Crystallography relevant to Mars and Galilean icy moons: crystal behavior of kieserite-type monohydrate sulfates at extraterrestrial conditions down to 15 K

The low-temperature crystallography of kieserite-type monohydrate sulfates, relevant for Mars and Galilean icy moons, has been investigated down to 15 K, revealing a pronounced anisotropy of their thermal expansion behavior. The present results provide a reliable basis for the proper assignment of sulfate-related signals obtained from remote-sensing data from orbiters at celestial bodies.


Introduction
The abundance of sulfate minerals, in particular hydrated magnesium sulfates (MgSO 4 ÁnH 2 O), has been confirmed on Mars and elsewhere in the solar system starting with the first in situ discovery in martian soils by the Viking Mars probe (Keil et al., 1978;Clark & Van Hart, 1981). The ability of these materials to release or absorb water and the prevailing humidity, even forming liquid brines (Peterson & Wang, 2006), ranks them among critical components governing the water cycles on the surface of Mars and other celestial bodies. The influence of temperature, humidity and pressure on dehydration and rehydration capabilities has been investigated in various laboratory studies (e.g. Wang et al., 2009;Nakamura & Ohtani, 2011;Fortes et al., 2017a). Among these MgSO 4 ÁnH 2 O phases, covering a wide range of hydration states (n) from 0 to 11, the monohydrate sulfate kieserite and its solid solutions with transition metal (TM) counterparts (Papike et al., 2007) dominate the deposits at lower martian latitudes, where it is assumed to be an essential H 2 O carrier (e.g. Christensen et al., 2004;Arvidson et al., 2005;Clark et al., 2005;Bishop et al., 2009).
However, there is some ongoing dispute regarding the stability of kieserite on the martian surface. On the one hand, Chipera & Vaniman (2007) argue that the relative humidity should lead to its rehydration to starkeyite, MgSO 4 Á4H 2 O, which is thermodynamically stable under the prevailing conditions. On the other hand, Wang et al. (2009) found that the presence of Ca-and Fe-sulfates and Fe-oxides or Fehydroxides enhances the dehydration of higher Mg-sulfates down to the monohydrate. Smectite minerals were shown to have a stabilizing effect on the hydration states, allowing lower sulfate hydrates to exist well outside their stability range (Wilson & Bish, 2012). Furthermore, there is evidence that martian kieserite actually has an intermediate composition between the end members kieserite, MgSO 4 ÁH 2 O, and szomolnokite, FeSO 4 ÁH 2 O (Bishop et al., 2009). The respective synthetic materials used for the investigation of the given solid solution  show no sign of rehydration even after several years under ambient conditions as soon as the Fe content exceeds 0.2 apfu.
A related debate concerns the existence and nature of a postulated second structural polytype of kieserite with a broader stability range but similar spectroscopic properties, the so-called low-humidity (LH) kieserite, presumed by several authors to prevail on Mars (Wang et al., 2009;Jamieson et al., 2014). While elucidation of the actual structure and properties of this enigmatic 'LH phase' is still pending to date, it is one of the goals of the present study to provide reliablehitherto missing -low-temperature reference data of 'classical' MSO 4 ÁH 2 O compounds, to serve as a benchmark for comparison and as a basis for spectroscopic and thermodynamic modeling purposes.
Spectroscopic data recorded by orbiters reveal sulfate-rich regions to be present not only on the martian surface, but also appear to be an abundant feature on many other celestial bodies in our solar system, i.e. on the icy moons of Jupiter and Saturn (Kargel, 1991;McCord et al., 2001a). Higher hydrated Mg-sulfates (n = 6-11) are assumed to be relevant non-ice mineral constituents on the surface but also in the ice mantles of the Galilean moons (e.g. McCord et al., 2001a;Dalton et al., 2005Dalton et al., , 2012. While McCord et al. (2001b) did not find any indication of Mg-sulfate radiolysis when subjecting epsomite (n = 7) to a 100 eV electron beam, Tani et al. (2012) determined the formation of hydrogen and sulfite radicals at 90 K by exposure to gamma radiation. More importantly, several studies have shown that the interaction of UV or other energetic radiation with highly hydrated sulfates significantly catalyzes and speeds up their dehydration. Under such influence, epsomite is converted to hexahydrite [the mineral name for the hexahydrate, MgSO 4 Á6H 2 O (Cardell et al., 2007)], the further dehydration of which has also been proven to be catalyzed by UV radiation (Cloutis et al., 2007). This processes could lead to the formation of sulfate monohydrates on the icy surface of the Galilean moons.
