Objectives

The general objective of this project is to test the usage of the Rietveld refinement on particular case: using Rietveld refinement software refine structure of a given compound using given data sets. Particularly:

• to test usage and limitations of a different range of data for structure refinement;
• to verify the importance of using the correct space group for structure refinement;
• to demonstrate the use of alternative settings of a space group;
• to acquire practical experience in the use of the archival data format CIF; and
• to practice publishing Web-based scientific reports on Rietveld refinement.

Generally, the project aims to show participant's general crystallographic knowledge and practice gained during the course on a real case of crystal structure refinement from angle-dispersive powder neutron diffraction data by the Rietveld method.
Paricularly, to find the correct low temperature (of 5 K) structure (and its refinement) of lead sulphate, PbSO₄, starting with the given orthorhombic room-temperature structure model.

Project Tasks

Task No.1

Refinement the structure of lead sulphate, PbSO₄, using data collected at 5 K on high resolution neutron powder diffractometer (D2B, at the ILL, Grenoble, France, with a nominal wavelength, l, equal to 1.594 Å); using the orthorhombic room-temperature structure of lead sulphate as a starting model; and using the range of data provided by data set A (Data file A (2q range from 8° to 62.475° with step 0.025°, data file has been provided by the tutor). The instrumental function parameters, UVW, of this diffractometer has been provided by the tutor and equals 0.05, -0.15, 0.20 (°²), respectively. Concerning the peak shape for D2B device, following information has been provided by the tutor: almost Gaussian shape but with some imperfections due to defects in the original monochromator.

The refinement calculations through the project have been all done using LHPM-Rietica 1.7.7 freeware program package (LHPM-Rietica Rietveld for Win95/NT by Brett Hunter (e-mail: bah@ansto.gov.au), downloaded form CCP14 Homepage at http://www.ccp14.ac.uk).

The background has been in all following cases fited manually. As the reflection peak shape function the Pseudo - Voigt function corrected for asymmetry ( with FCJ Asym function) has been used. The calculated parameters for assessment of the refinement have benn by Rietika calculated according to these definitions.

Task No. 1 - Solution
Setting of Instrumental preferences, creation of a new information file and crystallographic model
Before starting, I settled the preferences used: measuring instrument (D2B at ILL Grenoble), typical data limits and step size (8 - 62.475 - 0.025°(2th)), typical resolution parameters, U, V, W, (0.05, -015, 0.20), peak shape (is set to pseudo-Voigt with the FJC (Finger, Cox, Jephcoat) asymmetry correction for low-angle peaks; Gam0 is set to 0.05, which corresponds to a peak shape that is 95% Gaussian and 5% Lorentzian).
Then, the crystallographic model has been edited in the model tab page using data provided by the tutor (the "rough" orthorhombic room-temperature structure of lead sulphate).
note: in the atom type box there are possibilities for oxygene O to be e.g. O neutral, O^-, O^2-, but no choise for sulphur (formally S⁶⁺ in suphate anion). This should influence the form factors of S in case of XRD. Because this task concern with neutron scattering, this should not play any role. I used neutral atoms in all cases.

Background treatment
From two general approaches to background treatmet (firstly, to estimate the background level by graphical methods at several points where there are no diffraction peaks in the diffraction profile and to assume a linear function between them; and secondly, to use a background function (the polynomial function) with refinable parameters), I used the former access. This method works well for angle-dispersive data sets, for unit cells of a moderate size where no extensive peak overlap (which seems to be my case).
The background count of the diffraction data was estimated graphically as shown in this figure.
 

Refinement of the scale factor
Its initial value was left arbitrarily set to 0.01 and it have been run 4 cycles to refine its value (see this Figure). At this point, the R_w-factor dropped to only 58.05% with the fit. Large differencies can be observed between calculated and experimantal peak positions (even for the large d spacings). Moreover, the program placed observable reflections in places, where no reflection was measured (e.g., at 22, 37, and 59°(2q)). This indicate that the starting strucutral model I is too crude and inaccurate. This is acknowledged in the next steps: refinement of only the a,b,c unit-cell parameters (see here for result), or simultaneously the unit-cell parameters and a 2q zero error (here is the result), gave in every case very poor results, suggesting that the refinement diverges.
It seemes to me acceptable to take for granted that the use of the incorrect space group together with very crude structural model (originally designed for refinement of room temperature data) would hardly give any better result. I stopped at this stage using the model I as too rough and tried to start with already refined room temperature PbSO₄ structure model.

