Systematical study on superconducting radio frequency elliptic cavity shapes applicable to future high energy accelerators and energy recovery linacs

Elliptic cavities at mediumand high-β range are receiving broader use in the particle accelerator applications. Optimizing the shape of these cavities is a complex and demanding process. In this paper we propose an optimization approach to minimize the ratio of peak magnetic field to the acceleration field Hpk=Eacc while keeping the ratio of peak surface electric field to the accelerating field Epk=Eacc, aperture radius and wall slope angle α at some permitted values. We show that it is possible to substantially vary the cavity geometry without violating the constraints or deteriorating the objective of the optimization. This gives us freedom in designing the geometry to overcome problems such as multipactor while maintaining the minimal Hpk=Eacc. The optimization is then performed to find a set of optimized geometries with minimum Hpk=Eacc for different β’s ranging from 0.4 to 1, different peak surface electric fields, wall slope angles and aperture radii. These data could be generally used as a suitable starting point in designing elliptic cavities.


I. INTRODUCTION
As it was stated in [1] the interest in proton accelerators has necessitated the development of structures that bridge the gap between β ¼ 1 cavities for electrons and low-β resonators for heavy ions.All the superconducting radiofrequency (SRF) cavities in operation today for the speedof-light particles are of the same design: they are so-called elliptic cavities.Their geometry can be extended to lower β by reducing the length of the cells.
The attractive feature of the elliptic cavities is their simple geometry and therefore relative cheapness compared to other types of SRF cavities used for lower β [2], like different λ=4 or λ=2 transmission line cavities or multielement spoke cavities.So, the attempt to expand the area of elliptic cavities to lower β's is natural, and it is realized in many laboratories.
Any superconducting cavity is an expensive structure and it should be properly optimized to obtain maximal acceleration rate E acc .A hard limiting barrier in the SRF cavities is the maximal (or peak) surface magnetic field H pk .So, minimization of the value of H pk =E acc is the first task for the cavity optimization.Minimization of this value is closely related to the decrease of rf losses in the cavity which defines the cost of operation.However, some circumstances make necessary deviations from the optimal shape: higher order modes, multipactor, Lorentz force detuning and others.
In this paper, minimization of the H pk =E acc is done for given values of E pk =E acc , where E pk is the peak surface electric field.The proposed algorithm is interchangeable, and, conversely, minimization of the E pk =E acc can be done for given values of H pk =E acc if the field emission is the dominated factor for the operation field gradient.However, we see a little reason to do this work because the peak electric field, or, more precisely, E pk =E acc , to prevent field emission, is already limited from above as a primary parameter in the present paper.
In this work we try to systematically compare the properties of elliptic cavities at different β, to understand limitations from the lower values of β and to suggest some solutions of the multipactor problem.
We found that some substantial deviations of the shape can be done without deterioration of the value of H pk =E acc .This property opens the way to overcome above-mentioned problems with multipactor, higher order modes, and Lorentz forces keeping minimal peak magnetic field and cryogenic losses.

