Magnetic ﬁeld enhancement at a pit on the surface of a superconducting accelerating cavity

The thermal magnetic breakdown due to the magnetic ﬁeld enhancement at a pit is the major obstacle to achieving a high accelerating ﬁeld using a superconducting radio-frequency cavity. The well-type pit model has attracted attention as a model of the magnetic ﬁeld enhancement at a pit, whose enhancement factor depends only on an edge radius normalized by the pit width. However, many of the pits found on cavity surfaces are not well type but have gentle slopes, and the impacts of the slope angle on the enhancement factor have not yet been well under-stood. In the present study, we introduce a model that can describe a pit with an arbitrary slope angle and evaluate its enhancement factor. A two-dimensional model is enough to describe the magnetic ﬁeld enhancement at the pit. To evaluate the enhancement factor, an analytical method based on the conformal mapping was developed. The results are compared with those obtained by a simulation method, developed in this paper, that utilizes two-dimensional “electrostatics.” A general formula to evaluate the enhancement factor was derived, which agreed well with the simulation results. We found that not only the normalized edge radius but also the slope angle had a substantial impact on the enhancement factor. A pit with a gentle slope angle yields a much smaller enhancement factor than a well-type pit. The results can be applied to the calculation of the enhancement factors of real pits with arbitrary slope angles.


Introduction
The superconducting (SC) radio-frequency (RF) cavity has been developed as one of the core components of modern particle accelerators [1]. Improvements in cavity fabrication technologies have significantly pushed up the accelerating field, E acc , over the last few decades [2]. Some TESLA-shape nine-cell cavities made of pure niobium have achieved E acc > 45 MV m −1 [3], which is close to the fundamental limit of the niobium cavity in terms of the surface magnetic field [4]. Such a high field, however, cannot regularly be achieved due to thermal magnetic breakdown, so-called quench, triggered by normal-metal contaminants or geometrical defects. At present, the possibilities of involving metallic contaminants during fabrication processes are thought to be significantly reduced by modern surface preparation processes, i.e., electropolishing [5,6], high-pressure rinsing [7][8][9], and clean assembly [10]. On the other hand, a geometrical defect such as a pit accompanying electron-beam welding remains as one of the major concerns even today [11]. An example of a quench-inducing pit is shown in Fig. 1 (see also Refs. [11][12][13][14][15]).
The magnetic field enhancement (MFE) effect is the key to understanding the quench due to a pit. The surface magnetic field in the vicinity of a pit is generally written as PTEP 2015, 073G01 T. Kubo (a) (b) Fig. 1. A quench-inducing pit found on an equator weld of a cavity [15]. The cavity performance was limited by a quench at E acc = 31 MV m −1 . (a) An optical image taken by the Kyoto camera [11]. (b) A profile of the replica extracted by laser microscopy.
where r is a position, H 0 is the surface magnetic field far from the pit, and β(r) is a coefficient introduced to reflect the effect of the pit geometry. β(r) reaches its maximum value, β M , at the edge [16][17][18][19][20]. If the enhanced field, β M H 0 , is large enough, the edge becomes normal conducting due to thermal and magnetic effects, which triggers a thermal runaway [21,22]. A difference of edge shape affects the β M factor and thus the breakdown field. To reveal the relation between the β M factor and the geometry of the pit is the first step to understanding the quench due to the pit. An important contribution to understanding the MFE at a pit was made by Shemelin and Padamsee [19]. They numerically studied the β M factor of the "well-type" pit and proposed a simple relation, β M ∝ (r e /R) − 1 3 , where r e is the radius of the edge and R is half the well width. Their result enabled quantitative discussions on the MFE at a pit and furthermore opened the way to evaluating β M for pits with only the two measurable parameters, r e and R, which was then used for evaluating β M of well-type artificial pits in experimental studies [22]. However, many of the pits found on cavity surfaces are not well type but have gentle slopes, as shown in Fig. 1, where the slope angle is 38 • . It is not clear whether discussions based on the well-type pit model are applicable to such a pit with gentle slopes. The impact of the slope angle on the β M factor have not been well understood.
The present study aims to qualitatively and quantitatively understand the MFE at a pit with any slope angle. We introduce a model that can describe a pit with an arbitrary slope angle. To evaluate its β M factor, an analytical method and a simulation method were developed. By using these methods, the β M factor of the model was evaluated. An extremely useful formula for the evaluation of the β M factor was derived, which agreed well with the simulation results. Based on the results, we discuss the impact of the slope angle on the β M factor and an application of the model to real pits.

