Introduction

To decipher metamorphic conditions and for understanding the crustal evolution, a valid geothermobarometry serves as a fundamental tool. In the last few decades, several geothermobarometers have been proposed for determining the metamorphic conditions of rocks ranging from greenschist to eclogite facies and at a range of pressures.

The validity of a thermometer is a prerequisite in the study of metamorphism. Several thermobarometric studies have been undertaken in the past few years, which led to the development of a range of thermometers and barometers, such as garnet–biotite thermometer (Wu and Cheng 2006) and garnet–clinopyroxene thermometer (Jahnson et al. 1983; Fu et al. 1998), and comparison of different conventional thermometers (Bohlen and Essene 1980; Ferry 1980; Essene 1989; Essene and Peacor 1995; Rathmell et al. 1999).

The study of garnet–orthopyroxene geothermometry has a long history, and we now have several versions of these geothermometers. However, these diverse calibrations may be confusing to petrologists in choosing a suitable version. With the development of geothermobarometric studies, it appears necessary to do a review for a given thermometer or barometer every decade or so. Now in this paper, we summarize and compare the available garnet–orthopyroxene thermometer in order to recommend the best calibrations to geologists.

Garnet–orthopyroxene exchange thermometers

Quantification of the garnet–orthopyroxene Fe–Mg equilibrium is a widely applied thermometer for the assemblages of basic granulites/charnockite. The distribution of the Fe2+ and Mg between coexisting garnet and orthopyroxene is expressed by the exchange reaction using equations given by Dahl (1980), Raith et al. (1983), Harley (1984), Sen and Bhattacharya (1984), Lee and Ganguly (1984), Perchuk et al. (1985), Lee and Ganguly (1988), Aranovich and Podlesskii (1989), Perchuk and Lavrente’va (1990), Bhattacharya et al. (1991), Lal (1993), Carson and Powell (1997), Aranovich and Berman (1995, 1997), Berman and Aranovich (1996), and Nimis and Grütter (2010) which are as follows:

$$ {\displaystyle \begin{array}{l}1/2\ {\mathrm{Fe}}_2{\mathrm{Si}}_2{\mathrm{O}}_6+1/3\ {\mathrm{Mg}}_3{\mathrm{Al}}_2{\mathrm{Si}}_3{\mathrm{O}}_{12}\rightleftharpoons 1/2\ {\mathrm{Mg}}_2{\mathrm{Si}}_2{\mathrm{O}}_6+1/3\ {\mathrm{Fe}}_3{\mathrm{Al}}_2{\mathrm{Si}}_3{\mathrm{O}}_{12}\\ {}1/2\;\mathrm{Ferrosilite}+1/3\ \mathrm{Pyrope}\rightleftharpoons 1/2\ \mathrm{Enstatite}+1/3\mathrm{Almandine}\end{array}} $$

The partitioning of the Fe2+ and Mg, expressed by the distribution coefficient between coexisting garnet and orthopyroxene, has clearly shown that this distribution is a function of both physical conditions and compositional variations of the phases involved:

$$ {K}_D={\displaystyle \begin{array}{c}{\left({\mathrm{Fe}}^{2+/}\mathrm{Mg}\right)}^{\mathrm{Gt}}\\ {}-------\\ {}{\left({\mathrm{Fe}}^{2+/}\mathrm{Mg}\right)}^{\mathrm{Opx}}\end{array}} $$

For reversed phase equilibrium, the Al2O3 content in orthopyroxene coexisting with garnet in various rock types is recognized as a sensitive indicator of the pressure–temperature conditions at which rocks equilibrated, represented by following reaction (Aranovich and Berman 1995, 1997; Berman and Aranovich 1996):

$$ {\displaystyle \begin{array}{l}{\mathrm{Fe}}_3{\mathrm{Al}}_2{\mathrm{Si}}_3{\mathrm{O}}_{12}={3\mathrm{FeSiO}}_3+{\mathrm{Al}}_2{\mathrm{O}}_3\\ {}\mathrm{Almandine}=\mathrm{Ferrosilite}+\mathrm{Orthocorundum}\end{array}} $$

Exchange thermometers mainly imply the exchange of Fe and Mg between coexisting silicates, characterized by small ∆V compared to large ∆H so that equilibrium constant isopleths have rather steep slope. This Fe–Mg exchange equilibrium has been formulated by a number of workers by obtaining standard state thermodynamic data either by empirical method or by experiments involving the crystalline solutions (garnet and orthopyroxene) between the two cations Fe and Mg. A brief description of various models considered for this comparative study is summarized as follows.

