Introduction

Jeffbenite (ideal formula Mg3Al2Si3O12) is a rare mineral that so far was only found as mineral inclusion in super-deep diamonds1. It was discovered in 19972,3 and since then it has been referred to as TAPP, an acronym from “Tetragonal Almandine Pyrope Phase” for its stoichiometry, which is coincident with that of the pyrope-almandine garnet series. However, the crystal structure of jeffbenite is different from that of garnet, thus garnet and jeffbenite are actually polymorphs. In 2016, TAPP was finally given a mineral name approved by IMA, which is “jeffbenite” (IMA 2014–097)1 to honour Jeffrey W. Harris and Ben Harte, two eminent experts in the field of diamond research. From its first discovery 26 years ago, only 23 natural jeffbenites were reported in literature and 7 of them were identified only by chemical analysis; 2 further jeffbenites reported in the literature are synthetic. So, only 16 natural jeffbenite inclusions in super-deep diamonds have been identified by X-ray diffraction and/or micro-Raman spectroscopy (see Table 1, which summarizes all jeffbenites reported so far in the literature). Despite its rarity, jeffbenite inclusions in diamonds are considered to be an indicator of a super-deep origin for the diamond hosts and therefore it is an important mineral. Its super-deep origin is indeed well accepted in literature and jeffbenite is generally considered a transition zone or lower mantle mineral by the diamond research community1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20.

Table 1 Chemical composition (in wt% oxides), method of identification and name of inclusions of all jeffbenites reported so far in literature.
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However, excluding a very Ti–rich synthetic jeffbenite11, at present no pressure–temperature stability fields of jeffbenite are published and this mineral remains a geological enigma: (1) at what depth in the mantle does jeffbenite actually form? (2) is the Mg-end member of jeffbenite a higher- or lower-pressure polymorph of pyrope garnet?

To answer these questions, here we constrain the pressure–temperature stability field of pure jeffbenite, Mg3Al2Si3O12. As no thermodynamic data for jeffbenite are available in literature, we compute the data from first principles, at the hybrid Hartree–Fock/Density Functional Theory (HF/DFT) level, within the limit of the quasi-harmonic approximation and in the framework of statistical thermodynamics. This allows us to comprehend the actual nature of the pure Mg end-member of jeffbenite.

Results

Jeffbenite versus pyrope molar volume: an evident discrepancy

Indeed, the present work was driven not only by the need to constrain the pressure–temperature stability field for a mineral taken to be characteristic of super-deep diamonds, but also because already in 19972 study of the first natural jeffbenite (at that time indicated as TAPP) revealed inconsistencies in terms of volume and density with respect to garnet. In detail, the natural jeffbenite discovered in 19972 (sample 244B, on which the authors refined the crystal structure) had an approximate composition equal to [Mg2.64Fetot0.27(Ca + Na + Mn)0.08][Al1.85 Cr0.15][Si2.91Al0.09]O12 and a Mg# [Mg/(Mg + Fe)] = 0.91; its unit-cell volume was V = 774.35(± 0.77) Å3. Such unit-cell volume can be converted in a molar volume equal to 11.657(± 0.012) J/bar. Comparing this molar volume with that of a garnet with similar composition along the pyrope-almandine series [Mg2.70Fe0.30]Al2Si3O12 and Mg# = 0.91 (e.g., see Table 3 of ref.21), it is evident that the molar volume of the garnet, which is equal to 11.332(± 0.001) J/bar, is significantly smaller than that of jeffbenite.

This first simple calculation in terms of molar volume shows a strong discrepancy: the molar volume of jeffbenite seems significantly larger than its pyrope polymorph, thus jeffbenite should not be the higher-pressure polymorph of pyrope; this said on the basis of simple thermodynamical considerations.

To confirm this unexpected result with respect to what is believed in the super-diamond research, we need a complete set of thermodynamic parameters.

