Farringtonite (= Mg3(PO4)2-I), chopinite (= “Mg-sarcopside” = Mg3(PO4)2-II), ε-Mg2PO4OH, althausite, OH-wagnerite and P-ellenbergerite were synthesised from stoichiometric mixtures of MgO (Alfa Aesar, 99.95 %, fired at 1300 °C before weighing) added to NH4H2PO4 (Sigma-Aldrich, 99.999 %), reacted with water and heated at 600 °C for 2 h (see Brunet et al., 1998). The mixtures were enclosed, with 10 wt % deionised water for hydroxy-Mg-phosphate synthesis and only water traces for anhydrous Mg-phosphate synthesis, in gold capsules of 3 or 4 mm outer diameter, sealed by arc welding.

Farringtonite and ε-Mg2PO4OH were synthesised in externally heated Tuttle-type cold-seal pressure vessels (2 d at 550 MPa, 1000 °C, and 2 months at 60 MPa, 620 °C, respectively); the other higher-pressure phases were synthesised in a piston-cylinder apparatus, using a low-friction 1/2 in. (1.27 cm) diameter NaCl assembly (2 d at 2.2 GPa, 800 °C, for chopinite; at 1.2 GPa, 850 °C, for althausite; at 2.5 GPa, 850 °C, for OH-wagnerite; and at 2 GPa, 750 °C, for P-ellenbergerite).

Synthetic products were characterised by X-ray diffraction (cf. Sect. 2.3, “High-temperature X-ray diffraction”). All the samples were single-phase except the chopinite sample that contained traces of OH-wagnerite in spite of repeated synthesis attempts. Upon heating, the althausite batch used for thermal-expansion measurements also revealed traces of OH-wagnerite which had not been detected before.

In situ high-pressure X-ray diffraction experiments were carried out using a cubic multi-anvil high-pressure apparatus of the Max-80 type, installed at the DORIS III storage ring of HASYLAB (DESY, Hamburg, Germany). Pressure was generated quasi-isostatically by six tungsten-carbide anvils with 6 mm × 6 mm truncation in a cubic arrangement. Details of the apparatus were given by Peun et al. (1995).

The diffraction measurements were performed in energy-dispersive mode using a white synchrotron beam of 100 µm × 100 µm dimension at a fixed 2θ angle, determined for each run under ambient conditions from the diffraction patterns of NaCl obtained at the top, in the middle and at the bottom of the assembly. A radioactive 241Am source provided X-ray fluorescence Kα and Kβ lines from Rb, Mo, Ag, Ba and Tb targets for energy calibration of the Ge solid-state detector.

The sample assembly and the experimental technique were similar to that of Grevel et al. (2000a). The cell assembly contained two powdered samples sandwiched by pellets of NaCl or of a NaCl–BN mixture; each sample was mixed with vaseline. Grevel et al. (2000b) have shown that this procedure results in hydrostatic high-pressure conditions. The pressure was measured in the middle and at the ends of the cell using the NaCl or NaCl–BN pellets as calibrants (Decker, 1971). Pressure gradients (due to internal stress) were found to reach 0.1 to 0.2 GPa within the cell and are therefore the major source of pressure uncertainty (Table 2), as compared to those introduced by the refinement of NaCl cell volume (around 10-3 GPa) and by the determination of the 2θ angle (10-2 to 5×10-2 GPa).

The externally applied load was increased in most experiments by steps of 10 up to 80 t, which corresponds to sample pressures of 5–5.5 GPa. Diffraction measurements were performed after each loading increment, after a waiting time of at least 10 min which had enabled the sample pressure to reach a steady state. The first, “zero-load” measurement was made after the anvils were brought into contact with the cell, as indicated by an incipient rise of the load value, which was then left to relax. Depending on the extent of this relaxation, the zero-load pressure value measured in situ may depart more or less from the actual room pressure.

Several experimental runs under isothermal conditions at ambient temperature (Table 2) were carried out for each studied mineral (chopinite, althausite, OH-wagnerite and P-ellenbergerite) to check the reproducibility and the consistency of the measurements made over two campaigns (September 2001 and May 2002, R and S run series in Table 2, respectively).

