Perrierite-(Ce) from the Laacher See tephra, Eifel, Germany, and the modular character of the chevkinite group of minerals

Perrierite-(Ce) crystals from ejecta of the Laacher See volcano, Eifel, in Germany were studied by X-ray single-crystal diffraction and electron microprobe analysis. The composition and crystal structure of this sample is discussed in relation to the known properties of the chevkinite group minerals and related synthetic compounds. Taking into account the modular character of the chevkinite minerals, based on a rutile and a silicate module, the preferred formation of either the perrierite or the chevkinite structure type is correlated with the formal charge of the rutile and silicate modules. The rutile module is expected to carry a negative charge, compensated by a positive charge of the silicate module. On average, the charge modulus is observed to be larger for the chevkinite-type module stacking. It can drop to zero in perrierite-type structures rich in Sr or Ca. In such cases, it is generally not expected to rise above two. The perrierite-(Ce) described in this study crystallizes in space group C2/m. The anisotropic oxygen and rare-earth element displacements observed in this crystal indicate a local domain structure of P21/a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P2_1/a$$\end{document} symmetry, when compared to the P21/a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P2_1/a$$\end{document} symmetry of the synthetic perrierite La4Mg2Ti3O8(Si2O7)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {La}_4\hbox {Mg}_2\hbox {Ti}_3\hbox {O}_8(\hbox {Si}_2\hbox {O}_7)_2$$\end{document}.


Introduction
The chevkinite-group minerals (CGM) are common accessory minerals in a wide range of rock types. The chemical composition of the CGM conforms to a standard formula A 4 BC 2 D 2 O 8 (Si 2 O 7 ) 2 . The A-sites host relatively large (r > 1Å) predominantly di-and trivalent cations, i.e., rare-earth elements (REE), Ca and Sr. The REE content is dominated by the light REE (LREE) between La and Sm (Macdonald and Belkin 2002). While the A-sites are at least seven-coordinated by oxygen atoms, the remaining B-, C-and D-sites all have distorted octahedral coordination, hosting smaller cations with radii significantly below 1 Å. The site nomenclature of the CGM varies across publications by different authors (see Macdonald et al. (2019) and Nagashima et al. (2020) for summaries). We follow the site nomenclature of Macdonald et al. (2019) here, but emphasize that the site occupancies of the B-, Cand D-sites are extremely variable. Defining the structural sites by their site populations is, therefore, rather ambiguous. We are going to use the following definition instead: the BO 6 octahedra share all their corner oxygen atoms with SiO 4 tetrahedra of the sorosilicate groups. The CO 6 octahedra share two opposite corners with such silicate groups and the DO 6 are not connected to SiO 4 tetrahedra at all (Fig. 1).
The currently known CGM are subdivided into the chevkinite and perrierite structure types. In minerals of both types, space group symmetries C2/m and P2 1 ∕a(P2 1 ∕c) have been observed. Unit cell parameters a, b and c are generally similar, with b showing the smallest variability across different compositions. While there is systematic variation of the unit cell edges with structure type as well, the most pronounced difference is seen in the -angle. In the perrierite type, ≈ 114 • and in the chevkinite type, ≈ 100 • is observed. Ito and Arem (1971) reported high-temperature transformations from synthetic perrierite-type compounds of composition Pr 4 Mg 2 Ti 3 O 8 (Si 2 O 7 ) 2 and Pr 4 Ni 2 Ti 3 O 8 (Si 2 O 7 ) 2 to a chevkinite-type structure around 1070 and 1060 • C, respectively, but the transformation appeared to be irreversible. Other compositions of perrierite do not transform to chevkinite at all, according to this study. Calvo and Faggiani (1974) synthesized La 4 Mg 2 Ti 3 O 8 (Si 2 O 7 ) 2 with the perrierite structure as well as Nd 4 Mg 2 Ti 3 O 8 (Si 2 O 7 ) 2 and Nd 4 Co 2 Ti 3 O 8 (Si 2 O 7 ) 2 , both with the chevkinite crystal structure. All three compounds have been described in space group P2 1 ∕a , rather than C2/m. Starting from a large set of synthetic compounds, Ito (1967) obtained an empirical expression for the formation of either the chevkinite or the perrierite structure based on the average ionic radii at the B, C and A-sites. Ferraris et al. (2004) noted that the CGM are in fact modular materials, forming a polysomatic series. The modularity of the CGM is based on a rutile module and a silicate module, stacked along c * . The rutile module is topologically identical to a (110) layer of the rutile structure (R-module). It contributes the C 2 D 2 O 8 unit of the CGM formula. The other basic module is formed by the A 4 B(Si 2 O 7 ) 2 sorosilicate unit (S-module).