Moreover, lower-hydrated or even anhydrous sulfates are also expected to be present within the deeper mantle and core of these planetary bodies (Kargel, 1991;Nakamura & Ohtani, 2011;Meusburger et al., 2020). Depending on their thermodynamic stability, they play a key role for the formation of subsurface oceans, which could even contain extraterrestrial life, as discussed for the south-polar region of Saturn's moon Enceladus, where a subsurface body of liquid exists due to tidal heating (Solomonidou et al., 2011). However, no clear confirmation of the presence of sulfates on this latter object or in its hydrothermal plumes exists, in part because of significant overlaps of rather broad bands and combination modes of various sulfate minerals in the standard <5 mm spectral range covered by orbiters (Bishop et al., 2004). This problem becomes even more remarkable when the deposits are spatially restricted with other phases present, necessitating the use of complex spectral unmixing models (e.g. Combe et al., 2008). However, it is most probable that the rocky core of the moon corresponds to C1/C2 chondritic composition (Kargel, 1991;Sekine et al., 2015), where high sulfate contents are present in the soluble fraction (Fredriksson & Kerridge, 1988;Burgess et al., 1991), filling abundant brecciation cracks (Richardson, 1978); therefore the presence of sulfates also on Enceladus will probably be confirmed by future missions.
Following our investigations on the structural crystallography of kieserite-type solid solutions: M 2+ SO 4 ÁH 2 O (M 2+ = Mg, Fe, Co, Ni), including the temperature-dependent behavior of the end members at temperatures prevailing at equatorial martian latitudes (Bechtold & Wildner, 2016;Talla et al., 2020), we recently carried out investigations under high-pressure conditions relevant to the interior of individual icy moons (Meusburger et al., 2019(Meusburger et al., , 2020Ende et al., 2020;Wildner et al., 2021). These experimental studies revealed a second-order phase transition from the monoclinic -phase (space group C2/c) to a triclinic -phase (space group P1) at critical transition pressures ranging from 2.40 (Co) to 6.15 GPa (Fe). The findings from high-pressure crystallography show a partial rearrangement of the hydrogen-bonding scheme as the most obvious structural change. Earlier studies down to 110 K  gave no evidence of a comparable transition, however, it can not be ruled out to occur at temperatures even lower than the previously covered range. Although on Mars the temperatures in equatorial latitudes range between 280 and 170 K (Witzke et al., 2007), significantly lower temperatures can be expected on the surface of the icy satellites. The temperature changes during the seasonal and diurnal cycles on some of the icy moons of Jupiter and Saturn have been modeled, such as for Europa (Ashkenazy, 2019), where the mean annual temperature range spans (depending on the assumed internal heating rate) from 94 to 98 K in equatorial latitudes and from 35 to 62 K on the poles. The measurements during the Rosetta mission to comet 67P showed evidence of SO 2 in its plume and ammonium sulfate on its surface, suggesting the presence of sulfate on cometary bodies and asteroids (Poch et al., 2020). In such cases, they are exposed to research papers IUCrJ (2022). 9, 194-203 temperatures as low as those of the cosmic vacuum, amounting to a mere 2.7 K.
Since the reliable knowledge of phase relations and structural details of relevant sulfate minerals at astrophysically significant conditions is of utmost importance for their identification and discrimination in remote-sensing spectroscopic data from orbiters, we extend our investigations on endmember monohydrate sulfates well down to temperatures as low as 15 K. To reveal the structural changes, in situ lowtemperature X-ray crystallography was carried out at temperature conditions relevant to the icy moons of Jupiter and Saturn and comparable objects in the outer solar system, as well as the polar regions of Mars or dwarf planets and asteroids within the inner solar system. At the same time, this study aims to verify any potential structural phase transition equivalent to those reported earlier under compression of kieserite-type phases, following the analogy of 'inverse thermal pressure' on cooling.