Dahl (1980)

Dahl (1980) proposed an empirical calibration for the garnet–pyroxene geothermometer. The expression for geothermometer is as follows:

$$ T(K)=\left[\right(1391+1509{\left({X}_{\mathrm{Fe}}-{X}_{\mathrm{Mg}}\right)}^{\mathrm{Gt}}+2810\left({X}_{\mathrm{Ca}}^{\mathrm{Gt}}\right)+2855\left({X}_{\mathrm{Mn}}^{\mathrm{Gt}}\right)/\mathrm{R}\ln {K}_D\Big] $$

where KD = (XFe/XMg)Gt* (XMg/XFe)Opx = (Fe/Mg)Gt * (Mg/Fe)Opx; i = (Ca + Mg + Fe + Mn)Gt; \( {X}_{\mathrm{Mg}}^{\mathrm{Gt}} \) = (Mg/i)Gt; \( {X}_{\mathrm{Ca}}^{\mathrm{Gt}} \) = (Ca/i)Gt; \( {X}_{\mathrm{Mn}}^{\mathrm{Gt}} \) = (Mn/i)Gt; and \( {X}_{\mathrm{Fe}}^{\mathrm{Gt}} \) = (Fe/i)Gt; \( {X}_{\mathrm{Mg}}^{\mathrm{Opx}} \) = (Mg/Mg + Fe); \( {X}_{\mathrm{Fe}}^{\mathrm{Opx}} \) = Fe/(Fe + Mg).

Raith et al. (1983)

Raith et al. (1983) proposed an empirical calibration for granulite facies rocks. The expression for geothermometer is as follows:

$$ T(K)=\left[1684/\left(\ln {K}_D+0.334\right)\right] $$

where KD is same as defined by Dahl (1980) for this as well as the subsequently discussed geothermometer.

Harley (1984)

Harley (1984) experimentally investigated the partitioning of Fe and Mg between garnet and orthopyroxene and aluminous pyroxene in the P–T range of 5–30 kb and 800–1200 °C in the FeO–MgO–Al2O3–SiO2 (FMAS) and CaO–FeO–MgOAl2O3–SiO2 system. Within error of the experimental data, orthopyroxene can be regarded as microscopically ideal. The effect of calcium on the Fe–Mg partitioning between garnet and orthopyroxene can be attributed to non-ideal Ca–Mg interaction in the garnet described by interactions terms: WCa–MgGt–WCa–FeGt = 1400 ± 500 cal/mol per site. The expression for geothermometer is as follows:

$$ T(K)=\left[\left(3740+1400\left({X}_{\mathrm{Ca}}^{\mathrm{Gt}}\right)+22.86\;P\;\left(\mathrm{kb}\right)\right)/\left(\mathrm{R}\ln {K}_D+1.96\right)\right] $$

Sen and Bhattacharya (1984)

They formulated the orthopyroxene–garnet geothermometer for which the values of ∆H°, ∆S°, and ∆V° are as follows: ∆H1000K = − 2.713 K cal/mol, ∆S1000K = − 0.787 cal/K mol, and ∆V298K = − 0.221 cal/bar. The expression of geothermometer is as follows:

$$ T(K)=\left[\left(2713+3300\left({X}_{\mathrm{Ca}}^{\mathrm{Gt}}\right)+195{\left({X}_{\mathrm{Fe}}-{X}_{\mathrm{Mg}}\right)}^{\mathrm{Gt}}+0.022\left(P-1\right)\right)/\left(1.9872\times \ln {K}_D+0.787+1.5\left({X}_{\mathrm{Ca}}^{\mathrm{Gt}}\right)\right)\right] $$

Their expression for geothermometer includes interaction parameters (W) for Ca as well as Fe–Mg mixing in garnet.

