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GEODYNAMICS & TECTONOPHYSICS
PUBLISHED BY THE INSTITUTE OF THE EARTH'S CRUST SIBERIAN BRANCH OF RUSSIAN ACADEMY OF SCIENCES
2016 VOLUME 7 ISSUE 3 PAGES 459-476
ISSN 2078-502X
http://dx.doi.org/10.5800/GT-2016-7-3-0217
Thermodynamic properties of rock-forming oxides, a-aho3, cr2o3, a-fe2o3, and fe3o4 at high temperatures and pressures
P. I. Dorogokupets1, T. S. Sokolova1, A. M. Dymshits2, K. D. Litasov2, 3
1 Institute of the Earth's Crust, Siberian Branch of RAS, Irkutsk, Russia
2 V.S. Sobolev Institute of Geology and Mineralogy, Siberian Branch of RAS, Novosibirsk, Russia
3 Novosibirsk State University, Novosibirsk, Russia
Abstract: Equations of state of corundum (0C-AI2O3), eskolaite (Cr2O3), hematite (a-Fe2O3), and magnetite (Fe3O4) are constructed based on the Helmholtz free energy by simultaneous optimization of ultrasonic, X-ray diffraction, dilatometric, and thermochemical measurements. The magnetic contribution to Cr2O3, a-Fe2O3, and Fe3O4 Helmholtz free energy was determined via the A.T. Dinsdale model [Dinsdale, 1991]. The calculated thermodynamic properties of rock-forming oxides of aluminum, chromium, and iron are in good agreement with the reference data and experimental measurements at room pressure, as well as with P-V-T measurements at high temperatures and pressures. Thermodynamic functions (x, a, S, Cp, Cv, Kt, Ks, Yth, G) of corundum, eskolaite, hematite, and magnetite are calculated at different pressures (up to 80, 70, 50 and 20 GPa, respectively) and temperatures (up to 2000 K), and the results are tabulated. The calculated Gibbs energy of rock-forming oxides can be used to construct the phase diagrams of mineral systems, which include the oxides under the conditions of the Earth's mantle.
Key words: thermodynamics; equation of state; Helmholtz free energy; oxide; corundum; eskolaite; hematite; magnetite; mantle
Recommended by E.V. Sklyarov
For citation: Dorogokupets P.I., Sokolova T.S., Dymshits A.M., Litasov K.D. 2016. Thermodynamic properties of rock-forming oxides, a-AhO3, Cr2O3, a-Fe2O3, and Fe3O4 at high temperatures and pressures. Geodyna-mics & Tectonophysics 7 (3), 459-476. doi:10.5800/GT-2016-7-3-0217.
Термодинамические свойства породообразующих оксидов a-al2oз, cr2oз, a-fe2oз и feзo4 при условиях высоких
температур и давлений
П. И. Дорогокупец1, Т. С. Соколова1, А. М. Дымшиц2, К. Д. Литасов2, 3
1 Институт земной коры СО РАН, Иркутск, Россия
2 Институт геологии и минералогии им. В.С. Соболева СО РАН, Новосибирск, Россия
3 Новосибирский государственный университет, Новосибирск, Россия
Аннотация: На основе свободной энергии Гельмгольца построены уравнения состояния корунда (а-АЬОз), эсколаита (СГ2О3), гематита (а-Ре20з) и магнетита (Рез04) путем одновременной оптимизации ультразвуковых, рентгеновских, дилатометрических данных и термохимических измерений теплоемкости при атмосферном давлении и при повышенных температурах и давлениях. Магнитный вклад в свободную энергию Гельмгольца для СГ2О3, а-Ре20з и Рез04 определен с помощью модели А.Т. Динсдала [Dinsdale, 1991]. Предложенный подход к построению уравнений состояния хорошо описывает Л-видную аномалию в теплоемкостях эс-
колаита, гематита и магнетита, которая связана с изменением магнитных свойств. Полная термодинамическая модель уравнений состояния а-АЬОз, СГ2О3, а-Ре20з и Рез04 содержит группу из семи фиксированных параметров и группу из девяти подгоночных параметров, значения которых определяются методом наименьших квадратов. Рассчитанные термодинамические функции породообразующих оксидов алюминия, хрома и железа хорошо согласуются со справочными данными и экспериментальными измерениями при атмосферном давлении, а также с современными P-V-T измерениями в алмазных наковальнях и многопуансон-ных аппаратах высокого давления. Приведена табуляция термодинамических функций (объем, коэффициент термического расширения, изобарная и изохорная теплоемкость, энтропия, адиабатический и изотермический модули сжатия, термодинамический параметр Грюнейзена и энергия Гиббса) корунда, эсколаита, гематита и магнетита до температуры 2000 К при разных давлениях (до 80, 70, 50 и 20 ГПа, соответственно). Таким образом, полученные уравнения состояния уточняют термодинамику оксидных фаз от стандартных условий до температур и давлений, соответствующих условиям мантии Земли. Рассчитанная энергия Гиббса породообразующих оксидов алюминия, хрома и железа может быть использована для построения фазовых диаграмм минеральных систем с их участием, имеющих принципиальное значение для интерпретации глобальных и промежуточных границ в земной мантии.
Ключевые слова: термодинамика; уравнение состояния; свободная энергия Гельмгольца; оксид; корунд; эсколаит; гематит; магнетит; мантия
1. Introduction
Transition-metal sesquioxides have been already studied over a wide range of temperatures and pressures as their electrical and magnetic properties are widely variable. Corundum (a-A^Os), eskolaite (Cr2O3), hematite (a-Fe2Os) and iron-bearing compounds with related structures (Fe3O4) also play important role in the geology of the Earth interior, and ruby, (Al,Cr)2O3 was used as a pressure calibration scale for in situ diamond anvil cell (DAC) studies. Considering these applications, phase and magnetic transformations, as well as thermodynamic properties of corundum type oxides are of great interest both for fundamental and applied sciences.
Many experiments were conducted to study the high-pressure behavior of corundum-type compounds. It was shown that a-AhO3 transforms to the Rh2Os(II)-type structure (space group Pbcn) at ~80 GPa [Lin et al., 2004] and to the CaIrO3-type phase ("post-perov-skite", space group Cmcm) above 130 GPa [Oganov, Ono, 2005; Ito et al., 2009]. A further phase transition to a U2S3-type polymorph (space group Pnma) at ~370 GPa [Umemoto, Wentzcovitch, 2008] and new thermodynamically stable compounds in the system Al-O above 330 GPa [Liu et al., 2015] are predicted. To date, five different crystalline polymorphs of Fe2O3 have been discovered and described [Tucek et al., 2015]. At ambient conditions, a-Fe2O3 crystallizes in the rhombohedral corundum-type structure (space group R3c). At room temperature with increasing pressure, above ~50 GPa, Fe2O3 forms a novel monoclinic phase with space group P2i/n and, above 67 GPa, compression triggers the transition to a different HP phase with the orthorhombic unit cell and space group Aba2.
