Equation of State of a Natural Chromian Spinel at Ambient Temperature

A natural chromian spinel with the composition (Mg0.48(3)Fe0.52(3))(Fe0.06(1)Al0.28(1)Cr0.66(2))2O4 was investigated up to 15 GPa via synchrotron X-ray diffraction with a diamond-anvil cell at room temperature. No phase transition was clearly observed up to the maximum experimental pressure. The pressure–volume data fitted to the third-order Birch–Murnaghan equation of state yielded an isothermal bulk modulus (KT0) of 207(5) GPa and its first pressure derivative (K′ T0) of 3.2(7), or KT0 = 202(2) GPa with K′ T0 fixed as 4. With this new experimental result and the results on some natural chromian spinels in the literature, a simple algorithm describing the relation between the KT0 and the compositions of the natural chromian spinels was proposed. To examine this algorithm further, more compression experiments should be performed on natural chromian spinels with different chemical compositions.

The only major mineral source for metal chromium is the 2-3 natural chromian spinels (termed as Sp ss hereafter) [8].These chromian spinels are also among the most common mineral inclusions found in cratonic diamonds, and therefore can be used to estimate the formation P and T of the diamonds.To achieve this goal of the P-T estimation, the isothermal bulk moduli (K T0 ) of the natural chromian spinels and their compositional dependence should be accurately known.To date, there have been very limited experimental studies on the K T0 of the Sp ss [9][10][11].Considering the complicated correlations between the K T0 and compositions of the solid solution series of some minerals [12][13][14], more experimental investigations should be conducted to determine the K T0 of the Sp ss with different compositions.In this study, the K T0 of an Sp ss was determined by performing compression experiments at room T using a diamond-anvil cell (DAC) coupled with synchrotron X-ray radiation.Comparing this new result with the results in the literature, the correlation between the K T0 and the compositions of the Sp ss is tentatively discussed.

Experimental Method
The Sp ss -bearing sample was sourced from the Big Daddy Deposit, Sudbury, ON, Canada [15].The spinel crystals were black and exhibited good octahedral shape (grain size up to ~400 µm).Nineteen electron microprobe analyses were done with a JEOL JXA-8100 at the School of Earth and Space Sciences, Peking University and gave out the following compositional data, FeO 23.20 wt %, MgO 10.04 wt %, Cr 2 O 3 51.49wt %, and Al 2 O 3 14.86 wt %, leading to the chemical formula of (Mg 0.48(3) Fe 0.52( 3) )(Fe 0.06(1) Al 0.28(1) Cr 0.66(2) ) 2 O 4 .Some spinel crystals were picked up and ground into a fine powder, which was then loaded into a DAC.In the DAC experiments, we used a rhenium gasket and a Neon pressure medium, which was loaded by employing the GSECARS high-pressure gas-loading system.A flake of gold (~20 µm in diameter) was placed on the top of the sample to serve as pressure standard and position marker.In-situ high-pressure synchrotron X-ray diffraction experiments with the loaded DAC were performed up to ~15.00 GPa at the beamline 16-ID-B of HPCAT, the Advanced Photon Source of the Argonne National Laboratory.The sample was probed with a monochromatic X-ray beam (beam size 3 × 3 µm 2 , wavelength 0.37379 Å) and the data were collected with a 2-dimensional CCD detector.The sample-to-detector distance and orientation of the detector were calibrated by using a CeO 2 powder standard.

