Impact of geometry on the magnetic flux trapping of superconducting accelerating cavities
11 HOW CAN GEOMETRY IMPACT THE MAGNETIC FLUX TRAPPING OF SUPERCONDUCTING ACCELERATING CAVITIES?
David Longuevergne, Universitรฉ Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France. Akira Miyazaki, FREIA, Uppsala University, Uppsala, Sweden.
Abstract
Quality factor measurements of superconducting radio frequency cavities have been performed for different magnetic field configurations to address a geometrical effect on surface resistance caused by magnetic trapped flux. Field monitoring inside a cavity highlighted behaviours particularly important for the Meissner shielding at cold and flux trapping during cooling down. In this paper, we suggest that the ambient field orientation and cavity geometry have a significant impact on the cavity performance degradation due to trapped flux. Experimental data are presented, and a model to interpret the geometrical effect is introduced and assessed with experimental data. The results are compared to an elliptical cavity result in the literature. INTRODUCTION
Performances of a superconducting accelerating cavity made of bulk niobium can be strongly affected by the presence of a residual magnetic field while transitioning into its superconducting state. Magnetic vortices trapped by pinning centers of the cavity material during cool-down interact strongly with the radiofrequency (RF) electromagnetic fields inducing additional dissipations in the helium bath [1] . We denote these additional losses by a surface resistance R mag in nโฆ. The total surface resistance R s of a superconductor [2] is the sum of a strongly temperature T dependent contribution R BCS , derived from the linear response of the Bardeen Cooper Schrieffer theory of superconductivity [3] , and the other contribution, which is only weakly temperature dependent, defined as the residual resistance R res : ๐ s (๐, ๐) = ๐ BCS (๐, ๐) + ๐ res (๐, ๐), (1) with RF frequency f dependence. This R res consists of a component temperature-independent ๐ due to material imperfections (pollution, defects, grain boundariesโฆ) and R mag : ๐ res (๐, ๐) = ๐ + ๐ mag (๐, ๐). (2) Recent technical advances on mechanical process and surface treatment have reduced R , and therefore, understanding and controlling R mag becomes of critical importance in state-of-the-art superconducting RF cavities for many applications [4] . Previous studies [5, 6, 7, 8, 9] mainly focused on the specific elliptical cavities dedicated to high-energy electron accelerators. In this paper, we report on three more general shaped cavities developed for proton and heavy ion linear accelerators. The first is a Quarter-Wave Resonator (QWR) operating at a frequency of 88 MHz built for the Spiral2 project [10] . The second is a Double-Spoke Resonator (DSR) operating at 352 MHz for the ESS project [11] . The third is a Single-Spoke Resonator (SSR) operating also at 352 MHz for the MYRRHA project [12] . With the ambient residual magnetic field H res in mG present at transition, R mag can be decomposed into: ๐ mag = ๐ mag โ ๐ mag (๐, ๐) โ ๐ป res , (3) where ๐ mag is the dimensionless flux trapping efficiency coefficient with mag < 1 [13] , and S mag is the magnetic sensitivity [14] expressed in nโฆ/mG. From Eq. (3), it is evident that three independent factors, H res , ๏จ mag , and S mag play an important role to reduce R mag and thus to fulfil the requirement for total R s of each accelerator project. The purpose of this study is to address geometrical dependence of R mag when a condition ๏จ mag ~1 is satisfied. We propose that the origin of such geometrical dependence may be from a preferential flux trapping angle. This report is organized as follows. In the rest of the introduction, we briefly review the previous studies of H res , ๏จ mag , and S mag . Next, the experimental set-up in IJCLab is presented. The experimental results are followed by analyses using a model we propose to predict geometrical dependence of R mag . We compare the model to elliptical cavity results as well. The final section represents conclusions. Magnetic shield to reduce H res To protect niobium from environmental magnetic field H earth coming from the earth field and magnetic parts at the vicinity of a cavity, magnetic shields are usually installed around a superconducting cavity. High permeability material as permalloy (ยต-metal), Cryophiยฎ or A4K are used to funnel the magnetic field and thus strongly attenuate H res inside it. Historically, minimizing the attenuation factor ๐ป res ๐ป earth โ required to achieve R mag within the total budget of R s has been a major interest of the community [15] , and this has been successful for conventional elliptical cavities at 1.3 GHz. One technical difficulty appears when one tries to shield cavities with a different geometry, namely, low-beta structures [16] . Their relatively large dimensions can significantly increase the mechanical complexity and cost to fabricate ideal magnetic shields [17] and thus can practically limit field attenuations around the cavities. Therefore, reducing the other two factors in Eq. (3) becomes motivated to relax the mechanical and financial constraint of magnetic shielding. On top of this practical use, systematic studies of ๏จ mag , and S mag are of scientific interest for the applied superconductivity under strong RF fields [18]. Flux expulsion and trapping
The flux trapping efficiency coefficient ๏จ mag is typically evaluated with magnetic sensors installed in close proximity with a cavity [9] . The sensor probes the magnetic field distribution altered by the diamagnetic property of the Meissner state when the material goes through superconducting transition. Amongst all, bulk material history has a significant impact on flux trapping. Material re-crystallization by thermal treatment typically above 800ยฐC would ensure an almost complete flux expulsion beside some exceptions as reported [8] . Indeed, without any recrystallization process, close to 100% of the residual magnetic field is trapped as reported in [13, 19, 20] . Regarding the studies presented in this paper, the cavities are made of polycrystalline material without heat treatment above 650ยฐC and therefore we primarily assume, based on past observations, almost full flux trapping. The previous studies about cool-down dynamics sometimes showed contradictory results. For simple geometries like bare elliptical cavities in vertical cryostats [5, 8, 21] , the flux expulsion improves in proportion to the thermal gradient across the cavity. This was explained by two different models proposed by Kubo [22] and Checchin [7] . However, for more general configurations, a low thermal gradient is proposed in [23] due to the possible effect of the thermoelectric currents generated by bi-metallic junctions between the niobium cavity and the helium tank made of titanium [24, 25] . The experiments on the LCLS-II