Introduction

Garnets are an abundant group of minerals that are stable down to the top of the lower mantle (∼ 720 km). Their elastic properties are important for interpretation of seismological data as they constitute a substantial part of the upper mantle and mantle transition zone (MTZ). The fraction of garnet in peridotite and eclogite assemblages can increase up to 35 and 95 vol%, respectively, at MTZ conditions as pyroxenes progressively dissolve in garnet with increasing depth (Irifune and Ringwood 1993; Irifune et al. 1986; Litasov and Ohtani 2005; Ricolleau et al. 2010; Ringwood 1991; Wood et al. 2013).

Silicate garnets have the general formula X 2+3 Y 3+2 (SiO4)3 and crystallize in a cubic structure (space group \(Ia\bar {3}d\), Fig. 1). The distorted cubic site (X-site) is occupied by large divalent cations (Mg, Fe, Ca, Mn), while the octahedral site (Y-site) is populated by trivalent (Al, Fe, Cr) cations. Pyroxene dissolution in garnet at high-pressure high-temperature (HPHT) conditions leads to an excess of silicon that is incorporated in the Y-site. In such garnets, Si4+ is balanced either by divalent cations in the Y-site [mainly Mg, Mg3(Mg,Si)(SiO4)3] or by sodium in the X-site [(Na2,Mg)Si2(SiO4)3]. Therefore, the presence of such a majorite component is an unambiguous indication of a high-pressure formation of the garnet.

Fig. 1
figure 1

Crystal structure of skiagite–Fe–majorite solid solution. The corner-shared alternating Si tetrahedra (Z-site) and (Fe, Si)-octahedra (Y-site) form a framework in the garnet structure. The distorted cubic voids (X-site) are populated by divalent iron (depicted as yellow isolated atoms). The structure is visualized using the VESTA software (Momma and Izumi 2011)

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Up until now, HPHT studies of the elastic properties of garnets have focused on the end-members with Mg, Ca, and Fe2+ populating the X-site and Al, (Mg, Si) on the Y-site (Arimoto et al. 2015; Chantel et al. 2016; Gwanmesia et al. 2014; Kono et al. 2010; Zhou et al. 2014; Zou et al. 2012). However, there is strong evidence to support the hypothesis of reducing conditions and stabilization of Fe–Ni alloy below 250 km depth (Rohrbach et al. 2007; Woodland and Koch 2003), so one should consider the disproportionation reaction Fe2+ → Fe3+ + Fe0 and a subsequent incorporation of ferric iron into garnet. Recently, highly oxidized majoritic inclusions from deep mantle xenoliths were found (Xu et al. 2017) and majoritic inclusions from deep mantle diamonds (Kiseeva et al. 2017, accepted) revealed that the amount of Fe3+ in the Y-site increases considerably with depth (up to 25–30% of total iron at 500 km). These data indicate that garnets such as andradite [Ca3Fe2(SiO4)3] and skiagite [Fe 2+3 Fe 3+2 (SiO4)3] are potentially important for the Earth’s upper mantle and MTZ.

Here, we present an experimental high-pressure study of skiagite–iron–majorite [Fe3(Fe2+,Si)(SiO4)3] solid solution in a diamond anvil cell (DAC) at room temperature using Nuclear Inelastic Scattering (NIS, also known as NRIXS—Nuclear Resonant Inelastic X-ray Scattering).

Experimental methods

The single crystals of skiagite–iron–majorite were synthesized in a multi-anvil apparatus at 9.5 GPa and 1100 °C from a powdered mixture of chemically pure oxides Fe1−x O, 57Fe2O3 and SiO2 (Ismailova et al. 2015). The material that we studied has composition Fe 2+3 [Fe 3+1.532(2) Fe 2+0.234(2) Si0.234(2)](SiO4)3 based on single-crystal X-ray diffraction and microprobe analysis (Ismailova et al. 2015), and therefore, the samples contain approximately 23 mol% of iron–majorite component. High-quality crystals were selected based on the quality of their diffraction peak profiles using a three-circle Bruker diffractometer equipped with a SMART APEX CCD detector and a high-brilliance Rigaku rotating anode (Rotor Flex FR-D, Mo-K α radiation) with Osmic focusing X-ray optics.

