PHYSICAL REVIEW SPECIAL TOPICS - ACCELERATORS AND BEAMS 12, 114701 (2009)

Optimal choice of cell geometry for a multicell superconducting cavity

Valery Shemelin*
Cornell Laboratory for Accelerator-based Sciences and Education (CLASSE), Ithaca, New York 14853, USA
(Received 3 March 2009; published 11 November 2009)
An algorithm for optimization of the multicell cavity cells is proposed. Inner cells are optimized for
minimal losses or minimal magnetic field, when the aperture diameter, Epk =Eacc —the ratio of peak
electric field to the accelerating field, and the wall slope angle are given. Optimization of the end cells is
done for minimal losses or maximal acceleration in them. Two shapes of the end cells—with and without
the end irises—are analyzed. This approach facilitates further optimization for higher order modes
extraction because it permits keeping the achieved optimal values nearly the same while changing some
dimensions of the cells. Comparison of the proposed cavity geometry with the TESLA cavity geometry
illustrates the traits of the presented approach. It is also shown that lower values of the wall slope angle,
which lead to the reentrant shape for the inner cells, are also beneficial for the end cells. For the Cornell
Energy Recovery Linac most dangerous are dipole modes causing the beam breakup (BBU).
Minimization of power of higher order modes (HOMs) in a multicell cavity was done using derivatives
of the BBU parameter with respect to geometric parameters of the cavity cells. As a starting point of
optimization, the shape with minimal losses at the fundamental mode was taken. Further changing the
shape for better propagation of HOMs was done with degradation of the fundamental mode loss parameter
G
Rsh =Q within 1% while decrease of the BBU parameter was nearly 3 orders of magnitude. The BBU
threshold current tends to be inversely proportional to this parameter.

DOI: 10.1103/PhysRevSTAB.12.114701 PACS numbers: 29.20.c, 85.25.j

shape was remarkably well optimized as we will see in the

I. INTRODUCTION
following sections.
A superconducting (SC) cavity is an expensive device as Further development of the individual cells’ shape is due
regards to material, technology of mechanical and chemi- to progress achieved in developing clean cavity preparation
cal processing, and conditions of work: vacuum, liquid procedures that reduced significance of the peak electric
helium, adverse effect of outside magnetic fields, etc. field and led to understanding of the principal importance
On the first glance, the choice of a superconducting of the ratio of peak magnetic field to the accelerating field
cavity shape does not seem to be a complicated issue and [3] and proposal of usage of the reentrant (RE) shape [4],
a wide variety of cavity shapes can be fabricated using which decreases the value of Hpk =Eacc sacrificing to some
nearly identical manufacturing procedures. However, extent the ratio Epk =Eacc . This approach brought world
along with the surface treatment, the shape determines record accelerating gradient values of 47 MV=m [5] and
nearly all basic cavity figures of merit, such as achievable 52 MV=m [6] for a single-cell 1300 MHz SC cavity with
acceleration (accelerating field Eacc ), peak electric and 70 mm aperture, 20% higher Epk =Eacc , and 10% lower
magnetic field (Epk and Hpk ), and minimal losses (G
Hpk =Eacc than in the TESLA cavity.
Rsh =Q, geometric factor times specific shunt impedance). There is still a lot to be done to convince the SRF
It plays a major role in setting limits to maximal beam community that reentrant (RE) SC cavities should be ac-
current and minimal emittance, which strongly depend on cepted for broad use in accelerator applications. One of the
beam excitation of higher order modes (HOMs). A poorly most pressing issues is to demonstrate that the high accel-
chosen shape can be responsible for multipacting, which erating gradient can be achieved in a multicell RE cavity.
can limit the accelerating voltage, impair vacuum, and heat In spite of the success of single-cell reentrant cavities,
the cavity surface. some people expect problems with chemical treatment of
Many of the problems associated with the choice of multicell cavities. These problems are now being studied at
cavity shape were overcome early on by adopting spheri- Cornell. Behavior of HOMs is also to be studied for this
cal/elliptical cell shape with tilted end plates [1]. A cumu- geometry.
lative experience of the superconducting radiofrequency In what follows, the author wants to separate the prob-
(SRF) community was later applied to the development of lem of optimization for minimal losses of the cavity and
the cavities for the TESLA project [2]. The resulting cavity the problem of HOMs extraction. The inner cavity cells can
be optimized for maximal GRsh =Q, whereas the end cells
*vs65@cornell.edu can be tuned for extraction of HOMs. The end cells’

1098-4402=09=12(11)=114701(15) 114701-1 Ó 2009 The American Physical Society

VALERY SHEMELIN Phys. Rev. ST Accel. Beams 12, 114701 (2009)