II. ELLIPTIC GEOMETRY AND SURFACE FIELDS
Optimization of an elliptic cavity consists of the proper choice of geometrical parameters for a contour described by two elliptic arcs and a straight segment tangent to both of them, Fig. 1.
Three of these parameters, length of the half-cell L, the aperture radius R a , and the equatorial radius R eq are not a subject of optimization.They are defined by other physical requirements: L should be equal to the quarter of the wave length, so that the particle could be effectively accelerated in each cell of a multicell cavity, the aperture is defined by requirements to coupling between cells and by the level of wake fields that could be allowed for a given project, the equatorial radius R eq is used for tuning the cavity for a given frequency.
From the remaining six parameters shown in Fig. 1 (A,  B, a, b, d, γ), four only are needed to fully describe the geometry.Here A and B, also as a and b are the half-axes of the equatorial and iris constitutive ellipses, respectively.Some authors follow [3] and use combinations of these parameters: R ¼ B=A, the equator ellipse aspect ratio and r ¼ b=a, the iris ellipse aspect ratio, also as the wall distance from the iris plane d and the wall angle inclination γ.So, we have four dissimilar primary parameters: R, r, d, and γ.
The reason of this choice is explained [3] as follows: R defines a local minimum of the peak surface magnetic field, r defines a local minimum of the peak surface electric field, d allows to reduce the capacitive volume in favor of the magnetic volume and vice versa, in order to balance the peak surface magnetic and electric fields on the cavity walls, γ influences the mechanical behavior of the cavity and controls its inductive volume.
This explanation looks convincing and physically feasible.However, change of the angle γ or the permitted value of E pk =E acc leads to the necessity to optimize again the values of R and r.Besides, there is a dependence of the local minimum of the magnetic field on the iris aspect ratio though not so strong as on the equator aspect ratio, and conversely, the electric peak field minimum depends on the equator ellipse aspect ratio.On the other hand, the choice of four half-axes lengths (A, B, a, and b) as the primary parameters for optimization simplifies the description and does not increase the work for optimization.
Recalculation from A=B, a=b, d and γ to A, B, a and b can be easily done solving a system of algebraic equations.
Equations defining half-axes of the smaller ellipse, a and b, and also coordinates of the point of conjugation (x 2 , y 2 ) (point x2 in Fig. 2), taking as given the values of γ, L, R a , d and r, are the following: Half-axes of the bigger ellipse, A and B, and coordinates of the point of conjugation (x 1 , y 1 ) (x1 in Fig. 2), taking additionally as given the values of R eq and R, are the following: We will use this set of primary geometric parameters for optimization (A, B, a, and b), as well as it is used in many other projects [4][5][6].Some details of the geometry used in the future discussion are shown in Fig. 2. We will measure the angle of the wall inclination as the angle α between the axis of rotation and the straight segment of the wall, l ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , like it was done in our earlier publications.The cavity with α < 90°is known as the reentrant cavity.
As an example, three widely known geometries of the inner half-cells are shown in Fig. 3: TESLA [4] from DESY, the low-loss cavity [5] from JLab, and the reentrant cavity by Cornell [6].
The surface electric and magnetic fields in these cavities are shown in Figs. 4 and 5.One can see that the electric field has irregularities at the points P and Q on the surface of non-reentrant cavities-where the arc conjugates with the segment of the straight line.We believe that the best shape should have more regular dependencies for these curves, but for this purpose we have, possibly, to abandon the paradigm of an elliptic shape and to pass to a smoother contour, e.g., using some kind of splines [7].
In the case of the reentrant cavity, the maximum is not flat as it is on non-reentrant geometry.One can suppose that flattening of this maximum will decrease the peak electric field but the E acc should not change significantly.
Comparing the magnetic fields of the TESLA cell and the reentrant cell having the same aperture we can see that decrease of the maximal field is obtained through the lengthening of the maximal field region.Again, better flattening of the maximal values of this field could decrease the peak field though not much.The area under these curves is approximately the same in both cases.In the case of the low loss cavity the decrease of the field is obtained due to lesser aperture.As it will be shown below, H pk =E acc is a monotone function of the aperture radius and it is smaller here due to a smaller R a .
Because of high cost of superconducting cavities even a several percent further decrease of H pk =E acc is worth the investigations, and we see that this decrease is possible.

III. SOME DEFINITIONS
The accelerating field E acc is defined as the energy increment ΔU, in volts, after the passage of a particle through the inner cell of the cavity to the length of this cell.For β ¼ 1 this length (2L, Figs. 2 and 3) is equal to λ=2, L ¼ λ=4.In the case of β < 1 we have to define the accelerating field as E acc ¼ ΔU=ðβλ=2Þ since the length of the half-cell is L ¼ βλ=4.In the case when the particle velocity does not correspond to the length of the cell, the value of ΔU will differ from the value calculated for the geometrical β geo .For fear of causing confusion, we will use E acc corresponding to the geometrical beta.Usage of the particle velocity β eff higher than β geo is justified by the fact that the transit time factor becomes higher than for β eff ¼ β geo if this exceedance is about 2%-5%.The end cells do not have a physically defined exact length because the field propagates in the beam pipe exponentially decreasing.It appears logical for calculation of the accelerating field in the end cells to use the same definition: E acc ¼ ΔU=ðβλ=2Þ, so that the length of this cell is not required.
It is convenient to compare different cavities with a well known one.So, for the definition of the overvoltage E pk =E acc where E pk is maximal electric field on the cavity surface, we will use the value of e ¼ E pk =E acc =2 so that for the well-known TESLA cavity e ¼ 1.
Analogously, the values of H pk =E acc are normalized to corresponding values of TESLA [42 Oe=ðMV=mÞ] so that h ¼ H pk =E acc =42 is equal to 1 for TESLA cells.{According to our calculations, the normalized magnetic field appears about 1.5% less than this value [41. 4 Oe=ðMV=mÞ].In publications [8] of 1992 (Haebel) H pk =E acc ¼ 41.7, later [9] ( Edwards, 1995) this value is shown as 42, the last publication [4] (Aune, 2000) gives 42.6 Oe=ðMV=mÞ.Normalization for 42 is chosen because: (1) it is convenient to use round numbers, (2) some deviations in different references are about this value.} The dimensions of cavities (always in millimeters)unless specially noted otherwise-are given for the frequency of 1300 MHz, this is a "most popular" frequency of superconducting cavities.However, all the optimizations will be valid for any frequency because the discussed figures of merit, E pk =E acc , H pk =E acc , as well as R sh =Q-the effective cavity impedance, and GR sh =Qthe geometrical factor of losses, do not depend on frequency; dimensions should be scaled only, if a different frequency is used.Here R sh is the shunt impedance of a cavity, Q is the quality factor, and G is the geometrical impedance.