Lee and Ganguly (1984)

Lee and Ganguly (1984) experimentally determined the Fe–Mg exchange for garnet–orthopyroxene pairs. Empirical adjustment from natural data in terms of Ca and Mn leads to the following expression.

$$ T(K)=\left[\left(2187+1510{\left({X}_{\mathrm{Ca}}-{X}_{\mathrm{Mn}}\right)}^{\mathrm{Gt}}+8.6\times P\left(\mathrm{kb}\right)\right)/\left(\ln {K}_D+1.071\right)\right] $$

Perchuk et al. (1985)

The calibrated value for Fe–Mg exchange reaction given by Perchuk et al. (1981) is as follows: ∆H°970 K = − 4766 cal, ∆S°970 K = − 2.654 cal/K, and ∆V298 K = − 0.0234 cal/bar. Besides the non-ideal mixing of Ca in garnet, they also considered the interaction parameters of alumina and Fe+2–Mg mixing in orthopyroxene in their equation for geothermometer.

$$ T(K)=\left[\left(4766+2533\times {\left({X}_{\mathrm{Fs}}-{X}_{\mathrm{En}}\right)}^{\mathrm{Opx}}-5214\times {X}_{\mathrm{Al}}^{\mathrm{Opx}}+5704\times {X}_{\mathrm{Ca}}^{\mathrm{Gt}}+0.023\times P\left(\mathrm{bar}\right)\right)/\left(\mathrm{R}\ln {K}_D+2.65+1.86\times {\left({X}_{\mathrm{Fs}}-{X}_{\mathrm{En}}\right)}^{\mathrm{Opx}}+1.242\times {X}_{\mathrm{Ca}}^{\mathrm{Gt}}\right)\right] $$

where i = (Fe + Mg + Al/2) \( {X}_{\mathrm{Fe}}^{\mathrm{Opx}} \) = Fe/i; \( {X}_{\mathrm{Mg}}^{\mathrm{Opx}} \) = Mg/i and \( {X}_{\mathrm{Al}}^{\mathrm{Opx}} \) = (Al/2)/i.

Lee and Ganguly (1988)

They refined their earlier equation proposed in 1984, through their experimental data Fe–Mg exchange reaction between garnet and orthopyroxene.

$$ T(K)=\left[\left(1981+1509.66\times {\left({X}_{\mathrm{Ca}}-{X}_{\mathrm{Mn}}\right)}^{\mathrm{Gt}}+11.91\times P\left(\mathrm{kb}\right)\right)/\left(\ln {K}_D+0.97\right)\right] $$

Aranovich and Podlesskii (1989)

They proposed an empirical calibration for garnet–orthopyroxene assemblage:

$$ T(K)=\left[\left(\left(P-1\right)\times 0.02342+4766-A+{\left({X}_{\mathrm{Fe}}-{X}_{\mathrm{Mg}}\right)}^{\mathrm{Opx}}\times 2372-5204\times {X}_{\mathrm{Al}}^{\mathrm{Opx}}\right)/\left(\mathrm{R}\ln {K}_D+2.654+B+1.69\times {\left({X}_{\mathrm{Fe}}-{X}_{\mathrm{Mg}}\right)}^{\mathrm{Opx}}\right)\right] $$

where

$$ {\displaystyle \begin{array}{l}A=\left[-626\times {X}_{\mathrm{Ca}}^2-6642\times {X}_{\mathrm{Fe}}\times {X}_{\mathrm{Ca}}-8100\times {X}_{\mathrm{Mg}}\times {X}_{\mathrm{Ca}}+{X}_{\mathrm{Ca}}\times \left({X}_{\mathrm{Mg}}-{X}_{\mathrm{Fe}}\right)\times 1051.5\right]\\ {}B=\left[1.266\times {X}_{\mathrm{Ca}}^2+2.836\times {X}_{\mathrm{Fe}}\times {X}_{\mathrm{Ca}}+3.0\times {X}_{\mathrm{Mg}}\times {X}_{\mathrm{Ca}}+{X}_{\mathrm{Ca}}\times \left({X}_{\mathrm{Mg}}-{X}_{\mathrm{Fe}}\right)\times \left(-0.908\right)\right]\end{array}} $$

Perchuk and Lavrente’va (1990)

They proposed the following expression for garnet–orthopyroxene geothermometer based on experimental data and thermodynamics.

