The magnetic properties of metastable chemically ordered ilmenite-rich Fe2O3-FeTiO3 solid solutions in the temperature range 500–0 K represent a complex amalgam of the incompatible properties of the end-members. While hematite is controlled by adjacent layer antiferromagnetic interactions, ilmenite is controlled by double-layer antiferromagnetic interactions. The complex geometry of the exchange-coupled ions is visualized, and tentative explanations of property changes with temperature are provided by schematic 2-D ionic images along a common (1 1 −2 0) plane showing the exchange interactions in the end-members and in Ilm 90, 80, and 70 solid solutions, accompanied by parallel temperature- and frequency-dependent susceptibility diagrams. To provide a larger-scale visualization of the extent of exchange clusters and networks, 3-D images of Ilm 95, 90, 85, 80, 75, and 70 containing 2592 atomic positions were also constructed. Both types of images together provide a conceptual overview of the physical mechanisms governing this complex solid-solution system and its low-temperature magnetic behavior. Even though the geometric approach does not in itself provide a quantitative model of physical properties, it is an indispensable prerequisite for designing realistic quantitative models, judging their validity, and interpreting model results. It also supplies information on the distributions, numbers, and orientations of magnetic exchange interactions, which can become guidelines for more sophisticated calculations.
 Earth's most important magnetic minerals belong either to the cubic oxide series magnetite (Fe2+Fe3+2O4), ulvöspinel (Fe2+2Ti4+O4), or to the rhombohedral oxide series, hematite (Fe3+2O3), ilmenite (Fe2+Ti4+O3). Both series have important solid solution versus temperature relationships and magnetic complexities of high interest in the magnetism of rocks. The hematite-ilmenite phase diagram has major consequences for magnetic properties of rocks, both in its disequilibrium form [e.g., Nord and Lawson, 1989], involving only ordering phenomena, and its stable form involving major phase separation (exsolution), favoring separate hematite- and ilmenite-rich compositions with falling temperature [McEnroe et al., 2007].
 Fifty-two years ago Ishikawa and Akimoto  published a key paper on the hematite-ilmenite solid solution. This was followed by many other contributions by Ishikawa and coworkers [Ishikawa, 1962; Ishikawa et al., 1985; Arai and Ishikawa, 1985; Arai et al., 1985a, 1985b], that gave crucial groundwork for the low-temperature phase diagram of this solid solution, by combining magnetic, Mössbauer and neutron-scattering experiments with sophisticated theoretical interpretation. The synthetic samples discussed here were originally produced by Burton [1982, pp. 117–118] and were annealed at temperatures low enough for Fe-Ti ordering, but high enough, or cooled rapidly enough, to avoid true exsolution. Using these results, and the strong framework provided by Ishikawa and coworkers, we produced a more rigorous low-temperature phase diagram for mainly metastable Fe-Ti ordered phases in the composition range X Ilm 0.60–1.00 and temperature range 0–500 K [Burton et al., 2008]. During this work it became more evident, that the magnetic properties of the ordered hematite-ilmenite solid solutions depend critically on the details of the interplay between the different exchange interactions which are solely controlled by the random distribution of Fe3+ ions in the Ti4+ layers. Here we do not consider low-temperature magnetic properties that are the result of Fe-Ti disorder, partial order, or intergrowths produced by quench, or rapid cooling, from conditions above the Fe-Ti ordering transition. Such materials have been the subject of extensive and continuing research [e.g., Meiklejohn and Carter, 1959; Nord and Lawson, 1989, 1992; Lagroix et al., 2004].
 A first step toward understanding the magnetic behavior of metastable Fe-Ti ordered ilmenite-rich members of the hematite-ilmenite solid solution series is to visualize the distribution of Fe2+, Fe3+ and Ti4+ ions and their magnetic exchange interactions. Here, a series of models are constructed showing all positions occupied by Fe atoms over a discrete area within a plane parallel to (1 1 −2 0), as well as all interacting Fe atoms in the two adjacent (1 1 −2 0) planes. The (1 1 −2 0) planes are ideal for these images [see Burton et al., 2005] because such planes contain the centers of all ions, and two dimensional images can show accurate vertical and horizontal positions of the ions.
 We index features of the rhombohedral system using hexagonal axes c, a1, a2, and a3. The geometry of the (1 1 −2 0) plane with respect to these axes is illustrated in Figure 1, with c vertical and the a axes arrayed in the horizontal plane (0001). The footprint of a hexagonal unit cell on (0001) is a diamond (outlined) with edges of length a, and with the trace of the (1 1 −2 0) plane (red) running along the long diagonal of the diamond. A rhombohedral plane with Miller indices (1 −1 0 1) makes an angle α = arc tan c/a ∼ 69.89° with (0001) in the a-c plane for hematite, for ilmenite the same angle is α = 70.14 o. The same rhombohedral plane makes an angle β = arc tan c/(a/2 sqrt3) ∼ 72.40° with (0001) in (1 1 −2 0) plane (red) of for hematite, β = 72.63° for ilmenite. In the 2-D images the corresponding value is β ∼ 70°. An additional plane of interest is the plane of octahedral vacancies (see Figure 2). This has the intercepts (−2, 2, , 1) reducing to Miller indices (−1 1 0 2). The plane of vacancies makes an angle γ = arc tan c/(a sqrt3) ∼ 57.61° with (0001) in the (1 1 −2 0) plane (red) of hematite, γ = 57.97° for ilmenite. In the 2-D images the corresponding value is γ = 54°.
Figures 3 and 5 show the structure of the strong exchange interaction network of end-member hematite and ilmenite, based on exchange interaction strengths inferred from inelastic neutron scattering and model calculations [Samuelsen and Shirane, 1970; Ishikawa et al., 1985; Harrison et al., 2007]. To monitor the transition between these end-members within the solid-solution series, similar images were then prepared for compositions Ilm 70, 80 and 90 portraying the occupancy of 612 atomic positions and their predicted magnetic interactions. Details of these constructions are given below.