It is worthwhile to consider the charge balance of both modules separately. If both the C-and D-sites of the R-module are occupied by tetravalent cations like Ti 4+ , the module is formally charge neutral (as it would be in TiO 2 ). If D is a trivalent metal cation and C is tetravalent, as for example in chevkinite-type polyakovite-(Ce), (Ce,Ca) 4 MgCr 3+ 2 Ti 2 O 8 (Si 2 O 7 ) 2 (Popov et al. 2001), the R-module would carry two negative charges. The compensating positive charge of the S-layer mainly depends on the ratio of trivalent REE and divalent earth alkaline metal cations occupying the A-sites. For a 2:2 ratio and a divalent B-site occupancy, the S-module would be charge neutral. For compensation of the negatively charged R-layer in polyakovite-(Ce), the REE/ Ca ratio has to be larger than one and/or a tri-or tetravalent cation has to at least partially substitute at the B-site. In fact the ideal composition is Ca-free, REE 4 MgCr 3+ 2 Ti 2 O 8 (Si 2 O 7 ) 2 (Popov et al. 2001). On the other hand, perrierite-structured matsubaraite, Sr 4 Ti 5 O 8 (Si 2 O 7 ) 2 , has its A-sites entirely occupied by a divalent cation. This is compensated by tetravalent Ti at the B-site, yielding a zero module charge.
As is commonly the case in polysomatic series, the stacking of different modules determines the stoichiometry as well as the unit cell dimensions of the resulting compound. The chevkinite structure is formed by the sequence RSRSRS.... The perrierite-type structure is formed by RSRS'RSRS', where the S'-layer is shifted by ∕2 with respect to the S-layer. Other polysomes have been synthesized, in particular the sequence RRSRRS... (Wang et al. 1995). In general, the minerals of the CGM are able to form polytypes due to their modular character. In fact chevkinite-(Ce) and perrierite-(Ce) could be considered different polytypes of the same mineral species. The different topology of the interface between R-and S-modules resulting in both polytypes is shown in Fig. 1. While the RSRS sequence of chevkinite causes a staggered connectivity of SiO 4 tetrahedra and CO 6 octahedra along [010] at the module boundary, the shifted S'-modules of the perrierite sequence cause an eclipsed configuration of the SiO 4 tetrahedra and of the A-cations on either side of the R-module. While a very large amount of compositional data for the CGM exists in the literature, detailed crystal structure data are not so common. In part, the lack of crystal structure data is due to the scarcity of good quality natural crystals, as the CGM frequently occur in the metamict or at least partially metamict state (Sokolova et al. 2004) and might also be subject to secondary chemical alterations due to this metamict state. To increase crystallinity, such samples often are annealed at temperatures of 750 • C or higher (Sokolova et al. 2004;Stachowicz et al. Fig . 1 Configuration of the R-module boundary in a) the perrierite structure type (this work), b) the chevkinite structure type (structure data from Sokolova et al. (2004)) 2019b). This may lead to cation ordering, redox reactions or other transformation processes that are likely to render the crystals in a different state than that prior to metamictization. Hence, well crystallized, non-metamict natural samples and synthetic samples are needed to obtain reliable information about structural properties of well-equilibrated CGM. A detailed study of cation distribution in three non-metamict chevkinite-(Ce) samples has been published very recently (Nagashima et al. 2020). One aim of the present work is to identify possible mechanisms that could help to explain the empirically known compositional controls ) over the formation of either the chevkinite-or perrierite-type structure in CGM.

