pdf_lecture
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Characterization of Materials
using the PDF
Thomas Proffen
Manuel Lujan Jr. Neutron Scattering Center
Los Alamos National Laboratory
tproffen@lanl.gov
LA-UR 05-0111
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Why do we care about the atomic structure?
The atomic structure has a profound influence on the
properties of materials. Consider carbon ...
Diamond
– hard
– transparent
– insulating
– expensive
Graphite
– soft
– black
– metallic
– cheap
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Bragg‟s world
The average atomic structure
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Bragg‟s world: Structure of crystals
Bragg‟s law
n 2d sin
Assumes periodicity
Average structure from Bragg
peak positions and intensities
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Bragg‟s world: Theory
The condition for a Bragg-peak to appear
is:
n 2d sin
or
Q k k ' K hkl
The intensity of the Bragg peak is given by
the square of the “Structure factor”:
FK bi e iQ .ri
i
The sum running over atoms in the unit
cell.
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Bragg‟s world: Powder Diffraction
(220)
(200)
Sample
All orientations of (111)
crystallites Incident beam
x-rays or neutrons
possible.
Powder Diffraction gives Scattering
on Debye-Scherrer Cones
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
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.
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
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.
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Bragg‟s world: Information beyond the average structure
Bragg profiles: size,
size distribution and shape
of crystallites, and strain.
Intensity along powder rings: texture
and preferred orientation.
Texture of Ti wire plate
(Lujan Center)
Accessible using modern Rietveld
refinement programs. From Ungár, et al, Carbon 40, 929 (2002)
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Diffuse scattering
Local atomic structure
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
The challenge of real materials: Knowing the local structure
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 Nanostructures: Science (290) 2000
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.
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Total scattering ?
Cross section of 50x50x50 u.c. model crystal consisting of 70% black atoms and 30% vacancies !
Properties might depend on vacancy ordering !!
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Bragg peaks are blind ..
Bragg scattering: Information about the average
structure, e.g. average positions, displacement
parameters and occupancies.
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Diffuse scattering to the rescue ..
Diffuse scattering: Information about two-body
correlations, i.e. chemical short-range order or
local distortions.
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
See http://www.totalscattering.org/teaching/
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
How about powder diffraction ?
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Finally the Pair Distribution Function (PDF)
The PDF is the
Fourier transform of
the total scattering
diffraction pattern !
Proffen, Z. Krist, 215,
661 (2000)
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
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
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
More theory ..
Rewrite the scattering factor equation
substituting Ra and change the order of
integration:
“Structure factor” “Form factor”
For neutrons:
d c 1 iQ.r r
bi b j e i j
and d N i , j
b b2
2
1 d c
S (Q)
b d
2 2
b
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Even more theory ..
The atomic pair distribution function, G(r) is the Fourier couple of S(Q):
2
G (r )
Q[S (Q) 1] sin QrdQ
0
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
What is a PDF?
Pair distribution function
4.26Å
(PDF) gives the probability of
2.84Å
finding an atom at a distance
“r” from a given atom.
1.42Å
2.46Å
3.76Å
4.92Å
5.11Å
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
What is a PDF?
Example:
C60 - „Bucky balls‟
Intra-domain
The PDF (similar to
the Patterson) is
Inter-domain
obtained via Fourier
transform of the
normalized total
scattering S(Q):
2
G(r )
Q[S (Q) 1] sin( Qr )dQ
0
Q 4 sin /
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Examples
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Local atomic strain in
ZnSe1-xTex
Simon Billinge
Thomas Proffen (LANL)
Peter Peterson (SNS)
Facilities: IPNS, Lujan
Funding: DOE, NSF
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
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 !
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
ZnSe1-xTex : Total scattering
Behaves like
average structure
Behaves like
Peterson et al., Phys. Rev. B63, 165211 (2001) local structure
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
ZnSe1-xTex : Nearest neighbors and Z-plots ..
Local
bond length
BLUE: XAFS from Boyce et al., J. Cryst.
Growth. 98, 37 (1989); RED: PDF results.
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
ZnSe1-xTex : Potential based “supercell” modeling
Kirkwood potential
a 1
( Lij L0ij ) (cos ijk ) 2
2
V
2 i, j 2 i , j ,k 3
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Local structure of WS2
Simon Billinge
Thomas Proffen (LANL)
Peter Peterson (SNS)
Valeri Petkov (CMU)
Facilities: Chess
Funding: DOE, NSF
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
WS2 : Structure of the “restacked” material
25
S
NSLS, =0.413 Å WS2 useful as a
20 lubricant, catalyst,
Pristine WS2
Intensity, a.u.
3
15 solid-state electrolyte.
2
10 1
0
Exfoliated and
5 40 50 60 70
restacked WS2 has a
W
0 metastable disordered
0 10 20 30 40 50 60 70 80 90
Bragg angle, 2 structure. Disorder
10
CHESS, =0.202 Å precluded a full
8 structural solution.