The first steps were analogical to that done in previous trial: Setting of Instrumental preferences, creation of a new information file and crystallographic model, Background treatment, and Refinement of the scale factor. After the 3rd cycle of the scale factor refinement I obtained this results: R_w = 48.67, c² = 280.91, and graphical output here.
The mismatch in observed and calculated peak positions indicates that refinement of the unit-cell parameters and/or any peak-displacement parameters, such as a 2q zero error, should be the next step in this refinement.

Refinement of the lattice parameters, and 2q zero error
The observed, calculated, and difference profile plots can be used to indicate the next step in the refinement. Firstly only a,b,c unit-cell parameters, and then also 2q zero error were refined. After 9 cycles of refinement of (scale factor +) lattice parameters, the weighted - profile R_w-factor reduced to 9.72%, (c² = 10.63).
Further, after 6 cycles of refinement of (scale factor, lattice parameters) + zero error the weighted-profile R_w- factor reduced to 8.913%, (c² = 9.526) and the profile fit shown in this Figure has been obtained. In the Figure can be seen that a good match has been obtained between observed and calculated peak positions. So, attempts can be made to improve the match in peak intensities.

Refinement of the peak intensities (refinement of the atomic coordinates)
The key parameters now are the atomic coordinates since refinement of these will affect the relative intensities of neighbouring peaks in the diffraction profile. The approach which I used here had been to vary the atomic coordinates of the strongest scatters with the most degrees of freedom (O3, in general position), then the atomic coordinates of the less stronger scatters, with less degrees of freedom (Pb, O2, O1, which are in special positions). The actual sequence was as follows (in each step I released the coordinates of only one atom): 1. O3(x,y,z); 2. Pb(x,y); 3. O2(x,y); 4. O1(x,y); and 5. S(x,y). At the end, after 6 - 9 cycles (for each particular atom) of refinement of (scale factor, lattice parameters, zero error +) atomic coordinates the weighted - profile R_w- factor was reduced to 8.32%, (c² = 8.29) and the profile fit shown in this Figure has been obtained. The expanded section shown in this Figure demonstrates the improvement in the match between observed and calculated peaks positions.
In the previous figure can be seen that the peak shape does not match the shape of measured reflections precisely. There seems to be two types of dissaccord: the first one can be observed in the upper part of reflection profile (it seemes to be due to the in project instructions mentioned problems with some imperfections due to defects in the original monochromator), the second one can be observed at the tail of reflection profiles (especially at higher angles). This type indicate that peak shape functions/temperature factors should be refined next.

Refinement of the peak intensities (refinement of peak shape parameters/temperature factors)
I suppose that the valuse of resolution-function parameters UVW stated in the instruction to the project were gained on experimentally on a standard sample. As they take into account a sample - dependant contribution to the peak widths that can vary from one sample to the next. So, they deserve the (gradual) release in order to be refined.
Firstly, only the W parameter was released, then V parameter. Refinement of  values of V and W parameters reduced after 11 cycles of calculation the weighted-profile R_w - factor to 5.438% (c² = 3.572). At this stage, also the overall displacement factor was released for refinement. Refinement of  its value reduced after 10 cycles of calculation the weighted-profile R_w - factor to 5.39 % (c² = 3.52). The fit is shown in this Figure.
The most useful part of the plot is now the difference profile (for two examples - low and high angle region see this Figure), which still shows some differences between the observed profile and that calculated from the refined model. Compared to the previous step (UVW refinement), the calculated peak shape in its tails matches better the observed reflections, but still the dissaccord in the upper part (near the maxima) of reflection profile can be observed (marked in the Figure with green arrows).
After the refinement the overall temperature factors, B_iso, had negative value (it equalled to -0.29Ų). Trying to constrain the  B_iso at a physically meaningfull value (positive, slightly above zero), and at the same time, refinement of remaining peak shape parameter U, gave almost identical R_w factor with the refined value of U parameter to be equal to 0.66 and gama0 value to be equal to 0.111. This is more then ten times higher then originally proposed values (and perhaps, obtained experimentally on a standard sample). Even if these parameters can display a minor deviation from starting values, these differences in my opinion indicate that the refinement at this stage goes astray. I did not refine the model for the individual displacement factors. I would say there is no sense to continue further in more detailed refinement. I stopped at this stage.

