## 3 Results and Discussion
### Thermodynamic Properties  
The thermodynamic properties of Co\({}_{2}\)Zr and Co\({}_{2}\)Ti are analyzed with plane wave pseudo-potential density functional theory (DFT) based on quasi-harmonic Debye model (QHDM) [37] and generalized gradient approximation (GGA) method, concentrating on their Debye temperature \(\Theta\), Heat capacity C\({}_{\rm V}\) and thermal expansion coefficient \(\alpha\). The quasi-harmonic Debye model (QHDM) is used to study the physical properties of intermetallic compounds under high temperature [24; 38]. Figure 2 shows that the V/V\({}_{0}\) of Co\({}_{2}\)Zr and Co\({}_{2}\)Ti are functions of pressure at 0, 500, 1000, and 1500 K. Under a certain temperature, pressure increases while the V/V\({}_{0}\) decreases monotonously. When the pressure is over 10 GPa, Co\({}_{2}\)Zr decreases faster than Co\({}_{2}\)Ti. When P\(<\)10 GPa, the two crystals have the same V/V\({}_{0}\), and will not be affected by temperature. When P \(>\) 10 GPa, V/V\({}_{0}\) decreases with increasing temperature at a certain pressure but the amount of variation is very small. What is more, the V/V\({}_{0}\) of Co\({}_{2}\)Ti is always larger than that of Co\({}_{2}\)Zr, which indicates the pressure has a greater impact on volume of Co\({}_{2}\)Ti than that of Co\({}_{2}\)Zr. Figure 3 illustrates the relationship between Bulk modulus B and pressure at a certain temperature (0 and 1500 K), as well as the relationship of bulk modulus B and temperature at 0 GPa. As shown in Figure 3, under the same temperature and pressure, the B of Co\({}_{2}\)Ti is always larger than Co\({}_{2}\)Zr. With the increasing of pressure, the B of these two crystals increases in the same trend under a certain temperature. At a certain pressure, the B value is inversely proportional to the temperature, and Co\({}_{2}\)Ti cells more drastically than Co\({}_{2}\)Zr does within the same temperature difference. Therefore, for both Co\({}_{2}\)Zr and Co\({}_{2}\)Ti, increasing the pressure and decreasing the temperature have the same effect on the B value. Figure 1: The C\({}_{\rm ij}\), E, G and B\({}_{\rm s}\) of (**a**) Co\({}_{2}\)Zr and (**b**) Co\({}_{2}\)Ti versus pressure at 0 K.  
In the Debye model, the vibration of the elastic wave of the crystal is the main factor contributing to the heat capacity that reflects the ability to absorb heat from the surrounding environment. Figure 4a-c, respectively, show how the C\({}_{\rm V}\) of Co\({}_{2}\)Zr and Co\({}_{2}\)Ti vary with temperature under certain pressures (0, 50, and 100 GPa). Due to the anharmonic approximations of the Debye model used, the C\({}_{\rm V}\) values of Co\({}_{2}\)Zr and Co\({}_{2}\)Ti rely on temperature and pressure when T < 1300 K and a small amount of temperature difference will cause a great change in heat capacity. At a high temperature, the curve tends to be horizontal, which means that the heat capacity is close to constant, and the effects of temperature and pressure can be ignored. The heat capacity increases with the increasing temperature, but decreases with the increasing pressure, which indicates that the decrease in temperature and the increase in pressure have the same effect on the heat capacity of C\({}_{\rm V}\). In addition, under the same conditions, the heat capacity of Co\({}_{2}\)Zr is always greater than Co\({}_{2}\)Ti. Figure 3: (**a**) The bulk modulus B versus pressure P at 0 and 1500 K, and (**b**) the bulk modulus B\({}_{0}\) versus temperature T from 0 up to 1500 K at P = 0 GPa. Figure 2: The V/V\({}_{0}\) of Co\({}_{2}\)Zr and Co\({}_{2}\)Ti versus pressure from 0 to 100 GPa at 0, 500, 1000 and 1500 K.  
In the Debye model, the Debye temperature \(\Theta\) often reflects the vibrational frequency of crystal atoms [38]. Figure 5 illustrates the change curve of Debye temperature \(\Theta\) that varies with pressure at the given temperature (0, 1500 K). As the pressure goes up, the Debye temperatures of two crystals increase monotonously in the similar trend.