For pressure generation, we used panoramic DACs designed and manufactured in Bayerisches Geoinstitut. The size of the diamond culets was 250 µm. The isometric crystals of the garnet with typical linear dimension of ∼ 15 µm and a small ruby sphere were loaded into the pressure chamber in beryllium gaskets (the indentation thickness and the hole diameter were 30 and 120 µm, respectively). Several garnet crystals were loaded, and all measurements expect the one at 56 GPa were performed on the single crystal that gave the strongest NIS signal. The data point at 56 GPa was collected from another garnet crystal in a separate another DAC. Neon was used as pressure transmitting medium. Pressure in the DAC was determined by ruby fluorescence (Dewaele et al. 2008). It was measured before and after each data collection and the average value was used.

The single-crystal NIS experiments were performed at ambient temperature at the Nuclear Resonance Beamline (ID18, Rüffer and Chumakov 1996) of the European Synchrotron Radiation Facility. The synchrotron ring was operated in hybrid mode (one clean 4 mA single bunch diametrically opposed to a ∼ 196 mA multi-bunch beam composed of 24 groups of bunches spread over 3/4 of the storage ring circumference). The data were collected at ambient temperature over a range of − 20 to 100 meV relative to the 57Fe nuclear resonance energy (14.4 keV) with 0.5 meV step. The energy bandwidth and the beam spot size were 2.3 meV and 7 × 13 µm2, respectively. The data were processed using the DOS software (Kohn and Chumakov 2000).

Experimental results

The NIS technique provides information about lattice dynamics via nuclear resonant inelastic absorption and has its roots in the Mössbauer effect (see Chumakov and Rüffer 1998 for a review of the development of the method). The NIS spectrum consists of the elastic (the recoilless absorption, i.e., the Mössbauer effect) and inelastic part (which appears due to inelastic interactions with phonons in the studied sample). Data processing involves removal of the elastic peak and subtraction of background multi-phonon contributions, followed by determination of the partial (atomic) phonon density of states (pDOS). The raw NIS spectrum of the studied solid solution at ambient conditions is shown in Fig. 2 (for pressure evolution of the NIS spectra, see the Supplementary Information). A visual illustration of all the main steps in NIS data analysis can be found in Hu et al. (1999). For the detailed mathematical treatment, the reader is referred to Kohn et al. (1998).

Fig. 2
figure 2

Raw NIS spectrum of skiagite–iron–majorite solid solution at ambient conditions. The elastic peak at 0 meV is surrounded by phonon creation (positive energy) and annihilation (negative energy) wings. The inset shows the same spectrum in logarithmic scale

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Partial phonon density of states

Garnet has four formula units (80 atoms) per primitive unit cell. The phonon density of states (DOS) of garnet is, therefore, related to 240 vibrational branches. Garnet has a phonon bandgap between 80 and 100 meV and the high-energy band is composed of 48 modes mainly related to the vibrations of tetrahedral subunit (Baima et al. 2016; Mittal et al. 2001; Papagelis et al. 2002). The measured iron pDOS (Fig. 3), therefore, consists of 192 vibrational branches (3 acoustic and 189 optical).

Fig. 3
figure 3

Pressure evolution of the extracted pDOS. Vertical grey bars are the standard deviations

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The studied crystals have cubic symmetry; therefore, the obtained pDOS does not have directional dependence (Kohn et al. 1998) and represents the average pDOS of the crystal, not the projected one. The thermodynamic and vibrational parameters extracted from the iron pDOS are presented in Table 1. Because iron populates two crystallographic sites (X and Y), the parameters are the averaged values in the first approximation. However, it is not an exact averaging, because contributions of the X- and Y-sites are weighted by the corresponding Lamb–Mössbauer factors which are distinct (see discussion in Sturhahn and Chumakov 1999).

