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, modulationvectors 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.

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, 100&101, 1-100.

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-2O2x+1.

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.

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.

..... all rational commensurate structure ..... at least one irrational incommensurate structure 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.

Superspace Modulated and composite crystals can be described in a (3+d) 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 (3+d) dimensional superspace:

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 :

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:

But the real diffraction pattern is a projection of the 4d pattern into R3 which means that the term has to be constant. Conclusion : A real 3d density can be found as a section through the generalized density.

R3 Example : positionally modulated structure

 basic property unitary operator matrix representation Symmetry in the superspace Basic property of 3+d dimensional crystal - generalized translation symmetry : Trivial symmetry operator - translation symmetry :

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 3+1 reducible. This allows to derive possible rotations and translation is same way as for 3d case.

Examples 1. Inversion centre There is no modulation for the second case and therefore the inversion centre has to have

monoclinicplanar case monoclinic axial case 2. Two-fold axis along z direction

3. Mirror with normal parallel to z direction monoclinic axial case monoclinicplanar case

Translation part Symbol Reflection condition for

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. 

Basic modulation types Occupational (substitutional) modulation One harmonic wave

Form factor of changes from to Only main reflections and first order satellites will appear:

Occupational modulation – only 1st harmonic Fourier map

Occupational modulation – only 1st harmonic Diffraction pattern

Occupational modulation – 1st and 2nd harmonic Fourier map

Occupational modulation – 1st and 2nd harmonic Diffraction pattern

Crenel line modulation

Occupational modulation – crenel function Fourier map

Occupational modulation – crenel function Diffraction pattern

Positional modulation - longitudinal Modulation vector : Modulation wave :

Positional modulation longitudinal 1st harmonic 0.1Å Fourier map

Positional modulation longitudinal 1st harmonic 0.1Å Diffraction pattern

Positional modulation longitudinal 1st harmonic 0.5Å Fourier map

Positional modulation longitudinal 1st harmonic 0.5Å Diffraction pattern

Positional modulation - transversal Modulation vector : Modulation wave :

Positional modulation transversal – 1st harmonic 0.5Å Fourier map

Positional modulation transversal – 1st harmonic 0.5Å Diffraction pattern l=0

Positional modulation transversal – 1st harmonic 0.5Å Diffraction pattern l=1

Composite structure

Composite structure – without modulation Fourier map

Composite structure – without modulation Diffraction pattern

Composite structure – with modulation Fourier map

Composite structure – with modulation Diffraction pattern

R3 Modulation Functions The periodic modulation function can be expressed as a Fourier expansion:

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.

Hexagonal perovskites Sr1.274CoO3 and Sr1.287NiO3 M. Evain, F. Boucher, O. Gourdon, V. Petříček, M. Dušek and P.Bezdíčka, Chem.Matter.10, 3068, (1998).

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.

Sr at octahedral site Sr at trigonal site

Ni at octahedral site Ni at trigonal site

Special modulation function Crenel function V.Petříček, A.van der Lee & M. Evain, ActaCryst., A51, 529, (1995). Fourier transformation 

Contents

Contents

Presentation Transcript

Contents

Contents

Contents

Contents

Contents

Contents

Contents

CONTENTS

Contents

Contents