1. Refinement of the short data set (A) of leadsulfate measured at 5K

To get the refinement starting a crude starting model of leadsulfate measured at 300K was chosen. This included the room temperature spacegroup of leadsulfate Pbnm, diffractometer parameters U, V, W and pseudo-voigt peak shape function.
The refinement was then started with refining the overall scale factor and followed by background function (shifted cheby i), zero shift, cell parameters containing a,b,c and atomic coordinates. For the latter one atom after the other was selected and refined. After obtaining a stable refinement B_iso was the next parameter refined. Finally U, V, W, Uanis and Asy.1 were refined. Table 1 lists R values, cell parameters and atomic coordinates with ESD`s in brackets. A plot of the final refinement is shown under refinement plot 1 .
 

Taking a closer look the obtained values show two strange values. The negative B_iso for lead and the large value for sulphur. Isotropic temperature factors should be small at temperatures of 5K. But they can´t be negative! With B_iso = 0 the atom wouldn´t move at all. Negative numbers would mean that the an atom becomes smaller than it is in reality! In the opposite there is the large value for sulphur, which indicates a strong movement about the atomic origin. Having at least some knowledge of the structure one would expect that the sulphur is tightened by his four oxygen neighbours and hence shows only a small B_iso value, especially at 5K. This indicates, that something might be wrong with the refinement, which is absolutely not obvious form the pretty good looking R-factors.

2. Refinement of the long data set (B) of leadsulfate measured at 5K with the refined settings from above

With the same settings as above one obtains much higher values for R_p and R_wp. R_exp even becomes slightly smaller, as the data set is longer and high angle data are usually more precise. It is nearly amazing that all the cell axes and atomic coordinates remain the same as before with the only difference that the ESD´s are much smaller for the cell parameters but remain almost completely the same for the atomic coordinates. R_p and R_wp are obviously too large which can clearly be seen by taking a closer look at refinement plot 2. At high angle data the simulated diffractogram differs significantly from the experimental. That is a clear sign that there is something wrong. B_iso for each atom now makes much more sense as they are all positive and lead as the heaviest of all atoms in the cell has the smallest value.
 

3. Change of spacegroup Pbnm to P2₁/n

As all the cell parameters and atomic coordinates remain nearly the same the only solution for a better refinement is to change the spacegroup to lower symmetry. Coming from orthorhombic Pbnm the two lower symmetry spacegroups are P2₁/n and P2₁/a, which are both monoclinic. For this an extra set of oxygen atoms must be introduced into the refinement because O(3) is not longer mirrored and hence has to split into two sets of four oxygen atoms. The slight distortion also implicates that the special positions of Pb, S, O(1) and O(2) have to be given up and are accessible for the refinement. Obviously the multiplicity becomes 1.0 for each included atom. Trying to refine the structure from the settings above with the new parameters gave a highly unstable refinement. To avoid this problem the refinement was started from the begin using the crude data from the room temperature structure of leadsulfate with the necessary changes for the lower symmetry space group.
The most important figures are displayed in the following table and a plot of the best refinement can be seen under refinement plot 3. The refinement with the lower symmetry spacegroup gave much  better results, which is clearly visible from the R values. The only black dot is the negative B_iso value. Even setting a starting vlaue between 0.5 and 1.0 yielded a negative figure. Site occupancy of the sulphur is not exactly 1.0 which suggests, that a few sulphur atoms are missing in the structure. From the sight of chemical composition that has to be ABX₄ this looks very weird.
 

4. Change of spacegroup Pbnm to P2₁/a

The last refinement was done with spacegroup P2₁/a. As the coordinate system is different in this spacegroup in comparison to P2₁/n, a and b, x and y and a and b have to be exchanged. Again the refinement didn´t work so it was started anew as done for P2₁/n step by step from the crude starting model. In the begin the refinement improves slowly, but after a certain point it does not converge any more. The R values went up and down and no stable refinement was accessible. Even after addition of all sensemaking variables R values didn´t improve very much. As can be seen from the following table and the plot (refinement plot 4) it wasn´t very successful. The plot reveals significant differences of experimental and simulated data. There are measured peaks at positions where there is no simulated intensity and vice versa. Hence errors for cell parameters and atomic coordinates are of course much higher for the fit with spacegroup P2₁/a than with spacegroup P2₁/n. Only B_iso is in an expected range for each atom. Site occupancy has been tried to refine, but no stable refinement could be obtained. In comparison to P2₁/n atomic coordinates and cell parameters nearly remain constant, but the overall fit is very bad.
 