IV. METHOD OF OPTIMIZATION
First of all it should be noted that for calculation of the secondary values for any geometrical parameters we used the SUPERLANS code by D. Myakishev and V. Yakovlev [10] and the TUNEDCELL envelope code written by Dmitri Myakishev [11].These codes have the necessary accuracy needed for our optimization [12].
From the previous optimization [13] for the minimum h we know that this minimum is a monotone function of the overvoltage E pk =E acc , it is decreasing when this value increases, and a monotone function of the wall slope angle α, it is decreasing when the angle becomes smaller.So, the minimal value of h will be at the given permitted values of E pk =E acc and α.
The goal of the cavity optimization is to find this minimum of h under given constraints: the value of e is less than or equal to a given one, and the value of α is bigger than or equal to a given one.We are going to minimize the function hðA; B; a; bÞ under these conditions.The "brute force" or the "grid search" approach is to calculate h on a 4-dimensional grid around some point, to find the minimum of h between these points, to make this point of minimum the central point and repeat the procedure.When the central point is the best point for a given steps in all 4 directions, the value of this step can be decreased, so the minimum can be found with a needed accuracy.However, the example presented in Fig. 6 shows that this method is inefficient when the optimized function belongs to a class of so called ravine functions.
In order to prove this, we tried to solve the optimization problem with the Levenberg-Marquardt algorithm [14] which is gradient based.And in fact, as predicted by literature, due to the ravine shape of the problem, it did not show good convergence compared to a Monte-Carlo based algorithm.Eventually we used a Monte-Carlo scheme that uses information from precomputed values, which is often referred to as Markov-Chain Monte-Carlo in the literature [15].Therefore we pick a starting point from which, by Monte-Carlo, a better point is searched.
At the points marked with crosses around the point A (Fig. 6), the level of the surface is higher than at the point A. This level is decreasing to points B, C, and D. In this example steps along Y should be two times bigger than in the direction X to find the correct minimum.
As it will be shown below (Figs.8-11), we have exactly the case of a ravine function: a rapid descent to the boundary and gently inclined way along the boundary.
The Monte Carlo method to find the minimum was faster than the grid method but it also "stuck" very often not reaching the final minimal point.To check the correctness of the found minima hðA; B; a; bÞ for each value of the slope angle α, as presented in Figs. 17 and 19 to 24, the FIG. 6. Brute force optimization does not lead to the minimum.dependences of A, B, a, and b on α were plotted, an example is in Fig. 18.Very often these dependences appeared to be not smooth, in these cases a more thorough search for the minimum was needed.
In the search for the minimum h we can easily come to the boundary where the limiting value of e and α are reached.To move along the "bottom" of the ravine we can solve the system of equations: Here, Δh is a negative value to be added to h to decrease it.Now, we have three equations and four unknowns: ΔA, ΔB, Δa, Δb.The fourth equation can be added if we want to go to the minimum by the shortest way, so that the length of the vector of increments is minimal: For any Δh we can choose s, preferably small and solve this system of equations.So, we can consider this approach as an implementation of the gradient method in the case of the ravine function and the function under some constraints.
We can solve the system of equations ( 3) and (4) for any negative value of Δh and find a lower point than the point we started of.It looks like we can decrease h indefinitely long time.Of course, this is possible only as long as the absolute value of Δh is small enough so that the partial derivatives do not change significantly.
Nevertheless, it is not clear from this system that the minimum is reached.It becomes clear if we consider the dependencies of h or e on A, B, a, and b.In Fig. 7 the dependencies of h and e on a and b are shown for the case of β ¼ 1, R a ¼ 35 mm, e ¼ 2, α ¼ 100°(see this point in Table II and Fig. 17).The minimum h appears near the point of break of this dependence.When the derivative is considered for the left side of the point of break (Δa < 0), the result of solution of the system (3)-( 4) gives Δa > 0, and vice versa, considering Δa > 0, the solution gives Δa < 0. This contradiction shows that further decrease of h is impossible, so the minimum is reached.
The same consideration can be done for all other partial derivatives in (3).However, in practice, only derivatives of h and e with respect to a and b can change their behavior near the point of minimum h.
The cause of this break is the following: there are two local maxima on the curves of H vs L=L 0 , Fig. 5; one maximum is at L ¼ 0, another one is in the right part of each curve, like the maximum near the A point in the RE cavity.There can be also two local maxima on the flat part of the curve E vs L=L 0 , Fig. 4. The second local maximum E at L=L 0 ¼ 1 appears by thorough optimization and practically cannot be seen on the flat part of the curve 2, Fig. 4.These maxima are not very well pronounced on these graphs because the curves become very flat between them in the process of optimization.When changing, for example, the half-axis a, trying to minimize H pk (and so the h), the bigger maximum of H decreases but the smaller one grows and can overtake the other one.At the point of break (Fig. 7), these maxima become equal, and if we continue to change a, the value of H pk is defined now by the other hump and its dependence on a changes.So, the minimum h is reached when the humps on the curve H vs L=L 0 or on the curve E vs L=L 0 become equal by height.In the last case the decrease of h ¼ H pk =E acc due to equalization of local maxima E becomes not because of minimization of H pk but because of increase of E acc when the force lines are reaching their maximum uniformly around the iris tip.