The final fit (together with the details of lower and higher parts of the pattern) is illustrated in this Figure.The refined parameters and esd's are listed in the tables below:

Cell parameters : a = 6.92618(44) Å,  b = 8.41516(52) Å,  c = 5.36835(35) Å

Atomic coordinates:

Atom	x	y	z
Pb	0.1675(2)	0.1895(2)	1/4
S	0.1921(9)	0.4312(9)	3/4
O(1)	0.0924(4)	0.5922(4)	3/4
O(2)	0.0392(6)	0.3070(4)	3/4
O(3)	0.3068(3)	0.4179(3)	0.9755(3)

Instrumental : U = 0.05°²,  V = -0.0693(2)°²,  W = 0.201(3)°², gama = 0.111(2), zero error 0.069°;

R-Factors :R_wp = 5.44%,  R_exp = 2.88%,  R_p = 4.31%

Structural Parameters :

Bonds Lengths	Value		Bonds Angles	Value
S-O(1)	1.520(6) Å		Ð O(1)-S-O(2)	107.61(41)°
S-O(2)	1.487(6) Å		Ð O(1)-S-O(3)	108.51(30)°
S-O(3)	1.451(3) Å		Ð O(1)-S-O(3)¢	108.51(30)°
S-O(3)¢	1.451(3) Å		Ð O(2)-S-O(3)	109.61(35)°
			Ð O(2)-S-O(3)¢	109.616(35)°
			Ð O(3)-S-O(3)¢	112.84(43)°

The " ¢ " indicates that the coordinates of this atom are symmetry related (by a mirror plane in this instance) to the coordinates of a refined atom with the same name.

Here is for the reference the main output file after the last cycle of refinement, here the output files summarising the distance and angle relations between constituing atoms. I consider that the Bragg R factor (given at the end of the main output file) is identical to the R_I (intensity factor). In the current case it reaches the value of 2.94, indicating suprisingly high quality of used structural model.
Using the *.cif file constructed manually (see here), I was able to draw the picture of the structure of refined model, usinf the freeware demo version of DIAMOND program (Diamond demonstration version 2.1e, copyright 1998-2001 Crystal Impact GbR, Germany, author Klaus Brandenburg, see also http://www.crystalimpact.com/diamond/). The result is shown here.

Task No.1 - Comments
As it is seen above, the refinement of structure model of PbSO₄ at 5 K using "short" data set and under presumption of (incorrect) Pbnm space group gives a quite nice coincidence between the calculated ond observed pattern, plausible structural parameters together with not too bad values of R factors. The most remarkable characteristic of the this results is the finding that it is easily possible to make erroneous statements about a final structure, when one uses only narrow data set, and takes into account only the numerical output of refinemets (the hight of R factors).

Task No. 2
Using the refined model obtained above, I calculated the diffraction pattern for the whole range of the data set B (pbso4_b5.dat). ( I did not vary any of the structural or instrumental parameters at this point, except that I manually edited the bachgroung to be able to subtract it in the whole data range.
I followed two ways. A) I tried to change manually the Rietika input file, so as to be able to do only the calculation of the pattern (by changing 3 for 1 as a control variable in the second line of the input file (see here). The graphical output of this procedure is here. Unfortunately, I was not able to find any way how to get the observed and calculated pattern to compare them.
B) I decided to start a new refinement in which I manually edited the "General", "Phase", "Histogram", and "Sample" dialog window with final values obtained in the Task 1 refinement (I also used background in the form of user's input data). After that I locked up all refineable variables (with the only exception of scale factor), I obtained this output file, the graphical output is illustrated in this Figure. Here is for the reference the main output file after the last cycle of calculation, here the output files summarising the distance and angle relations between constituing atoms.
 

Task No.2 - Comments
Using the same strucutral model, the same values of peak shape parameters as in the task 1, but now, contrary to the task 1  using more then twice longer data range (up tu 142°(2q)), the erroneous presumption about the space group is clearly evident. Whereas seeing only the low and middle angle region (up to approx. 65°, see the graphical output Figure) one could exert an effort to explain the observable differencies in both the positions, and the area of reflections between calculated and observed pattern by common and acceptable differencies derivable from the iterative character of calculation. From the results gained on the high angle region (above 100°(2q)) is clear that something is wrong with the structure model. Neither the positions, nor the areas of calculated and observed reflections match acceptably, also the R factors increased remarkably.
The reason is in my opinion similar to that mentioned in the chapter about simple methods of Indexing. There is derived the equation:

Task No. 3
For the remaining task in my Rietveld refinements I used the full data set B. As it is notised in the instructions ..."one possible reason for the result observed in the preceding refinement is that a slight distortion of the unit cell occurred with consequent loss of the mirror-plane symmetry on cooling the sample to 5 K." It is theefore necessary to lower the symmetry of the strucutre model. There are two lower-symmetry space groups of Pbnm that are recommended in the instructions: (i) P2₁/n and (ii) P2₁/b . The task was to try and refine the 5 K structure in each of these space groups.
The space groups P2₁/n, and P2₁/b are different notation of the unique space group No. 14 (this notation said which type of possible unit cell selections - i.e. settings - is used). So, in priciple, there is no other reason for usage of paricular setting, except of convenience. Therefore, all the remaining settings can be used. To acquire a knowledge about the possible settings used in case of space group No. 14 (I have no access to Int. Cryst. Tables) I made use of freeware program PowderCell (PowderCell for Windows, version 2.3, authors: W.Kraus, G.Nolze, Federal Inst. for Mater. Research and Testing, Rudower Chausee 5, 12489 Berlin, Germany; see also http://www.ccp14.ac.uk/tutorial/powdcell/). Using this program I found out that there are defined 18 (!) different settings for this space group. I summarized examples of a few of these settings, together with the axis transformation relations in the following table:
 

In the first case, I started the refinement with the setting No.7 (short notation P2₁/n). In making the reduction of space group symmetry from orthorhombic to monoclinic I made use of two hints provided by the tutor (concerning the atomic coordinates the space-group symmetry: atoms that were on the mirror plane in Pbnm group will have generalized coordinates x,y,z; and the eight O(3) atoms with generalized coordinates in Pbnm group will split into two lots of 4, where the coordinates of O(3) and O(4) are approximately related by the mirror symmetry of the orthorhombic space group).

Setting of Instrumental preferences; creation of a new information file and crystallographic model
By lowering the symmetry from orthorhombic to monoclinic the structure model will now contain six atoms (Pb, S, O1, O2, O3, O4), all occupying general positions (all coordinates are refineable). It is necessary to adjust properly the starting coordinates of "new" O4 atom with reference to the mirror of the orthorhombic Pbnm space group. As can be seen here, there are three different mirror planes, namely m (symmetry operator x, y 1/2 - z), n (1/2 + x, 1/2 - y, 1/2 +z), and b (1/2 - x, 1/2 + y, z). I calculate the coordinates of O4 by applying of symmetry operators of these planes on O3 coordinates. Therefore I obtained three sets of coordinates for O4(1) ( I related the coordinates by the m plane), O4(2) ( I related the coordinates by the n plane), and O4(3) ( I related the coordinates by the b plane) atoms. The values are summarised in the following Table
 

	x	y	z
O4(1)	0.3068	0.4179	0.5245
O4(2)	0.8068	0.0821	0.4755
O4(3)	0.1932	0.9179	0.9755

In the next step I input gradually the coordinates of O4(1), then O4(2), and finally O4(3) into Rietika input file a did the refinement.

Refinement using P2₁/n space group with the atom O4(1) (0.3068, 0.4179, 0.5245)
I used the strategy briefly descibed by sequence of steps given in the following able with description and number of refined parameters:
 

When refining the B_iso parameters, I firstly refined the displacement parameter for lead, and for following stages of refinements I kept its value locked up at given value (0.02). Otherwise, the refinement became from an unknown reason unstable. Here one can see the Figures of calculated and observed patterns, together with the difference plot illustrating the particular steps of refinement: the scale factor refinement; refinement of lattice a,b,c,b parameters + zero error , refinement of coordinates  of all atoms. The final fit (together with the details of lower and higher parts of the pattern) is illustrated in this Figure.
The survey of input values, starting model, together with the results of refinement are given for reference in the main output file.
The refined parameters and esd's are listed in the tables below:

Cell parameters : a = 6.92832(6) Å,  b = 8.41889(5) Å,  c = 5.37146(5) Å, b = 89.7631(7)

Atomic coordinates:

Atom	x	y	z	Biso	N
Pb	0.1687(1)	0.1895(1)	0.2422(2)	0.02	1.0
S	0.1835(4)	0.4338(3)	0.7453(8)	0.12(4)	1.0
O(1)	0.0933(2)	0.5923(2)	0.7402(3)	0.32(2)	1.0
O(2)	0.0410(2)	0.3058(2)	0.7560(3)	0.21(3)	1.0
O(3)	0.3069(4)	0.4214(3)	0.9729(3)	0.09(3)	1.0
O(4)	0.3088(3)	0.4158(3)	0.5256(3)	0.24(3)	1.0

Instrumental : U = 0.044(1)°²,  V = -0.141(3)°²,  W = 0.211(2)°², gama0 = 0.153(6), zero error 0.059(5)°

R-Factors :R_wp = 5.68%,  R_exp = 2.80%,  R_p = 4.38%, R_B = 4.21% (equivalent to R_I)

Structural Parameters :

Bonds Lengths	Value		Bonds Angles	Value
S-O(1)	1.4738(6) Å		Ð O(1)-S-O(2)	112.43(17)°
S-O(2)	1.46256) Å		Ð O(1)-S-O(3)	108.77(23)°
S-O(3)	1.4984(3) Å		Ð O(1)-S-O(4)	109.13(24)°
S-O(4)	1.4697(3) Å		Ð O(2)-S-O(3)	107.68(23)°
			Ð O(2)-S-O(4)	110.60(23)°
			Ð O(3)-S-O(4)	108.11(18)°

Using the *.cif file constructed manually (see here), I was able to draw the picture of the structure of refined model, usinf the freeware demo version of DIAMOND program. The result is here .