GRsh =Q can be sacrificed to some extent because their cells only, so the total relative increase will be smaller for a
contribution to the total losses is limited. The similar multicell cavity. (3) If necessary, change the shape of the
procedure was done for the TESLA cavity optimization inner cells, keeping in mind their bigger contribution into
(Haebel, 1992 [2]). total losses.
This paper consists of two major parts. In the first part,
optimization for maximal GRsh =Q (minimal losses) is II. OPTIMIZATION OF A MULTICELL CAVITY
done for inner and end cells. The description of tuning FOR MINIMAL LOSSES
the end cells for HOMs extraction is the subject of the
A. Inner cell shape: The geometry for optimization
second part.
As a preliminary work, a thorough study of both non-RE We employ the construction of the cell profile line as
and RE geometries of the inner cells was performed and two elliptic arcs with half-axes A, B, a, and b, separated by
optimal shapes for these cells were found in [7]. Any wall a straight segment of length l, Fig. 1, conjugated to arcs.
slope angle, the Epk =Eacc ratio, and aperture radius can be We talk about a nonreentrant shape if the angle
is more
given depending on the project requirements. All other than 90. The reentrant cell can also have a straight seg-
geometrical dimensions of the inner cell in the elliptic ment. In earlier optimization [4] the length of this segment
approximation for both equatorial and iris region can be appeared to be zero after consecutive steps of optimization.
found using the tabulated data or following the procedure The radius of the iris aperture Ra is chosen by some
described in these papers. additional considerations; it is not the task of this optimi-
Shape and parameters of the end cells of a multicell zation and should be taken as an independent parameter.
cavity are mostly responsible for propagation of HOMs out The length L of the half-cell is taken as a quarter of the
of the cavity. At the same time they should not reduce the wavelength, and boundary conditions correspond to the 
total value of acceleration and not be the weakest link in mode. The procedure of search for the best shape consists
the sense of Epk or Hpk . in calculating Hpk =Eacc or G
Rsh =Q for all suitable sets of
Under preset limitations on the aperture, Epk =Eacc , and the half-axes; the value of Req is used for tuning the inner
cells to the operating frequency. (In further discussion the
the wall slope angle, the optimization of the inner cells
tuning is done to the resonance frequency f ¼ 1300 MHz,
consists in minimization of losses for a given Eacc .
which is the fundamental  mode of the cavity).
Acceleration in the end cells is not necessarily the same
Sure, a more intricate profile line can give a better
as in the inner ones. One can optimize the end cells in two
eventual result, and we used earlier a description of the
different ways: either for maximal acceleration regardless
profile with six circular arcs [4]. However, an improvement
of the power loss ( min Epk =Eacc , because Epk should be
of Hpk =Eacc was not more than 1% in the case of six circle
the same as in the inner cells), or for maximal acceleration
arcs in comparison to two elliptic arcs though this optimi-
per unit power ( max G
Rsh =Q for the end cells). The first
zation can be incomplete because of its complexity.
optimization brings maximal acceleration for a given num-
Adoption of an elliptic arc for the equatorial area is
ber of cells, the second one—for a given power. While the crucial. The problem of cavity electric strength led to the
difference in results can be small, it is important to under- iris edge to take the shape of an elliptic arc a long time ago.
stand what optimization is being performed. We apply an ellipse to the inductive part of the cell because
A short recollection of the inner cell shape optimization now we have a problem of magnetic strength.
is presented here. More details can be found in [7]. Some
aspects of the end cell optimization are also discussed in
the present paper.
Optimization of a SC cavity for minimal losses of the
fundamental mode power is necessary because these losses
define the major part of total power needed for cryogenics
in the continuous wave operation. On the other hand, the
current in the accelerator is limited by HOMs excited in the
cavities by the electron bunches, and to minimize this
detrimental effect one should change this initially found
‘‘best’’ shape.
We suppose to resolve this contradiction in the following
way. (1) Find the best shape of the inner and end cells of the
cavity from the viewpoint of minimal losses. (2) Change
the shape of the end cells, even end half-cells only, to
improve coupling between the cavity and the beam pipes
keeping the increase of fundamental losses in the end cells FIG. 1. (Color) Geometry of the inner cell: nonreentrant (left)
at some limited level. The losses will increase in the end and reentrant (right) shapes.

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OPTIMAL CHOICE OF CELL GEOMETRY FOR A . . . Phys. Rev. ST Accel. Beams 12, 114701 (2009)

In the optimization with two elliptic arcs without a gives 42:6 Oe=ðMV=mÞ. Normalization for 42 is chosen
straight segment we have three independent parameters because: (1) it is convenient to use ‘‘round’’ numbers,
for optimization: three half-axes (A, B, and a), the fourth (2) some deviations in different references are about this
one (b) is defined by geometrical restrictions. value.} Another defining parameter, Epk =Eacc which is
If we introduce the limiting angle of slope we need to close to 2 for the TESLA cells, was kept for the upper
search the minimum (of Hpk =Eacc or losses) in a 4D space: curve and increased for the next ones. (Again, our calcu-
A, B, a, and b under two limiting conditions: Epk =Eacc is lations give for the TESLA regular cells Epk =Eacc ¼ 1:99.
less and the angle
is bigger than definite values. As a This is why this point slightly falls out of the curve.) 10%
result of these conditions the value of l can be not a zero higher electric peak field decreases the magnetic peak field
anymore. by 7% as can be seen from the end point of the second solid
Calculations were done with TUNEDCELL code that is a curve. A sacrifice of the next 10% in electric field de-
wrapper code for SLANS and was developed especially for creases h more only by 2% [4] giving in sum 9% in h
fast optimization [8]. The SLANS code [9] is known as a for þ20% in Epk =Eacc . The aperture radius Ra ¼ 35 mm
code with high accuracy [10] that is necessary for our goal. for the first group of curves is the same as in TESLA inner
cells while it is 30 mm for another group. Influence and a
B. Results of optimization inner cells possible benefit for higher gradient from decreasing the
aperture is much higher than from increasing the over-
Results of optimization for a minimal magnetic peak
field are presented in Fig. 2 (solid lines). For easier com- voltage Epk =Eacc . Smaller aperture causes smaller cou-
parison with the well-known TESLA cavity [2], with
¼ pling and hence worst field flatness, also as higher
103:2 which is a prototype for the ILC, the values of wakefields. However, it is shown that ILC will tolerate
Hpk =Eacc on the graph are normalized to corresponding the cavities with the new (reentrant) shape and the smaller
values of TESLA (42 Oe=ðMV=mÞ) so that h ¼ iris diameter [11].
Results of optimization for maximal G
Rsh =Q are pre-
Hpk =42Eacc is equal to 1 for TESLA cells. {According to
sented in Fig. 3. They are also normalized for the TESLA
our calculations, the normalized magnetic field appears value: g
r=q ¼ ðG
Rsh =QÞ=ð30 800 Ohm2 Þ.
about 1.5% less than this value [41:4 Oe=ðMV=mÞ], as The extreme left points of curves in Figs. 2 and 3
shown on the graph, Fig. 2. In publications [2] of 1992 correspond to minimal length of the straight segment:
(Haebel) Hpk =Eacc ¼ 41:7, later (Edwards, 1995) this l ¼ 0 when the cell presents two conjugated elliptic arcs,
value is shown as 42, the last publication (Aune, 2000) the geometry discussed earlier [4,12].

