Introduction
The project is based on refining different data sets of lead sulphate at low temperature, with the aim of becoming fully competent with
the method of Reiveld refinement and usage of Rietica software. The procedures of refining different data sets of lead
sulphate at low temperature, which leads to different model arrangements of initial structures, are presented and illustrated step by
step in full detail. The discussion on effects of results of each successful refinement in different models are pointed
out and explained in this project. Finally, the information is given on the correct low-temperature structure of lead sulphate in a form that complies with requirements for
journal submission.
The refinement of the structure of PbSO4
1. Data set A
The lead sulphate data set A was obtained from the high-resolution powder diffractometer (i.e. Neutron diffractometer) at wavelength l = 1.594Å and low temperature 5K.
The space-group symmetry Pbnm of the crystal structure of lead sulphate at room temperature was used as a starting model for refinement of data set A.
The symmetry operators for this setting are given in the table below:
Cell parameters and approximate coordinates for the structure of PbSO4
under space group symmetry Pbnm are in the table below:
Cell parameters : a = 7.0 Å, b = 8.5 Å,
c = 5.4 Å
Stage 1 Stage 2
Stage 3
Stage 4
Stage 5
Stage 6
Stage 7
Stage 8
Stage 9
Stage 10
Stage 11
Stage 12
Cell parameters : a = 6.9257(7) Å, b = 8.4137(9) Å,
c = 5.3669(6) Å
R-Factors : Rp = 6.2%,
Rwp = 8.07%, Rexp = 2.87%
Structural Parameters :
Conclusions:
The successful refinement of the crystal structure of lead sulphate at low temperature(5K) with 2q range
8°-62.475° using space group Pbnm as a model for room temperature was achieved using the above stages. A plot having both value
of Rw factor and refinement stage is given below in Plot 1, which shows that the Rw factor fell
rapidly in the early stages of the refinement, but changed less towards the end of the
refinement. Each stage of decrease is different that indicates different parameters have significantly different
effect.
Space group 62 : Pbnm
x,
y,
z
1
-x,
-y,
-z
-1 at (0,0,0)
1/2-x,
1/2+y,
z
b at (1/4,y,z)
1/2+x,
1/2-y,
-z
21 at (x,1/4,0)
1/2+x,
1/2-y,
1/2+z
n at (x,1/4,z)
1/2-x,
1/2+y,
1/2-z
21 at (1/4,y,1/4)
x,
y,
1/2-z
m at (x,y,1/4)
-x,
-y,
1/2+z
21 at (0,0,z)
The sulphate ion with 4 oxygen atoms is tetrahedral, so the calculation of
the sulphate ion geometry using the above coordinates gives values of 1.1 to 2.2Å for the S-O bond lengths and 73°-114°
for the Ð O-S-O bond angles. In space group there is a mirror plane upon which the crystallographic atoms sit.
So the site occupation number, n, is 0.5 for Pb, 0.5 for S, 0.5 for O(1), 0.5 for O(2) and 1 for O(3).
Atom
x
y
z
n
Pb
0.15
0.20
1/4
0.5
S
0.20
0.50
3/4
0.5
O(1)
0.10
0.60
3/4
0.5
O(2)
0.00
0.30
3/4
0.5
O(3)
0.30
0.40
0.9
1.0
In addition, the resolution function of the diffractometer is characterised by the UVW values given below,
and in the diffraction pattern, the peak shape is nearly Gaussian with G = 0.05.
Initially, the diffraction data background count was estimated without the structural model to contribute diffraction
intensity to the calculated profile, so the weighted profile R-factor is 100%.
Starting parameters of the crystal structure of lead sulphate using the room temperature model and instrument were set in the Rietica program, which gave a Rw factor of
63.65%.
In the first Rietveld refinement, the procedure is to refine the overall scale factor, c, which includes all instrumental effects. Because scale factor is the
only linear parameter in the least square, the scale factor was refined in only one cycle of least square giving a value of
Rw = 60.27% as shown in Figure 1-1.
As precisely unknown the unit cell, the refinement of 2q zero error must be achieved. At
this stage, 2q zero was refined and the Rw factor dramatically reduced from 60.27% to 41.86%
shown in Figure 1-2.