Intensity, a.u.
“Restacked” WS2
6
0.5
? 4
0.4
0.3
0.2
PDF can help …
2 0.1
0.0
40 50 60 70
0
10 20 30 40 50 60 70 80 90
Bragg angle, 2
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
WS2 : PDF to the rescue
3
S
2
G ( Å)
Pristine WS2:
-2
1
0 Hexagonal
P63/mmc
-1 W
0 2 4 6 8 10 12 14 16 18
D is ta n c e r (Å )
1 .5
1 .0
“Restacked” WS2:
G ( Å)
-2
0 .5
0 .0
Monoclinic
P1121
-0 .5 (disordered derivative of WTe2)
-1 .0
0 2 4 6 8 10 12 14 16 18
D is ta n c e r (Å )
Petkov et al., J. Am. Chem. Soc. 122, 11571 (2001)
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Domain structures
Katharine Page
Thomas Proffen
Facilities: Lujan
Funding: DOE, NASA
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
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
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Domain structures : Pair Distribution Function
M-M
M-M
D-D
r > Domain size:
r < Domain size:
NO D-D and M-M pairs
MainlyD-D contribution.
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Domain structures : R-dependent refinements
Top: Refinement of O=15%
single-phase model with
blue/red fractional
occupancies (O).
Bottom: Refinement of
same model for 5Å wide
sections.
O=29% O=16% O=15% O=15% O=15%
Extensions:
– Multi phase models
– Modeling of boundary
– R-dependent refinable
mixing parameters
Domain radius
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
High temperature
local structure of
LaMnO3
Xiangyun Qiu
Simon Billinge
Thomas Proffen
Facilities: Lujan
Funding: DOE-BES, NSF
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
LaMnO3 : Local structure vs. electronic state
JT orbitals are ordered at low-temperature in a checker-
board pattern:
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
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
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
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).
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
LaMnO3 : T-dependence of Mn-O bond distribution
Long-bonds
Short-bonds
Local structure Average structure
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).
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
LaMnO3 : Crossover from local to average structure
O
O'
Local
Average
Intermediate??? R
Varying range refinement
– Fix rmin
– Vary rmax
– x axis is rmax
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
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
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
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).
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
“Complete” Structure of
Gold Nanoparticles
Katharine Page
Thomas Proffen
Ram Seshadri
Tony Cheetham
Facilities: Lujan
Funding: DOE, NASA
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Au nanoparticles : Why PDF ?
2nm
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.
50 nm
This study: 50
Total = 148
Average grain size = 3.6 nm
40
– 5nm monodisperse Au nanoparticles
30
Number
– 1.5 grams of material 20
– Neutron measurements on NPDF 10
0
1 - 1.9 2 - 2.9 3 - 3.9 4 - 4.9 5 - 5.9 6 - 6.9
Grain size [nm]
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Au nanoparticles : Nano vs. bulk
100Å
Experimental PDFs of gold nanoparticles and bulk gold, measured on NPDF.
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Au nanoparticles : Structural refinements
PDF from nano- and 16
Au-capping layer distance (Au-S)
bulk gold refined using
PDFFIT. 12
Nanoparticles show 8
G(r) (Å-2)
“normal” gold
structure. 4
No indication of 0
surface relaxations.
-4
abulk < anano 2 3 4 5 6 7 8 9 10
r (Å)
10 K FCC Model
Indication of Au-cap
distances
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).
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Local structure in
sandstone
Katharine Page
Christina Herrera
Thomas Proffen
Sylvia McLain
Tim Darling
Jim TenCate
Facilities: Lujan
Funding: DOE, NSF
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
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
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
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.
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
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)
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
Elastic properties of
Bulk-Metallic-Glasses
Katharine Page
Thomas Proffen
Bjorn Clausen
Ersan Ustundag
Seung-Yub Lee
Facilities: Lujan
Funding: DOE, NSF
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
BMG : Properties
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
http://www.its.caltech.edu/~matsci/wlj/wlj_research.html
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
BMG : Experiment
The amorphous BMG matrix does
not give rise to Bragg peaks =>
PDF !
+90
° Experiment on SMARTS
The BMG is compressed along
Beam one axis, causing atoms along
the other to expand
Detector Banks at +90 and –90
degrees receive scattering from
separate distortions
-90°
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
BMG : Result
2.5
2
10 MPa
-90°
500 MPa
1.5
1500 MPa
G(r) (Å -2)
1
+90°
0.5
0
2.4 2.45 2.5 2.55 2.6 2.65 2.7 2.75 2.8 2.85
-0.5
r (Å)
Work in progress ..
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
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
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
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
Characterizing materials using the PDF – Thomas Proffen
7th Canadian Powder Diffraction Workshop, May 2007