Experimental
2.1. Temperature-dependent single-crystal X-ray data collection Single crystals of synthetic kieserite-type compounds M 2+ SO 4 ÁH 2 O (M 2+ = Mg, Fe, Co, Ni) for the present data collections were extracted from the respective end-member batches prepared earlier (Bechtold & Wildner, 2016;Talla et al., 2020). Data collections between 313 and 113 K were performed in steps of 40 K on a Bruker Apex II diffractometer equipped with a CCD area detector and an Incoatec Microfocus Source ImS (30 W, multilayer mirror, Mo K), in a dry stream of nitrogen (Cryostream 800, Oxford Cryosystems). Several sets of 'and !-scans with a 2 scan width were measured at a crystal-detector distance of 40 mm up to 80 2 full sphere. Absorption was corrected by evaluation of multi-scans. Measurements and correction procedures were performed with the SAINT software package (Bruker AXS, 2012). Data collections between 75 and 15 K (and at room temperature) were performed in steps of 15 K with Mo K radiation on an Oxford Diffraction Gemini R Ultra diffractometer with a CCD detector. The open-flow helium cryostat Helijet from Oxford Diffraction with helium coaxial flow shielding was used to maintain low temperatures of the samples. Data collections and treatments were performed using CrysAlisPro software (Rigaku Oxford Diffraction, 2016). During several of the scheduled measurements we encountered a problem of excessive formation of an 'ice'-cover on the sample crystals, composed from solidified air components, mainly O 2 and N 2 , in the open-flow helium cryostat, leading to interference of sample reflections with the emergent multi-grain powder rings (see Zakharov et al., 2021). These datasets were excluded from consideration: respective data collections were aborted and/or deleted from the further analyses. Data collection was resumed after the problem was fixed or, at least, partly solved by changing the crystal holder type and other preventive measures.

Crystal structure refinements
All structure refinements were performed on F 2 with SHELXL (Sheldrick, 2015) in the 'traditional' non-reduced kieserite cell setting (cf. Wildner & Giester, 1991) in the space group C2/c. Scattering curves for neutral atoms were used. Note that, for the purposes of data consistency and comparability, the cell axis parameters from the temperature-dependent measurements were corrected in such a way that respective values at 293 K (interpolated for Apex II measurements) match those from earlier room-temperature (RT) studies, obtained on a Nonius Kappa CCD diffractometer (see Bechtold & Wildner, 2016;Talla et al., 2020), since the latter RT data proved to most closely match corresponding high-precision lattice parameters obtained with the 8-position centering mode (King & Finger, 1979) on a Stoe AED II diffractometer. The respective corrections were in the range 2%. Further note that the structures of the Mg and Fe compounds at 193 and 113 K have already been published by  and those of NiSO 4 ÁH 2 O at 273, 193 and 113 K by Talla et al. (2020). Crystal structure drawings throughout the paper were prepared with the program VESTA3 (Momma & Izumi, 2011).

Thermal equation of state calculations
Due to the limited number of temperature-dependent unitcell data points (8-10 at most) obtained from full single-crystal structure determinations, we preferred to keep the thermal equation of state (EoS) analyses as simple as possible and with a minimum of variable parameters. The unit-cell axes lengths and cell volumes between 15 and 313 K were fitted therefore to a Fei-type (Fei, 1995) thermal EoS using the EoSFit7c and EoSFit7-GUI programs by Angel and coworkers (Angel et al., 2014;Gonzalez-Platas et al., 2016). The reference temperature was chosen at T ref = 293 K; 2 was fixed at 0 (see also Angel et al., 2014), i.e. simplifying the thermal EoS to the linear approach with two coefficients following = 0 + 1 T. Strain and thermal expansion tensors were calculated from the thermal EoS with the Strain utility in the EoSFit7c program.

