Diffuse Scattering (print) - Spallation Neutron Source · 2019-06-27 · Measuring X-ray Diffuse...

Single Crystal Diffuse Scattering

Ray Osborn Neutron and X-ray Sca0ering GroupMaterials Science DivisionArgonne Na;onal Laboratory

Outline‣ What is diffuse scattering?

• What does it look like?• What causes it?• Who started it?

‣ What is it good for?• A random walk through disordered materials

‣ How do I model it?• A few equations• Rules of thumb

‣ Case Study 1: Diffuse scattering from vacancies in mullite‣ Case Study 2: 3D-ΔPDF in sodium-intercalated V2O5‣ How do I look at static disorder? Hint: Corelli

• Neutrons vs X-rays• Diffuse scattering with elastic discrimination

‣ Diffuse scattering - the musical

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National School on Neutron & X-ray Scattering - 2019


Diffuse Scattering

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Diffuse ScatteringDeviations from the Average Structure

Bragg ScatteringAverage Structure


Single Crystal Diffuse Scattering in 3D

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Simple Example of Disorder‣ In these examples, 30% of atoms (blue dots) have been replaced by vacancies (green dots)

• Left-Hand-Side: random substitution• Right-Hand-Side: high probability of vacancy clusters

- Thanks to Thomas Proffen

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Bragg Scattering‣ Bragg scattering is determined by the average structure.

• Since the average vacancy occupation is identical, both examples have identical Bragg peaks

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Diffuse Scattering‣ The diffuse scattering is quite different in the two examples

• Random vacancy distributions lead to a constant background (Laue monotonic scattering)• Vacancy clusters produce rods of diffuse scattering connecting the Bragg peaks

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An Ultra-Short History of Advances in Diffuse Scattering

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Yttria-Stabilized Zirconia

T. Proffen and T. R. Welberry J. Appl. Cryst. 31, 318 (1998)

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What is it good for?


Science Impacted by Diffuse Scattering

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‣ Subjects identified at the Workshop on Single Crystal Diffuse Scattering at Pulsed Neutron Sources

• Stripes in cuprate superconductors• Orbital correlations in transition metal oxides (including CMR)• Nanodomains in relaxor ferroelectrics• Defect correlations in fast-ion conductors• Geometrically frustrated systems• Critical fluctuations at quantum phase transitions• Orientational disorder in molecular crystals• Rigid unit modes in framework structures• Quasicrystals• Atomic and magnetic defects in metallic alloys• Molecular magnets• Defect correlations in doped semiconductors• Microporous and mesoporous compounds• Host-guest systems• Hydrogen-bearing materials• Soft matter - protein configurational disorder using polarization analysis of spin-incoherence• Low-dimensional systems • Intercalates• Structural phase transitions in geological materials

Diffuse Scattering from Metallic Alloys

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Short-range Order in Null Matrix 62Ni0.52Pt0.52

J. A. Rodriguez, S. C. Moss, J. L. Robertson, J. R. D. Copley, D. A. Neumann, and J. MajorPhys. Rev. B 74, 104115

0 2 40

2

4

0 qy

0 (r

.l.u.

) qx 0 0 (r.l.u)

Diffuse Scattering from a Fast-Ion Conductor

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National School on Neutron & X-ray Scattering - 2019M. T. Hutchings et al J. Phys. C 17, 3903 (1984)

CaF2

Diffuse Scattering from Molecular Solids

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T. R. Welberry et al J. Appl. Cryst. 36, 1400 (2003)

Diffuse Scattering from Relaxor Ferroelectrics

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T. R. Welberry et al J. Appl. Cryst. 38, 639 (2005)

Lead Zinc-Niobate

G. Xu, P. M. Gehring, G. Shirane, Phys. Rev. B 72, 214106 (2005).

Lead Magnesium-Niobate

E=0 E||[111]

Diffuse Scattering from Jahn-Teller Polarons

0 1-1-2 2

1.7

2.0

2.3

l

h

0

28

29

212

214

216

210

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4+4+ 4+3+

t2g

eg

ΔCF

ΔJTdMn

T = 115 K

d(3x2-r2)d(3y2-r2)

T = 300 K


Incommensurate Modulations in Sr0.5Ba0.5NbO6

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Acknowledgements: Bixia Wang and Daniel Phelan

Magnetic Diffuse Scattering from Geometric Frustration

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S.-H. Lee et al Nature 418, 856 (2002)

ZnCr2O4

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How do I model it?