Model
A typical wavelength of an SCRF cavity, O(10 −1 ) m, is much larger than the size of pits, O(10 −5 -10 −4 ) m; the RF wavelength can be regarded as infinity. Thus, in the following, we study a model in the framework of magnetostatics.
The magnetic field is enhanced at edges perpendicular to the direction of the surface magnetic field [19][20][21][22], which correspond to A and C in Fig. 1. The detailed shapes of other parts of the edges are not essential. Thus the MFE of a pit is well approximated by a groove perpendicular to the direction of the surface magnetic field. In fact, the β M factor of a three-dimensional well-type pit PTEP 2015, 073G01 T. Kubo   Fig. 2. Model of pit. Gray and white regions represent the SC in the Meissner state and the vacuum, respectively. r e is the radius of the edge, R is half the width of the pit aperture, and πα (0 < α < 1/2) is the slope angle. The arcs of the edges are smoothly connected to the slopes and the flat surfaces. has the same functional form as the two-dimensional one, as shown in the previous studies [19,22]. Furthermore, since the magnetic field attenuates toward the bottom [19][20][21][22], the bottom shape is not essential for describing the MFE. We developed the two-dimensional model shown in Fig. 2. Its geometry is parametrized by r e , R, and α, where r e is the radius of the edge, R is half the width of the pit aperture, and πα (0 < α < 1/2) is the slope angle. Note that a pit with α → 1/2 corresponds to a well-type pit with an infinite depth.

Formulation for analytical evaluations
Let us formulate the MFE in the vicinity of a general two-dimensional irregularity [see Fig. 3(a)]. The Maxwell equations for two-dimensional magnetostatics in the vacuum are given by rot H = 0 and div B = 0, where H is a magnetic field, B = μ 0 H is a magnetic flux density, and μ 0 is the magnetic permeability of the vacuum. By introducing a magnetic scalar potential φ(x, y), the problem is reduced to a Laplace equation, (2) Then the magnetic field is given by H = −grad φ = (−∂φ/∂x, −∂φ/∂y, 0). On the other hand, by introducing a vector potential A = (0, 0, −μ 0 ψ(x, y)), where ψ is a real function, the problem is reduced to the Laplace equation (2) ψ(x, y) = 0. The magnetic field is given by H = μ −1 0 rot A = (−∂ψ/∂y, ∂ψ/∂x, 0). Since both approaches should lead to the same magnetic field, we find which are the Cauchy-Riemann conditions. Thus a function defined by is an holomorphic function of z = x + iy, called the complex potential. When (z) is given, the components of the magnetic field are derived from where the property of the holomorphic function that d (z)/dz = ∂φ/∂x + i∂ψ/∂x is used. Thus the two-dimensional magnetostatics problem is reduced to the problem of finding (z).
The complex potential (z) can be derived from a complex potential (w) on the w-plane with a flat surface [see Fig. 3(b)] that is connected to the physical z-plane with a conformal mapping, z = F(w). The map is given by the Schwarz-Christoffel transformation [23], where f (w) is a function on the w-plane, and P and Q are constants. Here we introduce a complex potential on the w-plane, Then the complex potential on the z-plane is given by where F −1 is the inverse function of F. By using Eqs. (4)-(6), we find The β M factor is calculated by the maximum value of β(x, y). In principle, the β M factor of any two-dimensional irregularity can be evaluated by using Eq. (7), if an explicit form of the Schwarz-Christoffel transformation and its inverse are known.