V. MORE CONSTRAINTS TO THE SHAPE OF THE ELLIPTIC CAVITY
The proposed optimization can lead to shapes that are not realizable as an inner cell of a multicell cavity.So, for the reentrant cavity, and in most of cases for the non-reentrant cavity (except of the case of very big half-axes B), the value of A cannot be bigger than the cavity length.Moreover, taking into account the thickness of the wall, usually 3 mm, the restriction A þ 3 < L should be applied.If we consider the necessary gap between cells needed for welding them together, we should make this restriction even more severe.However, this restriction is valid for the frequency of 1300 MHz.If the same thickness 3 mm is used for a cavity, say, 650 MHz, the restriction due to the touch of the walls becomes A þ 1.5 < L.Here A and L are parameters of cavity at 1300 MHz (not at 650 MHz).So, the scaling factor should be taken into account.The absolute restriction is A < L.
Nowadays, the cavities are mainly stamped from niobium sheets.To guarantee accuracy in the process of stamping the half-cells, the curvature radius of the iris cannot be too small even if it does not increase the E pk .The reasonable minimal value of this radius is twice the thickness of the niobium sheet used in fabrication (usually 3 mm): r c > 6 mm.In case of the frequency different from 1300 MHz, the scaling should be also taken into account.
Even if the curvature is bigger than 6 mm (f ¼ 1300 MHz), the value of a cannot be smaller than 3 mm because the outer surface of the iris should be reachable.This gap should be possibly several millimeters wide to make possible welding of the cells.
For example the TRASCO-ASH cavity for β ¼ 0.47 and frequency f ¼ 704.4 MHz [16] has the radius of curvature at the iris tip r c ¼ 6.1 mm that corresponds to r c ¼ 3.31 mm at f ¼ 1300 MHz.The niobium sheet thickness for the cavities described in this paper was ranging from 3.3 to 4.3 mm.