 Because mean field or Monte Carlo models [Harrison, 2006; Ishikawa, 1962; Fabian et al., 2007] apparently provide more quantitative predictions than the graphical visualizations presented here, justification for our procedure is in order. The most direct reason in favor of a geometric approach is that mean field or Monte Carlo model predictions do not necessarily lead to a qualitative understanding. Even if a model correctly predicts the data, it does not help to understand by which mechanism the result occurs. On the other hand, none of the current models agrees exactly with the measurement data, leaving the possibility that some important feature of the real solid solutions is not yet recognized, and therefore also not represented in the models. A detailed visual study, as presented here, may lead to new ideas on how to improve the quantitative modeling leading to better agreement with the real physical situation. A second reason is that the random distribution of ions in the solid solution creates very complex clusters and networks of exchange interactions, which seem to have several hierarchies of length scales ranging from single into many hundreds and thousands of unit cell dimensions. Near the percolation threshold of the network, cluster size even diverges into macroscopic scale. It cannot be expected that typical numerical models are able to make exact predictions of the magnetic properties resulting from such complicated structures, and a thorough geometric understanding of the system is an indispensible prerequisite to assess the validity and meaning of numerical models.
3. Magnetic Interactions
Figure 2a shows horizontal and vertical positions of Fe3+ ions (red) and octahedral vacancies ([ ]) in a single (1 1 −2 0) ionic layer of hematite. The total array contains 10 octahedral layers (0–9) along c and 17 columns (0–16) in horizontal direction. Layers 0 and 9, and columns 0 and 16 are at the edges of the image where only a fraction of the magnetic interactions can be considered. The fully linked part of the image includes layers 1–8 and columns 1–15. Within each layer from columns 1 to 15 there are 10 filled octahedrons with Fe3+ ions and 5 vacancies. Selecting exactly 10 Fe3+ ions per layer simplifies the preparation of images for the “round” compositions Ilm 70, 80, 90. This is both a benefit and a curse; a benefit because we can actually look at an array of ions with an exact composition; a curse because we know that in the real world, with random selection of ionic locations, over a greatly extended volume, there will be a much greater variety of atomic clustering than it is possible to display in this limited image. The fully linked part of the image contains 80 atoms and 40 vacancies.
 In Figure 2a, the vertical distribution of filled and vacant octahedrons, and the rotational position of octahedrons in a column (the latter not showing in Figure 2a) determines that a repeat only occurs every six layers, thus a unit cell has six layers along c. Notice that the vertical positions of Fe3+ ions in each layer are “puckered,” lying alternately above and below the centers of the vacant octahedrons. This puckering is a crucial feature of the rhombohedral structure for both ionic and magnetic interactions. Where two Fe3+ ions are directly above and below each other in adjacent layers, the upper one is pucker up and the lower one down. This is a result of ionic repulsion across the unusual shared octahedral faces. Conversely, when an Fe3+ ion lies directly above or below a vacant octahedron, the ion is moved closer to the vacancy. The different Fe-O distances, produced by puckering, are crucial in consideration of ionic charge satisfaction of oxygen [Robinson et al., 2006]. Different Fe-Fe distances produced by puckering determine the abundances of short (very strong) and long (strong) interlayer magnetic interactions as discussed below.
Figure 2b differs from Figure 2a in that the magnetic moment orientations and the strongest interlayer magnetic interactions within the (1 1 −2 0) plane in hematite are illustrated. The strong interactions in hematite are AF interlayer interactions (marked by red lines) causing the magnetic moments to alternate between left (red arrows) and right (blue arrows) in alternate (0001) layers. The strongest interactions are indicated by short red lines, the next strongest by longer red lines. Note that each Fe3+ ion has three of these interactions associated with it, one strongest and two strong. These interactions form closed networks only along inclined layers parallel to rhombohedral planes with indices (−1 1 0 2). Due to crystal symmetry, each Fe3+ ion lies in three equivalent rhombohedral planes, each containing three strong interactions, so that the individual ion has associated with it 3 short interactions, and 6 longer strong interactions, making a total of 9 strong interactions.
Table 1 shows all the significant magnetic interactions in hematite. The magnetic interaction notations are from the diagrams of Ishikawa et al. , accompanied by a short description. The actual interaction parameters are those published by Harrison et al. . The last column shows the number of such interactions surrounding a single Fe3+ ion, of which 9 are strong, 1 is moderately strong, and 13 are weak, giving a total of 23. The important negative interlayer interactions are the J02 and J01 already emphasized in Figure 2b. The J03 is the one between atoms across the shared face. It is ferromagnetic and thus subtracts from the overall AF interaction, but only by 2.6%. Furthermore the total J11 intralayer interaction is 3 × +2.72 = +8.16 seeking to retain a consistent magnetization in each layer. All double-layer interactions, which are dominant in the low-temperature magnetization of ilmenite, are weak in hematite in comparison to the adjacent layer interactions.
Table 1. Magnetic Fe3+/Fe3+ Interactions in Hematite
 The cartoon of Figure 3a illustrates the strong interactions around a given Fe3+ ion. Ions within the (1 1 −2 0) plane of interest are shown with large circles. Positions of ions in a parallel layer, lying in front of that layer, are indicated by small black dots. Analogous positions of ions behind that layer are indicated by circles around the dots. Figure 3b shows the scheme of the cartoon carried out for a full image. Here, in addition to the 80 ions within the (1 1 −2 0) plane of the active part of the diagram, we also draw 80 ions within a plane in front and 80 ions within a plane in back. For magnetic interactions, however, we show only those involving ions within the chosen (1 1 −2 0) plane (large circles). The number of nine strong interactions of each of these ions is indicated. Total number of such interactions between rows in the active part of the model is 75. Thus for 8 rows of the model, there are 600 (8 × 75) interactions.
Figure 4a shows horizontal and vertical positions of Fe2+ ions (green), Ti4+ ions (blue) and octahedral vacancies ([ ]) in a single (1 1 −2 0) ionic layer of ilmenite. Like Figure 2a, the array contains 10 octahedral layers (0–9) along c, and 17 columns (0–16) in a horizontal direction though there is complete ordering of Fe2+ ions, Ti4+ ions in alternate layers.
 In Figure 4a, as in Figure 2a, the vertical distribution of filled and vacant octahedrons in a column determines that a repeat only occurs every six layers, thus a unit cell has six layers along c. The alternate Fe and Ti layers do not affect this feature. The vertical positions of Fe and Ti ions are also puckered as in hematite, and there is the same ionic repulsion across shared octahedral faces. Here each shared face has an Fe2+ ion on one side and a Ti4+ ion on the other, a feature essential to charge balance.
