A Novel Application of High Thermal Conductivity Material on Superconducting RF Cavities: The Inner-wall Thermal Conducting Film (ITCF)

Superconducting Radiofrequency (SRF) technology has developed rapidly for the requirements of high-energy accelerators in fundamental physics research and nuclear industry applications. Thus, the SRF community endeavored to improve the performance from every aspect for niobium and thin-film cavities and made significant breakthroughs in the past decades. However, the positive feedback effect from the RF loss seriously limits the cavity’s maximum acceleration gradient and causes the high field’s unstable operation. Mitigating this effect requires increasing the cavity thermal conductivity to reduce the temperature at the heating point. Conventional technology conducts the RF-loss heat from the surface to the liquid helium outside the cavity while the inner cavity remains vacuum. This paper introduces a new strategy which coats an inner-wall thermal-conducting film (ITCF) in the cavity for the first time globally. The COMSOL simulation results show that ITCF absorbs heat from the RF surface and transmits the heat to the distance on the inner wall, generating a different heat transfer route, thereby reducing the heating-point temperature more efficiently.


Introduction
Particle accelerators, such as the national large-scale scientific instrumentationthe China initiative Accelerator Driven System (CiADS) [1], have extensively adopted superconducting radiofrequency (SRF) cavities to achieve high accelerating gradients and low RF power dissipation [2].The most frequently used measure of cavity performance is the  0 vs.   plot (or the "Q vs. E curve"), where   is the cavity acceleration gradient, and  0 is the intrinsic quality factor given by: where  is the operation frequency,  is the storage energy,   is the power dissipation,  is the cavity geometry factor and   is the surface resistance.
Although their power losses are much lower than those of normal conducting RF cavities, superconducting cavities still suffer from multiple degrading effects that limit their performance, such as field emission (FE), multipacting, hydrogen Q-disease and thermal quench, resulting in a drop in Q as   increases (i.e.Q-slope).The mechanisms behind some of these degrading effects are still unclear, and a summary can be found in Ref. [3].Among those effects, the non-perfect surface always exists defects on the cavity's inner surface; they react with the RF field to generate heat, increase the local temperature, damage the status of superconductivity, and seriously limit the reachable  , .
1.1.The Role of Defects and the Thermal Conductivity The defect influence is vital in the SRF cavity because it has a positive feedback effect.Padamsee estimated the Q vs. E curve impact by the defects and changed with the field value [4].Figure1 black lines show that increasing the magnetic field from 503 Gauss (50.3 mT) to 723 Gauss (72.3 mT) would increase the maximum temperature from 6.5 K to 9.5 K.In the SRF cavity, the magnetic field is proportional to the acceleration gradient   , indicating that a higher field could break the superconductivity more easily.The relation between thermal conductivity κ and cavity performance is a hot topic in the SRF field.Initially, Ciovati and Halbritter deduced from the Ginzburg-Landau theory that the surface resistance   of a superconductor in the RF field is related to the  of materials.They explained the  0 vs   sharp decrease in the mid-field region.They refined this model later [5,6,7].Vines further modelled the relation between κ and cavity wall thickness [8], in which κ was calculated by Koechlin's model that involves RRR value, operating temperature, superconducting energy gap ∆ and phonon mean free path [9].Their works show that the defect could keep a lower   when applying a higher field if the κ increases.Figure1 blue dot lines represent the new temperature distribution values near defects under the same magnetic field with higher κ.Higher thermal conducting efficiency generates lower   for a particular defect.Thus, the cavity can suffer and reach a higher operation  , without quench.
Recently, the Nb 3 Sn thin-film cavity, the so-called "next generation cavity", is the frontier research of the SRF cavity.As shown in Table 1, Nb 3 Sn has a much higher critical temperature   (18 K,   of niobium is 9.2 K) and critical field   (535 mT,   of niobium is 200 mT).The SRF cavity could increase the operation temperature from 2 K to 4.2 K and abandon helium cryogenics to extend the whole SRF technology's application, opening a new era of particle accelerator technology.However, the performance of the Nb 3 Sn thin-film cavity has yet to reach the theoretical limit.One possibility is its low thermal conductivity (last column in Table 1   The ITCF has three requirements to be effective: (i) The ITCF should be able to be coated on the inner wall of the cavity to direct contact with the hot spots because the outer wall already has liquid helium to cool the whole cavity system.(ii) It should be dielectric so it would not interact with RF.Thus, the material loss-tangent needs to be low.(iii) Relatively high κ will be more practical for ITCF.Thus, we chose Al 2 O 3 and AlN as our simulation candidates.