Thermoelastic properties, entropy and Gibbs free energy of jeffbenite

We have determined a Birch-Murnaghan equation of state truncated to the third order (BM3-EOS22) for jeffbenite, which provides the following values of unit-cell volume, V0, bulk modulus, K0T and first pressure derivative, K´(at T = 298.15 K):

$$V_{0} = 766.033\;{\text{\AA}}^{3}$$
$$K_{0T} = \, 175.39\;{\text{GPa}}$$
$$K^{\prime } = 4.09$$

(V0 can be expressed in J/bar providing a value of 11.532, which is already rescaled by a 0.9787 factor to take into account the typical overestimation from the DFT calculation. The correction factor was estimated starting from the same identical overestimation on pyrope). The reason for the overestimation of the cell volume in ab initio calculations (at the DFT or HF/DFT level of the theory) is well understood23,24 and it has long been proved to be not an issue in the estimation of the second derivatives of the energy versus volume function on which, in turn, bulk moduli and vibrational frequencies are computed.

The full elastic constant tensor of jeffbenite have been computed at the static level (i.e. T = 0 K, P = 0 GPa and no zero point effects included) by fitting the second derivatives of the energy with respect to strain components, then using stress–strain relations25. Jeffbenite (tetragonal, space group I\(\overline{4}\)2d) has six independent elastic stiffnesses, calculated as follows: C11 = C22 = 319.2 GPa; C12 = 140.7 GPa; C13 = C23 = 123.5 GPa; C33 = 257.0 GPa; C44 = C55 = 100.5 GPa; C66 = 129.1 GPa. The aggregate elastic moduli (bulk and shear moduli) inferred by the elastic tensor through a Voigt-Reuss-Hill averaging scheme are KVRH = 184.2 GPa and GVRH = 98.4 GPa, respectively. The former value is in excellent agreement with that obtained at T = 0 K and P = 0 GPa from the static BM3-EOS (i.e. K0 = 182.8 GPa), which supports the internal consistency of ab initio elastic data computed for jeffbenite in this work.

The evolution of the bulk modulus as a function of temperature is shown in Fig. 1a. The temperature dependency of the bulk modulus is expressed as:

$$dK_{0T} /dT = \, - 0.0200\;{\text{GPa}}/{\text{K}}$$
Figure 1
figure 1

Dependency of bulk modulus, K0 (in GPa) (a) and that of the volume thermal expansion coefficient, α (in K−1) (b) as a function of temperature, T (in K), for jeffbenite in this study.

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The volume thermal expansion coefficient is given by:

$$a_{0V} = \, 1.717 \, \times \, 10^{ - 5} {\text{K}}^{ - 1} \left( {{\text{at}}\;298.15\;{\text{K}}} \right)$$

The thermal expansion coefficient evolution as a function of temperature is shown in Fig. 1b.

The values of entropy and Gibbs energy of formation (starting from pyrope) are given as:

$$S_{0} = \, 253.36\;{\text{J}}/{\text{mol}}\;{\text{K}}$$
$$DG_{0} = \, - 13360.85\;{\text{J}}/{\text{mol}}$$

Calculated versus experimental Raman spectrum of jeffbenite

To show that the thermodynamic properties of jeffbenite calculated in this work are reliable, we have compared the Raman spectrum calculated from our data with an experimental one for the holotype jeffbenite approved by IMA (Fig. 2)1. The spectra are nearly identical, with the calculated one showing a higher resolution (this is quite typical as it is unlikely that an experimental spectrum can reach the resolution of the computed one); the experimental spectrum appears to be slightly shifted toward higher wavenumbers: this could be related to the differences in chemical composition between the pure Mg3Al2Si3O12 jeffbenite used for the calculation in this work and the natural sample, which shows an average of 4.6 wt% of FeO (of which about 20% as Fe3+), some Cr3+ and slightly less Al than the pure end member. However, although such limited chemical differences, the overlap is satisfactory.

Figure 2
figure 2

Comparison between the calculated Raman spectrum of jeffbenite form this study and the experimental one1.

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As the entropy (at least the vibrational contribution to it, which is the only one if order/disorder phenomena are excluded, as it is in the present case) is uniquely determined by the phonon spectrum, the excellent match between the calculated and the experimental Raman spectrum in Fig. 2 provides assurance that the value of entropy we determined for jeffbenite is reliable (see next section).