Figure 4b adds the magnetic structure to Figure 4a, and therefore requires a larger array of 0–17 layers versus 0–16 columns to accommodate the magnetic unit cell of ilmenite, which contains 12 layers. Only for layers 1–16 and columns 1–15 all magnetic interactions between the ions can be drawn. The important magnetic interactions in ilmenite below its ordering temperature of 57 K (PM to AF) are indicated by blue lines. The strongest interactions shown in heavy blue are the intralayer ferromagnetic interactions that keep parallel magnetic moments within each Fe layer. Anisotropy aligns the spins, in this case parallel to c rather than parallel to (0001) as in hematite at room temperature. Slightly weaker interactions are the double-layer AF interactions which keep the moments pointing alternately down (dark blue arrows) and up (red arrows) in alternate layers. These are of two kinds, those connecting diagonally between Fe layers, and those connecting vertically between layers through a vacant octahedron in a Ti4+ layer. The alternate up and down magnetizations in alternating Fe2+ layers causes doubling of the 6 layer unit cell into the 12-layer magnetic cell. In combination, these magnetic interactions within the (1 1 −2 0) plane outline a rhombohedral plane with the Miller indices (1 −1 0 1) in the chemical cell. Unlike in hematite, these arrangements do not wrap around vacancies. Each Fe2+ ion in the plane has 1 intralayer interaction, and 3 double-layer interactions. In three dimensions there are three such rhombohedral planes, but all share the one vertical interaction through the vacancy. Thus, there are 3 ferromagnetic intralayer interactions and 7 AF double-layer interactions, giving a total of 10 important interactions.
Table 2 describes all the significant magnetic interactions in ilmenite, with magnetic interaction notations from the diagrams of Ishikawa et al. . The interaction parameters are those published by Harrison et al. , except the values for J16, J12 and J15 from Ishikawa et al. . The last column shows the 25 interactions surrounding a single Fe2+ ion, of which 3 are strong, 7 are weak, and 15 very weak,. The strongest interaction is the J11 ferromagnetic intralayer interaction that retains an identical moment within single Fe2+ layers. The other important interactions are the relatively weak J13 and J14 antiferromagnetic interactions that at low T keep the Fe layers with alternate magnetic moments. The relative weakness of these double-layer interactions is demonstrated by the fact that in a strong magnetic field (e.g., 5 T @ 4 K), the AF character can be overcome, so that ilmenite becomes ferromagnetic (metamagnetic state) with all moments in the same direction parallel to c in all layers.
Table 2. Magnetic Fe2+/Fe2+ Interactions in Ilmenite
 The cartoon of Figure 5a illustrates the important interactions around a given Fe2+ ion. Ions within the (1 1 −2 0) plane of interest are shown with large circles. Positions of ions in a parallel layer in front of that layer are indicated by small black dots, and analogous positions of ions behind that layer are indicated by circles around the dots. Of the 3 very strong ferromagnetic intralayer interactions, one is in the (1 1 −2 0) plane, and two are out of plane. Of the weaker AF interlayer interactions, 3 are in the (1 1 −2 0) plane, and 4 out, giving a total of 10 important interactions.
Figure 5b is a sketch of a fully linked ionic arrangement in ilmenite, where each Fe2+ ion is coupled by its 10 important magnetic interactions. For an image the size of Figure 4b with 80 Fe ions in the active part, there would be 400 important magnetic interactions. This compares to 1200 important interactions in a similar-sized image of hematite.
3.3. Hematite-Ilmenite Solid Solutions
Table 3 collects all the significant magnetic interactions in Fe-Ti ordered solid solutions with notations from Ishikawa et al. . The parameters are those published by Harrison et al. . The difference from hematite or ilmenite here is the group of Fe2+-Fe3+ interactions, which typically are somewhat weaker than the corresponding Fe3+-Fe3+ interactions. The value for the Fe2+-Fe3+ J11 interaction is currently under reconsideration by Harrison and coworkers, and an average of the Fe3+-Fe3+ and Fe2+-Fe2+ interactions yielding +6.75 K may be more reasonable. The solid solutions considered here are dominated by the strong Fe3+-Fe3+ and Fe2+-Fe3+ interactions. The much weaker Fe2+-Fe2+ interactions contribute significantly only below 57 K. There, however, they conflict with the dominant interactions and have major consequences for very low temperature magnetic behavior.
Table 3. Magnetic Interactions in Fe-Ti Ordered Solid Solutionsa
 In ordered ferri-ilmenite solid solutions, Fe2+ ions and Ti4+ ions chemically order into alternate layers, but Fe3+ ions maintain an even distribution between layers. This means that the adjacent layer magnetic interactions as in hematite, continue to dominate the magnetic properties, but in attenuated circumstances. Because of the even balance of Fe3+ ions in Fe and Ti layers, their magnetic moments cancel out, as they essentially do in hematite. The excess magnetic moment, which makes ordered ferri-ilmenites into strong ferrimagnets, resides then entirely in the Fe2+ ions. It is not known whether ferrimagnetic ilmenites retain some of the canting of magnetic moments in the (0001) plane as in hematite, and, if they do, whether the canting angle is the same or different in Fe-rich and Ti-rich layers. If canting occurs, then the net moment will be the vector sum of the ferrimagnetic and canted components, but the former would predominate. In well ordered solid solutions, the Fe layers are more or less completely Fe2+ and Fe3+. This means that wherever an Fe3+ ion appears in a Ti layer, it will have the full complement of 9 surrounding Fe atoms to interact with. By contrast, Fe ions in Fe layers, may have only a few, or even no adjacent Fe3+ ions in their adjacent Ti layers. Burton et al.  point out that this probability to have Fe2+ ions that are not magnetically linked, explains why attempts to use magnetization to measure the degree of Fe-Ti order have failed for Ti-rich compositions.
Figure 6a shows Fe3+ ions in Ti layers, where each Fe3+ ion is surrounded by 9 Fe ions, and each of these may be either an Fe2+ ion, or another Fe3+ ion. Fe3+-Fe3+ interactions are marked in red, Fe2+-Fe3+ in green. The total number of possible combinations of these interactions is 528 of which only 8 are shown. Combinations dominated by Fe3+-Fe3+ interactions are likely only in hematite-rich solid solutions, whereas those with only Fe2+-Fe3+ interactions are likely wherever an Fe3+ ion occurs in an ilmenite-rich solid solution. In the 2-D construction for Ilm 70, no Fe3+ ion in a Ti layer had more than 3 adjacent Fe3+ions and Fe3+-Fe3+ interactions.
