Refinement of the low temperature monoclinic structure (5K) of PbSO₄ using the room temperature (300K) orthorombic structure as starting model.

introduction

refinement of data set A

refinement of data set B

CIF information

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The room temperature structure of lead sulphate is orthorombic and it crystallises in the space group Pbnm (No. 62). Rietveld refinement of this orthorombic structure was performed using a rough basic model and the data collected at 300K on the powder diffractometer D1A at the ILL, Grenoble, see case study (use the back button on browser to return to this page). At very low temperature (5K), lead sulphate occurs in a lower-symmetry space group. This is probably due to the loss of mirror-plane symmetry on cooling. Therefore two lower-symmetry space groups of Pbnm were considered and refined (i.e. P2₁/n and P2₁/b which are both monoclinic).
Two data sets were used, collected at 5K on the high-resolution powder diffractometer, D2B, at the ILL, Grenoble, with a nominal wavelength, l, equal to 1.594 Å :

• data set A (8° to 62.475° in steps of 0.025°), which is a sub-set based on a limited 2q range of the data set B.
• data set B (8° to 142.975° in steps of 0.025°)

Programs used for this project:

• WINPLOTR
• FULLPROF
• GFOURIER
• DRAGON-WWW (generation of hkl and 2q)
• ORTEP3
• STRUPLO

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The initial model used for the refinement is given below. The values are obtained after the refinement of the rough room temperature orthorombic model structure in the case study, except for the initial UVW parameters. Approximate values for UVW were given in the project instructions, as well as the peak shape function (Gaussian) for data collected on D2B.

The subsequents steps in the refinement and comments are now listed:

• step 1: the overall scale factor, c. As expected, in the profile plot after this step, the position of the observed and calculated peaks differs greatly. Therefore, the next step is to refine the unit-cell parameters and the 2q zero-error.
• step 2: unit-cell parameters and zero-error. In the profile plot, the obtained position of the observed and calculated peaks is satisfying.
• step 3: atomic coordinates. Firstly, the coordinates of O(3), secondly, those of Pb, than O(1), O(2) and at last the coordinates of S. This is because it is generally better to increase the number of refined parameters progressively and in general the strongest scatterers with most degrees of freedom are refined first.
• step 4: Cagliotti parameters UVW and peak shape function. UVW are refined progressively. Also, at this point, the peak shape is changed from gaussian to pseudo-voigt and h is refined. The value of h is expected to be very small.
• step 5: individual isotropic temperature factors. Also these were refined progressively.

Results

The fit at this point, is shown in the illustration and the refined parameters and esd's are listed in the tables below figure 1.

Figure 1: Calculated (black line), observed (red circles) and difference plot (blue line)
of the PbSO₄ data (A) at 5K. The position of the reflections is also given.

Description of the structure

Pictures of the obtained structure are given next. The structure consists of rows of lead ions in between sulphate tetrahedra. In figure 2, the presence of a mirror plane through the sulphate tetrahedra is easily found. In figure 3, a view normal to the (001)-plane is given, with 50% probability thermal ellipsoids (spherical, only isotropic temperature factors are refined). In this presentation it is obvious that the temperature factors are very exceptional. The value of Pb (negative, see table 2) is not realistic at all. The larger value for S compared to O(3) and O(2) is indicative for mistakes in this model.

Figure 2: view on the (100) plane	Figure 3: view normal to (001)

Two tables (table 3 and 4), containing information of the geometric parameters derived from the final refined atomic coordinates of the structure are given below. Table 3 contains a selection of calculated bond lengths and table 4 contains a selection of bond angles (version without esd's, see e-mail note).

The next comments can be made:

• The isotropic temperature factors are very unlikely. The value for Pb is negative. This is a physically impossible value. The values for O(2) and O(3) are very small, though of the same magnitude as their error. The value of S is very large compared to the isotropic temperature factors of the other atoms. Though actually, the comparison is not very meaningful because of the large errors on the temperature factors. So, apparently the data set is insufficient to provide good quality isotropic temperature factors or the used model is not correct. It is crucial for structure refinement to possess good quality data over a large 2q range, because there is a lot of valuable information hidden in the larger angles reflections! (f.e. about the unit cell parameters -see unit cell refinement- and temperature factors)
• The bond distances for S-O are physically possible values and nearly similar to those of the room temperature structure. The tetrahedral structure of the sulphate anion seems slightly distorted due to the large O(3)-S-O(3a) angle. The typical angle for a tetrahedron is about 109°.