Experimental
The sample material used in this work is from ejecta of the Laacher See volcano (LSV), which erupted 12.9 ka ago (Schmitt et al. 2010) in the East-Eifel area, Rhineland-Palatinate, Germany. The sample was collected at the In den Dellen quarry near Mendig, Germany. A needleshaped crystal of perrierite-(Ce), extending along [010] and of dark reddish-brown color has been measured using X-ray single-crystal diffraction with a Nonius KappaCCD diffractometer equipped with graphite monochromated MoK radiation. Data collection and crystal details are given in Table 1. Pixel intensities were integrated using the Eval15 software package (Schreurs et al. 2010). Structure refinement was conducted using Jana2006 (Petříček et al. 2014). Scattering functions for the uncharged atoms are used throughout. All structure diagrams have been prepared using CrystalMaker (Palmer 2015).
Three fragments of similar crystals from the same location were measured using a Cameca electron microprobe (Cameca SX 100) in wavelength-dispersive mode. An accelerating voltage of 15 kV and a probe current of 20 nA were used. Standards used were andradite (Si K , Ca K , Fe K ), MnTiO 3 (Mn K , Ti K ), ZrSiO 4 (Zr L ), Nb 2 O 5 (Nb L ), Ta 2 O 5 (Ta L ), UO 2 (U M ), MgO (Mg K ), albite (Na K ), Al 2 O 3 (Al K ), and SrTiO 3 (Sr L ). Standards for REE and Th consisted of five different oxide glasses (Y L , La L , Ce L , Pr L , Nd L , Sm L , Gd  (Becker and Coppens 1974) L , Tb L , Dy L , Ho L , Dy L , Er L , Tm L , Yb L , Lu L , Th M ). An electron backscattering image of a crystal section normal to the elongation direction indicates chemical zonation, with a brighter area at the center of the crystal, surrounded by a darker rim. This correlates with the obtained REE profile, which indicates higher La concentration at the rim and elevated concentrations of Ce, Nd and Sm towards the core region.

Crystal structure refinement
Structure refinement of a perrierite-(Ce) crystal converged to R1=0.026 for 1859 reflections above the I > 3 (I) intensity level (Table 1). Refined structural parameters are compiled in Tables 2 and 3. A cif-file is made available as Online Resource 1. A small volume fraction of the crystal is a twin domain, rotated by 180 • about * , giving rise to some weak extra reflections (Online Resource 2). The domain boundary of such a twin can be associated with an R-layer (Fig. 2), as the twin axis is parallel to the layer stacking direction and the R-layer contains the twinoperation, disregarding minor distortions of the octahedra. It can be expected that this type of growth twinning is very common within the CGM, due to the resulting low energy of the twin boundary that does not alter the bond topology. Mashima et al. (2008) describe a microtwinned, orthorhombic polytype of rengeite, formed by a module sequence SRS T RSRS T ..., where S T is the twin equivalent 0.0195 (6) 0.0063 ( 0.0190 (9) 0.0171 (9) 0.0080 (7) 0 0.0058 (7) 0

Fig. 2
The R-layer as twin boundary of the chevkinite and perrieritetype structures. Unit cell outlines of both polytype structures and their twin equivalents are indicated in black and red color, respectively. SiO 4 tetrahedra are shown in blue, BO 6 octahedra in brown, A-site cations in yellow and C and D-site coordination polyhedra in light blue color of S. This orthorhombic structure is also found intergrown with monoclinic rengeite. Reflections potentially violating the condition h + k = 2n for C-centering are not observable by X-ray diffraction in perrierite-(Ce) from the LSV. Diffraction spots are sharp and the crystal shows no obvious signs of metamictization in spite of the significant Th content. This is not unexpected considering the relatively young age of these crystals.
