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..... all simple rational  commensurate structure. ..... at least one irrational  incommensurate structure. Commensurate structures.

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..... all simple rational commensurate structure

..... at least one irrational incommensurate structure

Commensurate structures

Proving of irrationality of modulation vector only from measured values can be difficult. 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.


R3

Translation periodicity


Modulation function of commensurate structure has limited number of definition points

6-fold superstructure of the delta phase of Na2CO3.

The modulation function (of carbon) is defined only in six points.

R3


Commensurate structures number of definition points

Example of a six-fold commensurate structure

Superspace description: 4d cell, atomic position + modulation function, superspace symmetry

Supercell description: 3d six-fold supercell, atomic positions in six cells, 3d symmetry

Both descriptions are equivalent.

Origin of R3 section may influence symmetry in the supercell.

Exact intersection

Gamma-Na2CO38.92, 5.25, 6.0590, 101.35, 90C2/m(0)q=(0.182,0,0.322)

Delta-Na2CO38.90, 5.24, 6.0090, 101.87, 90C2/m(0)q=(1/6,0,1/3)

170K

This angle follows from q-vector

Superspace

Supercell


The value of number of definition points t determines symmetry in the supercell


Example 1 - Na number of definition points 2CO3

3d  incommensurate  commensurate

Alpha9.02, 5.21, 6.5090, 90, 90P63/mmm

Beta8.98, 5.25, 6.2190, 90.33, 90C2/m

Gamma8.92, 5.25, 6.0590, 101.35, 90C2/m(0)q=(0.182,0,0.322)

Delta8.90, 5.24, 6.0090, 101.87, 90C2/m(0)q=(1/6,0,1/3)

757K

628K

170K

Change of rotation symmetry

Change of translation symmetry

Alpha

Beta

Delta

Gamma

The main change of the structure occurs during the alpha  beta transition which changes rotation symmetry. Although modulations in gamma and delta phase are very strong they only modify the beta structure keeping the rotation symmetry unchanged.


Example 2: Cr number of definition points 2P2O7

3d  incommensurate  commensurate

Beta6.97, 8.45, 4.6090, 107.90, 90C2/m

Alpha37.05, 8.41, 4.6490, 108.71, 90C2/m(0)0s q=(-1,0,0.5)

Alpha27.02, 8.40, 4.6290, 108.59, 90C2/m(0)0sq=(0.361,0,-0.471)

Alpha17.05, 8.41, 4.6490, 108.71, 90C2/m(0)q=(-1/3,0,1/2)

364K

345K

285K

In case of Cr2P2O7 no change of rotation symmetry occurs. The phases alpha1, alpha2 and alpha3 represent various ways how to resolve the disorder observed in the phase beta.

Beta

Alpha3

Alpha2

Alpha1

Lukas Palatinus et al., Acta Cryst. (2006). B62, 556–566


c* number of definition points

a*

Diffraction pattern of Cr2P2O7 at 140K


Data processing number of definition points

High order satellites can coincide with main reflections.

The same satellite reflection can be by mistake integrated two times

The safest way is to use the supercell and transform hkl file to hklm in Import wizard of Jana2006:


Setting up commensurate refinement in Jana2006 number of definition points

“Show supercell group” for sodium carbonate

“Select its origin” for sodium carbonate



Refinement : number of modulation parameters number of definition points

The number of parameters in a commensurate refinement must not exceed maximal number of structure parameters that would exist in a supercell with symmetry corresponding to the used t0 value.

An attempt to use more parameters then the limiting number leads to either singular matrix or unreasonable values of modulation parameters.

Tool for transformation of commensurate structure to supercell.

The conversion is fully automatic and the supercell structure can be immediately refined. This transformation can help in finding limiting number of refinable parameters.

Note: an inverse tool is not available.

For commensurate structures satellite order does not restrict possible number of modulation waves.


Example: comensurate phase of chromium diphosphate number of definition points

For t0=0 we obtain six-fold supercell I2/c

In the supercell chromium splits to three general positions:

Cr1-1 1 2 1.000000 0.178180-0.188608-0.010960

0.005924 0.002308 0.004063 0.000191-0.001323-0.000227 0111111111

Cr1-2 1 2 1.000000 0.490301-0.188056 0.012010

0.004413 0.002120 0.003815-0.000204-0.000828 0.000001 0111111111

Cr1-3 1 2 1.000000 0.835763-0.185488-0.003201

0.003966 0.002060 0.004144 0.000072-0.000664-0.000031 0111111111

We can refine 9 position parameters and 18 ADP parameters.

Possibility to save parameters: we can refine less parameters if the obtained R value is acceptable. By this way we can for instance find that ADP in the supercell are similar and need less than maximum number of modulation waves. This limitation would not be applicable in 3d refinement.


In commensurate refinement the corresponding position parameters are basic positions + position modulations; the corresponding ADP parameters are basic ADP's + ADP modulations.

