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### Characterization of Materialsusing the PDF

Thomas Proffen

Manuel Lujan Jr. Neutron Scattering Center

Los Alamos National Laboratory

LA-UR 05-0111

Why do we care about the atomic structure?

- Diamond
- hard
- transparent
- insulating
- expensive
- Graphite
- soft
- black
- metallic
- cheap

The atomic structure has a profound influence on the properties of materials. Consider carbon ...

Bragg’s world

The average atomic structure

Bragg’s world: Structure of crystals

Bragg’s law

- Assumes periodicity
- Average structure from Bragg peak positions and intensities

Bragg’s world: Theory

The condition for a Bragg-peak to appear is:

or

The intensity of the Bragg peak is given by the square of the “Structure factor”:

The sum running over atoms in the unit cell.

Bragg’s world: Powder Diffraction

(220)

(200)

(111)

Sample

Incident beam

x-rays or neutrons

All orientations of crystallites possible.

Powder Diffraction gives Scattering on Debye-Scherrer Cones

Rietveld refinement technique

Ic = Io{SkhF2hmhLhP(Dh) + Ib}

Io - incident intensity - variable for fixed 2Q

kh - scale factor for particular phase

F2h - structure factor for particular reflection

mh - reflection multiplicity

Lh - correction factors on intensity - texture, etc.

P(Dh) - peak shape function – includes instrumental resolution,

crystallite size, microstrain, etc.

Structure from powder diffraction

- Determination of the atomic structure using diffraction has revolutionized our knowledge about how materials work ..

- Zn insulin structure (> 1600 atoms in unit cell) determined from powder diffraction data (R.B. van Dreele)
- Average structure determined using Bragg reflections.

Bragg’s world: Information beyond the average structure

- Bragg profiles: size,size distribution and shapeof crystallites, and strain.
- Intensity along powder rings: texture and preferred orientation.
- Accessible using modern Rietveld refinement programs.

Texture of Ti wire plate

(Lujan Center)

From Ungár, et al, Carbon40, 929 (2002)

Diffuse scattering

Local atomic structure

The challenge of real materials: Knowing the local structure

Nanostructures: Science (290) 2000

- Traditional crystallographic approach to structure determination is insufficient or fails for
- Disordered materials: The interesting properties are often governed by the defects or local structure !
- Nanostructures: Well defined local structure, but long-range order limited to few nanometers (-> badly defined Bragg peaks)
- A new approach to determine local and nano-scale structures is needed.

Total scattering ?

Cross section of 50x50x50 u.c. model crystal consisting of 70% black atoms and 30% vacancies !

Properties might depend on vacancy ordering !!

Bragg peaks are blind ..

Bragg scattering: Information about the average structure, e.g. average positions, displacement parameters and occupancies.

Diffuse scattering to the rescue ..

Diffuse scattering: Information about two-body correlations, i.e. chemical short-range order or local distortions.

Finally the Pair Distribution Function (PDF)

- The PDF is the Fourier transform of the total scattering diffraction pattern !

Proffen, Z. Krist, 215, 661 (2000)

Theory again – no periodicity this time !

Elastic Scattering amplitude (from quantum mechanics)

The potential is given by

Where the sum is over all atoms in the sample and

More theory ..

“Form factor”

“Structure factor”

Rewrite the scattering factor equation substituting Ra and change the order of integration:

For neutrons:

and

Even more theory ..

The atomic pair distribution function, G(r) is the Fourier couple of S(Q):

What is a PDF?

4.26Å

2.84Å

1.42Å

2.46Å

3.76Å

4.92Å

5.11Å

Pair distribution function (PDF) gives the probability of finding an atom at a distance “r” from a given atom.

What is a PDF?

Intra-domain

Inter-domain

Example:

C60 - ‘Bucky balls’

The PDF (similar to the Patterson) is obtained via Fourier transform of the normalized total scattering S(Q):

Local atomic strain in ZnSe1-xTex

Simon Billinge

Thomas Proffen (LANL)

Peter Peterson (SNS)

Facilities: IPNS, Lujan

Funding: DOE, NSF

ZnSe1-xTex : Structure

- Zinc blend structure (F43m)
- Technological important : Electronic band gap can be tuned by the composition x.
- Bond length difference Zn-Se and Zn-Te strain.
- Local structural probe required !

