Title: Setup of high resolution thermal expansion measurements in closed cycle cryostats using capacitive dilatometers Show Chinese title

Title: 高分辨率热膨胀测量在闭循环制冷机中使用电容式膨胀计的设置

Abstract: We present high resolution thermal expansion measurement data obtained with high relative sensitivity of $\Delta$L$/$L = 10$^{-9}$ and accuracy of $\pm$2$\%$ using closed cycle refrigerators employing two different dilatometers. The data quality is in excellent agreement with those obtained using wet liquid helium based systems, demonstrating great technological possibilities for future thermal expansion measurements in view of the depleting resource of liquid helium. The cryogenic environment was achieved using two different cryostats that use pulse tube and Gifford-Mcmahon coolers as the cryocoolers. Both the dilatometers employ a spring movement for achieving the parallel movement of the capacitor plates. $Dilatometer \#1$ was built in-house based on a published design while $dilatometer \#2$ was obtained commercially. Cell calibration for $dilatometer \#1$ was done using copper and minimal deviation of the cell effect from the published values were found. Linear thermal expansion coefficient $\alpha$ obtained using both dilatometers was evaluated using two different techniques, namely, numerical differentiation and derivative of a polynomial fit. The resultant $\alpha$ obtained for metals silver and aluminium showed excellent match with published values obtained on systems using wet cryostats. Finite element method simulations were performed for understanding the spring movement in each dilatometer using which the effect of different forces$/$pressures on the displacement of the spring was studied. Finally, we report thermal expansion measurements done on single crystals of two high temperature superconductors YBa$_2$Cu$_{3-x}$Al$_x$O$_{6+\delta}$ and Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ along the c-axis and found very good match with published data obtained using wet liquid helium based cryostats. Show Chinese abstract

Abstract: 我们展示了使用两种不同膨胀计获得的高分辨率热膨胀测量数据，其相对灵敏度为$\Delta$L$/$L = 10$^{-9}$，精度为$\pm$2$\%$，采用闭循环制冷机。 数据质量与使用湿液氦系统的测量结果高度一致，表明在液氦资源日益减少的情况下，未来热膨胀测量具有巨大的技术潜力。 低温环境通过两种不同的恒温器实现，这两种恒温器使用脉冲管和Gifford-Mcmahon冷却器作为制冷器。 两种膨胀计均采用弹簧运动以实现电容器板的平行运动。 $Dilatometer \#1$是基于已发表的设计自行研制的，而$dilatometer \#2$是从市场上购买的。 对$dilatometer \#1$进行了单元校准，使用铜进行校准，发现单元效应的偏差最小。 使用两种不同的技术，即数值微分和多项式拟合的导数，评估了通过两种膨胀计获得的线性热膨胀系数$\alpha$。 对于金属银和铝得到的结果$\alpha$与使用湿式低温恒温器系统获得的 published values 非常吻合。 进行了有限元方法模拟，以了解每个膨胀计中的弹簧运动，从而研究不同力$/$压力对弹簧位移的影响。 最后，我们报告了对两种高温超导体单晶的热膨胀测量结果，分别是YBa$_2$Cu$_{3-x}$Al$_x$O$_{6+\delta}$和 Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$沿c轴的测量，并发现与使用湿液氦制冷装置获得的已有数据非常吻合。