The pressure-induced Fe3+ high-spin to low-spin transition was monitored accompanying the change in the crystal structure [Ono, Ohishi, 2005]. At ambient pressure, both a-Fe2O3 and Cr2O3 are insulators and anti-ferromagnetic below the Neel temperature (Tn ) of 960 and 311.5 K, respectively [Gronvold, Samuelsen, 1975; Worlton et al., 1968]. The electronic and magnetic properties of Cr2O3 under pressure were studied using a diamond anvil cell at pressures up to 55 GPa, and the evidence for two discontinuous transitions of electronic or magnetic nature, most likely associated with the change in magnetic ordering and charge transfer, were reported at ~15 to 30 GPa [Dera et al., 2011]. In the room-temperature compression experiment, the Cr2O3 remains the original rhombohedral structure up to 70 GPa [Kantor et al., 2012]. An orthorhombic phase was detected after heating at 30 GPa [Shim et al., 2004].
Magnetite (Fe3O4) at ambient conditions is a mixed-valence iron oxide and belongs to the spinel group of minerals and crystallizes in a cubic structure (space group Fd3m). At room temperature and under pressure of ~21 GPa, Fe3O4 transforms into distorted-cubic phase (h-Fe3O4) and this structure has been widely discussed. The recent in situ study showed that Fe3O4 forms a mixture of Fe4Os and hematite at 9.5-11 GPa and 973-1673 K, which must recombine to distorted-cubic phase at yet higher pressures [Woodland et al., 2012]. A following transition of h-Fe3O4 into FeO and hematite assemblage and a new structural transition to orthorhombic structure (space group Pnma) are debated [Ricolleau, Fei, 2016]. The conducting and magnetic properties of magnetite were studied by different methods (Mossbauer spectroscopic, X-ray diffraction, electrical resistivity measurements), which show that Fe3O4 and its high-pressure polymorph are ferromag-
netic in the stability field and to at least ~70 GPa, respectively [Shebanova, Lazor, 2003].
The properties of corundum-type compounds have been studied under various P-T conditions. Calorimet-ric measurements of enthalpies of formation are reported in [Moore, Kelley, 1944; Gronvold, Westrum, 1959; Westrum, Gronvold, 1969; Gronvold, Sveen, 1974; Gronvold, Samuelsen, 1975; Goto et al., 1989; et al.]. Results of the theoretical studies of elastic properties, crystal structure, thermodynamics and stability of corundum-type oxides are presented in [Sivasubramani-an et al., 2001; Wessel, Dronskowski, 2013]. Heat capacities at ambient pressure of a-AhO3, Cr2Ü3, a-Fe2Ü3, and Fe3O4 [Hemingway, 1990; Klemme et al., 2000; Gurevich et al., 2009; Snow et al., 2010] and thermal expansion [Wachtman, 1962; Schauer, 1965; Skinner, 1966; White, Roberts, 1983; Aldebert, Traverse, 1984; Hill et al., 2010; Dymshits et al., 2016; et al.] were studied experimentally in detail. Elastic properties of a-AhO3, Cr2O3, a-Fe2O3, and Fe3O4 were experimentally
studied as a function of P and T using X-ray diffraction and ultrasonic technique [Wilburn et al., 1978; Sato, Akimoto, 1979; Finger, Hazen, 1980; Nakagiri et al., 1986; Richet et al., 1998; Dubrovinsky et al., 1998; Kim-Zajonz et al., 1999; Grevel et al., 2000; Reichmann, Jacob-sen, 2004; Gatta et al., 2007; Dera et al., 2011; Kantor et al., 2012; Dewaele, Torrent, 2013; Dymshits et al., 2016], and their more detailed description is provided below (see section 3 and 4).
Based on the current experimental results, an internally consistent set of thermochemical and thermo-physical data can be consolidated. The thermodynamic analysis can provide internal consistency of different types of physical property measurements and give the complete set of data on a system, for example, based on the Helmholtz free energy. The aim of this paper is to set up the equations of state (EoS) of a-AhO3, Cr2O3, a-Fe2O3, and Fe3O4 and calculate the full set of ther-modynamic functions depending on temperature and pressure.
2. Thermodynamic model of EoS
The proposed thermodynamic model of equations of state of corundum, eskolaite, hematite, and magnetite is based on a modified formalism from our previous studies [Dorogokupets et al., 2012, 2014, 2015; Sokolova et al., 2013,2016] and takes into account the magnetic contribution.
The Helmholtz free energy of sesquioxides in classical form [Zharkov, Kalinin, 1971]:
F(V, T) = Uo + Eo(V) + Fth(y, T) - Fth(V, To) + Fanh(V, T) - Fanh(V, To) + Fmag(T) - Fmag(T0),
(1)
where Uo is the reference energy; E o(V) is the potential part of the Helmholtz free energy on reference isotherm To=298.15 K, which depends only on volume; Fth(V,T) is the thermal part of the Helmholtz free energy, which depends on volume and temperature; Fanh(V,T) is the contribution of intrinsic anharmonicity to the Helmholtz free energy, which depends on volume and temperature; and Fmag (T) is the magnetic contribution to the Helmholtz free energy, which depends only on temperature.
In physics of metals, a potential part of free energy is often described using the well-known Vinet equation [Vinet et al., 1987], which defines the potential components of EoS Eo(V), Po(V), Kto(V), and Ko' depending on volume, as:
Eo(V) = 9KoVor]-2(l - [1 - V(1 - y)]exp[(l - y)r]]),
Po(V) = 3Koy-2(1 -y)exp[(1 -y)n],
KTo(V) = Koy-2[1 + (ny + 1)(1 - y)]exp[(1 - y)rj\,
K0 = 3[2+ yri +
y(l-y)+2y2y i+(i-y)(i+yy)\'
(2.1) (2.2)
(2.3)
(2.4)
where y=(V/Vo)=xV3, and n=1.5(^-1).