$$ T(K)=\left[\left(4066-347\times {\left({X}_{\mathrm{Mg}}-{X}_{\mathrm{Fe}}\right)}^{\mathrm{Opx}}-17484\times {X}_{\mathrm{Al}}^{\mathrm{Opx}}+5769{X}_{\mathrm{Ca}}^{\mathrm{Gt}}+23.42\times P\left(\mathrm{kb}\right)\right)/\left(1.987\times \ln {K}_D+2.143+0.0929\times {\left({X}_{\mathrm{Mg}}-{X}_{\mathrm{Fe}}\right)}^{\mathrm{Opx}}-12.8994\times {X}_{\mathrm{Al}}^{\mathrm{Opx}}+3.846\times {X}_{\mathrm{Ca}}^{\mathrm{Gt}}\right)\right] $$

where i = (Fe + Mg + Al/2); \( {X}_{\mathrm{Fe}}^{\mathrm{Opx}} \) = Fe/i; \( {X}_{\mathrm{Mg}}^{\mathrm{Opx}} \) = Mg/i and \( {X}_{\mathrm{Al}}^{\mathrm{Opx}} \) = (Al/2)/i.

Bhattacharya et al. (1991)

The expression for Gt-Opx geothermometer proposed by them is as follows:

$$ T(K)=\left[\right(1611+0.012\left(P-1\right)+906{X}_{\mathrm{Ca}}^{\mathrm{Gt}}+A+477\left(2\times {X}_{\mathrm{Mg}}^{\mathrm{Opx}}-1\right)/\left(\ln {K}_D+0.796\right)\Big] $$

The above expression can be used to estimate temperature from orthopyroxene–garnet pairs subject to the following conditions: (i) Grossularite contents in garnet should be < 30 mol%, (ii) spessartine content should be low (XMn < 0.05) such as Mg/Mg binaries in garnet could markedly affect the lnKD-T relations.

Lal (1993)

Lal (1993) proposed the empirical calibration for garnet–orthopyroxene thermometer. His equation is applicable for crustal rocks containing garnet with low Mn content (< 5 mol%) as well as mantle derived rocks (Cr2O2 < 5 wt%).

$$ T(K)=\left[\left(3367+\left(P-1\right)\times 0.024-A-948\times {\left({X}_{\mathrm{Fe}}-{X}_{\mathrm{Mg}}\right)}^{\mathrm{Opx}}-1950\;{X}_{\mathrm{Al}}^{\mathrm{Opx}}\right)/\left(1.987\times \ln {K}_D+1.634+B-0.34\times {\left({X}_{\mathrm{Fe}}-{X}_{\mathrm{Mg}}\right)}^{\mathrm{Opx}}\right)\right] $$

where

$$ {\displaystyle \begin{array}{l}A=\left[-1256\times {X}_{\mathrm{Mg}}^2-2880\times {X}_{\mathrm{Fe}}^2+8272\times {\left({X}_{\mathrm{Mg}}\times {X}_{\mathrm{Ca}}\right)}^{\mathrm{Gt}}+812\times {X}_{\mathrm{Ca}}^{\mathrm{Gt}}\times {\left({X}_{\mathrm{Mg}}-{X}_{\mathrm{Fe}}\right)}^{\mathrm{Gt}}+90{X}_{\mathrm{Ca}}^2-2340\times {X}_{\mathrm{Ca}}^{\mathrm{Gt}}\times \left({X}_{\mathrm{Fe}}+{X}_{\mathrm{Mg}}\right)-3047\times {X}_{\mathrm{Mg}}\times {X}_{\mathrm{Ca}}-1813\times {X}_{\mathrm{Fe}}\times {X}_{\mathrm{Ca}}+{X}_{\mathrm{Ca}}\times \left({X}_{\mathrm{Fe}}-{X}_{\mathrm{Mg}}\right)\times \left(-4498\right)\right]\\ {}B=\left[1.0\times {X}_{\mathrm{Mg}}^2+1.7\times {X}_{\mathrm{Fe}}^2-5.4\times {X}_{\mathrm{Fe}}\times {X}_{\mathrm{Mg}}-0.35\times {X}_{\mathrm{Ca}}\times \left({X}_{\mathrm{Mg}}-{X}_{\mathrm{Fe}}\right)+1.5\times {X}_{\mathrm{Ca}}^2+1.666\times {X}_{\mathrm{Ca}}\times \left({X}_{\mathrm{Fe}}+{X}_{\mathrm{Mg}}\right)+0.332\times {X}_{\mathrm{Fe}}\times {X}_{\mathrm{Ca}}+{X}_{\mathrm{Ca}}\times \left({X}_{\mathrm{Fe}}-{X}_{\mathrm{Mg}}\right)\times 1.516\right]\end{array}} $$