cryomodules [26, 27] showed that such dynamic thermoelectric currents tend to vanish when the temperature is close to superconducting transition. Instead, independently of the cool-down rate, the intrinsic static thermoelectric currents remain and act as an additional external magnetic field to be expelled during the superconducting transition. We also observed a similar behaviour in our cryostat in the IJCLab [28] and thus we do not address the effect of thermoelectric current in this paper. It must be noted that the cavities installed in other cryostats or cryomodules may behave differently due to the different mechanical structure from our vertical test stand. Magnetic sensitivity S mag A static model concerning a normal conducting core in a trapped vortex gives a good approximation of S mag as formulated by [1] ๐ mag = ๐ n (๐, ๐)2 โ ๐ป c2 (๐) , (4) with R n the normal resistance and H c2 the upper critical field of the material. The same result is reproduced [29] by dynamic flux oscillation based on the model by Gittleman and Rosenblum [30] . Beyond this simple approximation, the magnetic sensitivity is extremely difficult to quantitatively predict and evaluate since it depends on many parameters [31] : โ Frequency of the cavity [14, 20] โ Temperature of operation [32, 33] โ Local heating due to trapped vortices [34, 35] โ Impurity content of material : dislocations, segregation, precipitates, and grain boundaries [36] โ Model of pinning potential [29] โ Interplay of various pinning centers [37, 38] โ Amplitude of RF fields [18, 29, 33, 37, 38, 39] โ Geometry of the cavity [18, 33, 37]
One must take care when studying the geometrical dependence, because the measurement observable R mag averaged over the cavity inner surface cannot directly separate the geometrical effects from either ๏จ mag or S mag . To study S mag , one needs to ensure ๏จ mag ~1 by very low thermal gradients generally associated with slow cooling speed. Our study fulfils this condition because the material is prepared not to efficiently expel the flux during cool-down. We show this by a dedicated experiment. The geometrical dependence of ๐ mag has been under debate. A group reported [40] that flux trapping happens uniformly over the surface and even preserves the orientation of the applied external field, if flux expulsion is suppressed by slow cooling down. The flux oscillation under RF fields depends on the orientation of the trapped flux versus RF currents and thus ๐ ๐๐๐ becomes non-uniform over the surface. We propose another hypothesis that flux trapping may happen preferentially for the normal component to the surface [41] , and the amount of flux to be oscillated by the RF fields becomes non-uniform. This results in ๐ ๐๐๐ being dependent on the geometry. EXPERIMENTAL
SETUP
The Vertical Cryostat
In an effort to fully qualify any new cavity design or a new surface treatment process or procedure, cavities are first tested in a vertical cryostat. The vertical cryostats are designed to provide optimal testing conditions to address cavity performances. The intrinsic quality factor (noted Q ) is evaluated at different accelerating gradients (noted E acc ) by measuring the power dissipation P c averaged over the cavity walls. The cryostat available on platform Supratech at IJCLab in operation since 1998 and upgraded in 2018 is capable of hosting two jacketed cavities (equipped with their helium jacket) in a volume constrained in a cylinder of 2m high and 1.15m in diameter as shown in Fig.1 The cryostat is externally shielded on the side by 1mm-thick permalloy sheets rolled around the vacuum vessel. The horizontal component of the magnetic field is significantly attenuated. Because of design constraints, the vertical component is not shielded by permalloy sheets installed on the top and bottom of the cryostat but by three compensating coils inserted in between the magnetic shield and the vacuum vessel as depicted in Fig. 1. This configuration allows the possibility to either reduce the magnetic field to a minimum value to optimize the cavity performances or to apply a uniform field to measure ๏จ mag and/or evaluate S mag with the field of any cavity. Figure 2 shows two examples of magnetic configurations of a residual field. Figure 1: Vertical cryostat in operation at IJCLab with its passive and active magnetic shields. The insert is loaded with two ESS Double Spoke Resonators. Magnetic Sensors
Regarding magnetic measurement, the very low magnetic field to be measured makes the fluxgate magnetometer the best technology in our test conditions in vacuum at low temperatures. As commercially available fluxgate sensors (at the time of these studies) are only available as single axis sensors, three of them are assembled on a 3D-printed support to measure each of the three axis. A home-made multiplexer has been built to read up to twelve type G sensors with only one controller (MAG01-H) from Bartington [42] . The magnetic field resolution is of 0.02 mG over a range of 20 G. Integration time imposes a minimum multiplexing rate of about six seconds. To avoid any crosstalk between sensors, both current and voltage leads are multiplexed. In normal operations, only one sensor is energized at a time.
Measurement capabilities
Several types of measurements are possible with the current set-up: โ
Evaluation of the magnetic shield efficiency. โ Evaluation of the flux trapping efficiency ๏จ mag (this is usually not precise as the cavities are equipped with a helium tank, which limits the accessibility for instrumentation). โ Evaluation of the magnetic sensitivity to a residual magnetic field of different types of superconducting cavities ( S mag ). โ Monitoring of the magnetic field behaviour during cooling down generated by thermoelectric currents because of the existence of bi-metallic junctions. โ
Detection of the magnetic field penetration in case of a quench
Figure 2: Example of residual magnetic field configuration applied in the vertical cryostat. The vertical and horizontal components are measured along the central axis of the cryostat
Cavities for this study
The three cavities dedicated for this study are shown in Fig. 3 with geometrical factor G [1] and operating frequency f in Table 1. The cavities are made of polycrystalline niobium and their surfaces have been prepared following the standard procedure with SUPRATECH facilities at IJCLab: โ Degreasing in an ultrasonic bath with detergent. โ Surface abrasion by Buffered Chemical Polishing of at least 200 um (BCP) โ Optional hydrogen degassing at 650ยฐC for 10h. โ High Pressure Rinsing with ultra-pure water โ Drying and assembly in ISO4 clean room
They are all manufactured out of the same polycrystalline (fine grain) bulk niobium material without any heat treatments above 800ยฐC. No low temperature baking [4] or nitrogen doping [43] were performed. In addition, all cavities for this experiment are equipped with a titanium helium tank.