As shown earlier [Dorogokupets, Dewaele, 2007; Dorogokupets, 2010], thermodynamic functions at temperatures above the reference isotherm (>298.15 K) can be calculated using the Debye or Einstein model. For a more accurate calculation of standard entropy, we use the Einstein model with two characteristic temperatures. The thermal part of the Helmholtz free energy is expressed as a sum of two Einstein temperature contributions and the contribution of intrinsic anharmonicity:
Fth(V, T) = miRTln (l - exp -!-) + n^Tln (l - exp -!■) + (-2nRaoXmT2),
where 01 h 02 are characteristic temperatures depending only on volume; m1+m2 =3n, n is number of atoms in a chemical formula of compound; a0 is an intrinsic anharmonicity parameter; m is an anharmonic analogue of the Gruneisen parameter; R the gas constant (R=8.31451 Jmol-1K-1).
Hereafter, for simplicity, let us limit ourselves to one characteristic temperature 0i (/=1, 2). Differentiating eq. (3) with respect to temperature at constant volume, we obtain entropy, and calculate internal energy:
5 = - (^)v = 3*R h* (1 - exp =?) + expl/kl} + 3nRa0*mT,
Eth = Fth + TS = 3nR [ezpl/¥-\ + InRaoX^T2.
(4)
(5)
Differentiating eq. (3) with respect to volume at constant temperature, we determine the thermal part of pressure:
P- = -6Fh)T = 3*R ilsiê^J + 2 "RaoXmT2=.
(6)
Differentiating eq. (5) with respect to temperature at constant volume and eq. (6) with respect to volume at constant temperature, we obtain isochoric heat capacity and isothermal bulk modulus:
* - m = 3"R [($
exp(0j/r)
[exp(0j/T)-1]2
+ 3nRa0xmT,
(7)
KTth - -V
dP,
th
dV
- 3nR
y (1+ Y-q)
0,-
Lexp(0j/T) - 1J
T
2
-v2 — r V
0
exp(0(/T)
TJ (exp(0j/T) - l)2
+
1 _ m
+ 2 aoXmT2- (1 -m)
- Pth(1 + Y-q) -Y2CvT~ + 2nRaoXmT
mT2 {mq-my-m2+2Y2) V '
(8)
Differentiating eq. (6) with respect to temperature at constant volume, we determine slope:
(dPth/dT)v - 3nR
i[(!)2
exp(0j/T)
(exp(0j/T)-1)2
+ 3nRa0xmT-
(9)
The volume dependence of the characteristic temperatures in eq. (3), and unknown parameters y and q in eq. (6, 8, 9) are taken from [Al'tshuler et al., 1987]:
0i - 0o;x-^exp [^ (1 -x?)},
y - - ©X - y™ + (yo- r«)xP,
* - (B)T - ^
(10.1) (10.2) (10.3)
where 0 0/ is characteristic temperature under standard conditions (/ = 1, 2); y 0 is the Gruneisen parameter under standard conditions; is the Gruneisen parameter at infinite compression, when x ^ 0; / is an additional parameter.
To calculate the magnetic contribution to the Helmholtz free energy, we use the formalism from [D/nsdale, 1991], which was modified in [Jacobs, Schm/d-Fetzer, 2010] to obtain the correct limit of the entropy at 0 K. The magnetic contribution is expressed as:
Fmaa(T) = RT\n(B0 + 1)(g(r) - 1), (11)
where B0 is average magnetic moment per atom and t=T/Tc, Tc is Curie temperature. The function g(r) is obtained as:
g(T) = 1 - ¡12111 + 414(1- i)(ll +1! + /D> if 1 < 1, (12.1)
J ¡140p 497 \p J\6 135 600)\' ' L J
g(r) = -(— + — + -—) /D, if T >1. (12.2)
a V J V 10 315 1500J' ' L J
The value D in eq. (12.1 and 12.2) is calculated as:
D = ^ +11621(1- 1), (13)
1125 15975 \p ' v J
where p is fraction of magnetic enthalpy.
The magnetic contribution in eq. (11) does not depend on pressure, so it will be equal for the Helmholtz free energy and Gibbs energy. The equations for the magnetic contribution to entropy, enthalpy and heat capacity are available in [Dinsdale, 1991]. The same formalism was used to construct the equation of state of bcc-Fe [Dorogo-kupets et al., 2014]. Eq. (11) is used to determine the magnetic contribution to one atom of chromium or iron. Therefore, the magnetic contribution in EoSes of eskolaite and hematite can be calculated with a multipler of 2, while the multiplier for magnetite is 3. Parameters B0 and p in eq. (11, 13) for Cr2O3, a-Fe2O3, and Fe3O4 are determined by fitting the heat capacity in the region A-anomaly.
The total pressure and isothermal bulk modulus are calculated as the sums of potential and thermal parts: P=P0(V)+Pth(V,T), KT=KT0(V)+KTth(V,T), respectively. Then we can calculate the volume coefficient of thermal expansivity a=(dPth/dT) v/Kt, heat capacity at constant pressure Cp=Cv+a2TVKT, adiabatic bulk modulus Ks=Kt+ +VT(aKT)2/Cv, and thermodynamic Gruneisen parameter y th=aVKT/Cv=aVKs/CP. Enthalpy and the Gibbs energy are determined from the linear relationships: H=E+PV, G=F(V, T)+PV, respectively.
Adding up the corresponding functions, we obtain a complete thermodynamic description of equation of state. Equations (1-13) contain fixed parameters U0, V0, K0, K', m1, m2, R, Tc, and the group of fitted parameters 001, 002, y0, y^, p, a0, m, B0, and p, which are derived by the least squares method. Table 1 shows the EoSes parameters of rock-forming oxides (a-A^O3, Cr2O3, a-Fe2O3, and Fe3O4), which are obtained by simultaneous optimization of experimental measurements of heat capacity, molar volume, thermal expansion, adiabatic bulk modulus at ambient pressure, and P-V-T measurements at the reference isotherm and at high temperatures.