Carson and Powell (1997)

Carson and Powell (1997) proposed a geothermometer based on the Fe–Mg exchange between coexisting garnet and orthopyroxene stated in the form of pressure as

$$ P=1/22.86\left[\mathrm{RTln}\left(1-{X_{\mathrm{Mg}}}^{\mathrm{Gt}}-{X_{\mathrm{Ca}}}^{\mathrm{Gt}}\right)\left({X_{\mathrm{Mg}}}^{\mathrm{Opx}}\right)/\left(\ {X_{\mathrm{Mg}}}^{\mathrm{Gt}}\right)\left(1-{X_{\mathrm{Mg}}}^{\mathrm{Opx}}\right)\right]+1.96T-3740-1400{X_{\mathrm{Ca}}}^{\mathrm{Gt}} $$

where P is pressure in kilobar, and T is temperature in kelvin.

Aranovich and Berman (1995, 1997), Berman and Aranovich (1996))

Aranovich and Berman (1997) incorporated reversed phase equilibrium data, collected over the P–T range of 12–20 kbar at 850–1100 °C to define the solubility of Al2O3 in ferrosilite in equilibrium with almandine garnet giving an equation:

$$ T(K)={\displaystyle \begin{array}{c}-\varDelta {H^0}_a\hbox{--} 3{H^x}_{\mathrm{Fs}}-{H^x}_{\mathrm{Ok}}+{H^x}_{\mathrm{Alm}}\hbox{--} P\left(\varDelta {V^0}_a\hbox{--} 3{V^x}_{\mathrm{Fs}}-{V^x}_{\mathrm{Ok}}+{V^x}_{\mathrm{Alm}}\right)\\ {}------------------------------------\\ {}\mathrm{RlnKa}-\varDelta {S^0}_a-3{S^x}_{\mathrm{Fs}}\hbox{--} {S^x}_{\mathrm{Ok}}+{S^x}_{\mathrm{Alm}}\end{array}} $$

Nimis and Grütter (2010)

Nimis and Grütter (2010) has proposed the following formula using mineral compositions of the best equilibrated natural ultramafic rocks in combination with P and T values estimated by two-pyroxene thermometer and Al in Opx thermometer.

$$ T(K)=\left[1215+17.4P+1495\left({X_{\mathrm{Ca}}}^{\mathrm{Gt}}+{X_{\mathrm{Mn}}}^{\mathrm{Gt}}\right)/\left(\ \ln {K}_D+0.732\right)\right] $$

Valid GOPX geothermometry in granulites

Granulites are typical rocks of the earth’s middle to lower crust under high-temperature conditions. They are found as xenoliths in basaltic volcanic rocks, mainly within continental rifts, but most granulites occur as complexes or terranes in orogenic settings. Orogenic granulites display a wide compositional range (Harley 1989) and are known from a variety of collisional belts that formed during different episodes since the Archaean. Determination of bulk rock and mineral compositions, calculation of peak equilibration conditions, and dating of orogenic granulites that provide us with important constraints on the thermal and chemical structure of the earth continental crust at different geological times. Harley (1989) showed that equilibration conditions deduced from the natural orogenic granulites cover a wide range. In particular, pressures are highly variable, and the granulite field may be divided into a low pressure, a medium pressure, and a high pressure facies according to Green and Ringwood (1967). Peak temperatures for many granulite terrain scatter around 800 °C (Bohlen 1987), but an increasing number of ultra-high-temperature granulite complexes (900–1100 °C and 0.7–1.3 GPa) are being recognized (e.g., Dasgupta and Sengupta 1995; Klemd and Bröcker 1999; Hokada 2001).

To test the validity and applicability of GOPX thermometry, we have collated 51 samples from the literature all over the world. Selection of samples fit the following criteria (Wu and Cheng 2006):

  1. 1.

    There is a clear description of textural equilibrium among garnet and orthopyroxene in the literature.

  2. 2.

