Contents

1
Contents

• Introduction
• Superspace approach
• Metric consideration
• Symmetry in the superspace
• Basic modulation types
• Occupational (substitutional) modulation
• Positional modulation
• Composite structure
• Examples

2
Introduction
Experiment Additional diffraction spots
satellites are present in the diffraction pattern
of modulated crystals. Satellites are regularly
spaced but they cannot be indexed with three
reciprocal vectors. One or more additional,
modulation vectors have to be added to index all
diffraction spots.
Consequence Diffraction pattern has no more 3d
lattice character ? the basic property of crystal
3d translation symmetry is violated but in a
specific regular way.
3
Additional satellite diffractions are often as
sharp as main spots and can be integrated and
used to describe modulation of aperiodic
crystals. The term aperiodic crystal covers
modulated, composite crystals and quasicrystals.
The effect of positional modulation was
described first by Dehling, (1927)
Z.Kristallogr., 65, 615-631. Occupational
modulation was described later by Korekawa
Jagodzinski (1967), Schweitz.Miner.Petrogr.Mitt.,
47, 269-278. Composite crystals approach was
introduced by Makovicky Hide, Material Science
Forum, 100101, 1-100.
4
Structural analysis of modulated crystals is
based on theoretical works of P.M. de Wolff,
A.Janner and T.Janssen. Modulated structures
were understood for long time as a curiosity not
having some practical importance e.g. Na2CO3
sodium carbonate. The number of studied modulated
crystals grew with improving of experimental
facilities imaging plate, CCD. Moreover the
real importance of modulations in crystals has
been demonstrated by studies of organic
conductors and superconductors (e.g.
(BEDT-TTF)2I3 ) and high temperature Bi
superconductors. The modulation can be present
even in very simple compounds as oxides PbO,
U4O9, Nb2Zrx-2O2x1.
5
In 1999 two papers reported incommensurate
composite character of pure metals under high
pressure Nelmes, Allan, McMahon Belmonte,
Phys.Rew.Lett. (1999), 83, 4081-4084. Barium IV.
6
Schwarz, Grzechnik, Syassen, Loa Hanfland,
Phys.Rew.Lett. (1999), 83, 4085-4088. Rubidium
IV.
Theory of aperiodic crystals and computer program
Jana2000 has been later applied to solve and
refine several analogical structures.
7
Superspace approach - Metric considerations
Additional diffraction spots
modulation vector q can be expressed as a linear
combination of
?
It is rather difficult to prove irrationality
only from measured values. The higher the
denominators the smaller difference between
commensurate and incommensurate approach. But
there are clearly distinguished cases 1/2,1/3
where commensurability plays an important role.
8
Superspace
Modulated and composite crystals can be described
in a (3d) dimensional superspace. The theory
has developed by P.M.DeWolff, A.Janner and
T.Janssen (Aminoff prize 1998). This theory
allows to generalize concept of symmetry and
also to modify all method used for structure
determination and refinement of aperiodic
crystals. The basic idea is that a real
diffraction pattern can be realized by a
projection from the (3d) dimensional
superspace
9
e
q
10

• The important assumption is that all satellites
  are clearly
• separated. This is true for the
  commensurate case or for the
• incommensurate case when the intensities
  diminish for
• large satellite index.
• 2. The additional vector e is perpendicular to
  the real space
• and plays only an ancillary role.
• All diffraction spots form a lattice in the
  four-dimensional
• superspace ? there is a periodic generalized
  (electron) density
• in the four-dimensional superspace.
• Reciprocal base

Direct base
11
Then the generalized density fulfill the periodic
condition
and therefore it can be expressed as a
4-dimensional Fourier series
where
From the definition of the direct and reciprocal
base it follows
12
Conclusion A real 3d density can be found as a
section through the generalized density.
13
Example positionally modulated structure
14
Symmetry in the superspace
Basic property of 3d dimensional crystal -
generalized translation symmetry
Trivial symmetry operator - translation symmetry
15
The rotational part of a general symmetry element
1. The right upper part of the matrix is a
column of three zeros. It is consequence of the
fact that the additional ancillary vector e
cannot be transformed into the real space.

• From the condition that symmetry operator has to
  conserve
• scalar product it follows

