1 Lecture 9: Worked Example

Crystallography Supplementary Subject Lecture 9: Worked Example 1 Lecture 9: Worked Example Massicot is a naturally-occurring form of lead(II) oxide produced by the oxidation of lead ores. Unfortunately, it is not one of the most visually spectacular minerals known to mankind. But it is one of the most dense: a cubic centimetre of the material weighs in at just under 10 g, which is only slightly less than metallic lead itself. Together with its dimorphic cousin litharge (the more common and more colourful form of PbO), massicot has been used for centuries in the manufacture of paints, ceramics, glasses, and – more recently – in batteries. The idea of this exercise is to use x-ray diffraction images to determine the space group of massicot and to draw some conclusions regarding the arrangement of Pb and O atoms in its structure. Unit cell (i) Diffraction photographs taken parallel to the three crystal axes are shown overleaf. The axis labels are arbitrary, but consistent across the three images. To what point group does the diffraction pattern belong, and hence in which crystal system does massicot crystallise? (ii) It is known from independent physical measurements that the crystal structure of massicot must be centrosymmetric. To what point group must its structure belong? Crystallography Supplementary Subject Lecture 9: Worked Example 2 Crystallography Supplementary Subject Lecture 9: Worked Example 3 (iii) The central squares drawn on each diffraction photograph are scaled such that they cut the axes at points in the diffraction pattern exactly 1 A˚ −1 away from the central (000) peak. Use this to calculate the (real-space) lattice parameters of massicot. (iv) Recalling that the density of PbO is close to 10 g cm−3, calculate how many formula units must be contained within each unit cell. (Atomic masses: Pb = 207.2 g mol−1; O = 15.999 g mol−1) (v) Illustrate on the diagrams which are the (020) and (102)¯ reﬂections. Space group (vi) List the reﬂection conditions for each of the three planes shown in the diffraction photographs, and decide the minimum number of rules needed to account for all systematic absences. (vii) Hence determine the space group, taking into account the fact that massicot is centrosymmetric. (viii) Draw the corresponding space group diagram, using a view that includes the a and b axes within the plane of the paper. Include on your diagram the rotation or screw axes that are generated by other symmetry elements, and also indicate the locations of the centres of symmetry. (ix) Quickly re-draw the diagram placing a centre of symmetry at the origin. Check that your diagram corresponds to that given in the “International Tables”. (x) Which acentric spacegroup would have been equally consistent with the given diffraction patterns? (xi) The particular space group labels at which you have arrived will have been deter- mined by the labelling of a∗, b∗ and c∗ given on the diagrams overleaf. These labels are of course arbitrary, since the crystal does not know which axis is which. Desig- nate the axes differently, and then derive the corresponding space group symbol for your new assignment of axes. Convince yourself that the two space group symbols you have obtained are equivalent. (xii) How many general positions are there in the original centrosymmetric space group? Give the coordinates of the general positions and relate each to one of the symmetry operations of the space group. What implication does the number of general positions have for the possible positions of the Pb and O atoms in massicot?.

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