Simulation method
A simulation code for the "electrostatic" problem is useful to evaluate the β M factor of our model, because the β M factor of a two-dimensional irregularity on an SC is equivalent to the electric field enhancement factor (β E factor) of the same irregularity on a conductor. Let us consider Fig. 3 again. This time, we regard the gray and white regions as the conductor and the vacuum, respectively, and replace the magnetic field far from the irregularity H = (H 0 , 0) by the electric field E = (0, −E 0 ). In much the same way as the magnetostatics problem, the electrostatics problem on the x y-plane is also reduced to the problem of finding a holomorphic complex potential on the z-plane, E (z) [24,25]. The components of the electric field are given by The complex potential E (z) can be derived from a complex potential E (w) on the w-plane with a flat surface that is connected to the physical z-plane with a conformal mapping given by Eq. (5). We introduce a complex potential on the w-plane, E (w) = −i E 0 w with E 0 ≡ P E 0 , which reproduces the uniform electric field on the w-plane; T. Kubo On the other hand, the electric field is enhanced and reaches its maximum near the edge; Then the complex potential on the z-plane is given by By using Eqs. (8) and (9), we find which completely corresponds with that of the magnetic field. Then we find Thus the β M factor can generally be evaluated by calculating the β E factor of the same irregularity. We use POISSON, a two-dimensional electrostatic simulation code [26][27][28]. Figure 4 shows an example simulation domain, which corresponds to the right half of Fig. 2. The left boundary corresponds to the plane of symmetry. The lower conductor has a pit with a geometry parametrized by α, r e , and R. Another conductor plane is located at the upper boundary. The difference of electrostatic potentials between the upper and lower conductors yields an electric field, which is uniform, E = (0, −E 0 ), far from the pit. In the vicinity of the edge of pit, the electric field is enhanced and reaches its maximum value, E = E max . Then the β E factor is given by |E max |/E 0 , and we find where Eq. (11) is used. In an implementation of the simulation, we must pay attention to locating the positions of the right and upper boundaries well away from the pit and adopting a fine-enough mesh near the pit. Otherwise, the simulation results depend largely on these parameters. We placed the right and upper boundaries at x = 10R and y = 8R, respectively, which are far enough away from the pit. The mesh size was set at 0.003R (< 10 −2 R), which is small enough for the evaluation of T. Kubo the model with an edge radius r e > O(10 −1 )R. For the evaluation of the model with r e < 10 −1 R, a simulation with smaller meshes is preferred.

Evaluation of the β M factor of the model 3.1. Sharp-edge model
Let us begin with a sharp-edge model as shown in Fig. 5, which can be regarded as the limit, r e /R → 0, of the original model. This simplified model is not only an instructive exercise of the formulation developed in the last section, but also a rough sketch that provides us with a rudimentary understanding of the MFE at the pit [29,30]. The mapping that connects the physical z-plane [ Fig. 5(a)] to the w-plane [ Fig. 5(b)] is given by Constants P = p 0 R and Q = q 0 R are obtained by imposing the conditions that A and B on the w-plane are mapped into A and B on the z-plane, respectively (see Appendix A): Then β(x, y) can be evaluated by using the general formula, Eq. (7): Note that β(∞) = β| w→∞ = 1, which is consistent with the fact that the magnetic field is not enhanced far away from the pit. By using Eq. (16), β at A and B are immediately obtained: The magnetic field is enhanced at the edge and vanishes at the bottom. These behaviors qualitatively correspond with the observations so far [19][20][21][22]. In order to evaluate β at an arbitrary z = x + iy, an explicit form of f 0 (F −1 (z)) is required [see Eq. (16)]. This task is approximately achieved by focusing on a small region around an edge [24,25]. Let us look at the vicinity of the edge A, which corresponds to the vicinity of A on the w-plane, w = 1 + (| | 1). Then f 0 (w) can be approximated by f 0 (w) 2 α α , and Eq. (5) is calculated as where r ≡ |z − R| = (x − R) 2 + y 2 is the distance from the edge A. By using Eq. (19), we can evaluate β for arbitrary (x, y) near the edge, in the range given by | | 1 or r 2 α p 0 R/(1 + α). The round-edge model shown in Fig. 2, which we call the original model, can also be described by the sharp-edge model at some level. The original model has the same geometry as the sharp-edge model except for the arcs: the geometry of the outside of the arcs of the original model corresponds  with that of the sharp-edge model at r > r ≡ r e (1 − cos πα)/ sin πα [29]. Thus the surface magnetic field and β at the outside of the arcs of the original model should be approximated by those of the sharp-edge model at r > r . Then β at the boundary of the arc and the flat surface can be evaluated by substituting r = r into Eq. (19), which is expected to be a rough estimate of the β M factor when r e /R is small: In the following, we evaluate the original model with a finite r e /R, where we found the relation Eq. (20) is reproduced at the sharp-edge limit, r e /R → 0.