Gibbs free energy, entropy and pressure–temperature stability field of jeffbenite

To analyse the possible phase transition between the two polymorphs jeffbenite and pyrope we need first to compute the Gibbs energy. The differences in Gibbs energy between jeffbenite and pyrope, as functions of pressure, at different (fixed) temperatures is described as follows:

$$\Delta G\left( P \right) = G_{jeff} \left( P \right) - G_{py} \left( P \right)$$

The free energy of pyrope has been evaluated in several different ways:

  1. (1)

    Full quantum–mechanical evaluation;

  2. (2)

    Quantum–mechanical evaluation but with a correction for the entropy at standard conditions (\(S_{0}\)) taken from the H&P 2011 database

  3. (3)

    From the H&P 2011 database

  4. (4)

    From the H&P 2002 database

  5. (5)

    From the Stixrude database

The entropy of pyrope at standard conditions is a critical parameter affecting the pressure of transition from jeffbenite to pyrope as the temperature increases. The quantum–mechanical evaluation of S0 is 276.31 J/mol K [the modified-Kieffer model26], has been used for the evaluation of the acoustic mode contribution, with frequencies taken from the original Kieffer publication27; the value of the entropy without such acoustic contribution is significantly lower: 263.82 J/mol K, about 4.5% lower]. The value of S0 we determined in this work by quantum–mechanical evaluation is nearly identical to a previous one28 (276.85 J/mol K), at the ab initio level, through a super-cell approach for the evaluation of both the phonon dispersion effects and the acoustic contribution. Furthermore, the vibrational entropy values calculated in this work for pyrope are consistent with previous computational investigations performed at the hybrid HF/DFT level29. In Fig. 3, the entropy as functions of temperature, at zero pressure and from different works, is shown.

Figure 3
figure 3

Entropy, S (in J/mol K), as a function of temperature, T (in K), for the pure end member pyrope (the experimental values are from ref.30 up to 350 K and from ref.31 for data at higher temperatures).

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The difference between the entropies from Baima et al.28 and the current work are negligible, at least up to a temperature of 600 K. Experimental values are generally lower than the corresponding ab initio values. Indeed, the experimental \(S_{0}\) is 266.27 J/mol K. The values of \(S_{0}\) for pyrope adopted in thermodynamic databases are: 266.30, 269.50 and 242.36 J/mol K for HP0232, HP1133 and Stx34, respectively.

In this work, the Gibbs energy of jeffbenite was evaluated by using the same methods and computational parameters as those employed for pyrope. In Fig. 4, a comparison of the \(\Delta G\left( P \right)\) between the two polymorphs, at 300 (Fig. 4a), 500 (Fig. 4b) and 700 K (Fig. 4c) is shown. In Fig. 4, the straight line \(\Delta G = 0\) (zero line) marks the transition from jeffbenite to pyrope, that occurs at 6.02 GPa at 300 K (Fig. 4a), as seen from the intersection of the solid line with the zero line. In this case, the Gibbs energy of pyrope is evaluated at the ab initio level. Dashed lines, in colour, refer to the evaluation of the Gibbs energy of pyrope by means of the thermodynamics databases HP02, HP11 and Stx. The estimated transition pressures are 6.25 GPa (HP02), 6.11 GPa (HP11) and 6.08 (Stx). At this relatively low temperature (300 K), the impact of entropy on the computed \(\Delta G\) is almost negligible. At higher temperatures, the situation significantly changes: in particular, at a temperature of 500 K (Fig. 4b), the transition pressure decreases to 4.27 GPa (red dashed line).

Figure 4
figure 4

ΔG (in KJ/mol) for jeffbenite in this study as a function of pressure, P (in GPa), at T = 300 K (a), at T = 500 K (b) and 700 K (c).