Result and Discussion
In total, 26 X-ray diffraction (XRD) patterns were collected at pressures from 0.90 to 15.00 GPa (Table 1), with the pressures determined using the Au equation of state (EoS) from Fei et al. [16].Some XRD patterns are shown in Figure 1.As pressure increases, all XRD peaks shift continuously toward higher 2θ angles.No apparent peak-splitting or new peak has been observed, indicating no phase transition for this spinel up to the maximum experimental pressure.However, several XRD peaks such as the 311 and 511 peaks show slight peak-broadening, potentially implying a structural instability which was somehow kinetically hindered due to the low experimental temperature.The used pressure medium neon can maintain a good hydrostatic experimental condition up to ~15 GPa [17], and should not be the major reason for the observed minor peak-broadening.To ensure a high accuracy in the unit-cell volume refinement, we excluded the peaks 311 and 511 because of their slight broadening.The P-V data are summarized in Table 1 and shown in Figure 2.They were fitted to the third-order Birch-Murnaghan equation of state (BM-EoS; [18]):  The P-V data are summarized in Table 1 and shown in Figure 2.They were fitted to the third-order Birch-Murnaghan equation of state (BM-EoS; [18]): where P is the pressure (GPa), K T0 the isothermal bulk modulus (GPa), K T0 the first pressure derivative of the K T0 , V 0 the volume at zero pressure, and V the volume at high pressure.Using the software EosFit 5.2 [19], we obtained K T0 = 207(5) GPa, K T0 = 3.2 (7), and V 0 = 571.69(15)Å 3 , or K T0 = 202(2) GPa with K T0 fixed to 4 and V 0 = 571.79(10)Å 3 .The quality of our derived BM-EoS can be evaluated by a linear fitting of the normalized pressure (F) as a function of the Eulerian strain (f E ) (i.e., the f E -F plot) [20].The two variables, F and f E , are defined as: (2) and The third-order BM-EoS can then be rewritten as: where the slope of the line defined by the experimental data is equal to 3/2K T0 (K T0 − 4), and the intercept value is the K T0 .Accordingly, a slope of zero means K T0 = 4, a negative slope K T0 < 4, and a positive slope K T0 > 4. Figure 3 clearly shows that the K T0 of our Sp ss should be close to 4, supporting a 2nd-order truncation of the BM-EoS fit.
The third-order BM-EoS can then be rewritten as: where the slope of the line defined by the experimental data is equal to 3/2 ( − 4), and the intercept value is the .Accordingly, a slope of zero means = 4, a negative slope < 4, and a positive slope > 4. Figure 3 clearly shows that the of our Spss should be close to 4, supporting a 2nd-order truncation of the BM-EoS fit.The third-order BM-EoS can then be rewritten as: where the slope of the line defined by the experimental data is equal to 3/2 ( − 4), and the intercept value is the .Accordingly, a slope of zero means = 4, a negative slope < 4, and a positive slope > 4. Figure 3 clearly shows that the of our Spss should be close to 4, supporting a 2nd-order truncation of the BM-EoS fit.Following Matsukage et al. [11], our Sp ss can be viewed as a complicated solid solution made of the following six end-members, spinel (Sp, MgAl 2 O 4 ), hercynite (He, FeAl 2 O 4 ), magnesiochromite (Mg-Ch, MgCr 2 O 4 ), chromite (Ch, FeCr 2 O 4 ), magnesioferrite (Mg-Fe, MgFe 2 O 4 ), and magnetite (Ma, Fe 3 O 4 ), with their mole percentages calculated as 13.44%, 14.56%, 31.68%,34.32%, 2.88%, and 3.12%, respectively.The K T0 values of these six end-members were experimentally constrained, and are summarized in Table 2. On the assumption of K T0 = 4 for all spinel oxides, as proposed by Liu et al. [13,14], the K T0 of our Sp ss (Sp 13 He 15 Mg-Ch 32 Ch 34 Mg-Fe 3 Ma 3 ) was approximated as 193.4 (7) GPa with the following simple algorithm: where K T0−Sp ss denotes the K T0 of the Sp ss , x i the mole fraction of the i-th end-member, and K T0−i the K T0 of the i-th end-member.The absolute difference and relative difference between the approximated bulk modulus and our experimentally measured value were only ~9 GPa and ~4.5%, respectively.a All compression experiments were done with the diamond-anvil cell (DAC), with some of the