To approximate the site scattering of the A-, B-, C-and D-sites, site population with the element pairs Ca/Ce (A1 and A2), Fe/Zr (B), Ti/Al (C) and Ti/Nb (D) was refined, assuming full occupancy of all structural sites. Good agreement of the refined site occupancies with the results of the EMPA (Tables 4, 5) is obtained at the A-and D-sites. At the A1-and A2-sites, the Ce-scattering function serves as an approximation of average scattering by the dominant LREE La, Ce, Pr and Nd, with the concentrations of all other REE either close to or below the detection limit (Tables 4, 5). The A1-site is seven coordinated by oxygen atoms with A1-O distances ranging between 2.47 and 2.532 Å, with a mean distance of 2.511 Å, if we assume that bonds have to contribute a bond valence of at least 4% of the central cation charge (Brown 2002). There are three further A1-O distances (3.026 and 3.094 Å, to O4 and O7, respectively), that do not match this criterion (Table 6). Following the same definitions, A2 is coordinated by eight oxygen atoms at distances ranging between 2.431 and 2.680 Å with a mean distance of 2.592 Å. Two more distant oxygens (O6)  (2)  at 2.976 Å contribute a little less than 4% to the central cation charge. The refined Ca occupancy is slightly higher at the A1-site than at A2 ( Table 2). The refinement results also confirm that Ti and Nb populate the D-site. Initial refinement of the B-site occupancy was attempted with Fe as the only scatterer. This yielded smaller displacement parameters than for the other octahedrally coordinated cations, indicating the additional presence of a stronger scatterer at the site. The B-site is coordinated by two opposite oxygen ligands at 2.021 and by four oxygen ligands at 2.129 Å distance in a distorted octahedron. The average bond length of 2.093 Å is smaller than expected for entirely ferrous iron, but too large for entirely ferric iron population. It is, however, well suited for Zr 4+ . Mixed occupancy by the pair Fe and Zr slightly lowered R1 and rendered B-site displacement parameters similar to those obtained at the C-and D-sites.
Site population of the C-site is more complicated, as this site is expected to accommodate the remaining cation species Al, Mg and Mn, in addition to Ti not accommodated at the D-site, as well as Zr and Fe not accommodated at the B-site. The scattering power significantly decreases in the sequence from Zr to (Ti, Mn, Fe) to (Mg, Al). This renders site occupancy by the pair Ti/Al less comparable to the results of the EMPA than the refined site occupancies at the A and D positions. Assuming uncharged atoms and ignoring the different -decays of the individual scattering functions, the total electron count of 40.4 for the refined Ti and Al population is only 1% smaller than the electron count obtained by EMPA (40.83) from the elements attributed to the C-site (Tab. 5). If we consider that the Zr occupancy at the B-site is 0.14 apfu from structure refinement, corresponding Fe substitution for Zr has to be considered at the C-site. This would yield an electron count of 38.87 at the C-site, 4% smaller than the refined site scattering.
The displacement parameters of several oxygen atoms (O1, O3, O6, O7), but also those of the A-cation sites are unusually large and anisotropic (Table 3). This is not due to thermal displacement, but an indication of local distortions from the C2/m space group symmetry, as will be discussed in section Space group symmetry of perrierite-(Ce).

Chemical composition, Fe oxidation state and module charge
In terms of chemical composition, the three crystal fragments are bordering on perrierite-(La), but Ce is still the , which also originates from the LSV, is 60% (Chukanov et al. 2012b). In the present sample, the bond valence sum (BVS) at the B-site indicates at least partial substitution of Fe 2+ by Fe 3+ too (Table 6). We, therefore, assume that a relative Fe 3+ content of 60% is reasonable. This also avoids unphysical overpopulation of the A and Si cation sites (Table 5) when calculating the empirical composition based on 22 oxygen anions.
It is instructive to compare the formal charges of the two modules. As the R-module contains substantial amounts of di-and trivalent cations and not enough pentavalent Nb to compensate for this, it is expected to be negatively charged. For the combined composition given in the third row of Table 5, the R-module can be attributed a formal charge of -1.37, with the S-module carrying a corresponding positive charge. Calculation of this module charge crucially depends on the particular configuration of metal cations and on their oxidation state and is, therefore, associated with an estimated uncertainty of 10%. If for example, we assume Fe 2+ at the C-site, the R-module charge would become -1.57. On the other hand, if Mg is relocated to the B-site and replaced by Fe 3+ , the module charge would become -1.23.