A Few Equations

‣ Laue Monotonic Diffuse Scattering

‣ Cowley Short-Range Order

‣ Warren Size Effect

‣ Borie and Sparks Correlations

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I =X

i

X

j

bibjexp(iQ · rij)

I = b̄2X

ij

exp(iQ · rij) +N(b2 � b̄2); b̄2 = (cAbA + cBbB)2; b2 = cAcB(bB � bA)

2

J. M. Cowley, J. Appl. Phys. 21, 24 (1950)

I =X

i

X

j

bibjexp (iQ · (Ri �Rj))

1 + iQ · (ui � uj)�

1

2(Q · (ui � uj))

2 + ...

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V. M. Nield and D. A. Keen Diffuse Neutron Scattering From Crystalline Materials (2001) T. R. Welberry Diffuse X-ray Scattering and Models of Disorder (2004)

;Idiffuse = NcAcB(bB � bA)2 +P

ij ↵ijcBcA(bB � bA)2exp(iQ · rij)↵ij =

⇣1� Pij

cj

⌘

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Idiffuse = NcAcB(bB � bA)2 +P

ij ↵ijcBcA(bB � bA)2exp(iQ · rij)↵ij =

⇣1� Pij

cj

⌘

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;Idiffuse = NcAcB(bB � bA)2

0

@1 +X

ij

↵ijexp(iQ · rij) + �ijexp(iQ · rij)

1

A

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�ij = f(✏ijAA, ✏ijBB)

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Thermal Diffuse Scattering‣ Lattice vibrations produce deviations from the

average structure even in perfect crystals‣ X-ray scattering intensity is given by the integral

over all the phonon branches at each Q

�20


M. Holt, et al, Phys Rev Lett 83, 3317 (1999).

Some Rules of Thumb (thanks to Hans Beat Bürgi)

Reciprocal space

‣ Only sharp Bragg reflections

‣ Sharp diffuse rods

‣ Sharp diffuse planes

‣ Diffuse clouds

�21


Direct space

‣ 3D-periodic structure ‣ no defects

‣ 2D-periodic structure ‣ perpendicular to the streaks

‣ disordered in streak direc;ons

‣ 1D-periodic structure ‣ perpendicular to the planes ‣ disordered within the plane

‣ 0D-periodic structure ‣ no fully ordered direc;on

�22

Case Study 1: Mullite


Mullite - A Case Study‣ Mullite is a ceramic that is formed by adding O2+ vacancies to Sillimanite

• Sillimanite has alternating AlO4 and SiO4 tetrahedra• Mullite has excess Al3+ occupying Si2+ sites for charge balance

‣ This results in strong vacancy-vacancy correlations

�23


Sillimanite: Al2SiO5 Mullite: Al2(Al2+2xSi2-2x)O10-x

AlO6 Octahedra

AlO4 Tetrahedra

SiO4 Tetrahedra

B. D. Butler, T. R. Welberry, & R. L. Withers, Phys Chem Minerals 20, 323 (1993)

Measuring X-ray Diffuse Scatteringwith Continuous Rotation Method

�24


Incident Beam

Sample

Detector

Cryocooler

Measuring X-ray Diffuse Scatteringwith Continuous Rotation Method

�25


Pilatus 2M Detector

‣ The sample is continuously rotated in shutterless mode at 1° per second‣ A fast area detector (e.g., a Pilatus 2M) acquires images at 10 frames per second

• i.e., 3600 x 8MB frames ~ 30GB every 6 minutes ‣ The detector needs low background, high dynamic range, and energy discrimination

• Ideally, this is performed with high-energy x-rays, e.g., 80 to 100 keV

Experiment Workflow

‣ Powder calibration to determine detector distance, centers, and tilts‣ Bragg peak search to optimize the

sample/detector geometry ‣ Determine the orientation matrix‣ Perform coordinate transformation at each detector position‣ Merge the three transforms