FIG. 2. (Color) Normalized magnetic peak field for different FIG. 3. (Color) Normalized loss parameter for different angles
angles of slope. Solid lines present optimization for min h, of slope. Solid lines are for max G
Rsh =Q, dashed lines are for
dash lines are for max G
Rsh =Q. minimal h. (Graphically both lines nearly overlap).

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VALERY SHEMELIN Phys. Rev. ST Accel. Beams 12, 114701 (2009)

When optimizing for max G
Rsh =Q, the values of h

slightly increase, these dependences are shown in Fig. 2
by dashed lines. When we optimize for min h, values of G

Rsh =Q become somehow smaller than immediately by
maximization of G
Rsh =Q. There is an attempt to show
this in Fig. 3 by dashed lines but actually these lines
graphically coincide with the solid ones. This means that
we do not need to optimize for max G
Rsh =Q—optimiza-
tion for min h gives us the shapes that have practically
minimal losses. When we try to optimize for max G

Rsh =Q, the maximal magnetic field shifts to smaller radius
because the losses depend not only on the value of the field
but also on the value of area where it exists. Smaller radii FIG. 5. (Color) Geometries of the cells (left picture) with min h
can give a smaller contribution to losses even if they have a (red solid line) and max G
Rsh =Q (blue dashed line) practically
higher field. However, this change is negligible if we coincide. Difference in magnetic fields along the profile lines of
optimize for max G
Rsh =Q. A minimal peak magnetic the cell (right) is also negligible. A, B, a, and b are half-axes of
field secures low losses in the whole cavity. elliptic arcs.
Another remarkable fact is that the cell-to-cell coupling optimized geometry takes its place on our graphs corre-
for the inner cells optimized for minimal Hpk =Eacc in- sponding to its aperture and slope angle (98.0 ). Its posi-
creases when passing from non-RE to RE geometry, tion on the graphs is defined also by Epk =Eacc ¼ 2:22 and
Fig. 4. Even though this benefit is not very significant, Ra ¼ 30:49 mm (recalculated to 1300 MHz). These ex-
about 0.1%, it denies the anxiety that the coupling can be amples, both TESLA and LL, show that cavity cells can be
lower for the RE case. compared only taking into account their Epk =Eacc ratio,
Distributions of the magnetic field along the profile line aperture radius Ra , and the wall slope angle. After the
of the cells with Ra ¼ 30 mm, Epk =Eacc ¼ 2:2 and with choice of these values, the correct optimization should be
minimal slope angle (lowest curves in Fig. 2 and the uppers done, and other figures of merit can be obtained. There is
in Fig. 3, extreme left points with l ¼ 0) for both cases of no necessity to optimize specially for low losses because
optimization are shown in Fig. 5. optimization for lowest peak magnetic field successfully
There is also shown on the graphs of Figs. 2 and 3 the serves to both goals: highest gradients and lowest losses.
cell of the low-loss (LL) cavity of JLab [13]. This well- Calculation of elliptic arc parameters, namely A, B, a,
and b, for both cases of optimization, appeared a time-
consuming task not only because of four-dimensional
space of these parameters but also because of a very small
change of h in some cases when this parameters are varied.
Gradients of the functions could not be calculated because
the computational noise becomes higher than accuracy of
calculations for small steps. To avoid false local minima
the dependences of these parameters on
were also ana-
lyzed. First results of these calculations gave smooth
curves hð
Þ but points for dependences Að
Þ, Bð
Þ, and
so on were scattered. After more accurate calculation most
points fell on smooth curves though corrections of hð
Þ
were mainly in the fourth digit. Results for these depen-
dences and other details are presented in our inner report
[7] (second paper).
Further in this paper we will use a seven-cell cavity
being developed for the Cornell Energy Recovery Linac
(ERL) as an example. During preliminary discussions of
this project it was decided to choose an optimized inner
cell with Epk =Eacc ¼ 2,
¼ 95 , and Ra ¼ 35 mm.
These numbers, according to [7], lead to the following
geometry and parameters: A ¼ 43:99, B ¼ 35:06, a ¼
12:53, b ¼ 20:95 (all in mm), G
Rsh =Q ¼ 31 838 Ohm2
FIG. 4. (Color) Cell-to-cell coupling vs angle of slope for inner (3% higher than for the TESLA cavity), and Hpk =Eacc ¼
cell optimized for minimal h. 40:23 Oe=ðMV=mÞ (96% of the TESLA cavity). Small

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OPTIMAL CHOICE OF CELL GEOMETRY FOR A . . . Phys. Rev. ST Accel. Beams 12, 114701 (2009)

