Characterization of the pressure coefficient of manganin and temperature evolution of pressure in piston-cylinder cells

We report measurements of the temperature- and pressure-dependent resistance, $R(T,p)$, of a manganin manometer in a $^4$He-gas pressure setup from room temperature down to the solidification temperature of $^4$He ($T_\textrm {solid}\sim$ 50 K at 0.8 GPa) for pressures, $p$, between 0 GPa and $\sim$0.8 GPa. The same manganin wire manometer was also measured in a piston-cylinder cell from 300 K down to 1.8 K and for pressures between 0 GPa to $\sim$2 GPa. From these data, we infer the temperature and pressure dependence of the pressure coefficient of manganin, $\alpha(T,p)$, defined by the equation $R_p = (1+\alpha p) R_0$ where $R_0$ and $R_p$ are the resistance of manganin at ambient pressure and finite pressure, respectively. Our results indicate that upon cooling $\alpha$ first decreases, then goes through a broad minimum at $\sim$120 K and increases again towards lower temperatures. In addition, we find that $\alpha$ is almost pressure-independent for $T\gtrsim$60 K up to $p\sim$2 GPa, but shows a pronounced $p$ dependence for $T\lesssim$60K. Using this manganin manometer, we demonstrate that $p$ overall decreases with decreasing temperature in the piston-cylinder cell for the full pressure range and that the size of the pressure difference between room temperature and low temperatures ($T=1.8$ K), $\Delta p$, decreases with increasing pressure. We also compare the pressure values inferred from the magnanin manometer with the low-temperature pressure, determined from the superconducting transition temperature of elemental lead (Pb). As a result of these data and analysis we propose a practical algorithm to infer the evolution of pressure with temperature in a piston-cylinder cell.