experimental details listed in the following order: P range (GPa), pressure scale, pressure medium, and XRD method.Gold, pressure scale of gold from Fei et al. [16] or Heinz and Jeanloz [21]; Ruby, pressure scale of ruby from Mao et al. [27]; Mo, pressure scale of Zhao et al. [28]; Quartz, pressure scale of quartz from Angel et al. [29].Ne, pressure medium of neon; He, pressure medium of helium; MEW, pressure medium of a 16:3:1 methanol-ethanol-water mixture; ME, pressure medium of a 4:1 methanol-ethanol mixture.Powder, powder XRD; SC, single-crystal XRD.b Numbers in parentheses represent one standard deviation.c If the reference did not provide the value of K T0 at K T0 = 4, we calculated it from the P-V data.
It is rather interesting that the simple algorithm of Equation ( 5) could reproduce so well the K T0 values of the Sp ss established by the properly-performed experiments, but reject the K T0 values of the Sp ss yielded by the malfunctioned experiments (Figure 4); According to our experience, a relative difference of ~5% (or even much larger) among the K T0 values determined for certain using similar experimental techniques is not unusual [7,13].If this observation is confirmed by further experimental studies on the Sp ss of different compositions, Equation ( 5) may provide a convenient method to estimate the K T0 of those Sp ss inclusions hosted by the diamonds.Meanwhile, we should keep in mind that Liu et al. [14] discovered a non-monotonic correlation between the K T0 and composition for the (Mg 1-x Mn x )Cr 2 O 4 spinel solid solutions.Nevertheless, it is highly possible that the cations in the Sp ss might more randomly enter the spinel structure, which should then result in a generally ideal mixing behavior, whereas the Mg and Mn cations in the (Mg 1-x Mn x )Cr 2 O 4 spinel solid solutions can compete for the T-site only, which eventually leads to a significant deviation from an ideal mixing.vs V0 of some Spss and their six end-members Sp, He, Mg-Ch, Ch, Ma, and Mg-Fe.The plotted experimental results and their sources for the six end-members, plus some experimental details, are listed in Table 2.The symbols for the Spss are the same as those in Figure 2, with the filled symbols standing for the experimental results whereas the empty symbols represent the approximated results.Note that the Spss from [9] can be recasted as Sp2He5Mg-Ch25Ch56Mg-Fe3Ma7Mn2; that from [10] as Sp19He8Mg-Ch51Ch22; that from [11] as Sp38He12Mg-Ch35Ch11Mg-Fe3Ma1; and that from this study as Sp13He15Mg-Ch32Ch34Mg-Fe3Ma3.The Spss from [9] has 2% manganochromite (Mn, MnCr2O4), which has = 199.2(106)GPa with a fixed of 4 [14].
Following Matsukage et al. [11], our Spss can be viewed as a complicated solid solution made of the following six end-members, spinel (Sp, MgAl2O4), hercynite (He, FeAl2O4), magnesiochromite (Mg-Ch, MgCr2O4), chromite (Ch, FeCr2O4), magnesioferrite (Mg-Fe, MgFe2O4), and magnetite (Ma, Fe3O4), with their mole percentages calculated as 13.44%, 14.56%, 31.68%,34.32%, 2.88%, and 3.12%, respectively.The values of these six end-members were experimentally constrained, and are summarized in Table 2. On the assumption of = 4 for all spinel oxides, as proposed by Liu et al. [13,14], the of our Spss (Sp13He15Mg-Ch32Ch34Mg-Fe3Ma3) was approximated as 193.4(7)GPa with the following simple algorithm: where denotes the of the Spss, the mole fraction of the i-th end-member, and the of the i-th end-member.The absolute difference and relative difference between the approximated bulk modulus and our experimentally measured value were only ~9 GPa and ~4.5%, respectively.2. The symbols for the Sp ss are the same as those in Figure 2, with the filled symbols standing for the experimental results whereas the empty symbols represent the approximated results.Note that the Sp ss from [9] can be recasted as Sp 2 He 5 Mg-Ch 25 Ch 56 Mg-Fe 3 Ma 7 Mn 2 ; that from [10] as Sp 19 He 8 Mg-Ch 51 Ch 22 ; that from [11] as Sp 38 He 12 Mg-Ch 35 Ch 11 Mg-Fe 3 Ma 1 ; and that from this study as Sp 13 He 15 Mg-Ch 32 Ch 34 Mg-Fe 3 Ma 3 .The Sp ss from [9] has 2% manganochromite (Mn, MnCr 2 O 4 ), which has K T0 = 199.2(106)GPa with a fixed K T0 of 4 [14].

