DIFFRACTION METHODS IN MATERIAL SCIENCE. PD Dr. Nikolay Zotov Lecture 10

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1 DIFFRACTION METHODS IN MATERIAL SCIENCE PD Dr. Nikolay Zotov Lecture 10

2 OUTLINE OF THE COURSE 0. 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 2

3 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

4 STRUCTURAL CLASSIFICATION OF SOLIDS Amorphous Quasicrystals Crystalline (Lecture 2) Cl Be Cl x n = x n-1 + x n-2 0 x n a x n = an = x n-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 T g (Glass Transition Temperature) Cleavage Melting Temperature T m 4

Classification of Amorphous Materials Composition Silicate

Ionic-Covalent Bonding Phosphate glasses (NaPO 3,.

.. ) " Chalcogenide Glasses (Se, Te, As-Se, Ge-Te; ) Covalent

5 Classification of Amorphous Materials Composition Silicate Glasses (SiO 2 ; NaSiO 4, Na 2 Si 2 O 5, ) Bonding Ionic-Covalent Bonding Phosphate glasses (NaPO 3,... ) " Borate glasses (BO 3 ; Na 2 O.B 2 O 3,... ) " Chalcogenide Glasses (Se, Te, As-Se, Ge-Te; ) Covalent Bonding Amorphous Carbon; amorphous Si (a-c; a-si; ) " Amorphous Polymers Metal Glasses Metal bonding 5

6 Number Density (Lecture 2) r(r,t) Atomic number density (at/å 3 ); r(r,t) = S d(r r j (t)) Time average: r(r) = <r(r,t)> Local density For isotropic (homogeneous) systems r o = N/V Macroscopic density 6

7 Density Density Correlation Function C(r) = < r(r i ) r(r i + r)> V For a system of N atoms: C(r) = (V/N)S i r(r i )r(r i +r); r(r i ) = d(r r i ) C(r) = (V/N)S i S j d(r - r ij ); r ij = r j - r i 7

8 Pair Correlation Function g(r) = C(r)/r o 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)/r o 4pr 2 dr lim[g(r)] = 1 r 8

9 (Monoatomic) Pair Correlation Functions (Examples) (Lecture 2) Liquid Ar Thermal Motion/Disorder Peak Broadening g(r) rapidly convergens to 1 A. Leach (2001) 9

10 fcc Au (Monoatomic) Pair Correlation Functions (Examples) 10

11 Pair Correlation Function Multicomponen Systems System containing 2 components (a, ß) Al-Zr Metallic Glass (a = Al, ß = Zr) # Partial pair correlation function g aß (r) g aß (r) = (N/r o N a N ß ) S i a S j ß δ(r r ij ) # Total pair correlation function g(r) = (1/N 2 ) S i S j N i N j g aß (r) 11

12 Pair Correlation Function Multicomponen Systems Ge-Ge GeO 2 Glass Ge-O O-O Marrocchelli et al. (2010) 12

13 Related Functions Reduced PDF G(r) = 4prr o [g(r) 1] g(r) = 1 + G(r)/4prr o Radial Distribution Functions RDF(r) = 4pr 2 r o g(r) = 4pr 2 r o + rg(r) T(r) = 4prr o g(r) = 4prr o + G(r) Zhilegi 13

14 Coordination Numbers Fcc Au N = RDF(r)dr 14

15 Relations between g(r) and Thermodynamic Properties Statistical Mechanics U = - ln(z)/ ß; ß = 1/k B T; Z Partitioning function p = k B T ln(z)/ V T ; g(r) = (1/r o2 ) [N(N-1)/Z]... dr 3...dr N exp[-ßu(r 1, r 2,... r N )] System described by (purely) pair-atomic interactions u(r) potential energy between pair of atoms <U> = 2pNr o g(r) u(r) r 2 dr; (0, ) p = k B Tr o [1 (2p/3) r o k B T g(r) (du/dr) r 3 dr ] 15