General aspects
Selected crystal data and details of the data collections and structure refinements are summarized in Table S1 of the supporting information. Final atomic parameters are listed in Table S2, and results of the thermal EoS and strain calculations in Table 1. The most relevant interatomic distances and angles are given in Table S3. Under ambient conditions, kieserite and isotypic compounds M 2+ SO 4 ÁH 2 O (M 2+ = Mg, Fe, Co, Ni) crystallize with monoclinic symmetry in the space group C2/c (Hawthorne et al., 1987;Wildner & Giester, 1991). The crystal structure of the Mg-representative kieserite is illustrated in Fig. 1  tetrahedra (point symmetry 2), intra-linking adjacent octahedra within the chains by common O2 corners. These octahedral-tetrahedral chains are aligned parallel to the c axis (i.e. along the viewing direction in Fig. 1) and interlinked to a framework by sharing the remaining polyhedral O1 corners as well as by moderately strong O3-HÁ Á ÁO2 hydrogen bonds.

Thermal expansion of the unit cell
The lattice dimensions, depicted in Fig. 2, as well as the structural properties of the kieserite-group representatives deviate more or less from a linear fashion with decreasing temperature. While approximately linear shifts of lattice parameters are usually observed in high-temperature studies from RT upwards, the tendency to flatten below around 100 K towards absolute zero is well known for many crystalline solids (e.g. Drebushchak, 2020), and even an inversion of the trends leading to negative bulk volume expansion may be observed below $50 K; respective examples among sulfates are higher hydrates like meridianiite [MgSO 4 Á11H 2 O (Fortes et al., 2008)] or mirabilite [Na 2 SO 4 Á10H 2 O (Brand et al., 2009)].
The thermal expansion parameters and Fei-type EoS coefficients of the title compounds are summarized in Table 1 , showing a pronounced negative thermal expansion (NTE) effect. The present NTE behavior is even more remarkable since, with b,293 ranging between À4.4 Â 10 À5 K À1 in kieserite (Mg) and À0.8 Â 10 À5 K À1 in NiSO 4 ÁH 2 O, it amounts to a magnitude comparable to the positive thermal expansion of the a and c axes, and it furthermore shows no sign of turning positive or at least flattenning towards higher temperatures. NTE behavior of single lattice axes in otherwise 'thermally normal' compounds is not uncommon, but usually at a much lower rate. For example, in epsomite (MgSO 4 Á7H 2 O) and meridianiite (MgSO 4 Á11H 2 O) each one of their lattice axes shows NTE (in the former case with an already high of at most À2 Â 10 À5 K À1 ), both turning positive at $250 K (Fortes et al., 2006(Fortes et al., , 2008; in chalcanthite, CuSO 4 Á5H 2 O, a very weak axial NTE is observed which flattens towards $300 K (Schofield & Knight, 2000). The strong NTE effect of the b axes in the title compounds is further discussed below (Section 3.4).
Regarding the bulk volume thermal expansion V [ Table 1, Fig. 2(c)], the Fe compound with the largest cell volume also shows the strongest volume expansion (4.7 Â 10 À5 K À1 ), while, within limits of error, the Mg and Co compounds (similar cell volumes) but also the Ni phase with the smallest cell volume, all have the same smaller volume expansion ($3.4 Â 10 À5 K À1 ). The flattening of the cell volumes towards low temperatures seems to be somewhat more pronounced in the Ni and Mg phases compared with those of Co and especially Fe. MgSO 4 ÁH 2 O consistently exhibits the strongest changes of the individual cell-edge lengths with temperature, especially concerning the excessive negative thermal expansion of the b axes, whereas NiSO 4 ÁH 2 O shows a significantly smaller average change of the axial lengths [Figs. 2(a) and 2(b)]. Among the four compounds, the a axes show a slightly higher expansion than the c axes, apart from the insignificant inversion in NiSO 4 ÁH 2 O (Table 1).    (Angel et al., 2014).

