# Improvement of Surface Roughness and Cracking Study of Nb\({}_{3}\)Sn SRF Films
## 4 Study on Strain Limits of Nb\({}_{3}\)Sn Film Coated by Tin Vapor Diffusion Method
### Simulation of Nb\({}_{3}\)Sn Cavity Tuning at Room Temperature  
Considering the past research on Nb cavities, the cracks may bring problems. Niobium substrate is exposed to RF field when cracks happen; these cracks are larger than the coherence length of Nb\({}_{3}\)Sn, so extra surface resistance would be introduced. During cooling, these low-Tc regions would act as flux pinning sites and degrade the Q value. In addition, the electromagnetic field may be enhanced in the crack region. We simulated the electromagnetic field at a single crack located at the cavity equator using multiphysics simulation software. Figures 17 and 18 present the magnetic field enhancement with different widths and depths. The simulations showed that the magnetic field at both ends of the cracks is generally 4-5 times larger than the field at the smooth surface. Avoiding cracks is necessary to achieve a higher acceleration gradient of Nb\({}_{3}\)Sn cavities. Additionally, the risk of the field emission may be increased in these sharp cracks and, thus, needs to be investigated in future Nb\({}_{3}\)Sn cavity applications.  
We obtained the upper limit of the initial crack occurrence, and then corresponded to it on the cavity to obtain the stress that can be borne when the single cell cavity is tuned; and then we finally corresponded to the tuning frequency to obtain the theoretical tuning range of the single-cell cavity at room temperature. The situations were calculated by multiphysics simulation software, and the curves of the maximum strain (black) and  
Figure 17: The H field simulated by multiphysics simulation software.  
Figure 18: Surface field enhancement by 2 \(\upmu\)m deep and 3 \(\upmu\)m deep cracks with different widths.  
stress (red) versus tuning frequency are shown in Figure 19. At 4.2 K, the yield strength of niobium is very large (over 600 MPa [15]), and the cavity is considered to be elastic within yield strength. The thermal expansion during cooling down was considered. The maximum strain is no larger than 0.25% when the frequency changes within \(\pm\)5 MHz, which means no cracks should happen when tuning at 4.2 K. However, at room temperature, the plastic deformation needs to be considered, and the yield strength is set to 68 MPa. The strain gets larger, and the maximum residual strain reaches 0.9% when the frequency changes \(\pm\)5.8 MHz, which means the possibility of cracks. The cracks may exist after cooling since the coefficients of thermal expansion of Nb\({}_{3}\)Sn and Nb are close [16].  
The tuning safety range given in the above calculation assumes that the crack safety threshold is not affected by temperature. Since the mechanical behavior of Nb\({}_{3}\)Sn films at 4.2 K is unknown, the calculated results are not applicable to 4 K on-line tuning. This experiment is a sample experiment, which has good mechanical uniformity, while the superconducting cavity may have residual strain in the preparation process, and the local stress state is different. The safety range of the experimental results is larger than the actual, so the direct application to the tuning of the superconducting cavity may need to be further narrowed. There might be other factors that affect the ability to defend the tensile strain of Nb\({}_{3}\)Sn, which could be controlled by changing the coating parameters, like annealing time.