Figure 3: From left to right, RF magnetic field distribution of: Spiral2 QWR, MYRRHA Single Spoke Resonator (SSR) and ESS Double Spoke Resonator (DSR). Black rectangles represent the position of magnetic field probes installed during experiments. Vertical and transverse residual magnetic fields were applied on Spiral2 QWR. For MYRRHA SSR and ESS DSR, only the vertical fields were applied in the cryostat. This corresponds to the field transverse to the beam axis in MYRRHA and along the beam axis of ESS DSR. At the time of these studies, the ESS DSR could only be loaded vertically in the cryostat as shown in Fig.3. The field orientations are also summarized in Table 1. Table 1: Geometrical factors and frequencies Type of cavity project G [Ohm] f [MHz] Applied magnetic field orientation QWR SPIRAL2 33 Vertical & Transverse SSR
MYRRHA 109
Vertical (transverse to the beam axis) DSR
ESS 133
Vertical (along the beam axis) FLUX
TRAPPING
STUDY
We first validate our assumption about uniform and almost full flux trapping in our fine grain material without substantial heat treatment above 650ยฐC. The conventional experiment on ๐ mag by fluxgate sensors around a cavity is not reliable in our experimental setup because of the complicated cavity geometries studied in this paper compared to the conventional elliptical cavities. . The maximum field enhancement even by the ideal flux expulsion is only a few percent around the cavity. Instead, a magnetic sensor is installed to monitor the vertical component of the magnetic field in the stem, the inner conductor of the Spiral2 QWR as depicted in Fig. 3. A particular advantage of using QWR for this study, is that the magnetic sensor can be installed inside the cavity structure, unlike the conventional elliptical cavities, in which the beam vacuum side is not suitable for sensor installation. In case of complete flux trapping during superconducting transition, no change of magnetic field would be observed whereas in the case of complete flux expulsion, the magnetic field would drop to zero as shown in Fig. 4 (a). As an intermediate case, Fig. 4 (b) shows the partial flux expulsion by the outer conductor when the inner conductor is still normal conducting. The particular case of complete flux expulsion with the concentrated flux trapped at the bottom of the stem is also considered as shown in Fig. 4 (c). Indeed, during regular cool-down, the SC/NC interface moves from the bottom of the outer conductor, then to the top of the cavity and finally down to the bottom of the inner conductor. This would result in a non-zero field measured in the inner conductor even with complete flux expulsion. The sensor inside the stem can distinguish uniform flux trapping from finite ๏จ mag because Fig.4 (c) must show a finite flux jump at the transition where the size is however smaller than the case in Fig.4 (a). Figure 4: Vertical component of the magnetic flux distribution calculated with CST Studio Suite considering (a) full expulsion, (b) full expulsion with the inner conductor normal conducting and (c) full expulsion with trapping at the bottom of the stem. The magnetic fields measured at the probe (red dot) are respectively 0%, 38% and 48% of the ambient residual field. The ambient vertical field is vertical. The flux trapping experiment has been performed as follows with the experimental data summarized in Fig.5: 1. The cavity has been cooled down to 4 K in a non-optimal ambient magnetic field resulting in a residual field measured by the probe of โ55 mG shown as a blue region noted Outer SC and Inner SC in Fig 5.
Figure 5: Magnetic field variations during thermal cycling up to 50K of the prototype ESS DSR (Romea). Two tri-axial magnetic sensors have been installed on the bottom (dashed dotted blue curve) and on the top (dashed green curve). The cavity is set vertical in the cryostat as depicted in Figure 3. Figure 6: Evolution of the magnetic field change at transition versus the thermal gradient across the ESS DSR during thermal cycling. The magnetic jump at transition (See Figure 6) is clearly fostered by the thermal gradient across the cavity although no degradation has been observed on the quality factor and thus surface resistance.
Evidence of full flux trapping and shielding capabilities of non-recrystallized material
Magnetic flux trapping experiment have been pursued on Spiral2 QWR to emphasize the amount of flux expelled or not specifically for a non-recrystallized Niobium (no thermal treatment above 150ยฐC have been performed). A magnetic sensor, probing the vertical component of the ambient magnetic field is installed in the stem (inner conductor) acting as a dead-end as depicted in Figure 3. Thus, in case of complete flux trapping during superconducting transition, no change of magnetic field would be observed whereas in the case of complete flux expulsion, the magnetic field would drop to zero (See Figure 7 (a)). The particular case of complete flux expulsion with the expelled flux trapped at the bottom of the stem is also considered (Figure 7 (b)). Indeed, during regular cool-down, the SC/NC interface is moving from the top of the cavity down to the bottom of the stem. This would result in a non-zero field measured in the inner conductor even with complete flux expulsion. Figure 7: Vertical component of the magnetic flux distribution calculated with CST Studio Suite considering (a) full expulsion, (b) full expulsion with trapping at the bottom of the stem and (c) full expulsion with the inner conductor normal conducting. The magnetic fields measured at the probe (red dot) are respectively 0%, 48% and 38% of the ambient residual field. The ambient vertical field is vertical. Flux trapping experiment has been performed as follow: โ The cavity has been cooled down in a non-optimal ambient magnetic field resulting in a residual vertical field measured by the probe of -55mG (See Figure 8, blue region noted Outer SC, Inner SC). โ The ambient magnetic field is changed (Helmholtz coils are off) to generate a magnetic field of +52mG at t = 0:00 (black arrow). โ The external conductor (body of the cavity) is warmed up with heaters above transition (light green region noted Outer NC, Inner SC). โ The outer conductor is cooled below transition โ The inner conductor (stem) is warmed up with a heater above transition (orange region noted Outer SC, Inner NC) โ The outer conductor is warmed up above transition. At this stage, the entire cavity is normal conducting (red region noted Outer NC, Inner NC). โ The compensating coils are set to optimize ambient magnetic field (-15mG). โ The cavity is then cooled down below transition with a cooling rate of 112 mK/s and a temperature gradient of 4K between top and bottom of the stem. Figure 8 is summarizing the full experiment, showing how the magnetic field inside the stem is changing The ambient magnetic field is changed (Helmholtz coils are off) to generate a magnetic field of +52 mG at t = 0:00 indicated by a black arrow still in the blue region. No reaction of the magnetic probe inside the stem proves the perfect flux expulsion by the Meissner effect at 4 K. The external conductor is warmed up with heaters above transition shown as the light green region noted Outer NC and Inner SC. No reaction of the magnetic probe indicates complete shielding by the inner conductor (stem). The outer conductor is cooled below transition again. Some amount of flux could be trapped during transition of the outer conductor but did not affect the sensor inside the stem because of perfect shielding by the inner conductor. The inner conductor (stem) is warmed up with a heater above transition shown as the orange region noted Outer SC and Inner NC. The magnetic field measured by the probe is changing significantly to reach +41 mG, indicating inefficient Meissner shielding of the outer conductor. The outer conductor is warmed up above transition. At this stage, the entire cavity is normal conducting shown as the red region noted Outer NC and Inner NC. The magnetic field finally reaches +52 mG, which is the true ambient field without any field distortion by flux expulsion at the cavity walls. The compensating coils are set to optimize ambient magnetic field ( โ15 mG). The cavity is then cooled down below transition with a cooling rate of 112 mK/s and a temperature gradient of 4 K between top and bottom of the stem, shown as a blue region noted Outer SC and Inner SC. No reaction of the magnetic probe was observed. This indicates full flux trapping in both inner and outer conductors.