Mg
Fe  (6) The respective thermal expansion tensors depicted in Figs. 3(a) and 3(b) (derived from the strain components in Table 1) exhibit pronounced anisotropy also in the plane perpendicular to the NTE-affected b axes, not evident from the more or less similar thermal expansion values a,c of the a and c axes ( Table  1). As illustrated in Fig. 3(b), the respective smallest positive eigenvalues are oriented approximately parallel to the short cell diagonal in the ac plane, the largest ones approximately parallel to [201]. For the Ni compound, these directions deviate slightly ($9 ) from the quite consistent orientation in the other compounds. The relation between the anisotropy of thermal expansion and stereochemical changes with temperature is further discussed in Section 3.4.
As a major result of the present investigation, we observed no evidence suggesting a structural phase transition in any of the four investigated compounds within the studied temperature range, i.e. all four title compounds keep the monoclinic C2/c symmetry that the kieserite structure type features under ambient conditions. Note that a magnetic order-disorder transition was reported for FeSO 4 ÁH 2 O at 29.6 K using Mö ssbauer spectroscopy (Van Alboom et al., 2009), which is, however, not expected to be detectable in X-ray diffraction studies. Bearing in mind that high-pressure phase transitions in the investigated compounds occur at critical pressures P c ! 2.40 GPa (  Unit-cell dimensions of kieserite-type compounds M 2+ SO 4 ÁH 2 O (M 2+ = Mg, Fe, Co, Ni) in the temperature range 15-313 K with lines representing the Fei-type EoS results (Table 1) Tables  1, S1 and S2. volume ratios V/V 0 0.96, whereas the intrinsic 'thermal pressure' (e.g. Anderson, 1995) induced on cooling from RT to 15 K merely results in a volume decrease to roughly V/V 0 ' 0.993 (Fig. S1 of the supporting information), it becomes clear that phase transitions corresponding to those at high pressures were not to be expected. This also means that kieserite-type compounds and very probably also their solid solutions will not undergo structural transitions at or close to the surface of sulfate-bearing celestial bodies. The relationships of changes in cell and crystal chemical parameters on cooling versus pressure increase are further discussed in detail in the respective sections below.

Unit-cell thermal expansion versus compressibility
Compared with the temperature-dependent changes in lattice dimensions discussed above, the high-pressure behavior of the title compounds, investigated recently by Meusburger et al. (2019Meusburger et al. ( , 2020, Ende et al. (2020) and Wildner et al. (2021) is characterized by a continuous second-order ferroelastic phase transition from the present monoclinic -phase (C2/c) to a triclinic -phase (space group P1) at pressures of 2.72, 6.15, 2.40 and 2.47 GPa for M 2+ = Mg, Fe, Co and Ni, respectively. Nevertheless, since the cells of the -polymorphs are based on the respective reduced cells of the C2/c -phase, a direct comparison of the lattice dimensions even across the phase transition is still feasible. However, such a comparison [see Tables S1, 1 and Figs. 2, S1 in the present work, and Table 2 in the work by Wildner et al. (2021)] shows that the ideally expected inverse relationship between compression and thermal expansion (e.g. Hazen & Finger, 1982a) is approximately fulfilled for selected aspects only (Fig. S1).
On the one hand, the Fe compound revealing the highest volume thermal expansion V (Table 1) shows, as expected, the highest compressibility and thus the smallest bulk modulus of K 0 ' 45 GPa (Meusburger et al., 2019), albeit close to K 0 of the Mg and Co phases. The Ni compound with the clearly highest bulk modulus K 0 ' 60 GPa  also exhibits the smallest thermal changes of the axial lengths (h |a,b,c| i ' 2.2 Â 10 À5 K À1 ) but a volume thermal expansion comparable to the Mg and Co phases (Table 1). Among the individual cell axes, the softest c axis upon compression (hM c i ' 125 GPa) most closely follows the expectation of equal slopes in V/V 0 versus l/l 0 plots for pressure and temperature (h c i ' 3.3 Â 10 À5 K À1 ) in Fig. S1. On the other hand, in contrast to the expected inverse relationship, the clearly stiffest a axes upon compression (hM a i ' 395 GPa) exhibit the highest thermal expansion (h a i ' 4.4Á10 À5 K À1 ) and, finally, the NTE behavior (h b i ' À2.4 Â 10 À5 K À1 ) of the rather 'soft' b axes (hM b i ' 145 GPa) is in harsh contradiction to all respective expectations.
