Displacement Short Range Order and the

Determination of Higher Order Correlation

Y. S. Puzyrev, D. M. Nicholson, R. I. Barabash, G. Stocks, G. IceOak Ridge National Laboratory

R. McQueeneyAmes National Laboratory

Yingzhou DuIowa 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

46.5 53.5Fe Ni Fe80.0

Ga20.0

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 Theory

Major 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)

Think in terms of momentum transfer

 

p 0 =h

lˆ d 0

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

• Defects of 1st kind– Bragg width unchanged

– Bragg intensity decreased

– Diffuse redistributed in reciprocal space

– Displacements remain finite

• Defects of 2nd kind– No longer distinct Bragg peaks

– Displacements continue to grow with crystal size

No defects

Defects of 1st kind

Defects of 2nd kind

Intensity

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

• Small Big reflections

• e-2M shrinks (bigger effect) with (q)

 

2M =16p2

us

2 sin2 q

l

Low Temperature

High temperature

Displacements, us depend on Debye Temperature D –

Bigger D smaller displacements !

TDS makes beautiful patterns reciprocal space

• Iso- intensity contours– Butterfly

– Ovoid

– Star

• Transmission images reflect symmetry of reciprocal space and TDS patterns

Experiment Theory

sodium Rock salt

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 1defect-site substitution

• Long range

– Ordering (unlike neighbors)

– Phase separation (like neighbors)

• Short ranged

– Ordering

– Clustering (like neighbors)

Cu3AuL12

CuAuL10

Alternating planes of Au and Cu

Each Au has 8 Cu near-neighbors

Redistribution depends on kind of correlation

Clustering intensity fundamental sites

Short-range ordering superstructure sites

Random causes Laue monotonic

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+ cbfb

2 = 3

• Bragg scattering (cafa+ cbfb)2 = 0

• Laue (diffuse) scattering

cacb(fa- -fb)2 =3

Isotopic purity important as different isotopes have distinct scattering cross sections- only one experiment ever done!

+1 +1

+1 +1

+1

+1

- 3

-3 0 0

0 0

0

0

0

0

+1 +1

+1 +1

+1

+1

- 3

-3

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

• Metals and alloy phases (Hume-Rothery)

– Ratio of radii

– Valence electron concentration

– Electrochemical factor a

c

a

r110

FeFe

r110

FeNi

r110

NiNi

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 direct link to first-principle

Max Planck integrates diffuse x-ray scattering elements!

Experimental measurement of short range structures

Crystal growth

Rapid measurementESRF

Data analysis

Measurements show competing tendencies to order

• Both L12 and Z1 present

• Compare with first principles calculations

Au3Ni -001 plane

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 221493 (1954) -SRO

• Krivoglaz JETP 34 139 1958 chemical and spatial fluctuations

• 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.

Other references:

Diffuse scattering done by real people-small community

• Warren school

• Krivoglaz school

Cullie SparksORNL

Bernie Borie,ORNL

Simon Moss,U. Of Houston

Rosa BarabashIMP, ORNL

Jerry Cohen, Northwestern

S. Cowley, Arizona St.

A. Gunier

Peisl, U. München

Gitgarts,Minsk

Rya Boshupka, IMP

B. Schoenfeld, ETH Zurich

W. Schweika, KFA Jülich

Gene IceORNL

Advanced Photon Source Argonne National Laboratory, Chicago IL

Upcomingin 2015: NSLS 2 in Brookhaven

Diffuse Scattering

pf

kIkIeffkI

p

diffuseBragg

qp

rrk

qptotalqp

atom offactor scattering atomic -

2

,

*

kIkIkIkI TDSstaticSROdiffuse

Reciprocal SpaceReal Space

29

Short-range order

Real space Reciprocal space

= correlation length

30

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

31

Fe-Ga structures and reflection conditions

Reflection 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

• 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

Experimental Method

33

-0.5 0.0 0.5 1.0 1.5 2.00.1

1

10

100

1000

10000

100000

(111)

Co

un

ts/m

onito

r[1,L,L]

26% quenched

LRO

18.7% slow-cooled

SRO

(100)

Data

Short-range orderThermal diffuse

Thermal diffuse scattering

Short-rangeorder

Conversion of the Data to Electron Units

• Scattering factors, - corrected for x-ray absorption fine structure (XAFS) and

Lorentzian hole width of an inner shell.

• Compton and Resonant Raman inelastic scattering is separated and removed from

integrated intensities.

•Ni standard integrated intensities are obtained to compute conversion factor.