To conclude

The standard difference plot looks nice, however the data range was rather limited. And when viewing the calculated pattern at higher angles (until 142°, see figure 4), it is obvious that the structure of lead sulphate at low temperature is not Pbnm, though pseudo orthorombic. It is hardly impossible to conclude that from only the small angle range data. In the next part, I will discuss the results for the monoclinic space groups P2₁/n and P2₁/a.

Figure 4: Calculated pattern (black) and observed pattern (red)
for the low temperature structure of lead sulphate.

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The relation between the reflections present in the higher symmetry crystal system Pbnm and the monoclinic crystal systems P2₁/n or P2₁/a (both n°14) is given in a the table below (Table 5). Reflections of the type hkl, with both h and l not equal to zero, split up into -hkl and hkl when going from the orthorombic setting (n°62) to the monoclinic setting (n°14).

The differences aren't really very clear. Comparing these and looking closer to the symmetry elements (in the flat projection of Pbnm, P2₁/n and P2₁/a) and the refined orthorombic room temperature structure of lead sulphate, I firstly considered the refinement of the 5K monoclinic structure of lead sulphate using a model based on the P2₁/n space group.

The mirror plane at z=1/4 perpendicular to the ab-plane gets lost due to a slight rotation of the sulphate tetrahedra. That's why another O(4) needs to be included in the new asymmetric unit of lead sulphate (the number of symmetry equivalent positions is decreased from 8 to 4.) The initial coordinates of O(4) were taken from the room temperature structure (the values of the mirror O(3) atom of the asymmetric unit). Because there can only exist one mirror plane in the monoclinic system, it is either one n or the a glide plane. Comparison of figure 3 and the flat projections and keeping in mind the orientation of the screw axis and glide plane compared to the used unit cell of the room temperature model, made me choose P2₁/n first. The two unit cells are possible, of course. The choice I have made is just a matter of the direct and easy relation to the unit cell in the Pbnm space group.

The input for the starting model used for the refinement of P2₁/n and P2₁/a is given in the next tables. The initial parameters for the unit-cell, angles a,g, overal isotropic temperature factor, instrumental parameters UVW and the gaussian peak shape function are identical as the initial input for the refinement of data set A. The number of background points was extended. (As a try out, I have used a polynomal background for P2₁/n too, comments on this will be given later on.)

Results

The final fit for the refinement of the model based on the space group P2₁/n using data set B is shown in the illustration below (figure 5).

Figure 5: Calculated (black line), observed (red circles) and difference plot
of data set B at 5K using the model P2₁/n.
The position of the reflections is also given.

A 'fit', which is not good at all, for the refinement of the model based on the space group P2₁/a using data set B is shown in the next illustration (figure 6).

Figure 6: Calculated (black line), observed (red circles) and difference plot
of data set B at 5K using the model P2₁/a.
The position of the reflections is also given. There is a clear difference
in the low angle range compared to P2₁/n for the occuring reflections.

The matrix for the 'P2₁/n-model' was much more stable then for the 'P2₁/a-model'. The stability of this matrix could have been enhanced by using restraints and constraints such as for example the S-O distances or the O-S-O angles (because of the tetrahedra). But, the instability of this matrix is an indication that the best solution is the 'P2₁/n-model'. The R-factors that can be obtained for the 'P2₁/n-model' are much lower then those for the 'P2₁/a-model'. In addition, the precision of the refined parameters are better for this model. Also the temperature factors are more likely.
In comparison to the room temperature structure, the unit cell parameters a,b and c are smaller which is rather logical due to the positive expansion coefficient of the material. The most obvious change is the slight rotation of the sulphate tetrahedra away from the 'mirror plane' at z=1/4. it is not possible to conclude something concerning the differences of bonds and angles within these tetrahedra going from the 5K to the room temperature structure.

