research papers
U_{4}O_{9}: atoms in general sites giving the hkl of special sites
^{a}Frank Laboratory of Neutron Physics, Joint Institute for Nuclear Research,141980 Dubna, Moscow Region, Russia, ^{b}National Institute for Materials Physics, PO Box MG7, Bucharest, Romania, and ^{c}Chemistry Research Laboratory, University of Oxford, Mansfield Road, Oxford OX1 3TA, England
^{*}Correspondence email: bertram.willis@chem.ox.ac.uk
In U_{4}O_{9}, most of the atoms are in general 48fold [48(e)] sites of the I3d and yet the hkl reflections obey the extinction rules for atoms occupying special 12fold [12(b)] sites. An explanation is given for this effect, which can be generalized to any space group.
1. Introduction
The β phase of U_{4}O_{9}, which is stable above 353 K, was determined by Bevan et al. (1986) from the analysis of singlecrystal neutrondiffraction data. They showed that most of the atoms in the occupy general 48fold [48(e)] sites of the I3d Nevertheless, experiments using Xray, neutron and electron diffraction have all confirmed that the hkl reflections conform to the of the special 12fold [12(b)] sites. How is it possible for most of the atoms to occupy general positions and yet to give systematic hkl absences corresponding to special positions? In the following section, we shall show that the answer to this question is that the structure consists of several types of atomic clusters that are centred on the special sites. The cluster associated with each type must possess just one kind of atom, have the pointgroup symmetry of the crystal, and have the same orientation on being translated between equivalent positions of the special site. Before proceeding to detailed calculations, we shall summarize the extensive data on the symmetry of the diffraction pattern of U_{4}O_{9}.
of theBelbeoch et al. (1961) studied U_{4}O_{9} using singlecrystal and powder Xray diffraction. They found that all the observed reflections, which covered an intensity range from 1 to 2500, obeyed the extinction rules of the 12(b) positions of the I3d On account of these rules, they stated that the structure consists of a `unique motif' placed around the 12(b) sites and that this motif preserves the same orientation wherever it is applied. We shall use a similar concept in the treatment below.
Single crystals of U_{4}O_{9} in the form of thin foils were examined by Blank & Ronchi (1968) employing electron diffraction. The diffraction pattern revealed numerous reflections including some not found by Xrays, especially at small angles of scattering. The conditions for possible reflections corresponded to the I3d, and all the were those associated with the 12(b).
Neutron diffraction studies of single crystals have been undertaken by Willis (1964), Masaki & Doi (1968), Bevan et al. (1986) and Lauriat et al. (1989). Masaki & Doi observed two weak reflections, 770 and 990, and so they reassigned the as I4_{1}32. The other workers failed to confirm the existence of these weak reflections, and they all agreed with Perio's assignment of I3d. Lauriat et al. (1989) stated: `We find that reflections whose indices are h, k, l = 8n ± 1, 8n ± 1, 4n and 8n ± 3, 8n ± 3, 4n are unobservable without exception. These indices correspond to the extinction rules of the 12(a), 12(b) positions of the I3d Bevan et al. (1986) found that these extinction rules are obeyed at a temperature of 503 K, but Cooper & Willis (2004) found that they break down at 773 K. We shall proceed on the assumption that the reflections obey the 12(b) extinction rules of I3d at low temperatures.
The complete set of rules for 12(b) [or 12(a)] is given in Table 1, where N denotes an integer. They imply that approximately one in four of the reflections that are expected for atoms occupying 48(e) Wyckoff sites are, in fact, forbidden.

