Electron Diffraction and the Structural Characterization of

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• Electron Diffraction and the Structural
  Characterization of
• Modulated and Aperiodic Structures

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Many crystalline materials are sensibly Daltonide
i.e. inflexible - locked to a lattice and in a
unique dominant free energy minimum in local
parameter space e.g. Si.
However not all materials are so boring. Enter
modulated structures!
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What do we mean by a modulated structure? The
standard answer is a material whose reciprocal
space exhibits sharp Bragg reflections in more
than 3-d
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In real space?
The differences between the average or parent
structure and the modulated structure for each
atom are given by finite, periodic functions of
the dot product of the modulation wave-vector q
with the position of this atom in the parent
structure I.e. with q.(r?t). These functions
are known as Atomic Modulation Functions or
AMFs. Knowledge of these (as well as the average
structure) is equivalent to refinement in the
case of modulated structures.
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Why restrict ourselves to only one modulation
wave-vector I.e. to sharp Bragg reflections,
albeit in (3n)-d?Are materials like these also
not modulated in a very real sense!?
There is a virtual continuum from conventional
3-d structures through (3n)-d incommensurately
modulated structures to materials that exhibit
sharp, highly structured diffuse intensity
distributions to virtually completely disordered
(random) structures ..?
.. We learn much more about the balance of
competing interactions/free energy terms giving
rise to complex crystalline structure in the
broadest sense from taking into account all
scattering and not just the average structure
Bragg reflections .. !?
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Why arent all crystalline materials describable
in conventional 3-d crystallographic terms?
Wrong question? What does the additional
scattering tell us about the balance of the
crystal chemical forces giving rise to our
structure? How is local crystal chemistry
reflected in reciprocal space?
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Outline of Talk
FeOF - a simple case study of order in
disorder Spectroscopically fully ordered but
3-d crystallographically disordered!
Structure factors and characteristic diffraction
signatures (extinction conditions, polarization
..) applied to FeOF and CaCu3Ti4O12
The various types of modulated structure and the
advantages of electron diffraction when seeking
to structurally characterize them.
Some conclusions
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Rutile type FeOF - a spectroscopically unique
(O3F3) Fe environment according to Mossbauer.
Conventional 3-d crystallography, however, says
there is no O/F ordering!! The resolution .. !?
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Structure factor expressions and characteristic
diffraction effects lt110gt rods of transverse
polarized diffuse intensity through hkl odd
reflections!?
Compositional and displacive modulation
waves ?f?(r?t) f?av ?q a?(q) exp 2?iq.(r?t) u?
(r?t) ?q e?(q) exp 2?iq.(r?t)
F(Gq) N?? f? exp(-2?iG.r?)? a?(q)
2?Gq.e?(q) ..
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Warning! Watch out for multiple scattering if
youre an electron microscopist and looking for
characteristic extinction conditions in diffuse
distributions
sometimes it pays to be off-axis!
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CaCu3Ti4O12, CCTO high ?, stable (above room
temperature)
F(Gq) N?? f? exp(-2?iG.r?)? 2?Gq.e?(q)
Structured diffuse through l even reflections
implies eTi2(q) eTi1(q) requires 1-d
correlated Ti shifts along lt001gt columns and
incipient ferroelectric behaviour
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Why is the TEM a good place to investigate
modulated structures?
very sensitive to weak subtle features of
reciprocal space, whether the modulations are
long or short range ordered!
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Can usually overcome problems associated with
micro-structure (pseudo-symmetry and twinning)
via either careful positioning of the incident
electron beam, micro-diffraction or imaging
e.g Sn1-xSb1x x 0.5
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e.g. Y0.33Sr0.67CoO3-?
Indexation without the subscript p is with
respect to the true C1c1, a 2ap  2cp, b
4bp, c 2ap  2cp (a 1/4 1,0,-1p, b
1/4 010 p, c 1/4 101p) supercell.
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If microstructure is too fine scale for
conventional electron diffraction, can use
micro-diffraction e.g Co2-xSe2, 0.06 lt x lt 0.25,
a (32)-d incommensurately modulated structure of
P-3m1, Cd(OH)2, average structure type.
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If finer scale still .. can use imaging e.g
(Ba1-xLax)2In2O5x, x 0.2
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Ni6Se5-xTex, x 0.5, Bmmb parent structure
Ordering of partially occupied Ni sites gives
rise to a q 1/2-101p (or 1/2101p)
modulation which is twinned perpendicular to c
on quite a fine scale
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Superspace symmetry characterization
e.g Sn1-xSb1x
-3m metric and Laue symmetry F(hklm)h 0 unless
-hkl 3J only condition Superspace group is
R-3m(0,0,1.311) (No.166.1)
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Co2-xSe2, 0.06 lt x lt 0.25.
Indexed with respect to the five basis vectors
M ap, bp, 1/3cp, q1 ?ap, q2 ?bp, ?
0.45-0.47 .
F(HKLMN) 0 unless L-MN 3J only condition
implies P x1, x2, x11/3, x4-1/3,
x51/3 Superspace group is P-3m1(??0,00,??0)11
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(1-x)Bi2O3.xNb2O5, x 0.2
F(HKLMNP) 0 unless HK, HL, KL, MN, MP, NP
2J plus F(HK0MN0) 0 unless MN 4J implies a
d hyperglide In conjunction with m3m Laue
symmetry Superspace group is PFm3mFd3m
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Atomic Modulation Functions e.g.
(1-x)Bi2O3.xNb2O5, 0.06 x 0.23