improvements are mainly due to a smaller value of
: the The equatorial radius of the end half-cells is now fixed to
TESLA cavity inner cells have
¼ 103:2 . A summary of be equal to the radius of the inner, optimized, cell. In
geometrical limitations for inner and end cells will be done contrast with the inner cells, where Req was used for
in Sec. III because some of them are connected with tuning, tuning of the end cell is now performed by chang-
propagation of HOMs. ing its length Le (all parameters of the end half-cells are
denoted by the index e, of the inner cell by i, and of the
C. End cells of a multicell cavity elliptic arc on the tube side by t). Dimensions Ae , Be , ae ,
Let us consider two shapes of the end cells presented in be , at , bt , and c can be used for optimization. Of course, so
Fig. 6. The left half-cells are taken to be of the same shape many free parameters make the problem of optimization
as the inner half-cells. However, the right half-cells should very difficult, but we can ease our task by making some
be different because addition of the beam pipe changes simplifications. For example, the value of the rounding
both the frequency and the field distribution. radius c influences the frequency and peak fields very
For the seven-cell cavity we have chosen asymmetric weakly and can vary in a broad range. On the other hand,
end cells: one of each type. The radius of the beam pipe for if this radius is small, a local minimum of the electric field
the type a end is Rbp ¼ 39 mm, like in the TESLA cav- occurs in this corner that can lead to multipactor [14] (in
ities. For the type b end Rae ¼ 37 mm, and Rbp ¼ 55 mm. spite of weakness of this field). A reasonable choice for c is
These values were adopted after several preliminary at- 2
ðRbp  Rae Þ as was checked in the cited paper.
tempts to optimize higher order modes. Some considera- The end cells can be optimized separately from the rest
tion about further optimization of these radii will be given of the structure if they are tuned to the same frequency.
later. However, even in this case, when the end cell is added to
several inner cells with the same frequency, the frequency
of the united structure is slightly different from the initial
one. This frequency shift depends on the cell-to-cell cou-
pling and was about 1


þ 3 kHz for the type a end cell
and þ4


þ 5 kHz for the type b. For optimization of the
end cells, analogous to inner cells, a special envelope code
TUNEDCELLEND was developed [8] on the basis of the
SLANS code for both end cell geometries discussed here.
This code saves time spent on tuning to the desired fre-
quency and, like the above-mentioned TUNEDCELL code,
makes it possible to analyze shapes for selected sets of
half-axes. Thus the tuning change of Le is moved ‘‘out of
brackets’’ and from now on we may not mention Le when
tuning the end cell separately of others.
In the following we will use Ae and Be as a pair of
variables for optimization. In principle, it is possible to
choose any other pair of values, e.g. ae and be , and make
compensation for one of them by changing the second one,
keeping all other variables (Ae and Be in this case)
constant.

D. Optimization of end cells for maximum acceleration

1. End cells of the type a
Curvature radius at the lowest point of the rightmost
ellipse [Fig. 6(a)] is chosen to be not less than 6 mm
because of possible technological problems in the process
of stamping the half-cells. Let us start with this curvature
and the simplest case of ae ¼ be ¼ 6 mm. Scanning Ae
and Be , one can plot the dependence of e ¼ Epk =2Eacc on
these variables, Fig. 7. The value of e should be minimal if
we want maximal acceleration in the end cell, according to
its definition and by the following reasons. The accelerat-
FIG. 6. (Color) Two possible end cells of a multicell cavity: with ing field of the last cell is defined as Eacc ¼ Vacc =ð=2Þ ¼
a simple transition to the beam pipe (a), and with an iris and a Vacc =ð2Li Þ like for the inner cell. This is because we are
broader beam pipe (b). actually interested in acceleration in the end cell and by

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VALERY SHEMELIN Phys. Rev. ST Accel. Beams 12, 114701 (2009)

FIG. 7. (Color) Normalized electric field vs Be and Ae for ae ¼

be ¼ 6 mm.

changing the length of the end half-cell for tuning we want

to increase the voltage Vacc without paying much attention
to its length. At the same time, the maximal electric field
Epk should be on the inner side of the end cell. In this case
it will be the same as on the other side of the inner iris due
to the same geometry of both sides of it. The right-hand
rising part of each curve in Fig. 7 corresponds to the peak
of electric field jump from the left side (iris) to the right
side (pipe rounding) of the cell in Fig. 6(a). This is why the
progress of the curves changes. One can see a very shallow
minimum of the envelope curve close to Ae ¼ 48 mm and
Be ¼ 33 mm (Fig. 7). When we move along the envelope
curve, changes of Ae are compensated by adjusting Be so
that each point on the envelope curve corresponds to the
minimum of e for an individual curve e vs Be .
By changing the value ae ¼ be , one can find that the
minimum of such envelopes occurs at the value of about
12 mm, Fig. 8(a). Values of Ae , providing minimal e for
each Be , are shown in Fig. 8(b). Figure 8(c) reveals that the
wall slope angle decreases very fast with the increase of
ae ¼ be . So we have to stop at ae ¼ be ¼ 10 mm having
e ¼ 1:0125, because of the agreement to keep the angle

 95 . [The curve with ae ¼ be ¼ 11 mm is not shown
because it overlaps with the curve for 10 mm in Fig. 8(a)
and gives no improvement in e.]
Further improvement in e can be made if we allow ae Þ
be , and make the iris elliptical. This phase of optimization FIG. 8. (Color) Normalized electric field e envelopes (a), length
can be done by small steps around the best point while of the half-axis Ae (b), and the wall slope angle
(c) for
checking for two conditions:
 95 and radius of cur- different circular roundings of the iris (ae ¼ be ) vs length of
vature Rc ¼ a2e =be  6 mm as it was mentioned above. the other half-axis, Be . The single point is the final result.
This optimization results in transforming the circle ae ¼
be ¼ 10 mm into an ellipse with ae ¼ 8:4 mm, be ¼
11:5 mm at Ae ¼ 50:9 mm, Be ¼ 45:3 mm for e ¼ The last step of the procedure can be called a fine-tuning
1:0103. The final result for the type a minimum e value for the minimum. It was not used from the very beginning
is shown as a separate point in Figs. 8(a)–8(c). because a very shallow minimum of e is difficult to find

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OPTIMAL CHOICE OF CELL GEOMETRY FOR A . . . Phys. Rev. ST Accel. Beams 12, 114701 (2009)