2. Diffraction from atomic clusters centred on 12(b) sites
Consider the case of a crystal with I3d in which all the atoms are of the same type and occupy general 48(e) positions. The number of atoms in the is a multiple of 48, so that there are 48n atoms, n ≥ 1. They can be divided into two groups, each of 24n atoms, which are related by a translation with the lattice vector C = (½,½,½) (body centring). The diffraction amplitude of the is then
where f is the (Xray, neutron, electron) scattering amplitude for a spherical atom, H(hkl) is the reciprocallattice vector, r_{m} is the position vector of the atom m and exp(−W_{m}) is the Debye–Waller temperature factor, which may be anisotropic. If we apply (1) to the special site 12(b), we need to take into account only the six Wyckoff sites
because the remaining six 12(b) sites are related to those in (2) by the lattice vector C.
The b) site is . If we start from a single atom in a general xyz position, giving this atom the coordinates (a,b,c) with respect to the site (,0,¼) chosen as origin of coordinates, we shall have a total of four atoms in a cluster of symmetry surrounding this site. These atoms will have the coordinates (a,b,c), (a,−b,−c), (−a,c,−b) and (−a,−c,b). The other 20 symmetryrelated general positions will yield similar fouratom clusters around the other five 12(b) sites in (2). Three of the fouratom clusters will be the same, but oriented along the three cellaxis directions (effect of threefold axes); the other three will be related via diagonal mirrors. Since we started with 24 atoms in general positions, there are no hkl selectionrule simplifications. We shall show, however, that the 12(b) selectionrule simplification arises if a copy of the atom with coordinates (a,b,c) is placed at each of the six sites in (2), where the origin of coordinates for each copy is the site itself. This procedure generates a total of 24 × 6 or 144 atoms, consisting of 24atom clusters at the six sites in the The clusters are identical to one another in geometry and orientation, have the 3m, and are related by a translation vector that is not a lattice vector.
of a 12(Although the 24atom clusters around 12(b) sites are identical to one another, they are formed from nonequivalent atoms. Thus only four atoms in every cluster are equivalent and these atoms correspond to the axis [the of the sites 12(b)]. The other 20 atoms are equivalent fourbyfour with atoms from different clusters. For example, in the cluster centred on (,0,¼), only the atoms (a,b,c), (a,−b,−c), (−a,c,−b) and (−a,−c,b) are equivalent and they correspond to the axis oriented along [100]. The atoms (b,c,a), (b,−c,−a), (−b,a,−c) and (−b,−a,c) from the same cluster are not equivalent with the previous four, but with the atoms (a,b,c), (−a,b,−c), (c,−b,a) and (−c,−b,a) from the cluster centred on (¼,,0), and so on. In other words, the clusters are identical and have the pointgroup symmetry 3m, not because all atoms of the cluster are equivalent but because an identical atom (a copy) was placed at the point (a,b,c) around each 12(b) site. By applying to these atoms all operations of the we obtain identical clusters in both geometry and orientation.
Let R_{i} be the position vector of the ith Wyckoff 12(b) site with respect to the origin of the and let r_{ij} be the position vector of the jth atom in the cluster S_{i} surrounding the ith site with respect to an origin at that site. The position vector in the of the atom j is R_{i} + r_{ij} and, denoting by exp(−W_{ij}) the corresponding Debye–Waller factor, in place of (1) we have:
Because the clusters are identical in both geometry and orientation, the quantities r_{ij}, W_{ij} in the inner sum of (3) are the same for each cluster S_{i} and so they are independent of the index i. Hence,
Equation (3) can then be factorized to yield the expression
The factor in (5) is the sum of the amplitudes of unit scatterers at the special positions 12(b), and follows the special selection rules for that site. On the other hand, there are no selection rules for the factor in (5).
Hitherto, we have considered only one type of atom in the _{4}O_{9}), there are different clusters for each type and equation (5) must be summed over each atomic species with the corresponding scattering factor and thermal parameters.
For several types of atom (as in UTo determine the xyz coordinates of the atoms in the clusters surrounding the special sites in (2), we start with the six atoms in different clusters with local coordinates (a,b,c). For these atoms,
where (x_{i}, y_{i}, z_{i}) are the coordinates of the special sites. The expression (a,b,c) ∈ S_{i} denotes the vector (a,b,c) translated to the centre of S_{i} as origin. The six atoms take the same exponent of the Debye–Waller factor:
Here the symbol 〈...〉 represents the average value and u = (u_{x},u_{y},u_{z}) is the atomic displacement vector due to thermal vibrations.