• ?fM (t1, t2, t3)/fMav
• aM(111) cos (2?(t1-?1)(t2-?2)(t3-?3))
• - cos (2? -(t1-?1)-(t2-?2)(t3-?3))
• cos (2? -(t1-?1)(t2-?2)-(t3-?3))
• cos (2? (t1-?1)- (t2-?2)- (t3-?3))
• .. higher order harmonics ..
• constant.
• t1 q1.(rMT) ?a.(rMT), t2 q2.(rMT)
• ?b.(rMT) and t3 q3.(rMT) ?c.(rMT)

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Interface modulated structures
For complete ordering on the local scale in real
space, compositional AMFs must of necessity take
an infinitely sharp (crenel type) form in
superspace which implies many higher order
harmonics or .. disorder .. (phason
fluctuations!) i.e variability of the spacing
between the boundaries in real space.
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Composite modulated structures
At first glance EDPs looks like conventional
modulated structure EDPs but composite
modulated structures exhibit characteristic
intensity asymmetries. Why? Two or more
intergrown mutually incommensurable
sub-structures.
Mutually intergrown I2/m framework and Ba
sub-structures with bBa 2/x bf, bBa x/2 bf
and I2/m (0,x/2,0)1, bBa x/2 bf. 2110 211
of the framework sub-structure, 2011 211 of the
Ba sub-structure etc.
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Where are the Sn atoms in LaSb2Snx, 0.1x0.75?
M aL,bL,cL, aS, cS 3/2 cL 0K0MN
reflns. correspond to MKN reflections of the Sn
sub-structure. Implies B-centred Sn
sub-structure with cS 2/3 cL
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Displacively modulated structures induced by
compositional ordering
Just because the scattering is dominated by
displacive effects doesnt mean there isnt an
underlying compositional origin!
Gives rise to transverse polarized Glthk1/3gt
sheets of diffuse and diffuse streaking along all
lth0lgt directions of reciprocal space.
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Pure displacively flexible framework
structures e.g silicas, perovskites, zeolites,
fresnoites etc.
Polyhedra are rigid but their relative
orientation is not - often gives rise to
incommensurately modulated structures at lower
temperatures arising from condensed RUM modes of
distortion.
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Evidence for inherent orientational flexibility
at higher temperatures
These sharp, continuous diffuse intensity
distributions map out the zero energy cost, or
zero frequency, Rigid Unit Mode (RUM) phonon
modes of distortion of these various inherently
flexible framework structures.
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Real space origin. Rigid Unit Mode (RUM) modes
of distortion?
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Usually ?? 0 or 1/2 for perovskites but .. in
e.g Sr0.63Ca0.37TiO3 or NaNbO3
Pbcm, v2 x v2 x 4 supercell i.e a ap bp, b
-apbp, c 4cp a 1/2110p, b
1/2-1,1,0p, c 1/4001p.
There are 2 types of satellite reflections in the
FOLZ ring 1) Gp 1/2,1/2,1/4p reflections
such as 14,1,1 and 1,14,1. Their lack of
azimuthal intensity variation shows that they are
associated with octahedral -- rotations around
001. 2) Gp 001/4p type reflections such
as 14,0,1 and 0,14,1 above. Associated with
displacements primarily along the 010
direction!
u?(r?t) ?q e?(q) exp 2?iq.(r?t) F(Gq) N??
f? exp(-2?iG.r?)? 2?Gq.e?(q)
Confirmed by neutron Rietveld refinement Chris
Howard
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Conclusions
.. Close inspection of a large and
ever-increasing number of (flexible) phases shows
that they are modulated in one form or another ..
.. The TEM is an extremely well-adapted
instrument for the detection of such modulated
structures, for the unambiguous indexation of
their reciprocal spaces, for the determination of
their superspace group symmetries and
(ultimately, in simpler cases) for the full
structural determination thereof ..
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Acknowledgements
Manu Perez-Mato Thomas Höche Gustaaf van
Tendeloo Chris Howard Lasse Norén Frank
Brink Yun Liu Valeska Ting - looking for a
post-doc soon Richard Welberry
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Atomic modulation Functions (AMFs)
e.g Co1Co1-xSe2, 0.06 lt x lt 0.25. The
compositional AMF describes the distribution of
Co-occupied and vacancy occupied sites in
alternate metal atom layers.
Superspace group is P-3m1(??0,00,??0)11 ????????
???????????? constrains the form of the
occupational AMF.
?fM(T mapnbppcp)/fMav a(-1,1)cos
2?(?a1/3c.t-?1)   cos 2?(-?a?b1/3c.t
?1-?2)  cos 2?(?b1/3c.t?2) ...
higher order harmonic terms ....
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e.g. Ba2TiGe2O8 (BTG)
Indexation wrt M a 1/21,-1,0p, b
1/2110p, c 1/2001p, q 0.62
b Cmm2(0,0.62,1/2)s00 implies a condensed q
0.31110p displacive RUM mode