The unit cell parameters with orthorhombic structure(Pbnm) with the room temperature model were refined and the Rw factor reduced from 41.86%
to 34.93% with the profile fit as shown in Figure 1-3.
A good fit between the observed and calculated peak position tends to improve the matching peak position. The main effect
on match in peak position is the refinement of the atomic coordinates. According to that, the usual approach will be to
vary the atomic coordinates of the strongest scattering with the most degrees of freedom (i.e. refine atoms on
general xyz positions first in preference to atoms on special positions), the O(3) atomic coordinates firstly were
refined and the value of Rw factor was reduced from 34.93% to 23.94% as shown in Figure 1-4.
The next refinement of atomic coordinates is Pb with a Rw factor of 22.83% as shown in Figure 1-5.
In this instance, z coordinate is limited and is fixed as ¼ due to special position.
The refinements of O(1) and O(2) with fixed z coordinates were achieved as a value of 16.23% decreased from 22.83% of the Rw factor shown in Figure 1-6.
In contrast, X-ray, for neutrons sulphur is the weakest scatterer, so the last refinement of atomic coordinates is performed for sulphur(S).
The refinement remained a value of Rw factor as 12.87% shown in Figure 1-7.
The Rw factor is still too high and the peaks fit is unsatisfied. There are many remaining variable factors in the refinement process.
Refinement of an overall temperature factor and the values of UVW associated with the peak width and shape now reduced the Rw factor from 12.87% to 9.86%
as shown in Figure 1-8.
In this case, all ions have no behavior of librational motion, so as to cause their atomic displacement to be isotropic.
Therefore, the isotropic temperature factors of each atom were fixed at a default value 0.6; These have to be applied
to refine the 6 anisotropic temperature factors, bij. In addition, there is a mirror
plane that is perpendicular to the z direction in space group Pbnm and upon which the atoms Pb, S, O(1) and O(2) lie, so anisotropic temperature factors of atoms Pb, S, O(1) and O(2),
bij, must satisfy the condition b13 = b23 =0.
The value of the Rw factor was reduced from 12.87% to 8.92% as shown in Figure 1-9.
Finally, the refinement focuses on peak shape as Gaussian factor. The final fit with a value 8.07% of the Rw factor is
illustrated in Figure 1-10, and the refined parameters and esd's (estimate standard deviation) are listed below. A standard CIF file was produced also.
Atom
x
y
z
B(iso)
Pb
0.1684(6)
0.1912(3)
1/4
0.60(0)
S
0.1557(23)
0.4440(22)
3/4
0.60(0)
O(1)
0.0917(9)
0.6037(10)
3/4
0.60(0)
O(2)
0.0547(15)
0.3029(8)
3/4
0.60(0)
O(3)
0.3067(10)
0.4144(5)
0.9745(10)
0.6
Atom
B11
B22
B33
B12
B13
B23
Pb
0.303(1968)
0.0163(1333)
0.0362(3277)
-0.0038(11)
0
0
S
0.0820(1967)
-0.0300(1332)
0.0029(3276)
-0.0035(47)
0
0
O(1)
0.0591(1968)
0.0024(1333)
0.0755(3276)
-0.0073(17)
0
0
O(2)
0.0489(1967)
-0.0182(1333)
0.0808(3276)
0.0073(17)
0
0
O(3)
0.0275(1968)
0.0243(1333)
0.0380(3277)
-0.0106(17)
-0.0020(21)
-0.0034(16)
Instrumental :
U = 2.0624°2, V = -1.645°2,
W = 0.4961°2, 2qzero = 0.058
Bonds Lengths
Value
Bonds Angles
Value
S-O(1)
1.414(13) Å
Ð O(1)-S-O(2)
131.24(1.25)°
S-O(2)
1.377(15) Å
Ð O(1)-S-O(3)
110.45(82)°
S-O(3)
1.615(12) Å
Ð O(1)-S-O(3)¢
110.45(82)°
S-O(3)¢
1.615(12) Å
Ð O(2)-S-O(3)
101.3(8)°
Ð O(2)-S-O(3)¢
101.3(8)°
Ð O(3)-S-O(3)¢
96.52(8)°
From the above refined parameter, the angle Ð O(3)-S-O(3)¢(96.52°) is significantly different from the ideal tetrahedral value of 109.5° and
S-O bonds length calculated(1.377Å) are significantly smaller than the ideal value of 1.6Å. Probably, a incorrectly chosen refinement model for this crystal limited the
reasonable arrangements of bonds and angles of each atom causing the observed deviations from the ideal value.