Table 1. Parameters of equations of state of rock-forming oxides Таблица 1. Параметры уравнений состояния породообразующих оксидов
Parameters а-ЛЬОз СГ2О3 a-Fe2O3 Fe3O4
U 0, kJmol-1 -1690.49 -1161.25 -851.78 -1158.10
Vo, cm3mol-1 25.575 29.057 30.274 44.580
Ko, GPa 252.3 211.7 202.5 181.2
K' 4.14 5.35 3.21 4.90
0io, K 956 718 554 613
m1 7.5 7.5 11.9 10.5
020, K 475 376 201 252
m2 7.5 7.5 3.1 10.5
Yo 1.323 1.388 2.084 1.341
В 1.725 0.5 1.3 0.9
a 0, 10-6К-1 8.2 - - 43.7
m 1.0 - - 1.0
Tc , K - 307 950 845.5
B 0 - 1.034 1.927 3.751
p - 0.378 0.304 0.321
3. Equation of state of a-AhOs
The equation of state of corundum is constructed according to the thermodynamic model using eq. (110). Figures 1-3 show the calculated thermodynamic functions of corundum depending on temperature at 1 bar pressure in comparison with various experimental measurements. The calculated heat capacity of a-A^O3 (Fig. 1) is in good agreement with measurements from [Goto et al., 1989; Archer, 1993] and reference data from [Gurvich et al., 1981; Chase, 1998]. The calculated coefficient of volumetric thermal expansivity (Fig. 2) is in close agreement with measurements from [Wacht-man et al., 1962; Schauer, 1965; Kirby et al., 1972; White, Roberts, 1983], but slightly differs from the measurements from [Aldebert, Traverse, 1984; Goto et al., 1989; Saxena, Shen, 1992] at temperatures above 1200 K. The calculated adiabatic and isothermal bulk modulus (Fig. 3) is consistent with the data from [Chung, Simmons, 1968; Goto et al., 1989; Anderson, Isaak, 1995].
The calculated parameters of the reference isotherm of corundum are based on the quasi-hydrostatic measurements in a helium pressure-transmitting medium from [Dewaele, Torrent, 2013]. Pressure is measured using the ruby scale from [Dorogokupets, Oganov, 2007], which slightly underestimates pressure compared to the new calibration of the ruby pressure scale in [Dorogokupets et al., 2012; Sokolova et al., 2013]. The parameters of the equation of state of corundum according [Dewaele, Torrent, 2013] are as follows: V0=25.64 cm3moH, K0=254.1 GPa, and K"=4.00 (Ryd-berg-Vinet equation, the ruby scale from [Dorogokupets, Oganov, 2007]). All the experimental P-V-T measurements are given in the form of x=V/V0, where V0 is the proposed reference value. Experimental data from [Dewaele, Torrent, 2013] are calculated based on the ruby pressure scale from [Sokolova et al., 2013] and using V0=25.575 cm3mol-1 from [Robie et al., 1978]. We have obtained for corundum: K0=252.3 GPa, K=4.14, and these values are in good agreement with the ultrasonic measurements reported in [Chung, Simmons, 1968; Goto et al., 1989; Anderson, Isaak, 1995]. Figure 4 shows that the calculated reference isotherm of corundum is consistent with the recalculated data from [Dewaele, Torrent, 2013], but earlier measurements [Richet et al., 1988; Dubrovinsky et al., 1998; Jephcoat et al., 1988; Funamori, Jeanloz, 1997] give a higher pressure. It should be noted that our calculations are in good agreement with the first studies of corundum compressibility [d'Amur et al., 1978; Finger, Hazen, 1978] in the classic hydrostatic conditions (4:1 volume mixture of methanol-ethanol). Figure 5 shows the deviation of the high-temperature measurements of compressibility [Dubrovinsky et al., 1998; Grevel et al., 2000] from our calculations. In [Grevel et al., 2000], pressure was calculated using NaCl pressure marker from
[Decker, 1971]. According to [Strassle et al., 2014], this scale underestimates pressure up to 0.5 GPa (at 1015 GPa) in comparison with NaCl EoS from [Brown, 1999]. Values more consistent with our EoS calculations for corundum can be obtained by recalculating the data from [Grevel et al., 2000] using NaCl scale from [Brown, 1999]. The data from [Dubrovinsky et al., 1998] significantly overstate the pressure probably due to the pressure measurement method.
The calculated thermodynamic functions of a-Al2O3 depending on temperature at pressures 0.0001, 50, and 80 GPa are presented in Table 2. The last two columns show the increments of the Gibbs energy of corundum at given temperatures and pressures, which are calculated from our data and database in [Holland, Powell, 2011] (G and G*, respectively). The Gibbs energy under standard conditions is calculated with regard to AH298=-1675.33 kJmol-i [Holland, Powell, 2011] and standard entropy S298=50.841 Jmol-1K-1 (Table 2) as follows: G298 =AH298-S298*298.15. In our equation of state, the Gibbs energy under standard conditions is defined by parameter U0, therefore G298=U0=-1690.49 kJmol-1. The calculated P-V-T relations of corundum can be used for pressure calculations at given temperatures and volumes, which is important for practical aspects of the high-pressure experiments.
4. EQUATIONS OF STATE OF Cr2Os, a-Fe2Os, AND FesO4
The magnetic contribution to the Helmholtz free energy is considered in the equations of state of esko-laite, hematite and magnetite using eq. (1-13). A specific A-type anomaly in the heat capacity of minerals is indicative of the magnetic transition of atoms or rare structural changes as shown for a-SiO2 ^ coesite transformation [Dorogokupets, 1995]. Curie point (Tc) determines the critical state that marks a sharp change in the magnetic properties of minerals.
Figures 6-8 show the calculated heat capacities of Cr2O3, a-Fe2O3, and Fe3O4 in comparison with the experimental measurements. The shape of the A-type anomaly in the heat capacity of oxides is well described by the EoS suggested in this study. The calculated heat capacity of Cr2O3 is in good agreement with measurements reported in [Bruce, Cannel, 1977; Klemme et al., 2000; Ziemniak et al., 2007; Gurevich et al., 2009; Aristo-va, Gusarov, 2008]. The heat capacities of a-Fe2O3 and Fe3O4 are consistent with the data from the experiments described in [Gronvold, Westrum, 1959; Westrum, Gronvold, 1969; Gronvold, Sveen, 1974; Gronvold, Samuelsen, 1975; Hemingway, 1990; Shebanova, Lazor, 2003; Snow et al., 2010] and the reference data [Robie et al., 1978]. Cr2O3 undergoes antiferromagne-tic-paramagnetic transition at Tc=307 K. The temperature of this magnetic transition is close to 300 K,
О
80
1-1-1-1-1— \ -p
c„
AGurvich et al., 1981
-Goto et al., 1989
□Archer, 1993
+ Chase, 1998
........... ........