The relationship referred to (Hazen & Finger, 1982a) is primarily expected for high-temperature structural studies, but the trends in Fig. 2 can evidently be extrapolated also to moderately higher temperatures, presumably up to the decomposition temperatures [!300 C under dry conditions (Chipera et al., 2006)]. Besides, the relationship is expected to fail for structures comprising polyhedra with grossly different ratios of expansivity versus compressibility (e.g. Sharp et al., 1987). While this is seemingly not the case in kieserite-type compounds with tetrahedral S-O bond lengths hardly responding to changes either in temperature (see below) or pressure (Wildner et al., 2021), the O-HÁ Á ÁO hydrogen bonds in these compounds run within structural voids which in the topologically related structure of titanite, CaTiO(SiO 4 ), are occupied by the Ca atoms (Hawthorne et al., 1987). Hence, the hydrogen bonds in MSO 4 ÁH 2 O might play a major role for the clear deviation from respective expectations.  to the H 2 O molecules [ Fig. 4(a) and Table S3]. For the Fe compound, the bonds to O1 and O2 also seem to show a faint trend to lengthen and shorten, respectively, with reduced temperature; thus, for the FeO 6 octahedron the aberrant tendency towards a [2+2+2]-coordination (cf. Wildner et al., 2021) still increases, compared with the clear and stable [4+2]-elongation of the other representatives. On average, the volumes of the MO 6 octahedra shrink with an Voct of roughly 2.9 Â 10 À5 K À1 ; in the Mg compound, for example, V oct decreases by 0.7% from 11.90 to 11.82 Å 3 from RT to 15 K, a rate comparable in magnitude to the MgO 6 octahedra in other magnesium sulfates, as listed by Fortes et al. (2008) -but excluding data from ultra-low T < 5 K. Along with the volume reduction, the octahedral bond length and angle distortions also decrease slightly with decreasing temperature for all four compounds.

Polyhedral thermal expansion and crystal chemical changes
S-O bond lengths and the tetrahedral volume show an artificial increase on cooling owing to changes in thermal motion (Table S3), a phenomenon typically found for uncorrected bonds in small coordination polyhedra [e.g. data for sulfate groups listed by Fortes et al. (2008)]. In the present kieserite-type compounds, a 'simple rigid bond' correction according to Downs et al. (1992) reveals almost constant hS-Oi bond lengths over the full temperature range. Anyway, the observed difference of around 0.004 Å in the hS-Oi distances in the Mg compound (shorter S-O bonds) compared with the TM representatives (longer S-O bonds) is maintained over the investigated temperature range. In the same range the O3-HÁ Á ÁO2 hydrogen bond lengths reduce by roughly 0.03 Å , in case of the Ni phase by half that value [ Fig. 4(b) and Table S3]. Somewhat surprisingly, the two O3-HÁ Á ÁO1 contacts (around 3.3 Å under ambient conditions), which shorten under pressure (one of them forming a new hydrogen bond in the high-pressure -MSO 4 ÁH 2 O polymorphs), significantly lengthen when the temperature is reduced. Since the O3-HÁ Á ÁO1 contacts run nearly parallel to the b axis, this unexpected finding can be attributed to a side-effect of the axial NTE behavior. Polyhedra-linking angles [Fig. 4(c) and Table S3] also show a parallel response to temperature in all four compounds, despite the fact that both M-O-S angles (but especially M-O1-S) are substantially larger in MgSO 4 ÁH 2 O than in the TM phases, leading to, among other things, a volume mismatch of octahedral versus cell volume, especially eye-catching when comparing the Mg and Co compounds [as discussed in detail by Bechtold & Wildner (2016)]. While the octahedral chain angle M-O3-M remains almost constant on cooling to 15 K, the M-O-S angles become smaller by 0.4-1.1 . The resulting rotation of the SO 4 tetrahedron around its twofold symmetry axis [Figs. 4(d) and 5] has the same direction as that found for increasing pressure (Wildner et al., 2021) or increasing x TM in (Mg,TM)SO 4 ÁH 2 O solid solutions (Bechtold & Wildner, 2016;Talla et al., 2020). Within the studied temperature range, it amounts to roughly 1.1 for both O1-O1 0 and O2-O2 0 tetrahedral edges, but again clearly less in the Ni phase. Substantial edge tiltings or rotations of the four other tetrahedral (O1-O2) edges, identified as the most probable driving force of the highpressure phase transition to triclinic symmetry (Wildner et al., 2021), are either forbidden or at least strongly hampered by the monoclinic symmetry prevailing also at very low   Tables S1, S2 and S3. temperatures [e.g. relative rotations of O1-O2 edges are <0.1 and therefore not shown in Fig. 4(d)].