VI. AN EXAMPLE OF OPTIMIZATION FOR THE TESLA CAVITY
The TESLA cavity [4,9] is thoroughly studied because it was proposed to be used in the International Linear Collider.In spite of "round" values of the elliptical halfaxes (A ¼ B ¼ 42, a ¼ 12, b ¼ 19 mm) it is very close to the optimized one in the way of optimization proposed here.In Fig. 8 are presented the values of normalized parameters h and e, and the wall slope angle α for fixed values of a ¼ 12 mm and b ¼ 19 mm and for different values of A and B around the nominal point.The pink area for e and the blue area for α are "forbidden" areas, i.e., areas beyond the limiting values of e ¼ 0.997 and α ¼ 103.17°.(We call these areas "forbidden" in the sense described in Sec.IV: to find the minimal h for e less than or equal to a given one and α bigger than or equal to a given one, so the values e and α violating these inequalities are "forbidden."Also see the second paragraph in this section).On the lower right picture all three parameters are presented, "twice forbidden" area is colored green.One can see that there is a possible improvement of the value of h from h ¼ 0.99, being in the white area.
After further optimization we will come to the Fig. 9.We have chosen the angle α ¼ 100°that is a little bit less than in the original cavity (103.17°) and have chosen e ¼ 1 (we also prefer "round" values.)Both of these changes improve the value of h from 0.99 to 0.97, but this is not the main result.
We believe that the main result consists in the fact that moving along the colored line and staying in the white area, we can change both A and B in a wide range changing the value of h not more than 0.5%.As will be shown later, this gives a possibility to create a geometry of a cavity, practically optimal, but free of multipactor or tune off the dangerous higher order modes.One can understand that this fact of a very flat minimum where some values can be changed in a broad range is a reverse side of the difficulty to find an exact minimum as described in the section "Method of optimization."

VII. AN EXAMPLE OF OPTIMIZATION FOR THE
SNS ELLIPTIC CAVITY WITH β geo = 0.81 As an example of an elliptic cavity with β ¼ 0.81 let us analyze the shape and fields of the Spallation Neutron Source (SNS) cavity [16].According to the data from this paper the inner half-cell of the cavity for frequency f ¼ 805 MHz has the wall distance d ¼ 15 mm, wall angle γ ¼ 7°, length L ¼ 75. 5  From these data, using formulas ( 1) and ( 2) we calculated the half-axes of ellipses and the normalized peak fields: This discrepancy can be explained by rounded values of geometrical parameters given in the paper (we do not have direct values of half-axes A, B, a, and b), we also believe that the software SUPERLANS used in our calculations has better accuracy [12] than SUPERFISH used in the cited work.
We will perform the optimization for the effective value of beta, because the used beam has this value of speed.For further analysis we will scale the dimensions of this cavity from 805 MHz to 1300 MHz.Let us analyze first the original geometry after scaling.In Fig. 10 are presented the values of normalized parameters h and e, and the wall slope angle α for fixed values of a and b and different values of A and B. The pink area for e and the blue area for α are "forbidden" areas, i.e., areas beyond the limiting values of e ¼ 1.0614 and α ¼ 97°.On the lower right picture all three parameters are presented, "twice forbidden" area is colored green.One can see that there is a possible improvement of the value of h from h ¼ 1.076, being in the white area.
Figure 11 shows the same parameters after optimization using the same limiting values of e and α.The value of h is improved to 1.069, though not so much: 0.6%.But the more interesting feature of this result, as well as in the previous section, is the fact that moving from the point near the lower left corner to the point near the upper right corner and remaining in the noncolored area we will have the value of h practically the same as at the point of optimum (it is in the center) but changing significantly the values of A and B. These changes give us freedom to avoid multipactor or/and to tune the most dangerous higher order modes to a nondangerous position being near the lowest possible value of h, i.e., lowest magnetic field and lowest losses.

