DIFFRACTION METHODS IN MATERIAL SCIENCE

PD Dr. Nikolay Zotov

Tel. 0711 689 3325

Email: zotov@imw.uni-stuttgart.de

Room 3N16

Lecture 9

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

Diffraction Studies of Amorpous Materials

Classification of Amorphous 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)

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; ) "

Calcogenide 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; r(r) = S d(r – rj(t))

Time – average and/or average over the volume of the sample (<>)

r(r) = <r(r,t)>

For isotropic (homogeneous) systems

r(r) = ro = N/V

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Density – Density Correlation Function

C(r) = < r(ri ) r(ri + r)>i

For a system of N atoms:

C(r) = (1/N)Sir(ri)r(ri+r); r(ri) = d(r – ri)

C(r) = (1/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 for Liquids and Glasses

(Lecture 2)

Liquid Ar

Thermal Motion/Disorder

Peak Broadening

g(r) rapidly convergens to 1A. Leach (2001)

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Crystalline fcc Au

10

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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Scattered Intensity

for Random Angular Orientations of Atoms

I(Q) = <I(Q)>|Orientaions = Σ Σ fifj* <exp[iQ .(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 Debey 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

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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

# Correct for Background, Lorentz-Polarization (LP) and Absorption (A)

IScat = (IM – IBKG)/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)

f2

I(Q)

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

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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)

Phospate Glasses: (MnO)x (NaPO3)1-x

Neutron Diffraction, LLB, France

Zotov et al (2004)Short-range order

Medium-range Order

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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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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!

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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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STARTING CONFIGURATIONS

Random models

Computer-machanics models

Molecular Dynamics models

Crystalline Structures

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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)/S(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)

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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

Ge - Se

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Anomalous X-ray ScatteringDifferential Structure Factors

And

Differential Radial Distribution Functions

DAI(Q) = DA[<f2> - <f>2] + DA<f>2DSFA(Q)

DAI(Q) = I(Q,E1) - I(Q,E2)

DA[<f2> - <f>2] = [<f(E1)2> - <f(E1)>

2] - [<f(E2)2> - <f(E2)>

2]

DA <f2> = <f2 (Q,E1)> - <f2 (Q,E2)>

DSFSe

DSFGe

Armand et al., JNCS 167 (1994) 3732

Composite Materials (Polymers)

Degree of Crystallinity

Degre 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)

Scattered Intensity for amorphous materials distributed in the

whole reciprocal space:

WcR ~ ∫ IC(Q)Q2dQ / ∫ (IC(Q) + IA(Q)]Q2dQ; Ruland (1961)

(does not require standards, but carefull fitting and separation of crystalline peaks from amorphous

Background)

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