Figure 5: Flux trapping experience on a Spiral2 QWR. From Fig.4 (b), whatever the flux expulsion efficiency during cool down is, the outer conductor acts as only a weak magnetic shield at 4 K for the inner conductor. This is due to the presence of the three coupler ports at the bottom of the cavity and 2 ports on the top. Therefore, even in case of full expulsion by the material, the outer conductor would shield 62% of the ambient field. However, in reality, because of the poor flux expulsion by the polycrystalline niobium, the shielding is only about 10% (the difference of magnetic field measured between step 5 and 6). The shielding capability of the outer conductor could thus not explain the very low R mag of this cavity to the vertical magnetic field. Note that no finite flux jump was observed during the cool-down. This indicates that the flux trapping is uniform and not like Fig.4 (c). A related subject concerning quench induced trapped flux is summarized in the appendix. The following statements can be concluded. The inner conductor provides a very efficient magnetic shielding at 4 K, whereas, even at 4 K, the simulation shows that the outer conductor has a very weak shielding capability due to the presence of ports at the top and bottom of the cavity. From the experiment, flux trapping during cooling down is close to 100% in agreement with our assumption of the fine grain material. Even the inner conductor does not act as an efficient shield against permanent ambient fields present during superconducting transition. However, the inner conductor can very efficiently shield fields activated during accelerator operation from solenoids. In the next section, we reveal the apparent geometrical dependence on ๐ ๐๐๐ , which in fact originates from the flux trapping discussed in this section. MAGNETIC
SENSITIVITY
From now on, we assume that the flux is uniformly trapped in the surface of the cavity. For the precise experiment of S mag , one has to remove the components R BCS and R to extract pure R mag . This is accomplished by the following procedure: The cavity is slowly cooled down in an ambient residual magnetic field ๐ป as low as possible, the so-called โoptimal configurationโ depicted previously in Fig. 2. The vertical component stays below 10 mG as well as the horizontal component The total surface resistance R s at low field is estimated from the Q measurement ๐ ๐ = ๐บ๐ , (5) with G the geometrical factor of the cavity as listed in Table1. The cavity is warmed up slowly above transition during a night (>50 K) and then cooled down in a homogeneous vertical magnetic field ๐ป = 110 ๐๐บ . The horizontal component stays below 15 mG. R s is measured again, considering that the flux is fully trapped. By subtracting two surface resistances we can estimate S mag by ๐ mag = ๐ ๐ (๐ป ) โ ๐ ๐ (๐ป )๐ป โ ๐ป . (6) The sensitivities measured on the three types of cavities are summarized in Table 2. The simple model calculation based on Eq. (4) overestimates the measured sensitivities of all the cavities. Apparently, recently proposed approaches with flux oscillation [29, 35, 36, 37, 38] based on the Bardeen-Stephen model [44] would improve the calculation but lack of precise information on material parameters especially on pinning centers prevents us from applying their models. Besides, the intrinsically complicated shape of the cavities would certainly limit the accuracy of these models, which were originally developed and validated for simpler geometries, such as 1-cell elliptical cavities. In the next section, we develop a novel way to take account of geometrical effects while keeping the local surface resistance the same as Eq. (4). This leads to dramatically better agreements with all the measurement results as shown in Table 2. Table 2: Comparison of measured and calculated sensitivities cavity type ๐ฏ ๐๐๐ orientation measurement (n ๏ /mG) uniform ๐บ ๐๐๐ (n ๏ /mG) Eq.(4) relative error (%) corrected ๐บโฒ ๐๐๐ (n ๏ /mG) Eq.(18) relative error (%) QWR Vertical 0.006 0.08 +93 0.011 +45
QWR Horizontal 0.05 0.08 +38 0.048 โ4 SSR Vertical 0.043 0.12 +64 0.047 +8.5
DSR Beam axis 0.06 0.12 +50 0.055 โ9 DISCUSSIONS
Geometrical dependence
In Table 2, the measured sensitivities are systematically and significantly lower than the calculated sensitivities based on the model which assumes uniform power dissipation over the inner surface for all the geometries. Moreover, the sensitivity difference of the QWR by a factor of 10 to a vertical or horizontal field suggests a very strong geometrical dependence. The fact that RF magnetic fields are mainly distributed around the inner conductor, where the surface is almost vertical, could explain these observations. Indeed, as the surface resistance is estimated from the power dissipations, a change in surface resistance could be measured if and only if it occurs in high RF magnetic field regions. Trapping flux on the bottom of the QWR where the RF electric fields dominate, for example, does not induce any Q drop as only RF magnetic fields are dissipating. The ๐ mag is no longer uniform over the cavity surface but shows position dependence. In this description, the angle between applied field ๐ป ๐๐๐ and the surface does play a major role on the local ๐ mag . The question is whether ๐ mag i.e. flux dynamics under RF fields is intrinsically angular dependent, or amount of the angular dependent flux trapping extrinsically results in angular dependent ๐ mag . In the latter case, ๐ mag calculated by Eq. (4) would be only artificially angular dependent. We compare these two somewhat similar but distinct scenarios. Angular dependence of flux oscillation
The contributions of S mag based on flux oscillation can be estimated by considering the Lorentz force interaction between the quantized flux line and the local RF current. As shown in Fig. 6, with a local spherical coordinate with an RF current density ๐ฑ ๐ ๐น aligned to the x-axis, ๐ the polar angle between the RF surface and the trapped flux ๐ฉ ๐๐ , and ๐ the angle between the trapped flux projected to the RF surface and the RF current, the Lorentz force density ๐ ๐ can be written as ๐ ๐ฟ = ๐ฑ ๐ ๐น โง ๐ฉ ๐๐ = (๐ฝ
00 ) โง (๐ cos ๐ sin ๐๐ sin ๐ sin ๐๐ cos ๐ ) = ( 0โ๐ฝ ๐ cos ๐๐ฝ ๐ sin ๐ sin ๐) (7) where ๐ is the flux quantum ( โ15 Wb) and ๐ฝ is amplitude of the RF current density. Figure 6: Local spherical coordinate system with RF current aligned in the x-axis and trapped flux pointing (๐, ๐) To calculate flux oscillation in a thick bulk niobium, Checchin et al. [36] introduced a 1-dimensional differential equation
๐ ๐ ๐ฆ๐๐ก + ๐ ๐๐ฆ๐๐ก + ๐ ๐ = โ๐ฝ ๐ cos ๐ , (8) with ๐ and ๐ the effective mass and viscosity of a flux, respectively, in the Bardeen-Stephen model and ๐ ๐ the pinning force. Note that Ref. [36] takes ๐ โ ๐ 2โ โ ๐ in their coordinate system. This model nicely explains frequency and mean free path dependence of ๐ ๐๐๐ . This equation originates from Gittleman and Rosenblum [30] who calculated flux oscillation in a thin film 12.7 um thick and thus ๐ = 0 was fairly applied in their case. In our local coordinate system, their oscillation model was restricted along the y-axis. However, Eq (7) implies that the flux vibration in a thick niobium used for the cavity application may also occur in the z-direction i.e. normal to the RF surface if ๐, ๐ โ 0 ๐ ๐ ๐ง๐๐ก + ๐ ๐๐ง๐๐ก + ๐ ๐ = ๐ฝ ๐ sin ๐ sin ๐ . (9) Providing that the material is uniform and isotropic, ๐ , ๐ and ๐ ๐ are similar in both y- and z-axis, and these two degrees of freedom contribute to two independent modes of the flux oscillation. Correspondingly, the sum of these two modes results in power dissipation. In both flux flow and pinning regimes, magnetic sensitivity is locally ๐ mag โ cos ๐ + sin ๐ sin ๐ . (10) In this model, (๐, ๐) is determined by the relative orientations of the magnetic residual field, the cavity surface and the direction of the RF currents in the particular position.