R-module octahedral distortions and site occupancy
The D-site of perrierite exhibits a strongly distorted oxygen coordination, with a short D-O bond of 1.822 Å and an opposite long D-O distance of 2.384 Å. Short metal-oxygen bond distances, i.e., those significantly below the sum of the respective Shannon-radii, are typical of early transition metal cations Ti 4+ and Nb 5+ , that frequently give rise to the formation of compounds and minerals (e.g., titanite, macedonite, natisite, KNbO 3 , LiNbO 3 ) with ordered arrangements of short Ti=O or Nb=O bonds. The distortion in perrierite has been explained as a result of repulsion away from the closest A-site cations (Gottardi 1960;Stachowicz et al. 2014). The respective D-A distances are 3.322 Å for D-A1 and 3.317 Å for D-A2 in the present refinement. If the D-cation would be located at the center of its coordination polyhedron, these distances would reduce to 3.207 Å and 3.147 Å, respectively. By contrast, in chevkinite-(Ce) in space group C2/m the two D-sites, e.g., M3 and M4 (Sokolova et al. 2004), are located at special Wyckoff-positions 2a and 2b and consequently Ti at this site cannot displace parallel [100] in this space group symmetry. The distances Ti3-Ce1 and Ti2-Ce2 are 3.223 and 3.183 Å in this case, only marginally longer than the equivalent Ti-Ce distances in perrierite-(Ce) for Ti assumed at the center position. Stachowicz et al. (2014) argue that in chevkinite-(Ce) the shortest Ce-Ti distances are trans-oriented with respect to the R-module, while they are cis-oriented in the perrierite structure, causing the shift of the D-site cation by repulsion. Following the same line of reasoning some other distortions of the R-module in chevkinite can be explained too. The DO 6 -octahedra are slightly tilted in the chevkinite structure, as the trans-standing oxygen ligands are pulled towards the respective A-site cations above or below them, while this tilting force is absent in perrierite with its symmetrical arrangement of A-cations above and below each R-module (Fig. 1). The slight kinking of the CO 6 chains parallel [010] in the chevkinite type can be similarly explained by the antisymmetrical interaction with the A-site cations. Regardless of these distortive effects based on nearest-neighbor attractive and next nearest-neighbor repulsive forces, the particularly strong distortion of the D-site coordination in the perrierite type is not likely to be entirely due to repulsive forces between the cations. In such a scenario, the ions would act like hard spheres and the deformation would propagate along the Ti-O bond, distorting the octahedron, rather than primarily displacing its central ion and forming a very short Ti-O bond. This particular behavior is promoted by the electronic properties of the Ti and Nb cations occupying this site. Gueho et al. (1995) took this into account by placing the Jahn-Teller active Mn 3+ cation onto the D-site in the perrierite-type La 4 Mn 5 O 8 (Si 2 O 7 ) 2 and Mn 4+ onto the C-site, thereby reversing the conventional distribution of formal charges inside the R-module. Table 7 compares the unit cell parameters of the LSV sample with those of CGM and synthetic chevkinites and perrierites from the literature. Apart from the apparent difference in -angle between the perrierite and the chevkinite-type phases, the a parameter ranges between 13.3 and 13.6 Å for the chevkinite-type structures, while it is slightly larger, between 13.55 and 14 Å for the perrierite type. The difference is more pronounced for the c parameter, which ranges between 10.5 and 11.4 for the chevkinites and between 11.6 and 12 for the perrierite-type phases. The transformation matrix suitable to convert from the perrierite unit cell setting to that of chevkinite is [-1 0 0][0 -1 0][1/2 0 1] (by rows). The corresponding chevkinite unit cell of the perrierite structure refined in this work would hence have c = 10.9317 Å and = 101.08 • , with the size of a and b unchanged.

Comparison with other CGM
To compare the metrics of the different members of the CGM, it is useful to calculate the distance between subsequent R-modules, which is tantamount to the (001) d-spacing, d = c sin . Figure 3 plots the resulting module separation against the calculated module charge for CGM and analogous synthetic compounds with known unit cell parameters. Calculating the module charge ideally requires knowledge of the chemical composition, the site occupancies and also of the oxidation state of the cations. While Sokolova et al. (2004) assumed for chevkinite-(Ce) that ferrous iron can only occupy the B position (or M1), even though they did not entirely rule out that mixed occupancy occurs also at the C-and D-sites, Nagashima et al. (2020) showed that ferrous and ferric iron can occupy the B-site (M1) and that ferrous iron is also present at the D-sites (M3, M4). If Fe 3+ is present, we have, therefore, equally distributed it over all Fecontaining sites. If necessary, empirical compositions have been recalculated from the oxide compositions based on 22 anions. CGM with unusually low oxide sums, significant vacancies or with potentially inaccurate unit cell dimensions have been omitted from Table 7.