One can see that the point where the electric field peak
jumps from the inner iris to the outer side of the cell is at a
different value of Be for the half-cavity as compared to the
end cell calculated alone, Fig. 10. However, at bigger
values of Be the angle
decreases below 95 , so one could
not increase Be , and even if one could, the decrease of e
with this change is negligible. The value of e for the half-
cavity can be found using formula
FIG. 9. Half-cavity with the type a end cell. 7Epk 7
e¼ ¼ : (1)
5Eacc;i þ 2Eacc;e 5=ei þ 2=ee
and the search often led to false minima because of lack of
accuracy and was very time consuming. The phase of For ei ¼ 1 (actually 1.0002) and ei ¼ 1:0103 we calculate
initial, rough tuning (Figs. 7 and 8) provides us some e ¼ 1:0031. The actual value from computer simulation is
additional clues: we could see that values of half-axes 1.0035. The discrepancy is due to a change of the end cell
can vary in a broad range having the optimized function by tuning and accuracy of calculation.
nearly the same. For example, Be can be changed within
10 mm while e increases by no more than 0.1%, see 2. End cells of the type b
Fig. 8(a). Here the change of Be is compensated by the
For the end cells of type b we could not search for
change of Ae , Fig. 8(b). This fact can be used for tuning
minimum e with arbitrary values of ae and at . This is
HOMs without noticeable decrease of e or G
Rsh =Q of
because the sum ae þ at in this optimization would grow
the fundamental mode.
infinitely resulting in a smaller radius of the beam pipe Rbp
Further calculations were done for the half-cavity with
and preventing propagation of some HOMs. So this value
an electric wall boundary condition at the left end plane,
has to be limited in some way. ‘‘Thickness’’ of the iris can
Fig. 9, to exclude influence of the other end cell. The upper
be defined by its curvature radius at the point nearest to the
curve in Fig. 10 is analogous to the V-like curves of Fig. 7
axis, for inner cells it is Rci ¼ a2i =bi ¼ 7:494 mm. For the
for the end cell when only Be is being changed, and the
end iris we have two curvatures: Rce ¼ a2e =be and Rct ¼
frequency is kept constant by adjusting Le . The lower
a2t =bt . Let us set the upper limit for the end iris to Rce þ
curve is for the half-cavity case. Addition of the end cell
Rct < 2Rci  15 mm. Technological limitations Rce 
disrupts flatness of the electric field on the axis of the inner
6 mm, Rct  6 mm are also valid as above.
cells and changes the frequency by about 1 kHz. Tuning by
Graphs analogous to the graphs for the type a cell are
changing the end half-cell length Le restores the flatness.
shown in Figs. 11 and 12. The same behavior of the peak
This tuning is similar to tuning field flatness of the cavity
electric field, jumping from one side of the cell to the other
in situ because it amounts for only a few microns of length
one when Be increases, is observed as for the type a
change and cannot be achieved in fabrication.
transition, Fig. 7. Reduction of the normalized electric field
with growing sizes of the end iris is shown in Fig. 12 in

FIG. 10. (Color) e ¼ Epk =2Eacc for the type a end cell alone and FIG. 11. (Color) Normalized electric field e vs Be and Ae for
for the half-cavity from Fig. 9. ae ¼ be ¼ 6 mm for the type b end cell.

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VALERY SHEMELIN Phys. Rev. ST Accel. Beams 12, 114701 (2009)

FIG. 14. (Color) e ¼ Epk =2Eacc for the type b end cell alone and
for the half-cavity from Fig. 13.
FIG. 12. (Color) Normalized electric field envelopes vs half-axis
length Be for the type b end cells. The single point is the final
result. kHz higher and the decrease of Le was needed to tune the
half-cavity. Figure 14 is similar to Fig. 10. Comparing
these two figures, one can see that now the jump of the
accordance with above-mentioned effect of the smaller peak electric field to the outer iris happens at a smaller
beam-pipe radius. Three upper curves in Fig. 12 are calcu- difference of Be values before and after adding than for the
lated for equal half-axes of the end iris. The fourth curve type a end cell.
shows the result of redistribution of the curvature radii in
the sum Rce þ Rct ¼ 15 ¼ 9 þ 6 mm: if the outer radius E. Optimization of end cells for minimal losses
is smaller the force lines are better attracted to it and Eacc
1. End cells of the type a
increases. The lowest curve represents replacement of the
circle of radius 9 mm by an ellipse with same curvature at Results of optimization for minimal losses of the type a
the lowest point. Further increase of ae and be is prevented end cells are presented in Fig. 15. Normalization of G

by the 95 limit on the wall slope angle. Change of the Rsh =Q (or Rsh , which is the same because we compare the
outer elliptic arcs of the end iris under condition a2t =bt  cells from the same material, and G=Q ¼ Rs , surface
6 mm leads only to a negligibly small decrease of e when resistance, is therefore constant) is made on this value of
both at and bt grow. This is also because of an effective the inner cell (G
Rsh =Q ¼ 31 838 Ohm2 ). As above, we
decrease of the beam-pipe diameter with the thicker iris. started from the circular iris with values ae ¼ be ¼ 6 mm:
Thus we decided to keep at ¼ bt ¼ 6 mm. the lowest group of four curves in Fig. 15. The curve
The final (‘‘fine’’) tuning of the type b end cell gives crossing the other three corresponds to the jump of Epk
min ee ¼ 1:0174 for Ae ¼ 52:1, Be ¼ 47:9, ae ¼ 9:9, from the inner iris to the outer side of the cell (now we do
be ¼ 11:3 mm, and
e ¼ 95:0 . This is shown as a sepa-
rate point in Fig. 12. Behavior of ee vs Be , when all other
cell dimensions but Le are constant, is shown in Fig. 14,
upper curve. The lower curve, for the half-cavity, has e ¼
1:0054, while from (1) we calculate 1.0051.
After adding the type b end cell to the chain of half-cells
(Fig. 13), the frequency of the whole cavity became several

FIG. 13. Half-cavity with the type b end cell. FIG. 15. (Color) Normalized Rsh for the type a end cells.