�26



Diffuse Scattering in the Relaxor PbMg1/3Nb2/3O3

�27

3D Diffuse Scattering in Mullite

‣ There is strong diffuse scattering throughout reciprocal space‣ The shape of the diffuse scattering is strongly dependent on the value of Ql‣ There are incipient superlattice peaks at Q = 0.5 c* + 0.31 a*

�28


Ql=0.16 Ql=0.5 Ql=0.75

Monte Carlo Analysis‣ In a classic analysis, Richard Welberry and

colleagues developed a set of interaction energies to model mullite disorder

‣ Interaction energies were initialized:‣ insights from chemical intuition‣ insights from the measured diffuse scattering

‣ The diffuse scattering was calculated using a Monte Carlo algorithm to generate vacancy distributions first in 2D slices and then in 3D

�29


B. D. Butler, T. R. Welberry, & R. L. Withers, Phys Chem Minerals 20, 323 (1993)


Monte Carlo Analysis Results

�30


Vacancy Short-Range Order in MulliteA First-Principles Approach (ab initio HRMC)

�31

(c)

32

10-1-2-3

b

a3210-1-2-3

(d)(e)

(a) (b)


Nearly-Commensurate Vacancy Stripes in Mullite

�32

c = 0 c = 1.0q = ±1

2c⇤ ± 1

3a⇤

ξ ~ 20Å

�33

Case Study 2: Sodium-Intercalated V2O53D-ΔPDF


Pair Distribution Function Analysis

�34


Emil Bozin (ADD 2013)

Three-Dimensional Pair Distribution Functions

‣ The ability to measure three-dimensional S(Q) over a wide range of reciprocal space provides the 3D analog of PDF measurements.• Total PDFs if Bragg peaks and diffuse

scattering can be measured simultaneously

• Δ-PDFs if the Bragg peaks are eliminated- using the punch and fill method

‣ This would allow a model-independent view of the measurements in real space.

�35



“Punch and Fill”

�36

Symmetrize Punch Fill


Sodium-Intercalated V2O5

�37

Na

VOx

M. J. Krogstad, S. Rosenkranz, J. M. Wozniak, G. Jennings, J. P. C. Ruff, J. T. Vaughey, and R. OsbornarXiv 1902.03318 (submitted to Nature Materials).


Diffuse Scattering in Na0.2V2O5 and Na0.4V2O5

�38

300KNa0.4V2O5

100KNa0.4V2O5

Qk=0.5

Sublattice Melting

�39


x = 0.4 100Kx = 0.4 300Kx = 0.2 100K


3D-ΔPDF Analysis of NaxV2O5

�40


Real Space vs 3D-ΔPDF

�41

A AAB

BBC CC C


Order-Disorder Transition Viewed in Reciprocal and Real Space

�42

40% Na-intercalation250K150K50K

3D-ΔPDF: Importance of High Energy

• Expanding the Q-range enhances the real-space resolution

�43


CHESS – A2 – 55keV APS – 11-ID-D – 18.5keV


Back to Mullite

�44

c = 0

3

2

1

0

-1

-2

-3

b

a3210-1-2-3

�45

How do I look at static disorder?


Comparison of Elastic Scattering and the Static Approximation

‣ Reference: Roger Pynn, National School of Neutron and X-ray Scattering, 2018

�46


since

Importance of Elastic Discrimination

�47

National School on Neutron & X-ray Scattering - 2019 T. R. Welberry et al J. Appl. Cryst. 36, 1400 (2003)

Q = 4⇡sin✓

�

Measuring Large Volumes of Reciprocal Space Conventional Time-of-Flight Neutron Methods

�48


White Beam: efficient

NO energy discrimination

Fixed ki : energy resolved

NOT efficient

M(t)

Sample with : elastic scattering

TOF Laue Diffractometer •highly efficient data collection •wide dynamic range in Q

Statistical Chopper •elastic energy discrimination •optimum use of white beam

inelastic excitations

Cross Correlation Chopper

�49


16

S. Rosenkranz and R. Osborn, PRAMANA- Journal of Physics, 71, 705 (2008).


�50


Corelli

�51

Instrument Scientists Feng Ye Yaohua Liu

Instrument Proposers Stephan Rosenkranz Ray Osborn

Cross Correlation in Action

�52


Reconstructed Scattering FunctionCross Correlation

elastic

Raw Data

inelastic 19


Elastic Discrimination with Cross Correlation Benzil C14H10O2

�53

T. R. Welberry and R. Whitfield, Quantum Beam Science 2, 2 (2018)


�54

0 1 2 3 4 5 6 [H,0,0]