Refinement using P2₁/n space group with the atom O4(1) - Comments
The structure of PbSO₄ is according to the results of refinement  composed at 5 K from distorted SO₄^2- tetrahedra, where only one angle (namely O(1)-S-O(4)) gets near to the ideal 109.5°. The values of S - O bond lenghts lie in the range from 1.46 to 1.50 Å, which are usual values for bonds with high proportion of ionic character. The individual isotropic B_iso displacement factors of atoms take values between 0.02 for Pb, and 0.32 for O1. The higher values for O atoms indicate either a larger inaccuracy in assessment of their coordinates, or their more pronounced movement. Since there atoms vibrate less then at room temperature, the values are genarally lower then their room temperature equivalent (as can be seen in the case study on PbSO₄ described in the text of the course).

Refinement using P2₁/n space group with the atom O4(2) (0.8068, 0.0821, 0.4755)
As in the case of refinement with O4(1), I started (after the manual fitting of background) with the refinement of scale factor. At this point, the R_w- factor dropped to 33.88% (c² = 144.79) with the fit. Contrary to the O4(1) case, large differencies can be observed between calculated and experimantal peak positions (even for the large d spacings). It seems that the structure model presses upon the refinement procedure to put strong reflections in angle regions, where no reflection is observed (see this Fig., illustrating the graphical output of refinement, and this main out put file for reference). Knowing that in case of neutron diffraction problems with preferred orientation are neglectable, these deviations indicate an erroneous structure model. The deviation did not vanish even when lattice parameters, or later atomic coordinates were refined. From that results that the structure model using the atom O4(2) is wrong.

Refinement using P2₁/n space group with the atom O4(3) (0.1932, 0.9179, 0.9755)
In this case, the refinement proceeds without striking deviations. At the beginning I followed the procedure described above: after the manual fit of the backdround, refinement of scale factor, a,b,c, and b lattice parameters + zero error , coordinates of atoms, overall displacement factor (B_overall), W,V,U, gamma0 instrumental peak shape parameters, I tried to refine the individual displacement parameters B_iso. After that I updated the input file with values obtained during this refinement a constrained the coordinates, the lattice parameters, the instrumental function parameters, and the B_overall + individual isotropic B factors to the previously gained values. Then I tried to refine (scale factor +) the anizotropic displacement factors for atoms with the highest B_iso values (in sequence O1, then O4, O2, O3, and finally S). At the end I released all the parameters locked up above, with the exception of B_overall, and refined also the wavelength l. The strategy I used is descibed in the following Table with description and number of refined parameters, and values of R-factors:
 

For the reference the main output file after the last cycle of refinement can be seen here, here are the output files summarising the distance and angle relations between constituing atoms. Using the *.cif file constructed manually (see here), I was able to draw the picture of the structure of refined model, using the freeware demo version of DIAMOND program. The result is here.
The final fit (together with the details of lower and higher parts of the pattern) is illustrated in this Figure.

The refined parameters and esd's are listed in the tables below:
Cell parameters : a = 6.928179(63) Å,  b = 8.418823(77) Å,  c = 5.371361(49) Å, b = 89.762(6)

Atomic coordinates:

Atom	x	y	z	Biso	N
Pb	0.1683(1)	0.1897(1)	0.2417(2)	0.08	1.0
S	0.1838(4)	0.4340(3)	0.7460(8)	0.18(4)	1.0
O(1)	0.0928(2)	0.5922(2)	0.7390(3)	0.38(2)	1.0
O(2)	0.0414(2)	0.3061(2)	0.7564(3)	0.27(3)	1.0
O(3)	0.3069(4)	0.4201(3)	0.9729(3)	0.15(3)	1.0
O(4)	0.1917(2)	0.9172(3)	0.9740(3)	0.30(3)	1.0

Cell parameters : a = 6.92832(6) Å,  b = 8.41889(5) Å,  c = 5.37146(5) Å, b = 89.7631(7)°

Anizotropic displacement factors, B_anizo:

Atom	B11	B22	B33	B12	B13	B23
S	-0.0027(5)	0.0001(3)	0.0041(8)	-0.00007(4)	-0.0005(6)	-0.0002(9)
O(1)	0.0004(3)	-0.0017(3)	0.0035(3)	0.0002(2)	0.0026(3)	0.0001(4)
O(2)	0.0001(2)	-0.0004(1)	0.0025(3)	-0.0002(3)	0.0037(3)	0.0009(3)
O(3)	0.0000(3)	0.0007(3)	0.0003(3)	0.0002(3)	-0.0016(3)	-0.0017(3)
O(4)	-0.0011(3)	0.0002(3)	-0.0007(3)	0.0000(2)	0.0023(3)	0.0009(3)

Instrumental : U = 0.0434(1)°²,  V = -0.1415(2)°²,  W = 0.2113(0)°², gama0 = 0.1495(3), zero error 0.0917(6)°;

R-Factors :R_wp = 5.32%,  R_exp = 2.80%,  R_p = 4.14%, R_B = 3.77%

Structural Parameters :

Bonds Lengths	Value		Bonds Angles	Value
S-O(1)	1.4741(5) Å		Ð O(1)-S-O(2)	112.75(21)°
S-O(2)	1.4615(6) Å		Ð O(1)-S-O(3)	109.36(27)°
S-O(3)	1.4947(3) Å		Ð O(1)-S-O(4)	107.80(27)°
S-O(4)	1.4672(2) Å		Ð O(2)-S-O(3)	107.31(23)°
			Ð O(2)-S-O(4)	110.73(23)°
			Ð O(3)-S-O(4)	108.48(18)°

Refinement using P2₁/n space group  - Comments
From the three refinements using the three different O4 atoms done above results that usage of only two of them, namely O4(1) and O4(3) gave satisfactory and in my opinion also meaningful results. The  picture simplyfies oneself futher, when we check the coordinates of O4 atoms in the elemental cell of these two structures. I conducted it by means of Diamond program and result obtained is summarised here. Virtually, atomic coordinates of all atoms in both types of unit cells are the same within the limits of calculation error, as it can be seen here. Consquently, what I meant that they are two different solution, gained by refined usind two different O4 atoms, is in reality the unique solution, giving the unique structure model, only using two different choises of asymmetry unit, as it can be seen comparing this, and this figure.
The structure of PbSO₄ is according to the results of refinement  composed at 5 K from distorted SO₄^2- tetrahedra, where only one angle (namely O(1)-S-O(4)) gets near to the ideal 109.5°. The values of S - O bond lenghts lie in the range from 1.46 to 1.50 Å, which are usual values for bonds with high proportion of ionic character. The tetrahedra are not connected each to other, with the Pb atoms in between.  The individual isotropic B_iso displacement factors of atoms take values between 0.08 for Pb, and 0.38 for O1. The higher values for O atoms indicate either a larger inaccuracy in assessment of their coordinates, or their more pronounced movement. Since there atoms vibrate less then at room temperature, the values are genarally lower then their room temperature equivalent (as can be seen in the case study on PbSO₄ described in the text of the course). Knowing that neutron diffraction measured on a high resolution diffractometer could bring information about the anizotropy character of atom displacement, I also tried to do refinement of B_aniso. Since the sample was measured at 5 K the anizotropy if B factors is almost neglectable, the values B_ii are of the order 0.00xx (positive or negative). Taken into account the inaccuracy caused by the statistical nature of initial data (i.e. the neutron counts), it can be concluded, that the B_anizo coefficients equal to zero.

Refinement using P2₁/a space group
Since all the defined settings for the space group No.14 belong to the solitary space group, the final picture gained after the refinement of data in whatsoever of these settings should be the same. I understand the use of differing setting as a way, how to test the correctness of the refinement (just use a different starting point). Instead of using of P2₁/b space group, which is not supported by the Rietika. (For monoclinic space groups, only b angle is refineable. In the case of P2₁/b, the a angle should be refined).
I have choosen the P2₁/a space group firstly. For the re-calculation of lattice parameters and atomic coordinates for the starting model from P2₁/n to P2₁/a I made use the utility of space group transformation build in the PowderCell program. By means of this I was able to get the needed values for any setting of space group No.14 (see here). For this calculation P2₁/n to P2₁/a I used the refined coordinates refinement "O4(3)". Using the same strategy as in the case of  P2₁/n - O4(3) refinement (two stages, at the end of the first stage only the individual isotropic displacement factors were refined; in the second stage, the anizotropy displacement factors of O1,O4, and O2 atoms were refined) I refined the structure model in P2₁/a. The calculation proceeded as it is illustrated in the following table:
 

For the reference the main output file after the last cycle of refinement can be seen here, here are the output files summarising the distance and angle relations between constituing atoms. Using the *.cif file constructed manually (see here), I was able to draw the picture of the structure of refined model, using the freeware demo version of DIAMOND program. The result is here.
The final fit (together with the details of lower and higher parts of the pattern) is illustrated in this Figure.

