Equation of State of a Natural Chromian Spinel at Ambient Temperature

Abstract

A natural chromian spinel with the composition (Mg_0.48(3)Fe_0.52(3))(Fe_0.06(1)Al_0.28(1)Cr_0.66(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 ($K_{T0}$) of 207(5) GPa and its first pressure derivative ($K_{T0}^{\prime }$) of 3.2(7), or $K_{T0}$ = 202(2) GPa with $K_{T0}^{\prime }$ 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 $K_{T0}$ 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.

1. Introduction

The 2-3 spinel oxides, with the general chemical formula AB₂O₄ (A = 2+ and B = 3+), as well as the so-called 4-2 spinel oxides (A = 4+ and B = 2+; ringwoodite (Mg,Fe)₂SiO₄-spinel for example), are geologically important minerals, and are frequently found in different types of rocks in different geological settings [1]. They all have the cubic Fd$\overline{3}$m structure, but show wide compositional variation [2]. Due to their compositional diversity, they have been used as key petrogenetic indicators of many geological processes [2,3,4] and played important roles in constraining pressure, temperature, oxygen fugacity, and other quantities [5,6,7].

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.

2. 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 %, Cr₂O₃ 51.49 wt %, and Al₂O₃ 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))₂O₄. 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 μm², 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 CeO₂ powder standard.

3. Result and Discussion

In total, 26 X-ray diffraction (XRD) patterns were collected at pressures from 0.90 to 15.00 GPa (

Table 1. Unit-cell parameters of chromium spinel at high pressures.

P (GPa)	a (Å)	V (Å3)
0.90(7) a	8.284(1)	568.57(14)
1.37(1)	8.281(1)	567.76(11)
1.46(2)	8.283(2)	567.76(38)
1.78(3)	8.275(2)	566.70(38)
2.07(1)	8.270(1)	565.59(26)
2.17(2)	8.270(1)	565.53(27)
3.06(1)	8.258(1)	563.10(25)
3.61(6)	8.249(1)	561.21(24)
4.46(4)	8.240(1)	559.45(22)
5.87(5)	8.223(2)	556.08(30)
6.92(6)	8.213(1)	553.97(19)
8.24(4)	8.196(2)	550.49(35)
9.36(2)	8.181(2)	547.58(46)
10.20(2)	8.172(3)	545.74(52)
10.96(9)	8.163(3)	543.89(52)
11.45(7)	8.159(2)	543.03(48)
12.37(1)	8.150(2)	541.25(46)
12.39(6)	8.147(2)	540.80(33)
12.42(8)	8.145(4)	540.26(69)
12.86(7)	8.139(3)	539.11(63)
13.10(8)	8.135(3)	538.44(59)
13.33(9)	8.133(3)	537.99(59)
13.62(10)	8.130(3)	537.43(67)
13.93(5)	8.124(4)	536.22(87)
14.46(4)	8.123(4)	535.89(74)
15.00(4)	8.117(3)	534.84(66)

^a Numbers in parentheses represent one standard deviation.

), 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]):

$$P=\frac{3}{2}{K}_{T0}\left[{\left(\frac{V_0}{V}\right)}^{\frac{7}{3}}-{\left(\frac{V_0}{V}\right)}^{\frac{5}{3}}\right]\left\{1+\frac{3}{4}\left(K_{T0}^{\prime }-4\right)\left[{\left(\frac{V_0}{V}\right)}^{\frac{2}{3}}-1\right]\right\},$$

(1)

where P is the pressure (GPa), K_T0 the isothermal bulk modulus (GPa), $K_{T0}^{\prime }$ the first pressure derivative of the $K_{T0}$, V₀ 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, $K_{T0 }^{\prime }$ = 3.2(7), and $V_0$ = 571.69(15) ų, or $K_{T0}$ = 202(2) GPa with $K_{T0}^{\prime }$ fixed to 4 and $V_0$ = 571.79(10) ų.

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:

$$F=\frac{P}{3f_E{\left(1+2f_E\right)}^{\frac{5}{2}}}$$

(2)

and

$$f_E=\frac{1}{2}\left[{\left(\frac{V_0}{V}\right)}^{\frac{2}{3}}-1\right].$$

(3)

The third-order BM-EoS can then be rewritten as:

$$F=\frac{3}{2}{K}_{T0}\left(K_{T0 }^{\prime }-4\right)f_E+K_{T0},$$

(4)

where the slope of the line defined by the experimental data is equal to 3/2$K_{T0}$($K_{T0}^{\prime }$ − 4), and the intercept value is the $K_{T0}$. Accordingly, a slope of zero means $K_{T0}^{\prime }$ = 4, a negative slope $K_{T0}^{\prime }$ < 4, and a positive slope $K_{T0}^{\prime }$ > 4. Figure 3 clearly shows that the $K_{T0}^{\prime }$ of our Sp_ss 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, 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 $K_{T0}$ values of these six end-members were experimentally constrained, and are summarized in