For each atom in (6), there exists another 23 equivalent atoms spread across the Their coordinates are calculated as follows. The general equivalent positions of the I3d are
and, as an example, we take the general position (½+y,½−z,−x) operating on the atom at (a,b,c) in the first cluster S_{1}, centred on . From (6), we have
and the unitcell coordinates of the transposed atom are
where the subscript on the lefthand side refers to the first cluster, S_{1}. The sixth special site in (2) is at (0,¾,) and so .
Similarly, for the atom at (a,b,c) in the third cluster S_{3}, centred on , . The general equivalent position (−y,½+z,½−x) operating on this atom gives the coordinates
By proceeding systematically in this way, we can calculate all the atomic positions of (6). n in equation (5) is equal to 6, and the complete set of coordinates is then given by expressions such as (9a), (9b) with the following permutations of (a,b,c):
This set of 24 values describes a hexatetrahedral cluster of 24 atoms centred on every 12(b) site. The cluster has the 3m, which is the of the I3d. The coordinates in (10) are those of the general positions of the primitive symmorphic group P3m
For a ≠ b ≠ c ≠ 0, there are 24 × 6 × 2 = 288 atoms in the (including the body centring condition) but there are fewer atoms [i.e. n < 6 in equation (5)] if constraints are imposed on a, b, c. For b = c ≠ a, there are 12 atoms in the cluster (144 in the unit cell); for a ≠ 0 and b = c = 0, there are six atoms (octahedral grouping); for a = b = c, there are only four atoms (tetrahedral grouping); and, for a = b = −c, there are similarly four atoms (inverse tetrahedral grouping). In these particular cases, the atoms in the cluster sit in special positions of the P3m. Thus, in the tetrahedral cluster, the atoms sit in the special position 3m of P3m and the multiplicity is that of the . Clearly, there is the possibility of accommodating many different clusters for a given type of atom by taking different sets of (a,b,c) and also by using the second special position 12(a). We shall see in the subsequent paper that clusters with tetrahedral, octahedral and hexatetrahedral symmetry all exist in U_{4}O_{9}.
It is an easy matter to find the Debye–Waller factors of the 24 atoms in (10). Indeed, the equivalents of the thermal displacement vector u are just those given by (10), changing only (a,b,c) into (u_{x},u_{y},u_{z}). Thus, for example, we have for the tenth atom in the list of coordinates in (10):
3. Generalization to any space group
The treatment presented in the previous section can be generalized to any n_{g} the number of general equivalent sites, by n_{s} the number of special equivalent sites, both without counting the lattice condition of face or body centring, and by n_{c} the number of vectors C_{m} giving the lattice centring (n_{c} = 1, 2 or 4, and n_{c} = 3 for the rhombohedrally centred hexagonal lattice).
if the atoms occupy a general position but the diffraction pattern shows selection rules for a special position. Let us denote byWe start by placing one atom close to each of the n_{s} sites. This atom is in a general position given by (xyz)_{i} = R_{i} + (a,b,c) ∈ S_{i} (i = 1, n_{g}).
(a,b,c) is an arbitrary vector r_{1} referred to an origin at the ith special site. The n_{g} equivalents of every one of these atoms are spread across all n_{s} subgroups S_{1},…,S_{ns}, with n_{g}/n_{s} atoms in every Thus there are n_{s} identical clusters, each containing n_{g} atoms centred on a special site. These clusters have the pointgroup symmetry of the and are related to one another by translation only. The coordinates and the Debye–Waller factors of all atoms from these clusters are then
where r_{j} = r_{1}·Φ_{j} and u_{j} = u_{1}·Φ_{j}. Here Φ_{j} is an operation of the point group.