Table 1 Extracted parameters from the NIS spectra and elastic moduli of skiagite–iron–majorite solid solution
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So far, there has been no theoretical investigation of the phonon properties of skiagite or iron–majorite. Nevertheless, the contributions of the different crystallographic sites to the iron pDOS at ambient conditions (Fig. 4) can be assigned using the theoretical iron pDOS of almandine [Fe3Al2(SiO4)3] and/or andradite. This is possible, because the potential wells of the X- and Y-sites in skiagite–iron–majorite solid solutions should be similar to those of X-site Fe2+ in almandine and Y-site Fe3+ in andradite, respectively. Hence, this approach is also valid for the iron pDOS. In the literature, only the pDOS of X-site Fe2+ in almandine has been reported (Mittal et al. 2000, 2001) as obtained from semi-empirical interatomic potential calculations. Comparing it with our experimental iron pDOS (Fig. 4), one can judge that the 15 meV peak is dominated by vibrations of X-site Fe2+. Accordingly, after subtracting the X-site contribution from experimental pDOS, one can see that the noticeable contribution of Y-site Fe appears above 18 meV and the low-intensity 55–80 meV band is formed only by its vibrations.

Fig. 4
figure 4

Iron pDOS of skiagite–iron–majorite solid solution at ambient conditions. The yellow line is the calculated pDOS of X-site Fe2+ in almandine (Mittal et al. 2000). Comparing it with our experimental pDOS, one can judge that the 15 meV peak is dominated by vibrations of X-site Fe2+. The dip on the almandine pDOS around 11 meV is a calculation artefact; see Fe pDOS in a subsequent paper of the same group (Mittal et al. 2001). The inset shows the example of the parabolic fit for the low-energy part of the experimental pDOS at 27 GPa

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Sound velocities

The determination of the sound velocities from the NIS data is based on the evaluation of the Debye sound velocity from the low-energy part of the pDOS (Hu et al. 2003). The latter depends quadratically on the energy and can be written in the following form (Achterhold et al. 2002; Hu et al. 2003):

$$\mathcal{D}\left( E \right)=\left( {\frac{{\tilde {m}}}{m}} \right)\frac{{{E^2}}}{{2{\pi ^2}{\hbar ^3}nV_{D}^{3}}},$$
(1)

where \(\tilde {m}\) is the mass of the nuclear resonant isotope (57Fe in our case), m is the average atomic mass, n is the density of atoms, and V D is Debye sound velocity.

Due to the small size of the garnet Brillouin zone, the acoustic modes can cross with optical modes even below 10 meV at ambient conditions (Baima et al. 2016; Papagelis et al. 2002). This leads to a narrow range for the parabolic fit of the low-energy part of pDOS, and thus, we carried it out from 4 to 7–10 meV depending on the pressure (Fig. 4). The data below 4 meV lie under the strong elastic peak in the NIS spectrum, so data points affected by the elastic peak subtraction are not used for the fitting. The upper limit of the fitted interval is determined by the deviation of the pDOS data from parabolic dependence. The phonon spectrum shifts to higher energies with increasing pressure, so the parabolic region is extended at higher pressures.

The Debye sound velocity can be expressed as follows:

$$\frac{3}{{V_{D}^{3}}}=\frac{1}{{V_{p}^{3}}}+\frac{2}{{V_{s}^{3}}},$$
(2)

where V p is the average velocity of the primary (compression) wave and V s is the average velocity of the secondary (shear) wave. There is additional relationship between V p and V s :

$$\begin{aligned} & {V_p}=\sqrt {\frac{{K+4/3G}}{\rho }} ,\quad {V_s}=\sqrt {\frac{G}{\rho }} \\ & =>~~~\frac{K}{\rho }=V_{k}^{2}=V_{p}^{2} - \frac{4}{3}V_{s}^{2}, \\ \end{aligned}$$
(3)

where K is the adiabatic bulk modulus, G is the shear modulus, and ρ is the density. The density was determined using the equation of state (EoS) from Ismailova et al. (2017), which was measured on the same material that we studied. Besides, the isothermal bulk modulus K T was used instead of adiabatic bulk modulus (Table 1). The latter introduces a negligible error, as K T is only slightly lower than K at ambient temperature (for example, for andradite, the difference is 0.7%, Jiang et al. 2004).