In contrast to the structure refinement of the data set at room temperature, the value of Rw factor at low
temperature is quite poor. The possible reason is that at low temperature (5K),
a slight distortion of the unit cell occurs with consequent loss of the perfect data set related to
space group Pbnm as a model of refinement. The refinement of isotropic temperature factor of the sulphate ion at low
temperature is also different from that at room temperature, the possible reason is that at low temperature,
the sulphate atom usually vibrates equally in all directions due to slight distortion of the unit cell.
2. Data set B
The first data set A is simply a subset based on a limited 2q range of the second data set B (8° to 142.975° in steps of 0.025°) without changing any test condition. In the procedures of refinement of data set B, the two lower symmetry space groups (P21/n and P21/b) of Pbnm need to be considered, because at low temperature in the structure of lead sulphate there is a slight distortion of the unit cell which occurs with consequent loss of the mirror plane symmetry, on cooling the sample to 5K. These happen to be different arrangements of the symmetry elements from data set A.
(1) P21/n
Approximate coordinates for the structure of PbSO4 at low temperature with respect to space group P21/n are give in the table below:
Cell parameters : a = 7.0 Å, b = 8.5 Å, c = 5.4 Å
| Atom | x | y | z | n |
|---|---|---|---|---|
| Pb | 0.15 | 0.20 | 0.25 | 1.0 |
| S | 0.20 | 0.50 | 0.75 | 1.0 |
| O(1) | 0.10 | 0.60 | 0.75 | 1.0 |
| O(2) | 0.00 | 0.30 | 0.75 | 1.0 |
| O(3) | 0.30 | 0.40 | 0.9 | 1.0 |
| O(4) | 0.30 | 0.40 | -0.4 | 1.0 |
In contrast to the refinement of data set A, there are coordinates of an extra oxygen atom in the above table. The reason is that when low symmetry with space group P21/n is considered in the procedure of refinement, i.e. space group Pbnm is transformed into space group P21/n, the structure of lead sulphate loses it's mirror plane between O(3) and O(4) due to the low temperature so that coordinates of O(3) are separated into two parts evenly. Also the refinement of the coordinations of atoms Pb, S, O(1) and O(2) along the z direction are considered, because the unit cell is thought to transform into the monoclinic space group. Other conditions of the structure of lead sulphate and the instrument conditions are the same as for data set A.
The initial background count of the diffraction data without any conditions such as instrument parameters
and crystal parameters gave a value 100% of Rw. After the parameters of structure model at room temperature and conditions
of the instrument were entered into the Rietica program, the value of Rw
was 63.16% with refinement shown in Figure 2-1.
The next stages of refinement were to refine the parameters in order of phase scale factor (c),
zero error and unit cell parameter(except angle), which reduced Rw to values of 59.03%, 52.29% and 47.19% respectively.
The usual approach will be to vary the atomic coordinates of the strongest scattering with most
degrees of freedom; The atomic coordinates in order of O(3) and O(4), Pb, O(1) and O(2), and S were
refined, giving Rw value of 32.09%, 30.42%, 26.08%, and 20.97% respectively.
At this stage, the refinement of angles has involved the transformation of the unit cell from high symmetry (orthorhombic)
to low symmetry (monoclinic). The angle b was refined and the value of Rw decreased dramatically from 20.97%
to 10.51%; a good fit is shown in figure 2-2.
So far the quality of Rw factor is reasonable, but the refinements of the thermal temperature factor
and the calculated peak widths as determined by resolution-function parameters UVW can further improve the quality of Rw. A
value of Rw factor was further reduced to 9.08% and the fit is shown in Figure 2-3.
At low temperature, atoms usually vibrate equally in all directions, but in this case the anisotropic temperature factors
were refined because of the transformation of the unit cell. So according to the above, the isotropic displacement parameters are limited in
the default value 0.6. The refinement of anisotropic displacement parameters has to be applied to the 6 anisotropic
temperature factors, bij. At this stage, Rw factor was further decreased to 8.52% with the profile shown in Figure 2-4.