The refined parameters and esd's are listed in the tables below:
Cell parameters : a = 8.78431(8) Å,  b = 8.41897(5) Å,  c = 5.37155(5) Å, b = 127.9350(5)°

Atomic coordinates:

Atom	x	y	z	Biso	N
Pb	0.8315(1)	0.8103(1)	0.0731(2)	0.02	1.0
S	0.8167(4)	0.5658(3)	0.5632(8)	0.01(4)	1.0
O(1)	0.9067(2)	0.4075(2)	0.6457(3)	0.30(2)	1.0
O(2)	0.9587(2)	0.6938(2)	0.7152(3)	0.18(3)	1.0
O(3)	0.6934(4)	0.5791(3)	0.6667(3)	0.05(3)	1.0
O(4)	0.6916(2)	0.5834(3)	0.2178(3)	0.22(3)	1.0

Anizotropic displacement factors, B_anizo:

Atom	B11	B22	B33	B12	B13	B23
O(1)	0.0005(3)	-0.0015(3)	-0.0011(3)	-0.0001(2)	-0.0016(3)	-0.0001(4)
O(2)	-0.0002(2)	-0.0003(1)	-0.0052(3)	-0.0002(2)	-0.0039(3)	-0.0009(3)
O(4)	-0.0014(3)	0.0009(3)	-0.0079(3)	0.0000(2)	-0.0044(4)	0.0006(3)

Instrumental : U = 0.039(1)°²,  V = -0.129(3)°²,  W = 0.204(2)°², gama0 = 0.186(6), zero error 0.0918(6)°;

R-Factors :R_wp = 5.66%,  R_exp = 2.80%,  R_p = 4.41%, R_B = 4.32%

Structural Parameters :

Bonds Lengths	Value		Bonds Angles	Value
S-O(1)	1.4717(5) Å		Ð O(1)-S-O(2)	112.57(17)°
S-O(2)	1.4599(6) Å		Ð O(1)-S-O(3)	109.51(24)°
S-O(3)	1.4950(3) Å		Ð O(1)-S-O(4)	108.50(23)°
S-O(4)	1.472(2) Å		Ð O(2)-S-O(3)	107.56(22)°
			Ð O(2)-S-O(4)	110.47(24)°
			Ð O(3)-S-O(4)	108.11(18)°

For the reference, here is the main output file, here is the file summarising the angle and distancies values. Figure of resulting structure refined in P2₁/a (i.e., setting 2) is here, using Diamond program and its cif file (see here).
Retracing the structural parameters of PbSO₄ refined in P2₁/a and comparing them with those obtained in refinement using P2₁/n, one can conclude the the results are the same within the calculation error due to the statistical nature of starting data. This is just an alternative way how to describe the structure (namely using another setting of space group).

Refinement using P2₁/c space group
For the re-calculation of lattice parameters and atomic coordinates for the starting model from P2₁/n to P2₁/c I made use the utility of space group transformation build in the PowderCell program together with the refined coordinates from refinement "O4(3)".Using the same strategy as in the case of  P2₁/a refinement (two stages, at the end of the first stage only the individual isotropic displacement factors were refined; in the second stage, the anizotropy displacement factors of O1,O4, and O2 atoms were refined) I refined the structure model in P2₁/c. The strategy I used is descibed in the following Table with description and number of refined parameters, and values of R-factors:

The final fit (together with the details of lower and higher parts of the pattern) is illustrated in this Figure. For the reference the main output file after the last cycle of refinement can be seen here, here are the output files summarising the distance and angle relations between constituing atoms. Using the *.cif file constructed manually (see here), I was able to draw the picture of the structure of refined model, using the freeware demo version of DIAMOND program. The result is here.

