## 2 Measurement Principles and Device
### The Structure of a Cryogenic Vacuum Apparatus
#### 2.2.3 Temperature Stability  
The measurement stability and repeatability of emissivity are closely related to the temperature control accuracy of the blackbody and the sample. To enhance temperature stability, a traditional commercial temperature controller was abandoned in favor of a customized high-precision PID controller (PID stands for proportion, integral, and differential coefficient, which represent the three types of control algorithms). By individually setting parameters for each temperature point, ultra-high temperature stability was achieved.  
The target temperatures were set to 40 \({}^{\circ}\)C, 60 \({}^{\circ}\)C, 80 \({}^{\circ}\)C, and 100 \({}^{\circ}\)C, respectively. The resulting temperature stability curves for the blackbody and the sample heating stage are shown in Figures 5 and 6:  
Figure 4: Results of the simulations presented at \(-20\)\({}^{\circ}\)C. (**a**) Schematic diagram of finite element analysis mesh division for heating stage temperature. (**b–d**) Temperature distribution at different viewing angles.  
As can be seen from Figures 6 and 7, the temperatures of both the blackbody and the sample heating stage are extremely stable at each temperature point, with a maximum variation of only \(\pm\)2 mK.  
Figure 5: Temperature stability curve of the blackbody.  
Figure 6: Temperature stability curve of the sample heating stage.  
According to the uncertainty propagation formula, assuming that the above physical quantities are uncorrelated, the combined relative uncertainty can be expressed as below:  
\[\frac{u_{\varepsilon_{r}}}{\varepsilon_{r}}=\sqrt{\begin{array}{l}\frac{1}{ \varepsilon_{b}^{2}}\cdot u_{b}^{2}+\frac{1}{\varepsilon_{r}^{2}}\cdot u_{l}^{2 }+\frac{C_{2}^{2}}{\varepsilon_{1}^{2}T_{1}^{4}}\cdot\frac{\varepsilon_{2}^{ \frac{C_{2}}{\varepsilon_{1}}}}{(\frac{C_{2}^{\frac{C_{2}}{\varepsilon_{1}}}}{ \varepsilon_{1}^{\frac{C_{2}}{\varepsilon_{1}}}}-1)^{2}}\cdot u_{T_{1}}^{2}+ \frac{C_{2}^{2}}{\varepsilon_{2}^{\frac{C_{2}}{\varepsilon_{1}}}}\cdot\frac{ \varepsilon_{2}^{\frac{C_{2}}{\varepsilon_{2}}}}{(\varepsilon^{\frac{C_{2}}{ \varepsilon_{1}}}-1)^{2}}\cdot u_{T_{2}}^{2}\\ +\frac{C_{2}^{2}}{\lambda^{4}}(\frac{1}{12}.\frac{C_{2}}{\varepsilon^{\frac{C_ {2}}{\varepsilon_{1}}}-1}-\frac{1}{11}.\frac{C_{2}^{\frac{C_{2}}{\varepsilon_{1 }}}}{\varepsilon^{\frac{C_{2}}{\varepsilon_{1}}}-1})^{2}\cdot u_{\lambda}^{2} \end{array}} \tag{15}\]  
Furthermore, since \(u_{T1}\) = \(u_{T2}\), we can let \(u_{T}\) = \(u_{T1}\) = \(u_{T2}\). When \(T\) = \(T_{1}\) = \(T_{2}\),  
\[\frac{u_{\varepsilon_{r}}}{\varepsilon_{r}}=\sqrt{\frac{1}{\varepsilon_{b}^{2} }\cdot u_{b}^{2}+u_{1}^{2}\frac{1}{\varepsilon_{1}^{2}}}+(\frac{2C_{2}^{2}}{ \lambda^{2}T^{4}}.\frac{\varepsilon^{\frac{2C_{2}}{\lambda T}}}{(\frac{C_{2}}{ \varepsilon^{\frac{C_{2}}{\lambda T}}}-1)^{2}})\cdot u_{T}^{2}} \tag{16}\]  
From the formula, it can be seen that the uncertainty of the measured emissivity result is directly related to the uncertainty of the emissivity value of the measured blackbody, the uncertainty of the directly measured emissivity result, and the temperature.