diffraction methods in material science · pair correlation function g(r) = c(r)/ro probability to...
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OUTLINE OF THE COURSE0. Introduction
1. Classification of Materials
2. Defects in Solids
3. Basics of X-ray and neutron scattering
4. Diffraction studies of Polycrystalline Materials
5. Microstructural Analysis by Diffraction
6. Diffraction studies of Thin Films
7. Diffraction studies of Nanomaterials
8. Diffraction studies of Amorphous and Composite Materials
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OUTLINE OF TODAY‘S LECTURE
Classification of Amorphous Materials
Diffraction Studies of Amorpous Materials
Pair Correlation Function g(r)
Structure Factor S(Q)
Relation between Structure factor and g(r)
Determination of Structure Factor
Examples of S(Q) and g(r)
Modelling Methods (RMC)
Degree of Crystallinity
3
a
0 xn
xn = an = xn-1 + a
No Long-range Order (LRO) LRO LRO
No Translational Symmetry (TS) No TS TS
Chemical Short-range Order (SRO) SRO SRO
(Magnetic)
------------------------------------------------------------------------------------------------------------------------
Isotropic Anisotropic Anisotropic
No cleavage Cleavage
Tg (Glass Transition Temperature) Melting Temperature Tm
STRUCTURAL CLASSIFICATION OF SOLIDS
Amorphous Quasicrystals Crystalline
xn = xn-1 + xn-2
Cl
Be
Cl
1 1 2
(Lecture 2)
4
Classification of Amorphous Materials
Composition Bonding
Silicate Glasses (SiO2; NaSiO4, Na2Si2O5, … ) Ionic-Covalent Bonding
Phosphate glasses (NaPO3, ... ) "
Borate glasses (BO3; Na2O.B2O3, ... ) "
Chalcogenide Glasses (Se, Te, As-Se, Ge-Te; …) Covalent Bonding
Amorphous Carbon; amorphous Si (a-C; a-Si; …) "
Amorphous Polymers
Metal Glasses Metal bonding
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Number Density(Lecture 2)
r(r,t) Atomic number density (at/Å3); r(r,t) = S d(r – rj(t))
Time – average:
r(r) = <r(r,t)> Local density
For isotropic (homogeneous) systems
ro = N/V Macroscopic density
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Density – Density Correlation Function
C(r) = < r(ri ) r(ri + r)>V
For a system of N atoms:
C(r) = (V/N)Sir(ri)r(ri+r); r(ri) = d(r – ri)
C(r) = (V/N)Si Sjd(r - rij); rij = rj - ri
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Pair Correlation Function
g(r) = C(r)/ro
Probability to find an atom at a radius-vector r from another atom at r = 0.
For a isotropic system in which the atoms (particles) are randomly oriented,
(no translational symmetry and long-range order)
C(r) will depend only on the distance between the atoms, but not on the
orientation of the radius vector r
g(r) = <g(r)>Angles Pair Correlation Function (PDF)
N(r) - Number of atoms between r and r+dr
from the centre, regardless of their orientation
g(r) = N(r)/ro4pr2dr
lim[g(r)] = 1
r →∞
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Pair Correlation Functions (Examples)
(Lecture 2)
Liquid Ar
Thermal Motion/Disorder
Peak Broadening
g(r) rapidly convergens to 1A. Leach (2001)
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(Monoatomic)
fcc Au
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(Monoatomic)
Pair Correlation Functions (Examples)
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Pair Correlation Function
Multicomponen Systems
System containing 2 components (a, ß)
Al-Zr Metallic Glass (a = Al, ß = Zr)
# Partial pair correlation function gaß(r)
gaß(r) = (N/ro Na Nß ) Sia Sjß δ(r – rij)
# Total pair correlation function
g(r) = (1/N2) Si Sj Ni Nj gaß(r)
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Pair Correlation Function
Multicomponen Systems
GeO2 GlassGe-Ge
Ge-O
O-O
Marrocchelli et al. (2010)
Related Functions
Reduced PDF
G(r) = 4prro [g(r) – 1]
g(r) = 1 + G(r)/4prro
Radial Distribution Functions
RDF(r) = 4pr2ro g(r)
= 4pr2ro + rG(r)
T(r) = 4prro g(r) = 4prro + G(r)