Fig. 1. Calculated isobaric (Cp) and isochoric (CV) heat capacity of a-Al2O3 (solid lines) in comparison with selected reference and experimental data [Gurvich et al., 1981; Goto et al., 1989; Archer, 1993; Chase, 1998].
Рис. 1. Рассчитанная изобарная (Cp) и изохор-ная (Cv) теплоемкость а-АЬОз (линии) в сравнении со справочными и экспериментальными данными [Gurvich et al., 1981; Goto et al., 1989; Archer, 1993; Chase, 1998].
300 600
900 1200 1500 1800 2100 2400
Temperature (K)
Fig. 2. Calculated coefficient of volumetric thermal expansion of а-АЬОз (solid line) in comparison with selected reference and experimental data [Wachtman et al., 1962; Schauer, 1965; Kirby et al., 1972; White, Roberts, 1983; Aldebert, Traverse, 1984; Goto et al., 1989; Saxena, Shen, 1992].
Рис. 2. Рассчитанный коэффициент термического расширения а-АЬОз (линия) в сравнении со справочными и экспериментальными данными [Wachtman et al., 1962; Schauer, 1965; Kirby et al., 1972; White, Roberts, 1983; Aldebert, Traverse, 1984; Goto et al., 1989; Saxena, Shen, 1992].
Temperature (К)
Fig. 3. Calculated isothermal (Kt) and adiabatic (KS) bulk moduli of a-Al2O3 in comparison with experimental data [Chung, Simmons, 1968; Goto et al., 1989; Anderson, Isaak, 1995].
Рис. 3. Рассчитанные изотермический (Kt) и адиабатический (Ks) модули сжатия а-АЬОз в сравнении с экспериментальными данными [Chung, Simmons, 1968; Goto et al., 1989; Anderson, Isaak, 1995].
co CL О
со С
О
10 15 20
Pressure (GPa)
60 80 100 120 Pressure (GPa)
X 300 K, ruby, d'Amour et al., 1978 x 300 K, ruby, Finger, Hazen, 1978
• 300 K, Pt, Funamori, Jeanloz, 1997 0300 K, ruby, Funamori, Jeanloz, 1997
300 K, ruby, Richet et al., 1988 x 300 K, Ar, Richet et al., 1988
♦ 300 K, ruby, Jephcoat et al., 1988
□ 300 K, ruby, Dubrovinsky et al., 1998 A 300 K, ruby, Kim-Zajonz et al., 1999 + 300 K, NaCI, Grevel et al., 2000
□ 300 K, ruby, Dewaele, Torrent, 2013
180
Fig. 4. Difference between observed pressure (Pobs) [Dewaele, Torrent, 2013; Richet et al., 1988; Kim-Zajonz et al., 1999; Grevel et al., 2000; Dubrovinsky et al., 1998; Jephcoat et al., 1988; Funamori, Jeanloz, 1997; d'Amour et al., 1978; Finger, Hazen, 1978] and calculated pressure (Pcal) from equation of state of corundum at 300 K isotherm. The measurements in [Dewaele, Torrent, 2013] are recalculated to the ruby scale from [Dorogokupets et al., 2012; Sokolova et al., 2013]. The upper figure is limited to a pressure 30 GPa.
Рис. 4. Разница между измеренным давлением (Pobs) [Dewaele, Torrent, 2013; Richet et al., 1988; Kim-Zajonz et al., 1999; Grevel et al., 2000; Dubrovinsky et al., 1998; Jephcoat et al., 1988; Funamori, Jeanloz, 1997; d'Amour et al., 1978; Finger, Hazen, 1978] и рассчитанным давлением (Pcal) по уравнению состояния корунда на изотерме 300 К. Измерения из работы [Dewaele, Torrent, 2013] были пересчитаны на основе рубиновой шкалы из работ [Dorogokupets et al., 2012; Sokolova et al., 2013]. Верхний рисунок приведен для области низких давлений до 30 ГПа.
making it difficult to precisely measure. Magnetic transitions for a-Fe2Û3 and Fe3Û4 are calculated at 950 K and 845.5 K, respectively.
The deviations in the pressures calculated from the EoS of eskolaite in this study and observed in [Dymsh/ts et al., 2016] are limited to AP/P=±1.5 % up to 1873 K (Fig. 9). The difference between pressures does to exceed ±1 GPa in the measurements at reference iso-
therm [Sato, Akimoto, 1979; Finger, Hazen, 1980; Kantor et al., 2012]. The difference between the observed pressure and calculated pressure from equations of
state of hematite and magnetite at the reference isotherm is AP/P=±0.6 % (Fig. 10 a, b). Thus, the proposed equations of state of Cr2O3, a-Fe2O3, and Fe3O4 are in good agreement with the experimental studies of these minerals.
□ 11-17 GPa, Dubrovinsky et al., 1998
□ 24-32 GPa, Dubrovinsky et al., 1998
□ 36-40 GPa, Dubrovinsky et al., 1998 ■ 42-54 GPa, Dubrovinsky et al., 1998
□ 62 GPa, Dubrovinsky et al., 1998 A 5 GPa, Grevel et al., 2000
A 7 GPa, Grevel et al., 2000
300 600 900 1200
Temperature (K)
1500
1800
I Fig. 5. Difference between observed pressure (Pobs) [Dubrovinsky et al., 1998; Grevel et al., 2000] and calculated pressure (Pcai) from equation of state of corundum at high temperatures.
I Рис. 5. Разница между измеренным давлением (Pobs) [Dubrovinsky et al., 1998; Grevel et al., 2000] и рассчитанным давлением (Pcal) по уравнению состояния корунда при высоких температурах.
The calculated thermodynamic functions of Cr2O3, a-Fe2O3, and Fe3O4 depending on temperature at different pressures are listed in Tables 3-5. The calculated Gibbs energy under standard conditions for Cr2O3
(G 298=U o=-1161.25 kJmol-i), a-Fe2O3 (G298=Uo=-851.78 kJmol-i) and Fe3O4 (G298=Uo=-1158.10 kJmol-i) agree well with the results reported in [Holland, Powell, 2011] (G* in Tables 3-5).