¯

ZnSe1-xTex : Total scattering

Behaves like

average structure

Behaves like

local structure

Peterson et al., Phys. Rev. B63, 165211 (2001)

ZnSe1-xTex : Nearest neighbors and Z-plots ..

BLUE: XAFS from Boyce et al., J. Cryst. Growth. 98, 37 (1989); RED: PDF results.

Local

bond length

Average

bond length

ZnSe1-xTex : Potential based “supercell” modeling

Kirkwood potential

Local structure of WS2

Simon Billinge

Thomas Proffen (LANL)

Peter Peterson (SNS)

Valeri Petkov (CMU)

Facilities: Chess

Funding: DOE, NSF

WS2 : Structure of the “restacked” material

S

Pristine WS2

W

“Restacked” WS2

?

- WS2 useful as a lubricant, catalyst, solid-state electrolyte.
- Exfoliated and restacked WS2 has a metastable disordered structure. Disorder precluded a full structural solution.
- PDF can help …

WS2 : PDF to the rescue

S

Pristine WS2:

Hexagonal

P63/mmc

W

“Restacked” WS2:

Monoclinic

P1121

(disordered derivative of WTe2)

Petkov et al., J. Am. Chem. Soc. 122, 11571(2001)

Domain structures : Simulated example

- Simulated structure of 20x20x20 unit cells.
- Matrix (M): blue atoms
- Domains (D): red atoms, spherical shape, d=15Å.
- Simulated using DISCUS.

Proffen & Page, Z. Krist. (2004), in press

Domain structures : Pair Distribution Function

r > Domain size:

NO D-D contribution.

r < Domain size:

Mainly D-D and M-M pairs

M-M

M-M

D-D

Domain structures : R-dependent refinements

- Top: Refinement of single-phase model with blue/red fractional occupancies (O).
- Bottom: Refinement of same model for 5Å wide sections.
- Extensions:
- Multi phase models
- Modeling of boundary
- R-dependent refinable mixing parameters

O=15%

O=29%

O=16%

O=15%

O=15%

O=15%

Domain radius

High temperature

local structure of

LaMnO3

Xiangyun Qiu

Simon Billinge

Thomas Proffen

Facilities: Lujan

Funding: DOE-BES, NSF

LaMnO3 : Local structure vs. electronic state

- JT orbitals are ordered at low-temperature in a checker-board pattern:

LaMnO3 : Crystallography

Rhombohedral

No JT distortion

Less-Orthorhombic-O‘

Virtually no JT distortion

Orthorhombic-O

Large JT distortion

JT distortion disappears at the O-O’ transition

LaMnO3 : T-dependence of Mn-O bond distribution

- Two Mn-O peaks persist up to the highest T measured
- Thermal broadening appears to be the ONLY contributor to peak profile changes
- Local JT distortion exists in both high T orthorhombic (pseudo-cubic) and rhombohedral phase
- Two Gaussian curves fit the data very well

Xiangyun Qiu, Th. Proffen, J. F. Mitchell and S. J. L. Billinge, Phys. Rev. Lett.94, 177203 (2005).

LaMnO3 : T-dependence of Mn-O bond distribution

- Mn-O bond lengths are invariant with temperature, right up into the R-phase
- JT distortions persist locally in the pseudocubic phase
- Agrees with XAFS result: M. C. Sanchez et al., PRL (2003).

Long-bonds

Short-bonds

Average structure

Local structure

LaMnO3 : Crossover from local to average structure

- Varying range refinement
- Fix rmin
- Vary rmax
- x axis is rmax

O

O'

Local

Average

Intermediate???

R

LaMnO3 : Crossover from local to average structure

- Assume the PDF “form-factor” for a sphere
- Take asymptotic values to be low-r result from peak fitting and the high-r result from Rietveld
- Three curves are self-consistently fit with one parameter – the diameter of the spherical domain

LaMnO3 : T-dependence of orbital clusters from PDF

rmax(Ǻ)

- Diameter of orbitally ordered domains above TJT is 16Ǻ
- Appears to diverge close to TJT
- Red lines are a guide to the eye (don’t take the fits too seriously!)

