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

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## 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**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**Example 1 - Na2CO3**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: Cr2P2O7**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***a* Diffraction pattern of Cr2P2O7 at 140K**Data processing**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**“Show supercell group” for sodium carbonate “Select its origin” for sodium carbonate**Refinement : number of modulation parameters**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**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**KNbB2O6 Cell parameters 7.3056 3.8954 9.1659 90 90 90 q-vector (0,3/8,0) Symmetry Pmn21(0β0)s**hk1l0**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**Incommensurate refinement results with overlap checking**When commensurate refinement is practical?**- 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 structure model allows for direct comparison**with incommensurate structure.**Commensurate families**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**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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