Table 2. Thermodynamic functions of a-AhO3 calculated at different pressures
Таблица 2. Рассчитанные термодинамические функции a-AhO3 при разных давлениях
P T x=V/Vo a 5 Cp Cv Kt Ks Yth G G*
GPa K 10-6K-i JmoHK-i GPa kJmol-i
0.0001 298.15 1.00000 16.223 50.841 79.702 79.196 252.30 253.91 1.32 -1690.49 -1690.43
0.0001 500 1.00396 22.034 99.557 106.494 104.956 246.82 250.44 1.33 -1705.86 -1705.85
0.0001 1000 1.01659 27.170 180.516 124.729 120.290 231.24 239.77 1.36 -1777.69 -1777.66
0.0001 1500 1.03135 30.449 232.521 131.778 123.910 214.50 228.13 1.39 -1881.70 -1881.54
0.0001 2000 1.04807 33.969 271.243 137.736 125.561 196.82 215.90 1.43 -2008.04 -2007.76
50 298.15 0.86330 7.179 38.199 68.078 67.928 439.98 440.94 1.03 -509.29 -507.54
50 500 0.86488 10.458 81.777 98.481 97.953 436.06 438.40 1.03 -521.51 -519.25
50 1000 0.87009 12.976 158.419 119.401 117.811 424.46 430.19 1.04 -583.17 -580.00
50 1500 0.87599 13.967 208.100 125.240 122.538 412.08 421.16 1.05 -675.54 -671.10
50 2000 0.88230 14.721 244.597 128.450 124.545 399.30 411.83 1.07 -789.12 -784.89
80 298.15 0.81198 5.245 34.264 63.917 63.824 541.42 542.20 0.92 132.51 136.95
80 500 0.81308 7.878 75.950 95.500 95.153 537.97 539.94 0.93 121.30 126.61
80 1000 0.81680 9.894 151.108 117.861 116.782 527.56 532.43 0.93 63.01 69.89
80 1500 0.82101 10.598 200.205 123.809 121.982 516.39 524.13 0.94 -25.55 -17.46
80 2000 0.82548 11.076 236.262 126.766 124.151 504.90 515.54 0.95 -135.06 -126.12
Note. G - data calculated in this study; G* - data from [Holland, Powell, 2011].
Примечание. G - данные, рассчитанные в настоящей работе; G* - данные из работы [Holland, Powell, 2011].
Fig. 6. Calculated isobaric and isochoric heat capacity of СГ2О3 (solid lines] in comparison with selected reference and experimental data [Bruce, Cannel, 1977; Klemme et al., 2000; Ziemniak et al., 2007; Aristova, Gusarov, 2008; Gurevich et al., 2009].
Рис. 6. Рассчитанная изобарная и изохорная теплоемкость Cr2O3 (линии] в сравнении со справочными и экспериментальными данными [Bruce, Cannel, 1977; Klemme et al., 2000; Ziemniak et al., 2007; Aristova, Gusarov, 2008; Gurevich et al., 2009].
Fig. 7. Calculated isobaric and isochoric heat capacity of a-Fe2Ü3 (solid lines] in comparison with selected reference and experimental data [Gronvold, Westrum, 1959; Gronvold, Samuelsen, 1975; Robie et al., 1978; Snow et al., 2010].
Рис. 7. Рассчитанная изобарная и изохорная теплоемкость a-Fe2Ü3 (линии) в сравнении со справочными и экспериментальными данными [Gronvold, Westrum, 1959; Gronvold, Samuelsen, 1975; Robie et al., 1978; Snow et al., 2010].
Fig. 8. Calculated isobaric and isochoric heat capacity of Fe3Ü4 (solid lines] in comparison with selected reference and experimental data [Westrum, Gronvold, 1969; Gronvold, Sveen, 1974; Robie et al., 1987; Hemingway, 1990; Shebanova, Lazor, 2003].
Рис. 8. Рассчитанная изобарная и изохорная теплоемкость Fe3Ü4 (линии] в сравнении со справочными и экспериментальными данными [Westrum, Gronvold, 1969; Gronvold, Sveen, 1974; Robie et al., 1978; Hemingway, 1990; Shebanova, Lazor, 2003].
0298 K, NaF, Sato, Akimoto, 1979 x298 K, ruby, Finger, Hazen, 1980 + 300 K, ruby, Kantor et al., 2012 □300 К, Au, Dymshits et al., 2016 □473-673 К, Dymshits et al., 2016 □873-1073 К, Dymshits et al., 2016
□ 1273-1473 К, Dymshits et al., 2016
□ 1673-1873 К, Dymshits et al., 2016
20 30
Pressure (GPa)
I Fig. 9. Difference between observed pressure in [Sato, Akimoto, 1979; Finger, Hazen, 1980; Kantor et al., 2012; Dymshits et al., 2016] and calculated pressure from equation of state of eskolaite at the isotherms 298 to 1873 K.
I Рис. 9. Разница между измеренным давлением в работах [Sato, Akimoto, 1979; Finger, Hazen, 1980; Kantor et al., 2012; Dymshits et al., 2016] и рассчитанным давлением по уравнению состояния эсколаита на изотермах от 298 до 1873 К.
□298 К, ruby, Wilburn et al., 1978 □298 К, NaCI, Wilburn et al., 1978 □298 K, NaF, Wilburn et al., 1978 + 298 K, NaF, Sato, Akimoto, 1979 x298 K, ruby, Finger, Hazen, 1980
4 6
Pressure (GPa)
Ь 0.2
CO cl О
-0.1 --
-0.2 --
-0.3
□298 К, ruby, Nakagiri et al., 1986 0298 K, Reichmann, Jacobsen, 2004 x298 K, ruby, Gatta et al., 2007
4 6 8
Pressure (GPa)
Fig. 10. Difference between observed pressure in [Wilburn et al., 1978; Sato, Akimoto, 1979; Finger, Hazen, 1980; Nakagiri et al., 1986; Reichmann, Jacobsen, 2004; Gatta et al., 2007] and calculated pressure from equation of state of hematite (a) and magnetite (b) at the reference isotherm 298 K.
Рис. 10. Разница между измеренным давлением в работах [Wilburn et al., 1978; Sato, Akimoto, 1979; Finger, Hazen, 1980; Nakagiri et al., 1986; Reichmann, Jacobsen, 2004; Gatta et al., 2007] и рассчитанным по уравнениям состояния гематита (а) и магнетита (b) на отсчетной изотерме 298 К.