Figure 6b shows examples of Fe2+ and Fe3+ ions in Fe layers. Here the number of possible structures is 1024 (n = 2 × 29), because (1) there are two choices of initial ion and (2) each of the nine adjacent sites can be occupied by Fe3+, or Ti4+. The examples shown are all from the models constructed here, where the number of adjacent Fe atoms ranged from zero to a maximum of 5. This means that not a single Fe atom in an Fe layer in our Ilm 70 model had more than 5 surrounding Fe atoms, and even this was rare.
Figure 7 shows two ordering models for the Ilm 90 composition involving 160 occupied octahedral positions in the central layer parallel to (1 1 −2 0). In Figure 7a there is an “orderly” placement of Fe3+ ions so that, in each layer parallel to (0001), one Fe3+ ion occurs within the 10 octahedral positions in each layer, also one in each (0001) layer in the (1 1 −2 0) layer in front and one in back. This ratio is consistent with the Ilm 90 composition, but may be less than realistic in nature. To develop Figure 7b, a larger model was made with 1000 atomic positions in each (0001) layer. 100 Fe3+ ions were then inserted at random within each layer, and a small part of this model was cut out and placed within the same framework as Figure 7a. The result of this exercise is to show that the random model has much more clustering of Fe3+ ions in some regions and a nearly total dearth of clusters in other regions, hence there is certainly an effect on the overall magnetic properties. This approach was not used for our detailed 2-D models of Ilm 70, 80, and 90 compositions, partly because it would not be possible to illustrate the exact compositions of interest. However, random placement of ions was used for the larger 3-D models in Figure 12.
4. Two-Dimensional Interaction Models
4.1. Link Geometry for Ilm 90, 80, and 70
Figures 8–10 show the complete 2-D models for the compositions Ilm 90, 80 and 70. In Figures 8a, 9a, and 10a, the Fe3+ and Fe2+ ions are coded by color and by the number of antiferromagnetic interactions. All the Fe3+ ions in the odd-numbered Ti layers have 9 interactions and are colored red. All Fe3+ ions in the even numbered Fe layers have 5, or less, interactions and are colored pink. All Fe2+ ions in the even numbered Fe layers have interaction numbers from 5 to 0, and are colored decreasingly dark shades of green (5) to yellow (0). Fe3+-Fe3+ interactions are marked with wide red lines. All Fe3+-Fe2+ interactions are marked with narrow green lines.
Figures 8b, 9b, and 10b use a different color scheme for the same model to emphasize the structure of linked clusters. For Ilm 90 (Figure 8b), ions with interaction numbers 2 and above are parts of clusters linked by adjacent layer magnetic interactions. Some of these groups are colored red, others are colored blue to emphasize that the clusters are not part of a global linkage system, and can behave independently, as in the PM′ region of the phase diagram (see below). Atoms with interaction numbers 1 and 0 are either at the edge of a cluster or cannot belong to a cluster, and represent regions nearly free of adjacent layer interactions, hence their spins are controlled by double-layer interactions at very low temperature, as in end-member ilmenite.
 For Ilm 80 (Figure 9b), ions with interaction numbers 2 and above are colored red or blue and those in red belong to very substantial linkages in two dimensions, which in three dimensions apparently become global, as indicated in Figure 12. However, probably not all the 2-D linked parts belong to the global linkage, and a few may still behave independently, at least at elevated temperatures below TC. The areas of ions with interaction numbers 0 and 1 are greatly reduced so that, although some pockets of dominant double-layer magnetic interactions may exist, the development of a global double-layer interaction network is suppressed.
 For Ilm 70 (Figure 10b), ions with interaction numbers 2 and above, colored red, form a complete network in two dimensions. There are no remaining two-dimensional clusters that are unlinked to the full network, nor is it necessary to resort to the third dimension to prove a complete linkage. However, even here, a few ions show only 1 linkage (only three show zero, and two of these are at model edges), but there are still some regions clear from adjacent layer interactions where very limited regions controlled by double-layer interactions, could occur at very low temperature.
4.2. Statistics and Interpretation
 The histograms in Figure 11 summarize the magnetic interactions in the images of Ilm 90, 80, and 70. In Ilm 90 (Figure 11a), 8 out of 80 sites in Ti layers are occupied by Fe3+ ions each with 9 interactions. Among ions in Fe layers, those with 0 or 1 interaction are most common and only those contain a total of 8 Fe3+ ions. Sites with 2 and 3 interactions are scarce and all are occupied by Fe2+ ions. It is no surprise that adjacent layer interactions produce mostly small weakly magnetized clusters.
 In Ilm 80 (Figure 11b), 16 out of 80 sites in Ti layers are occupied by Fe3+ ions each with 9 interactions. Among ions in Fe layers, those with 1 or 2 interactions are most common and these also contain most of the Fe3+ ions. Sites with 3 interactions are next most abundant and also contain some Fe3+ ions. Next in abundance are sites with 0 interactions and no Fe3+ ions. There are only 4 sites with 4 interactions (one with an Fe3+ ion), and only one Fe2+ site with 5 interactions. Thus, although magnetic linkages are there in some abundance, they are not strong. The 2-D graphics do not indicate the existence of the global linkage, which is known to exist in three dimensions. Even then, as seen by the blue-colored clusters in Figure 9b, there may be some unlinked clusters.
 In Ilm 70 (Figure 11c), 24 out of 80 sites in Ti layers are occupied by Fe3+ ions each with 9 interactions. Among ions in Fe layers, those with 2 or 3 interactions are most common and these also contain most of the Fe3+ ions. Sites with 1 or 4 interactions are next most abundant, and also contain some Fe3+ ions. There is only one site with 0 interactions and only two with 5 interactions (one with an Fe3+ ion). Thus, virtually all the ions in Fe layers are linked ferrimagnetically, though none with more than 5 out of a possible 9 interactions. Even in two dimensions one can recognize a global network of adjacent layer interactions, though pockets still remain that lack such linkages.
Figure 11d presents the total numbers of Fe2+-Fe3+ and Fe3+-Fe3+ interactions and total numbers of all interactions for the three compositions. For Ilm 90 only 8.3% of interactions are Fe3+-Fe3+, for Ilm 80, 19.7%, and for Ilm 70, 26.8%.
