Structure determination of incommensurate phases An introduction to structure solution and refinement Lukas Palatinus, EPFL Lausanne, Switzerland

Outline This 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 compositesquasicrystals structures

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

Incommensurate structures reciprocal space Reciprocal space is discrete despite of the aperiodicity

Incommensurate structures reciprocal space

Incommensurate structures reciprocal 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: Incommensurate structures reciprocal space

Incommensurate structures

Superspace

Construction of superspace in reciprocal space

Superspace Construction of superspace in reciprocal space

Superspace Construction of superspace in reciprocal space a* s1 a* s4 q b1b1

Superspace Embedding of the structure into superspace R3R3

Superspace Embedding of the structure into superspace R3R3

Superspace Structure 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:

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 a symmetry operation: Example of superspace group operations: x 1, -x 2, 1/2+x 3, -x 4 -x 1, -x 2, x 3, 1/2+x 4 Symmetry

How can the symmetry be determined? The first three rows are the components of the basic space group. The sign of R I depends on the action of the symmetry operation on the q-vector: 2-fold: -x 1, x 2, -x 3, -x 4 2-fold: -x 1, x 2, -x 3 mirror: x 1, -x 2, x 3, x 4 mirror: x 1, -x 2, x 3

Symmetry The translational part is determined from the extinction conditions in complete analogy to the 3D case: in general: hR = h, h.  = integer c-glide: x 1, -x 2, 1/2+x 3 : h0l, l=2n “superspace c-glide” with shift along x 4 : x 1, -x 2, 1/2+x 3, 1/2+x 4 : h0lm, l+m=2n

C2/m(  0  )0s Symmetry superspace 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 dimension s=1/2; t=1/3 q=1/4; h=1/6 Generators:-x 1, x 2, -x 3, (1/2)-x 4 x 1, -x 2, x 3, 1/2+x 4 Centering:1/2 1/2 0 0 Symmetry superspace group symbol

Symmetry The search for the superspace group is facilitated by the space group test of Jana2000

Symmetry Rational part of the q-vector

Symmetry Rational part of the q-vector

Symmetry  Rational part of the q-vector Centering vector: 0 1/2 0 1/2

Superspace symmetry

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 solution Case 1 - small modulations: a) Solve the average structure from main reflections b) Refine the modulations from small starting values

Structure solution Case 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 solution In 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

Two 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

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

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

Crenel function (block wave) Sawtooth function

Structure refinement Special functions + harmonic modulation Harmonic functions are mutually orthogonal on the interval. Shorter interval leads to severe correlation between the parameters.

Structure refinement

Evaluation of the structure

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

Evaluation of the structure t-plots R3R3

Conclusions Structure 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* q2q2 q1q1 (3+2)D -q 1 -q 2

Structure refinement Special functions Crenel function (step function, block wave)  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

Structure solution Case 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.