Angular dependence of trapped flux
In the above model, the flux trapped by pinning centers during superconducting transition preserves the original orientation of the applied field. On the other hand, flux trapping may also depend on the orientation of the RF surface. In a plate-like superconductor in the Meissner state, the screening currents are strongly constraint to flow parallel to the RF surface. This implies that the flux component normal to the RF surface is more efficiently pinned than the parallel component. Thus, the Lorentz force that oscillates such a typical trapped flux would be ๐ ๐ฟ = (๐ฝ
00 ) โง ( 00๐ cos ๐) = ( 0โ๐ฝ ๐ cos ๐0 ) (11) and the flux oscillation becomes locally one-dimensional. Correspondingly, the sensitivity can be expressed as ๐ mag โ cos ๐ (12) Therefore, the predicted angular dependence is different from the former model Eq. (10), in which all the flux components are trapped whichever the orientation to the RF surface is. Previous theoretical and experimental studies [41, 45] on the reversible ๐ด eq and irreversible magnetization ๐ด irr verified the angular dependence on trapped flux applied on isotropic type-II thin films. The magnetization is a macroscopic measure of trapped vorticesโ orientation. In equilibrium magnetization with ๐ป > ๐ป ๐1 , ๐ด eq and correspondingly vortex lines are normal to the surface at low fields and become aligned with an externally applied field ๐ฏ when it approaches the upper critical field ๐ป ๐2 . On the other hand, the pinning effect ๐ด irr is always normal to the surface regardless of the field strength if the ๐ is within certain value determined by the geometry (smaller than 70 degree in their samples). In summary, the sample experiment proved trapped flux is almost always normal to the surface. Although the experimental condition is apparently different between the sample magnetization measurement and superconducting RF cavities, the experimental results in the sample measurement of Pb Tl [41] can be applied to the trapped flux in cavities in the following way. The magnetizations on small samples are measured by cooling down the sample without external field (zero field cooling) followed by increasing ๐ป first to ๐ป ๐1 ~10 G (Meissner phase), further up to ๐ป ๐2 ~1500 ๐บ (Mixed state), and then decreasing ๐ป to 0 G. The hysteresis cycle of magnetization gives ๐ด eq and ๐ด irr . A similar magnetic environment was fulfilled also in the SRF cavities during cooling down. Cooling under a constant small external field ๐ป ๐๐๐ ~100 mG , a region around the NC/SC phase front in the cavity wall satisfies ๐ป ๐2 (๐ ๐ โ ๐ ) < ๐ป ๐๐๐ (13) ๐ป ๐1 (๐ ๐ โ ๐ ) < ๐ป ๐๐๐ < ๐ป ๐2 (๐ ๐ โ ๐ ) (14) ๐ป ๐๐๐ < ๐ป ๐1 (๐ ๐ โ ๐ ) (15) in the chronological order with ๐ < ๐ < ๐ . This is the same path as drawing the upper branch of the hysteresis cycle in the magnetization measurement. Also, near ๐ ๐ , the penetration depth is so long that the cavity wall can be relatively approximated as a thin film. The cavity cool-down virtually mimics the magnetization measurement of a small sample. Consequently, one can assume that the findings in the sample measurement are also relevant to the flux pinning during cavity cool-down. In the next section we adopt the angular dependence of flux trapping as a working ansatz, and develop a model to correct the magnetic sensitivity. Model building
As stressed previously, the measured magnetic sensitivities are systematically lower than the theoretical model based on oscillating flux trapped uniformly over the surface. Moreover, the discrepancy between the measured and theoretical sensitivities is changing with the type of cavity made of the same material and the magnetic residual field orientation. This indicates that the origin of the inconsistency is purely due to the geometry, not the material properties of niobium. The model proposed here is based on one strong assumption as described earlier: only the normal component to the cavity surface of the ambient magnetic field can be trapped in the material. This hypothesis leads to a very good agreement between measured and predicted sensitivities. The local spherical coordinate (๐, ๐) in the previous discussion is defined at each point over the cavity surface in the global Cartesian coordinate (๐, ๐, ๐) , and namely, the Lorentz force and ๐ mag depends on ๐(๐, ๐, ๐) . When we apply ๐ฏ ๐๐๐ to a cavity, ๐(๐, ๐, ๐) is determined by the three dimensional structure of the cavity. We take account of this effect as follows: 1. Numerically evaluate the normal component of the residual field to be trapped ๐ป โฅ (๐, ๐, ๐) = ๐ป ๐๐๐ cos ๐(๐, ๐, ๐) (16) from three dimensional models of cavities. 2. Evaluate the local surface resistance caused by the trapped flux oscillation ๐ mag (๐, ๐, ๐) = ๐ ๐ (๐, ๐) ๐ป โฅ (๐, ๐, ๐)2๐ป ๐2 (17) from Eq.(3) and Eq.(4) with ๐ mag = 1 as discussed before. Here, material dependence is included in ๐ ๐ (๐, ๐) and ๐ป ๐2 and fixed in this analysis which focuses on geometrical effects. 3. Similar to the previous work by one of the authors [46] , integrate Eq.(17) all over the cavity surface ๐ and obtain a new ๐โฒ ๐๐๐ with geometrical correction ๐ โฒ๐๐๐ = โฌ ๐ ๐๐๐ (๐, ๐, ๐)๐ป ๐ ๐น2 (๐, ๐, ๐)๐๐ ๐ ๐ป ๐๐๐ โฌ ๐ป ๐ ๐น2 (๐, ๐, ๐)๐๐ ๐ (18) with ๐ป ๐ ๐น (๐, ๐, ๐) the local RF magnetic field evaluated in the same model as for ๐ป โฅ . This model is computed in a NI LabVIEW software [47] with exported files generated by CST Microwave Studio, such as ๐ป ๐ ๐น (๐, ๐, ๐) and ๐ป โฅ (๐, ๐, ๐) distributions and surface mesh [48] . Figure 7 depicts the three kinds of graphical output generated by the code. Figure 7: Outputs from LabVIEW routine showing from left to right: the normal trapped magnetic field ( ๐ป โฅ ) under a vertical ๐ป ๐๐๐ , the RF field distribution ( ๐ป ๐ ๐น ), and the normalized power dissipations caused by trapped flux from vertical ๐ป ๐๐๐ (P mag_v ) and horizontal ๐ป ๐๐๐ (P mag_b ) for the Spiral2 QWR. The results of this model are shown in Table 2. The relative errors between the experimental and calculated values with the proposed model are dramatically improved by the geometrical correction. From this good agreement, we argue that our working ansatz on trapped flux orientation may reflect the reality, and the angular dependence of tilted flux oscillation may be ruled out at least to the geometry and material of our cavities. The flux in the coaxial-type cavities is almost fully trapped but the amount of the flux normal to the surface, which contributes to the RF power dissipation, depends on the relative angle between the applied field and the cavity surface. Model application to an elliptical cavity