Round-edge model
For a quantitative evaluation of the β M factor of the original model, the edge radius must be explicitly taken into account. An infinite number of triangles can express the arc of the round edge, as shown in Fig. 6. This approach can be regarded as the generalized version of the chamfered-edge model studied by the author [30]. The mapping that connects the physical z-plane [ Fig. 6(a)] to the w-plane [ Fig. 6(b)] is given by Eq. (5) with f (w) = lim n→∞ f n (w) [23], where and The constants Q = q R, P = p R, and are determined by the conditions that B , A 1 , and A 0 on the w-plane are mapped onto B, A 1 , and A 0 on the z-plane (see Appendix B). The first condition yields The second and the third conditions are somewhat complicated, but, under the assumption 1 (see Appendix B), they are reduced to    r e /R. Since p = 0, Eq. (24) yields → 0 at r e /R → 0, which is shown in Fig. 7(a). Then Eq. (22) is reduced to w k → 1 (k = 1, . . . , n), and Eq. (21) is reduced to f n (w) → f 0 (w). Since Then Eq. (25) yields p → p 0 , which is shown in Fig. 7(b). Thus, when r e /R → 0, f n (w) → f 0 (w), p → p 0 , and q = q 0 , which shows that the mapping is reduced to the sharp-edge model.
The MFE is expected to reach its maximum near the tip of the edge, which corresponds to w w M ≡ (1 + w n )/2 on the w-plane. Then the β M factor is approximated by β M lim n→∞ |1/ f n (w M )|. After some calculations (see Appendix C), we find where Eq. (24) is used. Figure 8 shows β M calculated by Eq. (26) as functions of r e /R. At the sharpedge limit, r e /R → 0, as is obvious from Eq. (26), log 10 β M is a linear function of log 10 (r e /R) with a negative slope −α/(1 + α); the result of the sharp-edge model, Eq. (20), is reproduced. As r e /R increases, β M continues decreasing, but the linearity of log 10 β M is gradually lost due to the r e /R dependence of p [see Fig. 7(b)]. As r e /R further increases, the formula ceases to be applicable. Not only the assumption 1, but also the postulate that β M reaches its maximum in the vicinity of w M are not valid in these regions, where the analytical approach is not applicable.

Discussion
As α → 1/2, our model approaches the well-type pit with an infinite depth. By using the formula Eq. (26), we find β M = 1.16 × (r e /R) − 1 3 for r e /R 1, which agrees well with the simulation result of the three-dimensional well-type pit model, β M = 1.17 × (r e /R) − 1 3 , by Xie and Liepe [22]. The two-dimensional and three-dimensional models have a good agreement not only in the functional form ∝ (r e /R) − 1 3 but also in the coefficient. This agreement supports the validity of modeling the pit with the two-dimensional model and shows how effective the formula is. It should be emphasized that the agreement is not surprising; the magnetic field is enhanced at edges perpendicular to the direction of the surface magnetic field, and thus, modeling a cross-section parallel to the direction of the surface magnetic field is enough to describe the MFE, as mentioned in Sect. 2.
Not only the edge radius r e /R, but also the slope angle πα has a substantial impact on β M . This behavior cannot be derived in the framework of the well-type pit model with the fixed slope angle πα = π/2. As shown in Fig. 8, the dependence on the slope angle becomes large at r e /R 1. Let us consider the worst case. As discussed in previous studies [16,19], the effective radius of the edge is limited below by the order of the normal conducting skin depth ∼1 µm for RF frequencies of O(1) GHz. Thus we can set r e = 1 µm as the worst case. For a typical pit size, R ∼ 10 2 µm, the formula, Eq. (26), is available because r e /R 1. Figure 9 shows β M factors in the worst case as functions of slope angle. The larger the slope angle, the larger the β M . A pit with a slope angle of 90 • and R = 50 µm can reach β M > 4. Pits with the smaller slope angles, however, cannot yield such a large value. For example, when the slope angle is 10 • , even in the worst case, β M = 1.4-1.5 for R = 50-200 µm. The previous study on the well-type pit model by Shemelin and Padamsee [19], which concludes that β M > 4 is the worst case for R = 50 µm, is valid only when the slope angle is 90 • . Real pits, which have more gentle slopes in general, yield much smaller β M than the well-type pit.
The present model and the results for the β M factor are directly applied for the evaluation of the real pits found on the surface of the cavity. This can be done by extracting the model parameters r e , R, and πα and substituting them into the formula, Eq. (26), or reading from Fig. 8. The parameters can be extracted from a cross-section of the pit parallel to the surface magnetic field obtained by using, for example, laser microscopy combined with the replica technique [14,18,31]. Taking Fig. 1(b) as an example, R 1.0 × 10 2 µm and πα 0.21π (38 • ) are extracted. r e is given by the minimum value of the curvature radius along the edges. Figure 10 shows the profile and the curvature radius near the edge C, where the curvature radius is calculated by r cur = − 1 + h 2 3 2 / h and h(x), the polynomial corresponding to the fitting curve. Then r e = min[r cur ] 3.0 × 10 1 µm is obtained. By using the formula, Eq. (26), we obtain β M 1.5. We should comment on the study by Xie et al. [22], where β M factors of artificially formed pits on the surface of an SCRF cavity were evaluated based on the well-type geometry, and substitute r e and R extracted from the laser microscopy into their simulation result, β M = 1.17 × (r e /R) − 1 3 . As shown in the figures of Ref. [22], however, the profile of the artificial pits is not well type. Actually, it has a slope angle 80 • . Thus an application of the well-type pit model might overestimate the β M factor of the artificial pit. Since the dependence on the slope angle becomes significantly large at r e /R 1, the overestimation becomes worse as r e /R decreases. For example, when r e /R = 0.01, we obtain β M (80 • ) = 4.8 and β M (89.9 • ) = 5.3; the well-type pit model overestimates by 10% in this example.
The viewpoint based on the model formulated in this paper might reduce considerably the time needed for local mechanical grinding [15,[31][32][33]; the grinding removes pits one by one and thus 10 Fig. 1(b). The solid curve represents the fitting curve of the profile data extracted from laser microscopy. The red dashed curve represents the curvature radius calculated from the fitting curve. takes a long time if there are many pits. Adopting our model, we can evaluate the β M factors of pits by using experimentally observed model parameters. Then it becomes possible to select only those with large β M . The time required for the curing of the SCRF cavity should be reduced, in particular for the case when a lot of pits with shallow slope angles exist. It is important to note that the present model and results are applicable not only to pits on current SCRF cavities based on the niobium technology, but also to pits on future SCRF cavities based on new technologies such as a multilayer SC [34,35] or an alternative bulk SC material [36,37]. Furthermore, the present model and results can also be applied to pits on normal conducting cavities, although the motivation of this study originated from one of the challenges that SCRF technology has been facing.
In this paper, the relation between the β M factor and the geometry of a pit is analyzed. This is, however, merely the first step to understanding quenching due to a pit. Thermal calculations taking the magnetic field distribution on the surface into account are the next step.