Because of the lack of experience it is not possible to state what is wrong with the refinement. From a theoretic point of view the obtained refinement should be very similar for P2₁/n and its transformed P2₁/a.

5. Differences in structure of leadsulfate at 300K and 5K

A graphical image of leadsulfate at 300 K and 5 K along aixs a is displayed by clicking on it. The thermal movement of the atoms is not shown in the pictures.
Obviously a distortion in crystal lattice can lead to changes in space group symmetry, cell axes, atomic coordinates, bond lengths and angles between atoms. Furthermore all other parameters related to atoms such as isotropy or anisotropy can be influenced by the distortion. A change in spacegroup from higher to lower symmetry enforces a change in crystal parameters. Hence in this case a,b,c and b must undergo a slight change. For a better visualisation the data of high (300K) and low temperature (5K) structure were combined in Table 5.
 

The cell parameters decrease about 0.04 Å in maximum and also decrease the unit cell volume. Most visible sign for a change is the introduction of the angle b for the monoclinic spacegroup. Because of the fact that pure atomic coordinates are not very useful for a direct interpretation the S-O distances and angles between S-O atoms are displayed in table 6. The former O(3) atom splits up into two different oxygen atoms in the low temperature structure. Each of them has its own individual bond length and angle.
 

6. CIF file of low temperature structure of leadsulfate with spacegroup P2₁/n

A asked here is an incomplete CIF file of leadsulfate refined with spacegroup P2₁/n. It contains atomic coordinates and cell parameters and a few other lines that I snapped up from the internet site of www.iucr.org. pbso4 at 5K.cif

7. Overall conclusions

Rietveld refinement is a powerful method of analysing powder diffraction data. Most important to start with a refinement are accurate data of high quality and a good starting model. Low quality data sets might lead to an overall good refinement in some cases but with a high error probability. Without knowledge or any hints about the structure it would be nearly impossible for larger structures to get a solution at all. ICSD is a good source for starting structures. Also a structure with substitution of only one or two atoms might give an appropriate starting system. The ideal case would be an analysed single crystal structure, but latter are sometimes not accessible.
Having started the refinement it is important to add the variables in a slow way and not randomly. With good data and easy samples a fast introduction of variables might work but usually two much in one step will end up in a mess. The general variables scaling factor, background and zero shift and cell parameters should come first before the atomic coordinates and further variables related to the atoms are added. If all works well the refinement will converge to a low value, that depends on data quality, length of the data set and the amount of possible variables. High data quality decreases the absolute minimum that is possible to reach, whereas the length of the data sets can lead to an increase, when analysing a short and a long data set of the same sample. But long data sets have a very important advantage. The refinement gives much smaller ESD´s which makes the output data more reliable. The best data set would be of high quality, is measured over a long range (more than 120°) and a long collecting time so that the signal to noise ratio is good.
The more variables are refined the better the fit will be but it is necessary to avoid an overrefinement. This would give good final values for R_wp, but might lead to absurd data in the refinement. During the refinement one should keep an eye on the variables and try to deduce if a variable goes astray. Poor quality data inhibit a great danger for this case.
Another advantage is the sensitive reaction of the rietveld refinement towards the chosen spacegroup. A wrong spacegroup will instantly lead to a bad refinement.
A problem that can occur are the dumping values which give important information for the refinement. They limit the range in which a variable can be changed in each step. When dumping values are very small the refinement will converge only slowly, whereas larger values might give unstable cycles. In general dumping values should be decreased as the refinement improves.

Rietveld refinement is a highly valuable method to extract nearly all possible information from a powder diffraction pattern. It has a lot of advantages but also some footholes that can make a refinement really complicated. As good as the method is there remains one basic problem. Powder diffraction data are only 1D data. Therefore a refinement is always only a fit of a model to experimentally measured data and no absolute structure solution, as sometimes stated!