A 3-dimensional structure of the low temperature monoclinic PbSO₄ (P2₁/n) structure is given in figure 7 to show clearly the alternation of lead-ions and sulphate groups. (Pb in grey color, S in blue and O in red). Figure 8 is to be compared to figure 2. When looking very carefully, this illustrates how the sulphate tetrahedra are orientated away from the 'mirror plane'. And thus the mirror plane gets lost. In figure 9, the same view as in figure 3 is given. It is immediately obvious that the thermal ellipsoids (spherical) are more relevant. This was also concluded from the difference fourier maps (only shown in figure 10 for the final P2₁/n refined model, with max-min electron density range set between 0.15 and -0.15). The plot was found to be much smoother then that for the Pbnm model.

Figure 7: 3-dimensional crystal structure of PbSO4 at 5K along the b-axis. The unit cell of the structure is included in the diagram.	Figure 8: view on the (100) plane

Figure 9: view normal to (100)	Figure 10: Fourier difference map. The max/min range of electron density was set between -0.15 and 0.15

The following tables (table 10 and 11) have information of a selection of geometrical parameters (bond distances and angles) derived from the refined 'P2₁/n-model'. As mentioned above, the changes in angles and bonds between the room temperature and 5K structure are very small. Because I have not included esd's, I am not able to make conclusions. (Perhaps, if differences are to be found, it will be in the lost symmetrical arrangement of O(3) and O(4) in the tetrahedra, resulting in slightly different bonds and angles.)

Note on neutron versus X-ray diffraction

When low temperature XRD (Bragg Brentano) data would have been collected instead of neutron data, the higher angle reflections would have been less intensive. Furthermore, the resolution of the high angle data with XRD is much less and the scattering factors are completely different: the ion lead, would be a very intense scatterer in comparison to O. So information on the coordinates and parameters of the weaker scatterers would be more difficult to refine. Also the used wavelenght is a little different. As a result, the pattern would be highly different. As far as I can guess now, I predict that the obtained XRD pattern would not have allowed to receive the same conclusions concerning the space group and atomic coordinates. Because, as mentioned before, a lot of information was hidden by the high angle data and furthermore the scattering factors are completely different. I guess that it would have been very hard to obtain the same quality with XRD as with neutron data because of the loss of the high angle information and the presence of the large scatterer lead compared to O and S.

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_data_PbSO4

_symmetry_cell_setting monoclinic

_symmetry_space_group_name_H_M 'P 21/n'

_symmetry_Int_Tables_number 14

loop_

_symmetry_equiv_pos_site_id

_symmetry_equiv_pos_as_xyz

1   x,y,z

2   1/2-x,1/2+y,1/2-z

3   -x,-y,-z

4   1/2+x,1/2-y,1/2+z

_diffrn_radiation_wavelength  1.594

_pd_meas_number_of_points  5390

_pd_meas_2theta_range_min  8

_pd_meas_2theta_range_max  142.725

_pd_meas_2theta_range_inc  0.025

_cell_length_a   6.9311(1)

_cell_length_b   8.4222(2)

_cell_length_c   5.3738(1)

_cell_angle_alpha   90.0000

_cell_angle_beta    90.2418(7)

_cell_angle_gamma   90.0000

loop_

_atom_site_label

_atom_site_fract_x

_atom_site_fract_y

_atom_site_fract_z

_atom_site_U_iso_or_equiv

_atom_site_thermal_displace_type

_atom_site_occupancy

_atom_site_type_symbol

Pb   0.1684(1)   0.18925(7)   0.2579(2)   .25(2)   Biso   1   Pb

S    0.1832(3)   0.4338(3)    0.7532(7)   .35(4)   Biso   1   S

O1   0.0938(2)   0.5925(1)    0.7601(3)   .62(2)   Biso   1   O

O2   0.0410(2)   0.3057(2)    0.7446(3)   .47(2)   Biso   1   O

O3   0.3086(2)   0.4155(3)    0.9745(3)   .53(3)   Biso   1   O

O4   0.3068(2)   0.4214(2)    0.5271(3)   .35(3)   Biso   1   O