    There is detailed and high-quality electron microprobe analyses of the minerals involved, at least SiO2, TiO2, Al2O3, FeO, MnO, MgO, CaO, Na2O, and K2O, and there stoichiometry of the analyzed minerals was confirmed.

  3. 3.

    Core composition of garnet and rim composition of orthopyroxene have been mostly used. If there is growth zoning in garnet, only the rim composition was used, and accordingly, only the rim composition of matrix orthopyroxene has been used.

  4. 4.

    Data is used where elemental oxide totals for the minerals analyses were 100 ± 1.5%.

The 51 samples listed in the table (supplementary data associated with this article can be obtained from the authors) fall in the following mineral composition ranges: XFe = 0.077–0.6918 (mostly between 0.30 and 0.50), XMg = 0.4090–0.7473 (mostly between 0.45 and 0.65), XAl = 0.0011–0.1422 (mostly between 0.01 and 0.10) in orthopyroxene; XFe = 0.222–0.7448 (mostly between 0.45 and 0.60), XMg = 0.071–0.6325 (mostly between 0.10 and 0.30), XMn = 0.0–0.182 (mostly between 0.01 and 0.03), and XCa = 0.01–0.2493 (mostly between 0.10 and 0.20) in garnet.

Result and discussion

For the validation of the software (Thomas 2003; Thomas and Paudel 2016) and comparative study of different models, 51 pairs of data have been processed through this software and results are shown in Table 1 in the Electronic Supplementary Material in a fixed format with temperature based on different workers along with KD, lnKD, XMg, XFe, and XAl for orthopyroxene and XFe, XMg, XMn, and XCa for garnet. To observe the effect of XFe, XMg, and XAl of orthopyroxene and XFe, XMg, XMn, and XCa of garnet on KD, we have plotted the graphs between XFe, XMg, and XAl of orthopyroxene vs KD and XFe, XCa, XMn, and XMg of garnet on KD (Fig 2a-g: supplementary figures associated with this article can be obtained from the authors) respectively and observed that XFe(OPX) and XMg(OPX) are showing a horizontal trend line as XFe(OPX) = 0.001/KD + 0.400 with R2 = 0.000 having a compositional range of 0.077–0.6918 (mostly between 0.30 and 0.50); XMg(OPX) = 0.004/KD + 0.540 with R2 = 0.000, having a compositional range of 0.4090–0.7473 (mostly between 0.45 and 0.65) and XAl(OPX) is showing negative slope as XAl(OPX) = − 0.006/KD + 0.059 with R2 = 0.015, having a compositional range of 0.0011–0.1422 (mostly between 0.01 and 0.10). In case of garnet, XFe(GT), XCa(GT), and XMn(GT) are showing positive slope as XFe(GT) = 0.020/KD + 0.518 with R2 = 0.02, having a compositional range of 0.222–0.7448 (mostly between 0.45 and 0.60); XCa(GT) = 0.026/KD + 0.018 with R2 = 0.069, having a compositional range of 0.01–0.2493 (mostly between 0.10 and 0.20) and XMn(GT) = 0.006/KD + 0.007 with R2 = 0.024, having a compositional range of 0–0.182 (mostly between 0.01–0.03) while XMg(GT) is showing negative slope as XMg(GT) = − 0.058/KD + 0.475 with R2 = 0.105, having a compositional range of 0.071–0.6325 (mostly between 0.10 and 0.30).