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at ¼ bt ¼ 6 mm—the lowest group of curves, then we

change these values to 7.5 mm, and then redistribute the
sum ae þ at ¼ 15 mm, so that to increase curvature at the
outer side of the iris: ae ¼ be ¼ 9, at ¼ bt ¼ 6 mm. In
these transformations, Epk again jumps from the inner iris
to the outer one. Finally, the best point under two condi-
tions: Epk on the inner iris and
 95 , is found by
stepwise search of the maximal Rsh .
Adding of the optimized end cell of both a and b types to
the uniform row of inner cells is done similar to the earlier
described procedure for maximal acceleration and requires
only small frequency tuning afterwards.

FIG. 16. (Color) Normalized Rsh for the type b end cells. F. Discussion of optimization results
Results of optimization for maximal acceleration and for
not have this simple indicator as the break of curves in minimal losses are summarized in Table I. In spite of
Figs. 7 and 11). Increase of the values of ae and be up to different goals of two optimizations, the results are very
10 mm increases the maximum of this separating curve close in terms of e and G
Rsh =Q. Only 0.03% to 0.04%
(see the group of four curves for ae ¼ be ¼ 10 mm, and gain in Rsh was obtained with optimization for minimal
Ae ¼ 48, 49, and 50 mm). A further increase of ae and be , losses versus optimization for maximum acceleration.
up to 12 mm, leads only to a small increase of the maxi- Even smaller gain was obtained for e, not more than
mum of the separating curve; the wall slope angle for ae ¼ 0.01%, which is at the level of accuracy of the calculations.
be > 12 mm becomes smaller than 95 and these points do The situation is similar to the optimization for minimal
not add to the maximal Rsh . losses and for minimal Hpk performed in [7] where the
Change to an elliptical cell-to-pipe transition (ae Þ be ) difference in results and in shapes of the optimal cavities
further improves Rsh , and the best point is shown in Fig. 15. was very small.
However, in the presented here case half-axes of the big
2. End cells of the type b ellipses differ by more than 0.5 mm, and the cells with
maximal Rsh are about 1% shorter than those with minimal
Results of optimization for maximal G
Rsh =Q for the
e. This is not a very strong but an additional argument to
type b cells are shown in Fig. 16. Here we again started
optimize for maximal Rsh : the whole cavity length be-
from minimal half-axes of the iris ellipses: ae ¼ be ¼
comes shorter.

TABLE I. Comparison of geometries. All dimensions are in mm. Rsh =Q is in Ohm, G
Rsh =Q is in Ohm2 . Angles
i or
e are in
degrees. Some dimensions for the TESLA [2] cavity (Req , Li and Le ) are tuned for 1300 MHz.

Cell ERL ERL, max acceleration ERL, min losses TESLA

Inner End a End b End a End b Inner End 1 End 2
Req 101.205 101.205 101.205 101.205 101.205 103.353 103.353 103.353
Ai or Ae 43.99 50.9 52.1 50.3 51.4 42 40.34 42
Bi or Be 35.06 45.3 47.9 44.8 47.3 42 40.34 42
ai or ae 12.53 8.4 9.9 8.4 10.0 12 10 9
bi or be 20.95 11.5 11.3 11.7 11.5 19 13.5 12.8
at





6


6









bt





6


6









Ra or Rai 35 35 35 35 35 35 35 35
Rae


39 37 39 37


39 39
Rbp


39 55 39 55


39 39
Li or Le 57.6524 59.988 62.665 59.421 62.094 57.6524 55.716 56.815

i or
e 95.0 95.0 95.1 95.0 95.0 103.2 106.0 107.3
Rsh =Q 116.1 110.0 108.1 110.3 108.5 113.6 110.0 109.4
G
Rsh =Q 31 837 31 404 30 987 31 408 30 991 30 800 29782 29976
G
Rsh =Q 1 0.9864 0.9733 0.9865 0.9735 1 0.9669 0.9732
e ¼ Epk =2Eacc 1 1.0103 1.0174 1.0104 1.0174 0.9938 1.0140 1.0108
Eacc , or Vacc 1 0.9898 0.9829 0.9897 0.9829 1 0.9801 0.9832

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VALERY SHEMELIN Phys. Rev. ST Accel. Beams 12, 114701 (2009)

FIG. 17. (Color) Geometry of the cavity combined from the two
half-cavities: shown in Fig. 6 (mirrored about the left end plane)
and 10, the fields along the profile line, and the electric field on FIG. 18. (Color) Geometry, fields along the profile line, and
axis. electric field on the axis of the TESLA cavity.

A minimum of e (or maximum of Rsh ) is very shallow have different maximal fields on axis in the inner and in the
and we can remain to be close to it even if one of the end cells. Our goal makes it unnecessary to have exactly
parameters (Ae , Be , ae , be ) is perturbed: the other three the same amplitude of the axial electric field in the inner
variables can be adjusted to compensate the initial devia- and end cells. Let us compare fields in the seven-cell ERL
tion, keeping the declared limitations. cavity optimized for minimum losses, Fig. 17, with the
The limiting angle
¼ 95 was reached in all optimi- TESLA cavity, Fig. 18. The ERL cavity optimized for
zations. This means that the abandonment of angle limita- maximum acceleration is not too different.
tions and transition to the RE shape will be beneficial for One can see that the maxima of both electric and mag-
the end cells as it is for the inner ones. netic fields on the surface along the profile line of the ERL
It should be noted that optimization for minimal cavity are equal in both inner and end cells. Maximal
Hpk =Eacc is not needed for the end cells, at least in the electric fields on the cavity axis are higher in the end cells
presented case. A maximum of the magnetic field always (2.3% and 2.6%, for the left and right side, respectively)
appears on the inner wall of the end cell and its value is than in the inner cells. This difference is big enough and
close to the maximum in the inner cells as it is seen in can be taken into account in the process of cavity tuning
Fig. 17. after fabrication. Optimization for maximal Rsh made it
We can also note that in the case of TESLA cavity, the possible to have the end cells only by 1.0% (type a) and
magnetic field in the end cells has nearly equal values at 1.7% (type b) less accelerating than the inner cells.
both sides of the end cells while the electric field is lower at The TESLA cavity has not identical end cells. This
the outer side, Fig. 18. asymmetry, however, is not seen in Fig. 18, because of
scale. For calculation of fields in the TESLA cavity, we
G. Field flatness used geometric data from [2] but ‘‘tuned’’ them to
The quality of the cavity tuning is usually characterized 1300.000 MHz by changing lengths of the cells and equa-
by so-called field flatness, which is defined as cell-to-cell torial radius, because accuracy given in the papers is not
equality of maximal fields on the cavity axis [15]. This sufficient to have equal frequencies of inner and end cells,
requirement is important from the practical point of view and, consequently, field flatness of the inner cells. These
and gives a clear guide for cavity tuning. changes, however, were within the precision range given in
In the presented here analysis, keeping Epk constant, and the papers, for example, the left cell length given as Le ¼
striving to increase Eacc (or to minimize losses), we can 56 mm was tuned to be 55.722 mm. The TESLA cavity

