3.1. Average crystal structure
The average crystal structure was solved in the space group
using direct methods (SIR
97; Altomare et al.
) and subsequently refined with JANA
2000 (Petříček et al.
). Attempts to solve and refine the structure using the non-centrosymmetric space group
resulted in strong correlation of atomic coordinates of sites which are equivalent in the space group
. Therefore, the centrosymmetric space group was assumed to be correct.
Four cation sites were identified: two sites with high coordination numbers (8 and 9) which were assigned to Ca atoms. The remaining two sites show tetrahedral and octahedral coordination, consequently these sites were assigned to iron and manganese/titanium, respectively. However, the tetrahedral site is a split position, as is one of its coordinating oxygen sites. The tetrahedra are disordered with respect to two distinct configurations.
Bond-valence sum (BVS; Brown & Altermatt, 1985
) calculations for Fe2 revealed a value of 3.01 (1) and therefore do not show any evidence for four-valent species on the tetrahedral position. Furthermore, refinement with a mixed (Mn,Fe)/Ti occupany showed that no significant amount of Ti is present on the tetrahedral site.
As iron and manganese cannot be distinguished in X-ray diffraction experiments, the sum of iron and manganese was refined as iron. The final refinement included mixed occupancy of Ti/(Fe,Mn) for the octahedral site only, showing that ca
25% of that site is occupied by Ti. As the structure requires two three-valent
cations per formula unit, it can be assumed that the octahedral site is filled with 50% iron and 25% manganese: giving a formula of Ca
. This is in good agreement with the energy-dispersive X-ray spectroscopy (EDX) analysis, which was carried out using an XFLASH 5010 detector (Bruker). As the contrast in scattering factors of (Mn,Fe) and Ti is small and the EDX analysis was not performed quantitatively, the chemical composition should be understood to be only an estimate. Furthermore, mixed-valence states of manganese and iron, as well as non-stoichiometric oxygen content, cannot be excluded. Details of the final refinement can be found in Table 1, atomic coordinates of the structural model can be found in the supplementary material
. Structure drawings were prepared using DRAWxtl
(Finger et al.
The structural model obtained is basically isotypic with the structure of Sr
(Barrier et al.
). However, Barrier et al.
) decided not to employ a split position for the tetrahedrally coordinated iron, which results in a high distortion of the tetrahedra and a noticeably high bond-valence sum (Brown & Altermatt, 1985
) of 3.18 for Fe2 in Sr
(Barrier et al.
Whereas the brownmillerite structure types exhibit octahedral (O
) and tetrahedral (T
) layers connected to form a framework, this structure type shows two-dimensional O
slabs (brownmillerite blocks) separated by distorted rock-salt type CaO layers (see Fig. 3). Therefore, we use the term layered brownmillerites
for this structure type. As pointed out by Barrier et al.
) this structure type can also be described as an intergrowth between the K
and brownmillerite-type structures.
The coordination of the two Ca atoms show a significant difference: Ca2 is located between the octahedral and the tetrahedral layers, and shows a ninefold coordination and a close to ideal BVS of 2.071 (3). The Ca1 site is within the CaO sheets between the brownmillerite blocks and is slightly under-bonded exhibiting a BVS of 1.845 (4).
3.2. Analysis of diffuse X-ray scattering
The observation of structured one-dimensional diffuse scattering reveals the presence of stacking faults in the structure of the material. However, the average structure shows orientationally disorder of the tetrahedral chains and thus does not allow stacking faults. Therefore, it can be assumed that the chains are not randomly disordered but well ordered. In fact, the type and pattern of the diffuse scattering reveals the ordering scheme of the tetrahedral chains: an alternating sequence of right- and left-handed tetrahedral chains causes the doubling of the
parameter. Therefore, each brownmillerite block may adopt one of two different configurations (Fig. 4), which allow stacking faults to occur. Two neighbouring tetrahedral layers can be related by two different vectors, namely
, with respect to the cell of the average structure.
Figure 4 Two neighbouring brownmillerite blocks showing an alternating chain sequence within the tetrahedral layers. Two possible translation vectors are shown: the right one relates the neighbouring layers, the left one would require a shifted position of the (more ...)
