Disorder in condensed matter

1
Disorder in condensed matter

• Disorder is everywhereand subtle
• Gas, liquid, glass, plastic, wood, fabric
• Screens, watches (liquid crystals)
• Metal (polycrystals, alloys)
• Where is order?
• Semiconductors ? (doped)
• Gems (color due to impurities)
• Salts NaCl (vacancies)
• almost nowhere, but a fundamental concept and
  also subtle
• These concepts are defined by diffraction!
• Disorder related to SYMMETRY
• Disorder is more symmetric
• Ordering is breaking a symmetry (liquid-solid)

2
Origin of order in condensed matter

• Order and disorder are due to a competition
• Potential energy of attraction V(r)
• Kinetic energy of thermal motion (temperature)
• Potential energy depends on the distance between
  atoms or molecules
• V(r) has a minimum corresponding to equilibrium
  distance bond
• Physical properties compressibility, thermal
  expansion, melting T
• Five types of bonds
• Ionic bond (heteropolar) giving electrons
• NaCl-1.5 Å-4.34 eV(kcal/mol)-strong-Not
  directional
• Covalent bond (homopolar) sharing electrons
• Diamond-CC-1.5 Å-6.5 eV-strong-Directional
• Metallic bond (sharing electrons throughout the
  crystal)
• Cu, Ni-few eV-less strong-non directional
• Van der Waals bond (induced dipole-dipole r-6)
• Ar, Xe, molecular materials,- 3-4 Å-0.01 eV-very
  weak
• Hydrogen bond (HA electronegative)
• Water, biologic systems (proteins, nucleic
  acids), 3-4 Å, 0.1 eV-weak

3
What type of structure?

• Knowing V(r) do we know the structure (ground
  state)? No in general
• Only solved for specific V(r) closest packing
  structures (12 neighbors)

Plato

• f.c.c Cu, Ni, Al, Ag, Au, Pt, C60, Ar, CO2
• h.c.p Co, Zn, Be But Ba, Rb, Cs are body
  centered (wrong V(r))
• SMALL Aggregates are icosahedral (Arlt1000
  CO2lt30)
• V(r) LOCAL ORDER (aggregates, glass,
  quasi-crystals)
• Not valid for infinite crystal.

4
Pair correlation functions

• Time-dependant pair correlation function G(r,t)
• Neutron scattering
• Probability that, given a particle at the origin
  at t0,
• Any particles is in the volume d3r a distance r
  at time t.
• G(r,t0)d(r)rag(r)
• g(r) static pair distribution function
• X-ray scattering

5
Molécules chirales

• Phases cholestériques
• Structure en hélice du
• directeur n
• Ordre nématique dans
• le plan (x,y)
• Pas P3000 Å

6
Phase Colonnaire hexagonale
Molécule discotique
n
n
Clichés de RX dhéxapentoxytriphenylène (n
indique le directeur)
Hexatic
7
Diffraction by an arbitrary structure

• Disorder of 1st type and of 2nd type, g(r)
• Liquid crystal have anisotropic g(r)

8
Coherence and diffraction

• Coherence of phase
• Incoherence random phases after a process, in
  time or space
• Coherence of the individual scattering process
• Compton scattering vs Thomson scattering
• Incoherent scattering in neutron
• No interferences can be built
• Coherence of the scattering
• Phase difference q.rij random, lte-iq.rij gtij
  0
• Coherence (Amplitude add) Incoherence
  (Intensities add)
• Coherence of the light beam
• Temporal or spatial coherence (Laser, x-rays)

back
9
Diffraction by a disordered structure
In a gas, for example, ltcos(q.(rn-rn))gtnn 0
(incohérence)
Intensity is the sum of the individual
intensities There IS interferences, but they
average to zero
10
Diffraction by an arbitrary structure

• Kinematic approximation (ideally imperfect)
• Multiple scattering very difficult for disordered
  structures
• Problems of coherence of the x-ray beam
• Longitudinal (temporal) coherence
• Dw, Dt Dw-1, LcDt (Synch. And tube 1 mm)
• Rem. Same in r-space/q-space (duality)
• Transverse coherence
• D distance of the source
• s size of the source
• lDl/s (Synch. 10 mm, less for tube, flux)
• Generally, ltgt statistical average (enough).
• 300 K 26 meV 6.3 THz, x-ray frequency 106
  THz
• X-ray instantaneous positions of atoms
• ltgtt on time of measurement (statistical
  ergodic theorem)