High-temperature diffraction data were collected on a 17 cm vertical Philips PW1050/25 goniometer in angle-dispersive mode, using Ni-filtered CuKα radiation, at the Laboratoire de Cristallochimie du Solide (Université Pierre-et-Marie-Curie, Paris, France), employing a platinum-alloy plate as the heating sample holder. Samples were ground, mixed with a little xylene, spread on the Pt plate to form a thin continuous film and then briefly heated to around 100 °C, at which temperature the xylene evaporates, leaving a thin smooth powder. The heating device and attached thermocouple used during data collection were calibrated in situ by bracketing various structural phase transitions (KNO3 (α→β), 139 °C; BaCO3 (γ→β), 811 °C) and melting reactions (KNO3, 337 °C; KCl, 776 °C; NaCl, 801 °C). Nominal temperatures were corrected using the calibration; the corrected temperatures reported below are believed accurate to within 1 % above 200 °C, within 2 °C below.

The diffraction patterns were recorded over the 20°<2θ<86° range. Four scans were obtained at each temperature and merged, five at room temperature so as to improve the counting statistics for the standard-volume determination. Alloy peaks were not taken into account during the refinement by ignoring some 2θ intervals ([39, 41], [45.5, 47.5], [67, 69.5] and [80.5, 83] at room temperature). The data were collected up to 900 °C for anhydrous phases and up to 400, 500 or 600 °C for hydrous phases, before or until incipient dehydration.

The lattice parameters of the samples were determined by the Rietveld method. Two different programs were run to refine the crystallographic and instrumental parameters: GSAS (Larson and Von Dreele, 1988) for the high-pressure diffraction patterns obtained by energy-dispersive methods (Sect. 2.2) and the DBW program of Young et al. (1995) for the high-temperature diffraction patterns, obtained in angle-dispersive mode (Sect. 2.3).

Available literature data were used as input values for cell parameters (Table 1) and atomic positions; the latter were kept constant throughout the refinement procedure. The refined data set included the cell parameters, the overall scale factor, a profile shape factor, coefficients of the FWHM (full width at half maximum) Caglioti's polynomial and background parameters. Given the number of atoms in the unit cells and the quality of our diffraction patterns, we did not refine the isotropic displacement parameters independently but refined a common Biso variable for all the oxygen atoms and another variable for the cations. The March–Dollase preferred-orientation factor was also refined for chopinite, ε-Mg2PO4OH, and althausite, which showed strong orientation effects. When the samples were multi-phase (i.e. with traces of a parasite compound), the cell parameters, the scale factors, the profile shape factors and the FWHM polynomial coefficients of both phases were refined simultaneously.

For the high-pressure experiments, the NaCl cell parameter was refined from the whole diffraction pattern for each pressure condition and served as the input value to determine pressure from Decker's equation of state (EoS) (Decker, 1971).

For the high-temperature experiments, carried out on a Bragg–Brentano diffractometer, a potential offset of the 2θ zero point had to be considered. We first measured diffraction patterns on the pure compounds and, following Launay et al. (2001), refined the instrumental zero shift simultaneously with the crystal parameters. The results at room temperature were then compared with those obtained under the same conditions with the same compounds mixed with α-Al2O3 as internal standard. Refinements indicated that the differences in the lattice parameters were negligible (e.g. P-ellenbergerite with α-Al2O3: a=12.426(5) Å, c=5.009(2) Å; P-ellenbergerite without α-Al2O3: a=12.426(9), c=5.008(4) Å). Therefore, we chose to go on performing the measurements without internal standard, refining the 2θ-zero shift at room temperature (e.g. P-ellenbergerite: zero shift = -0.185(9)°2θ) and keeping this value of the shift for the high-temperature refinements. The sample displacement (due to holder expansion) was then refined at each temperature step and showed a smooth variation with increasing temperature (e.g. from 0.126 at 101 °C to 0.135 at 405 °C for P-ellenbergerite).