The X-ray diffraction data show a smooth continuous intensity distribution (with periodic variations, see Figs. 1 and 2) along the lines of diffuse scattering, suggesting random stacking faults, with no preferred stacking sequences.
In order to reveal the structural features which result in the observed intensity variation of the diffuse rods, a computer simulation approach was employed.
The unit cell used to set up the model has a doubled
parameter in order to allow an alternating sequence of R
chains within the layers. Each unit cell contains two tetrahedral layers, therefore two random variables (0 or 1) are needed (per unit cell) to represent the configuration of the layers.
As a system of stacking faults (of perfect two-dimensional layers) represents a strictly one-dimensional disorder problem, very poor statistics are obtained unless a very large one-dimensional sequence can be used. This is not feasible in a relatively small computer model and this inevitably leads to very noisy calculated diffraction patterns. To overcome this problem the system was modelled instead using a two-dimensional disordered array of 512
256 pixels, as shown in Fig. 5. Each pixel in this figure represents a perfect one-dimensional column of two neighbouring tetrahedral chains along
. In the (horizontal)
direction a strong nearest-neighbour correlation (of 0.9) is used to induce long rows of like-coloured pixels and this approximates the stacking layers. Black represents the chain configuration RL
and white represents LR
. In the vertical
direction the sequence of white and black is random.
Figure 5 Part (100 200) of the ‘configuration array’ used to set up the model. Each pixel represents two tetrahedral chains, either RL (black) or LR (white).
The model crystal is built according to this two-dimensional array, using the atomic coordinates of the average structure. However, a few simplifications were made: iron was used for all octahedral and tetrahedral coordinated cations and the displacement parameters were neglected. The positions of atoms represented by split sites in the average structure were chosen in a way as to adopt the configuration required by the configuration array.
The diffraction patterns of the model were calculated utilizing the software DIFFUSE
(Butler & Welberry, 1992
). The calculation was performed using 1800 lots (randomly chosen subregions of size 1
6 unit cells) assuming periodic boundary conditions. Bragg scattering (average scattering) was determined from 5% of the model crystal and subtracted from the result.
The first model causes continuous rods of diffuse scattering, their positions corresponding to those observed in the experimental data. However, no periodic variation of the intensity along the rods (as present in the experimental data) was produced.
It can be assumed that the local environment of the tetrahedral chains is subject to changes depending on the configuration of the chain, thus leading to two different local environments, which are superimposed in the average crystal structure as derived from X-ray diffraction (XRD) data. Further analysis of the average structure reveals details of the two different configurations: the anisotropic displacement parameters of the Ca atoms, of the bridging (tetrahedra–octahedra) oxygen atoms (O3) and the oxygen atoms opposite (O4) the bridging ones are significantly elongated along a
. To reveal the magnitude of the displacements these atoms were split (isotropic displacement parameters were used) and another refinement was carried out. The distances between the pairs of split positions are: O3: 0.23, Ca2: 0.15, O4: 0.13 and Ca1: 0.11 Å. The sites closer to the T
atoms (O3 and Ca2) show the largest displacements, whereas the more distant sites (O4 and Ca1) show smaller separations. Atomic coordinates of the refined split positions are listed in the supplementary material
To work out the directions of the shifts (e.g. which of the two distinct positions of the split atoms corresponds to which chain configuration), considerations on polyhedral distortions were employed, as well as calculated diffraction patterns of the model crystal which were compared with experimentally determined diffuse scattering data. First a definition of the chain configuration/direction is needed. In a projection along a (Fig. 4), the two chain configurations can be easily distinguished: One type of chain has its intra-chain connecting oxygen atoms (O5) pointing along a (Fig. 4, red): these will be referred as type 1 chains; their chain direction is a. The type 2 chains have their connecting O atoms pointing in the opposite direction (−a, Fig. 4, blue); the chain direction is −a.
As shown by the split-position refinement the bridging oxygen between the tetrahedron and the octahedron (O3) exhibits the largest shift. We calculated distortion parameters (Robinson et al.