 The ordering of the charges of Fe ions takes place at temperatures far below those of Fe-Ti ordering. By movement of electrons, Fe2+ ions can be changed to Fe3+ ions and vice versa, while retaining overall charge balance. This subject was explored recently by bond valence calculations [see Robinson et al., 2006, Figure 4]; Monte Carlo simulations [Harrison, 2006], and Density- Functional Theory calculations [Pentcheva and Nabi, 2008]. When Ti4+ and Fe ions occur adjacent to each other across shared octahedral faces, the Fe ion assumes the Fe2+ state, as in end-member ilmenite. When an Fe ion occurs adjacent to an Fe3+ ion in a Ti layer across a shared octahedral face, the Fe ion assumes the Fe3+ state, as in end-member hematite. This charge ordering (same references) is opposite from the charge ordering found in contact layers at phase interfaces between hematite and ilmenite in exsolution intergrowths. There, perfect charge balance cannot be achieved, but charge ordering significantly reduces charge imbalance.
 Charge ordering, involving movement of Fe3+ ions in Fe layers toward the Fe3+ ions in Ti layers, has no effect on number and geometry of strong magnetic linkages between the Fe ions, which are solely controlled by the locations of Fe3+ ions in Ti layers. For simplicity, the models constructed here do not take charge ordering into account. Later we explored the consequences briefly and found that, although charge ordering does change some weaker Fe2+-Fe3+ interactions to stronger Fe3+-Fe3+ interactions and vice versa, it never changes the number of Fe3+-Fe3+ interactions in counts of the Ilm 90, 80, and 70 models by more than 1. Therefore, no consistent difference between charge-ordered and charge-disordered models could be found. Hence the graph in Figure 11d is hardly influenced by charge ordering.
5. Three-Dimensional Interaction Models
 To gain a better insight into geometric structures created by interactions in large-scale compositional arrays, three-dimensional models of different solid solutions (Figure 12) were created by randomly selecting positions for the Fe3+ ions substituted into the ordered ilmenite structure. The models are constructed to contain 6 × 6 × 6 unit cells. One of the practical problems in visualizing these large arrays is to see inside the volume to understand the three-dimensional structure of clusters formed by adjacent layer linkage, or lack thereof. Color schemes have been selected to maximize this visibility. In addition to models for Ilm 90, 80, and 70, three-dimensional models were also created for Ilm 95, 85, and 75.
5.1. Model Construction
 The 3-D models are built by first creating lists of 1296 Fe2+ sites and 1296 Ti4+ sites for the initial model of 6 × 6 × 6 ordered ilmenite unit cells. Using the target composition (e.g., Ilm70) the total number N3 of Fe3+ ions in Fe2+ and in Ti4+ layers is determined, e.g., as N3 = (1–0.7) 1296, which is rounded to 389. In both, Fe2+ and Ti4+ layers, N3 randomly selected positions are then replaced by Fe3+ ions, creating a globally charge-balanced model with approximately correct composition (e.g., Ilm 69.98 for Ilm70). All model boundaries are periodically continued to the opposite side, yielding an effectively infinite periodic crystal with a period of 6 unit cells in each direction. Ferrimagnetically linked clusters are constructed using a mathematical graph containing all Fe sites as vertices. The edges of this graph are the next-neighbor exchange links with bond lengths 3.42Å, 3.74Å and 3.77Å, originating from the substituted Fe3+ ions in the Ti4+ layers. Only these links transmit the ferrimagnetic order through the solid solution. A standard graph theoretical breadth-first search algorithm [e.g., Hopcroft and Tarjan, 1973] is then used to determine the connected components of this graph. These connected components correspond to the ferrimagnetic clusters in the solid solution. Above the percolation threshold (∼Ilm87) a variety of different disconnected clusters is found. Below the percolation threshold a single infinite cluster exists, which transports global ferrimagnetic order. Yet not all Fe ions are strongly linked to this cluster.
5.2. Cluster Visualization
 In the graphical representation in Figure 12 all Fe ions not linked to one of the Fe3+ in the Ti4+ layers have been removed to improve visibility of the cluster network. The first image for each composition shows all exchange links with bond lengths 3.42Å, 3.74Å and 3.77Å, originating from Fe3+ ions in Ti4+ layers. Fe3+ ions are shown as red dots; Fe2+ ions as brown dots and red and green connection colors represent the two bond lengths. Links across periodic boundaries are not shown. The diagrams in the center for each composition differentiate individual magnetic clusters. For Ilm 95 and 90 blue, green and red indicate three disconnected clusters assumed to behave as short-range ordered ferrimagnetic units. In Figure 12 the clusters can appear disconnected when some of the links cross the periodic boundaries; for example, the two large parts of the blue cluster for Ilm90 are coupled across the top-bottom periodic boundary. For Ilm 80, 75 and 70, nearly all Fe3+ ions are connected in a single long-range ordered ferrimagnetic unit. The third image separately shows the Fe ions not linked to any of the Fe3+ ions in the Ti4+ layers. It thereby complements the other two images by showing the abundance of “holes” in the ferrimagnetic structure. These holes carry the ilmenite-like double-layer AF order at low T. For Ilm 95 and 90 the abundance of such unlinked atoms permits a pervasive double-layer AF order to dominate at very low T. An interesting observation is that, even in Ilm 85, though it is inside the region where ferrimagnetic order percolates, may also contain a pervasive double-layer AF order at very low temperature, because in three dimensions, two infinite clusters of ferrimagnetic and antiferromagnetic order can be intertwined. For Ilm 80, a pervasive double-layer AF order network is probably not possible, in agreement with the neutron scattering experiments on Ilm 79.6 by Arai et al. [1985b] (see below), but such double-layer interactions probably create AF clusters within “holes” in the adjacent layer network (see “holes” in blue networks of center diagrams for Ilm 80, 75, 70). These double-layer interactions at very low temperature, appear able to create frustrations against the pervading adjacent layer FM network that has a significant effect on magnetic behavior even in compositions less than Ilm 50 (K. Fabian et al., manuscript in preparation, 2010). Another important outcome of the three-dimensional modeling is that the 6 × 6 × 6 models shown, are far too small to give a statistically correct representation even of the dilute ilmenite hematite solid solutions. Even when apparently small cluster sizes are dominant (Ilm 90 and Ilm 95), there may occur sporadic large ferrimagnetic clusters which are clearly visible in the magnetic properties [Ishikawa and Akimoto, 1957]. A satisfactory modeling of magnetic properties in intermediate solid solutions might therefore require substantially larger models, or sophisticated mean field approaches. The models also show that in solid solutions of high ilmenite content, the clusters tend to assume fractal forms with very high relative surface area. This may explain the efficiency of the spin glass phase which is probably related to frustrated exchange coupling across the cluster surfaces.
