Background

Insight into the accretion of the Earth and its subsequent differentiation can be gained through knowledge of the present-day Earth composition. The lower mantle in particular has been the focus of much attention as well as controversy, and questions such as how closely it approximates a chondritic composition have occupied geochemists for decades. Comparison of laboratory measurements of elastic wave velocities of lower mantle minerals with seismic data has played a crucial role in the discussion starting with Birch (1952), and experimental techniques such as inelastic X-ray scattering (e.g., Fiquet et al. 2004), Brillouin scattering (e.g., Sinogeikin and Bass 2002; Jackson et al. 2005; Murakami et al. 2007; Murakami et al. 2008), ultrasonics (e.g., Irifune et al. 2008; Gwanmesia et al. 2009; Liu et al. 2015), and impulsive stimulated scattering (e.g., Crowhurst et al. 2008) have provided important results on transition zone and lower mantle minerals. In situ measurements are important, because some transitions (for example, spin transitions) are not quenchable, and may influence the elastic properties of iron-containing minerals.

Nuclear inelastic scattering (NIS) offers the attractive possibility to determine elastic wave velocities of iron-containing minerals in the laser-heated diamond anvil cell through direct measurement of the partial density of states (e.g., Sturhahn and Jackson 2007), and results for mantle minerals have already been reported for ferropericlase (Lin et al. 2006; Wicks et al. 2010; Chen et al. 2012; Sinmyo et al. 2014), orthoenstatite (Jackson et al. 2009) and post-perovskite (Mao et al. 2006). So far, however, no investigations using NIS have been carried out on the Earth’s most abundant mineral that constitutes the bulk of the lower mantle, namely bridgmanite.

Computational studies provide an important complement to experimental measurements and allow a more extensive probe of parameter space. Methods have advanced substantially in recent years to enable densities and elastic wave velocities of candidate lower mantle assemblages to be calculated for the appropriate compositions at relevant pressure and temperature compositions (e.g., Wang et al. 2015 and references therein). Electronic transitions in iron present a challenge, however, and the apparent inconsistency between experimental and computational results on the stability of the intermediate spin state of Fe2+ in bridgmanite remains controversial (e.g., Stackhouse 2008; McCammon et al. 2013; Shukla et al. 2015). Nevertheless, a strong advantage of computations is the capability to calculate a wide range of properties including experimental observables based on vibrational density of states (DOS), for example infrared spectra of hydrous post-perovskite (Townsend et al. 2015). The combination of theoretical DOS calculations with experimental determinations using NIS provides a powerful new tool to understand the vibrational properties of iron-containing phases.

In this work, we present an experimental study using NIS to determine sound velocities of five different compositions of (Mg,Fe)(Si,Al)O3 bridgmanite at pressures between 0 and 89 GPa at room temperature, including a comparison with results for (Mg,Fe)(Si,Al)O3 majorite with the same composition. We present a parallel theoretical study to calculate the DOS of two bridgmanite end-members (MgSiO3 and FeSiO3) at comparable pressures that provides insight into the anomalous experimental results.

Methods

The starting materials for the DAC experiments are listed in Table 1. Sample S4883 was synthesized from orthopyroxene loaded into a Re capsule in a Kawai-type multianvil press with a LaCrO3 heater assembly and W75Re25/W97Re03 thermocouple at 26 GPa and 2000 °C for 40 min (Keppler and Frost 2005). The compositions of the starting materials were determined using the electron microprobe and conventional (radioactive point source) Mössbauer spectroscopy.

Table 1 Starting materials used for DAC experiments
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NIS measurements were made using panoramic DACs that were designed and constructed at Bayerisches Geoinstitut. For each experiment, a starting material in the form of powder was loaded into a Be gasket together with small ruby chips for pressure determination using the ruby scale of Mao et al. (1986). Generally, LiF or NaCl was used as a pressure medium. The diameters of the diamond culets and opening holes in the gaskets varied depending on the pressure range of the experiment. The samples were laser-heated in the DAC using a portable system mounted on the beamline incorporating either single-sided (Dubrovinsky et al. 2009) or double-sided (Kupenko et al. 2012) laser heating.

