Many techniques were used up to now in order to extract structure factors in an objective to determine a crystal structure from powder diffraction data. The simplest technique consists in weighting the intensity by hand. This is possible for isolated reflections. More or less sophisticated methods were applied. Peaks were cutted out of the paper representation (selecting the largest possible scale), yes yes, by chisel, and then the weight of the pieces of paper were measured, yes yes, with a balance ; also, the peak surfaces were carefully measured by a planimeter (a funny surface integrator for which you need good eyes and a stable hand if you expect to be able to reproduce the measurement with accuracy). I did both myself in the past. You can really solve simple problem by these methods. Nowadays, serious pieces of work are undertaken by whole profile analysis. Two approaches have been retained which are distinguishable by the fact that they make use or not of the cell parameters as a constraint for prediction of the angular reflection positions.
Methods working in this way represent an alternative to the derivative method used for peak position hunting. A by-product is of course the intensity. Profiles of individual reflections were approached by learnt representations (selecting an isolated reflection on the studied pattern or from a standard material) or analytical functions or convolutions of analytical functions (such as Gaussian, Lorentzian, Pearson VII possibly splitted, Pseudo-Voigt, Voigt...).
Some author names have stayed in memory corresponding to a few famous softwares : Taupin (1973), Huang and Parrish (1975) or Will, Sonneveld and Visser (1975)... Some programs are still in use. The main problem of this kind of software is how to make the decision without error about the number of peaks to be included in a compact block of overlapping peaks. The decision is made either by the user or by an algorithm designed for peak position hunting (eventually by the derivative method). One can imagine the disaster if you need really exactly an estimation for the intensity of each peak. On another hand, if one needs to extract say 1000 reflections (and this number is not large since we have mentioned that synchrotron data could soon consist in 10000 reflections !), these methods may ask for the refinement of up to 5 to 21 times more parameters (imagine you have to refine 5000 or 21000 parameters, or eventually 50000 or 210000 for synchrotron data !). Indeed, these methods refine the position, the intensity, the width and possibly several profile parameters for each reflection. As a conclusion, the use of these methods should be limited to simple cases, possibly to the estimation of peak positions at low angle for indexation purposes. None of the really complex or moderately complex structures gathered in the SDPD-Databank was solved from structure factors extracted by using these pionneering methods (taking complexity in the sense that many atoms had to be located simultaneously before to be able to start a refinement - see the top 30).
The ultimate evolution of these programs has been to make a list of the starting positions for the peaks to be hunted by using the cell parameters. These softwares are now very similar to those of the next chapter.
A revolution came with the Pawley method as described in a paper entitled Unit-cell refinement from powder diffraction scans, (G.S. Pawley J. Appl. Cryst. 14 (1981) 357-361). The main purpose was thus clearly to refine cell parameters from the whole pattern, however the possibility to use the extracted intensities as the starting point for the application of direct methods was offered. As well as for the Rietveld method, the Pawley method was not recognized as a revolution for a long time. In the Pawley method, profiles are analytical, their width is constrained to follow a Caglioti law with the three refinable parameters U, V, W as defined in most of the Rietveld-derived softwares. The main difference with the Rietveld method is that the intensities are considered as refinable parameters. Slack constraints were introduced for stabilization of the intensities of those reflections overlapping too much. So, the reflection positions are constrained by the cell knowledge. The cell parameters being themselves refined during the process if the user chooses to do so. Taking the case of 1000 reflections as an example, the number of parameters to be refined when applying the Pawley method is near of 1010 (1000 intensities + one to six parameters for the cell + the zeropoint + one or two profile parameters + the U,V,W ones). This leads to a pretty matrix which should be reversed in the refinement process (and consider 10010 for the 10000 reflections of the dreamed synchrotron pattern). The version I tried several years ago was limited to a maximum of 300 reflections on a pattern. I had to cut the pattern into several pieces for complicated crystal structures. Current versions have been improved.
More recently, a different process was proposed and applied (1988) by the author of this tutorial (be careful, this is an autopromotion ;-). The algorithm works by iteration of the Rietveld decomposition formula (which evaluates the so called "|Fobs|" used in any of the Rietveld programs in order to propose the Bragg reliability factor RB and to allow Fourier syntheses) so that the only parameters refined are the cell and profile parameters (10 to 15 parameters maximum whatever the number of reflections to be extracted). The starting proposition corresponds to a set of reflections as determined by the cell and space group, to which are allocated arbitrarily the same starting intensity (say 100). The intensities evolve by the simple iterative application of the Rietveld decomposition formula whereas the cell and profile parameters may be refined at each cycle. The "|Fobs|" become the |Fcalc| for the next iteration. With so few parameters, the method is quite fast, stable and efficient. Those reflections which are strictly overlapping are equipartitioned naturally by the process. Efficiency does not mean exactness. One cannot pretend recovering the information definetely lost in powder diffraction due to the monodimensional reduction of data which should be tridimensional for being complete (as it is the case for single crystal four-circle diffractometers data). The quotes in "|Fobs[" meaning that they are not really observed since they are estimated by a process which make them to depend on the calculated |F| : they are biased, you should be conscious of that.
Nowadays, various modifications of the Pawley and Le Bail methods are in use, representing the main approaches for extracting structure factors from a powder pattern. In the present scenario, the Le Bail method was applied exclusively :
At this stage, if there is still a doubt about the space group, the best would be to extract the structure factors for each of the space groups remaining in competition. A meticulous examination of the angular ranges where there should be extinction is required. Two different extractions with or without these reflections should lead to some difference allowing to conclude. In the case of Na2C2O4, the P21/a space group seems the more appropriate and the structure factor extraction is realized by using the Le Bail method (naoxa8.html). Afterthat we have more than 320 "|Fobs|" at our disposal (the .fou output file from FULLPROF) and we can go to the next step.
After a satisfying structure factors extraction in the P21/c space group (figpd6.gif), 1054 "|Fobs|" are at our disposal (pdcr4.html). With a cell volume of more than 1000 A3, we are really outpassing the limits which were given for a monoclinic P lattice at the chapter 3.1.1. Fortunately the presence of heavy atoms (palladium) should facilitate the realization of the next step. Locating them should lead to a minimal starting model allowing refinement and Fourier synthese. Please note that the published original paper corresponding to this experimental case made use of different programs (ARITB, ARIT4) than those applied here. So that you should not compare exactly the two studies, all was remade for the needs of this tutorial.
By using the more symmetrical space group P4/nmm, 447 "|Fobs|" were extracted up to 145° 2-theta (talf32.html). The cell dimensions suggested 16 formula units per cell. Clearly, the direct methods are adequate for t-AlF3 because Al3+ and F- are isoelectronic. The finding of half the independent atoms would not suffice for starting a refinement in such a case very similar to an organic structure problem characterized by the presence of ligth elements only. Almost the whole structure has to be found by the direct methods if one expect to succeed in the structure determination.
The structure factor extraction in the P21/n generates 1387 "|Fobs|" up to 147° 2-theta (baalf52.html). In principle Z = 8 so that 14 atoms in general position have to be located.
Extracting structure factors in the P21/n space group provides 924 "|Fobs|" (cim2a.html). With 4 formula units per cell, the C, N and S atoms represent presumably 17 independent atomic positions at least to be located.
And see also the 2 samples of the SDPD Round Robin.