In practice, we need not calculate r_{j} and u_{j} because they are listed in International Tables for Xray Crystallography (1969) as general positions of the corresponding primitive symmorphic having the same as the actual Including the lattice centring, the number of atoms in the for one set of a,b,c is n_{c} × n_{g} × n_{s} and the diffraction amplitude can be written as follows:
The terms within the first square brackets in (11) represent the amplitude due to the centring. The second square brackets contain the amplitude of the cluster, which has no selection rules. The terms in the last square brackets contain the sum of amplitudes of the special sites, which (together with the terms in the first parentheses) follow the selection rules of these sites, although the atoms sit in general positions. For more than one type of atom in the equation (11) must be summed over the different atomic species with corresponding scattering factors and thermal parameters.
4. Conclusions
We have shown that the International Tables for Xray Crystallography (1969): `It should be remembered that the special conditions only apply when the special positions are assumed to be occupied by spherical groups.' The related topic of the diffraction enhancement of symmetry has been treated by Iwasaki et al. (see PerezMato & Iglesias, 1977), in which the pointgroup symmetry of the diffraction pattern is higher than the pointgroup symmetry of the crystal. This treatment does not apply to the present case where the two symmetries are the same.
of a special Wyckoff site are unchanged when a single atom occupying that site is replaced by a cluster of the same type of atom which possesses the pointgroup symmetry of the It is also necessary that the clusters at the equivalent positions of the special sites are related by pure translation operators. More generally, if the pointgroup symmetry of the cluster is equal to the pointgroup symmetry of the or a of this then the are identical to those found when a single spherical unit is placed exactly at the special position. These statements are generalizations of the comment inIn the structure of U_{4}O_{9} described by Bevan et al. (1986), it was found that the contains clusters of atoms centred on the 12(b) sites. Each cluster possesses pointgroup symmetry 3m and contains just one kind of atom, U or O. Most of the atoms were allocated to 15 general xyz sites but the data were too few to allow of these 45 positional parameters individually. Using the present analysis, the number of independent parameters is substantially reduced, and in the following paper we shall describe the of neutron diffraction data for U_{4}O_{9}, taking into account the special extinction rules for 12(b) sites.
Acknowledgements
The authors are indebted to Professor D. W. J. Cruickshank for numerous suggestions for improving the manuscript.
References
Belbeoch, B., Piekarski, C. & Perio, P. (1961). Acta Cryst. 14, 837–843. CrossRef CAS IUCr Journals Web of Science Google Scholar
Bevan, D. J. M., Grey, I. E. & Willis, B. T. M. (1986). J. Solid State Chem. 61, 1–7. CrossRef CAS Web of Science Google Scholar
Blank, H. & Ronchi, C. (1968). Acta Cryst. A24, 657–666. CrossRef CAS IUCr Journals Web of Science Google Scholar
Cooper, R. I. & Willis, B. T. M. (2004). Acta Cryst. A60, 322–325. Web of Science CrossRef CAS IUCr Journals Google Scholar
International Tables for Xray Crystallography (1969). Vol. I. Birmingham: Kynoch Press. Google Scholar
Lauriat, J. P., Chevrier, G. & Boucherle, J. X. (1989) J. Solid State Chem. 80, 80–93. CrossRef CAS Web of Science Google Scholar
Masaki, N. & Doi, K. (1968). Acta Cryst. B24, 1393–1394. CrossRef CAS IUCr Journals Web of Science Google Scholar
PerezMato, J. M. & Iglesias, J. E. (1977). Acta Cryst. A33, 466–474. CrossRef IUCr Journals Web of Science Google Scholar
Willis, B. T. M. (1964). J. Phys. 25, 431–439. CrossRef CAS Google Scholar
© International Union of Crystallography. Prior permission is not required to reproduce short quotations, tables and figures from this article, provided the original authors and source are cited. For more information, click here.