Generally speaking, the equations (1),(2), (3) are valid for elastically isotropic media. The use of them is justified, as garnets have very small elastic anisotropy (Erba et al. 2014). The system of Eqs. (2) and (3) has an approximate solution according to the corrected formula from Sturhahn and Jackson (2007):

$${V_p}=\sqrt {1.002V_{k}^{2} - 0.104{V_k}{V_D}+1.208V_{D}^{2}} ,~~~~~~{V_s}=0.952{V_D} - 0.041{V_k}.$$
(4)

Figure 5 shows the obtained values of V p and V s as function of pressure. For the data point at 56 GPa, we report only Debye sound velocity (Table 1) as the bulk modulus of the sample is unknown due to the spin crossover of Y-site octahedral iron in this pressure range (Ismailova et al. 2017). A linear fit gives the following pressure dependencies of sound velocities at ambient temperature:

Fig. 5
figure 5

Sound velocities of skiagite–iron–majorite solid solution and some silicate garnet end-members as a function of pressure at ambient temperature. The black diamonds, cyan squares, and pink hexagons are estimated sound velocities of pure skiagite, khoharite, and JF-55A inclusion (correct composition with Y-site Fe3+), respectively (see main text). The solid lines conform to the pressure range investigated in the corresponding studies, while dashed lines designate regions of extrapolation

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$${V_p}=7.43\left( 9 \right)+0.039\left( 4 \right)~ \times ~P,~~~~~~~{V_s}=3.56\left( {12} \right)+0.012\left( 6 \right)~ \times ~P.$$
(5)

Here, V p and V s are in km/s and pressure is in GPa. The magnitudes of sound velocities are substantially lower than for all other garnet end-members considered so far, while the pressure derivatives of both V p and V s are comparable (Fig. 5).

Discussion

In petrological models of the upper mantle and MTZ, iron ions are conventionally assumed to be divalent. This is also a common assumption when reducing electron microprobe data of natural samples from the deep mantle. In the case of garnet, Fe2+ is assigned to the X-site. It is, therefore, important to estimate the extent to which the presence of Fe in the Y-site can influence sound velocities of the complex garnet solid solution. The most important consequences would be for the seismic profile in the 400–600 km depth interval where pyroxene completely dissolves into garnet and the volume fraction of the latter reaches a maximum (Wood et al. 2013).

Experimental studies have shown that, in the presence of both Mg and Fe in the majorite solid solution, there is a strong preference to balance Si4+ in the Y-site by Mg2+ (not Fe2+) (Kiseeva et al. 2017, accepted; McCammon and Ross 2003). Moreover, in the presence of Al, Fe2+ was not detected in the Y-site at all. Therefore, for further discussion, only end-members with ferric iron in the octahedral site are important.

Garnets with Y-site Fe3+: skiagite and khoharite

Among garnets of interest, only the elastic properties of andradite have been studied in detail (Jiang et al. 2004). Due to the lack of experimental data for pure skiagite, we will estimate its elastic moduli based on solid solution considerations.

For a solid solution formed by atomic substitution on multiple distinct crystallographic sites, the unknown elastic moduli and sound velocities of end-members can be estimated from solid solutions with a particular composition. Let us consider the hypothetical solid solution (Fe2.4Ca0.6)(Fe0.4Al1.6)(SiO4)3 for the estimation of sound velocities of pure skiagite. The composition has two equivalent representations using end-members: 20% And + 80% Alm ≡ 20% Gro + 20% Ski + 60% Alm [Gro refers to grossular—Ca3Al2(SiO4)3]. From Fig. 5 and a comparison of sound velocities of these end-members, one can see that the first representation should lead to sound velocities higher than almandine lines. As the second representation is equivalent, it is obvious that skiagite should balance grossular relative to the almandine line.

