Structure determination of incommensurate phases

An introduction to structure solution and refinement

Lukas Palatinus, EPFL Lausanne, Switzerland

OutlineThis tutorial will cover:

introduction to incommensurate structures (very

briefly)

determination of the symmetry

structure solution

structure refinement

validation of the structure

Incommensurate structures

Aperiodic structure is a structure that lacks periodicity, but exhibits a long-range order

Three main classes: Modulated composites

quasicrystals structures

Incommensurate structures

Modulated structure Composite

Incommensurately modulated structure has a basic 3D periodicity that is perturbed by an incommensurate modulation.

Incommensurate structuresreciprocal spaceReciprocal space is discrete despite of the aperiodicity

Incommensurate structuresreciprocal space

1-101-101-101-10

-120-120-120-120 120120120120

Incommensurate structuresreciprocal space

1-101-10001-101-1000

-120-12000-120-12000 1201200012012000

1-101-10221-101-1022

1-101-10-3-31-101-10-3-3

-130-13011-130-13011

Incommensurate structuresreciprocal space

Most current diffractometer softwares allow for indexing of an aperiodic diffraction pattern. However, the q-vector can be only refined, not found automatically. The result is indexing of the pattern by 4 integers: -6 -2 4 2 1970.51001 80.49380

-4 -2 2 0 116733.00000 327.45499 -4 -2 1 -1 280.85901 56.31390 -4 -2 1 -2 156.37300 51.69950 -4 -2 4 -2 135.81400 42.38190 -4 -2 1 0 50292.10156 214.59900 -4 -2 1 -3 21.82130 23.57890 -6 -2 -1 0 1678.30005 69.71670 -4 -2 1 1 372.96399 53.42990

Incommensurate structuresreciprocal space

Incommensurate structures

Superspace

SuperspaceConstruction of superspace in reciprocal space

SuperspaceConstruction of superspace in reciprocal space

SuperspaceConstruction of superspace in reciprocal space

a*s1

a*s4

q

b1

SuperspaceEmbedding of the structure into superspace

R3

eA =4

1A

1a

SuperspaceEmbedding of the structure into superspace

R3

eA =4

1A

1a

SuperspaceStructure model of a modulated structure consists

of:• Structure model of basic structure• Modulation functions for the parameters of the

basic structure:– Modulation of position– Modulation of occupancy– Modulation of displacement parameters

Modulation functions are most often modeled by a Fourier series:

€

u(x4 ) = An sin(2πnx4 ) + Bn cos(2πnx4 )n=1

m

∑

Superspace

Superspace symmetry

The symmetry is described by a (3+d)-dimensional space group. A 4D superspace group must be 3+1 reducible = the internal and external dimensions cannot mix together.

General form of asymmetry operation:

Example of superspace group operations:x1, -x2, 1/2+x3, -x4

-x1, -x2, x3, 1/2+x4

Symmetry

€

RE 0

RM RI

⎛

⎝ ⎜

⎞

⎠ ⎟τ Eτ I

⎛

⎝ ⎜

⎞

⎠ ⎟

SymmetryHow can the symmetry be determined? The first three rows are the components of the basic space

group. The sign of RI depends on the action of the symmetry

operation on the q-vector:

2-fold: -x1, x2, -x3, -x4 2-fold: -x1, x2, -x3

mirror: x1, -x2, x3, x4 mirror: x1, -x2, x3

SymmetryThe translational part is determined from the

extinction conditions in complete analogy to the 3D case:

in general:hR = h, h. = integer

c-glide:x1, -x2, 1/2+x3: h0l, l=2n

“superspace c-glide” with shift along x4:

x1, -x2, 1/2+x3, 1/2+x4: h0lm, l+m=2n

C2/m(0)0s

Symmetrysuperspace group symbol

C2/m(0)0s

Herman-Mauguin symbol

of the basic space group

Symbol of the

q-vector

Definition of the intrinsic shifts in the fourth

dimensions=1/2; t=1/3q=1/4; h=1/6Generators:-x1, x2, -x3, (1/2)-x4

x1, -x2, x3, 1/2+x4 Centering: 1/2 1/2 0 0

Symmetrysuperspace group symbol

SymmetryThe search for the superspace group is facilitated by the

space group test of Jana2000

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

SymmetryRational part of the q-vector

*c ( )210=q

*b

*b

Symmetry

*c ( )210=q

*b

*b

Rational part of the q-vector

Symmetry

*c ( )210=q ( )00=′q

*b

*b

*b′

Rational part of the q-vector

Centering vector: 0 1/2 0 1/2

Superspace symmetry

Structure solution

Structure solution

Structure solution means finding a starting model that is good enough to be refined by least-squares.

