# STRATEGIES IN STRUCTURE DETERMINATION FROM POWDER DATA

### 3- The structure determination from powder data is decided

3.1- Limits

In this kind of job, you cannot rush without to think a bit. Applicability limits have been estimated. See if your problem dimensions are not quite out of these limits and make use of your common sense.

3.1.1- Conventional in-laboratory diffractometer

One can estimate limits by considering the recognized maximum number of parameters refinable by the Rietveld method which is the normal ultimate stage of the whole process. The values proposed here are personnal estimations which you may try to pass beyond. Corresponding to minimal FWHM (Full Width at Half Maximum) of the order of 0.12° 2-theta somewhere on the pattern (usually in the 20-40° range), 50 to 70 free atomic coordinates (x,y,z) are refinable reasonably (without taking account of fixed coordinates due to special positions nor of thermal parameters). This corresponds to 17 up to 23 independent atoms in general position (or more if some atoms accupy special positions).

Corresponding to these limits, cell volumes can be more or less estimated, depending on the crystal system and on the Bravais lattice. For centrosymmetrical space groups, these maximal volumes are the following :

```   Vmax(Å3)    Multiplicity of the        Lattice     System
general position ```
```   500                2                               Triclinic
1000                4                     P         Monoclinic
2000                8                     C              ,,
2000                8                     P         Orthorhombic
4000               16                  A,B,C,I           ,,
8000               32                     F              ,,
etc for tetragonal, hexagonal and trigonal
12000               48                     P         Cubic
24000               96                     I         Cubic
48000              192                     F         Cubic
```

Translated in maximal number of reflections, any of these above maximal possibilities corresponds to approximately 1000 to 1500 reflections for a pattern extending from 5 to 150° 2-theta recorded with a ~1.5 Å wavelength. One will have approximately 20 reflections per xyz refined parameter. In a single crystal study, 10 reflections per parameter, including the thermal ones, is something considered as normal. The fact that a larger value is proposed for powder data is a consequence of reflection overlapping. I hope that you will find these limits impressive after all. You shall divide the above volumes by 2 if you are working with an acentric space group. Divide them also by two if you wish accuracy otherwise you may have to present dubious interatomic distances in your manuscript and the reviewers will not be happy. At the beginning of the real expansion of this new sub-discipline (1986-1987) I had to face a lot of incredulity. Some reviewers simply reply to the editor that such a job was impossible...

3.1.2- Synchrotron data

Now if your minimal FWHMs decrease down to 0.06 or 0.02° 2-theta, all you have to do is to multiply the maximum volumes listed at chapter 3.1.1 (given for minimal FWHM ~ 0.12° 2-theta) by 2 or 6. This is much more comfortable than with conventional X-ray data. It can be expected really to play with 150 independent atoms, corresponding to 450 xyz refinable parameters. Up to 9000 or 10000 reflections could be extracted from a synchrotron powder pattern. Triclinic centrosymmetrical cells with up to 3000 Å3 volume or cubic cells with F lattice as large as 300000 Å3 are the theoretical upper limits which one could expect to attain by using high resolution synchrotron data ! No study has approached such limits up to now. This is because no try has been done. Very recently, FWHMs as low as 0.008° 2-theta were obtained at the ESRF facility. These synchrotron high performances may need a counting step as low as 0.002° 2-theta corresponding to 75000 points if the pattern was measured in the 5-155° 2-theta range. If compared to the 10000 expected reflections, this gives one new reflection as a mean every 7 points. This appearss manageable, however the range is generally limited to 2-80°, selecting a short wavelength, because the sample fall down easily at larger angle if unpacked. Actually, the largest problems solved (up to 60 independent atoms, 180 xyz parameters refined) are considerable. However they are far from the limits suggested here. A list of complex experimental cases already published may be found as a Top 50 inside the SDPD-Databank. On another hand, the maximal limits suggested for conventional in-laboratory diffractometers have been already reached without too much difficulties for triclinic, monoclinic or orthorhombic cells. Consequently, one can predict exceptional results in a very near future from synchrotron data.

3.1.3- Neutron data

Neutron conventional theta-2theta diffractometers present at best minimal FWHMs ~ 0.12° or at worst ~ 0.25 to 0.30° 2-theta. As a consequence, the maximal cell volumes of chapter 3.1.1 would be respectively applicable or would need to be divided by 2 or 3. Nevertheless, the maximum number of parameters that one could expect to refine by the Rietveld method is not a sufficient criterion for an estimation of the feasibility limits of an ab initio structure determination from powder diffraction data. Indeed, winning the game depends on the successful application of the Patterson or Direct methods. One has to obtain a starting model sufficiently large for being able to start the refinement and then complete the structure by difference Fourier syntheses. With the presence of atoms distinctly heaviest than the others, in the sense of having distinctly higher diffusion factors or Fermi lengths, it is generally sufficient to locate them for the initial structure model building. Without these heavy atoms, one has to locate almost the whole structure before to be able to refine. Neutron data place you almost systematically in this later case. Indeed, the Fermi lengths are of the same order for all atoms. Therefore, a structure determination from exclusively neutron data is generally much more difficult than from X-ray data, however organic compounds are difficult whatever the data. It is advisable to make use of both X-ray and neutron data, playing with their complementary advantages. A structure can be determined partly from X-ray data by locating heavy atoms, and it can be completed and/or the accuracy on the light atom positions can be improved from neutron data. Finally, one can refine the structure simultaneously from both data, a few softwares allow this. In the SDPD-Databank, the top 30 lists the most complex structures according to the criterion of the largest number of atoms simultaneously located at the stage of applying the Patterson or Direct methods. The upper limit is relatively low with 18 atoms from X-ray data (6 only from neutron data). This rather small record should be broken soon by using synchrotron data with the highest resolution. More people have to be convinced that structure determination has become a quasi routine task by using powder diffraction data with some expertize. The difference with solving a structure from single crystal data is that much expertize is needed for powder data because several essential steps in the whole process are not 'automatized' : the goal is to locate goodies among garbages.

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