Summary
The magnetic field enhancement effect is the key to understanding the thermal magnetic breakdown triggered by a pit. As a model of the magnetic field enhancement at a pit, the well-type pit model had been studied strenuously so far. Real pits found on cavities, however, generally have gentle slopes, as shown in Fig. 1. The magnetic field enhancement at a pit with a realistic slope angle had not been well understood.
A two-dimensional model is enough to describe the magnetic field enhancement at pits, because only the edges perpendicular to the direction of the surface magnetic field are essential for the enhancement. In this paper, we studied a two-dimensional model, shown in Fig. 2, that can describe a pit with an arbitrary slope angle. The pit geometry is parametrized by r e , R, and πα, where r e is the radius of the edge, R is half the width of the pit aperture, and πα (0 < α < 1/2) is the slope angle. To evaluate the β M factor based on this model, an analytical method was developed based on the conformal mapping. A simulation method that utilizes a simulation code of the "electrostatics" problem was also developed. By using these analytical and simulation methods, the β M factor of a pit with a slope angle was evaluated. An extremely useful formula, Eq. (26), was analytically derived, and it agreed well with the simulation result. The results are summarized in Fig. 8.
We confirmed, at α → 1/2, that the formula reproduces the three-dimensional simulation results for the three-dimensional well-type pit model. This reflects the validity of modeling the pit with a two-dimensional model. Then the impact of the slope angle on the β M factor were discussed, which could not be considered in the framework of the well-type pit model. We found that a pit with a gentle slope angle yielded much smaller β M than that of the well-type pit; a pit with a slope angle = 10 • and R = 50-200 µm yields β M = 1.4-1.5, even at the worst case. The previous study [19], which concluded that a pit with R = 50 µm can reach β M > 4 at the worst case, is valid only when the pit has a slope angle 90 • . For the application of the present model, β M factors of real pits found on cavities can be evaluated by extracting the model parameters from pit profiles and substituting them into the formula or Fig. 8. The above application allows us to selectively remove pits with large β M factors and might reduce time to cure cavities by local mechanical grinding. The present model is applicable not only to the currently developed niobium cavities, but also to those made of a multilayer superconductor or an alternative material, and even to normal-conducting RF cavities.