Modulation and its Crystallographic Methodology II

Václav Petříček and Michal Dušek

Institute of Physics

Acadamy of Sciences of the Czech Republic

Praha

Contents

• Structure analysis of modulated crystalsSolution of the structureHeavy atom methodRefinement of modulated crystals

• Jana2000Main characteristisJana2000 for powders

Solution of modulated structures

The solution of modulated structures is usually divided into two steps.

Solution of the average structure

Only main reflections are used. Standard techniques can be applied, direct or heavy atom methodby using standard programs – SIR, SHELX, … Strongly modulated atoms have usually large atomic displacement parameters or they must be refined as split atoms.Such “wrong” atoms are good candidates to be used as modulatedones when we are start with solution of the modulated structure.

Solution of the modulated structure

Weak modulations < 0.1 Å

The refinement can be started just from small randomly chosen displacements of already known atoms of the basic structure.The similar situation as when we are going from isotropic atomic displacement parameter to anisotropic ones

Strong modulations > 0.1 Å

A special techniques such as heavy atom or direct methods in superspace are necessary to get proper starting phasesfor satellite reflection.

Direct methods in superspace

Q. Hao, Y.-W. Liu & Fan Hai-Fu, Acta Cryst, A43, 820 (1987)Fan Hai-Fu, S. van Smaalen, E.J.W. Lam & P.T. Buerskens, Acta Cryst, A49, 704 (1993)

Program DIMS written by Fan Hai-Fu

Heavy atom method in superspace

W. Steurer, ActaCryst., A49, 704 (1987).V. Petříček, Aperiodic’94, World Scientific, 388, (1995).J. Peterková, M. Dušek V. Petříček & J. Loub Acta Cryst. B54, 809 (1998).

Program JANA2000 written by V. Petříček and M.Dušek.

Application of heavy atom method to AsKF4(OH)2

superspace group a=4.818, b=16.001, c=6.374 Å, =99.360 Amplitudes of modulations ~ 0.3 ÅThe anion is modulated in the first approximation as a rigid body

1. Solution and refinement of the average structure

Composed from the octahedral anion [AsF4(OH)2]- and

The cation K+ both located at m

2. (3+1) dimensional Patterson map for As atom

Symmetrical maximum at between the original position and the position related by the two fold axis or by the inversion center.

02 cC

21,2,0 y

Modulation function restricted by site symmetry to :

413303

412202

411101

2sin

2cos

2sin

xuxx

xuxx

xuxx

s

c

s

013.012 cu 025.013 su

0.011 su no significant modulation visible in x-section

025.0013.0 1312 sc uuTwo possibilities :

Model map at x4=0.25 based on Fcalc

025.013 su 025.013 su

Real map at x4=0.25

The modulation of the heavy atom was included into the refinement and the subsequent Fourier synthesis allows to find estimation of modulation waves.

3. Fourier synthesis based on known modulation of heavy atom

For F atom

Refinement of modulated structures

kinematical theory of diffraction of modulated crystals the integrated intensity of diffractions is proportional to the square of the generalized structure factor :

H

H HRR 2exp)(~ F

Numerical methodsGaussian integration A.Yamamoto REMOSFFT integration W.Paciorek MSR

Analytical Bessel function expansion V.Petříček JANAGeneralized Bessel function W.Paciorek MSR

Modulation Functions

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

n

nc

n

ns nxAnxAAxp 4,4,04 2cos2sin

R3

eA 4

1A

1a

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 belarge. 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 describedas 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 displacementparameters.

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 sitex3=0.000,x4=0.000

-0.15 -0.05 0.05 0.15x1

-0.15

-0.05

0.05

0.15

x2

x3=0.000,x4=0.000

-0.15 -0.05 0.05 0.15x1

-0.15

-0.05

0.05

0.15

x2

Sr at trigonal site

x3=0.000,x4=0.250

-0.15 -0.05 0.05 0.15x1

-0.15

-0.05

0.05

0.15

x2

x3=0.000,x4=0.250

-0.15 -0.05 0.05 0.15x1

-0.15

-0.05

0.05

0.15

x2

Ni at octahedral site

Ni at trigonal site

x3=0.000,x4=0.000

-0.10 0.00 0.10x1

-0.10

0.00

0.10

x2

x3=0.000,x4=0.000

-0.10 0.00 0.10x1

-0.10

0.00

0.10

x2

x3=0.000,x4=0.250

-0.10 0.00 0.10x1

-0.10

0.00

0.10

x2

x3=0.000,x4=0.250

-0.10 0.00 0.10x1

-0.10

0.00

0.10

x2

Crenel function

0.0 0.4 0.8 1.2 1.6 2.0t0.0

0.4

0.8

1.2

occ

2,20

2,21

0044

0044

44

44

xxxxp

xxxxp

mmimxxPm sin2exp, 04

04

Fourier transformation

Special modulation function

V.Petříček, A.van der Lee & M. Evain, ActaCryst., A51, 529,(1995).

Example : TaGe0.354Te – F. Boucher, M. Evain & V. Petříček, Acta Cryst.,B52, 100, (1996).