6. Low-Temperature Phase Diagram for Ilmenite-Rich Solid Solutions
 In this section, we survey visually the changes in the geometry of the exchange interaction network, which occur by decreasing the temperature for three different compositions in the hematite-ilmenite solid solution. At high temperatures, thermal energy k T overcomes all exchange bonds, and the magnetic material behaves paramagnetically. With decreasing temperature, the various exchange constants J come stepwise close to k T, and Fe spins with different exchange neighborhoods start to freeze in. This leads to a complex sequence of blocking events in these solid solutions, which are intimately linked to the experimentally observed magnetic properties and transitions. Here we only intend to discuss the possible connections qualitatively. Several methods of quantitative analysis have been developed or are in preparation. However, all these methods are based on specific assumptions which include considerable mathematical simplifications, or are yet to be verified experimentally. It is therefore still necessary to crosscheck these methods against a more qualitative, but geometrically understandable visualization, especially to assess their scope, and to understand possible reasons for their failure.
Figure 13 shows a part of the low-temperature phase diagram covering the composition region Ilm 60–Ilm100 and temperatures from 0 K to 500 K. It is based on the data presented by Burton et al. , and only the interpolation of the equilibria around the bicritical point (intersection of 4 solid lines) differs slightly from that paper. Superimposed on the diagram are paths indicating the likely positions of the discussed states for compositions Ilm 70, 80 and 90. Highest temperature in the paramagnetic (PM) region is indicated by yellow dots in Figure 13. Red and blue dots correspond to tentative temperatures for increasingly blocked exchange interactions.
6.1. Hematite-Like Adjacent Layer Interactions
 For T > 57 K, hematite-like adjacent layer interactions dominate the magnetic properties for all compositions shown. Only above Ilm97 this might be different. At Ilm 60, these interactions produce the initial FM structure with moments parallel to (0001). FM prevails with increasing ilmenite content and decreasing TC until ≈Ilm 87, where the Fe3+ occupancy of the Ti-ordered layers is too low to produce a global three-dimensional magnetic network. Here at the percolation threshold, the average FM cluster size becomes finite. Beyond Ilm 87, along with falling TC, strongly interacting local FM clusters can produce frequency-dependent susceptibility (PM′) below the thermally dominated paramagnetic (PM) field. This behavior is commonly compared to the “superparamagnetism” of small particles. Within the FM region, below Ilm 87 and well below TC, weakly linked Fe atoms at the edges of the percolating FM cluster acquire hardened magnetization in diverse orientations and also are exposed to the conflicting weak double-layer interactions of previously unlinked Fe ions. This increases the coercivity and also decreases the overall magnetic susceptibility, which provides a tentative explanation for the experimentally observed crossover from FM to FM′.
6.2. Temperature- and Frequency-Dependent Susceptibility
 For ordered synthetic samples of compositions Ilm 71.1, 80.6 and 91.5, alternating current susceptibilities (AC) were measured on a MPMS2 from 5 or 15 K, to 300 K, or on a LakeShore Cryotronics AC Susceptometer from 20 to 300 K, with five or seven frequencies between 0.1 and 997 Hz [Burton et al., 2008]. The in- and out-of-phase AC magnetic susceptibilities have been plotted in the range 0.1–997 Hz to monitor time-dependent magnetic behavior. These frequency-dependent susceptibilities are particularly sensitive to thermal activation of interacting magnetic particles [e.g., Shcherbakov and Fabian, 2005]. In the ordered solid solutions, AC magnetic susceptibility is sensitive to ferrimagnetic clusters and to spin glass transitions, as will be discussed below for different temperature ranges.
6.3. Low-T Limits of the PM Region
Figure 14 shows the ionic models with magnetic linkages for Ilm 70, 80 and 90 compositions and the corresponding temperature susceptibility diagrams for Ilm 71.1, 80.6 and 91.5. The red arrows on the temperature susceptibility diagrams indicate the temperatures at the red arrows in Figure 13. Clusters in Ilm 90 are not strongly exchange-coupled to each other. In the 2-D model of Ilm 80 strong exchange linkage appears not to be complete, but our 3-D models indicate that most, but not all ions are strongly linked to each other. Available data indicate that the percolation threshold for 3-D linkage is about Ilm 87. In Ilm 70, linkage is almost complete already in 2-D, though even within the 3-D network there exist islands of unlinked regions.
 In Ilm 91.5, the PM to PM′ crossover with decreasing T is marked by an upward ramping of susceptibility indicating an increasingly coherent magnetization of the independent clusters. There is a peak in susceptibility and onset of frequency-dependent behavior near 80 K, corresponding to a peak in magnetization of the FM clusters which are unable to connect into a global FM network. Within the frequency-dependent region, absolute susceptibility falls as the magnetizations of clusters are hardened in incompatible directions by the blocking of ilmenite-like exchange links. In Ilm 80.6 and 71.1 there are subordinate cluster phenomena related to the high-susceptibility peak associated with initial FM magnetization (red arrows), however the frequency diagram for Ilm 71.1 extends only to 300 K. The TC is plotted in Figure 13 (reported by Burton et al. ). This indicates that although the material has nearly global adjacent layer linkage, there are still minor clusters behaving independently. This multispectral behavior is frozen out with falling temperature.
6.4. FM to FM′ Crossover
Figure 15 shows the ionic models for Ilm 70 and 80 compositions and parallel temperature susceptibility diagrams for Ilm 71.1 and 80.6 and the dark blue arrows correspond in temperature to the dark blue arrows in Figure 13. Here the models indicate a new phenomenon occurring on the surface of the percolating, possibly fractal-shaped, FM network. This has been highlighted by adding light blue color to ions with one magnetic linkage, and dark blue color to ions with only two linkages. The phenomenon, as best understood, is as follows. With decreasing temperature, magnetic linkages to ions at the boundaries of the FM network become more important, and these ion spins freeze. This has the anomalous effect of reducing the net susceptibility and leads to the fairly abrupt decrease in susceptibility with decreasing temperature above the positions of the dark blue arrows. This anomalous reduction in susceptibility with decreasing temperature marks the crossover from the FM to the FM′ region. This phenomenon takes place at a higher temperature in Ilm 80.6 (≈100 K) than Ilm 71.1 (≈90 K). Geometrically, the weakly linked blue ions on the edges of linked regions are more abundant in Ilm 80.6 than Ilm 71.1. However, to decide, whether these two observations are connected, requires detailed quantitative modeling and is beyond the scope of this article.
