## 4 Discussion
### Simulation of RF Performance of NS03 Cavity  
To examine the conduction-cooling RF performance of NS03 cavity when the film was not damaged or contaminated, electromagnetic-thermal simulations were performed using the following boundary conditions: (a) \(P_{l}\) as a function of \(H_{\tau}\), \(R_{res}\), and \(T_{s}\) (Nb\({}_{3}\)Sn film temperature), as shown in Equation (5) [33]; (b) the temperature-dependent load curve of SRDK-418D cryocooler provided by the manufacturer; and (c) 0.1 W heat leakage from RF cables and the base.  
\[P_{l}=\frac{1}{2}\times R_{s}\iint|H_{\tau}|^{2}dS=\frac{1}{2}\times\left(9.4 \times 10^{-5}\frac{f2}{T}e^{-\frac{2\gamma_{T}}{T}}+R_{res}\right)\times\iint|H_{ \tau}|^{2}dS \tag{5}\]  
There are a few assumptions made in the simulation: \(R_{BCS}\) is calculated using Equation (3) and does not change with \(E_{acc}\); the \(R_{res}\) at different \(E_{acc}\) is from the test in 4.2 K LHe; the contact thermal resistance (\(R_{c}\)) between Al and Al and Al and Cu is 2 \(\times\) 10\({}^{-4}\) (K\(\cdot\)m\({}^{2}\))/W and does not change with temperature. Since \(R_{res}\) comes from a limited number of gradient points, the goal of our simulations is to examine whether the conduction-cooled NS03 cavity can be stabilized at these gradient points. We calculated two cases, where the \(R_{res}\) under conduction cooling is exactly the same as the test in LHe and the \(R_{res}\) doubles due to insufficiently controlled cooldown or other causes. The \(Q_{0}-E_{acc}\) curves obtained for both cases are shown in Figure 18.  
Figure 19 contains the results of the simulation, showing that the maximum \(E_{acc}\) will be at least 10.86 MV/m and 9.91 MV/m in case 1 and case 2, and a larger \(E_{acc}\) will not be  
Figure 17: Contamination or defect in the vicinity of the cavity iris: before cold spraying (a), after alcohol sonication and HPR (b).  
Figure 18: \(Q_{0}-E_{acc}\) curves of Nb\({}_{3}\)Sn cavity obtained by vertical testing in 4.2 K LHe and simulations of different conditions. The diamonds are test results in 4.2 K LHe. The circles are case 1: \(R_{res}\) is identical to that measured in LHe. The triangles are case 2: \(R_{res}\) is twice that measured in LHe. The color of the points indicates the average body temperature of the cavity.  
able to converge the solution, i.e., the cavity will not be able to operate at the next gradient point. At low \(E_{acc}\), \(P_{l}\) is very small, and the \(T_{avg}\) is lower than 4.2 K. Therefore, in case 1, the \(Q_{0}\) will be higher than that in LHe. With the increase in \(E_{acc}\) and the rise in \(T_{avg}\), \(Q_{0}\) will decrease gradually, and when the \(T_{avg}\) is more than 6 K, the increase in \(P_{l}\) will be significantly faster than that of the increase in the cooling capacity, which leads to a sharp decrease in the \(Q_{0}\). In the simulation results of case 1 and case 2, the \(T_{avg}\) is higher than that measured at the same \(P_{l}\). This may be because the thermal conductivity of the Cu used in the simulation was measured on sample pieces that were sprayed without \(N_{2}\) cooling and with a lower purity of 99.9% Cu powder used. The actual Cu shell on the NS03 cavity has a lower oxygen content and higher thermal conductivity than the sample pieces. Figure 19 gives the temperature distribution of the cavity operating at the 9.91 MV/m in case 2, with a \(P_{l}\) of 8.1 W, \(T_{avg}\) of 5.7 K, and a maximum temperature difference of 0.18 K over the cavity. Since the Cu shell enhances the thermal stability of the cavity, and we have achieved the same temperature gradient as in LHe cooling with the synergistic control of the four heaters, we believe that this treatment on NS03 allows the Nb\({}_{3}\)Sn cavity to operate at 10 MV/m.