The different types
of order
What is order?
« 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
‘’An infinite set of points is
geometrically ordered,
if it is generated by a
determinist algorithm
of finite complexity’’
D. Gratias et al., Annu. Rev. Mat. Res. (2003)
Correlation functions
Time-dependant pair correlation function
𝑣𝑎 = 𝑉/𝑁
Average atomic
volume
𝑑3𝒓
𝒓, 𝑡
O
𝑑𝑛(𝒓, 𝑡) = 𝐺(𝒓, 𝑡)𝑑 3 𝒓
Temporal, statistical, volume means
𝐺(𝒓, 𝑡): Space and time Fourier transform by
Neutron scattering
𝑡=0
Pair distribution function (pdf)
instantaneous 𝐺(𝒓, 𝑡 = 0)
𝑑𝑛(𝒓, 𝑡 = 0) = 𝛿 𝒓
𝑑3𝒓
𝑔(𝒓) 3
+
𝑑 𝒓
𝑣𝑎
X-ray scattering: Fourier Transform of 𝑔(𝒓)
Density-density correlation function:
𝐺(𝒓, 𝑡)~ 𝜌(𝒓′ , 0)𝜌(𝒓 + 𝒓′ , 𝑡
Pair distribution function
𝑑𝑛 𝒓 = 𝛿 𝑟 𝑑3 𝒓 + 𝑔(𝒓)𝜌𝒂 𝑑3 𝒓
𝑔(𝒓)
Peaks:
First neighbour
Second neighbour
etc.
Peak width:
Distance fluctuation
Peak integral:
Number of neighbours
1
0
𝑟
Orientation correlation
𝑔(𝒓)
1
0
Here, 𝑔(𝒓) only depends on |𝒓|
It is not the general case!
𝑟
𝑔(𝒓)
𝜓6 𝒓 = 𝑒 𝑖6𝜃(𝒓)
Orientationnal correlation function :
𝑜 𝒓 = 𝜓6 (0)𝜓6 (𝒓)
𝜃
Three types of order
𝑑𝑛 𝒓 = 𝛿 𝑟 𝑑3 𝒓 + 𝑔(𝒓)𝜌𝒂 𝑑3 𝒓
• Large distance behaviour of 𝑔(𝒓)
defines three types of order :
• Short-Range Order (SRO)
•
𝑔 𝒓 ~ exp −
𝑟
𝜉
𝜉 : correlation length
• Ex: glass, liquids
• Maximum order in 1D
•
• Quasi Long-Range Order (QLRO)
𝑔(𝒓) ~ |𝑟|−𝜂
• No length scale
• Ex: Smectics, 2D crystals
•
Maximum order in 2D
•
• Long-Range Order (LRO)
•
𝑔(𝒓) has no limit
• Ex: Crystals
•
Bragg peaks
𝑔(𝒓)
1
0
𝑟
Experimental evidence of order
Long-Range Order: diffraction
X ray
Electrons
Crystal of C60
Quasi-crystal
Existence of Bragg peaks
width are resolution limited
Neutrons
Otherwise: diffuse scattering
Continuous scattering
Water
Smectic liquid crystal
Short-Range Order (SRO)
• Order in 1D
𝑎 + 𝛿𝑎
• Liquids, amorphous, glass
𝑛𝑎 + 𝑛𝛿𝑎
LRO is lost when 𝑛𝛿𝑎 = 𝑎 thus 𝜉 =
𝑎2
𝛿𝑎
• 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.
Melting and Quasi Long-Range Order
Melting in 3D
1er order
Phase transition
𝑔(𝑟)
LRO
exp(−𝑟/𝜉)
𝑜(𝑟)
LRO
exp(−𝑟/𝜉)
Solid
Liquid
Melting and Quasi Long-Range Order
• Melting in 2D
Unlike classical melting,
2D crystals melt through an intermediate phase:
the hexatic phase
2nd order
phase transition?
2D crystal
𝑔(𝑟)
𝑟 −𝜂
𝑜(𝑟)
LRO
1st order
phase transition?
Hexatic
Liquid
exp(−𝑟/𝜉)
𝑟 −𝜂
exp(−𝑟/𝜉)
Kosterlitz-Thouless
Transition
Evidence in liquid crystals
Brock, PRL57, 98 (1986), Colloids (Petukhov, 2006)
Chou, Science 1998
Hexatic phase in
Liq Xtal films
Quasi Long-Range Order
• 2D crystals (Orientionnal LRO)
• Vortices in
type II superconductors
h
• Order is lost very gradually
• Between Hc1 and Hc2 Abrikosov phase
• Bragg Glass (Giamarchi et al. 1994)
impu.