7. Lowest-Temperature Phase Diagram for Ilm 60–100 and 0–150 K
Figure 16 focuses on the very low temperature part of the phase diagram, 0–150 K for compositions Ilm 60–100. The data points are identical to those reported in Burton et al. . Again, the superimposed paths indicate the likely sequence of states for compositions Ilm 70, 80 and 90 in our models, beginning in the FM′ (dark blue dots and arrows) and PM′ (red dot) regions and extending to lowest temperature.
7.1. Ilmenite-Like Double-Layer Interactions
 At TN 57 K, the double-layer interactions overcome the thermal energy k T, and produce antiferromagnetism with moments parallel to c for the ilmenite end-member. With increasing hematite substitution TN decreases slightly, because Fe3+ ions in Ti layers produce adjacent layer interactions that counteract the double-layer Fe2+-Fe2+ interactions. Ilmenite double-layer interactions form the only long-range order until ∼Ilm 87 at ∼40 K. Neutron scattering data indicate that for Ilm 88.3 (see below) the anisotropy of the double-layer interactions is strong enough turn nearly all magnetic moments parallel to c. Below the FM percolation threshold near Ilm 87, the then prevailing global adjacent layer interactions suppress or prevent three-dimensional double-layer interaction networks, although these may still percolate simultaneously with the FM order over a very limited composition range. In any case, in randomly occurring pockets depleted of Fe3+ ions in the Ti layers, double-layer magnetization still occurs at ≈35 K for Ilm 80 and ≈30 K for Ilm 70. Conceptually this is “the back side of the percolation threshold.” Neutron-scattering evidence for Ilm 79.6 (see below) shows, that for this composition the anisotropy of the adjacent layer interactions keeps the magnetic moments parallel to (0001) down at least to 13 K. Locally, double-layer interactions may influence the very low temperature magnetic properties even of Ilm 60 and more hematite rich compositions [Fabian et al., 2008].
7.2. Ordering due to Double-Layer Interactions
Figure 17 shows the ionic models for Ilm 70, 80 and 90 compositions and parallel temperature susceptibility diagrams for Ilm 71.1, 80.6 and 91.5. The light blue arrows on the T susceptibility diagrams correspond in temperature to the light blue arrows in Figure 16. The models show features of magnetic linkage emphasized previously, but the nature of adjacent layer linkage is emphasized by using the red and orange color scheme for ions with 2 or more adjacent layer interactions, similar to the red and blue scheme used for Figures 8b, 9b, and 10b. In addition, light blue color is added to all ions with 0 or 1 adjacent layer linkage. These are the ions essentially free of adjacent layer linkage, and at very low temperature, 50–20 K, start to block into their double-layer ilmenite-like linkages. The model for Ilm 90 is dominated by light blue colored ions, showing that, at appropriate temperature, a global percolating double-layer interaction network forms the magnetic order. In the model for Ilm 80 the light blue colored ions form mainly isolated clusters, which locally can have considerable size. In 3-D it is geometrically possible that a percolating double-layer network resides in a small part of the solid solution simultaneously with the percolating FM network. However, the 3-D network of adjacent layer interactions greatly curtails and may entirely prevent a 3-D network of double-layer interactions, requiring instead clusters forming on the ilmenite-poor “back side” of the percolation threshold, much as adjacent layer interaction clusters formed on the ilmenite-rich “front side.”
 In Ilm 91.5, dispersion in susceptibility behavior is reduced in the region 50–40 K when the global double-layer networks orders into an AF structure. In the interval 35–25 K frequency dependence increases again, because of conflicting interactions between adjacent layer clusters and the double-layer AF network allows for several competing metastable spin configurations at the onset of the spin glass phase. In Ilm 80.6 and 71.1, where frequency-dependent susceptibility and cluster phenomena died out at about 90–70 K, they reappear at lower temperature because regions of incipient double-layer AF order form conflicting isolated clusters within the already established global FM network (“the back side of the percolation threshold”), which destabilize the spins at the volumetrically large fractal boundary.
7.3. Results From Neutron Diffraction
Figure 18 shows results from critical neutron scattering studies conducted by Arai et al. [1985a, 1985b] on compositions Ilm 79.6 and 88.3. They are presented in the context of Figure 16. These compositions are on opposite sides of the percolation threshold. In fact the data on composition 88.3 combined with evidence for ferrimagnetic behavior for Ilm 85 by Brown et al.  is the best direct evidence we have for placing the percolation threshold at ≈Ilm 87.
 The neutron-scattering technique used by these authors is very effective in detecting magnetic moments produced by adjacent layer interactions, but only indirectly useful in detecting magnetic moments related to double-layer magnetizations. For Ilm 88.3 at 50 K (Figure 18a) the phase is in the PM′ region and magnetic moments are parallel to (0001). By 40 K, at the onset of AF global double-layer interactions, all magnetic moments are rotated quasi-parallel to c. By 38 K, when cluster spin glass behavior is detected, as a result of frustration between adjacent layer interaction clusters and the global double-layer interactions, magnetic moments remain parallel to c and this orientation persists at least down to 25 K. It appears that the anisotropy of ilmenite-like double-layer interactions prevails so strongly that even local adjacent layer interactions cannot keep their moments in the (0001) plane.
 For Ilm 79.6 at 80 K (Figure 18b), recorded by this method as the FM to FM′ transition (a lower T than recorded using other measurements and samples) the magnetic moments are parallel to (0001). By 29 K, considered by this method to be the transition FM′ to SG, despite evidence for double-layer interactions, the moments remain parallel to (0001), and this orientation is retained at least to 13 K within the spin glass field. It appears that when ilmenite-like double-layer interactions do not become global, then the anisotropy of the prevailing adjacent layer interactions dominates and keeps the magnetic moments in the (0001) plane.