The interlayer spacing shown in Fig. 3 is close to the distance resulting from stacking four oxygen anions along c * , i.e., 10.88 Å (the horizontal dashed line in Fig 3). The perrierite structure type is found for smaller module Table 7 Comparison of CGM and isostructural synthetic compounds. References: 1. Stachowicz et al. (2014) , 2. Ito and Arem (1971), 3. Calvo andFaggiani (1974), 4. Miyajima et al. (2002), 5. Yang et al. (2012), 6. Miyajima et al. (2001), 7. Chukanov et al. (2012b), 8. Sokolova et al. (2004), 9. Popov et al. (2001), 10. Chen et al. (1995), 11. Chen et al. (1996) , 12. Xu et al. (2008), 13. Chukanov et al. (2012a), 14. Chen et al. (1994, 15. Gueho et al. (1995), 16. Nagashima et al. (2020) 17. Gottardi (1960, Ito (1967) C2/m 13.6348 (8) 5.6484 (5) 11.7194 (7) 113.736 (5)  charges at or below 2.0, while the chevkinite-type structure is observed at higher module charges, with the smallest value of 1.43 found for christofschäferite-(Ce). From this graph, it emerges that the perrierite module sequence is frequently associated with a smaller R-layer distance and smaller module charges than the chevkinite module sequence. An extreme example is matsubaraite, Sr 4 Ti 5 O 8 (Si 2 O 7 ) 2 , which has a nominal module charge of zero. The increase of the R-interlayer spacing from the perrierite to the chevkinite structure is also found for the two synthetic compounds studied by Ito and Arem (1971), with high-and low temperature polymorphs connected by vertical lines in Fig. 3. This indicates that, for a given chemical composition, the chevkinite module stacking inherently requires a larger size of the RSR layer stack than the perrierite module stacking. In addition, the separation of R-modules is of course influenced by other factors that influence the height of the S-module, especially the average size of the A-site cations. We might expect a positive correlation between R-module separation and unit cell volume. Plotting the latter against the former in Fig. 4 reveals a clear splitting between such positive correlations for the perrierite and the chevkinite module stacking. Chevkinites have an R-module separation that is, on average, about 0.14 Å larger than that of perrierites with similar unit cell volume. This difference in module height can be almost entirely attributed to the C-O bond that is normal to the (001) plane. This bond length ranges between 2.02 and 2.05 Å (Sokolova et al. 2004;Nagashima et al. 2020) in most chevkinite-type structures. Taking the LSV perrierite-(Ce) of this work as a representative of the perrierite-type structures, the equivalent C-O distance is just 1.966 Å. Additional shortening of the height of the perrierite CO 6 octahedra relative to the corresponding octahedra in chevkinite is caused by slight bending of the O-C bonds with respect to the (001) plane. The O-C-O angle is 174.1 • in the LSV perrierite, while it is 180 • in chevkinite-(Ce) (Sokolova et al. 2004). In total, a difference of at least 0.11 Å in d (001) results from the difference in CO 6 height in both structures, accounting for almost 80% of the observed difference in R-module separation alone. Ultimately the larger module separation in the chevkinite structure type can be associated with its higher module charge, which is negative for the R-module, caused by lower positive charge of the C-and D-cations at the center of this module, weakening the C-O bonds normal to (001) and causing a more elongated octahedral environment parallel to the module stacking direction.

