## 3 Results Analysis  
### NS04 Cavity Test Results  
In the first test, a total of approximately 100 h of cooling was performed, and the vacuum in the cryostat was as low as \(1\times 10^{-7}\) Pa at low temperatures. The end of the beam tube was cooled down to about 5.4 K (\(T_{1}\)), and the ellipsoidal portion was cooled down to 4.0 K (\(T_{2}\)). During the initial RF test, the cavity temperature rose fast after loading RF power, so we collected data points when adding instantaneous power. The average temperature (\(T_{avg}=(T_{1}+T_{2})/2\)) of the cavity was cooled to 5 K before each power load, and \(Q_{0}\) and \(E_{acc}\) were measured immediately after loading power, followed by power shutdown. When the cavity was re-cooled to 5 K, the power was increased and the above operation was repeated. Due to power limitations, the maximum instantaneous \(E_{acc}\) measured during the test was 5.09 MV/m and the instantaneous \(Q_{0}\) was 1.2 \(\times\) 10\({}^{9}\). After reaching the maximum transient \(E_{acc}\), the power loading was maintained and the cavity temperature was finally stabilized at 9 K for \(T_{1}\) and 6 K for \(T_{2}\), at which time \(E_{acc}\) was 1.25 MV/m and \(Q_{0}\) was 2.1 \(\times\) 10\({}^{8}\), and the RF loss in the cavity (\(P_{l}\)) was 0.936 W. The cavity was operated for more than 10 min at this operating point.  
After the test, a temperature cycle through \(T_{c}\) was performed by turning off the cryocoolers until \(T_{1}\) and \(T_{2}\) exceeded 25 K. Then, the cryocoolers were power-cycled on and off to slowly lower the cavity temperature to 17 K, below which the cryocoolers were kept on. The cavity was cooled to the base temperature, with \(T_{1}=5.4\) K, \(T_{2}=4.0\) K. During this cooling process, the thermal gradient at either side of the equator along the axial direction of the cavity temperature at 17-19 K was reduced from the original \(\sim\)40 K/m to 12 K/m. A subsequent RF test in CW mode was performed, which involved loading power and waiting for the cavity to thermally equilibrate, followed by a slow increase in power, with each equilibrium point stabilizing for more than 10 min. The \(E_{acc}\) of the cavity reached a maximum value of 1.75 MV/m when the forward power was 1.37 W. At this time, the  
Figure 11: (**a**) Cu sample and PPMS sample stands, (**b**) installation for thermal conductivity measuring of Al plates, (**c**) resistance measurements and electrical conductivity calculations for Cu sample, (**d**) thermal conductivity of Cu samples and 5N Al plates used in experiments.  
was 6.7 \(\times\) 10\({}^{8}\), the \(P_{l}\) was 0.573 W, and the \(T_{2}\) increased to 5.5 K. Continuing to increase the power, the \(Q_{0}\) decreased rapidly due to the increased temperature of the cavity, and the \(E_{acc}\) decreased as well. When \(P_{l}\) was increased to 0.937 W, \(E_{acc}\) and \(Q_{0}\) reached 1.31 MV/m and 2.3 \(\times\) 10\({}^{8}\), respectively, and the \(T_{2}\) increased to 5.9 K. Figure 12 shows the \(Q_{0}-E_{acc}\) curves for the NS04 cavity for these two tests, as well as the curve for the vertical test in 4.2 K LHe (for comparison, only test points below 5 MV/m are plotted).