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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
Diffuse Scattering
pf
kIkIeffkI
p
diffuseBragg
qp
rrk
qptotalqp
atom offactor scattering atomic -
2
,
*
kIkIkIkI TDSstaticSROdiffuse
Reciprocal SpaceReal Space
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
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