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However, by recognising the fact that the entropy at standard conditions (S0) of pyrope, computed at the ab initio level, is overestimated with respect to the experimental value, a correction could be applied that results in black solid line of the Fig. 4b (S0 corr., which refers to such a correction of the entropy in the standard state). The correction here adopted corresponds to the HP11 value of S0. In this case, the transition pressure is 4.94 GPa. Indeed, the higher value of the transition pressure in the latter case is due to a relative decrease of the Gibbs energy of pyrope (in turns, due to the decrease of its entropy). The transition pressures computed by employing the Gibbs energies of pyrope from the databases are 5.04 (HP02) and 4.93 GPa (HP11). The curve resulting from the Stixrude database is not reported, as the corresponding value of S0 for pyrope is too far from the experimental one to be considered reliable; in addition, at variance with the entropy reported in the other databases, with the experimental measurements and with the quantum–mechanical estimations, the Stixrude database reports a value for S0 that is lower than the corresponding value for jeffbenite computed in the present work; this leads to an increase of the transition pressure as the temperature is increased (6.45 GPa, at 500 K; this P–T point is not represented in Fig. 4b).

At higher temperatures (see Fig. 4c for T = 700 K) the observed trends are confirmed. The estimation of the transition pressure, if pyrope is dealt at the ab initio level (with the correction for S0 described above), is 3.20 GPa; otherwise, the pressure is 3.53 GPa (pyrope from the HP11 database) or 3.82 GPa (pyrope from the HP02 database). Even with the Stixrude database, at 700 K, the transition pressure decreases (5.92 GPa; again, this P–T point this is not shown in Fig. 4c).

By entering the thermodynamic data of Mg3Al2Si3O12 jeffbenite determined in this work into the Stixrude database34 as implemented in Perple_X software35, the phase diagram section for the Mg3Al2Si3O12 system can be calculated and is shown in Fig. 5. Surprisingly, and in total disagreement with respect to the literature, the stability field indicates that jeffbenite is a low pressure polymorph of pyrope; in detail, by using the Stixrude database34, jeffbenite stability field expands toward higher temperatures and lower pressures. At the maximum temperature, e.g., 1400 K, jeffbenite is stable at about 0.90 GPa, while it reaches a maximum of 6 GPa just above 400 K.

Figure 5
figure 5

Pressure–temperature stability field of the pure Mg end-members jeffbenite and pyrope calculated using Perple_X35.

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Discussion

Thermoelastic properties: a comparison between jeffbenite and pyrope

The main target of this work was to compute pressure–temperature stability field of jeffbenite and to establish its polymorphic relationship with pyrope. However, thermodynamic and thermoelastic data of jeffbenite were lacking in literature and thus we had to calculate them.

With respect to pyrope, in term of bulk modulus, jeffbenite shows a value of 175.39 GPa which lies within the average value through several data published in literature for pyrope, that spans between about 164 and 182 GPa, with an average value around 170.2 GPa25,36,37,38,39,40,41,42,43,44,45,46,47. The first pressure derivative, K´, is 4.09 for jeffbenite and appears to be slightly lower than the average value of all published values for pyrope, which is 4.63 (ranging between 3.2 and 6.4). The computed bulk modulus and K´ for pyrope (same computational parameters as those employed for jeffbenite) are 162.8 GPa and 4.36, respectively, and are consistent with the ranges of the experimentally measured values.

In terms of bulk modulus dependency with temperature, we obtained for jeffbenite a value equal to -0.020 GPa/K against an average (experimental) value of − 0.022 GPa/K of pyrope [although this value has a quite significant data scatter in literature going from − 0.0194(30)43, to − 0.021(9)48, up to − 0.026(4)47]. However, the computed dK0/dT for pyrope is -0.033 GPa/K; that is, it is slightly higher from that of jeffbenite.

The volume thermal expansion coefficient for jeffbenite is here calculated as α0V = 1.717 × 10–5 K-1 (at 298.15 K) and, differently with respect to what we observed for the bulk modulus, we find a significant difference between jeffbenite and pyrope, with this last showing an average value of 2.19 × 10–5 K-1 (with values ranging between about 2 and 2.5 × 10–5 K-1, ref.44,49,50). The ab initio computed value of α0V for pyrope is 3.0 × 10–5 K-1 (at 298.15 K).