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OPTIMAL CHOICE OF CELL GEOMETRY FOR A . . . Phys. Rev. ST Accel. Beams 12, 114701 (2009)

geometry and our results for fields on the cavity surface The basic geometric parameter which we will take as a
and on axis are presented in Fig. 18. Our ‘‘tuning’’ shows given one, is the iris aperture Ra . Smaller values of Ra
also a slight excess of the axial field in the end cells, 0.9% decrease losses of the fundamental mode but strongly
and 1.3%, respectively. The end cells have lower peak increase problems with HOMs. We will rely upon
fields on the cell-to-pipe transitions, 7.2% and 7.9%, re- TESLA experience and take for the inner cells Ra ¼
spectively. Because of this ‘‘easier regime,’’ the contribu- 35 mm. As it was said earlier, the curvature radius of the
tion to acceleration of the end cells is lower than in the iris cannot be too small even if it does not increase the Epk .
inner cells, by 2.0% and 1.6% (both of the type a in our This is due to difficulties to guarantee accuracy in the
designation). If this is a payment for better extraction of process of stamping the half-cells.
HOMs—and the TESLA end cells were designed with this The higher order modes should have a possibility to
purpose (Haebel [2])—it is, of course, acceptable. propagate to the load through the beam pipe. So, the radius
Each end cell has 4 degrees of freedom: their half-axes of the beam pipe should be above the cutoff value of the
Ae , Be , ae , and be , whereas Req is defined by inner cells and lowest HOMs. In the TESLA cavities (Edwards and Aune
Le is used for tuning to the fundamental frequency. A very [2]) the beam-pipe radius is Rbp ¼ 39 mm. This corre-
strong influence of the cell profile on the mode spectrum sponds to the cutoff frequency of the dipole mode equal
was pointed out in [16] that gives us a hint to use these to fc ¼ 2253 MHz. For the geometry chosen for the ERL
degrees of freedom—but mainly in the end cells—for cavity, only modes of the 3rd dipole band and higher can
better extraction of HOMs. propagate through this beam pipe. The lowest modes of the
first band have their frequency near 1600 MHz but can be
III. EXTRACTION OF HOMS FROM A tuned for our geometry to about 1700 MHz. To guarantee a
MULTICELL CAVITY possibility of their extraction, we choose the beam-pipe
radius Rbp ¼ 55 mm with a cutoff frequency of 1597 MHz
A. Limitations for the peak electric field and geometric and decided to make the beam pipes on different sides of
parameters the cavity with different inner radii: Rbpa ¼ 39 and Rbpb ¼
We should impose some limitation on the cell shape due 55 mm.
to computational, technological, and other conditions. This We will keep the smaller radius from one side of the
discussion will be also done on the basis of parameters cavity because in the case of a broad beam pipe we need to
chosen for the Cornell Energy Recovery Linac (ERL). place the HOM load further from the cavity to prevent
We should choose limitations for the cell wall slope degradation of the quality factor of the fundamental (ac-
angle. In spite of better loss properties of the reentrant celerating) mode. We are forced to use a broad pipe but can
shape [7], Fig. 1, this shape is still in a stage of detailed use it from only one side of the cavity to make the whole
investigations in our lab and elsewhere and now we will cavity shorter. The solution with a broad pipe was not used
discuss more traditional, nonreentrant shape. Nevertheless, in the TESLA cavity, possibly because the need to suppress
the angle
of the wall slope should be given, and we will HOMs was not as essential as it is in the case of the ERL.
take
¼ 95 , trying to come closer to the angles
< 90 Trying not to weaken accelerating properties of the end
but still to be on the traditional side of this barrier. cell with a broad pipe, we will use an iris between the
The next limitation is connected with normalized peak cavity and the broad pipe. So, the end cells will be of two
surface field Epk =Eacc , where Epk is maximal electric field kinds, Fig. 6, in Sec. II they are called ‘‘end cells of type a,
on the surface and Eacc is the acceleration Vacc in the cell in and type b’’.
volts divided by =2. This definition, Eacc ¼ Vacc =ð=2Þ
instead of Eacc ¼ Vacc =Lcell , where Lcell is the geometric B. Model of the HOM load for simulation
length of the cell, should be kept for the end cell also
because its active length is not defined: the field is pene- If we optimize the end cells for better propagation of
trating into the beam pipe and actually we are interested in HOMs whose frequencies are over the cutoff, we should
voltage on the cell, the length of the end cell is not very have a nonreflecting load at some distance from the end
important. Increasing the value of Epk =Eacc , one can de- cell at each side of the cavity. For the free space such a load
crease the maximal normalized magnetic field Hpk =Eacc is known: having the impedance of material Z ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
and losses in the cell. Minimization of Hpk =Eacc also gives 0 =""0 of the same value as the impedance of free
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a possibility to achieve the maximal accelerating rate Eacc space Z0 ¼ 0 ="0 , we should have relative permeability
in the cavity because the magnetic field is a hard limit for and permittivity of the material equal and having nonzero
the SC niobium and the electric field is a soft limit [3]. imaginary parts, for example,  ¼ " ¼ 1  i, we will
However, too high Epk =Eacc will lead to the field emission, have full absorption if the thickness of the absorber is big
and we should be limited by the reasonable value of it. In enough. Unfortunately, in the waveguide, the impedance
the case of the Cornell ERL we took a conservative value has a dispersion, and such a perfect absorber cannot be
Epk =Eacc ¼ 2. realized in simulation or in practice. The impedance has