Zhilegi
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Coordination Numbers
Fcc Au
N = ∫ RDF(r)dr
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15
Relations
between g(r) and Thermodynamic Properties
Statistical Mechanics
U = - ∂ln(Z)/∂ß; ß = 1/kBT; Z – Partitioning function
p = kBT ∂ln(Z)/∂V│T;
g(r) = (1/ro2) [N(N-1)/Z] ∫...∫dr3...drN exp[-ßU(r1, r2, ... rN)]
System described by (purely) pair-atomic interactions
u(r) – potential energy between pair of atoms
<U> = 2pNro ∫ g(r) u(r) r2 dr; (0,∞)
p = kBTro[1 – (2p/3) ro kBT∫ g(r) (du/dr) r3dr ]
Scattered Intensity
for Random Angular Orientations of Atoms
I(Q) = <I(Q)>|Orientaions = Σ Σ fifj* <exp[2piQ .(ri - rj)]>|Orient
I(Q) depends only on the difference between atomic positions
Structure Factor
S(Q) = I(Q)/<f2(Q)> = 1 + (I(Q) - <f2(Q)>)/ <f2(Q)>
due to the spherical symmetry (averaging over orientations) S depends
only on the magnitude Q of Q (Q = 4p/l sin(Q))
(Lecture 8)
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Structure Factor and Debye Equation
I(Q) = Σ fi2 + Σ Σ fifj* sin(Qrij)/Qrij
i ≠ j
Weighted sum of Sinc functions
with weights fifj* /<f2>
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Relation between S(Q) and g(r)
G(r) = 2/p ∫ Q [S(Q) – 1]sin(Qr)dQ, (Qmin, Qmax )
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Determination of Structure Factor
# Measure the Background scattering without sample (IBKG)
# Measure the scattering from the sample (IM)
IM = IScatLPA + IBKG + Isub
# Correct for Background, Lorentz-Polarization (LP) and Absorption (A)
IScat = (IM – IBKG - ISub)/LPA
# Normalize to electronic units
INor = ßIScat
by comparing with <f2> (High-angle method)
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0 5 10 15 20 25 30 35 400
5
10
15
20
25
30
35
40
45
Co
un
ts
2Q (degrees)
Sample+Substrate
Substrate
Air-Scattering
RDF Calculation Example:
Am-Al3Zr Thin Film; Mo Radiation, Q-Q Diffractometer
0 1 2 3 4 5 6 7 8 9 10 110
100
200
300
400
500
600
700
800
I(Q
)
Q (A-1)
I(Q)
f2
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
1.0
1.2
1.4
1.6
1.8
2.0
PO
L(s
)
s
Polarization
factor
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0 2 4 6 8 10 12-1
0
1
2
3
4
S(Q
)
Q (A-1)
0 5 10 15 20 25
0.0
0.5
1.0
1.5
2.0
g(R
)
R (A)
0 5 10 15 20 25-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
G(R
)
R (A)
RDF Calculation Example:
Am-Al3Zr Thin Film; Mo Radiation, Q-Q Diffractometer
Zotov et al., J. Non-Cryst. Solids 427 (2015) 104 21
EXAMPLES OF S(Q) and g(r)
Silicate Glasses: Na2Si4O9
Neutron Diffraction, Studswik, Sweden
Zotov (1998)
SiO4
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EXAMPLES OF S(Q) and g(r)
Phosphate Glasses: (MnO)x (NaPO3)1-x
Neutron Diffraction, LLB, France
Zotov et al. (2004)
Short-range order
Medium-range Order
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O
P
O
EXAMPLES OF S(Q) and g(r)
Chalcogenide Glasses: (Ag2Se)x (AsSe)1-x, x = 0.27, 0.39, 0.53
Zotov et al. (1997)
F(Q) = S(Q)-1
<N> = 2.5
<N> = 3.5
Ag-Se
Ag-As
As-Se
X-ray Diffraction
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F(Q) = S(Q)-1
EXAMPLES OF S(Q) and g(r)
Metallic Melts
Holland-Moritz (2002)
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Interpretation and Modelling of
Scattering from Amorphous Materials
Molecular Dynamics
Reverse Monte Carlo Simulations
Use of Complementary Structure-Sensitive Methods
(Raman & IR Spectroscopy)
Single Crystal 103–105 reflexions
Crystalline Powder 101–103 reflexions
Glass/Melt 100 ‚reflexions‘
We need modelling!26
RMC Simulations: Practical Aspects
Generation of Starting Configuartion (with periodic boundary conditions)
Metropolis Monte Carlo Algorithm
# Calculate g(r)
# Calculate c2 = S (gexp(r) – gcalc(r))2
# Move randomly selected atom in random direction at a distance less
than a predefined maximal distance Drmax;