The experimental high-pressure data are presented in Figs. 1 to 3. The unit-cell parameters contained in Table 2 were used to determine the room-temperature compressibility of chopinite, althausite, OH-wagnerite and P-ellenbergerite. The strong orientation effects along the (001) perfect cleavage of althausite, combined with the small number of peaks in its diffraction patterns, prevented us from refining the althausite cell parameters for pressures higher than 3.2 GPa.

We did not merge the several data sets into a single one for each phase because the measured volumes at room pressure (and temperature) differ by one or more cubic ångströms, e.g. by 4 Å3 in the case of P-ellenbergerite (Table 2). This indicates that the P–V data of different runs have their own systematic error. This would bias the results if a common V0 were fitted to a combined data set. Therefore we chose to use EosFit7c (Angel et al., 2014) version 7.6, which allows one to apply individual “scale factors” to the data set of each measurement run in combination with a fixed V0. The latter was calculated as the mean value of the measured volume data of the individual runs at room pressure and temperature. Such a procedure is equivalent to a fit with individual V0 values of each data set. The volume data were weighted according to the “effective variance method” (Orear, 1982) proposed by Angel (2000): w=σ-2, where σ2=σP2+σV2⋅[(∂P/∂V)T]2.

A Birch–Murnaghan equation of state truncated at second-order in the energy (K0′ set to 4; Angel, 2000) was fitted to the volume data by means of a weighted least-squares regression: P=3/2K0 [(V0/V)7/3-(V0/V)5/3], where V is the unit-cell volume at high pressure P (in GPa), V0 the volume at ambient pressure and K0 the bulk modulus. The data scatter and the need to merge data from different experiments and refine scale factors made it impossible to reliably refine a third-order Birch–Murnaghan EoS. For instance, application of a third-order instead of a second-order EoS to the chopinite data results in a slightly lower w-χ2 of 2.67 instead of 2.78 but in an unrealistic negative value for K′=-0.6(2). For the other compounds, the w-χ2 was increased or unchanged using a third-order EoS (althausite and P-ellenbergerite) and the uncertainty in the bulk modulus K0 was dramatically increased (althausite, OH-wagnerite and P-ellenbergerite). Therefore we decided to use a second-order Birch–Murnaghan EoS for the fits.

From the least-squares fit, the bulk moduli (Table 3) are found to range between 64.5(62) GPa (althausite) and 88.4(19) GPa (OH-wagnerite). They show the gross correlation that can be expected with the oxygen packing efficiency, the more compact structures being less compressible. Compared to other phosphate minerals, magnesium phosphates are less compressible than berlinite (K0∼41 GPa, calculated from Sowa et al., 1990) but more compressible than hydroxylapatite (K0=97.5 GPa; Brunet et al., 1999), monazite-(La) or xenotime (K0=144(2) and 149(2) GPa, respectively; Lacomba-Perales et al., 2010). Besides, whereas the silicates forsterite and ellenbergerite, homeotypical with chopinite and P-ellenbergerite, respectively, are relatively incompressible (K0(forsterite)=127.4 GPa – Kroll et al., 2014; K0(ellenbergerite)=133 GPa – Comodi and Zanazzi, 1993a) and display 3 % average reduction in their molar volume at 5.0 GPa, chopinite and P-ellenbergerite are more compressible with about 5 % volume reduction at 5.0 GPa (K0(chopinite)=81.6 GPa; K0(P-ellenbergerite)=86.4 GPa). The 25 % octahedral vacancies in chopinite with respect to forsterite (Berthet et al., 1972) may well account for its higher compressibility. Indeed, in the olivine-type phosphate series triphylite–heterosite, LiFe2+PO4 to □Fe3+PO4, the 50 % octahedral vacancies result in a drop of K0 from 106 to 61 GPa (Dodd, 2007, confirming the first-principle modelling of Maxisch and Ceder, 2006). However, this line of argument alone does not hold for the ellenbergerite series, in which the number of vacancies (in the octahedral single chain) is slightly lower in the softer phosphate (0.2 to 0.3 “atoms” pfu; Brunet et al., 1998; Brunet and Schaller, 1996) than in the stiffer silicate (0.7 to 1.0 “atoms” pfu; Chopin et al., 1986).