The refined parameters and esd's are listed in the tables below:
Cell parameters : a = 5.37234(6) Å,  b = 8.42020(7) Å,  c = 8.78556(4) Å, b = 127.9348(5)°

Atomic coordinates:

Atom	x	y	z	Biso	N
Pb	0.0730(3)	0.1896(1)	0.8316(1)	0.03	1.0
S	0.5626(9)	0.4343(3)	0.8166(4)	0.06(4)	1.0
O(1)	0.6457(4)	0.5925(2)	0.9068(2)	0.36(2)	1.0
O(2)	0.7149(4)	0.3060(2)	0.9585(2)	0.24(3)	1.0
O(3)	0.6665(3)	0.4210(3)	0.6932(2)	0.09(3)	1.0
O(4)	0.2176(4)	0.4165(3)	0.6916(2)	0.26(3)	1.0

Anizotropic displacement factors, B_anizo:

Atom	B11	B22	B33	B12	B13	B23
O(1)	-0.0006(3)	-0.0015(2)	0.0006(3)	0.0001(4)	-0.0014(4)	0.0000(2)
O(2)	-0.0048(8)	-0.0003(2)	0.0000(3)	0.0011(3)	-0.0037(4)	0.0000(2)
O(4)	-0.0065(8)	0.0008(2)	-0.0013(3)	-0.0005(3)	-0.0040(4)	-0.0001(2)

Instrumental : U = 0.041(6)°²,  V = -0.1359(31)°²,  W = 0.2075(18)°², gama0 = 0.205(6), zero error 0.091887(6)°;

R-Factors :R_wp = 5.61%,  R_exp = 2.80%,  R_p = 4.01%, R_B = 4.16%

Structural Parameters :

Bonds Lengths	Value		Bonds Angles	Value
S-O(1)	1.4723(5) Å		Ð O(1)-S-O(2)	112.59(18)°
S-O(2)	1.4613(6) Å		Ð O(1)-S-O(3)	109.42(22)°
S-O(3)	1.4982(3) Å		Ð O(1)-S-O(4)	108.68(24)°
S-O(4)	1.4701(2) Å		Ð O(2)-S-O(3)	107.44(24)°
			Ð O(2)-S-O(4)	110.49(22)°
			Ð O(3)-S-O(4)	108.12(19)°

Retracing the structural parameters of PbSO₄ refined in P2₁/c and comparing them with those obtained in refinement using P2₁/n, one can conclude the the results are the same within the calculation error due to the statistical nature of starting data. This is just an alternative way how to describe the structure (namely using another setting of space group).

Conclusions
The data of powder neutron diffraction measured on high resolution diffractometer (D2B, at the ILL, Grenoble, France) at 5 K on powderred sample of PbSO₄ were refined using Rietweld refinement procedure implemented in Rietika program. Several chioses of settings of space group No.14 were used. The final structure model is then described in either P2₁/n, P2₁/a, or P2₁/c setting, as it can be seen above.
The structure of PbSO₄ is  composed  from deformed SO₄^2- tetrahedra, where only one angle (namely O(1)-S-O(4)) gets near to the ideal 109.5°. The values of S - O bond lenghts lie in the range from 1.46 to 1.50 Å, which are usual values for bonds with high proportion of ionic character. The tetrahedra are not connected each to other, with the Pb atoms in between. The individual isotropic B_iso displacement factors of atoms take values between 0.08 for Pb, and 0.38 for O1. The higher values for O atoms indicate either a larger inaccuracy in assessment of their coordinates, or their more pronounced movement. The attempt to refine B_aniso showed that at 5 K the anizotropy of B factors is almost neglectable.

Powder XRD versus neutron diffraction
If appropriate low-temperature data had been collected using a conventional laboratory X-ray diffractometer in Bragg-Brentano geometry (instead of using the neutron powder diffractometer) I would expext to obtain results results, which would differ in the following manner:

• Since the data sets are collected with different wavelengths, peaks due to identical d spacings will obviously appear at different angles (due to Bragg's law). The relative intensities of adjacent peaks will be also very different in ND and XRD data sets. Reflections that exhibit a high intensity in the X-ray data set will be be relatively weak when measured by neutron diffraction, and vice-versa. These differencies constitute a possibility to gain a more complex picture of a powder sample when combination of these two methodologies is used.
• For the X-ray data set, the intensity of the reflections decays with increasing scattering angle due to the effect of the X-ray form factor, f, in constrast to that observed in the neutron data set, where the peaks are still intense at high scattering angle. Usually, the peaks are a lot broader in the neutron data set than in the X-ray data set due to the different instrumental resolution functions. It comes to this, that use of ND is particularly useful when the assessment of individual (or even anizotropy) displacement factors is needed, and contrary, XRD is more useful when one faces the problem of pattern indexing (narrower peaks, especially in a higher angle region).
• In case when a material with an pronounced shape of grains would be analysed, then definitely preferred orientation would (negatively) influence the picture gained by XRD, whereas it would has low influence on the results gained by ND.
• In case when a material containing ferromagnetic elements (Fe, Ni, Co...) would be analysed, then picture gained by means of ND would bring more complex information (using polarised neutrons).