Finally, the refinement of Gaussian parameter gave a good peak fit and stable value of Rw factor
7.61% with the profile shown in Figure 2-5, and the refined parameter and esd'
are listed in the table below. A standard CIF file was produced.
Cell parameters : a = 6.9275(2) Å, b = 8.4192(2) Å, c = 5.3704(1) Å, b = 90.2351°
| Atom | x | y | z | B(iso) |
|---|---|---|---|---|
| Pb | 0.1683(2) | 0.1900(1) | 0.2595(3) | 0.60(0) |
| S | 0.1847(5) | 0.4331(5) | 0.7556(11) | 0.60(0) |
| O(1) | 0.0928(2) | 0.5913(2) | 0.7620(5) | 0.60(0) |
| O(2) | 0.0415(3) | 0.3072(2) | 0.7437(5) | 0.60(0) |
| O(3) | 0.3085(3) | 0.4189(4) | 0.9724(4) | 0.60(0) |
| O(4) | 0.3053(3) | 0.4182(4) | -0.4734(4) | 0.60(0) |
| Atom | B11 | B22 | B33 | B12 | B13 | B23 |
|---|---|---|---|---|---|---|
| Pb | 0.0032(701) | 0.0026(475) | 0.0082(1166) | -0.0001(2) | -0.0023(3) | -0.0012(3) |
| S | 0.0014(701) | 0.0073(475) | 0.0088(1166) | -0.0039(6) | -0.0012(8) | 0.0070(12) |
| O(1) | 0.0055(701) | 0.0017(475) | 0.0137(1166) | 0.0006(2) | -0.0010(3) | -0.0013(5) |
| O(2) | 0.0052(701) | 0.0031(475) | 0.0104(1166) | 0.0014(3) | -0.0029(3) | -0.0003(5) |
| O(3) | 0.0046(701) | 0.0049(475) | 0.0067(1166) | 0.0021(3) | -0.0007(2) | 0.0037(4) |
| O(4) | 0.0057(701) | 0.0030(475) | 0.0070(1166) | -0.0008(3) | 0.0006(1) | 0.0055(4) |
R-Factors : Rp = 5.82%, Rwp = 7.61%, Rexp = 2.79%
Structural Parameters :
| Bonds Lengths | Value | Bonds Angles | Value | |
|---|---|---|---|---|
| S-O(1) | 1.4760(0) Å | Ð O(1)-S-O(2) | 111.38(0.24)° | |
| S-O(2) | 1.4533(0) Å | Ð O(1)-S-O(3) | 108.04(0.36)° | |
| S-O(3) | 1.4484(0) Å | Ð O(1)-S-O(4) | 109.73(0.29)° | |
| S-O(4) | 1.4953(0) Å | Ð O(2)-S-O(3) | 112.15(40)° | |
| Ð O(2)-S-O(4) | 113.22(36)° | |||
| Ð O(3)-S-O(4) | 108.97(57)° |
Conclusion
A plot of the weighted-profile R-factor during the different stages of the
refinement is shown in Plot 2. The plot clearly shows the different refinement of parameter
affect on the value of the Rw factor at each refinement stage. When the angle b of the unit cell
was refined, the value of Rw decreased steeply and approached 10.51%.
In contrast to the refinement of data set A, more parameters (coordinations z and angle b) are involved
in the refinement and a better fit is given.
Model P21/n indicates that most angles of O-S-O have
reasonable values in the range 113°-108° compared to the ideal tetrahedral value of 109.5°, and S-O bond lengths have values in the range 1.44-1.49Å
compared to the ideal value of 1.48Å.
The above result gave three types of value R factor, i.e. Rp, Rw and Rexp. The low value
of Rexp(2.79%) indicates that the data are of high quality, while the relatively high value for the weighted-profile
R-factor(7.61%) indicates the structural parameters are reasonable and that the profile parameters are poor, e.g.
a wrong peak-shape function may have been used.
(2) P21/b
Another low symmetry space group of Pbnm is P21/b with the symmetry elements relating to the
x-axis. As previously discussed, at low temperature, the space group of lead sulphate crystal may be
transformed into P21/b, so in the next stage the crystal structure of lead sulphate with space group
Pbnm at low temperature will be involved in the discussion.