For each of the Mg-phosphates studied, the pressure dependence of unit-cell volumes normalised to Vref=V0 ⋅ scale factor was also fitted using a Berman-type polynomial (Brown et al., 1989) for specific thermodynamic applications (Table 3). We chose a second-order regression (except for chopinite, first-order, v4 being not significant): V(P)/Vref=1+v3(P-1)+v4(P-1)2; P is expressed in bars, and Vref is the mean value of the room-pressure volumes V0 of the different measurement runs multiplied by the refined scale factor derived from the fit using EosFit7 7.6. The v3 and v4 Berman coefficients are listed in Table 3.

The experimental high-temperature data are presented in Table 4 and Figs. 4 to 6. For the hydrous phases, the measurements were limited by the breakdown of the minerals at high temperature: both althausite and ε-Mg2PO4OH began to alter and to form the anhydrous-phase farringtonite at 700 °C. The high-pressure phases OH-wagnerite and P-ellenbergerite showed no evidence of decomposition at 500 and 400 °C, respectively.

The unit-cell volumes all smoothly increase with a parabolic dependence on temperature. The temperature dependence of the unit-cell volume has been fitted to a Berman second-order polynomial (Brown et al., 1989; Berman, 1988) for each Mg-phosphate: V(T)=V0(1+v1(T-T0)+v2(T-T0)2), where T0 is the standard temperature (293.15 K) and V0 the standard volume. The fits were done using EosFit7c 7.6 with the implemented Berman EoS V0(1+α0(T-T0)+1/2α1(T-T0)2) using V0 as well as α0 and α1 as fit parameters. Then the coefficients of the second-order Berman polynomial are v1=α0 and v2=1/2α1. Their values are reported in Table 3, together with those obtained from a fit of Hu et al. (2022) data. As can be realised from Fig. 4, these authors have measured a slightly higher thermal expansion of chopinite = Mg3(PO4)2–II.

Compared to other phosphate minerals, Mg-phosphate thermal expansions are higher than those of monazite (v1∼1×10-5 K-1; e.g. Perrière et al., 2007), berlinite (v1=1.9×10-5 K-1; Troccaz et al., 1967) and hydroxylapatite (v1=1.79×10-5 K-1; Brunet et al., 1999). They are comparable to the expansion of compact silicates like grossular (v1=2.38×10-5 K-1) or pyrope (v1=2.15×10-5 K-1) (calculated from Thiéblot et al., 1998) but exceed those of their silicate structural equivalents: v1(chopinite)=3.51×10-5 K-1 > v1(forsterite)=3.04×10-5 K-1 (calculated from Kroll et al., 2012), and v1(P-ellenbergerite)=3.1×10-5 K-1 > v1(ellenbergerite)=1.96×10-5 K-1 (calculated from Comodi and Zanazzi, 1993b).

The anisotropy of compressibility and thermal expansion of crystals can generally be described by the eigenvectors Ev1,2,3 of the second-rank tensors representing these properties (Nye, 1985). The components of the tensors were determined by temperature- and pressure-dependent X-ray diffraction measurements of the lattice parameters. The corresponding data of the Mg-phosphates and additional minerals are compiled in Tables 5 and 6. In the case of orthorhombic and higher-symmetry compounds (Table 5), the eigenvectors Ev1,2,3 are oriented parallel to the crystallographic axes a, b and c. Therefore, in these cases the anisotropy of the corresponding property can be evaluated by determining the linear expansion αi and the linear moduli Mi with i = a, b, c along the crystallographic axes. As the linear moduli Mi=-xi(∂P/∂xi)T with xi=a, b, c lattice parameter are the inverse of the linear compressibility βi=-1/xi(∂xi/∂P)T, this procedure results directly in diagonalised tensors with αa=α11, βa=β11 and so on.

The Mi values were retrieved using EosFit7c 7.6 from the pressure dependence of the lattice parameters and were calculated in the same way as the bulk moduli K0 of the volume data.

