Heat capacity of Mg ₃ Sb ₂ , Mg ₃ Bi ₂ , and their alloys at high temperature

The thermoelectric figure of merit reported for n-type Mg ₃ (Sb,Bi) ₂ compounds has made these materials of great engineering significance, increasing the need for accurate evaluations of their thermal conductivity. Thermal conductivity is typically derived from measurements of thermal diffusivity and determination of the specific heat capacity. The uncertainty in this method (often 10% or more) is frequently attributed to measurement of heat capacity such that estimated values are often more accurate. Inconsistencies between reported thermal conductivity of Mg ₃ (Sb,Bi) ₂ compounds may be attributed to the different values of heat capacity measured or used to calculate thermal conductivity. The high anharmonicity of these materials can lead to significant deviations at high temperatures from the Dulong-Petit heat capacity, which is often a reasonable substitute for measurements at high temperatures. Herein, a physics-based model is used to assess the magnitude of the heat capacity over the entire temperature range up to 800 K. The model agrees in magnitude with experimental low-temperature values and reproduces the linear slope observed in high-temperature data. Owing to the large scatter in experimental values of high-temperature heat capacity, the model is likely more accurate (within ±3%) than a measurement of a new sample even for doped or alloyed materials. It is found that heat capacity for the solid solution series can be simply described (for temperatures: 200K≤T≤800K) by the polynomial equation: c _p [Jg ⁻¹ K ⁻¹ ]=[Formula presented](1+1.3×10 ⁻⁴ T−4×10 ³ T ⁻² ),where 3NR=124.71Jmol ⁻¹ K ⁻¹ , M _W is the molecular weight [gmol ⁻¹ ] of the formula unit being considered, and T is temperature in K. This heat capacity is recommended to be a standard value for reporting and comparing the thermal conductivity of Mg ₃ (Sb,Bi) ₂ including doped or alloyed derivatives. A general form of the equation is given which can be used for other material systems.