In order to investigate our model further, the same calculations are applied to the elliptical cavity geometry of ๐ป ๐ ๐น shown in Fig. 8 a, considering an ambient magnetic field in both directions transversally (vertically) and longitudinally (along the beam axis) as shown in Fig. 8 respectively b and d. Figure 8: Outputs from LabVIEW routine showing from left to right: (a) the RF magnetic field distribution ( ๐ป ๐ ๐น ), (b and c) the trapped magnetic field ( ๐ป โฅ ) and normalized power dissipations under a transverse ๐ป ๐๐๐ , and (d and e) the same for a longitudinal ๐ป ๐๐๐ , for an elliptical cavity. Trapping transverse field produces two dissipating areas aligned on the equator and placed at 180ยฐ from each other (Fig. 8 c). Trapping magnetic field along the beam tube creates two dissipating rings in-between the equator and the iris (Fig. 8 e). These simulation results are in very good agreement with temperature mapping data published in [40, 49] . Indeed, temperature mapping can localize highly dissipating areas and thus reveal the area with trapped flux. When an axial magnetic field is trapped, two โhotโ bands appear in-between the equator and the two iris on the full circumference. When a transverse magnetic field is applied, two โhotโ regions sit at 180ยฐ centred on the equator. We compare our geometrically corrected ๐โฒ ๐๐๐ and the experimental data taken by the same group of temperature mapping [40] . Their experiment was conducted in a similar configuration as ours. They deliberately applied 100 mG to a cavity with different angles, and several thermal cycles ensured ๐ ๐๐๐ higher than 0.5 and even close to 0.9. Note that this group concluded homogeneous trapping without any angular preference, contrary to our working ansatz at the first glimpse. Their conclusion was based on the magnetic field mapping outside the cavity. We will come back to this point in the next section and claim that their results do not contradict our model. We remove a common residual resistance ๐ = 5 ๐ฮฉ [50] from published ๐ ๐๐๐ and divide it by 100 mG to obtain ๐ ๐๐๐ . Table 3 compares the extracted ๐ ๐๐๐ in the case of axial field (=90ยฐ) and transverse field (=0ยฐ) with our model prediction ๐โฒ ๐๐๐ . The absolute values do not match because material properties and history of their particular cavity, which influence for example the mean free path [36] and thus ๐ ๐๐๐ , are not included. In order to eliminate the material effect, we introduce the ratio of the axial and transverse sensitivities. The ratios of experiment and our model are respectively 1.42 and 1.49. Thus, remarkably, they are in very good agreement, indicating our geometrical correction based on preferential trapped flux orientation accurately reproduces the geometrical dependence of the experiment. Table 3: Calculated sensitivities for elliptical cavities
Field orientation measurement (n ๏ /mG) [40] corrected ๐บโฒ ๐๐๐ (n ๏ /mG) Eq.(18) Transverse 0.325 0.092 Axial 0.46 0.137 Ratio 1.42 1.49
The same analysis is performed for several field angles and compared with results presented in [40] . For the correction of the material effect, we linearly scale the model prediction ๐โฒ ๐๐๐ to fit the data. As depicted in Fig. 9, quantitative agreement is obtained between our model and their experimental data. The RF power dissipation is more sensitive to higher angle closer to the magnetic field parallel to the beam axis. Although the RF field between the iris and the equator is lower than that at the equator, the total area of trapped flux normal to the cavity surface is large so that Eq. (18) leads to higher sensitivity. Figure 9: Angular dependence of the magnetic sensitivity for an elliptical cavity. The model (blue line) predicts the magnetic sensitivities measured in [40] (Red dots). The sensitivities calculated by the model is scaled by a factor of 3.6 to fit experimental values within 5% error. It has to be pointed out that experimental data published in [6] showed on the contrary that the transverse sensitivity (=0ยฐ) is higher than the axial sensitivity (=90ยฐ). This difference can be explained, on one hand, by the capability of the material to expel the magnetic flux instead of fully trapping it, and on the other hand, by the cooling configuration. Indeed, the expelled vortices are pushed by the SC/NC interface and can be concentrated on the last remaining normal conducting region. In [6] , the cavity is set horizontally and cooled from the bottom to the top. The last remaining normal region where all vortices are concentrated is at the top of the equator, the most sensitive region. They achieved an excellent flux expulsion by heat treatment above 800ยฐC with nitrogen doping and also by a fast cool-down. On contrary, the study by [40] was dedicated for almost full flux trapping and flux expulsion was suppressed. The material for our coaxial cavities is not prepared for efficient flux expulsion. Thus, the conclusions can be different in each case. Surface barrier and the flux orientation
When the fluxes are carefully trapped on purpose, our working ansatz of preferential flux trapping normal to the surface can reproduce experimental results reported in [40] . However, their magnetic field measurement outside the cavity apparently showed good agreement with a homogeneous trapping scenario without any preferences in flux orientation. In this section, we consider that the flux can be trapped in parallel to the surface but does not contribute to the power dissipation which happens at the inner surface. We apply the surface barrier model by Bean and Livingston [51] to our case. We first assume that our cavity is a local superconductor but is still relatively clean, because we do not perform any low temperature baking and/or nitrogen doping and infusion. The trapped flux is also assumed to be sufficiently smooth. The magnetic field of one vortex trapped parallel to the surface can be obtained by the modified London equation [20, 52] โ ๐ป(๐ฅ, ๐ง) โ 1๐ ๐ป(๐ฅ, ๐ง) = โ ๐ ๐ ๐ [๐ฟ(๐ฅ)๐ฟ(๐ง โ ๐ง ) โ ๐ฟ(๐ฅ)๐ฟ(๐ง + ๐ง )] (19) where ๐ is the London penetration depth and ๐ง is the depth of the flux. We take the coordinate system as shown in Fig. 10. The image force method is used to fulfil the boundary condition at the surface. If such a vortex exists within the RF penetration, where the RF current is along the x-axis, it vibrates in the z-direction and contributes to additional power dissipation. Figure 10: Flux line trapped in parallel to the surface. The solution of Eq. (19) is a sum of one particular solution of the Green function of the two-dimensional inhomogeneous Helmholtz equation and a general solution of the homogeneous equation without the source term. The former gives the image force per unit length, attractive to the surface ๐ (๐ง ) = ๐ ๐ ๐พ (2๐ง ๐ ) (20) with ๐พ the modified Bessel function of the second kind. The latter is a repulsive force per unit length from the interaction to the external magnetic field ๐ (๐ง ) = ๐ ๐ป ๐ exp (โ ๐ง ๐ ) (21) with a constant ๐ป which satisfies continuity of the magnetic field at the interface between niobium and the vacuum. Bean and Livingston determined the surface barrier by relating these counter-acting forces: ๐ ๐ (๐ง ) = ๐ (๐ง ) โ๐ (๐ง ) . As is well known, even a small ๐ป generates a finite surface barrier, which prevents a trapped flux from escaping toward the vacuum. In our configuration, a very small ๐ป of maximum 100 mG results in the peak of the surface barrier i.e. ๐ ๐ (๐ง ) = 0 deep inside the bulk. We compare [20] the image force to another force from the pinning centers. This pinning force further prevents the escaping trapped flux as a frictional force. In general, estimating the pinning effect is very difficult [53] and its strength can vary by several orders of magnitudes with impurity contents, dislocations, precipitates, grain boundaries etc. Here, we evaluate this in two ways. In the following discussion, we take the parameters of clean niobium: lower critical field ๐ ๐ป ๐1 (0) = 170 mT, thermodynamic critical field ๐ ๐ป ๐ (0) = 200 mT, upper critical field ๐ ๐ป ๐2 (0) = 240 mT, coherence length ๐ =39 nm, ๐ = 32 nm [1] . First, we estimate the lower bound of the pinning force using our results. The almost full trapping of the flux during cooling down was observed in our cavities. This implies that the pinning force is at least stronger than the thermal force [7, 54] ๐ ๐ = ๐ฮ๐ < ๐ ๐ , (22) with the transport entropy ๐ per unit length ๐ = โ๐ ๐๐ป ๐1 ๐๐ (23) If we take an empirical formula ๐ป ๐1 (๐)~๐ป ๐1 (0)[1 โ(๐ ๐ ๐ โ ) ] , we get ๐ = 2๐ ๐ป ๐1 (0) ๐๐ ๐2 ~5.6 ร 10 โ10 ๐๐ ๐2 (24) The maximum ฮ๐ was 80 K between a typical cavity size of 1 m at transition ๐~๐ ๐ = 9.25 K, and we can estimate ๐ ๐ > ๐ ๐ ~4.8 ร 10 โ9 N mโ . (25)
Since the cavity is cooled down along the wall, this force is in either x or y directions in Fig. 10. We assume isotropic pinning effects and apply this lower bound also to the z-direction along which the parallel fluxes migrate. Next, we estimate the pinning force from the sample experiments on critical de-pinning current density ๐ฝ ๐ , at which trapped vortex starts to escape from the pinning centers by Lorentz force; thus, ๐ ๐ = |๐ฑ ๐ ร ๐ |. (26) On low-purity niobium, Das Gupta et al obtained [55] ๐ฝ ๐ ~5 ร 10 A/m near the surface and ๐ฝ ๐ ~2 ร 10 A/m at 3 ๏ญ m deep inside the bulk. Recently, a more relevant experiment on clean fine grain niobium for the cavity application showed [56] ๐ฝ ๐ = 10 โ 10 A m โ and therefore ๐ ๐ = (2 ร 10 โ7 โ 2 ร 10 โ6 ) N mโ (27) The large uncertainty of a factor 10 comes from different models to estimate ๐ฝ ๐ from DC magnetization. These results are consistent with the estimation Eq. (25). Figure 11 compares the above three forces ๐ , ๐ , and ๐ ๐ , with a region with RF field penetration by . Since the external field is as small as 100 mG, ๐ is also small so that the surface barrier of Bean and Livingston exists around 800 nm deep inside the bulk. Therefore, this effect does not keep parallel flux inside the RF penetrating region. Although the ๐ ๐ estimate shows huge uncertainty, the image force is still stronger than the pinning force inside the RF region. This indicates that parallel fluxes, which can contribute to the RF power dissipation, are totally expelled from the surface. This supports our working ansatz, which shows good agreement with the geometrical effect of both coaxial and elliptical cavities as discussed so far. Figure 11: Comparison of image force ๐ (red solid line) external field interaction ๐ (black dashed line), pinning force ๐ ๐ (horizontal red hatch) and RF penetration region (vertical blue hatch) From this consideration, we can argue that the observation in [40] does not contradict our statement. The outer surface of a cavity is usually not as clean as the inner surface and thus the pinning force must be stronger than the above estimation. The flux at the equator may be trapped in parallel at the outer surface and does not contribute to the RF power dissipation. The magnetic sensor placed outside the cavity with some distance may not resolve the parallel flux at the inner surface. If the parallel fluxes were trapped in the inner surface within a few penetration depths, the other oscillation mode in Eq. (9) would change the spatial distribution of power dissipation. Finally, we stress that this discussion is for relatively clean niobium at the inner surface. For cavities after low temperature baking or nitrogen doping, the impurity content just underneath the inner surface within penetration depth is known to be substantial [4, 57] . This would enhance the surface barrier to protect the RF field penetrating into the bulk but the same barrier would prevent the parallel ambient flux escaping from the bulk. From this consideration, such cavities might keep some amount of parallel trapped flux, which may show additional heat dissipation in Eq. (10). CONCLUSIONS