The linear thermal-expansion coefficients αi were calculated using the implemented Berman EoS of EosFit7c of first-order V(T)=V0(1+α0(T-T0)) using V0 as well as α0 as fitting parameters.

In the case of the monoclinic compounds farringtonite, chopinite and OH-wagnerite, the orientations of the orthogonal eigenvectors Ev1,2,3 and therefore the axes of the representation quadric (Nye, 1985) differ from those of the crystallographic axes. One eigenvector must lie parallel to the diad axis, which is the b axis in the settings used here. The other two are oriented within the a–c plane. In order to quantify the anisotropy, the values of the properties along the orthogonal eigenvectors were compared. The eigenvalues and eigenvectors of the tensors of expansion and compressibility were calculated using the STRAIN routine in EosFit7c in the range of highest- and lowest-temperature or highest- and lowest-pressure data for each compound (Tables 2 and 4) under the condition that the eigenvectors are unit vectors. In Table 6 the elements of the diagonalised tensors are given together with the orientation of the eigenvectors towards the nearest crystallographic axis (Tables 5 and 6).

In order to compare our results to the recently published lattice parameters of Mg3(PO4)2-I (farringtonite) and Mg3(PO4)2-II (chopinite) by Hu et al. (2022), the monoclinic angle was calculated from their published lattice parameters a, b, c and unit-cell volumes, as the mentioned authors did not give data for the angle β.

The compression is slightly anisotropic for the investigated magnesium phosphates, with a maximum compressibility contrast for chopinite (β33/β11=2.02). Figure 7 shows a projection of the olivine-like chopinite structure along the b axis (modified after Grew et al., 2007) together with the representation quadrics of compressibility and the thermal-expansion tensor. The lengths of the axes of the ellipsoids equal the inverse of the square root of the property in that direction (Nye, 1985; Knight, 2010). The directions of both the highest and the lowest compressibility do not match the crystallographic axes. The highest compressibility appears near the [101] direction, whereas the lowest is near [-102]. Compared to forsterite, the tensor is rotated within the (010) plane by 54°. The anisotropy of chopinite in this plane is about β33/β11=2.02, whereas in forsterite it is (β11=βa)/(β33=βc)=1.40 (forsterite in the Pnam setting). The vacancies of 50 % of the M1 sites increase the anisotropy of chopinite compared to forsterite. The anisotropy in P-ellenbergerite (βc/βa=1.40) is about the same as in the homeotypical silicate (Mg1/3,Ti1/3,□1/3)2Mg6Al6Si8O28(OH)10 (βc/βa=1.33) and can be related to the properties of the two main structural elements, the octahedral single and double chains running parallel to c. The dimers of face-sharing octahedra in the double chain have their common face parallel to c, with short Mg–Mg (or Mg–Al) distances across it; they should therefore be least compressible perpendicular to c. This effect combines with the presence of vacancies in the face-sharing octahedra of the single chain on the 6-fold screw axis, which also make this chain easily compressible parallel to c.

Thermal expansion is slightly to moderately anisotropic in the dense phases OH-wagnerite (α11/α33=1.23), chopinite (α33/α22=1.36) and P-ellenbergerite (α33/α11=1.63), but it is more anisotropic in the low-pressure phases ε-Mg2(PO4)OH (α33/α11=2.63), althausite (α33/α11=2.86) and farringtonite (α22/α33=13.9). Whereas the near-isotropic behaviour of OH-wagnerite may reflect the three-dimensional nature of its Mg polyhedral framework (Raade and Rømming, 1986b), the high expansion along c in althausite can be related to the existence of a perfect (001) cleavage plane (space group Pnma, hence the preferred-orientation problems). The strong anisotropy of farringtonite thermal expansion is essentially due to the low expansion along c. Along this direction, the Mg(2)O6 octahedra are connected to each other by two PO4 tetrahedra and form rather straight (Mg(PO4)2O2) chains, in which the sum of the octahedral and tetrahedral edge lengths controls the c repeat and cannot be much increased by polyhedral tilting.