The Ge position is either fully occupied or empty:

x2=0.250,x3=0.249

0.30 0.40 0.50 x10.0

0.2

0.4

0.6

0.8

1.0

x4

This is typical map for crenel like occupational wave.

Te atom is also strongly modulated but the modulation is positional

x2=0.117,x3=0.486

0.05 0.15 0.25 x10.0

0.2

0.4

0.6

0.8

1.0

x4

Difference Fourier shows that the continuous function does not describe real modulation completely.

x2=0.117,x3=0.486

0.05 0.15 0.25 x10.0

0.2

0.4

0.6

0.8

1.0

x4

Splitting of the modulation wave into two parts each circumscribedby crenel function allows to account for discontinuity

x2=0.116,x3=0.474

0.05 0.15 0.25 x10.0

0.2

0.4

0.6

0.8

1.0

x4

The superspace approach allows to analyze behaviour of atoms inin the modulated structure. But it is rather cumbersome to presentthe result in this form to non-specialists. Therefore we should makesome 3d pictures showing how the modulation affects arrangementof atoms in the real 3d space.

Average structure

Only occupational modulation

Final result

x2=0.660,x3=0.072

-0.30-0.10

0.100.30x1

0.0

0.4

0.8

1.2

1.6

2.0

x4

Saw-tooth function

Bi2Sr2CaCu2O8 - V.Petříček, Y.Gao, P.Lee & P.Coppens, Phys.Rew.B, 42, 387-392, (1990)

Oxygen atom at Bi layer

00

442 xxuu

The displacement u is a linear function of x4 coordinate:

04x

0xu

1x

4x

2,2 00444

xxxfor

2,2 00444

xxxnot occupied

mmimxxLm sin2exp,, 04

04

Fourier transform:

where 02 uh

The saw-tooth modulation changes the original periodicity andit can indicate some composite character of the compound.

Commensurate modulations

Modulation vector : nnnnnn 321 ,,q

The crystal structure may be described as a regular one in the n-fold supercell.

The superspace approach can reduce the number of used parameters and moreover it can help to make a systematic study either of different phases of one structure or of different structures of one structural type.

The structure factor calculation is reduced to the summation over n different cells having internal coordinates :

nntnttt )1(,,1, 000

0.0

0.2

0.4

0.6

0.8

1.0x2

0.0

0.2

0.4

0.6

0.8

1.0

x4

`̀

0.00.2

0.40.6

0.81.0

x2

0.0

0.2

0.4

0.6

0.8

1.0

x4

161

25.00 tt

75.00 tt

0tt

5.00 tt

The superspace symmetry operator IEIE ssRR

represents a 3-dimensional operator only if :

ltt IIE 00 Rslsq

The supercell 3-dimensional symmetry derived from the same superspace group can be different for different modulation vectors and different values of . 0t

Example : Hexagonal perovskitessuperspace group : smR 0003

Hexagonal perovskites together with analogous sulphides (Sr1.143TiS3, ...) belong to the same structure type defined in

superspace.  

t0

0

0.25

General

np 3np 3

nq 2 nq 2nq 2 nq 2

cR3

cR3

3R

3R

3R

32R

3P 3P

13cP

13cP

3P

321P

qp

Rigid body option

The JANA2000 system allows to refine selected groups of atoms as rigid bodies. Each group can be put and refined in several positions in the crystal. This option makes possible :

• to fix shape of the group • to use one rigid body group at different positions • to apply TLS approximation to temperature parameters • to reduce number of parameters necessary to describe

modulation of positional and temperature parameters – rectilinear approximation (OK for rotations < 5deg)

• to apply local non-crystallographic symmetry ( for C60) 53m

The actual position of the i-th atom from the rigid body group is calculated from the relevant position of the model atom : mx

ix

txxRx 0mi

where R and t are respectively rotation (proper or improper) and translation of the rigid body and is a point chosen as the rotation center. The rotational matrix can be expressed either with Eulerian or axial angles. The second choice has meaning in cases where the first choice is close to singularity e.g. .0,0,0

Modulation of PO4 tetrahedrons in Na9{Fe2[PO4](O,F)2}

B.A.Maximov, R.A.Tamazyan, N.E.Klokova, V.Petříček, A.N.Popov & V.I.Simonov, Kristallografija, 37 (1992) 1152-1163.

Restrains of distances and angles

Minimized function:

j

jcalcjj

i

ncalci

nobsii ffwFFwS 2

0,,

2

,,

stand either for distances or angles 0,, , icalci ff

• the second summation runs over selected distances or angles and for modulated structures also over t

• for modulated structures the target value need not be specified. Then the restrain will just keep the selected distance or angle constant over t

• rotational modulations of a rigid group having restrained distances are not limited by 5 deg

Conclusions: Superspace description allows to make generalisation of the standard methods of the structure analysis:

collection techniques data reduction and determination of the crystal symmetry direct and heavy atom methods for determination of modulation

wave refinement technique calculation of distances, angles and bond valences

Comparing to the standard crystal structure analysis

It is necessary to collect more reflections Satellite reflections are usually weaker than main reflections The solution and refinement is more time consuming Methods for solution are not yet “well established” The International Tables for Crystallography vol.C contains

just basic information on superspace groups

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

• 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

JANA2000 for powders

M. Dušek, V.Petříček, 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.