Vortices decorated
by Fe clusters,
observed by SEM
(Kim et al., PRB60, R12589)
Bragg glass are
dislocation free
Map of vortices displacements
with respect to perfect lattice
106 µm, 37003 vortices
QLRO and macroscopic quantum systems
Suprafluids, Supraconductors (BCS) and Bose-Einstein condensates (BEC)
are described by a macroscopic wave function: 𝝍(𝒓)
Order can be studied by 𝒈𝟏 𝒓 = 𝝍(𝟎)𝝍(𝒓)
Measure of 𝒈𝟏 𝒓 in 2D confined 6Li ultracold gas
Bosons (Li2 molecules) :
Bose-Einstein condensate
BEC
Fermions Li :
Cooper pairs
Low T
𝒈𝟏 𝒓 ~𝑟 −𝜂
QLRO
Kosterlitz-Thouless
Transition
Li-Li
Interactions
High T
𝒈𝟏 𝒓 ~𝒆−𝒓/𝝃
SRO
BEC
BCS
BCS
Murthy et al. PRL 115, 010401 (2015)
Fractal structures
• Self-similarity
Sierpiński carpet
• Scale invariance
D=log(3)/log(2)= 1,5849...
• Hausdorff dimension
of fractal (1918):
n(k)=kD
von Koch snowflake
D=log(4)/log(3) = 1,261...
Menger sponge
D=log(20)/log(3) = 2,7268...
Regular fractals
do not exist un nature...
Irregular fractals
Gold nanoparticle clusters
𝑑𝑓 = 1,75 ± 0,05
• Fractal dimension
• Minkowski-Bouligant
𝑑𝑓
Structure of a 2D lattice of Ising spins at its critical temperature
𝑑𝑓 = 1,75 ± 0,05
𝑛 𝑟 ~𝑟
𝑔(𝑟) ~ 𝑟 𝑑𝑓 −𝐷
Brownian motion boundary (W. Werner)
𝑑𝑓 = 4/3
Broccoli
𝑑𝑓 = 2,33
Lichtenberg figures
Periodical crystals
• A crystal is a basis associated to a lattice
Na
= *
Atom
NaCl
Atoms
C60
Molecule
Nucleosom
Macromolecule
Basis
Crystal
• Incommensurate modulated crys.
Aperiodic crystals
• Local property (ex: polarisation) has a
periodicity 𝜆, incommensurate with lattice period 𝑎.
• Ex: Charge density wave, NaNO2

• Long-range order
2𝜋
𝑛𝒂 + 𝒖𝑛 = 𝑛𝒂 + 𝒖0 sin
𝑛𝑎
𝜆
un
a
• No periodicity
• Incommensurate composite crystals
• Compounds with at least two subsystems with lattices parameters
mutually irrational.
• Ex: Rb, Ba, Cs under pressure, Hg3-dAsF6
a
𝑎
𝑏
b
• Quasicristals
• Systems with long-range order
and forbidden symmetry (5, 8, 10...)
Penrose tilling
irrational number
Incommensurate modulated crystals
• Tantalum dichalcogenide 1T-TaSe2: Charge density wave
• Modulation of the electron density at 2kF (twice the kF Fermi vector)
Atomic force microscopy:
Average lattice
Scanning tunneling microscopy:
Charge density wave
E. Meyer et al. J. Vac. Sci. Technol. 8, 495 (1990)
Composite crystals
• Alkane/Urea
• Inclusion of alkane in urea channels
Entanglement of periodic crystals
with incommensurate
lattice parameters
B.Toudic et al., Science 319, 69 (2008)
• Ba under 12 GPa (120000 atm.)
• Ba in Ba channels! (𝑐𝑔/𝑐ℎ irrational number)
R.J. Nelmes et al. Phys. Rev. Lett. 83, 4081 (1999)
Quasicrystals
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.