11
Calculation of the Intensity Diffracted
Scattering amplitude
Atomic scattering factor

• Distinction between the effects of
• finite size and Pair-correlation function
• SAXS-WAXS

Form factor
Phase q.r gt p to built interferences
q4psin(q)/l r large q small (small
angles) r small q large (large angles)
12
Calculation of the Intensity Diffracted II
Pair correlation
Finite size
13
u and ur are inside the object u inside the
object u inside the object-r
14
Calculation of the Intensity Diffracted III
AND
Small angle scattering Wide angle scattering
Case g(r)1 (gas) same result
15
Pair-correlation functions
Dphq, DEhw
The scattering function S(q,w) is the Fourier
transform in TIME and SPACE of the time
dependent pair-correlation function
This definition is valid for x-ray and neutron
scattering, Classical x-ray scattering
impossible to perform an energy analysis 10-6
(ESRF ID 28, ID 16). X-rays integrate in
energy
16
Small Angle X-ray Scattering
Identical small particles (r) placed at
random Suspended or embedded in a medium (r0)
Intensity add
r0
r
Guiniers law
Ln(I)f(q2)
Slope R2/3
Radius of gyratio
q2
17
Simple example
x-1
-2p/a
2p/a
The WAXS is formed of a ring of radius a-1
Example of Argon X/Neutron comparison/ drop of
lead
18
1D exemple harmonic calculation
film
n2 n1 n0 n-1 N-2
Chains position Incoherent Intensities add
2q
asin2q nl
Rnmaun lt(un1- un)2gts2 kBT
Exemple 1 Exemple 2
Intensity can be calculated exactly I(s)
19
Application to liquid crystals

• Isotropic liquid
• Nematic
• Smectic
• Lamellar

I
20
Diffraction by disordered crystals

• Disorder of 1st type
• Position disorder
• Substitution disorder
• Phase transitions

21
Diffraction by disordered crystals
Bragg scattering (LRO)
Diffuse scattering disorder
Precession method
22
Calculation of the diffuse scattering
Disorder all cells different
Deviation to perfect order
Bragg scattering Average lattice (N2)
Diffuse scattering disorder
coherence
23
Phonons
uk (r,t)eku0cos(w(k)t-k.r)
Transverse acoustic
First Brilluoin zone
w
3N-3 optical modes
LA
TA1,2
3 acoustical modes
vs
k
0
p/a
-p/a
24
Debye-Waller factor
The second term is zero, for small
displacements Exact in the harmonic
approximation. This expression show that
ltu2gt can be measured by an x-ray diffraction
experiment. 0.05-0.1 Å and 0.5 Å in organic
crystals. Lindemann criteria a crystal melts
when is about ¼ of the unit cell
parameter. The intensity of a Bragg reflection
is
Debye-Waller factor (qualitative explanation)
Prop. To T and to q2
25
Thermal diffuse scattering
The diffuse scattering is given by the Fourier
transform of
26
Thermal diffuse scattering II
High angles anisotropic
Acoustic modes dominate
Temperature
Photon contribution k and -k

• In 3D, Bragg spots are not broadened by thermal
  agitation (LRO)
• Upon heating Bragg intensity decreases and TDS
  increases
• First measured by Laval (1939-1941), then Curien
  (1952), then
• MUCH better with neutron scattering because
  energy analysis