The striking anisotropy of the thermal expansion tensor in the ac plane [Figs. 3(a) and 3(b)] can be correlated to the tetrahedral rotations described above. The sense of rotation on heating [indicated in Fig. 3(b) by black arrows] leads to, apart from comparable expansions of the a and c axes (Table  1), an opposite shift of neighboring octahedral-tetrahedral chains along the c axes [red arrows in Fig. 3(b)], resulting in a widening of the cell angle (Fig. 2) and maximum expansion roughly along the long cell diagonal, while minimum expansion occurs near the [101] direction.
Although the anisotropy of thermal expansion in the ac plane can be readily explained, an attempt to elucidate reasons for the peculiar NTE behavior of the crystallographic b axis [ Fig. 3(a), Table 1] needs to consider the mutual interactions of the various temperature-dependent structural changes observed in the present study. It appears that only a complex, cooperative effect involving polyhedral rotations, changes of the interpolyhedral M-O-S angles and shortening of the O3-HÁ Á ÁO2 hydrogen bond length, can provoke this clear trend in contrast to the decrease in cell volume and other lattice directions on cooling. Fig. 5 illustrates the main contributing factors: firstly, the sense of major polyhedral rotations on cooling is shown, i.e. on the one hand the rotation of the SO 4 tetrahedron around its twofold symmetry axis [as discussed above; Figs. 3(b) and 4(d)], on the other hand a rotation of the MO 6 octahedron (point symmetry 1) approximately around its O3-O3 axis; secondly, it indicates the most relevant components of resulting positional shifts of the tetrahedral atoms in the plane of projection. Altogether, the SO 4 tetrahedron as a whole is shifted up along the b axis, thus involving an elongation of this axis on cooling, while at the same time the a axis, including the roughly parallel hydrogen bond, and to a lesser extent the c axis, are shortened. The comparatively weak NTE of the b axis in NiSO 4 ÁH 2 O (Table 1) can then be correlated, at least qualitatively, with the respective smallest temperature-dependent changes of the hydrogen bond lengths [ Fig. 4(b)], of both M-O-S angles [Fig. 4(c)] and of the tetrahedral rotations [ Fig. 4(d)] among the four title compounds.

Polyhedral thermal expansion versus compressibility
In analogy with the present low-temperature results, the decrease of the octahedral volume on pressure increase is also dominated by shortening of the longest M-O3 bonds to the water molecules, to a lesser extent also the M-O2 distances, while the M-O1 bonds are the stiffest and shorten the least [thus contributing to the high axial hM a i values (Wildner et al., 2021, and references therein)]. When the octahedral volume thermal expansion values V,oct (in the range 2.8-3.1 Â 10 À5 K À1 ) are compared with the respective octahedral volume compression values V,oct (1.1-1.4 Â 10 À6 bar À1 ), we find a ratio V,oct / V,oct of $25 bar K À1 , i.e. for the MO 4 (H 2 O) 2 polyhedra a pressure change of approximately 25 bar offsets a temperature change of 1 K. Hence, judging from the influence of temperature and pressure on the octahedral units in kieserite-type compounds, cooling from RT to 15 K corresponds to a 'thermal pressure effect' of $0.7 GPa, thus far below the magnitude of critical pressures mentioned earlier, and a further corroboration of the absence of structural phase transitions on cooling.