The NIS data were collected at ambient conditions on the nuclear resonance beamline ID18 at the European Synchrotron Radiation Facility (Rüffer and Chumakov 1996) between 2008 and 2010. Further details of the NIS setup and data analysis are given in Glazyrin et al. (2013) and references therein. Nuclear forward scattering (NFS) data were collected at the same time as the NIS data to monitor the electronic state of iron, and X-ray diffraction was used to confirm the identity of bridgmanite off-line after laser heating. The NIS data were collected typically over a range of −80 to 80 meV around the 57Fe nuclear resonance energy of 14.4 keV in steps of 0.2 meV. The energy resolution of the X-ray beam was 1 meV. Debye sound velocities (V D) were determined from the reduced partial density of states (pDOS) using the “homogeneous model” described in Sinmyo et al. (2014). The densities required for the calculation of V D of bridgmanite were determined using a third-order Birch-Murnaghan equation of state based on volumes and elastic parameters reported in Boffa Ballaran et al. (2012) (sample S4850) and Glazyrin et al. (2014) (samples U1219, SL16, SL18, S4883), where the generalized model in the latter work was used to calculate values relevant to the different compositions. Equations of state parameters for majorite were taken from McCammon and Ross (2003) (cell volume) and Kavner et al. (2000) (elastic parameters). The adiabatic bulk modulus can be used to calculate longitudinal wave (VP) and transverse wave (VS) velocities from V D and vice versa (e.g., Sturhahn and Jackson 2007).

First-principles calculations were performed based on the local density approximation of DFT to determine the DOS for MgSiO3 bridgmanite at 60 GPa and antiferromagnetic FeSiO3 bridgmanite at 20, 40, and 60 GPa. We computed the dynamical matrices on a regular grid of 4 × 4 × 4 special q points (Monkhorst and Pack 1976). We used Fourier interpolation techniques to obtain the interatomic force constants on a dense grid in the reciprocal space, from which we obtained the phonon DOS (Gonze et al. 2005). We determined both the total and partial DOS for all the elements and calculated V D using the same approach as for the experimental data. The equation of state parameters were taken from Caracas and Cohen (2005). We also determined seismic wave velocities from the full elastic tensor where elastic constants were obtained from the linear relation between stresses and strains. Uniaxial and/or pure shear strains on the order of ±1 % and ±2 % were applied, and then the positions of the atoms were relaxed. The elastic constants were determined from the residual stresses divided by the strains. Voigt-Reuss-Hill schemes were employed to average the elastic tensor, and seismic wave velocities were determined based on homogeneous aggregates.

Results and discussion

We measured room temperature NIS spectra of one majorite composition and five bridgmanite compositions as a function of pressure. In the case of majorite, sample U1219 was first measured at 33 GPa and then laser-heated to produce bridgmanite and measured again without changing pressure. The NIS spectra show a clear change between majorite and bridgmanite (Fig. 1a) that leads to different pDOS (Fig. 1b). Subsequent compression of the U1219 bridgmanite sample produced systematic changes in the NIS spectra (Fig. 2a) that are reflected in the pDOS (Fig. 2b). A similar behavior on compression was observed in bridgmanite with lower iron content (SL18, Fig. 3) and Al-containing bridgmanite with lower (S4883) and higher (S4850) amounts of Fe3+ (Figs. 4 and 5, respectively). Finally, we collected data for the five different samples at ambient conditions for comparison (Fig. 6).