At this point P21/b, the symmetry elements are along the y-axis direction. In order to enable
symmetry elements of space group P21/b to be along the x-axis direction with respect to space group
symmetry Pbnm, it is simple to interchange both the a and b cell parameters and the x and
y coordinates of all of the atoms, the space group then becomes P21/a with the screw
axis along the y direction.
Approximate coordinates for the structure of PbSO4 with space group symmetry P21/a are
given in the table below:
Cell parameters : a = 7.0 Å, b = 8.5 Å, c = 5.4 Å
| Atom | x | y | z | n |
|---|---|---|---|---|
| Pb | 0.15 | 0.20 | 0.25 | 1.0 |
| S | 0.20 | 0.50 | 0.75 | 1.0 |
| O(1) | 0.10 | 0.60 | 0.75 | 1.0 |
| O(2) | 0.00 | 0.30 | 0.75 | 1.0 |
| O(3) | 0.30 | 0.40 | 0.9 | 1.0 |
| O(4) | 0.30 | 0.40 | -0.4 | 1.0 |
In the above table, the atom O(4) is also considered and the coordinations of each atom along the z direction were altered. Other conditions for the refinement of the structure of PbSO4 at low temperature are the same as for the model P21/n and the procedure of refinement is the same with P21/n as well. The stages of refinement with a relative value of Rw factor are tabulated below:
Stage 1 2 3 4 5 6 7
Rw 100% 64.45% 60.06% 53.15% 48.06% 34.45% 31.31%
Stage 8 9 10 11 12 13
Rw 26.53% 21.81% 20.53% 15.93% 14.76% 13.91%
Cell parameters : a = 6.92729(19) Å, b = 8.41872(17) Å, c = 5.36865(16) Å, b = 90.0731°
| Atom | x | y | z | B(iso) |
|---|---|---|---|---|
| Pb | 0.1686(4) | 0.1907(2) | 0.2570(9) | 0.60(0) |
| S | 0.1815(9) | 0.4284(9) | 0.7545(21) | 0.60(0) |
| O(1) | 0.0954(5) | 0.5884(4) | 0.7560(16) | 0.60(0) |
| O(2) | 0.0398(6) | 0.3080(5) | 0.7467(15) | 0.60(0) |
| O(3) | 0.3132(7) | 0.4209(5) | 0.9640(11) | 0.60(0) |
| O(4) | 0.2970(8) | 0.4146(8) | -0.4847(10) | 0.60(0) |
| Atom | B11 | B22 | B33 | B12 | B13 | B23 |
|---|---|---|---|---|---|---|
| Pb | -0.0029(274) | -0.0041(404) | 0.0140(673) | -0.0006(3) | 0.0073(7) | 0.0000(0) |
| S | 0.0052(273) | -0.0071(404) | -0.0181(672) | 0.0042(12) | 0.0049(22) | 0.0000(0) |
| O(1) | -0.0049(274) | 0.0045(404) | 0.0058(672) | 0.0018(6) | 0.0044(12) | 0.0000(0) |
| O(2) | -0.0024(274) | 0.0007(404) | -0.0035(673) | -0.0006(5) | -0.0107(11) | 0.0000(0) |
| O(3) | -0.0096(274) | -0.0114(404) | 0.0332(672) | -0.0041(5) | -0.0006(9) | 0.0000(0) |
| O(4) | 0.0047(273) | 0.0080(404) | -0.0282(673) | 0.0079(9) | 0.0043(7) | 0.0000(0) |
R-Factors : Rp = 10.61%, Rwp = 13.91%, Rexp = 2.80%
Structural Parameters :
| Bonds Lengths | Value | Bonds Angles | Value | |
|---|---|---|---|---|
| S-O(1) | 1.4736(0) Å | Ð O(1)-S-O(2) | 112.02(0.45)° | |
| S-O(2) | 1.4114(0) Å | Ð O(1)-S-O(3) | 106.94(0.69)° | |
| S-O(3) | 1.4496(0) Å | Ð O(1)-S-O(4) | 106.676(0.67)° | |
| S-O(4) | 1.5175(0) Å | Ð O(2)-S-O(3) | 115.41(0.68)° | |
| Ð O(2)-S-O(4) | 107.00(0.39)° | |||
| Ð O(3)-S-O(4) | 108.72(0.49)° |
Conclusion
A plot of the value of Rw in the different stages of the refinement is illustrated in
Plot 3. It clearly indicates a significant effect in the different stage of the refinement.