VIII. MULTIPACTOR CONSIDERATION
Multipactor is a parasitic rf phenomenon limiting the accelerating rate of a cavity.There are several varieties of multipactor depending on the geometry of an rf device.For a superconducting elliptical cavity the 2-point multipactor on the cavity equator occurs most often.We have a method of predicting the appearance of this type of multipactor without time consuming simulations but immediately from the field calculations done with a 2D program [17].
Zone of existence of multipactor on the cavity equator, Fig. 12, is defined by two coefficients: the geometrical coefficient p ¼ β=ðα þ βÞ and the field coefficient M ¼ eB 0 =mω, where e=m is the specific charge of the electron, ω ¼ 2πf is the angular frequency of oscillations and α and β are factors of proportionality for the longitudinal and radial components of the electric field near the equator, E z ¼ αd and E r ¼ βd, respectively.The points of definition of these factors are at a distance d ¼ 1 mm, Fig. 13, this is a characteristic size for the orbit of an electron in this kind of multipactor at any frequency as follows from [17].For a frequency lower than f ¼ 1300 MHz this value of d can be proportionally increased to have a better accuracy of the calculated fields.Actually, p weakly depends on d because the area of linearity for E z and E r is bigger than 1 mm at f ¼ 1300 MHz.
The value of B 0 in the expression for the field coefficient M is the amplitude of the magnetic field at the equator where the multipactor takes place.B 0 is proportional to the accelerating field E acc in the cavity and usually is maximal or close to the maximal magnetic field in the cavity, B pk .
The value of B pk =E acc is always known when the cavity is being designed.So, knowing B 0 when the multipactor starts, we can easily find E acc for this condition.
Each cavity has its own geometrical coefficient p. Moving along the line p ¼ const, as shown in Fig. 12, we can increase the magnetic field at the point of the equator, i.e., the field coefficient M. When the energy of the primary electrons E p (see the graph) becomes high enough for the secondary emission yield > 1, the multipactor starts.This happens usually near the point M ¼ 2 as can be seen from the figure.The results with different cavities analyzed in the paper [17] and presented in Fig. 12 show that the critical value for multipactor is about p ¼ 0.3, so that we have a very weak and usually easily processed multipactor in the TESLA cavity (p ¼ 0.286), a well pronounced multipactor in the Mark I cavity fabricated specially for studying this phenomenon (p ¼ 0.303), and practically no multipactor at the equator of the Cornell ERL cavity (p ¼ 0.276).More details and references can be found in the cited paper.
The force lines of the electric field near the equator have a shape of elliptic arcs (Fig. 14) as follows from here r is measured from the equator to the axis.The solution of this differential equation is So, the ratio of the half-axes of the elliptic force lines near the equator is b fl =a fl ¼ ffiffiffiffiffiffiffiffi β=α p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p=ð1 − pÞ p and the condition p > 0.3 corresponds to b fl =a fl > ffiffiffiffiffiffiffi ffi 3=7 p ≈ 0.655.As it was stated above in the section "An example of optimization of the TESLA cavity," we can change the dimensions A and B so that the value of normalized magnetic field h will only slightly change in a wide range of A and B. For the example presented in Figs. 8 and 9, let us present values of p in the same coordinates, Fig. 15.We will not overlap this additional graph with the previous ones but it is easily seen that (i) the optimized cavity has significantly decreased p, so that there will be no multipacting, and (ii) this parameter can be further decreased increasing the value of h not more than 0.5%.

IX. HIGHER ORDER MODES OF THE MODIFIED SHAPES
To avoid the higher order modes (HOM) cavity frequencies falling on dangerous frequency spectrum lines of beam is an important design task for the ERL cavity designer.The presented approach to change the cavity shape keeping the optimal value of H pk =E acc for given E pk =E acc , R a , and α will also change the spectrum of the cavity HOMs.We calculated the lowest monopole and dipole modes of the TESLA cavity when the bigger half-axes of it are changed from 42 to 41.1 mm for A and from 42 to 38 mm for the B half-axis.According to Fig. 8 the value of h will not change compared to the original shape.The data are presented in Table I.These data are calculated for an infinitely long multicell cavity consisting of identical cells.
One can see that the change in frequencies is present though not so big as wished it to be.However, the range of changes of these half-axes can be increased, so will be increased the shift of the HOMs.

X. TABLES AND GRAPHS FOR DIFFERENT VALUES OF β
A. Elliptic cavities with β = 1 Results of optimization for β ¼ 1 are presented in Table II and in Figs. 17 and 18. Curves in Fig. 18 are plotted in accordance to data in Table II.
B. Elliptic cavities with β = 0.9 Results of optimization for β ¼ 0.9 are presented in Fig. 19 and Table III.C. Elliptic cavities with β = 0.8 Results of optimization for β ¼ 0.8 are presented in Fig. 20 and in Table IV.

D. Elliptic cavities with β = 0.7
Results of optimization for β ¼ 0.7 are presented in Fig. 21 and in Table V.
Figure 16 shows for one of the points from Fig. 21 that the choice of half-axes A and B can also be done in a very large range with a deviation from the optimal value not more than a fraction of a percent.Here, it is also shown the geometrical parameter p.It is clearly to be seen, also as it was earlier (Fig. 15), that the lowest p corresponds to the maximal radius of curvature at the equator.However, TABLE II.Values of H pk =E acc , GR=Q and dimensions of optimized inner cell (two elliptic arcs and a straight segment) vs the slope angle α.Optimization for min H pk =E acc (or min h).