Magnetic sensitivity measurements performed at IJCLab on several type of resonators (QWR, single and double spoke) have confirmed the same behaviour as what has been observed in the literature. Unlike during cooling down, once the cavity becomes superconducting, magnetic shielding is complete. A magnetic shield made of superconducting material is very efficient to shield any magnetic field absent during cool-down, such as fields generated by coils or solenoids for accelerator operation. However, a superconducting shield made of poorly expelling material, such as reactor grade niobium, is totally inefficient to shield any ambient magnetic field. Our measurements reveal a strong geometrical dependence of surface resistance to magnetic field. The real sensitivity, evaluated indirectly and globally by RF power measurements, is consistently lower than the theoretical sensitivity under an assumption of uniformly dissipating trapped flux. Assuming that only the normal component of the residual magnetic field is trapped at the inner surface during the superconducting transition appears to be a reasonable hypothesis. A very good agreement between calculated and measured sensitivities has been obtained for several types of geometries. The magnetic sensitivity would be determined by the orientation of the surface versus the ambient field during superconducting transition. The ambient flux parallel to the surface would be expelled by the image force effect over a depth as short as the London penetration depth at the inner surface of the cavities. The outer surface can be substantially dirtier than the inner surface and magnetic sensor measurement outside the cavity may not resolve the preferential orientation of the trapped flux inside a cavity.
Acknowledgments
The authors would like to thank all the experimental works and efforts of Supratech team and the accelerator department at IJCLab to allow such specific studies to happen on prototypes cavities dedicated to projects. Our special thanks goes to R. E. Laxdal for the useful discussions. Part of this study (MYRRHA SSR) happened during MYRTE project, which has received funding from the EUR-ATOM research and training programme 2014-2018 under grant agreement Nยฐ662186.
APPENDIX Q UENCH AND FLUX TRAPPING
Quenching a cavity during operation could potentially lead to the degradation of its quality factor due to fast flux entry into a normal conducting quench spot. This degradation is fully extrinsic and it only depends on the external residual magnetic field around the cavity during quench [7] . Also, it is of importance to study the evolution of a quench spot by using the dynamics of flux penetration followed by flux rearrangement. The sensitivity of the fluxgate sensor enables us to address these phenomena. So as to study this on Spiral2 QWR, we installed a magnetic sensor probing the vertical magnetic component at the quench location as shown in Fig. A1 and also indicated in Fig. 3 as a probe named quench. Before this experiment, previous works [28, 58] had localized the quench location. In this particular experiment, only one of the fluxgate sensors is read out without multiplexing in order to catch the fast quench events. For signal amplitudes less than 10 mG, the response time of the fluxgate sensor is faster than 30 ms [59] while the data acquisition rate limits the time resolution to 50 ms. Figure A1: Magnetic sensor at the quench location (bottom of the port) and second sound transducer (top of the port) installed on a Spiral2 prototype QWR. Figure A2 depicts how the magnetic field at the quench location is changing after several quenches. Before quenching the cavity, the compensating coils are switched off at time 5 s to change the magnetic background as indicated by the dashed black line. As presented in the main text, the inner conductor completely shields this magnetic field and results in no change of the measured field by the probe. After the first quench around time 38 s, the field promptly drops significantly, indicating the flux penetration into the quench spot. Such a flux entry is possible because of the weak shielding provided by the outer conductor. The measured magnetic field reaches saturation around 3 mG after several quenches around time 70 s. After the saturation, the coils are switched back on and the field is compensated again at time 95 s. Then, after a couple of quenches, the measured magnetic field saturated back to the initial level around time 130 s. The cavity has to undergo at least three quenches to reach saturation Figure A2: Magnetic flux trapping during cavity quench. The transient response of the magnetic field to the quench event is characterized by a narrow peak of width 50 ms (standard deviation), followed by an exponential relaxation with time constant 75 ms and is eventually stabilized to a constant floor as shown in the enlarged plot in Fig. A2. The observed peak is due to the demagnetization effect at the opening of a normal conducting quenched area, in which the magnetic field contained between the outer and inner conductor can tunnel. According to our simulation, the demagnetization factor becomes the maximum when the radius of the quench spot is around 10 mm. Our previous study [60] showed that the time scale of hot spot expanding is less than 1 ms; therefore, this phenomenon is smeared by the time resolution of the detector response. On the other hand, the quench zone cools down with a characteristic time of the order of 50 ms [60] and eventually collapses to be superconducting again. The flux lines penetrating the wall during the quench are pushed inwards by the phase front. As we discussed in the main text, almost all the flux would be trapped by pinning centers of the polycrystalline material to relax the demagnetization at the phase front. Therefore, opening and closing the quenched spot is an irreversible process and results in exponential relaxation in the measurement. Finally, the trapped flux is frozen and results in the constant floor. This determines the total number of trapped flux during a single quench event. Multiple quenches let magnetic flux quanta occupy all the potential pinning centers, similar to an observation in an elliptical cavity [7] . In conclusion of this appendix, flux trapping happens during quench and could be reversible. Several cycles of quench are necessary to reach saturation. It is thus possible to recover a ๐ degradation triggered by a quench event by re-quenching the cavity in a re-optimized magnetic environment instead of warming up a full cryomodule above transition as reported in [7]. The ๐ degradation by quench (if caused by flux trapping) is caused by a non-optimal magnetic shielding. REFERENCES [1] H. Padamsee, J. Knobloch, and T. Hays,
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