Anomalous (resonant) scattering

Warren-Cowley Short-Range Order Parameters

1 , ' 'AB

ABlmnlmn lmn

A

Pwhere P probability to find atom Ain shell lmn away fromB

c

Short-Range Order Parameters

Large Scattering Factor Contrast12 eV below Fe absorption edge

E = 7.100 keV

Small Scattering Factor Contrast50 eV below Ga absorption edge

E = 10.320 keV

Large atomic displacements are instrumental for understanding of occurrence of DO3 phase

0.00

0.25

0.50

0.75

0.00

0.25

0.50

0.75

1.00

8.0

4.0

2.7

2.0

1.6

1.3

1.1

1.0

12 13 14 15 16 17 18 19 200.00

0.25

0.50

0.75

8.0

4.0

2.7

2.0

1.6

1.3

1.1

13 14 15 16 17 18 19 20

8.0

4.0

2.7

2.0

1.6

1.3

1.1

SR

O P

ea

k W

idth

(U

nits o

f 2

/a)

(100)

(100)SC

(100)Q

(011)SC

(011)Q

(111)SC

(111)Q

(1/2,1/2,1/2)

(3/2,1/2,1/2)

(111)

Co

rrela

tion

Le

ng

th (U

nit C

ells

)

(3/2,3/2,3/2)

x, Ga Composition (%)

(5/2,1/2,1/2)

Fe100-x

Gax

• Correlation lengths

– Slow-cooled more ordered than

quenched

– At low x, ~ 2-4 unit cells

• Slow-cooled

– dramatic increase in correlation length

above 17.7%

• Quenched

– Gradual increase in correlation length

• Anisotropy

– Long correlation lengths along (111)

– Anisotropy disappears at high Ga%

• Atomic displacements – increase

before magnetostriction decreases

Magnetistriction EffectExperiment at high energy E= 80-120 keV, R. McQueeney

X-ray diffraction diffuse scattering experiment

Warren-Cowley SRO parameters

Atomic displacements lmn

where lmn include 20 or more shells

Based on this information one can reconstruct atomic configuration

If interactions are accurately described by

Pair Hamiltonian, the higher order correlationsare reproduced correctly from pair correlations which can be shown using DFT

lmn

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

Determine pair and higher

Order correlations

Construct a configuration

By Reverse Monte-Carlo

From pair correlations

Determine pair and higher

Order correlations

Compare

correlations

• Set 1 - pair interactions to 4th neighbor

• Set 2 -pair +multi-atom interactions

Nearest Neighbor distribution

•Equilibrium•Constructed•Random

• Pair correlations determine all higher order correlations for pair potential Hamiltonians.

• For many body Hamiltonians they do not.

• 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

F. H. Stillinger,S. Torquato, “Pair Correlation Function Realizability”, J. Phys. Chem. B 108, 19589

Inappropriateness of so called “reverse Monte CarloMethod” that uses a candidate pair correlation functionas a means to suggest typical many-body configuraitons

( )g r process used to check realizability of pair correlation function

Iso-

Lattice Model

configuration of n particles distributed on N sites,n N

Class collects all members that differ only by

•Translation

•Inversion

j

j

m number of configurations

w weight

j

For example

11

3

N

n

Enumeration of states

1 2 1 2 3

1 3 2 4 5

1 4 1 3 5

2 2 2

2 2

2 2

...

w w w k

w w w k

w w w k

etc

The solution of linear equations

• Exists and not unique•1,2, n dimensional manifold

• Exists and unique

• Doesn’t exist

( )g rThe weights have to be positive

Given

101 N001 R100 L111 N011 L110 R000 N

,j j

• 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

Displace 0 in the middle to the right if 0 is to the left and 1 is to the right

Displacements SRO in 1D Model with PBC

0

0

1

Displacement table

Displacements SRO in 1D Model with PBC

d1d

4= w

1+ w

3+ 2w

5= f

etc...

s1s

2= 2w

1+ 2w

2+ 2w

3= k

s1s

3= 2w

2+ 2w

4+ w

5= k

s1s

4= w

1+ 2w

3+ 2w

5= k

etc...

This puts additional restrictionsOn the possible configurations

The displacements in the real system must give an indication whether the particular local environment is realizable or not, i.e. restrict the higher order correlations in the system

Average correlations: point, pair, triplet

1

2 2

1

3

,1

1

1

1 2 1

1 4( 1 ) 2 1

1

i

N

ii

N

jj i i ji

N

j k i i j i ki

A

B

cN

c c c cN

N aq( ) = 1+ 2cAcB

Vq( )

kBT

æ

èç

ö

ø÷

-1

Real alloy system

46.5 53.5Fe Ni

Configurations with the same pair correlations

Vary triplet correlations

Compute displacements from first principles

Compare the displacements withMeasured values

1-st

2

1

1N

j i i ji

N

Configurations with triplet correlation variation

11 0.1AAAP 11 0.4AAAP

Fe 4.09e-4 Ni 1.70e-4 Fe 7.09e-4 Ni 0.70e-4

A Ni

Experiment

Fe 0.028 Ni 0.012

128 atomVASP

Displacements

ReverseMonte-Carlo

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.