given time, no perfect periodicity• Periodicity is recovered on average

• Orientational disorder• Ex : C60, plastic crystals

a

b

c

C60Kroto et al. 1985

T=300 Kfcc

• Average structure is periodic• Statistical average time average

(Ergodic hypothesis)

Real crystals: 2-Defects

• Topological defects• Deformations which change the

local atomic environment, such as the number of neighbors

• Dimension 0• Vacancies, intersticials

• Dimension 1• Dislocations (metal plasticity)

• Disclinations (2D, liquid crystals)

• Dimension 2• Surfaces, stacking faults• Grain boundaries, twins

Vacancy• Always present (2.10-4 Cu at 300 K)

• Diffusion, colored centers

Intersticial• Plasticity

(Impurety)• semi-cond. doping• Colors of jewels

• Plasticity

Surface Stacking faults Grain boundary

Dislocation Disclination

www.techfak.uni-kiel.de/matwis/amat/def_en/makeindex.html

Dislocation creep

• GP shear by an edge dislocation • High resolution electron microscopy

• GP zone (Guinier-Preston) • Clusters of atom• Hardening of Al alloys (Concorde)• Platelets in Al-1.7at.%Cu

From M. Karlík et B. Jouffrey, J. Phys. III France, 6 (1996) 825

Dislocation: Cottrell atmosphere

D. Blavette, E. Cadel, A. Fraczkiewicz and A. Menand. Science 286 (1999) 2317.

GPM UMR 6634 CNRS, Université de Rouen

• Visualization of an edge dislocation• Field Ion Microscope• B-doped FeAl alloy• Dislocation pinning• Aging

Grain boundaries

• Subgrain boudaries: • Formed by array of dislocations

• Interface between two grain in a polycrystal• When the angle is smaller than 15° ou 20°: subgrain boundary• When the angle is larger than 20°: grain boundary

• Grain boundary: • Structure not well known, ordered or disordered (amorphous)

Example :(110) gold grains on Ge(100)

At the interface, parameters are and Interface ordered and even quasiperiodic !

F. Lançon et al. EPL49, 603 (2000)

Read et Shockley model (1950)

Tereptal-bis(p-butylanilin) TBBA

Isotropic liquid

T=236 °C

Nematic

T=200 °C

Smectic A

T=175 °C

Smectic C

Intermediate states: thermotropic liquid crystals

• anisotropic

• Phase transitions depend on temperature

Nematic order

• Positional SRO• In n direction

• Normal to n

• Orientation LRO• In n direction

Nematic

n

• Positional SRO• Normal to n

• QLRO• In n direction• Quasiperiod a

Smectique A

an

Smectic order

Hexatic order

• SRO in position • QLRO in orientation

Normal à n

• Orientational disorder of molecules

• 3D crystalline order• Plastic order

Smectic A

a

Smectic B (plastic crystal)

Hexatic

a

a

Columnar phases

• Positional LRO• Between columns

• Positional SRO • Along the columns

Discotic molecules

Cholesteric phases

• Long and chiral molecules• Nematic-based helicoidal structures

• T-dependant pitch P: 100 nm to 800 nm

Thermometers

Lyotropic liquid crystals• Phases depend on solvent concentration• Amphiphilic molecules (soap)

• Air bubbles with facets

• Phase diagram

• Hydrophilic headHydrophobic

tail• Crystals• Micelles • Tubes • Layers

• Cubic phase

From P. Sotta, J. Phys. France,

• Cubic phase

Dotera 2014

Liquid… quasicrystals: Q12 and Q18