Fig. 1
figure 1

Comparison of U1219 sample data for the majorite and bridgmanite phases at 33 GPa. a NIS spectra. b pDOS (upper) and reduced pDOS (lower). Majorite was measured first and then transformed to bridgmanite by laser heating, after which the NIS data for bridgmanite were collected

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Fig. 2
figure 2

Influence of pressure on U1219 bridgmanite. a NIS spectra. b pDOS (upper) and reduced pDOS (lower). Data were collected on compression during the same beamtime experiment and follow on from the spectra shown in Fig. 1

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Fig. 3
figure 3

Influence of pressure on SL18 bridgmanite. a Selected NIS spectra. b Selected pDOS (upper) and reduced pDOS (lower). The SL18 pyroxene starting material was transformed to bridgmanite by laser heating in the DAC, and data were collected during three different beamtime experiments: #1—17, 36, 62, and 75 GPa; #2—50 GPa (not shown); and #3—65 GPa

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Fig. 4
figure 4

Influence of pressure on S4883 bridgmanite. a NIS spectra. b pDOS (upper) and reduced pDOS (lower). The sample was loaded as bridgmanite into the DAC, and data were collected on compression during the same beamtime experiment

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Fig. 5
figure 5

Influence of pressure on S4850 bridgmanite. a NIS spectra. b pDOS (upper) and reduced pDOS (lower). The sample was loaded as bridgmanite into the DAC, and data were collected during two different beamtime experiments: #1—50 GPa and #2—53 and 72 GPa. The sample was laser-heated at 50 GPa, and data were collected both before and after laser heating

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Fig. 6
figure 6

Comparison of majorite and bridgmanite sample data at 0 GPa. a NIS spectra. b pDOS (upper) and reduced pDOS (lower). Majorite was measured as the starting material in air while bridgmanite samples were measured after decompression

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Debye sound velocities (V D) were calculated from the reduced DOS in the limit as energy goes to zero by averaging D(E)/E2 in the low-energy range as described in Sinmyo et al. (2014). V D values for majorite (Table 2) are slightly lower than the results obtained for end-member MgSiO3 majorite and solid solutions containing Fe and/or Al that were obtained using ultrasonics (Mg0.59Fe0.04Ca0.18Na0.03Al0.23Cr0.01Si0.90O3: Irifune et al. 2008; Mg0.875Al0.25Si0.875O3 and Mg0.85Al0.3Si0.85O3: Gwanmesia et al. 2009; Mg0.95Al0.1Si0.95O3: Liu et al. 2015) and Brillouin scattering (MgSiO3 and Mg0.875Al0.25Si0.875O3: Sinogeikin and Bass 2002, Mg0.79Fe0.08Al0.30Si0.84O3: Murakami et al. 2008) (Fig. 7). These lower values are consistent with the higher iron content of our majorite (Mg0.82Fe0.18SiO3). In contrast, the V D values for bridgmanite (Table 2) are unrealistically low compared to the results from Brillouin scattering (Fig. 8). We can rule out the influence of local clustering as a significant factor that lowers velocities since the effect is too small to account for the large discrepancy (e.g., Sinmyo et al. 2014).

Table 2 Debye sound velocities at high pressure determined from NIS at room temperature and DFT at 0 K
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Fig. 7
figure 7

Influence of pressure on V D for majorite. Values from the present study are indicated by large green circles, while literature data are indicated by black symbols according to the legend

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Fig. 8
figure 8

Influence of pressure on V D for bridgmanite. Experimental values from the present study are indicated by large colored circles, while Brillouin scattering data from the literature are indicated by small black triangles (Jackson et al. 2005) and small black circles (Murakami et al. 2007). Values from a previous computational determination of the elastic tensor from finite differences (Caracas and Cohen 2005) are indicated as red triangles for MgSiO3 bridgmanite and red squares for antiferromagnetic FeSiO3 bridgmanite