16 Scattered Intensity for Random Angular Orientations of Atoms (Lecture 8) I(Q) = <I(Q)> Orientaions = Σ Σ f i f j * <exp[2piq.(r i - r j )]> Orient I(Q) depends only on the difference between atomic positions Structure Factor S(Q) = I(Q)/<f 2 (Q)> = 1 + (I(Q) - <f 2 (Q)>)/ <f 2 (Q)> due to the spherical symmetry (averaging over orientations) S depends only on the magnitude Q of Q (Q = 4p/l sin(q)) 16

17 Structure Factor and Debye Equation I(Q) = Σ f i2 + Σ Σ f i f j * sin(qr ij )/Qr ij i j Weighted sum of Sinc functions with weights f i f j * /<f 2 > 17

18 Relation between S(Q) and g(r) G(r) = 2/p Q [S(Q) 1]sin(Qr)dQ, (Q min, Q max ) 18

19 Determination of Structure Factor # Measure the Background scattering without sample (I BKG ) # Measure the scattering from the sample (I M ) I M = I Scat LPA + I BKG + I sub # Correct for Background, Lorentz-Polarization (LP) and Absorption (A) I Scat = (I M I BKG - I Sub )/LPA # Normalize to electronic units I Nor = ßI Scat by comparing with <f 2 > (High-angle method) 19

20 RDF Calculation Example: Am-Al 3 Zr Thin Film; Mo Radiation, Q-Q Diffractometer Sample+Substrate Substrate Air-Scattering Counts Q (degrees) Polarization factor I(Q) f 2 POL(s) I(Q) s Q (A -1 ) 20

21 RDF Calculation Example: Am-Al 3 Zr Thin Film; Mo Radiation, Q-Q Diffractometer S(Q) g(r) Q (A-1) R (A) G(R) Zotov et al., J. Non-Cryst. Solids 427 (2015) 104 R (A) 21

$9 Neutron Diffraction,$

22 EXAMPLES OF S(Q) and g(r) Silicate Glasses: Na 2 Si 4 O 9 Neutron Diffraction, Studswik, Sweden SiO 4 Zotov (1998) 22

$Diffraction, LLB, France O P O$

23 EXAMPLES OF S(Q) and g(r) Phosphate Glasses: (MnO) x (NaPO 3 ) 1-x Neutron Diffraction, LLB, France O P O Short-range order Medium-range Order Zotov et al. (2004) 23

24 EXAMPLES OF S(Q) and g(r) Chalcogenide Glasses: (Ag 2 Se) x (AsSe) 1-x, x = 0.27, 0.39, 0.53 X-ray Diffraction Ag-Se Ag-As As-Se F(Q) = S(Q)-1 <N> = 3.5 <N> = 2.5 Zotov et al. (1997) F(Q) = S(Q)-1 24

25 EXAMPLES OF S(Q) and g(r) Metallic Melts Holland-Moritz (2002) 25

Interpretation and Modelling of Scattering from Amorphous Materials Single Crystal 10 3 10 5 reflexions Crystalline Powder 10 1 10 3 reflexions Glass/Melt 10

26 Interpretation and Modelling of Scattering from Amorphous Materials Single Crystal reflexions Crystalline Powder reflexions Glass/Melt 10 0 reflexions Molecular Dynamics Reverse Monte Carlo Simulations Use of Complementary Structure-Sensitive Methods (Raman & IR Spectroscopy) We need modelling! 26

27 RMC Simulations: Practical Aspects Generation of Starting Configuartion (with periodic boundary conditions) Metropolis Monte Carlo Algorithm # Calculate g(r) # Calculate c 2 = S (g exp (r) g calc (r)) 2 # Move randomly selected atom in random direction at a distance less than a predefined maximal distance Dr max ; # Calculate c 2 n; # If c 2 n < c 2 o the move is accepted, if c 2 n > c 2 o it is accepted with probability exp(-dc 2 ); GOF Alternatively, GOF = c 2 = S i [F(q i ) exp F(q i ) cal ] 2 F(q) = S(q) 1 = (4pr 0 /q) r[g(r) 1]sin(qr)dr x x x x x10 4 Accepted Moves 27