Across high-temperature and high-pressure studies on structure types of relevant rock-forming minerals, the ratio of V,oct / V,oct for most M 2+ O 6 octahedra is about three times higher than in kieserite, scattering between 65 and 90 bar K À1 (e.g. Hazen & Prewitt, 1977;Hazen & Finger, 1982a,b;Hazen et al., 2000). On the one hand, this difference can be attributed to the fact that poly increases with temperature: for hightemperature studies from RT up to 1000 C, Hazen & Prewitt (1977) found h V,oct i ' 4.2 Â 10 À5 K À1 (from axial oct ' 1.4 Â 10 À5 K À1 ) for MgO 6 octahedra (and similar values for TM 2+ O 6 units). On the other hand, the compressibility V,oct of the MgO 6 octahedron in kieserite (Meusburger et al., 2020;Wildner et al., 2021) is approximately twice as large compared with the values usually found for MgO 6 units in oxides and silicates [where the polyhedral modulus K oct ' 150 GPa (e.g. Hazen et al., 2000)], a discrepancy which can be reasonably explained by the presence of H 2 O ligands and polyhedral connectivity through shared corners in the kieserite structure type.

Conclusions
The results for the in situ investigation of the crystal behavior of kieserite and three sulfate analog phases revealed that the structural topology remained unchanged within the experimental temperature range, i.e. down to 15 K. Neither the bond-length evolution nor the overall symmetry of the lattice is subject to a discontinuity that can be assigned to a phase transformation similar to that observed at high pressures for any of the four investigated M 2+ SO 4 ÁH 2 O (M 2+ = Mg, Fe, Co, Ni) end-member phases. The monoclinic C2/c lattice of this so-  called -form shows a rather uniform volume expansivity with values for V between 3.3 and 4.7 Â 10 À5 K À1 , approximately following the inverse relationship to the molar volume and the size of the M 2+ cation, respectively. On the other hand, the thermally induced lattice expansion is subject to a rather pronounced anisotropy, with an NTE along the [010] direction, and anisotropic positive eigenvalues for the strain tensor within the ac plane. The NTE effect can be related to cooperative polyhedral rotations associated with the chain motifs of tetrahedral SO 4 groups and octahedrally coordinated M 2+ cations, which has also been observed for other sulfates exhibiting equivalent topological octahedral chain units.
As a major result of the present investigations, there is no evidence for a structural transition for any of the four investigated end-member compounds. This suggests that, for the kieserite-type solid solution series known to exist, there is no evidence for the occurrence of a low-temperature polymorphic transformation or dissociation, and phase stabilities are not affected even by the effect of cation substitution on temperature variations at 1 bar 'atmospheric' pressure. In the context of evaluating structural aspects of phase-stability issues with respect to potentially changing hydration states, the experiments clearly show that there is no evidence for any changes with respect to the amount of water per formula unit.
The knowledge of phase relations and structural details at astrophysically relevant temperature conditions is an important fundament to derive, model and analyze other physical properties, e.g. by using density functional theory calculations. Based on the low-temperature structural behavior presented, ab initio calculations allow us to determine key thermodynamic properties of the phase of interest, such as heat of formation, enthalpy, entropy and the heat capacity (e.g. Deffrennes et al., 2019;Dinsdale et al., 2019;Bao et al., 2021). In turn, such as in the case of kieserite-type sulfate monohydrates, knowledge of these properties for end members allows us to extrapolate their values even for intermediate compositions, given the linear Vegard-type behavior along the solid solutions between kieserite and all isotypic TM end members investigated here (Bechtold & Wildner, 2016;Talla et al., 2020). Knowledge of these parameters will contribute to future modeling of thermodynamic equilibria and stable surface mineral assemblages on various objects in our solar system even at very low temperatures. Last but not least, the absence of lowtemperature structural phase transitions or dehydration effects on the one hand, and the smooth variation of stereochemical details with temperature on the other, also forms an important fundament to evaluate kieserite-related signals in remote-sensing spectroscopic data from orbiters at celestial bodies with lower surface temperatures than prevailing on Mars, e.g. on the surface on the icy Galilean moons.