Xiangyun Qiu, Th. Proffen, J. F. Mitchell and S. J. L. Billinge, Phys. Rev. Lett.94, 177203 (2005).

“Complete” Structure of Gold Nanoparticles

Katharine Page

Thomas Proffen

Ram Seshadri

Tony Cheetham

Facilities: Lujan

Funding: DOE, NASA

Au nanoparticles : Why PDF ?

2nm

50 nm

- Nanoparticles often show different properties compared to the bulk.
- Difficult to study via Bragg diffraction (broadening of peaks).
- PDF reveals “complete” structural picture – core and surface.
- This study:
- 5nm monodisperse Au nanoparticles
- 1.5 grams of material
- Neutron measurements on NPDF

Au nanoparticles : Nano vs. bulk

100Å

Experimental PDFs of gold nanoparticles and bulk gold, measured on NPDF.

Au nanoparticles : Structural refinements

- PDF from nano- and bulk gold refined using PDFFIT.
- Nanoparticles show “normal” gold structure.
- No indication of surface relaxations.
- abulk < anano
- Indication of Au-cap distances

Au-capping layer distance (Au-S)

K.L. Page, Th. Proffen, H. Terrones, M. Terrones, L. Lee, Y. Yang, S. Stemmer, R. Seshadri and A.K. Cheetham, Direct Observation of the Structure of Gold Nanoparticles by Total Scattering Powder Neutron Diffraction, Chem. Phys. Lett. , accepted (2004).

Local structure in sandstone

Katharine Page

Christina Herrera

Thomas Proffen

Sylvia McLain

Tim Darling

Jim TenCate

Facilities: Lujan

Funding: DOE, NSF

Sandstone: Crystalline quartz ?

- Measured on NPDF
- High statistics data (24 hrs)
- Solid rock sample
- Ambient conditions – sealed to avoid taking up of water
- Motivation: Structural explanation for non-linear acoustic properties

Sandstone: Local structure

- Refinement of single phase quartz model.
- Good agreement above r > 3Å.
- Missing “intensity” in first two PDF peaks corresponding to Si-O and O-O NN distances.

Sandstone: Local structure

- Refinement of two phases :
- Crystalline quartz
- “Amorphous” quartz up to 3Å
- Good agreement over complete range
- Amorphous regions “stress formed” by point like contacts at grain contacts ?

K.L. Page, Th. Proffen, S.E. McLain, T.W. Darling and J.A. TenCate,Local Atomic Structure of Fontainebleau Sandstone: Evidence for an Amorphous Phase ?, Geophysical Research Lett.31, L24606 (2004)

Elastic properties of

Bulk-Metallic-Glasses

Katharine Page

Thomas Proffen

Bjorn Clausen

Ersan Ustundag

Seung-Yub Lee

Facilities: Lujan

Funding: DOE, NSF

BMG : Properties

http://www.its.caltech.edu/~matsci/wlj/wlj_research.html

- High Specific Strength
- Light Weight
- High Elastic Strain
- High Hardness
- Excellent Wear Resistance
- Excellent Corrosion Resistance
- BMG’s are prone to catastrophic failure during unconstrained loading due to the formation of macroscopic shear bands
- Crystalline reinforcements to suppress the formation of macroscopic shear bands

BMG : Experiment

+90°

Beam

BMG

-90°

- The amorphous BMG matrix does not give rise to Bragg peaks => PDF !
- Experiment on SMARTS
- The BMG is compressed along one axis, causing atoms along the other to expand
- Detector Banks at +90 and –90 degrees receive scattering from separate distortions

BMG: Phase analysis on composite sample

- Ability to distinguish between phases
- Difference between measured composite PDF and calculated Tungsten PDF agrees well with measured BMG PDF

Summary and more information

- Analysis of total scattering gives valuable insight in structure properties relationship
- High-resolution instruments open the door to medium-range order investigations
- Obtain structural information from disordered crystalline, amorphous of composite materials
- Fast powder measurements allow systematic exploration of local structure as function of T, x, P

http://www.totalscattering.org

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