) for the tetrahedra and the octahedra depending on the displacement of O3 along the chain direction. Fig. 6 shows that the tetrahedral distortion has a minimum for a shift slightly below −0.3 Å (negative is against the chain direction). However, the minimal distortion for the octahedra is close to a displacement of 0. Consequently the displacement of O3 is opposite the chain directions, its magnitude being determined by a distortion equilibrium between the octahedra and the tetrahedra.
Distortion parameters for displacement of the bridging oxygen O3 (between octahedra and tetrahedra).
Given the fact that the direction of the displacement of O3 is well determined by the tetrahedral distortion parameters, we tried to use BVS calculations to reveal the direction of the Ca2 displacement. However, the derived BVS for the two options are not significantly different. Therefore, the displacement of Ca2 has to be determined by analysis of the diffuse scattering.
Four computer models (all combinations of positive and negative displacements for O3 and Ca2, see Table 2) have been derived and their diffraction patterns were compared with the experimentally derived line profiles. It turns out that all major intensity peaks are well reproduced by model A
using opposite displacements for O3 and Ca2 (with Ca2 displaced along the chain direction), which shows the best fit to the experimental data. Models B
fail to produce the most intense peaks, whereas model C
does not reproduce the intensities of the peaks and is unlikely because of the tetrahedral distortion parameters. A constant scaling factor is applied to the calculated data as presented in Fig. 1, which shows the 3
1.5 and 3
2.5 line profiles as an example. Fig. 2 compares the experimental and calculated data of the (3kl
) plane. All other measured line profiles and those derived from the four models are included in the supplementary material
Displacements (in Å) for models A–D
Compared with the average structure the coordination shell of Ca2 is considerably changed. The Ca2 site lies on a mirror plane (
) in the average structure, which is reflected by the Ca2—O distances, as shown in Table 3. However, in the local structure this mirror plane is lost and the coordination polyhedron exhibits stronger distortion. Fig. 7(a
) shows the coordination of Ca2 in the average structure, whereas Fig. 7(b
) depicts the local environment of Ca2. Black arrows show the direction of the displacements of Ca2, O3 compared with their positions in the average structure. The largest changes are found in the distances to the neighbouring O3 atoms in the
direction. Even though the distances change significantly (see Table 3), the BVS of Ca2 is just slightly larger.
Ca2—O distances (Å) in the average and local structure
The displacements of the atoms in the rock-salt layer (O4, Ca1) are smaller and depend not only on one tetrahedral chain, but also on the chain configuration in the next brownmillerite block. Furthermore, a shearing distortion can be assumed due to displacements of O atoms in the equatorial planes of the octahedra. Attempts to model these features of the local structure failed, owing to the limited quality of the data (poor counting statistics, especially in the
= 0.5, 1.5, 2.5, see supplementary material
). Some features in the diffuse lines remain unexplained, e.g.
. However, the derived model of the local structure explains the dominant intensity variation along the diffuse lines. A future study using synchrotron radiation and an improved data treatment (Lp and absorption correction) might be able to reveal more details of the local structure.
3.3. Transmission electron microscopy
In contrast to the XRD results, selected-area electron diffraction revealed that the samples show additional sharp peaks located on the diffuse streaks, which indicates a certain degree of long-range order with respect to the corresponding stacking sequences (see Fig. 8).
Figure 8 ED patterns of the  zone axis, showing two different sets of superstructure reflections. (a) Commensurate with and (b) incommensurate with . The incommensurate satellites are dominant in the investigated sample.
Two different patterns of superstructure reflections were observed along the  zone axis, as shown in Fig. 8. The electron-diffraction patterns were recorded from different regions of the same grain. Both show the same pattern of main reflections and diffuse rods parallel to
. However, satellite reflections located on the diffuse streaks show different positions. For the pattern shown in Fig. 8(b
) a q
can be used to index the superstructure reflections. In Fig. 8(a
) a commensurate q
is sufficient to index the pattern. However, the incommensurate q
vector seems to be dominant in the investigated sample.
All observed electron diffraction (ED) patterns show a
, resulting in an alternating intra-layer chain sequence. However, two different values for
, each of them corresponding to a certain stacking sequence.
