Figure 1: Comparison of data (black crosses), and the fit (red line) for the small data set assuming space group Pbnm . The calculated hkl positions (blue bars) and the difference spectrum (green line) are also shown.

From the quality of the fit shown in Figure 1 and the low R-factors obtained, this would appear to be an excellent result. The parameters obtained are reasonable; in particular, the cell dimensions have decreased from the room temperature values as would be expected on cooling. The only surprising note is that the sulfate bond lengths and angles are significantly different from those at  room temperature. This is unexpected, since the geometry would not be expected to change dramatically with temperature  because the intramolecular forces are much stronger than the intermolecular ones. This is a hint that perhaps something is not right. This is confirmed in Figure 2 which was obtained using the full data set with the same parameters.

Figure 2: Comparison of data (black crosses), and the fit (red line) for the full data set assuming space group Pbnm . The calculated hkl positions (blue bars) and the difference spectrum (green line) are also shown.

R-Factors : R_p = 17.74,  R_wp =  29.49 and c ²  = 106.32

The R-factors and the fit displayed in Figure 2 show that the model is actually very poor, despite the excellent fit to the small data set. This clearly demonstrates the need for high angle data. The reason this is essential is also shown in Figure 2; there are very many more reflections per degree at high angles than at low angles so most of the  information in the diffraction pattern is at high angles.

Part 2: Structure refinement using P2₁/n

The room-temperature structure of lead sulfate has the space-group symmetry Pbnm in which the sulfate anion has mirror-plane symmetry. One possible reason for the result observed in the preceding refinement is that a slight distortion of the unit cell occurs with consequent loss of the mirror-plane symmetry on cooling the sample to 5 K. The two lower-symmetry space groups of Pbnm that will need to be considered are therefore (i) P2₁/n and (ii) P2₁/b (with the symmetry elements relating to the x-axis). These are actually the same space group, but with different arrangements of the symmetry elements with respect to the unit-cell axes.

In both of these space groups, the atoms that formerly occupied special positions (Pb, S, O(1) and O(2))  in Pbnm, now occupy general positions, thus x,y,z are all refinable for them. In addition, the loss of the mirror plane means that the symmetry equivalent position of O(3) (called O(4) from now on) now needs to be specified. As starting parameters, I have used the refined positions and instrumental parameters from the Pbnm refinement above with the additional atomic position for O(4):

O(4) atomic coordinates : x = 0.3070(1),   y = 0.4184(1),   z = 0.5250

The refinement followed the strategy used previously. A three-term polynomial background followed by sequential addition and refinement of: the scale factor, a, b and c cell parameters, zero error, (x,y,z) of O(3) and O(4), (z) of Pb, S, O(1) and O(2), (x,y) of Pb, S, O(1) and O(2), b, overall B factor, UVW and the Lorentzian contribution to the lineshape and isotropic B factors  were carried out. The final values are given in Table 2, a  plot of the final fit is shown in Figure 3 and the unit cell in Figure 4. A powder CIF file is attached.

Figure 3: Comparison of data (black crosses), and the fit (red line) for the full data set assuming space group P2₁/n . The calculated hkl positions (blue bars) and the difference spectrum (green line) are also shown.

Figure 4: Unit cell of PbSO₄ at 5 K assuming spacegroup  P2₁/n .

Part 3: Structure refinement using P2₁/a

I was unable to refine the structure in P2₁/b, so I adopted the suggestion to refine in P2₁/a by interchange of the x and y coordinates. The starting parameters were therefore: