Displacement Short Range Order and theDetermination of Higher Order Correlation Y. S. Puzyrev, D. M. Nicholson, R. I. Barabash, G. Stocks, G. Ice Oak Ridge National Laboratory R. McQueeney Ames National Laboratory Yingzhou Du Iowa State University
Outline• Short Range Order and Diffuse Scattering• Experimental Method• Data Analysis and Atomic Structure• Pair and Many-body Correlations• Displacement SRO and Higher Order Correlations in One-dimensional Model• First Principles Calculation of Atomic Displacement in Fe46.5 Ni53.5 Fe80.0Ga20.0
Diffuse scattering poised for a revolution!• Synchrotron sources /new tools enable new applications – Rapid measurements for combinatorial and dynamics – High energy for simplified data analysis – Small (dangerous) samples• Advanced neutron instruments emerging – Low Z elements – Magnetic scattering – Different contrast• New theories provide direct link beween experiments and first-principles calculations Experiment TheoryMajor controversies have split leading scientists in once staid community!
Diffuse scattering due to local (short ranged) correlations/ fluctuations• Thermal diffuse scattering (TDS)• Point defect – Site substitution – Vacancy – Interstitial• Precipitate• Dislocations• Truncated surface- more All have in common reduced correlation length!
Bragg reflection occurs when scattering amplitudes add constructively• Orientation of sample• Wavelength• Bragg Peaks sharp! ~1/N (arc seconds) h ˆ Think in terms of momentum transfer p0 = d0 l
Length scales are inverted!• Big real small reciprocal• Small real big reciprocal• We will see same effect in correlation length scales – Long real-space correlation lengths scattering close to Bragg peaks If you remember nothing else!
Krivoglaz classified defects by effect on Bragg Peak Intensity• Defects of 1st kind No defects – Bragg width unchanged – Bragg intensity decreased – Diffuse redistributed in reciprocal space – Displacements remain finite Defects of 1st kind• Defects of 2nd kind – No longer distinct Bragg peaks Defects of 2nd kind – Displacements continue to grow with crystal size
Thermal motion-Temperature Diffuse Scattering- (TDS) -defect of 1st kind • Atoms coupled through atomic bonding • Uncorrelated displacements at distant sites – (finite) • Phonons (wave description) – Amplitude – Period – Propagation direction – Polarization (transverse/compressional)Sophisticated theories fromJames, Born Von Karmen, Krivoglaz
A little math helps for party conversation • Decrease in Bragg intensity scales like e-2M, where sin2 q 2M = 16p u2 2 s l Low Temperature • Small Big reflections • e-2M shrinks (bigger effect) with (q) High temperatureDisplacements, us depend on Debye Temperature D –Bigger D smaller displacements !
TDS makes beautiful patterns reciprocal space sodium Rock salt• Iso- intensity contours – Butterfly – Ovoid – Star• Transmission images reflect symmetry of Experiment Theory reciprocal space and TDS patterns Chiang et al. Phys. Rev. Lett. 83 3317 (1999)
Often TDS mixed with additional diffuse scattering• Experiment• Strain contribution• Combined theoretical TDS and strain diffuse scattering TDS must often be removed to reveal other diffuse scattering
Alloys can have another type 1 defect-site substitution Each Au has 8 Cu near-neighbors• Long range – Ordering (unlike neighbors) Cu3Au – Phase separation (like neighbors) L12• Short ranged – Ordering CuAu – Clustering (like neighbors) L10 Alternating planes of Au and Cu
Redistribution depends on kind of correlationClustering intensity fundamental sitesRandom causesLaue monotonicShort-range ordering superstructure sites
Neutron/ X-rays Complimentary For Short-range Order Measurements• Chemical order diffuse scattering proportional to contrast (fA-fB)2• Neutron scattering cross sections – Vary wildly with isotope – Can have + and – sign – Low Z , high Z comparable• X-ray scattering cross section – Monotonic like Z2 – Alter by anomalous scattering
Neutrons can select isotope to eliminate Bragg scattering• Total scattering cafa2+ cbfb2 = 3 +1 +1 +1 -3 +1 +1 +1 -3• Bragg scattering (cafa+ cbfb = 0 )2 +1 +1 +1 -3 +1• Laue (diffuse) scattering +1 +1 -3 0 0 0 cacb(fa- -fb)2 =3 0 0 0 0 0 Isotopic purity important as different isotopes have distinct scattering cross sections- only one experiment ever done!
X-ray anomalous scattering can change x-ray contrast • Chemical SRO scattering scales like (fa-fb)2 • Static displacements scale like (fa-fb) • TDS scales like ~faverage2
Atomic size (static displacements) affect phase stability/properties • Ionic materials (Goldschmidt) – Ratio of Components – Ratio of radii – Influence of polarization FeFe r 110 • Metals and alloy phases (Hume-Rothery) c – Ratio of radii r FeNi 110 – Valence electron concentration – Electrochemical factor NiNi r 110 a a Grand challenge -include deviations from lattice in modeling of alloys
Measurement and theory of atomic size are hard!• Theory- violates repeat lattice approximation- every unit cell different!• Experiment – EXAFS marginal (0.02 nm) in dilute samples – Long-ranged samples have balanced forces Important!