Optimization for min
Normalized peak magnetic field (also norm.to 42) Geometrical dimensions in mm Loss parameter (also normalized to TESLA) making the equator flat can lead to the one-point multipactor.
E. Elliptic cavities with β = 0.6 Results of optimization for β ¼ 0.6 are presented in Fig. 22 and in Table VI.Here, for comparison, data from [16] are also shown for the SNS cavity with β ¼ 0.61.After scaling from 805 MHz to 1300 MHz the aperture radius R a changed from 43 to 26.627 mm.Point a for the normalized magnetic field is taken from this paper: B peak =E acc ¼ 5.73 mT=ðMV=mÞ that corresponds to h¼1.364.Our calculations, however, give for this point h ¼ 1.294 (point b), and after optimization this shape was slightly improved-to h ¼ 1.277 (point c).
The values of R a and E pk =E acc are chosen so that the data for the SNS cavity are between these values, and the difference in β is not large.

FIG. 3 .
FIG. 3. Comparison of three known geometries of the elliptic half-cells.

FIG. 7 .
FIG. 7. Dependencies of h and e on a and b near the point of min h.

FIG. 14 .
FIG. 14. Electric force lines for the entire half-cell of the TESLA cavity and near the equator.Force lines near the equator are elliptic arcs with ratio of half-axes b fl =a fl ¼ 0.633 < 0.655, see text.

FIG. 16 .
FIG. 16.Map for one of the points from Fig. 21 for β ¼ 0.7.Wide choice of A and B keeping the optimal h.A possibility to get rid of multipactor choosing p < 0.3.

FIG. 18 .
FIG. 18. Elliptic half-axes for the previous figure.Designation of points are the same.See details in Sec.X A.

F
FIG. 21.Normalized magnetic peak field for different angles of the wall slope for β ¼ 0.7.

FIG. 23 .
FIG. 23.Normalized magnetic peak field for different angles of the wall slope for β ¼ 0.5.Points a, b, and c, also as d, e, f are shown for comparison with data from [16], see text.
376 mm, a ¼ 18.677, b ¼ 33.619 mm, E pk =E acc ¼ 2.145, and B pk =E acc ¼ 4.572.The equatorial radius was calculated by tuning the cavity to the given frequency.Our results are 2.3% lower for the E pk =E acc and 3.3% lower for the B pk =E acc value.These values were calculated for β geo ¼0.81.If β eff ¼ 0.83 is used, the deviation of E pk =E acc and B pk =E acc from the values presented in the paper is even bigger: E pk =E acc ¼2.1228 and B pk =E acc ¼ 4.519, or e ¼ 1.0614 and h ¼ 1.076.

TABLE I .
Frequencies of higher order modes for geometries specified in Fig.8: original TESLA cavity and its modification without increasing magnetic field.Smaller elliptic half-axes are kept same: a ¼ 12 mm, b ¼ 19 mm.Bigger half-axes A and B are seen in the Table.

TABLE III .
Values of h ¼ H pk =E acc =42, and dimensions of optimized inner cell (two elliptic arcs and a straight segment) vs the slope angle α.Optimization for min H pk =E acc (or min h).R a ¼ 35 mm, E pk =E acc ¼ 2 (e ¼ 1)

TABLE IV
. Values of h ¼ H pk =E acc =42, and dimensions of optimized inner cell (two elliptic arcs and a straight segment) vs the slope angle α.Optimization for min H pk =E acc (or min h).Optimization for min h,β ¼ 0.8 R a ¼ 30 mm, E pk =E acc ¼ 2 (e ¼ 1)

TABLE V .
Values of h ¼ H pk =E acc =42, and dimensions of optimized inner cell (two elliptic arcs and a straight segment) vs the slope angle α.Optimization for min H pk =E acc (or min h).R a ¼ 35 mm, E pk =E acc ¼ 2.4 (e ¼ 1.2) [16]22.Normalized magnetic peak field for different angles of the wall slope for β ¼ 0.6.Points a, b, and c are shown for comparison with data from[16], see text.

TABLE VI .
Values of h ¼ H pk =E acc =42, and dimensions of optimized inner cell (two elliptic arcs and a straight segment) vs the slope angle α.Optimization for min H pk =E acc (or min h).Optimization for minh, β ¼ 0.6 R a ¼ 30 mm, E pk =E acc ¼ 2.8 (e ¼ 1.4) ¼ 25 mm, E pk =E acc ¼ 2.6 (e ¼ 1.3)