COMSOL Simulation
To evaluate the ITCF effect, we did a COMSOL simulation for the new structure.COMSOL adopts the heat conduction equation to simulate the conduction process: where  is the density of the material,  is the specific heat,  is the temperature,  is the time,  is the thermal conductivity and the  ̇ is the power of heat source.

Simulation I: Nb with Al 2 O 3 ITCF
We first simulate the Nb with Al 2 O 3 ITCF.A disk-shaped local area with a 1 mm radius and 1 mm thickness was selected to represent the cavity wall.A disk-shaped constant-power heat source was set in the center of the inner surface, with a power equal to 10 16 W/m 3 , a radius equal to 0.01 mm (close to a defect size) and a thickness of 50 nm (close to the London penetration depth   , as shown in Table 1).The cavity material was niobium (Solid, polycrystalline, from the COMSOL material library, with a broadened temperature range κ from Ref. [12]), and the wall outer boundary temperature was set to 4 K (close to the operation cooling temperature).Figure 3 (a) and (b) show that the maximum temperature of this simulation at steady-state (after 100 seconds) is 31 K.
We then add 1 µm Al 2 O 3 ITCF to the inner wall with the κ from Ref. [13]. Figure 3 (c) shows that the maximum temperature drops to 16 K after 100 seconds.We also found that the ITCF conducts the heat into the distance and flattens the isotherms (Figure 3 (d)).In this case, a micrometer thickness ITCF generates an adequate new thermal path.From the simulation process, we also suppose the ITCF would improve more for worse baseline-κ materials such as Nb 3 Sn.We then simulate the Nb 3 Sn with AlN ITCF.Currently, the Nb 3 Sn film is always coated on the Nb inner surface to fabricate a thin-film cavity, so we keep the Nb wall part the same as section 3.1 but add a 2 µm Nb 3 Sn layer, where the Nb 3 Sn properties are from COMSOL library and the κ from Ref. [11] for a broader temperature range.We adjusted the heat source power to 10 13 W/m 3 to close to the cavity operation temperature.Figure 4 shows the simulation results that the maximum temperature at steadystate (after 10 seconds) reaches 8.97 K.After adding 1 µm AlN ITCF (κ from Ref. [14]), the maximum temperature drops to 4.5 K.According to Figure1, this new structure would increase about 30% of the  , and highly widen the application of Nb 3 Sn thin-film cavity.

Conclusion
In summary, we started with the cavity degradations' appearance and found one strategy to increase the cavity's thermal conductivitythe inner-wall thermal conducting film (ITCF).We adopted Al 2 O 3 and AlN as heat conduction films and performed COMSOL simulations on pure niobium and Nb 3 Sn.We found that the ITCF can not only transfer the hot spot heat to the heat conduction layer and decrease the temperature but also transfer the heat to the surroundings and "flatten" the isotherm.
Soon, we will start the experimental investigation of the ITCF on superconductors, verify the ITCF strategy, increase the cavity's   limit, and eventually build a new type of SRF cavity with a better performance.

Figure 1 .
Figure 1.Temperature profile (vertical) at the RF surface near a defect for varied magnetic fields (left figure black lines, from Ref. [4], where 1 T = 1000 mT = 10000 Gauss = 10000 Oe), and a schematic diagram of the relative positions of a defect on the Nb surface with coolant liquid helium (right figure, the orange arrow defines the vertical direction).The blue dots on the left graph represent the new temperature distribution values near defects under the same magnetic field with higher thermal conductivity.

2 .
The New Thermal-conducting Path: the ITCF Strategy Based on the significance of thermal conductivity, we encountered a new strategy for cavity performance: the inner-wall thermal conducting film (ITCF).The idea is to add a high κ film on the cavity inner wall to generate an extra heat conduction path.Figure2shows the diagram.Initially, two defects have temperatures of  1 and  2 .After adding ITCF, the heat generated by the two hot spots remains unchanged; however, the temperatures decrease to  1,ℎ and  2,ℎ , by ∆  1 and ∆  2 (Figure2, left).Also, due to the limited film thickness, the heat transfers to distant non-heating regions along the ITCF, decreasing the overall temperature near the heating point (as shown in Figure2 right).

Figure 2 .
Figure 2. The ITCF schematic diagram of temperature relief at the defect locations.

Figure 3 .
Figure 3. COMSOL thermal simulation results of Nb with a constant-power heat source without (a and b) and with (c and d) Al 2 O 3 ITCF.(a) and (c) are the surface temperature distributions; (b) and (d) are the isotherms.

Figure 4 .
Figure 4. Surface temperature distribution of 2 µm Nb 3 Sn thin-film on Nb before (a) and after (b) AlN ITCF by COMSOL thermal simulation.