Pressure–temperature stability field of jeffbenite

Thanks to the above calculated thermodynamic data, here we have reported, for the first time, the pressure–temperature stability field of jeffbenite, at least for its Mg end-member with composition Mg3Al2Al3O12, which represents the ideal formula of jeffbenite reported in the official mineral list updated to July 2022 by the International Mineralogical Association. The stability field definitively indicates that jeffbenite is not a high pressure mineral and is the lower pressure polymorph of pyrope (see Fig. 5).

Although, based on what is now well accepted in literature, jeffbenite is considered only as a super-deep inclusion in diamonds and thus the higher pressure polymorph of pyrope, our results contradict this statement and, at the same time, they are totally consistent with the analysis of the molar volumes of the two phases. Indeed, as we also mentioned in the first section of the Results, even without any of our calculation and completely neglecting our work, a clear contradiction was already evident at the experimental level by comparing the molar volume of jeffbenite, which has V0 = 11.532 (J/bar), and that of pyrope, which has a V0 = 11.316 (J/bar)51.

By taking into account the thermoelastic parameters of jeffbenite and pyrope as calculated in this work, jeffbenite has a larger molar volume with respect to pyrope in the whole P–T stability range investigated in this work (see Fig. 6). This definitively demonstrates that the former cannot be a higher pressure polymorph of the Mg3Al2Si3O12 end-member phase but instead it is the lower pressure polymorph. This is consistent with our pressure–temperature stability field for jeffbenite in Fig. 5.

Figure 6
figure 6

Primitive unit cell volume difference (in Å3) between jeffbenite and pyrope, in the indicated pressure–temperature ranges.

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As to the other thermodynamic properties, it is interesting to note that some attempts was made in the past to empirically estimate the thermodynamic properties of jeffbenite in such a way to reconcile the phase relations observed in type III inclusions in diamonds from Brazil with an hypothetical stability field of this garnet phase52,53. In particular, a minimum pressure and temperature of 25 GPa and 2273 K were suggested for the formation of jeffbenite on the basis of observations on the Ca content of type III inclusions in diamond where jeffbenite coexists with two silicate perovskites52. Nevertheless, the purely hypothetical HP-HT stability field of jeffbenite in the predicted phase diagram for the enstatite (MgSiO3)—pyrope (Mg3Al2Si3O12) join, besides not being supported by any experimental evidence, is clearly flawed by physical unsoundness of the thermodynamic data assumed for jeffbenite. In fact, the former thermodynamic assessment53 was forced to assume a much higher compressibility of jeffbenite with respect to pyrope as well as huge entropy values for jeffbenite to stabilize this phase at high pressures and temperatures, respectively. This clearly contradicts our first principles results, which define both internally- and physically-consistent thermoelastic parameters and reliable entropy values for jeffbenite and pyrope as well. As an example, the entropy value assessed for jeffbenite at T = 970 K53 is S0 970 = 827.35 J/mol·K, which is overestimated by roughly 10% as compared to our ab initio value (i.e. S0 970 = 754.5 J/mol·K). This discrepancy is well beyond the level of confidence by which DFT is able to predict vibrational entropy of silicate minerals54.

Jeffbenite is a rare silicate only found in super-deep diamonds. Although jeffbenite is a rare mineral, however, it covers a crucial role in super-deep diamond research as it is considered a very high pressure mineral marker2 stable at least at the transition zone depths between 410 and 660 km. However, before our study, no pressure–temperature stability fields for jeffbenite was published (with the exception of one very Ti–rich synthetic jeffbenite11) because no thermodynamic data were available for such rare mineral. Here we have calculated all thermoelastic and thermodynamic data and determined the first pressure–temperature stability field for the Mg jeffbenite end-member. Of the 16 natural analysed jeffbenite known in literature, well 8 jeffbenites show a Mg# between 0.90 and 0.92, 7 jeffbenites have a Mg# between 0.81 and 0.89 and only one jeffbenite shows a Mg# equal to 0.43. Thus, such data indicate that the Mg jeffbenite end-member is the most abundant and critical one to understand the behaviour of jeffbenite and this is why we focused on it.