Relation to empirical compositional trends
The empirical compositional boundary between the perrierite and the chevkinite module sequence for CGM  Table 7. Red symbols denote perrierite-type structures (circled = this work), blue symbols denote the chevkinitetype structures. The horizontal dashed line indicates the distance corresponding to the sum of four oxygen diameters. Vertical lines connect samples of identical composition transformed from the perrierite to the chevkinite structure by heat treatment (Ito and Arem 1971) Fig . 4 Unit cell volume as a function of R-module separation. Red symbols denote perrierite-type structures (circled = this work), blue symbols denote the chevkinite-type structures. Red and blue lines show the best fit to both sets of data, respectively. Black lines connect samples of identical composition transformed from the perrierite to the chevkinite structure by heat treatment (Ito and Arem 1971) occurrences in rocks has been located along a line in a plot of FeO versus (CaO+SrO) content (Macdonald and Belkin 2002;Macdonald et al. 2019). The line extrapolates to an FeO content of about 4 wt% at zero wt% (CaO+SrO) and reaches 15 wt% FeO at about 10 wt% (CaO+SrO). Compositions above this line apparently crystallize as chevkinite, those with lower FeO and higher (CaO+SrO) contents as perrierite. If we assume a simplified composition Ce 4−x Ca x Fe y Ti 5−y Si 4 O 22 , the charge balance for the coupled substitution scheme Ca S + Ti R ↔ Ce S + Fe R can be evaluated. Towards this aim, we assume that the B-site is fully occupied by Fe, with Ti substitution occurring for y < 1 . Equations for Fe (y) as a function of Ca content (x), based on charge balance for ferrous and ferric iron, respectively, can be calculated. Thus, we obtain y = 2 − x∕2 for Fe 2+ and y = 4 − x for Fe 3+ . The two resulting lines are plotted in Fig. 5 for 0 ≤ x ≤ 4 . The module charge above an Fe-content of 1 apfu increases linearly along these lines from bottom right to top left. All compositions corresponding to the above simplified formula can be expected to fall onto the two lines, with compositions containing Fe of mixed oxidation state expected to plot in the area between them. In the CGM, the presence of other metal cations, different substitution schemes or cation vacancies are obviously going to cause deviations, but generally the fields of perrierite and chevkinite structures shown in Fig. 5 are in qualitative agreement with those observed in natural samples. The crosses indicate positions of species shown in Table 7, excluding those like polyakovite-(Ce), that have Fe significantly substituted by other cations. The location of the dashed line in Fig. 5 is similar to that drawn by Macdonald et al. (2019). The oxide compositions at the intersections 1 and 2 amount to 3.65 wt% CaO and 9.61 wt% FeO and to 8.8 CaO and 14.6 FeO, respectively. At both intersection points, the charge modulus of the R-and S-modules is close to 1.25. Strontiochevkinite (Haggerty and Mariano 1983) does defy categorization by the above, however. To the authors' knowledge, no crystal structure refinement of this mineral exists to this day. The Debye-Scherrer powder diffraction data were indexed with a chevkinite-type unit cell by Haggerty and Mariano (1983). The electron microprobe analysis suggests an empirical composition of (Sr 2.27 Ca 0.37 REE 1.29 Na 0.02 ) ∑ =3.95 (Fe 0.913 Mn 0.02 Al 0.02 Zr 0.99 Cr 0.07 Ti 3.28 ) ∑ =5.29 Si 3.77 O 22 . This composition plots in the perrierite field of Fig. 5, in spite of the chevkinite-type unit cell data. There is also a deficit of Si and an excess of small cations beyond five apfu. This may indicate Ti substitution at the Si sites. But more importantly, it is impossible to predict which cation species occupy the B, C and D-sites, without structure analysis. Miyawaki et al. (2002) pointed out that the available powder   data of strontiochevkinite could be indexed with a perrierite-type unit cell as well and that the material analyzed by Haggerty and Mariano (1983) might have been an Fe-rich rengeite.
We have so far observed that the module charge is larger in chevkinites and smaller in perrierites. One possible reason is that the perrierite structure with its strong out of center distortion of the D-site cation is stabilized if enough Ti or Nb is present to completely fill this site. An R-module charge at or below -2, as we postulate for the chevkinite structure increases the probability that some of the D-site Ti is substituted with a trivalent or even divalent cation. Lesser positive charge of the metal cations in the D-position would in turn also reduce repulsion with respect to the A-site cations and might thus stabilize the chevkinite structure type with its smaller, symmetrical D-A distances. Lower positive charge at the C and D-sites of the R-module in the chevkinite structure type, corresponding to its higher absolute module charge, causes the expansion of this R-module, leading to the observed larger R-module separation in the chevkinite structure (Fig. 4).

