# Nb\({}_{3}\)Sn Cavities Coated by Tin Vapor Diffusion Method at Peking University
## 3 Results Analysis  
### Simulation Analysis of the Vapor Pressure in the Nucleation Stage and Coating Stage  
In order to find out the reason that causes the variations in grain size along cavity axes in previous experiments with a single tin source, tin and tin chloride evaporation was analyzed theoretically.  
First of all, in the nucleation stage, the vapor pressure of tin was quite small, about \(10^{-6}\) torr at 800 \({}^{\circ}\)C, and tin chloride was dominant. The saturated vapor pressure of tin chloride was larger than several kPa-s [22]. Then, the evaporation rate of the tin chloride could be derived using the Langmuir Formula (1) and the Knudsen number \(K_{n}\) could be calculated by Formulas (2) and (3), where \(\frac{dM}{dt}\) is the evaporation rate of the tin chloride, \(A\) is the evaporation area, \(P\) is the vapor pressure, \(m\) is the atom mass, \(k_{B}\) is the Boltzmann constant, \(T\) is the vapor temperature, \(\lambda\) is the mean free path, \(d\) is the diameter of the atom and \(D\) is the characteristic length of 300 mm [23; 24]. The results are shown in Figure 4a. It can be seen that \(K_{n}\)\(<\) 0.01, which means viscous flow. The evaporation rate is above 300 g/s at 800 \({}^{\circ}\)C, which means 3 g tin chloride would turn to gas in the front stage of nucleation. This derivation assumes the tin chloride vapor is the ideal gas and evaporates at saturated pressure, while actual evaporation can be faster due to a low level of the initial vacuum. Then, the pressure of the tin chloride vapor is supposed to be distributed by gravity after diffusion, as shown in Figure 4b. The pressure gradient between the top and bottom of the cavity is small, allowing us to conclude that the nucleation agent is evenly distributed inside the cavity.  
\[\frac{dM}{dt}=AP\sqrt{\frac{m}{2\pi k_{B}T}} \tag{1}\]  
\[\lambda=\frac{k_{B}T}{\sqrt{2}\pi d^{2}P} \tag{2}\]  
\[K_{n}=\frac{\lambda}{D} \tag{3}\]  
The vapor pressure of tin is negligibly low in the nucleation stage, but tin is the main part of the gas in the coating stage while tin chloride vanished. [PERSON] simulated the tin particle flux on the cavity wall at a temperature of 1400 \({}^{\circ}\)C using the free molecular flow module and revealed the importance of the high temperature of the tin source [25]. We simulated the tin evaporation at 1100-1300 \({}^{\circ}\)C using the slip flow module in COMSOL Multiphysics. The tin vapor follows a transitional flow, corresponding to the Knudsen number presented in Figure 5a. The evaporation rate used is calculated by Formula (1), shown in Figure 5b. In this model, the cavity upper port is open and only one tin source  
Figure 4: Calculations for tin chloride: (**a**) Knudsen number and evaporation rate at 450–800 \({}^{\circ}\)C of tin chloride, calculated from Formulas (1)–(3); (**b**) presents the vapor pressure at different temperatures and heights, showing that the vapor pressure of tin chloride is uniform inside the cavity.  
is used. It is difficult to measure the absorption rate of tin at the cavity wall; we only consider that little tin is absorbed into the walls and most tin vapor flows out through the top end, which may happen in the initial state of the coating stage or after a long time of coating. From the simulation results in Figure 6, pressure at the bottom is nearly 10 Pa while pressure in the upper region is only 2 Pa, which could result in a smaller grain on the upper cavity wall.