A comparison of the calculated lnKD and 1/T for different geothermometric models has been done. The plots of lnKD vs 1/T are shown in Fig. 1a–n. The data selected in this way was used to check the temperature dependence of the distribution coefficient: Aranovich and Berman (1997), Fig. 1a graph of lnKD vs 1/T has been plotted as lnKD = − 2907/T (°C) − 1.502 with R2 = 0.947; Raith et al. (1983), Fig. 1b as lnKD = 983.3/T (°C) − 0.053 with R2 = 0.79; Harley (1984), Fig. 1c as lnKD = 1061/T (°C) − 0.434 with R2 = 0.720; Nimis and Grütter (2010), Fig. 1d as lnKD = 1209/T (°C) − 0.336 with R2 = 0.682; Sen and Bhattacharya (1984), Fig. 1e as lnKD = 832.9/T (°C) + 0.000 with R2 = 0.654; Carson and Powell (1997), Fig. 1f as lnKD = − 192.9/T (°C) + 1.565 with R2 = 0.600; Lee and Ganguly (1984), Fig. 1g as lnKD = 1253/T (°C) − 0.387 with R2 = 0.577; Lee and Ganguly (1988), Fig. 1h as lnKD = 1120/T (°C) − 0.287 with R2 = 0.576; Bhattacharya et al. (1991), Fig. 1i as lnKD = 1032/T (°C) − 0.318 with R2 = 0.505; Lal (1993), Fig. 1j as lnKD = 591.2/T (°C) + 0.303 with R2 = 0.280; Aranovich and Podlesskii (1989), Fig. 1k as lnKD = 410/T (°C) + 0.569 with R2 = 0.216; Perchuk and Lavrente’va (1990), Fig. 1l as lnKD = 235.4/T (°C) + 0.775 with R2 = 0.206; Perchuk et al. (1985), Fig. 1m as lnKD = 336.1/T (°C) + 0.686 with R2 = 0.167; and Dahl (1980), Fig. 1n as lnKD = 76.43/T (°C) + 1.0204 with R2 = 0.080.

Fig. 1
figure 1figure 1figure 1

The plots of lnKD vs 1/T. a lnKD vs 1/T (Aranovich and Berman 1997). b lnKD vs 1/T (Raith et al. 1983). c Relationship between lnKD and 1/T (Harley 1984). d Relationship between lnKD and 1/T (Nimis et al. 2009). e Relationship between lnKD and 1/T (Sen and Bhattacharya 1984). f Relationship between lnKD and 1/T (Powell et al. 1997). g Relationship between lnKD and 1/T (Lee and Ganguly 1984). h Relationship between lnKD and 1/T (Lee and Ganguly 1988). i Relationship between lnKD and 1/T (Bhattacharya et al. 1991). j Relationship between lnKD and 1/T (Lal 1993). k Relationship between lnKD and 1/T (Aranovich and Podlesskii 1989). l Relationship between lnKD and 1/T (Perchuk and Lavrente’va 1990). m Relationship between lnKD and 1/T (Perchuk et al. 1985). n Relationship between lnKD and 1/T (Dahl 1980)

Full size image

Although, the data’s used for the validation has given a very prominent range of temperature in order to the GOPX thermometer, except a few like Fareeduddin et al. (1994), Pal and Bose (1997), Sanjeev et al. (2004), Joshi et al. (1993), Mouri et al. (1996), Nurminen and Ohta (1993), and Smithies and Bagas (1996) which are of retrograde metamorphism showing little lowering of temperature.

On the basis of different plots, it is observed that Aranovich and Berman (1995, 1997), Berman and Aranovich (1996)), Raith et al. (1983), Harley (1984), Nimis and Grütter (2010), and Sen and Bhattacharya (1984) are showing very good relation between lnKD and 1/T and maximum points are coming in best fit lines and have high regression values.

Conclusion

On the basis of applying the 14 versions of the GOPX thermometers to the empirical data collected from literature considered for this comparative study, we conclude that the five GOPX thermometers (Aranovich and Berman 1995, 1997; Berman and Aranovich 1996; Raith et al. 1983; Harley 1984; Sen and Bhattacharya 1984; and Nimis and Grütter 2010) are nearly equally valid. These models are showing the highest regression values and maximum points are coming in best fit lines. So, these models can be considered the most appropriate ones to be used for the calculation of temperature.

However, Aranovich and Berman (1995, 1997), Berman and Aranovich (1996) is the best among them as the regression correlation coefficient value, R2, is close to 1 which indicates that the maximum points are coming in the best fit line. Therefore, the temperature value obtained by the Aranovich and Berman (1995, 1997), Berman and Aranovich (1996) model is more accurate compared to others.

It is necessary to do further experimental work to calibrate GOPX thermometer using garnet and orthopyroxene with chemical compositions comparable to natural minerals. However, it is equally important to improve our knowledge of the various chemical interactions within minerals such as garnet and orthopyroxene, so that we can improve the activity models for these minerals. It is also suggested that the ongoing papers in calibrating geothermobarometry be constructed in a way that is friendly to readers, that is, the formalisms be explicitly expressed. A writing style in which no clear expression of a thermometer is presented should be avoided in order to provide an easy way for geologists, especially field geologists, to use for geothermometers conveniently.