# 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
### Strain Limit of Flat Nb\({}_{3}\)Sn Film Samples  
Considering that the average grain size of Nb\({}_{3}\)Sn films we prepared is only about 1.8 \(\upmu\)m, the stress was almost loaded on the substrate instead of the Nb\({}_{3}\)Sn film, but the strains of the substrate and the Nb\({}_{3}\)Sn film should be the same as long as the grain links are not destroyed by tension or compression. Figure 10 plots the curves of the strains and stresses applied on the samples, indicating that those samples have different initial strain states. It is not convincing to measure the yield strength of the film in this way; however, only the strain limit that the Nb\({}_{3}\)Sn film could withstand can be analyzed. As the curves described, some of the samples reached the yield stage, resulting in irreversible strain in the samples after the tensile or compression experiments. The residual strain and transversal strain were calculated and are shown in Table 3, and the Poisson ratio of Nb is 0.38 [14].  
As for tension experiments, severe cracks perpendicular to tensile stress were found on Nb\({}_{3}\)Sn surface when the residual strains were larger than 0.99%, as shown in Figure 11. Most of the cracks are along the grain boundary, and some cracks are across the grain, which is consistent with early arguments about the weak link of grain boundaries from  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & **No.** & **Stress/MPa** & \begin{tabular}{c} **Residual** \\ **Strain/\%** \\ \end{tabular} & \
\begin{tabular}{c} **Transversal** \\ **Residual** \\ **Strain/\%** \\ \end{tabular} &
\begin{tabular}{c} **Amounts of** \\ **Cracks** \\ \end{tabular} \\ \hline \multirow{4}{*}{Tension} & T1 & 25 & 0 & 0 & 0 \\ \cline{2-6}  & T2 & 50 & 0 & 0 & 0 \\ \cline{2-6}  & T3 & 75 & 0.99 & \(-\)0.38 & 1-5 \\ \cline{2-6}  & T4 & 100 & 2.54 & \(-\)0.97 & \(>\)5 \\ \cline{2-6}  & T5 & 152 & 6.34 & \(-\)2.42 & \(>\)5 \\ \hline \multirow{4}{*}{Compression} & C1 & \(-\)15 & 0 & 0 & 0 \\ \cline{2-6}  & C2 & \(-\)30 & 0 & 0 & 0 \\ \cline{2-6}  & C3 & \(-\)45 & \(-\)0.44 & 0.16 & 0 \\ \cline{2-6}  & C4 & \(-\)60 & \(-\)0.76 & 0.29 & 0  
Siemens [13]. The distance between cracks is 50-100 \(\upmu\)m, and the width of the gap is 100-300 nm from the sample of 0.99% residual strain. We tested the composition of the samples using EDS, and the results show very low tin contents of 5-6% at the crack regions, meaning that the Nb substrate is under exposure, and the surface resistance would be increased by the cracks. The cracks become denser and longer under larger residual strains, the conditions of 75 MPa and 100 MPa are shown in Figure 12a,b.  
In terms of compression situations, the transverse strain limit is first reached, and the direction of cracks is parallel to the compressive stress. Samples with 1.13% and 1.48% transversal residual strain caused cracks, displayed in Figure 12. Like the tension, cracks also become denser and longer under larger transversal residual strains, and most cracks are along the grain boundary. Due to the good ductility of Nb sheet, when the tensile or compressive residual strains are large enough, transversal and longitudinal cracks both happen, as shown in Figure 12c,d.  
For the 1-2 \(\upmu\)m Nb\({}_{3}\)Sn film grown on niobium substrate, it is a complicated matter to perform a complete strength test on it. The purpose of this experiment is to conduct a preliminary exploration of its tensile and compression endurance, because this step is an important step before the use of niobium cavity, so we think that the study of Nb\({}_{3}\)Sn film tensile and compression is very necessary. From the above analysis, the safe residual strain range of 1.8 \(\upmu\)m Nb\({}_{3}\)Sn films on 3 mm thick Nb substrate may close to (\(-2.6\%\), 1.0%) at room temperature, which is verified in next section.  
Figure 11: Cracks generated by tension (**a**) and compression (**b**), measured by SEM.  
Figure 10: Tension curves and compression curves.