Bogdan Palosz

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Institute of High Pressure Physics Polish Academy
of Sciences Warsaw, Poland
Bogdan Palosz Svitlana Stelmakh
Effect of thermal atomic vibrations on total and
characteristic (Bragg) scattering of
nanocrystaline materials
European Powder Diffraction Conference,
EPDIC-10 Geneva, Switzerland, September 1-4,
2006. Session Total Scattering Analysis
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"Effect of thermal atomic vibrations on total and
characteristic Bragg scattering of nanocrystaline
materials Bogdan PALOSZ Svitlana
STELMAKH Institute of High Pressure Physics,
Polish Academy of Sciencesul. Sokolowska 29/37,
01-142 Warsaw Equilibrium conditions in
nano-crystalline phases are different than those
in large volume samples of the same composition.
Also, physical properties of nanomaterials
deviate significantly from those in the bulk
materials. Undoubtedly, a nano-crystal cannot be
considered as a fragment of a larger size solid
it is a unique piece of matter with the
properties specific for this individual object.
In a tentative model, a nanocrystal is a sort of
a two-phase system formed by the grain core and
the surface shell, both having their own
characteristic dimensions. Determination of
thermal expansion coefficients (anharmonic
thermal motions) and atomic temperature factors B
(harmonic motions) for crystalline materials is a
routine task. Neither of these parameters can be
determined for a nanocrystalline material based
on a conventional diffraction experiment.
Information on the effect of thermal vibrations
on the scattered intensity is contained in Bragg
intensities, but the same information is
contained also in TDS scattering. The Wilson
method uses only attenuation of Bragg
intensities. The alternate approach which is
proposed is based on a simultaneous (coupled)
examination of Wilson plots and TDS intensities.
Accounting for (examination of) TDS scattering
for complex structures is not an alternative to
the Wilson method but a necessity which has to be
a part of the basic elaboration procedure. The
Wilson method was originally proposed for
monoatomic structures, for which atomic
temperature factor is identical with the "overall
temperature factor", thus only for such
structures the Wilson method yields
unquestionable results. For polycrystalline
materials with complex structures containing
atoms vibrating with different amplitudes, like
in nanocrystals, Wilson plot is no longer a
straight line with a unique slope.
Determination of the atomic temperature factors
of such structures requires a simultaneous
analysis of attenuation of the Bragg intensities
and Thermal Diffuse Scattering in a large
diffraction vector range. The unique
interpretation of Wilson plots for crystals with
weak vibration component(s) requires measurements
performed up to a very large diffraction vector
Q, (gt 25 Å-1), larger for weaker vibrating
atoms. For nanocrystalline SiC and diamond two
different temperature atomic factors which
describe vibrations of the atoms in the grain
interior (Bcore) and at its surface (Bshell) were
determined experimentally. In situ measurements
were performed at LANSCE with HIPPO (from RT up
to 1100oC) and NPDF (from 15 K up to 225 oC)
instruments. Differences between atomic
vibrations in powders and nano-ceramics of
diamond and SiC sintered under high-temperature
high-pressures conditions, which are related to
transformation of free surfaces into grain
boundaries, will be discussed.
Acknowledgments Support of this work by the
Polish Ministry for Education and Science grant 3
T08A 066 28 and US Department of
Energy/LANL-LANSCE (project 2004135), and by NSF
(grant NSF DMR 0502136) is greatfully
acknowledged.
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• outline
• Specific problems of diffraction studies of
  nano-crystals
• Do we need examination of total scattering?
• Specific problems
• - separation of coherent and incoherent
  scattering
• inelastic scattering
• Application of Wilson method
• Experimental determination of Si and C
  temperature factors in SiC
• micro- and nano-crystalline

