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short communications
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reflections in tilted perovskites
aDepartment of Physics, University of Warwick, Coventry CV4 7AL, UK
*Correspondence e-mail: robin.sjokvist@warwick.ac.uk
The
spots that appear in diffraction patterns of tilted perovskites are well documented and easily calculated using crystallographic software. Here, by considering a distortion mode as a perturbation of the prototype perovskite structure, it is shown how the structure-factor equation yields Boolean conditions for the presence of reflections. This approach may have some advantages for the analysis of electron diffraction patterns of perovskites.Keywords: perovskites; electron diffraction; structure factors; octahedral tilting.
1. Introduction
Conditions for the appearance of ABO3 perovskites with BO6 octahedral tilting were outlined almost exactly 50 years ago in the seminal work of Glazer (1972, 1975
). In X-ray and neutron diffraction, where many diffracted beams are routinely collected over a wide range of crystal orientations, these reflections, and changes in unit-cell dimensions, may be used to determine the extinction conditions and thus the However, in (TEM) and electron diffraction (ED), excluding 3D-ED methods (Gemmi et al., 2019
), it is usual to examine a few low-index zone axes in individual crystals or domains, which provides only partial data. Nevertheless, this can provide crucial information that is sufficient to distinguish between alternative structures (Woodward & Reaney, 2005
). In addition, the sensitivity of electron scattering to low (i.e. oxygen) atoms and the ability to probe nanoscale regions gives ED an important role in the characterization of perovskite oxides.
With incomplete knowledge of the 3D i.e. doubled lattice translations produced by octahedral tilting give reflections at half-order positions. The different patterns of reflections produced by different Glazer tilt systems have been determined by inspecting simulations (Woodward & Reaney, 2005) for some low-index ED patterns. Here, we revisit this question and derive general equations for reflections. The emphasis on tilt system (or other distortion mode), rather than avoids the need to change reference frame according to different choices of and the conversion of describing the reciprocal-lattice vectors, and this entails. The approach is therefore convenient when analysing diffraction patterns of perovskites exhibiting different distortion modes, and provides a result for any zone axis.
2. Calculation
Ignoring thermal factors, the structure-factor equation that gives the complex amplitude of a diffracted beam g in a crystal with a static distortion mode can be written
where the sum is taken over all j atoms in the each having fg(j), fractional coordinates in the prototype structure, and static displacement from these prototype coordinates (due to a distortion mode, such as an oxygen octahedral tilt system)
.
Here, we are not interested in the precise value of Fg for a reflection. Rather, our main concern is whether a distortion mode produces a reflection or not. The answer to this question is simply that a reflection will be present when the result of equation (1) is not exactly zero and, as is shown below, this can be determined most easily by allowing
to be arbitrarily small. This approach also means that any second-order effects (e.g. octahedral distortions) can be neglected. Using the approximation e(a+b) = eaeb = ea(1 + b) for small b, and noting that the for reflections in the prototype structure is precisely zero, the of a reflection with infinitesimal
is
On first sight, equation (2) does not appear to be much more informative than equation (1)
, but further simplification can be obtained, as follows.
We choose a i.e. integer indices in the pseudo-cubic frame). This expanded cell is eight times the size of the prototype cell and has 24 oxygen atoms with coordinates given in Table 1. Here, for reasons that will shortly become apparent, we write the oxygen coordinates ri, which all have positions that are multiples of a quarter of the lattice parameter of the with an integer form si = 4ri. The exponential term can then be written
using the substitution ,
and
. The gi are integers and thus the terms A, B and C take values of ± 1 for even gi and ± i for odd gi.
|
We are now ready to consider specific distortion modes. Fig. 1 shows the direction of (infinitesimal) oxygen-atom displacements for in-phase octahedral rotations about c, a0a0c+ in Glazer notation, which are listed in Table 2
. Using equation (3)
and substituting into equation (2)
we obtain
which nicely reduces to
This equation can be interpreted as a set of Boolean conditions, all of which must be satisfied for a A = ±1 for even g1 and A = ±i for odd g1, (1 − A2) is only non-zero, and a reflection will only be present, when the first index of the reflection, g1, is odd. Similarly, (1 + C2) is only non-zero for even g3 and therefore equation (5) indicates that reflections of the a0a0c+ tilt system must have the form odd–odd–even in the frame of the doubled cell. As for the other two terms in equation (5)
,
is never zero, while (g2B − g1A) = 0 when |g1| = |g2|. We thus obtain the result that reflections occur with pseudo-cubic indices
, |g1| ≠ |g2|. The latter condition describes the that result from the b-glide plane in the of the P4/mbm a0a0c+ structure.
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![]() | Figure 1 Displacements of oxygen atoms due to a BO6 octahedral rotation about the c axis. Labels correspond to Table 1 ![]() ![]() |
A similar procedure can be performed for the a0a0c− tilt system, in which the displacements of oxygen atoms with even-numbered labels in Table 2 are reversed, with the result
indicating that with the same this time from the c glide in the I4/mcm of the a0a0c− structure.
Other distortion modes, such as antiferrodistortive cation displacements, can also be considered in a similar manner using their coordinates and displacements.
A particularly elegant aspect of this approach is that rules for e.g. antiferrodistortive displacements, distorted oxygen octahedra etc.) – can be constructed simply by adding equations, giving a straightforward and quick method of calculation. The results are summarized for the 14 crystallographically distinct oxygen octahedral tilt systems (Howard & Stokes, 1998) in the Appendix
.
3. Conclusions
Equations governing the appearance of ) and Woodward & Reaney (2005
). The emphasis on distortion mode, rather than allows the interpretation of ED patterns without the need to rewrite vectors in real and for different unit cells.
APPENDIX A
For completeness we compile in Table 3 the conditions governing the existence of spots in the pseudo-cubic reference frame for the 14 crystallographically distinct Glazer tilt systems as listed by Howard & Stokes (1998
). These may be obtained by adding equations similar to equations (5)
and/or (6)
for the appropriate tilt system.
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Some general rules become apparent from Table 3. The rules for in-phase tilting are quite straightforward, with each tilt system a+, b+, c+ producing its own set of spots with pseudo-cubic indices
,
,
with no dependence on the presence of any other distortions. Conversely, all antiphase tilt systems produce pseudo-cubic
spots, and are only distinguished by their Furthermore, because in these cases the type of reflections is always the same, tilts of equal magnitude operating about different axes can result in changes to the set of Accordingly, are most apparent for the a−a−a− system. This means that determining antiphase tilting systems is less straightforward than in-phase tilt systems. For investigations using ED, it may thus be important to explore in 3D since can readily be `filled in' by double diffraction where the possibility exists, particularly in the zero-order Laue zone. Access to higher-order Laue zones, or zone axes where no double diffraction pathways are present, is generally necessary.
Table 3 shows that reflections of the form
,
and
do not result from oxygen octahedral tilting. They may, however, be produced by antiferrodistortive displacements of cations. Calculation of extinction rules for these distortion modes is left as an exercise for the reader.
Acknowledgements
We thank Professor A. M. Glazer for constructive comments.
Funding information
This work was funded by EPSRC grant EP/V053701/1.
References
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