Question

how to find thermal coefficient

Answer 1

, the simplest experimental way to find the (linear or volume) coefficients of thermal expansion for any solid material is TMA or Thermal Mechanical Analysis. Small (a few mm in size) single crystals are ideal samples for such TMA measurements. One needs (1) to put a sample onto a TMA instrument measuring platform, (2) lower the instrument pirex probe having flat tip onto the sample surface under a minimal applied static load (a few mN are sufficient); (3) to place the assembly into the TMA furnace; (4) to select the heating program and (5) to accomplish the heating run. The so obtained TMA-curve (usually – a line steadily raising with temperature respective the horizontal baseline) is all one needs for calculating the expansion coefficients α (T1-Tref.). The average expansion coefficient for the temperature interval Tref ...T1 is mathematically defined as: α (T1-Tref.) = [dL/L0 (T1) - dL/L0 (Tref.)] / (T1 – Tref.), where L0 is the sample length measured at ambient temperature; T1 = upper temperature limit; Tref. = lower temperature limit named as reference temperature; dL/L0 – is the sample length (or thickness) change relative to L0.

Excellent TMA instruments (with nanometers linear displacement resolution) are produced for many years by, e.g., Perkin-Elmer, TI Instruments, Netzsch and others. Nowadays the TMA makers software usually permits to extract a lot of useful info (physical alpha curve and values, technical alpha curve and values, etc.) from the single TMA run. Also, the TMA curve will register the phase transitions (if any) within the sample in the temp range chosen. Good Luck!