7.4. Incompatible Interaction Schemes
 The adjacent layer hematite and double-layer ilmenite antiparallel interaction schemes lead to incompatible magnetic moment directions. Thus, it is inevitable that conflict develops at low temperature, leading to frustration of the spins of many atoms on the boundaries between regions of adjacent layer and double-layer interactions. This phenomenon produces spin glass behavior at low temperature (<40–20 K) from near end-member ilmenite to very hematite-rich compositions. In compositions ≥ Ilm 87 and below TN, the conflict comes from frustration between the global double-layer antiferromagnetism and the adjacent layer FM of small Fe-enriched clusters. For compositions ≤ Ilm 87 it comes from frustration between the adjacent layer ferrimagnetic network and the antiferromagnetism below TN from the double-layer interactions that dominate within local Ti-rich pockets.
7.5. Evolving Spin Glass Behavior
 The low-temperature spin glass behavior of hematite-ilmenite solid solutions has been extensively studied by numerical models [see Harrison, 2009, and references therein] and geometric consideration adds little to its explanation. However, Figure 19 shows the 2-D ionic models for Ilm 70, 80 and 90 compositions and parallel temperature susceptibility diagrams for Ilm 71.1, 80.6 and 91.5. The violet arrows correspond in temperature to the violet arrows in Figure 16. The models emphasize the adjacent layer linkage using the red and orange color scheme for ions with 2 or more adjacent layer interactions. Ions with 0 or 1 adjacent layer linkages that could develop double-layer ilmenite-like linkages are shown in light blue. An additional violet color is added for ions with 2 or 3 adjacent layer linkages to indicate ions in locations where frustration is likely to develop with falling temperature and overall magnetic hardening. This happens because adjacent layer and double-layer magnetization schemes are completely incompatible, both in terms of anisotropy and layer systems. Because of the small size of adjacent layer clusters in Ilm 90, they are overtaken by many frustrated ions with falling temperature. As the size and abundance of these clusters decreases toward Ilm 100, the temperature of onset of frustration decreases. Because of the dense adjacent layer linkage in Ilm 80, and even more in Ilm 70, the abundance of violet-colored frustrated ions is increased. At 15 K, a temperature which is well within the spin glass field in all three compositions dispersion and susceptibility become very low due to frustration of competing interactions.
 The magnetic behavior of metastable Fe-Ti ordered ilmenite-rich members of the hematite-ilmenite solid solution series shows a complexity, which is beyond the current ability of comprehensive numerical models. Improving the physical interpretation requires a thorough geometrical understanding of this solid-solution system. Starting from a visualization of the likely distribution of Fe2+, Fe3+ and Ti4+ ions and their magnetic interactions, we try here to lay the foundations for further quantitative modeling. A series of 2-D images were prepared showing all positions occupied by Fe atoms over a discrete area within a plane parallel to (1 1 −2 0), as well as all interacting Fe atoms in the two adjacent (1 1 −2 0) planes. Based on published interaction data, they show features of end-member hematite and ilmenite, as well as of compositions Ilm 90, 80 and 70, portraying indirectly the occupancy of 612 atomic positions and all the predicted strong magnetic interactions between them. In addition, larger simplified 3-D models were created with 2592 atomic positions in a 6 × 6 × 6 unit cell volume for compositions Ilm 95, 90, 85, 80, 75, 70. These provide a more complete visual assessment of the effects of random atomic distribution and extent of magnetic networks for both adjacent layer and double-layer interactions.
 Hematite-like adjacent layer interactions produce an initial ferrimagnetism (FM) with moments parallel to (0001). A global FM network prevails with increasing ilmenite content and decreasing TC until, at Ilm 87, the Fe3+ occupancy of the Ti-ordered layers is too low to maintain a three-dimensional “infinite” FM network. Beyond this “percolation threshold,” only finite FM clusters prevail. At sufficiently low T below these clusters produce frequency-dependent susceptibility (PM′) within a paramagnetic (PM) field, in a behavior commonly compared to the “superparamagnetism” of small particles. Within the FM region, <Ilm 87, the percolating infinite FM network has a complex, probably fractal, geometry with high surface/volume ratio. Fe atoms with weak and asymmetric linkages at the boundary of this percolating FM cluster, acquire a hardened magnetization only well below TC, which decreases the overall magnetic susceptibility, explaining the transition FM to FM′.
 Double-layer interactions, producing antiferromagnetism with moments parallel to c, order for the ilmenite end-member at TN 57 K. The observation that TN decreases with increasing hematite substitution is explained by the fact that sparse Fe3+ ions in Ti layers produce adjacent layer interactions which disrupt and even counteract the double-layer Fe2+-Fe2+ interactions. Thereby, they delay the formation of global AF order. Ilmenite double-layer interactions dominate the global order until the FM percolation threshold at ∼Ilm 87. There is neutron scattering evidence for Ilm 88.3, indicating the double-layer interactions are so strong that all magnetic moments are turned nearly parallel to c. Between Ilm 87 and ∼Ilm 80, the prevailing global adjacent layer FM interactions appear to interfere with an intertwined three-dimensional double-layer interaction network, but may not suppress its effects completely. Neutron-scattering evidence for Ilm 79.6 shows, for this composition, that adjacent layer interactions produce magnetic moments parallel to (0001) down at least to 13 K and no evidence for a complete double-layer interaction network. Below ∼Ilm 80, double-layer magnetization still takes place, but only in pockets or “holes” within the prevailing adjacent layer network. Conceptually this is “the back side of the percolation threshold.” Although no longer the dominant global link, double-layer interactions may be locally significant to Ilm 50 and probably beyond, even to very hematite-rich compositions.
 The hematite and ilmenite magnetic interaction schemes and magnetic moment directions are incompatible, and they come in conflict at very low T, leading to frustration of the spins of many atoms on the boundaries of regions of adjacent layer and double-layer interactions. This phenomenon produces spin glass behavior at very low T, and elsewhere we show this persists to well below Ilm 50. Above Ilm 87 and slightly to greatly below TN, depending on composition, the conflict comes from frustration between the global double-layer antiferromagnetism and the adjacent layer magnetism of small Fe-enriched clusters. Below Ilm 87 it comes from frustration between the adjacent layer ferrimagnetic network and the antiferromagnetism below TN from double-layer interactions that dominate within local Ti-rich clusters.
 The ionic/magnetic interaction images provide a conceptual overview toward an understanding of low-temperature magnetic behavior of hematite-ilmenite solid solutions, even though they do not provide a quantitative basis for the properties. These models also supply information on the distribution, numbers and orientations of magnetic interactions that can become guidelines for more sophisticated calculations.
 This research was supported by the Research Council of Norway (RCN grants 189721/S10 and 169470/S30) and by NGU.