the different types of order. what is order? ‘’an infinite set of points is geometrically...
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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