The monoclinic stacking sequences allow for twinning according to a mirror plane
, which was frequently observed. In Fig. 8(b
) very weak satellites can be seen which cannot be indexed with
, but with a mirror-related vector
As demonstrated by other authors (Krekels et al.
; D’Hondt et al.
) the  zone axis represents a special direction, which gives (for geometrical reasons; see Fig. 17 of Krekels et al.
, for details) a strong contrast for different tetrahedral chain configurations. The same is true for the layered brownmillerite
structure, since the same geometry applies for the brownmillerite blocks. Consequently, high-resolution electron microscopy (HREM) images were recorded along the  zone axis (Fig. 9). The image shows smooth lines perpendicular to
, which are likely to be caused by the rock-salt layers. The distance between these corresponds to
. In between these lines the dotted contrast represents the alternating tetrahedral chains. Their spacing (dot-to-dot) is ca
4.8 Å, which agrees with the doubled
lattice in this projection. The straight line is a projection of
as in the average structure), which intersects a distinct tetrahedral chain at each tetrahedral layer. As can be seen along the line, it intersects either bright or dark spots in an irregular sequence. The staggered line follows the stacking vector layer by layer and every change in the stacking vector is marked with a circle. By correlating five HREM images, we were able to map the stacking vectors between 45 brownmillerite blocks: − − + − + + + + + − − − + + − + + + + − + + + − − + − − − + + − − + + + − + + − + − + + (where + and − denote the two possible stacking vectors). As shown later, this sequence is not compatible with any of the observed superstructures and thus represents a region with stacking faults.
Figure 9 HREM image recorded along the  zone axis. The length of the arrow corresponds to . The Fourier transform is derived from a larger area [ImageJ (Abràmoff et al., 2004 ) was used for image processing].
3.4. Superspace model
As the observed electron-diffraction patterns can be indexed using different q
vectors, it is possible to describe the structures utilizing the (3 + 1)-dimensional superspace approach (van Smaalen, 2007
; Janssen et al.
). In order to find a unified superspace model for the different superstructures a monoclinic subgroup of
has to be considered as the basic three-dimensional space group. The direction of the unique monoclinic axis has to be parallel to the chain direction. Furthermore, the (3 + 1)-dimensional space group should exhibit an internal phase shift of
connected with the mirror plane, the effect of this will be described later. The superspace group
(which is a non-standard setting of No. 11.2; Janssen et al.
) exhibits the desired properties. The disorder of the tetrahedral chains, as implied by the
mirror plane, is now solved by the fact that the two mirror-related chains (R
) are subject to an internal phase shift
. The two atoms (Fe2, O5) which exhibit split positions due to the
mirror plane are modelled with crenel-type occupational modulation functions (Petříček et al.
) of width
. The two symmetry-equivalent positions of the two mentioned atoms are phase-shifted by
. To ensure that they cannot occur in any real-space section at the same time the centres of the crenel functions (
) have to be carefully chosen: let us consider two consecutive T
atoms in the tetrahedral chain, that is for example Fe2 and its symmetry-equivalent position generated by the
-screw axis along
. The Fe2 atom takes the position (
), whereas that generated by the screw axis is located at (
) (note the effect of the superspace symmetry on
). These two atoms shall coexist in every real-space section, therefore the phases of their modulation functions need to be the same. As the phase
is given by
is the position vector of the atom in the basic structure), the phase of the first atom is
. The phase of the next T
atom along the chain is
. Putting these into an equation and solving it for
yields the values to be used as the
parameters of the crenel functions. The values of
) are: 0.34135 and 0.3174 for Fe2 and O5, respectively.
A possible stacking sequence resulting from this structural model (for
) is − + + + + − + + + − + + + − + + + + − + + + − + + + − (where + and − denote the two possible stacking vectors). The sequence exhibits groups of 3 or 4 identical vectors interrupted by single vectors of the other direction. For
35% of the groups have four identical vectors. If this is compared with the sequence obtained by HREM, it can be noted that the observed sequence also shows groups of two or five identical stacking vectors, which cannot occur in an incommensurate sequence if
. Consequently, the observed sequence includes stacking faults as supported by the Fourier transform of the HREM images, which show diffuse streaks. However, as evident by the ED images showing sharp satellite reflections, long-range ordered regions exist.