Space group symmetry of perrierite-(Ce)
While the Laacher See perrierite-(Ce) crystallizes in C2/m space group symmetry, many examples of perrierites (Table 7) with SG symmetry P2 1 ∕a exist, notably the synthetic La 4 Mg 2 Ti 3 O 8 (Si 2 O 7 ) 2 (Calvo and Faggiani 1974). The P2 1 ∕a space group symmetry would manifest itself by the formation of reflections violating the condition for C-centering, h + k = 2n . This is clearly not the case with the perrierite-(Ce) of this study. However, the distortions of La 4 Mg 2 Ti 3 O 8 (Si 2 O 7 ) 2 from C2/m symmetry are small. Stachowicz et al. (2019b) showed that C2/m chevkinite-(Ce) can be annealed at 750 • C to obtain the P2 1 ∕a symmetry, with some oxygen atoms showing the largest shifts away from the mirror plane that is present in C2/m. In Fig. 6, we show that the displacement ellipsoids obtained for the perrierite-(Ce) structure at room temperature encompass the atom positions of the distorted La 4 Mg 2 Ti 3 O 8 (Si 2 O 7 ) 2 structure at the 95% probability level. This indicates that the distortions present in the P2 1 ∕a structure of perrierite, notably a kinked Si-O-Si bridging bond, off the pseudo-mirror plane, as well as slight rotations of the SiO 4 tetrahedra and displacements of the A-site cations parallel [010], are locally present within the S-modules of perrierite-(Ce). The observed C2/m structure is the result of averaging over many such S-layers with out-of-phase distortions. Only if the distortions in a sufficiently long sequence of S-layers are in-phase, the h + k = 2n + 1 reflections develop and the structure can be refined in SG P2 1 ∕a . Locally, each R-layer may act as an antiphase domain boundary, separating S-layers with different sense of distortion. An ordered arrangement of the distortions across many layer packets could be obtained via heating, as it was shown by Stachowicz et al. (2019b) for similarly disordered chevkinite-(Ce), even though the partially metamict nature of their samples has to be considered in this context. In terms of group-subgroup relations, C2/m constitutes the parent symmetry and P2 1 ∕a(P2 1 ∕c) constitutes the symmetry of the distorted phase. The distortions can be decomposed into mode amplitudes using the Isodistort program (Stokes et al. 2019). The results are shown in Table 8, where the LSV perrierite has been used as the parent structure and La 4 Mg 2 Ti 3 O 8 (Si 2 O 7 ) 2 as the distorted structure. The largest mode amplitudes occur for the Y 2+ mode at the zone boundary point Y, corresponding to the h + k = 2n + 1 positions of reciprocal space. We emphasize the importance of domain formation of the distorted P2 1 ∕a crystal structure. The transition from C2/m to P2 1 ∕a symmetry would be displacive, with Y 2+ as the active irrep and O7, the bridging oxygen of the sorosilicate group, showing the largest displacement away from the C2/m parent symmetry. As the phase transition would occur at the Brillouin zone boundary, it would be accompanied by formation of antiphase domains, related by a change of origin of [1/2, 1/2, 0] according to the results of the Isodistort program. Figure 6 superimposes the coordination environments surrounding the bridging oxygen atom of two Si 2 O 7 -groups of the distorted La 4 Mg 2 Ti 3 O 8 (Si 2 O 7 ) crystal structure related by such a vector [1/2, 1/2, 0]. As these two different orientations of the kinked Si-O-Si bridge closely resemble the displacement ellipsoids obtained in the C2/m average structure, it is extremely likely that this average structure results from averaging over many such antiphase domains, related by the [1/2, 1/2, 0] phase shift. Large domains, extending over tens or hundreds of RS layer packets, would give rise to reflections violating the C-centering condition. The domains have to be rather small for these reflections to be absent, probably extending over just a few ordered RS layer packets, followed by an antiphase boundary. Formation of such microdomains could be attributed to rapid crystal growth in the stability field of the P2 1 ∕a phase, or to rapid quenching through the phase transition. The latter case requires that the temperature of the phase transition is lower than the formation temperature of the perrierite crystals. For the chevkinite-(Ce) crystals studied by Stachowicz et al. (2019b), the equivalent phase transition temperature has to be higher than 750 • C, as this was the annealing temperature that stabilized the P2 1 ∕a symmetry. In both scenarios, crystal growth and quenched microdomains, the domain boundaries are probably stabilized by cation disorder, especially REE-Ca disorder occurring at the A-sites.

Conclusions
In terms of composition, perrierite-(Ce) from the Laacher See volcano borders on the previously described perrierite-(La) from the same locality. It has C2/m space group symmetry that is the macroscopic average of local antiphase domains of P2 1 ∕a symmetry. The preferred formation of the perrierite-type module stacking for Ca-rich and Fe-poor compositions can be related to their lower module charge.
The modular character of the CGM is of important consequence for the variability of the crystal structure and the composition of these minerals. While other polysomes than those conforming to the 1:1 proportion of R-and S-modules have so far only been synthesized, the