Systematic study of bond distances in Fe-Ni alloys raises interesting questions• Why is the Fe-Fe bond distance stable?• Why does Ni-Ni bond swell with Fe concentration?• Are second near neighbor bond distances determined by first neighbor bonding?
Diffuse x-ray scattering measurements of AgPd alloy finds surprising lack of local ordering• Negative deviation from Vegard’s law indicates ordering.• First principle theory predicts ordering.- Good system for theory.• Semi-empirical theory predicts clustering.
High-energy x-ray measurements revolutionize studies of phase stability • Data in seconds instead of days • Minimum absorption and stability corrections • New analysis provides Crystal growth direct link to first- principle Experimental measurement of short range structuresMax Planck integrates diffuse x- Data analysisray scattering elements! Rapid measurement ESRF
Measurements show competing tendencies to order Au3Ni -001 plane• Both L12 and Z1 present• Compare with first principles calculations Reichert et al. Phys. Rev. Lett. 95 235703 (2005)
New directions in diffuse scattering• Microdiffuse x-ray scattering – Combinatorial – Easy sample preparation• Diffuse neutron data from every sample• Interpretation more closely tied to theory – Modeling of scattering x- ray/neutron intensity Experiment Model
Intense synchrotron/neutron sources realize the promise envisioned by pioneers of diffuse x-ray scattering • M. Born and T. Von Karman 1912-1946- TDS • Andre Guiner (30’s-40’s)-qualitative size • I. M. Lifshitz J. Exp. Theoret. Phys. (USSR) 8 959 (1937) • K. Huang Proc. Roy. Soc. 190A 102 (1947)-long ranged strain fields • J. M Cowley (1950) J. Appl. Phys.-local atomic size • Warren, Averbach and Roberts J. App. Phys 22 1493 (1954) -SRO • Krivoglaz JETP 34 139 1958 chemical and spatial fluctuations Other references:• X-ray Diffraction- B.E. Warren Dover Publications New York 1990.• http://www.uni-wuerzburg.de/mineralogie/crystal/teaching/dif_a.html• Krivoglaz vol. I and Vol II.
Diffuse scattering done by real people-small community S. Cowley, Arizona St. • Warren school Bernie Borie,ORNL Cullie Sparks ORNL Jerry Cohen, Northwestern B. Schoenfeld, ETH ZurichA. Gunier W. Schweika, KFA Jülich Simon Moss,U. Of Houston • Krivoglaz school Peisl, U. München Gitgarts,Minsk Rosa Barabash IMP, ORNL Gene Ice Rya Boshupka, IMP ORNL
Advanced Photon SourceArgonne National Laboratory, Chicago IL
Diffuse Scattering 2 * k rp rq I total k fp f e q I Bragg k I diffuse k p ,q f p - atomic scattering factor of atom p I diffuse k I SRO k I static k ITDS kReal Space Reciprocal Space
Short-range orderReal space Reciprocal space = correlation length 29
Short-range atomic ordering in Galfenol• Position of doped Ga atoms influences magnetostriction – Quenched material has larger MS – MS is suppressed in D03 ordered samples – Proposed that uniaxial Ga pairing enhances MS (R. Wu, JAP 91, 7358, 2002)• Ga atom distributions measured using X-ray scattering – Long-range order (Ga ordering, D03, B2, L12) – Short-range order (Ga clustering, pairing)• Can study variation of Ga clustering with … – Sample composition – Heat treatment – Temperature – Stress/strain state – Magnetic field 30
Fe-Ga structures and reflection conditionsReflection A2 (BCC) B2 (SC) D03 (FCC) (1,0,0) X (2,0,0) (1,1,0) (1,1,1) X (1/2,1/2,1/2) X X (1/2,1/2,0) X X X 31
Experimental Method• Measurements performed at UNI-CAT (Sector 33ID-D APS) – Absorption edge energy monochromatic x-rays • (E=7.1 and 10.32 keV) – Measure weak diffuse scattering signal – Point detector for quantitative data• Sample Fe80Ga20 – Crystals prepared at Ames Lab by Bridgman technique – Samples cut Size ~ 5x5x0.5 mm – “Quenched” - rapidly cooled – Etched for x-ray scattering
Conversion of the Data to Electron Units• Scattering factors, - corrected for x-ray absorption fine structure (XAFS) andLorentzian hole width of an inner shell.• Compton and Resonant Raman inelastic scattering is separated and removed fromintegrated intensities.•Ni standard integrated intensities are obtained to compute conversion factor.