Table 3. Thermodynamic functions of Cr2O3 calculated at different pressures
Таблица 3. Рассчитанные термодинамические функции Cr2O3 при разных давлениях
P T x=V/Vo а S Cp Cv Kt Ks Yth G G*
GPa K 10-6K-I JrnoHK-i GPa kJmol- ■1
0.0001 298.15 1 21.211 80.2160 123.793 122.968 211.70 213.12 1.06 -1161.25 -1162.08
0.0001 500 1.0049 25.942 138.785 115.587 113.560 206.32 210.00 1.38 -1183.87 -1185.12
0.0001 1000 1.0191 29.951 223.073 126.685 121.584 192.02 200.08 1.40 -1276.44 -1279.12
0.0001 1500 1.0352 32.669 275.479 131.894 123.358 177.27 189.53 1.41 -1401.87 -1406.11
0.0001 2000 1.0530 35.586 314.057 136.553 123.984 162.18 178.62 1.42 -1549.66 -1555.56
30 298.15 0.8986 12.133 67.1730 115.728 115.317 358.64 359.92 0.99 -338.36 -335.46
30 500 0.9012 15.427 122.491 110.858 109.755 353.95 357.51 1.30 -357.97 -356.14
30 1000 0.9088 17.672 204.185 123.270 120.454 341.43 349.41 1.32 -441.63 -443.61
30 1500 0.9171 18.642 255.037 127.392 122.827 328.63 340.84 1.33 -557.22 -563.66
30 2000 0.9259 19.446 292.070 130.097 123.673 315.72 332.12 1.34 -694.41 -705.81
70 298.15 0.8207 7.8670 57.5000 108.557 108.323 531.94 533.09 0.92 657.16 663.17
70 500 0.8223 10.459 109.899 106.693 106.004 527.65 531.08 1.24 639.85 644.49
70 1000 0.8270 12.141 189.579 121.136 119.308 516.02 523.93 1.26 563.11 562.46
70 1500 0.8322 12.702 239.588 125.241 122.291 504.14 516.30 1.27 455.04 448.14
70 2000 0.8376 13.083 275.943 127.464 123.364 492.18 508.53 1.27 325.75 311.88
5. Discussion and conclusion
In this study, the equations of state of rock-forming oxides (a-Al2O3, Cr2O3, a-Fe2O3, and Fe3O4) are developed based on optimization of variety experimental measurements and can be used for calculation of different thermodynamic properties in a wide range of pressures and temperatures. The proposed approach for constructing of equation of state perfectly approximates the thermochemical data of the heat capacity and allows one to describe the A-type anomaly in case of changes in magnetic properties of Cr2O3, a-Fe2O3, and Fe3O4.
Nonetheless, other studies concerning equations of state (especially for corundum) are noteworthy. In [Dubrovinskaya et al., 1997], the equation of state is based on the Helmholtz free energy with a single set of common parameters and the Birch-Murnaghan equation on zero isotherm. In this study, the extrapolations of the heat capacity, thermal expansion, elastic moduli and P-V-T properties to high pressures and temperatures are in good agreement with the experimental data. In the computational study reported in [Dorogokupets et al., 1999], an empirical model is constructed for the joint optimization of data on isobaric heat capacity, volume, thermal expansion coefficient, and
Table 4. Thermodynamic functions of a-Fe2O3 calculated at different pressures
Таблица 4. Рассчитанные термодинамические функции a-Fe2O3 при разных давлениях
P T x=V/Vo а S Cp Cv Kt Ks Yth G G*
GPa K 10-6K-1 Jmol-iK-i GPa kJmol- 1
0.0001 298.15 1 33.873 87.695 102.870 100.773 202.50 206.71 2.06 -851.78 -851.65
0.0001 500 1.0076 40.449 146.916 125.411 120.511 196.37 204.35 2.01 -875.78 -875.81
0.0001 1000 1.0301 46.982 248.732 151.340 138.892 180.85 197.05 1.91 -975.50 -977.81
0.0001 1500 1.0560 52.498 307.882 147.287 125.514 164.74 193.32 2.20 -1116.21 -1119.53
0.0001 2000 1.0858 59.146 351.684 158.678 124.668 147.88 188.22 2.31 -1281.57 -1285.98
20 298.15 0.9175 23.348 71.4180 93.8630 92.673 263.70 267.09 1.85 -272.74 -270.04
20 500 0.9224 29.062 126.977 119.836 116.791 258.25 264.98 1.79 -293.02 -291.12
20 1000 0.9372 33.595 225.228 145.669 137.842 244.42 258.30 1.69 -381.78 -384.52
20 1500 0.9537 36.394 281.513 138.242 125.032 230.30 254.63 1.94 -510.03 -516.75
20 2000 0.9719 39.250 321.999 143.954 124.391 215.79 249.73 2.00 -661.41 -672.81
50 298.15 0.8314 15.352 56.4180 83.2700 82.6550 347.78 350.36 1.63 518.99 534.37
50 500 0.8344 20.335 107.672 113.547 111.755 342.98 348.48 1.58 502.25 516.41
50 1000 0.8439 23.846 202.484 141.166 136.363 330.66 342.31 1.48 424.15 431.64
50 1500 0.8544 25.431 256.725 132.330 124.347 318.15 338.57 1.68 307.79 308.70
50 2000 0.8656 26.790 295.184 135.486 123.997 305.43 333.73 1.73 169.30 162.50
Table 5. Thermodynamic functions of Fe3O4 calculated at different pressures Таблица 5. Рассчитанные термодинамические функции Fe3O4 при разных давлениях
P T x=V/V0 a S Cp Cv Kt Ks Yth G
GPa K 10-6R-1 JrnoHK-i GPa
0.0001 298.15 1 24.251 146.246 151.503
0.0001 500 1.0054 28.235 233.239 186.151
0.0001 1000 1.0209 32.553 387.280 205.181
0.0001 1500 1.0386 36.341 468.424 201.166
0.0001 2000 1.0588 40.832 527.662 211.941
10 298.15 0.9522 18.916 136.960 147.615
10 500 0.9562 22.177 222.341 183.605
10 1000 0.9677 25.214 374.643 202.257
10 1500 0.9805 27.470 454.289 196.413
10 2000 0.9947 29.893 511.790 204.276
20 298.15 0.9149 15.603 129.825 144.433
20 500 0.9181 18.440 213.912 181.640
20 1000 0.9272 20.849 364.985 200.368
20 1500 0.9373 22.435 443.709 193.608
20 2000 0.9482 24.036 500.223 200.070
bulk moduli of minerals at room pressure. The internal energy and isochoric heat capacity in this study are approximated by the Nernst-Lindeman function. The comparison of the Nernst-Lindeman function and our model with two characteristic temperatures (eq. 3) shows that the approximations look almost identical, so in both cases they provide good smoothing and approximation of the heat capacity, thermal expansion coefficient, and bulk moduli at reference pressure. Besides, other EoSes are available to calculate thermody-namic properties of corundum at high temperatures and pressures, such as described in [Jacobs, Oonk, 2001; Jacobs, Shmid-Fetzer, 2010; Jacobs et al., 2013; Stixrude, Lithgow-Bertelloni, 2005; Brosh et al., 2008; Otero-de-la-Roza, Luana, 2011].