From the above structural parameter, the angles of Ð O(2)-S-O(3) = 115° deviates significantly form the ideal tetrahedral value of 109.5° and the S-O(4) bonds length of 1.51Å compares to an ideal 1.48Å. Apparently, the above refinement is not particularly accurate, while the Rw gave a high value. So the model with space group P21/b is not suitable for refinement of lead sulphate at 5K.
3. Discussion and conclusions
The above three models, which are part of the diffraction pattern of lead sulphate at 5K with Pbnm space group; whole diffraction pattern of lead sulphate at 5K with P21/n; and whole diffraction pattern of lead sulphate at 5K with P21/b, have been refined by the Rietica program. The first model indicates that at low temperature, the diffraction data of lead sulphate with angles between 8°-62° possesses the same structure as for room temperature and temperature does not affect X-ray data at low angle for lead sulphate with respect to low Rw 8.07. Moreover, there is significant noise in the deference plot (Figure 1-10) between 40°-45° of 2q angle. Presumably, changing temperature alters the unit cell slightly. As a part of X-ray data collected from low temperature, the structure solved does not represent an actual structure at low temperature due to the lack of information of the whole structure, although the low value of Rw is given. So using space group Pbnm as a model to solve the structure of lead sulphate at low temperature is incorrect.
The second result (Rw) of refinement with space group P21/n using data set B shows that at low temperature, the unit cell of lead sulphate is distorted, transforming from high symmetry space group Pbnm to low symmetry space group P21/n, and the thermal displacement of each atom in the low symmetry model behaves anisotropically due to the offset for distortion of the unit cell. In the final refinement plot, the difference plot (Figure2-5) is nearly a straight line with little noise, so the refinement under model P21/n is successful and gives a reasonable Rw value.
The third result (Rw) of refinement with space group P21/b using data set B indicates that a unsuccessful refinement was achieved due to a high value of Rw In Figure 3-2, the difference plot is rough with more noise. So, lead sulphate at low temperature has only one structure with respect to space group P21/n. In summary, the refinement of materials at low temperature is more complex than at room temperature, and refinable elements are notably different at the different temperature.
The above refinement of lead sulphate based on the high resolution powder diffractometer at low temperature using three different refinement models was carried out and resulted. Comparing the refinement of lead sulphate at room temperature, the refinements demonstrate: [1]. the same material at different temperature has different structures; [2].the refinement model at room temperature is different in comparison to low temperature. At low temperature, atoms tend to be packed close so that the symmetry elements at room temperature are lost; [3]. the structure at low temperature is more complex and the space group at low temperature forms subsets of the space group at room temperature.
When using a conventional laboratory X-ray diffractometer in Bragg-Brentano geometry (angle-dispersive X-ray diffractometer) instead of using the neutron powder diffractometer for collecting the
appropriate low temperature data, initially some modifications to the basic parameters are required:
(1). X-ray form factors, f(2q), must be used instead of the neutron scattering lengths, b, in the
calculation of the structure factors; (2). A polarization correction, P(2q), must be included in
addition to the Lorentz factor data collected with laboratory X-rays, together with an absorption
correction , A(2q), for data collected in capillary geometry; (3). A different (non-Gaussian) peak-
shape function is required.
According to laboratory X-ray data, the peak-shape function is mainly dependent on the instrumental
set-up. As with a narrow-band pass primary-beam monochromator for the laboratory X-ray diffractometer, the
pseudo-Voigt peak-shape function is empirically found to be most suitable. For the X-ray diffractometer
with only nickel filters or graphite monochromators, peak shapes based on the less symmetric function are
preferred.
Benefits of using the X-ray diffractometer are a better quality and higher resolution of the collected X-ray data
together with improvements in the design of the X-ray powder diffractometer.
|
yuan@img.cryst.bbk.ac.uk School of Crystallography Birkbeck College, University of London Malet Street, London WC1E 7HX, UK |
©Zhanhui Yuan |