PMN, T=6K, HK0 plane

4

3

2

1

0

[0,K

,0]

0 1 2 3 4 5 6 [H,0,0]

PMN-20PT, T=6K, HK0 plane

4

3

2

1

0[0

,K,0

]

0 1 2 3 4 5 6 [H,0,0]

PMN-30OT, T=6K, HK0 plane

4

3

2

1

0

[0,K

,0]

0 1 2 3 4 5 6 [H,0,0]


4

3

2

1

0

[0,K

,0]

0 1 2 3 4 5 6 [H,0,0]


4

3

2

1

0

[0,K

,0]

0 1 2 3 4 5 6 [H,0,0]


4

3

2

1

0

[0,K

,0]

PMN-20PT PMN-30PT

PMN-35PT PMN-40PT PMN-50PT

PMN

M. J. Krogstad, P. M. Gehring, S. Rosenkranz, R. Osborn, F. Ye, Y. Liu, J. P. C. Ruff, W. Chen, J. M. Wozniak, H. Luo, O. Chmaissem, Z.-G. Ye, and D. Phelan, Nat Mater 48, 1 (2018).

Relaxor Ferroelectrics - Pb(Mg1/3Nb2/3)O3-xPbTiO3


Complementarity of Neutrons and X-raysPb(Mg1/3Nb2/3)O3-30%PbTiO3

�55

CHESS 55keV X-raysCorelli Neutrons

Sublattice Melting in Superionic Cu1.8Se

‣ It has been proposed that Cu2-xSe is a Phonon Liquid-Electron Crystal thermoelectric• zT > 1.5 at high temperature

�56



3D-ΔPDF from Cu1.8SeCorelli Data

�57

Symmetrized Corelli Data

c=0z=1/16cz=7/16cz=1/2c

Alex Rettie

The Future‣ High-Energy X-rays

• Absorption lengths similar to neutrons• Detectors now have sensors optimized for high energies, e.g. Pilatus 2M CdTe

‣ Micro-diffuse scattering• Benefiting from increased brightness of, e.g., APS Upgrade

‣ Increasing use of ab initio computational modeling• Allowing more complex systems to be investigated• Less dependence on intuition in modeling

‣ Enhanced analysis tools• Machine learning• Correlated data analysis• Easier co-refinement of neutrons

and x-rays

�58


A Few References

‣ T. R. Welberry & B. Butler, Chem Rev 95, 2369–2403 (1995).‣ F. Frey, Acta Cryst B 51, 592–603 (1995).‣ T. R. Welberry & D. J. Goossens, Acta Cryst A 64, 23–32 (2007).‣ D. A. Keen & A. L. Goodwin, Nature News 521, 303–309 (2015).

�59


Diffuse Scattering Song‣ Come eager young scholars - so tender and new

I’ll teach you diffraction - what I says mostly trueBetween the Bragg Peaks lies a world where you seeFluctuations and defects- they stand out plane-ly

‣ ChorusFor its dark as a dungeon between the Bragg peaksBut here in the darkness - each defect speaksIt gathers- from throughout- reciprocal spaceAnd re-distributes all over the place.

‣ Between the Bragg peaks - one thing that we seeIs TDS on our CCDIntensity totals are conserved- you can’t winIt steals from the Bragg peaks that stay very thin

‣ Substitutional alloys can cause quite a stirThe shorter the length scale the greater the blurWith care you can find out the bond length betweenEach atom pair type-the measurements clean

‣ Dislocations and other- type 2 defectsDestroy the Bragg peaks -they turn them to wrecksBut near the Bragg peaks- you still can seeIntense diffraction continuously

‣ Many -are- the defects you findBetween the Bragg peaks where others are blindSo go tell your friends and impress your bossYou’ve new understanding -with one hours loss

�60


Gene Ice

No defects

Defects of 1st kind

Defects of 2nd kind

θ

Krivoglaz Classifications

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