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specific problems of structural studies of
nano-crystals
relaxed lattice
?
?
it looks like a two-phase system !
under pressure
?
lattice parameters
no !
at different temperatures
?
B0(core) ? B0(shell)
BT(core) ? BT(shell)
?T(core) ? ?T(shell)
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questions faced by a crystallographer a
nano-crystalline material
core
core
core
shell
shell
shell
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thermal motions
thermal motions
thermal motions
thermal motions
thermal motions
an-harmonic atomic vibrations lead to
lattice expansion
they affect positions of Bragg reflections
harmonic vibrations do not change average atomic
positions, but they
affect Bragg intensities
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What kind of information we like to witdraw from
the diffraction data
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components of
experimental diffraction data
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examination of elastic scattering
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structural information contained in a
diffraction pattern
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effect of thermal atomic motions of a diffraction
pattern
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principles of evaluation of thermal atomic
motions in crystals with application of
diffraction techniques
Debye-Waller Factor
M B (sinQ/l)2
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effect of atomic thermal motions on a diffration
pattern
the components
the pattern
to be examined by Wilson method
?
?
?
?
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Wilson procedure of determination of overall
temperature factor B
division of reciprocal space into spherical
shells
experimental
theoretical (no motions)
Wilson plot
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5 nm SiC nanocrystal with a perfect lattice
calculated patterns
neutrons
static lattice
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effect of thermal atomic motions (disorder) on a
diffraction pattern
scattering amplitude f(Q)
S(Q)X-rays I(Q) / ltf(Q)gt2
intensity
ltf(Q)gt S ni fi (Q)
scattering amplitude b
S(Q)neutrons I(Q) / ltbgt2
structure function
ltbgt S ni bi
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the diffraction data elaboration
Mo
Cu
intensity I(Q)
structure function S(Q)
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Debye-Waller Factor vs overall temperature
factor
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a meaning of Debye-Waller Factor ?
Underneath the Bragg Peaks Egami Billinge
(2004)
D-WF exp (?ltu2gt Q2 ) exp?2B (sin?/?) 2
ua 0.05 Å ub 0.20 Å
the model of vibrations
ca 0.6 cb 0.4
D-WF ca exp (-ltua2gtQ2) cb exp (-ltub2gtQ2)
ltu2gt1/2 caua2 cbub21/2 0.132 Å (average)
D-WF refined back from attenuation of Bragg
intensities (as an overall temperature factor)
Mo
Cu
D-WF as a function of Q range when using a
single Gaussian to a two-component vibration
mode......... (calculated/refined from
attenuation of Bragg reflections)
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with use of coherent scattering Wilson
method how to identify and evaluate vibrations
of more than one amplitude
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coherent scattering
Underneath the Bragg Peaks Egami Billinge
(2004)
D-WF exp (?ltu2gt Q2 ) exp?2B (sin?/?) 2
incoherent scattering
TDS 1 - D-WF
real TDS
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model of incoherent scattering ?
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Einstein model TDS
first order TDS
transfer of scattered intensity from Bragg to a
diffuse TDS
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models of vibrations
Thermal Vibrations in Crystallography B.T.M.Willis
, A.W.Pryor Cambridge University Press, 1975
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the components
the pattern
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the components
the pattern
?
?
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TDS scattering
one atom structure TDS(Q) Nf(Q)2(1-exp(-Q2ltu2gt
) f(Q)2 b2 for neutrons
complex structure TDS(Q)total S ni
fi(Q)2(1-exp(-Q2ltui2gt)
ni atomic fraction
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Thermal Diffuse Scattering, TDS of complex
atomic systems
TDS(Q)total S ni fi(Q)2(1-exp(-Q2ltui2gt)
neutron scattering
TDS TDSC TDSSi
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application of Wilson method to examination
of atomic thermal motions in complex materials
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(No Transcript)
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experimental determination of Si and C
temperature factors in micro-crystalline SiC
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Si and C vibrations in micro-crystalline SiC at
50 K
Baverage S ni bi2 Bi S bi2
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Si and C vibrations in micro-crystalline SiC at
300 K
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Si and C vibrations in micro-crystalline SiC at
1373 K
ltUC2gt1/2 ltUSi2gt1/2
50K 0.037 Å 0.025 Å
300K 0.048 0.034
1373K 0.080 0.045
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BT(shell)
BT(core)
atomic temperature factor (s) of a
nano-crystalline material ?
?
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• model of vibrations in a nano-crystal
• 1. the core and shell atoms are represented by
  same
• average atom, ltSi-Cgt,
• the core and the surface shell atoms vibrate with
  
• different amplitudes

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Wilson plots for models of 10 nm SiC nano-crystal
with different ratio of core to shell
atoms vibrating with different amplitudes
shell
core
core atoms
U2
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U1
42
25
13
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Experimental Wilson plots of three
nano-crystalline SiC powders
at 300 K
ltu2gt1/2 core 0.045 Å
ltu2gt1/2 shell 0.107 Å
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what about inelastic scattering ?!
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do we need a correction for inelastic scattering
?
in principle
for amorphous materials - necessary !
for crystals - not needed
for nano-crystals - ? ? ?
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total elastic S(Q) intensity ?
total elastic intensity ? S(Q)
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total (elastic) S(Q) intensity ?
is there an extra intensity measured with the
elastic scattering ?
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what does one really measure in an experiment ?
needed !
measured
detectors
corrections
inelastic effects
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the shape of the total S(Q)
Placzek correction
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effect of presence of inealastic scattering on
Wilson plot (derived atomic temperature factors)
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experimental data
effect of presence of inealastic scattering on
Wilson plot (derived atomic temperature factors)
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do we need a correction for inelastic scattering
?
in principle
for amorphous materials - necessary !
for crystals - not needed (?) wellll..
for nano-crystals - ? ? ?
YES !
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Summary
Analysis of thermal atomic motions in
nanocrystals requires examination of Bragg
scattering together with TDS The core and
surface atoms vibrate with different
amplitudes but to measure this effect one needs
the diffraction data measured up to very large
diffraction vector Q, much above that which can
be reached with use of laboratory radiation
sources Analysis of atomic thermal motions is
potentially a very powerful tool which can serve
for examination of structural and elastic
properties of grain boundaries Inelastic
scattering brings important information on
non-crystallinity of nano-crystals
.............
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Collaboration
Stanislaw GIERLOTKA
Ewa GRZANKA
LANSCE, Los Alamos Thomas PROFFEN Sven VOGEL
TCU Waldek ZERDA BAE System -
Witold PALOSZ