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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.