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

Spinel	V0	${\mathit{K}}_{\mathit{T}0}$	${\mathit{K}}_{\mathit{T}0}^{\prime }$	Experimental Details a	Reference
Spss	571.7(1) b	207(5)	3.2(7)	0–15; Gold; Ne; Powder	This study
	571.8(1)	202(2)	4
Spss	579.6(9)	179(10)	3.9(9)	0–29; Ruby; He; SC	[9]
	579.6(9)	179(1)	4 c
Spss	557.86	209(9)	7(1)	0–26.8; Mo; MEW; Powder	[10]
	556.5(8)	242(7)	4 c
Spss	560.6(2)	192(7)	4(1)	0–10.19; Ruby; ME; Powder	[11]
	560.6(2)	192(7)	3.6(13)
MgAl2O4	529.37	196(1)	4.7(3)	0–65; Gold; ME; powder	[22]
Sp		529.37	201.6(6)	4 c
FeAl2O4	542.58	193.9(2)	6.0(5)	0–7.5; Quartz; ME; SC	[23]
He		542.58	200.9(7)	4 c
MgCr2O4	573.9	182.5(4)	5.8(4)	0–8.2; Quartz; ME; SC	[24]
Mg-Ch	573.9	189.6(7)	4 c
FeCr2O4	588.47	184.8(2)	6.1(5)	0–9.2; Quartz; ME; SC	[24]
Ch		588.47	193(1)	4 c
MgFe2O4	590.7	179(2)	3.3(2)	0–53; Ruby; He; Powder	[25]
Mg-Fe	590.7	170.5(8)	4 c
Fe3O4	591.5	181(2)	5.5(15)	0–4.5; Ruby; ME; SC	[26]
Ma	591.5	187(11)	4 c

^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 $K_{T0}^{\prime }$ = 4, we calculated it from the P–V data.

. On the assumption of $K_{T0}^{\prime }$ = 4 for all spinel oxides, as proposed by Liu et al. [13,14], the$K_{T0}$ of our Sp_ss (Sp₁₃He₁₅Mg-Ch₃₂Ch₃₄Mg-Fe₃Ma₃) was approximated as 193.4(7) GPa with the following simple algorithm:

$$K_{T0-S{p}_{ss}}={{\displaystyle \sum }}^{\text{}}{x}_i{K}_{T0-i},$$

(5)

where $K_{T0-S{p}_{ss}}$ 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.

Matsukage et al. [11] compressed another Sp_ss (Sp₃₈He₁₂Mg-Ch₃₅Ch₁₁Mg-Fe₃Ma₁) with the chemical formula (Mg_0.77Fe_0.23)(Cr_0.46Al_0.5Fe_0.04)₂O₄ using similar experimental techniques (Table 2; Figure 2). The experimentally-obtained $K_{T0}$ was 192 GPa, and the approximated $K_{T0}$ with the above algorithm was 195 GPa, suggesting a relative difference of only 1.5%. Shu et al. [9] studied another Sp_ss (Sp₂He₅Mg-Ch₂₅Ch₅₆Mg-Fe₃Ma₇Mn₂; Mn standing for manganochromite, MnCr₂O₄) with the chemical formula (Mn_0.02Mg_0.30Fe_0.68)(Al_0.07Fe_0.10Cr_0.83)₂O₄. The experimentally-obtained $K_{T0}$ was 179 GPa (Table 2) and the approximated $K_{T0}$ was 193 GPa, suggesting a large relative difference of 8%, which was presumably caused by the limited and scattering data points (seven data as shown in Figure 2). Fan et al. [10] investigated another Sp_ss (Sp₁₉He₈Mg-Ch₅₁Ch₂₂) with the chemical formula (Na_0.01Mg_0.68Fe_0.28)_0.97(Cr_1.49Al_0.54)_2.03O₄ (Figure 2). Fitting their P–V data at room T to Equation (1), we obtained $K_{T0}$ = 242(7) GPa as $K_{T0}^{\prime }$ fixed at 4 or $K_{T0}$ = 194(17) GPa with $K_{T0}^{\prime }$ = 9(2). In comparison, the approximated $K_{T0}$ was only 192 GPa, being 50 GPa smaller than the experimentally-obtained value (with a large relative difference of ~21%; $K_{T0}^{\prime }$ fixed at 4). As pointed out by Fan et al. [10], the 16:3:1 methanol–ethanol–water mixture pressure medium used in their experiments resulted in somewhat non-hydrostatic experimental condition at P > ~10 GPa, which in turn should have led to the nominally much larger $K_{T0}$ [30].

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 minerals 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-xMn_x)Cr₂O₄ 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-xMn_x)Cr₂O₄ spinel solid solutions can compete for the T-site only, which eventually leads to a significant deviation from an ideal mixing.