The formalism in studies by Gerya et al. [1998, 2004] is considered thermodynamics of minerals based on the Gibbs energy. It is of special interest and may be used to construct the equations of state of corundum and phases with A-type anomaly in the heat capacity, including Cr2O3, a-Fe2O3, and Fe3O4.
The thermodynamic properties of rock-forming oxides, which are established in this study, can provide a very useful contribution for geobarometry and modeling of the mineral composition of the Earth. The presence of sesquioxide stabilizes in the solid-state systems at depth and plays an important role in understanding the mantle mineralogy and structural transformations, which associated with the global and intermediate boundaries in the Earth's mantle (olivine ^ wadsleyite ^ ringwoodite; kyanite ^ corundum + stishovite; garnet ^ perovskite + corundum-ilmenite) [Pushcharovsky, Pu-shcharovsky, 2010]. Modeling of solid solutions based on iron oxide can be highly important for describing pos-
G*
kJmol-1
150.087 181.20 182.91 1.31 -1158.10 -1158.27
183.001 176.33 179.37 1.22 -1196.85 -1197.39
197.300 163.43 169.96 1.23 -1355.10 -1357.68
187.424 149.83 160.82 1.35 -1570.44 -1575.12
190.604 135.56 150.74 1.37 -1820.04 -1827.81
146.581 228.34 229.95 1.25 -723.45 -724.10
181.259 223.82 226.72 1.17 -760.15 -760.04
196.446 211.88 218.15 1.17 -912.48 -911.40
186.547 199.41 209.95 1.28 -1121.13 -1118.73
189.499 186.46 201.00 1.30 -1363.24 -1359.93
143.626 272.65 274.18 1.21 -307.59 -308.91
179.772 268.38 271.16 1.13 -342.69 -342.29
195.749 257.06 263.13 1.13 -490.46 -486.60
185.870 245.28 255.49 1.24 -694.05 -686.11
188.684 233.12 247.19 1.26 -930.63 -918.69
sible associations in the Earth's core and its solubility in the mantle phases. Chromium oxide with small concentrations of iron and aluminum was discovered as inclusions in diamond from the Udachnaya kimberlite pipe, Yakutia [Logvinova et al., 2008]. Because correct thermal equation of state of this phase is lacking it is impossible to calculate the released pressure of such inclusions.
The rock-forming oxides may form solid solutions with spinel structure. These spinel phases may contain variable amounts of trivalent cations Fe3+, Al3+ and Cr3+ and divalent cations Fe2+ and Mg2+. The properties of such materials are interesting and important for not only geology and geophysics. The refractoriness of some of these oxides is recognized as an extremely valuable characteristic in certain special applications in steelmaking furnace industry. Therefore, the temperature dependence of elastic and thermodynamic parameters, such as cell volume, bulk modulus, thermal expansion and the Gruneisen parameter of oxides, are significant for understanding the material dynamics with respect to thermal and pressure stress. The phase relations of the Cr2O3-Fe 2O3-AI2O3 system at reference pressure have been known for more than 50 years [Muan, Somiya, 1959; Schultz, Stubican, 1970], but the knowledge of high-pressure behavior is still lacking.
The thermodynamic parameters of corundum, esko-laite, hematite, and magnetite, which are calculated in this study, specify the thermodynamics of pure oxide phases according to the current P-V-T measurements obtained in diamond anvils and multianvil apparatus [Dewaele, Torrent, 2013; Dymshits et al., 2016], and can be applied in calculations of more complex mineral systems and solid solutions at high temperatures and pressures.
6. Acknowledgments
We thanks PhD B.S. Danilov for his help in calculations of the Gibbs energy from the database [Holland, Powell, 2011]. This study was supported by the Russian
Scientific Foundation (Project no 14-i7-0060i) and the Russian Foundation of Basic Research (Projects no. 15-35-20556 and 16-35-00061) and conducted under the program of the Ministry of Education and Science of the Russian Federation (Project no 14.B.25.31.0032).
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Dorogokupets, Peter I., Doctor of Geology and Mineralogy, Head of Laboratory Institute of the Earth's Crust, Siberian Branch of RAS 128 Lermontov street, Irkutsk 664033, Russia El e-mail: [email protected]
Дорогокупец Петр Иванович, докт. геол.-мин. наук, зав. лабораторией Институт земной коры СО РАН 664033, Иркутск, ул. Лермонтова, 128, Россия И e-mail: [email protected]
Sokolova, Tatiana S., Candidate of Geology and Mineralogy, Researcher Institute of the Earth's Crust, Siberian Branch of RAS 128 Lermontov street, Irkutsk 664033, Russia Tel.: 8(3952)511680, e-mail: [email protected]
Соколова Татьяна Сергеевна, канд. геол.-мин. наук, н.с. Институт земной коры СО РАН 664033, Иркутск, ул. Лермонтова, 128, Россия Тел.: 8(3952)511680, e-mail: [email protected]
Dymshits, Anna M., Candidate of Geology and Mineralogy, Senior Researcher V.S. Sobolev Institute of Geology and Mineralogy, Siberian Branch of RAS 3 Academician Koptyug ave., Novosibirsk 630090, Russia Tel.: 8(3833)303581, e-mail: [email protected]
Дымшиц Анна Михайловна, канд. геол.-мин. наук, с.н.с. Институт геологии и минералогии им. В.С. Соболева СО РАН 630090, Новосибирск, пр. Академика Коптюга, 3, Россия Тел.: 8(3833)303581, e-mail: [email protected]
Litasov, Konstantin D., Doctor of Geology and Mineralogy, Chief Researcher, Professor of RAS V.S. Sobolev Institute of Geology and Mineralogy, Siberian Branch of RAS 3 Academician Koptyug ave., Novosibirsk 630090, Russia Tel.: 8(3833)332517; e-mail: [email protected]
Литасов Константин Дмитриевич, докт. геол.-мин. наук, г.н.с., профессор РАН Институт геологии и минералогии им. В.С. Соболева СО РАН 630090, Новосибирск, пр. Академика Коптюга, 3, Россия Тел.: 8(3833)332517; e-mail: [email protected]