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For a better understanding of the DOS for end-member bridgmanite, we calculated the total DOS for MgSiO3 bridgmanite and antiferromagnetic FeSiO3 bridgmanite as well as the pDOS for each element at 60 GPa (Figs. 9 and 10). For consistency with previous finite difference computations (Caracas and Cohen 2005), we calculated the full elastic tensor to confirm that we obtained the same elastic velocities within error. We then applied the same approach used to obtain V D from the experimental DOS obtained from NIS data to the theoretical DOS shown in Figs. 9 and 10. We found that the V D values calculated from each pDOS as well as the total DOS for each phase were identical within error, i.e., the same result is obtained irrespective of which element pDOS is used. In addition, we found that the V D values calculated using this approach (Table 2) are consistent with values that we obtained from the full elastic tensor (8.23 and 6.78 km/s for MgSiO3 and FeSiO3, respectively).

Fig. 9
figure 9

DOS for MgSiO3 bridgmanite at 60 GPa calculated using DFT. The total DOS is shown in the upper left panel, while the pDOS are shown in the other panels for the indicated element

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Fig. 10
figure 10

DOS for antiferromagnetic FeSiO3 bridgmanite at 60 GPa calculated using DFT. The total DOS is shown in the upper left panel, while the pDOS are shown in the other panels for the indicated element

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A Birch plot for bridgmanite data from both calculations and experiments highlights the anomalous behavior of sound velocities determined using NIS (Fig. 11). To obtain further insight into the origin of the low velocities of bridgmanite, we compare the pDOS for experiments and calculations. The breadth of vibrational states is smaller for bridgmanite calculated at 0 K compared to the experimental data at room temperature, but it is similar for the experimental data for both bridgmanite and majorite at room temperature (Fig. 12, upper). In the reduced pDOS, the enhanced intensity at low energies in the experimental bridgmanite data leads to a significantly higher intercept and hence lower velocity compared to both calculated bridgmanite and experimental majorite (Fig. 12, lower). While particle size (e.g., Marquardt et al. 2011) or anisotropy (Chumakov et al. 2009) could be a factor, we note that this effect is present in all the bridgmanite compositions that we measured using NIS, irrespective of sample synthesis method or history (i.e., whether they were synthesized in the multianvil press or by laser heating in the DAC). While it is clear from our results that sound velocities for bridgmanite determined using NIS cannot be used for comparison with seismic data, there is clearly scope for further work to identify the contributions to the vibrational DOS derived from NIS data in order to more fully understand the lattice vibrational properties of bridgmanite.

Fig. 11
figure 11

Birch plot of V D versus density for bridgmanite. Symbols are the same as for Fig. 8

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Fig. 12
figure 12

Comparison of pDOS (upper) and reduced pDOS (lower) determined by experiment and calculation. Experimental data are from the room temperature NIS spectra of S4850 bridgmanite at 53 GPa (red circles) and U1219 majorite at 33 GPa (blue line) while the black line indicates DFT calculations of antiferromagnetic FeSiO3 bridgmanite at 60 GPa and 0 K

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Conclusions

Our work has presented an extensive dataset for bridgmanite with different compositions based on NIS measurements in a DAC as well as first-principles calculations:

  1. (1)

    NIS measurements of majorite give sound velocities that are consistent with literature data from other methods, while NIS measurements of bridgmanite give sound velocities that are significantly lower than literature data.

  2. (2)

    Sound velocities of bridgmanite determined from NIS data are consistently low, irrespective of composition or sample history.

  3. (3)

    Treatment of the DOS calculated using DFT for MgSiO3 and FeSiO3 bridgmanite using the same approach as for the NIS data gives sound velocities that are consistent with each other for each phase irrespective of which partial DOS is used.

  4. (4)

    Treatment of the DOS calculated using DFT for MgSiO3 and FeSiO3 bridgmanite using the same approach as for the NIS data gives sound velocities that are consistent with the values obtained from the calculation of the full elastic tensor.

  5. (5)

    Comparison of the calculated and experimental DOS shows enhanced intensity at low energies in the latter that leads to the lower sound velocities.