3.
These conditions show that any superspace group
is a four-dimensional space group but on the
other hand not every four-dimensional space
group is a superspace group. The superspace
groups are 31 reducible.
This allows to derive possible rotations and
translation is same way as for 3d case.
16
Examples
   1. Inversion centre
There is no modulation for the second case and
therefore the inversion centre has to have
17
2. Two-fold axis along z direction
18
3. Mirror with normal parallel to z direction
monoclinic axial case
monoclinic planar case
19
Translation part Symbol
Reflection condition for
20
Super-space group symbols
There are three different notation
The rational part of the modulation vector
represents an additional centring. It is much
more convenient to use the centred cell instead
of the explicit use of the rational part of the
modulation vector.
?
21
Basic modulation types Occupational
(substitutional) modulation
One harmonic wave
22
Form factor of changes from
to
Only main reflections and first order satellites
will appear
23
Occupational modulation only 1st
harmonic Fourier map
24
Occupational modulation only 1st
harmonic Diffraction pattern
25
Occupational modulation 1st and 2nd
harmonic Fourier map
26
Occupational modulation 1st and 2nd
harmonic Diffraction pattern
27
Crenel line modulation
28
Occupational modulation crenel function Fourier
map
29
Occupational modulation crenel
function Diffraction pattern
30
Positional modulation - longitudinal
Modulation vector
Modulation wave
31
Positional modulation longitudinal 1st harmonic
0.1Ĺ Fourier map
32
Positional modulation longitudinal 1st harmonic
0.1Ĺ Diffraction pattern
33
Positional modulation longitudinal 1st harmonic
0.5Ĺ Fourier map
34
Positional modulation longitudinal 1st harmonic
0.5Ĺ Diffraction pattern
35
Positional modulation - transversal
Modulation vector
Modulation wave
36
Positional modulation transversal 1st harmonic
0.5Ĺ Fourier map
37
Positional modulation transversal 1st harmonic
0.5Ĺ Diffraction pattern l0
38
Positional modulation transversal 1st harmonic
0.5Ĺ Diffraction pattern l1
39
Composite structure
40
Composite structure without modulation Fourier
map
41
Composite structure without modulation Diffracti
on pattern
42
Composite structure with modulation Fourier map
43
Composite structure with modulation Diffraction
pattern
44
Modulation Functions
The periodic modulation function can be expressed
as a Fourier expansion
45
The necessary number of used terms depends on the
complexity of the modulation function. The
modulation can generally affect all structural
parameter occupancies, positions and atomic
displacement parameters (ADP). The set of
harmonic functions used in the expansion fulfils
the orthogonality condition, which prevents
correlation in the refinement process. In many
cases the modulation functions are not smooth and
the number of harmonic waves necessary for the
description would be large. In such cases
special functions or set of functions are used to
reduce the number of parameters in the
refinement.
46
Hexagonal perovskites Sr1.274CoO3 and
Sr1.287NiO3 M. Evain, F. Boucher, O. Gourdon,
V. Petrícek, M. Dušek and P.Bezdícka,
Chem.Matter. 10, 3068, (1998).
47
The strong positional modulation of oxygen atoms
can be described as switching between two
different positions. This makes octahedral or
trigonal coordination of the central Ni/Sr atom
and therefore it can have quite different atomic
displacement parameters. The regular and
difference Fourier through the central atom
showed that a modulation of anharmonic
displacement parameters of the third order are
to be used.
48
Sr at octahedral site
Sr at trigonal site
49
Ni at octahedral site
Ni at trigonal site
50
Special modulation function
Crenel function
V.Petrícek, A.van der Lee M. Evain, ActaCryst.,
A51, 529, (1995).
Fourier transformation ?
51
Example TaGe0.354Te F. Boucher, M. Evain V.
Petrícek, Acta Cryst.,B52, 100, (1996). The Ge
position is either fully occupied or empty
This is typical map for crenel like occupational
wave.
52
Te atom is also strongly modulated but the
modulation is positional
53
Difference Fourier shows that the continuous
function does not describe real modulation
completely.
54
Splitting of the modulation wave into two parts
each circumscribed by crenel function allows to
account for discontinuity
55
The superspace approach allows to analyze
behaviour of atoms in in the modulated structure.
But it is rather cumbersome to present the result
in this form to non-specialists. Therefore we
should make some 3d pictures showing how the
modulation affects arrangement of atoms in the
real 3d space.
Average structure
56
Only occupational modulation
Final result
57
Saw-tooth function
Bi2Sr2CaCu2O8 - V.Petrícek, Y.Gao, P.Lee
P.Coppens, Phys.Rew.B, 42, 387-392, (1990)
Oxygen atom at Bi layer
58
The displacement u is a linear function of x4
coordinate
for
not occupied
59
Fourier transform
where
The saw-tooth modulation changes the original
periodicity and it can indicate some composite
character of the compound.
60

• Main characteristics of Jana2000
• it can be installed on PC under Windows (W98, NT,
  XP)
• and on most of UNIX machines
• it is written mainly in Fortran. C language is
  used just to make
• connection to basic graphic functions
• it uses own graphic objects which makes the
  program almost
• independent of the used system
• applicable for regular, polytype, modulated and
  composite
• structures
• superspace approach for modulated structures even
  for
• commensurate cases
• allows to make data reduction and merging data
  from different
• diffractometers (but not different radiation
  types)
• Fourier maps (up to 6d), p.d.f., j.p.d.f.,
  deformation maps
• distance calculation and distance plots up to 6d

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• twinning up to 18 twin domains, meroedry,
  pseudo-meroedry,
• twin index 1 or different from 1, overlap of
  close satellites
• Rietveld refinement multiphase up to 6d
• charge density studies only 3d 
• symmetry restrictions following from a site
  symmetry can be
• applied automatically for most refined
  parameters
• restrains of distances and angles
• rigid-body option to reduce number of regular
  and/or modulated
• parameters, TLS tensors, local
  non-crystallographic symmetry
• CIF output for regular, modulated structures
  refined either from
• single or powder data

62
JANA2000 for powders
M. Dušek, V.Petrícek, M.Wunschel, R.E.Dinnebier
and S. van Smaalen, J. Appl. Cryst. (2001), 34,
398-404.
JANA2000 allows to Rietveld refinement against
powder diffraction data. All features
(modulated structures, rigid body option, ADP,
...) of Jana2000 are usable. It provides a
state-of-the-art description of the peak profiles.
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