27
Substitution disorder I Alloy AxB1-x
Bragg intensity
Diffuse scattering, total disorder (incoherent)
Laue formula Small angle N
Diffuse scattering, correlations (local order)
Warren-Cowley parameters where PA(m) is
the probability of having an A atom at rm of a B
atom. In the case of total disorder,
PA(m?0)x and PA(m0)0. This gives a(m?0)0 and
a(m0)1 which gives the Laue equation.
General formula Modulation of DD
28
Substitution disorder II
One-dimensional alloy local order correspond to
the ABA configuration PA(1) the probability of
having an A atom at a of a B atom (PA(1) gt x).
The intensity reads
S(s)
1
ABABBBAABAABAB
h
0
1
2
3
1/2
Position of the ABABABABA Bragg
scattering toward the study of phase
transitions
29
Diffuse scattering in Pt3V alloy
D. Le Bolloch, R. Caudron, ESRF ID 15
f.c.c. (hkl) same parity
k
l
h
Diffuse scattering corresponds to (1,0,0)
position simple cubic position
E60 keV
PPt0.75 PV0.25 Probability does not depend on
the site AVERAGE
Disordered f.c.c. structure
Local order corresponds to s.c. structure Cu3Au
(L12)
30
Diffraction et diffusion diffuse des rayons X
Section efficace de diffusion faible (10-24 cm2)
? théorie cinématique ? transformée de Fourier
qkd-ki
kd
ki
Fn(q) ( f e-iq.u)
I
Cristal désordonné
Diffraction Ordre à grande distance
Diffusion diffuse Ordre local
q
I
u
Désordre de déplacement
Phonons, transitions displacives, Peierls,
spin-Peierls
q
I
Désordre de substitution
Diffusion de Laue
Transition ordre-désordre, ordre de charge
q
I
Couplage substitution- déplacement
Diffraction holographique, asymétrie
dintensité Distorsions autour des défauts,
accrochage des ondes de densité de charge
q
31
Diffuse scattering study of phase transition

• Order parameter (symmetry breaking) h0, TgtTC
  h?0, TltTc
• - Above the phase transition TgtTC h h(x)
  fluctuations (disorder diff.diff.)

?
order parameter
Below Tc New LRO, new Bragg spots (satellite),
prop. h2 Above Tc Ornstein-Zernike law (mean
field) Diff. Diff. Lorentzian
- Critical exponents
For TltTc, h (Tc-T)b and for TgtTc, x (Tc-T)-n
, c(0) (Tc-T)-g.
32
Determination of potential of interaction
33
Diffuse scattering and potential of interaction
order parameter
Ising model Spins, molecular orientation, atomic
positions in alloys
i j
kc
-kc
kBTc(0)
x -1
Qhkl

• Diffuse scattering ?
• Thermodynamical quantitties x, c(q), h
• Anisotropy of driving forces
• Potentiel of interaction (model)

34
Intermolecular interactions in C60
P. Launois, R. Moret, S. Ravy
Précession Tc 300 K
Face- centred cubic
Tc 260 K
Simple cubic

• Diffuse scattering at
• X, L, G of Brillouin zone
• Competing local orders
• Mean-field analysis of scattering
• from models of intermolecular
• interactions

H-DL
P-DL
Tg 80 K
frozen
No reorientations of C60
35
Modelling of intermolecular interactions in C60
K. H. Michel, K. Parlinski
Spherical harmonics adapted to cubic symmetry l
0, 6, 10, 12

• Van der Waals or Born-Mayer interactions between
  atoms ou additionnals sites (DB, SB)
• Interactions between effective charges (C, DB,
  SB)
• Mean field
• Guo al, 1991 Cheng al, 1992
• Sprikal,1992 Lual,1992 Burgosal,1993,1994
• Lamoen, Michel, Copley, 1994, 1997
• Pintschovius, Chaplot, 1995
• Savin, Harris, Yildirim, 1997

36
Intermolecular interactions in C60
hk9
hk9

• Modèle Lamoen-Michel (1994)
• Born-Mayer interactions ( C1exp(-C2r) - Br -6 )
• between sites dinteractions placed on
• atoms, single and double bonds.
• Orientational pair correlation fonctions
• at T300 K (P-DB)
• No model account for
• the Bragg reflections intensity,
• diffuse scattering
• and phonon dispersion curves.

hk11
37
Spin-Peierls dans (BCPTTF)2AsF6 Effect des
fluctuations sur le magnétisme
Q. Liu B. Dumoulin, C. Bourbonnais, S. Ravy,
J.-P. Pouget, C. Coulon, PRL 76, 1360 (1996)
S
S
Correlation length 1D fluctuations
a
S
S
BCPTTF
Satellite reflections in (1/2, 1/2, 1/2)
S0
2a
38
Bronze bleu K0.3MoO3 (or Rb0.3MoO3)
J.-P. Pouget, S. Girault, H. Moudden
Potassium (Rubidium)
// 2a-c
a
b
a2c (2ac)
c
MoO6 octahedra
39
Metal-isulator transition / non-linear
conductivity
- Non-linear conductivity (Monceau et al. 1976,
NbSe3)
Tp183 K
- Sliding of CDW pinned to impurities
Champ seuil
E. Bervas, thèse (1984)
40
Blue bronze Rb0.3MoO3 ou K0.3MoO3
Experimental results
Theory