Short-Range Order Parameters Anomalous (resonant) scatteringWarren-Cowley Short-Range Order Parameters AB Plmn ABlmn 1 , where Plmn probability to find atom Ain shell lmn away from B cA
Large Scattering Factor Contrast12 eV below Fe absorption edge E = 7.100 keV Small Scattering Factor Contrast 50 eV below Ga absorption edge E = 10.320 keV
Large atomic displacements are instrumental for understanding of occurrence of DO3 phase
Magnetistriction Effect Experiment at high energy E= 80-120 keV, R. McQueeney• Correlation lengths 1.00 1.0 Fe100-xGax 1.1 – Slow-cooled more ordered than 0.75 1.3 1.6 quenched 0.50 2.0 2.7 – At low x, ~ 2-4 unit cells 0.25 4.0 SRO Peak Width (Units of 2 /a) 8.0• (1/2,1/2,1/2) (3/2,1/2,1/2) Correlation Length (Unit Cells) Slow-cooled 0.00 1.1 – dramatic increase in correlation length (100)SC (100)Q 0.75 (011)SC 1.3 (011)Q 1.6 above 17.7% 0.50 (111)SC (111)Q 2.0• 2.7 Quenched 0.25 4.0 (100) (3/2,3/2,3/2) 8.0 – Gradual increase in correlation length 0.00 1.1• Anisotropy 0.75 1.3 1.6 – Long correlation lengths along (111) 0.50 2.0 2.7 – Anisotropy disappears at high Ga% 0.25 4.0 (111) (5/2,1/2,1/2) 8.0 0.00 12 13 14 15 16 17 18 19 20 13 14 15 16 17 18 19 20 x, Ga Composition (%)• Atomic displacements – increase before magnetostriction decreases
X-ray diffraction diffuse scattering experimentWarren-Cowley SRO parametersAtomic displacements lmn lmn where lmn include 20 or more shellsBased on this information one can reconstructatomic configurationIf interactions are accurately described byPair Hamiltonian, the higher order correlationsare reproduced correctly from pair correlationswhich can be shown using DFT Classical DFT: R. Evans, Advances in Physics 28, 2, 143 (1979)
Test multi-atom real-space correlations in Reverse Monte- Carlo model against equilibrium configurations Create equilibrium configurations With realistic interactions• Set 1 - pair interactions to 4th neighbor Determine pair and higher• Set 2 -pair +multi-atom Order correlations interactions Construct a configuration By Reverse Monte-Carlo From pair correlations Compare Determine pair and higher correlations Order correlations
Nearest Neighbor distribution • Pair correlations determine all higher order correlations•Equilibrium for pair potential Hamiltonians.•Constructed • For many body Hamiltonians they do not.•Random • Derivatives provide information about higher order correlations: pressure, magnetic field, stress,…
SRO in 1D Model with PBC •Show that displacements SRO limits higher order correlations •Simple analytical model Iso- g (r ) process used to check realizability of pair correlation function Inappropriateness of so called “reverse Monte Carlo Method” that uses a candidate pair correlation function as a means to suggest typical many-body configuraitonsF. H. Stillinger,S. Torquato, “Pair Correlation Function Realizability”, J. Phys. Chem. B 108, 19589
Lattice Model n, N configuration of n particles distributed on N sites For exampleClass collects all members that differ only by •Translation •Inversion jm j number of configurations N 11w j weight n 3
Enumeration of states GivenThe weights have to be positive g (r ) The solution of linear equations 1 2 2w1 2w2 2w3 k • Exists and not unique •1,2, n dimensional manifold 1 3 2w2 2w4 w5 k w1 2w3 2w5 k • Exists and unique 1 4etc... • Doesn’t exist
Displacements SRO in 1D Model with PBC • Suppose that there is an average ‘lattice’ and introduce displacements off the ideal positions • Create a simple RULE for the displacement dependence on chemical environment Displacement table 101 N Displace 0 in the 001 R1 middle to the right if 100 L j , j 0 is to the left and 1 111 N0 is to the right 011 L 0 110 R 000 N
Displacements SRO in 1D Model with PBC d1d4 = w1 + w3 + 2w5 = f This puts additional restrictions On the possible configurationsetc... s 1s 2 = 2w1 + 2w2 + 2w3 = k The displacements in the real system must give an indication s 1s 3 = 2w2 + 2w4 + w5 = k whether the particular local environment is realizable or not, s 1s 4 = w1 + 2w3 + 2w5 = k i.e. restrict the higher order correlations in the systemetc...
Average correlations: point, pair, triplet 1 Ai 1 B N 1 2c 1 N i 1 i N 2 1 4(c 2 c 1 c ) 2c 1 j N i i j j i 1 ö-1 N æ V (q) a ( q ) = ç1+ 2cA cB ÷ 3 1 j ,k N i i j i k i 1 è kB T ø
Real alloy systemFe46.5 Ni53.5 N 2 1 Configurations with the same pair correlations j N i 1 i i j Vary triplet correlations 1-st Compute displacements from first principles Compare the displacements with Measured values
Configurations with triplet correlation variation P AAA 11 0.1 A Ni P AAA 11 0.4 Reverse Monte-Carlo 128 atom VASP Displacements ExperimentFe 4.09e-4 Ni 1.70e-4 Fe 0.028 Ni 0.012 Fe 7.09e-4 Ni 0.70e-4
Conclusion• One-dimensional model shows that displacement SRO further restricts higher order correlations• For realistic system calculation of displacements can give qualitative prediction of higher order correlations.