Assuming the validity of Vegard’s law, we obtain a = 11.73 Å for pure skiagite (unit cell parameters of almandine, andradite, and grossular were taken from Arimoto et al. (2015), Jiang et al. (2004), and Rodehorst et al. (2002), respectively). The unit cell parameter is in excellent agreement with experimental data (11.73 Å, Woodland et al. 1999; Woodland and Ross 1994).

A widely used assumption for the pyralspite (pyrope-almandine-spessartine) series (Erba et al. 2014) is that elastic moduli of the solid solution depend linearly on those of the end-members. A recent ab initio investigation verified this assumption for the grossular–andradite solid solution (Lacivita et al. 2014). Using this assumption for skiagite gives an aggregate adiabatic bulk modulus of 156.6 GPa, a shear modulus of 76.4 GPa, and Poisson’s ratio of 0.29. For these calculations, we used the aggregate elastic moduli known from the literature for almandine (Arimoto et al. 2015) and andradite (Jiang et al. 2004), as well as the average of values for grossular published by Kono et al. (2010) and Gwanmesia et al. (2014). The calculated bulk modulus is in good agreement with the experimental isothermal bulk modulus of 157(3) GPa (Woodland et al. 1999); hence, we consider our approximation to be valid and use the calculated moduli to obtain the values V p  = 7.52 km/s and V s  = 4.09 km/s for skiagite at ambient conditions (plotted in Fig. 5 as black diamonds). Comparing our experimental data of skiagite–iron–majorite solid solution and estimated sound velocities of pure skiagite (Fig. 5), one can see that the presence of iron-majorite strongly decreases V s . This fact suggests non-linear behavior for the studied solid solution (the isothermal bulk modulus values also show non-linear behavior for this composition, see Ismailova et al. 2017). The cause of this behavior requires further investigation.

Recently reported majoritic inclusions in host garnet from an eclogite xenolith (Xu et al. 2017) contain a considerable amount (from 40 to 48% depending on the particular end-member representation) of khoharite, Mg3Fe2(SiO4)3. This Y-site Fe3+ end-member may, therefore, also be relevant for the Earth’s mantle. However, there are no experimental data at all for this garnet; only a single theoretical study (Milman et al. 2001).

We can estimate khoharite sound velocities in a similar way as performed above for skiagite. The hypothetical solid solution with composition (Mg2.4Ca0.6)(Al1.6Fe0.4)(SiO4)3 can be represented as 20% And + 80% Pyr ≡ 20% Gro + 20% Kho + 60% Pyr [Pyr refers to pyrope—Mg3Al2(SiO4)3]. Elastic moduli and the unit cell parameter of pyrope were taken from Chantel et al. (2016) (results of “global” fit) and Hazen and Finger (1978), respectively. We obtain the following values for khoharite: a = 11.66 Å, ρ = 3.859 g/cm3, K = 153.9 GPa, G = 73.4 GPa. The sound velocities for khoharite at ambient conditions are then V p  = 8.08 km/s and V s  = 4.36 km/s (plotted in Fig. 5 as cyan squares). As seen in the figure, khoharite also has lower sound velocities compared to other silicate garnets, but occupies an intermediate position between andradite and skiagite in the series of garnets with Y-site Fe3+.

Skiagite and khoharite are particularly relevant to the properties of Ca-depleted solid solutions. For instance, such garnets can be formed in harzburgite rock whose existence in the MTZ was proposed by Irifune et al. (2008). Indeed, the garnet inclusions from Xu et al. (2017) that contain a large proportion of khoharite have a low amount of CaO, but their genesis is unclear. In the following discussion, we will, therefore, consider only the inclusions from Kiseeva et al. (2017, accepted).