Figure 1 .
Figure 1.Some X-ray diffraction patterns of the natural chromian spinel (cubic Fd3m) at 1.4, 4.4, 8.2, 12.4, and 15.0 GPa.All major peaks can be assigned to the structures of our 2-3 natural chromian spinels (Sp ss ) and Au (marked with asterisks).

Figure 3 .
Figure 3. Eulerian strain-normalized pressure (fE-F) plot.Standard deviations were calculated following the method from Heinz and Jeanloz [21].The solid line represents a weighted linear fit through our data.

Figure 3 .
Figure 3. Eulerian strain-normalized pressure (fE-F) plot.Standard deviations were calculated following the method from Heinz and Jeanloz [21].The solid line represents a weighted linear fit through our data.

Figure 3 .
Figure 3. Eulerian strain-normalized pressure (f E -F) plot.Standard deviations were calculated following the method from Heinz and Jeanloz [21].The solid line represents a weighted linear fit through our data.

Figure 4 .
Figure 4.vs V0 of some Spss and their six end-members Sp, He, Mg-Ch, Ch, Ma, and Mg-Fe.The plotted experimental results and their sources for the six end-members, plus some experimental details, are listed in Table2.The symbols for the Spss are the same as those in Figure2, with the filled symbols standing for the experimental results whereas the empty symbols represent the approximated results.Note that the Spss from[9] can be recasted as Sp2He5Mg-Ch25Ch56Mg-Fe3Ma7Mn2; that from[10] as Sp19He8Mg-Ch51Ch22; that from[11] as Sp38He12Mg-Ch35Ch11Mg-Fe3Ma1; and that from this study as Sp13He15Mg-Ch32Ch34Mg-Fe3Ma3.The Spss from[9] has 2% manganochromite (Mn, MnCr2O4), which has = 199.2(106)GPa with a fixed of 4[14].

Figure 4 .
Figure 4. K T0 vs. V 0 of some Sp ss and their six end-members Sp, He, Mg-Ch, Ch, Ma, and Mg-Fe.The plotted experimental results and their sources for the six end-members, plus some experimental details, are listed in Table2.The symbols for the Sp ss are the same as those in Figure2, with the filled symbols standing for the experimental results whereas the empty symbols represent the approximated results.Note that the Sp ss from[9] can be recasted as Sp 2 He 5 Mg-Ch 25 Ch 56 Mg-Fe 3 Ma 7 Mn 2 ; that from[10] as Sp 19 He 8 Mg-Ch 51 Ch 22 ; that from[11] as Sp 38 He 12 Mg-Ch 35 Ch 11 Mg-Fe 3 Ma 1 ; and that from this study as Sp 13 He 15 Mg-Ch 32 Ch 34 Mg-Fe 3 Ma 3 .The Sp ss from[9] has 2% manganochromite (Mn, MnCr 2 O 4 ), which has K T0 = 199.2(106)GPa with a fixed K T0 of 4[14].

Table 1 .
Unit-cell parameters of chromium spinel at high pressures.
a Numbers in parentheses represent one standard deviation.

Table 2 .
Birch-Murnaghan equation of state (BM-EoS) parameters of some Sp ss , and their six end-members (ambient P and T).