Two cases:1) small to medium modulations (weak to moderately strong satellites)

2) strong modulations = satellites comparable to or stronger than main reflections

Structure solutionCase 1 - small modulations:

a) Solve the average structure from main reflections

b) Refine the modulations from small starting values

Structure solutionCase 2 - large modulations: no reasonable average structure

exists

The structure can be solved by two methods:• superstructure approximation: the components of a q-

vector are approximated by commensurate values and the structure is solved as superstructure:

q=(0.345, 0, 0.478) ==> q(1/3, 0, 1/2) => 6-fold supercell

• directly in superspace by charge flipping (lecture tomorrow, 13:30). Both the average structure and modulation functions can be obtained at the same time.

Structure solutionIn Jana2000 you can:• Directly call Sir97/Sir2004. The data are prepared, sent

to Sir2004, and the model is imported back.

• Manually export data into SHELX format, solve the average structure by SHELX and import the structure back to Jana2000.

• Prepare input files for the charge flipping calculation with Superflip and EDMA. Superflip returns the density map and a list of structure factors in Jana2000 format, EDMA can provides a structure model of the average structure.

Structure solution

Structure refinement

Structure refinementTwo step procedure:• Refine the average structure against the main

reflections using standard crystallographic methods.

• Refine the modulation parameters of the atoms, namely:

– Occupational modulation (1 function)– Positional modulation (1 function for the x, y and z

components)– Modulation of ADP’s (1 function per parameter = up

to 6 functions)

Structure refinement Initial modulation refinement cookbook

Recommended:• Start with the heaviest atoms or with atoms

with largest modulation• If you suspect strong occupational modulation

of some atoms, start with occupational modulation, otherwise refine positional modulation first.

Structure refinement Initial modulation refinement cookbook

Recommended II:• Watch the R-values of

the satellites AND the Fourier maps of the modulation functions

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Structure refinement Initial modulation refinement cookbook

Discouraged:• Don’t use more

modulation waves than you have satellite orders

Reason: The contribution of the higher harmonics to low-order satellites is negligible. If it were there, high-order satellites would be observed.

Structure refinement Initial modulation refinement cookbook

Discouraged II:• Don’t switch off automatic refinement keys and automatic

symmetry restrictions of Jana2000 unless you are sure it is necessary. For temporary fixing of some parameters use Refine commands/Fixed commands

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Structure refinement Initial modulation refinement cookbook

Discouraged III:• Don’t refine the ADPs in the initial stages of the

refinement unless you see the evidence in the difference Fourier map

Structure refinement Special functions

Structure refinement Special functions

Crenel function (block wave) Sawtooth function

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Structure refinement Special functions + harmonic modulation

Harmonic functions are mutually orthogonal on the interval <0; 1>. Shorter interval leads to severe correlation between the parameters.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Structure refinement

Evaluation of the structure

Evaluation of the structureFourier maps

Fourier maps are indispensable: Check, if the modulation functions match the shape of the electron density:

Evaluation of the structuret-plots

R3

eA =4

1A

1a

Evaluation of the structuret-plots

ConclusionsStructure solution and refinement of an incommensurately modulated structure can be a relatively straightforward undertaking if:• The symmetry is determined correctly• The modulation is not too strong• The modulation is refined step by step from the most significant to the least significant waves

If becomes less straightforward if:• The modulation is very strong• Special functions are needed for description of the modulation

Acknowledgement: Special thanks to Michal Dusek for providing me his set of lectures on modulated structures

Incommensurate structures

How many q-vectors?Each rationally independent q-vector counts as

one q-vector = one additional dimension in superspace

b*

a*q2

q1

(3+2)D

-q1

-q2

Structure refinement Special functions

Crenel function (step

function, block wave)

04x

0xu

1x

4x

Saw-tooth function

Structure refinement Setting of special functions

Find the parameters in the Fourier map.

Structure refinement Setting of special functions

Check the function in the Fourier map after setting.

Structure refinement Special functions

Special functions allow to describe discontinuous modulation functions with few parameters

3 harmonic waves = 6 parameters; crenel function = 2 parameters

0.0 0.4 0.8 1.2 1.6 2.0t

-0.2

0.2

0.6

1.0

occ

Structure solutionCase 1 - small modulations:

a) Solve the average structure from main reflections

b) Refine the modulations from small starting values

The basic structure often gives a hint on

the nature of the modulation.