Cr1 1 2 0.500000 0.500000-0.187392 0.000000 010 0 3 3

0.004770 0.002192 0.003999 0.000000-0.000953 0.000000 0010111010

0.000000-0.001805 0.000000 0.016057 0.000000-0.020145 010101

0.005771 0.000000-0.005884 0.000000-0.000649 0.000000 101010

0.025921 0.000000-0.007967 0.750000 1.000000 0.000000 101000

0.0011420.000144-0.000135 0.000000-0.000384 0.000000 111010

0.000000 0.000000 0.000000 0.000218 0.000000-0.000110 000101

0.000000 0.000000 0.000000 0.000103 0.000000-0.000066 000101

-0.000382-0.000108-0.000092 0.000000 0.000081 0.000000 111010

0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 000000

0.000000 0.000000 0.000000-0.000066 0.000000-0.000020 000101

Parameters fixed by symmetry are not counted. We can refine basic position + two complete position harmonic waves + sawtooth function (9x1); basic ADP values + three ADP harmonic waves (18x1). The cos or sin part of the last ADP wave must be fixed by user command.


Refinement: Overlaps due to commensurate modulation vector parameters are basic positions + position modulations; the corresponding ADP parameters are basic ADP's + ADP modulations.

KNbB2O6

Cell parameters 7.3056 3.8954 9.1659 90 90 90

q-vector (0,3/8,0)

Symmetry Pmn21(0β0)s


hk parameters are basic positions + position modulations; the corresponding ADP parameters are basic ADP's + ADP modulations.1l0

hk2l0

hk3l0

hk4l0

KNbB2O6 refined incommensurately

The 4th order satellites are overlapped when q-vector is close to 3/8. In incommensurate refinement they must be treated like reflections with common intensity. In commensurate refinement the program calculates structure factors as a summation over really existing t sections and the overlap does not need a special setting.


Incommensurate refinement results without checking overlaps parameters are basic positions + position modulations; the corresponding ADP parameters are basic ADP's + ADP modulations.

Incommensurate refinement results with overlap checking


KNbB parameters are basic positions + position modulations; the corresponding ADP parameters are basic ADP's + ADP modulations.2O6 refined commensurately


When commensurate refinement is practical? parameters are basic positions + position modulations; the corresponding ADP parameters are basic ADP's + ADP modulations.

- for large supercells in offers stability of refinement and saving of parameters- for comparison with incommensurate phases- for finding relationship between compounds

Ephedrine: 1x1x4 supercell would not allow comparison of incommensurate and commensurate structure model

Chromium diphosphate: can be easily refined in 3x1x2 supercell but we would lose comparison with other phases

Phenantin:

Allows to work with one molecule instead of five


Comparison of incommensurate and commensurate structures for sodium carbonate

The Na3-O distances plotted as a function of the internal coordinate t

for (a) - gamma and (b) - delta phase of sodium carbonate.

Bold lines indicate Na3-O2 distances; other lines correspond to Na3-O1 distances. The distance indicated by the arrow is the only permanent contact.

The vertical dotted lines indicate the values of t where the modulation functions of phase are defined. The open circles show the shortest distances found in delta phase .



Commensurate families with incommensurate structure.

Commensurate family can be derived from the parent incommensurate structure by changing q-vector and t-zero. By this way we can relate three-dimensional structures which are at the first glance very different.

Example: M2P2O7 diphosphates are derived from the parent phase alpha2.

Alpha2-Cr2P2O77.02, 8.40, 4.6290, 108.59, 90C2/m(0)0sq=(0.361,0,-0.471)

Alpha1-Cr2P2O77.05, 8.41, 4.6490, 108.71, 90C2/m(0)q=(-1/3,0,1/2)

285K


cell 10.5136 12.027 10.2225 90 101.7477 90 with incommensurate structure.

pgroup 1

scdist C-C 1.4

N1t 3 2 0.100000 0.406214 0.455483 0.773915

N2t 3 2 0.100000 0.329818 0.667726 0.819365

C7t 1 2 0.100000 0.432334 0.350141 0.750675

H1C7t 2 1 0.100000 0.467392 0.304285 0.824945

C8t 1 2 0.100000 0.411084 0.302699 0.622600

H1C8t 2 1 0.100000 0.428542 0.226223 0.611035

C9t 1 2 0.100000 0.364944 0.368714 0.515490

H1C9t 2 1 0.100000 0.351362 0.338801 0.427730

C10t 1 2 0.100000 0.337350 0.481102 0.534525

C11t 1 2 0.100000 0.293388 0.556509 0.425820

H1C11t 2 1 0.100000 0.279248 0.529615 0.336645

C12t 1 2 0.100000 0.272042 0.663486 0.447610

H1C12t 2 1 0.100000 0.247242 0.712255 0.373710

C13t 1 2 0.100000 0.285914 0.706376 0.581125

C14t 1 2 0.100000 0.259474 0.817602 0.607730

H1C14t 2 1 0.100000 0.234566 0.868812 0.536355

C15t 1 2 0.100000 0.269782 0.851593 0.736620

H1C15t 2 1 0.100000 0.254190 0.926938 0.756420

C16t 1 2 0.100000 0.303882 0.773225 0.839695

H1C16t 2 1 0.100000 0.308586 0.797339 0.929050

C17t 1 2 0.100000 0.324568 0.634411 0.691580

C18t 1 2 0.100000 0.356180 0.520487 0.667710

Phenantin – bug in input M45!!!


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