# Calculate c2n;
# If c2n < c2
o the move is accepted, if c2n > c2
o it is accepted with
probability exp(-Dc2);
5.0x103
1.0x104
1.5x104
2.0x104
2.5x104
310
315
320
325
330
335
GO
F
Accepted Moves
GOF = c2 = Si[F(qi)exp – F(qi)cal]2
F(q) = S(q) – 1 = (4pr0/q) ∫r[g(r) – 1]sin(qr)dr
Alternatively,
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McGreevy (2001)
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2D crystal, Bragg peaks
disordered 2D crystal,Bragg peaks + diffuse scattering
Glass
STARTING CONFIGURATIONSRandom models
Computer-machanics models
Molecular Dynamics models
Crystalline Structures
‚All good [runs] must come to an end, but
all bad [runs] could continue for ever‘
old Arab wisdom
Convergence Time
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RMC EXAMPLES
0 5 10 15 20 25
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0Cu
0.20As
0.25Te
0.55 Glass
Neutron G(r)
RMC Fit
G(r
)
r (Å)
Zotov et al. (2000)
3000-atoms model Random Starting Models
Chalcogenide glasses;
Metallic Glasses;
Metallic Melts with
non-directional bonds
Pair Distance(Å) Compound
Cu - Cu 2.63 CuTe
2.70 Cu1.4Te
2.64 Cu2As
Cu - As 2.55 Cu2As
Cu - Te 2.68 CuTe
2.66 Cu1.4Te
As - As 2.44 c-As
As - Te 2.78 As2Te3
Te - Te 2.83 Te
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RMC EXAMPLES
Computer-Mechanics Models
Zotov & Keppler (1998)
0 5 10 15 20 25 30
-0.5
0.0
0.5Na
2O.4SIO
2 Glass
Neutron F(q)
RMC Fit
F(q
)
q (Å-1)
Network glasses/Melts;
(silicate, phosphate, borate)
Molecular liquids, which have
directional bonds (SRO)
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Effect of Starting Configuration
0 5 10 15 20 25
-1
0
1
2
3
4M5
M2
M3
M4
M1
Fx(q
)
Q (Å-1)
ModelM1 M2 M3
_____________________________GOF(XRD) 0.056 0.056 0.056GOF(ND) 0.008 0.008 0.008 Time (hours) 36 75 63_________________________________M1 RandomM2 Crystalline Te2BrM3 Another RMC
Te2Br0.75I0.25 Glass
X-ray + Neutrons; 3000 atoms
Zotov et al., (2005)
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Constraints in RMC Simulations
Experimental Density
Coordination Constraints
Bond-Angle Constraints (three-body constraints)
Different Scattering Data Sets (Neutrons + X-rays; Anomalous Scattering*)
Limited diffraction quotient
G(r)/F(q) One-dimensional projections of the 3D Structure;
weighted sums of several partial pair correlation functions
G(r) = SiSj gij(r)fifj*/<f2> (double sum over all types of atoms
gij – partail pair correlation functions)
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Anomalous X-ray Scattering
f(Q,E) = fo(Q) + f'(E) + if"(E)
Ge - Se
Structure Factor
S(Q,E) = I(Q,E)/<f (Q,E)>2 = 1 + [I(Q,E) - <f2(Q,E)>]/ <f (Q,E)>2
Armand et al., JNCS 167 (1994) 37
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Anomalous X-ray ScatteringDifferential Structure Factors
And
Differential Radial Distribution Functions
I(Q,E1) - I(Q,E2) =
{[<f(E1)2> - <f(E1)>
2] - [<f(E2)2> - <f(E2)>
2]} +
+ [<f2 (Q,E1)> - <f2 (Q,E2)>] DSFA(Q)
DSFSe
DSFGe
Armand et al., JNCS 167 (1994) 3735
Composite Materials (Polymers)
Degree of Crystallinity
Degree of Crystallinity (Wc)
# Density measurements Wc = (r – rA) / (rC – rA)
# X-ray diffraction
# Calorimetry
# IR Spectroscopy
Mo and Zhang (1995)
WcO = 100(IA – I)/(IA – IC); Ohlberg (1962)
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Composite Materials (Polymers)
Degree of Crystallinity
Scattered Intensity for amorphous materials distributed in the
whole reciprocal space:
R(Q) ~ ∫ IC(Q)Q2dQ / ∫ (IC(Q) + IA(Q)]Q2dQ; Ruland (1961)
R(Q) = 1/WcR + kQ
(does not require standards, but carefull fitting and separation of crystalline peaks
from the amorphous ‚background‘)
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Crystalline Quartz + SiO2 Glass
Composite Materials (Polymers)
Degree of Crystallinity
o Ohlberg+ Ruland
Zotov et al. (1994)