28 2D crystal, Bragg peaks disordered 2D crystal, Bragg peaks + diffuse scattering Glass McGreevy (2001) 28

29 STARTING CONFIGURATIONS Random 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 29

30 RMC EXAMPLES 3000-atoms model Random Starting Models G(r) Cu 0.20 As 0.25 Te 0.55 Glass Chalcogenide glasses; Metallic Glasses; Metallic Melts with non-directional bonds Neutron G(r) RMC Fit Zotov et al. (2000) r (Å) Pair Distance(Å) Compound Cu - Cu 2.63 CuTe 2.70 Cu 1.4 Te 2.64 Cu 2 As Cu - As 2.55 Cu 2 As Cu - Te 2.68 CuTe 2.66 Cu 1.4 Te As - As 2.44 c-as As - Te 2.78 As 2 Te 3 Te - Te 2.83 Te 30

31 RMC EXAMPLES Computer-Mechanics Models Network glasses/melts; (silicate, phosphate, borate) Molecular liquids, which have directional bonds (SRO) 0.5 Na 2 O.4SIO 2 Glass F(q) Neutron F(q) RMC Fit Zotov & Keppler (1998) q (Å -1 ) 31

32 Effect of Starting Configuration X-ray + Neutrons; 3000 atoms Te 2 Br 0.75 I 0.25 Glass Model M1 M2 M3 GOF(XRD) GOF(ND) Time (hours) M1 Random M2 Crystalline Te 2 Br M3 Another RMC F x (q) M5 M4 M3 M2 M Q (Å -1 ) Zotov et al., (2005) 32

33 Constraints in RMC Simulations Limited diffraction quotient G(r)/F(q) One-dimensional projections of the 3D Structure; weighted sums of several partial pair correlation functions Experimental Density Coordination Constraints Bond-Angle Constraints (three-body constraints) Different Scattering Data Sets (Neutrons + X-rays; Anomalous Scattering*) G(r) = S i S j g ij (r)f i f j */<f 2 > (double sum over all types of atoms g ij partail pair correlation functions) 33

34 Anomalous X-ray Scattering Ge - Se f(q,e) = f o (Q) + f'(e) + if"(e) Structure Factor S(Q,E) = I(Q,E)/<f (Q,E)> 2 = 1 + [I(Q,E) - <f 2 (Q,E)>]/ <f (Q,E)> 2 Armand et al., JNCS 167 (1994) 37 34

35 Anomalous X-ray Scattering Differential Structure Factors And Differential Radial Distribution Functions I(Q,E 1 ) - I(Q,E 2 ) = {[<f(e 1 ) 2 > - <f(e 1 )> 2 ] - [<f(e 2 ) 2 > - <f(e 2 )> 2 ]} + DSF Se + [<f 2 (Q,E 1 )> - <f 2 (Q,E 2 )>] DSF A (Q) DSF Ge Armand et al., JNCS 167 (1994) 37 35

36 Composite Materials (Polymers) Degree of Crystallinity Degree of Crystallinity (W c ) # Density measurements W c = (r r A ) / (r C r A ) # X-ray diffraction # Calorimetry # IR Spectroscopy Mo and Zhang (1995) W co = 100(I A I)/(I A I C ); Ohlberg (1962) 36

37 Composite Materials (Polymers) Degree of Crystallinity Scattered Intensity for amorphous materials distributed in the whole reciprocal space: R(Q) ~ I C (Q)Q 2 dq / (I C (Q) + I A (Q)]Q 2 dq; Ruland (1961) R(Q) = 1/W cr + kq (does not require standards, but carefull fitting and separation of crystalline peaks from the amorphous background ) 37

38 Composite Materials (Polymers) Degree of Crystallinity Crystalline Quartz + SiO 2 Glass o Ohlberg + Ruland Zotov et al. (1994) 38

Structural characterization. Part 1

Structural characterization. Part 1 Structural characterization Part 1 Experimental methods X-ray diffraction Electron diffraction Neutron diffraction Light diffraction EXAFS-Extended X- ray absorption fine structure XANES-X-ray absorption