Very surprisingly, our results definitively show that the Mg end-member of jeffbenite is not a high-pressure polymorph of pyrope and is stable at low pressure and temperature conditions (Fig. 5). However, although several Mg-rich jeffbenites were indicated by authors as super-deep inclusions, using our data on pure Mg3Al2Si3O12 jeffbenite, we cannot speculate about the thermodynamic stability of Fe-richer and Ti-richer jeffbenites (and in some jeffbenites even the Cr content could be significant); actually, for Fe-rich and Ti–rich jeffbenites authors reported synthesis experiments demonstrating that they can be obtained at high pressure (Ti–rich up to 13 GPa11, Fe-dominant jeffbenite at 15 GPa17).

Combining our results and those from experiment laboratories11,17, we would suggest to be cautious in using extremely Mg-rich and extremely Ti-poor jeffbenites to claim super-deep origin for their diamond hosts.

Methods

Computational details

Structures (unit cell parameters and atomic fractional coordinates), static energies and vibrational frequencies at the Γ point of the Brillouin zone of jeffbenite were computed at different values of the primitive unit cell volume, in the [359, 406 Å3] range (10 points in the range). The full elastic tensor of jeffbenite, at the equilibrium static volume was also computed. The calculations were performed at the ab initio level by using the CRYSTAL17 code55. The hybrid Hartree–Fock/Density Functional WC1LYP56,57,58,59 was employed. The localized basis sets chosen for the atoms were of the type 85-11G(1d) for Mg, 85-11G(2d) for Al, 88-31G(2d) for Si, and 8-411G(2d) for O. The thresholds controlling the computation of the Coulombic and exchange integrals (ITOL1 to ITOL5 in the CRYSTAL17 input56) were set to 9, 9, 9, 9 and 22. The shrinking factor (IS) controlling the sampling of points in the BZ where the electronic Hamiltonian is diagonalized was set to 4, resulting in 13 independent k points in the BZ. An XXL grid56 for the numerical evaluation of the integrals of the DFT functionals of the electron density was chosen, which corresponded to 219,069 points in the unit cell; the very high accuracy of such numerical evaluation can be measured by the integration of the electron density over the unit cell, resulting in 399.999958 electrons in the cell, out of 400. Quantum–mechanical results are included in the manuscript related files.

By using the QM-thermodynamic software59, which implements a standard statistical thermodynamics formalism, in the limit of the Quasi-Harmonic Approximation (QHA), vibrational frequencies and static energies at each unit cell volume were employed to compute (i) the Equation of State (third-order Birch-Murnaghan) parameters (V0, K0, and K´) at 298.15 K; (ii) the dependence of K0 by the temperature (dK0/dT); (iii) the specific heat at constant pressure (CP) and its temperature dependence; (iv) the entropy at standard conditions (S0); (v) the Gibbs free energy at standard conditions (G0); (vi) the thermal expansion and its temperature dependence. A correction to CP, S0 and G0, in order to take into account the contribution of the acoustic phonons to those quantities, was made by employing the modified Kieffer-model as described in previous work26. This method allows to define shear and longitudinal seismic velocities along different propagation (and polarization) directions of the single crystal from the ab initio elastic constant tensor by solving the Christoffel determinant. A set of directionally-averaged seismic velocities are then used to calculate the acoustic contributions to thermodynamic properties according to a sine wave dispersion relation assumed for the three acoustic branches in the phonon spectrum27.

Thermodynamics quantities were estimated up to a temperature of 1700 K, and a maximum pressure of about 20 GPa at a temperature of 298.15 K (24 GPa at 1700 K). Results are provided with the deposited files (see the below Data availability statement).

In order to consistently compare the stability of jeffbenite with that of pyrope, in the appropriate P/T region, identical ab initio calculations at the same HF/DFT level, which employed the same parameters as those already set for jeffbenite, where also performed for the pyrope case. The Gibbs free energy of jeffbenite at standard conditions (G0), in the appropriate scale, was computed from the difference of the ab initio energies of jeffbenite and pyrope (energies estimated at the same ab initio level) and rescaled to the G0’s found in the thermodynamic databases used for subsequent computations33,60. The Perple_X program35 was then used to compute pseudosections for fixed global stoichiometries of the system corresponding to several significant mineral assemblages.