Decagonal Al-Ni-Co :
10-fold symmetry
10
1
2
9
8
3
7
4
6
5
Sharp diffraction peaks
Long-range order
AND
5-fold symmetry
(not consistent with periodicity)
www.cbed.rism.tohoku.ac.jp/saitoh/saitoh.html
Penrose tiling
2D quasicristals can be modelled
by a Penrose tiling
Al-Fe-Cu alloy
(Marc Audier)
36°
72°
• Two types of ‘‘tiles’’
• Matching rules
Penrose tiling
• Quasiperiodic tilings
before Penrose…
Darb-i Imam temple
Isfahan, Iran, XVe
• Non periodic tilings
• Long-range order WITHOUT periodicity
• N-fold symmetry for any N
12-fold symmety
Origin of order
• Interaction potentials
• Interaction potential 𝑈(𝑟): 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)
Energie (eV)
10
5
0
Van der Waals
Ionique, Covalent, Metallic
1
2
3
4
5
6
7
8
r(Å)
-5
-10
• Shape of potential determines properties:
• Equilibrium distance given by 𝑑𝑈(𝑟)/𝑑𝑟 = 0: structure.
• Rigidity given by 𝑑2𝑈(𝑟)/𝑑𝑟2: elasticity, dynamics (phonon dispersion),
Thermal conductivity, specific heat.
• Anharmonicity 𝑈(𝑟): thermal dilatation.
Five types of bondings
• 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 THz
208.5 cm-1
48 µm
From interaction to order-1
• 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)
Icosahedral order
HCP

Feg(fcc).
CFC
1
3
B
C
6
A
A
3
B
B
5
5
1
Icosahedra
Cuboctahedra
Growing
a crystal
atom by atom…
c
b
a
Structure of the elements
cfc
hc
cc
From R.K Vainshtein, Structure of Crystals
From interactions to order-2
• 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
7.36°
• 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 order
not consistent with infinite system.
Frustration produces defects (liquids, glass)
From interactions to order-3
• Small clusters of icosahedral symmetry
more 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
• Average structure is periodic
• Statistical average  time average
(Ergodic hypothesis)
• Orientational disorder
• Ex : C60, plastic crystals
T=300 K
fcc
c
C60
Kroto et al. 1985
a
Real crystals: 2-Defects
www.techfak.uni-kiel.de/matwis/amat/def_en/makeindex.html
• Topological defects
• Dimension 0
• Vacancies, intersticials
• Deformations which change the
local atomic environment,
such as the number of neighbors
Vacancy
• Always present
(2.10-4 Cu at 300 K)
• Diffusion, colored centers
Intersticial
• Plasticity
(Impurety)
• semi-cond. doping
• Colors of jewels
• Plasticity
• Dimension 1
• Dislocations (metal plasticity)
• Disclinations (2D, liquid crystals)
Dislocation
Disclination
• Dimension 2
• Surfaces, stacking faults
• Grain boundaries, twins
Surface
Stacking faults
Grain boundary
Dislocation creep
• 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
• GP shear by an edge dislocation
• High resolution electron microscopy
Dislocation: Cottrell atmosphere
• Visualization of an edge dislocation
•
•
•
•
Field Ion Microscope
B-doped FeAl alloy
Dislocation pinning
Aging
D. Blavette, E. Cadel, A. Fraczkiewicz and A. Menand.
Science 286 (1999) 2317.
GPM UMR 6634 CNRS, Université de Rouen
Grain boundaries
• 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
• Subgrain boudaries:
• Formed by array of dislocations
Read et Shockley model (1950)
• Grain boundary:
• Structure not well known, ordered or disordered (amorphous)
Example :
(110) gold grains on Ge(100)
At the interface, parameters are 𝑎 and 𝑎√2
Interface ordered and
even quasiperiodic !
F. Lançon et al. EPL49, 603 (2000)
Intermediate states: thermotropic liquid crystals
• Phase transitions depend on temperature
• 𝑔(𝒓) anisotropic
Tereptal-bis(p-butylanilin) TBBA
Isotropic liquid
T=236 °C
Nematic
T=200 °C
Smectic A
T=175 °C
Smectic C
Nematic order
• Orientation LRO
• In n direction
n
Nematic
• Positional SRO
• In n direction
• Normal to n
• QLRO
• In n direction
• Quasiperiod a
a
n
Smectique A
Smectic order
• Positional SRO
• Normal to n
Hexatic order
a
Smectic A
• SRO in position
• QLRO in orientation
a
Normal à n
Hexatic
• Orientational disorder of
molecules
a
• 3D crystalline order
• Plastic order
Smectic B (plastic crystal)
Columnar phases
• Positional SRO
• Along the columns
Discotic molecules
• Positional LRO
• Between columns
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)
• Hydrophilic head
•
•
•
•
Crystals
Micelles
Tubes
Layers
Hydrophobic
tail
• Cubic phase
• Cubic phase
• Phase diagram
• Air bubbles with facets
From P. Sotta, J. Phys. France,
Liquid… quasicrystals: Q12 and Q18
Dotera 2014