## 4 Discussion
### Equilibrium and Nonequilibrium among Minerals and Coexisting Fluid  
Equilibrium between minerals A and B, or among minerals with mineral-forming fluids, is important in the ore formation environment. Equilibrium and nonequilibrium in mineral formations were studied on the epithermal quartz-calcite veins at the epithermal Kushikino deposit in Kyushu, Japan [22]. Assuming that quartz and calcite precipitate from a fluid in equilibrium, they are supposed to have the texture of the assemblage of fine-grained crowded quartz and calcite grains under the microscope as a working hypothesis. In contrast, when quartz and calcite precipitate from a fluid not in equilibrium, they are supposed to have the assemblage of unevenly distributed coarse-grained quartz and/or calcite under the microscope, in which it is common to lack one mineral. Let us denote the former, \"equilibrium assemblage,\" and the latter, \"nonequilibrium assemblage.\" Figure 6 shows a set of \(\delta^{18}\)O values for quartz and/or calcite bearing small pieces from a hand specimen. In the equilibrium assemblage (solid circles in Figure 6), both quartz and calcite precipitated in equilibrium with the coexisting fluid. The equilibrium assemblage is on the isotopic equilibrium line (broken line in Figure 6). The broken line denotes the temperature dependence of the equilibrium fractionation of oxygen isotope for quartz and calcite. The \(\delta^{18}\)O value of the Kushikino hydrothermal fluid was \(-7.0\%\)[22], which is stable throughout the fluid evolution, and it was assumed that the fluid was of meteoric water origin. The temperature on the broken line denotes the oxygen isotope equilibrium temperature for the quartz-calcite pair calculated using a corresponding calibration equation (either quartz-water [20] or calcite-water [23]).  
Figure 6: Plot of the \(\delta^{18}\)O values of quartz vs. the \(\delta^{18}\)O values of calcite from the veins of the Kushikino deposit (produced from the data in [22]). The solid and open circles refer to the equilibrium and nonequilibrium assemblages, respectively. The open squares represent quartz or calcite in a small piece from the hand specimen, resulting in only one \(\delta^{18}\)O value in the analyzed sample (W222V-1 has only quartz, but W222V-2 has quartz and calcite, both in the same hand specimen). The data points from the same hand specimen are connected by a tie line, implying that there are two mineralization stages in a hand specimen. The \(\delta^{18}\)O values of quartz that precipitates from the fluid having \(-7.0\%\) are calculated as a function of temperature using the experimentally determined equilibrium fractionation factors between quartz and water [20]. Similarly, the \(\delta^{18}\)O values of calcite that precipitates from the fluid having \(-7.0\%\) are calculated as a function of temperature using the equilibrium fractionation factors between calcite and water [23]. Then, the two calculations are merged into the broken line. When a sample containing quartz and calcite is on the broken line, the quartz and calcite are considered to be in equilibrium with the fluid.  
A set of \(\delta^{18}\)O values for the equilibrium assemblage of quartz and calcite moves toward the upper right on the isotopic equilibrium line with decreasing temperature. Conversely, the \(\delta^{18}\)O values of quartz and/or calcite are plotted off the line for the nonequilibrium assemblage (open symbols in Figure 6). They precipitated separately at different stages; thus, it was considered that the temperature or forming fluid for quartz and calcite differed. Thus, the isotopic measurements confirmed that it is possible to distinguish the equilibrium assemblage from the nonequilibrium assemblage via the microscopic observation of veins (Figure 6), which proves the working hypothesis.