IB Mineral Sciences
Module B: Reciprocal Space, Symmetry and Crystallography
BD3
Solving the crystal structure of massicot (PbO)
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.
In this demonstration we are going to attempt to solve the crystal structure of massicot using x-ray diffraction data. The idea of the exercise is to bring together the different aspects
of the course: we are going to need to use our knowledge of symmetry and reciprocal
space and our understanding of diffraction experiments themselves. Hopefully, by the end
of the exercise you will have an appreciation for how all the techniques we have learned
come together to enable us to determine the arrangement of atoms in minerals.
This demonstration uses a java applet, which is located at:
http://rock.esc.cam.ac.uk/∼alg44/teaching ib bd3.html
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?
IB Mineral Sciences
Module B: Reciprocal Space, Symmetry and Crystallography
BD3
IB Mineral Sciences
Module B: Reciprocal Space, Symmetry and Crystallography
BD3
(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 Å−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 (102̄) 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) How many general positions are there in this space group? What general implication
does this have for the positions of the Pb and O atoms?
Atom positions
(x) The atomic number of Pb is 82 while that of O is 8. What does this tell us about the
sensitivity of x-ray diffraction patterns to the positions of the Pb and O atoms?
(xi) We know that the Pb atoms must lie on a symmetry element. Using your space group
diagram, see if you can determine the four different sets of positions possible for the
Pb atoms. As a hint, two of these can be obtained by placing Pb atoms on centres of
symmetry, and the other two will involve other symmetry elements.
In order to distinguish between these four possibilities, we are going to use the Patterson
method. This involves calculating the Fourier transform of the observed intensities, which
gives us a map of the different interatomic vectors (the Patterson map). Instead of calculating the entire three-dimensional Fourier transform, we are going to restrict ourselves to
the (hk0) reflections, which will give us information about the structure as if it had been
projected onto the (a,b) plane.
(xii) For each of the four possible sets of Pb atom positions, sketch the distribution of the
first few interatomic vectors projected onto the (a,b) plane. Use these to suggest what
the corresponding Patterson map would be expected to look like in each case.
IB Mineral Sciences
Module B: Reciprocal Space, Symmetry and Crystallography
BD3
Now open the web page
http://rock.esc.cam.ac.uk/∼alg44/teaching ib bd3.html
to access the java applet we are going to use:
The way this applet works is that we can enter the intensities of (hk0) reflections and it will
automatically calculate the Patterson map as we go. To enter intensities, you have to select
the appropriate reflection first. This can be done by clicking the individual reflection in the
top right-hand panel, or by entering the h and k values in the top left-hand panel. We then
enter the intensity value in the “|F|” box (making sure the phase is set to zero), and select
“Set SF”. You should notice that the relevant square on the reflection diagram becomes
coloured red (the intensity being proportional to the value just entred), and the Patterson
map in the bottom right-hand corner should change. The program already knows Friedel’s
law, so entering an intensity for (hk0) will add the same value for (h̄k̄0); however, you will
need to apply any additional Laue symmetry yourself.
(xiii) The strongest measured intensities for |h,k| ≤ 4 are listed overleaf. Enter these into
the applet and sketch the Patterson map it calculates. Does this correspond to any of
the maps you were expecting? Can you now deduce the positions of the Pb atoms?
(xiv) What additional information would you need to be able to determine the positions of
the O atoms?
(xv) Draw a quick sketch of the crystal structure of massicot, illustrating just the Pb atoms.
IB Mineral Sciences
Module B: Reciprocal Space, Symmetry and Crystallography
Reflection
(000)
(010)
(200)
(020)
(220)
(030)
(310)
Intensity (arb. units)
1000
88
938
901
481
132
115
Reflection
(230)
(400)
(040)
(420)
(240)
(440)
Intensity (arb. units)
60
152
151
152
144
103
BD3