Influence of Y-site Fe3+ on sound velocities of garnet from MTZ

As a representative example of Y-site Fe3+ influence, we consider the JF-55A inclusion (formation depth of 440 km) with pyroxenitic composition from the Jagersfontein kimberlite (Kiseeva et al. 2017, accepted). This is the most oxidized garnet from the series, so it is ideal as a limiting case. Based on electron microprobe and Mössbauer data, the composition of JF-55A from single-crystal structure refinement is (Na0.06Ca0.57Fe 2+0.42 Mg1.97)(Al1.01Fe 3+0.15 Si0.44Mg0.38)(SiO4)3 (Kiseeva et al. 2017, accepted). Therefore, the endmember representation is 3% Na-maj + 38% Mg-maj + 27% Pyr + 13% Alm + 11.5% Gro + 7.5% And.

In the original publication, all iron was assumed to be ferrous in the reduction of electron microprobe data (Tappert et al. 2005). There is no unique chemical formula that can be written from these data. Indeed, it is impossible to derive a composition without excess cations or the presence of vacancies based on normal assumptions about site charges. To proceed, we consider the following end-member representation: 3% Na-maj + 38% Mg-maj + 27% Pyr + 16% Alm + 16% Gro.

Using the experimentally determined unit cell parameter of JF-55A (Kiseeva et al. 2017, accepted), we obtain densities of 3.673 and 3.638 g/cm3 for the actual and “Fe2+ only” compositions, respectively. Elastic moduli and pressure derivatives of Mg–majorite were taken from Sinogeikin and Bass (2002). We neglect the Na–majorite contribution to the elastic moduli as there are no experimental data on the pressure derivative of its shear modulus. This term enters symmetrically in both representations, so it will not appreciably influence the relative difference between them.

The differences between the two representations are the amount of almandine and grossular, and the presence of andradite. From Fig. 5 one can see that consideration of Y-site Fe3+ should decrease the resulting sound velocities. Indeed, the calculations for composition with Fe3+ give V p  = 8.88 km/s and V s  = 4.98 km/s (Fig. 5), while for the “Fe2+ only” composition, the result is V p  = 8.96 km/s and V s  = 5.03 km/s at ambient conditions. Therefore, Y-site ferric iron decreases sound velocities by 1% in this case. At 20 GPa and 300 K, the difference remains 0.08 and 0.05 km/s for V p and V s , respectively. In terms of the 400—600 km depth interval in the preliminary reference Earth model (Dziewonski and Anderson 1981), it corresponds to a 16 km depth difference.

However, if instead of using the experimental unit cell parameter for both compositions, the values calculated using Vegard’s law are used [average unit cell parameters of Na– and Mg–majorite were taken from Bindi et al. (2011) and Angel et al. (1989), respectively], the difference in the velocities decreases to 0.05 and 0.03 km/s at ambient conditions for V p and V s , respectively. Moreover, the non-uniqueness of the “Fe2+ only” composition substantially spreads the range of this difference. We also note that current uncertainties in the determination of elastic moduli and their derivatives exceed such small differences. A striking example is the shear modulus of pyrope. In recent HPHT ultrasonic interferometry experiments in a multi-anvil press, the obtained values are 89.1(5) and 93.2(1) [Chantel et al. (2016) and Zou et al. (2012), respectively].

Conclusions

In this study, we performed an ambient temperature high-pressure NIS investigation of skiagite (77 mol%)–iron–majorite solid solution. The determined sound velocities are significantly lower than sound velocities of the silicate garnet end-members, grossular, pyrope, Mg–majorite, andradite, and almandine. We also estimated the sound velocities of two end-members with Fe3+ in the Y-site: skiagite and khoharite. Comparison of the NIS-data-derived values of sound velocities with those estimated for pure skiagite demonstrates that the iron–majorite component decreases the sound velocities, especially V s . The neglect of Y-site Fe3+ may decrease sound velocities of garnet solid solution relevant to MTZ up to 1% relative to a composition with only Fe2+ in the X-site.