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VALERY SHEMELIN Phys. Rev. ST Accel. Beams 12, 114701 (2009)

different dependences on frequency for TE waves, ZTE ¼ lengths in the waveguide and in the free space, respec-
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tively, a is the radius of the waveguide, Em is the maximal
Z0 = "  ð=c Þ2 , and for TM waves, ZTM ¼ ðZ0 ="Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi electric field on the magnetic wall at the butt of the
"  ð=c Þ2 , where  is the wavelength in the free waveguide, He is the maximal magnetic field on the elec-
space and c is the cutoff wavelength of an empty tric wall at the butt of the waveguide, and J1 ð011 Þ is the
waveguide. Bessel function of the first kind at the point of the first root
The reflection coefficient from the interface between an of the derivative J10 ðxÞ.
empty and a filled waveguide can be found for the TE wave In the model, the load is a disk at the butt of the pipe
as filled with the lossy material, Fig. 20. A half-cavity with a

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  magnetic wall at the left boundary was used for this

 1  ð=Þ2  1  2 
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


; simulation. The structure of the electric field of a mode

¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 
 1  ð=Þ2 þ 1  2   with a low coupling with the load is also presented in the
picture.
where  ¼ 0 =c ; for the TM waves  in the equation for Comparison of results with the lossy load in the beam

should be changed by ". If we take the loss tangent equal pipe of radius Rbp ¼ 39 mm having  ¼ " ¼ 1  i and
to 1, we can see, Fig. 19, that
very weakly depends on the results with calculated Qext according to above-mentioned
absolute value of " and . For simplicity of the mesh in the procedure is shown in Fig. 21. The relevant values of the
simulation of the lossy stuff, we will take  ¼ " ¼ 1  i. beam breakup (BBU) parameter p are also presented. Its
One can see that for the 10% shorter wavelength than the change with the transition from the modeled load to the
cutoff wavelength, the reflection is equal to
¼ 0:5 or ideal one is practically the same as of Q because R=Q
only 25% in power. Absorption of three-quarters of power weakly depends on Q. The BBU parameter on this figure is
propagating into the pipe will secure very low Q of the big because this calculation is done before its optimization.
mode if the coupling with the pipe is big enough. Here the modes of the 3rd dipole band were examined.
The ideal absorption can be found if we calculate the One can see that the ideal Qext is about 2 times lower
external quality factor Qext of the cavity. Calculation of the than the loaded QL at the lowest frequency of this band,
Qext is analyzed in [17]. For the case of the round wave- and only 20% lower at the highest frequency. Let us remind
guide with a TE11 wave, we can find Qext ¼ QE þ QH , that the cutoff frequency is fc ¼ 2253 MHz, less than 10%
where QE and QH are defined when different boundary lower than the lowest frequency of this band. It is clear that
conditions are imposed at the end of the waveguide: for highest bands of HOMs the load with  ¼ " ¼ 1  i
can be treated as a good one.
2U
QE ¼ ;
"0 a2 2 E2m ð1  1=02 2 0
11 Þ
J1 ð11 Þ
2U
QH ¼ 2
;
0 a He2 ð1  1=02 2 0
11 Þ
J1 ð11 Þ

where U is total energy in the cavity,  and  are wave-

FIG. 20. A half-cavity with a disk shape load at the end of the
FIG. 19. (Color) Coefficient of reflection from the lossy material beam pipe for the data presented at Fig. 21 and a dipole mode
in the waveguide. with high Qext .

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OPTIMAL CHOICE OF CELL GEOMETRY FOR A . . . Phys. Rev. ST Accel. Beams 12, 114701 (2009)

FIG. 22. (Color) Matrix of derivatives @p=@q and associated

frequencies, p’s, and q’s.

FIG. 21. (Color) Qext and BBU parameter p [Ohm=cm2 =GHz]

for the cavity optimized for minimum losses, before HOM had minimal fundamental losses, Fig. 23. A very high
optimization. BBU threshold [18], about 10 A, corresponds to this new
geometry. The value of G
Rsh =Q, defining fundamental
For the 1st dipole band, we have a smaller margin than losses in the cavity, decreased in this optimization only by
for the 3rd one, approximately 1700 MHz versus 4.6% for the type a cell and by 1.1% for the type b
1600 MHz, i.e. about 6%. However, as we will see later, (broader) cell. Since the losses in the inner cell did not
there is no problem with coupling of these modes with the change, the total drop of G
Rsh =Q, i.e., increase of losses,
pipe, and, moreover, the ideal Qext (and, hence, p) is in a seven-cell cavity will be 0.8% only. Unfortunately,
always less than in the model. Of course, we cannot find small deviations of the shape lead to a dramatic increase of
the best solution with the nonideal load but we still can find p [19] and a further decrease of maximal p is desirable.
a geometry which has a significant coupling with the load. From a general point of view, a decrease of the BBU
Final optimization will be done with the model of a real parameter p should lead to a decrease of its derivatives
HOM load which (1) is far from the ideal in the shape, it @p=@q because the value of p is limited from below. This
cannot fill the whole pipe; and (2) is far from ideal in the should lead to a weaker sensitivity of dimensions to dis-
electromagnetic properties of the lossy material. However, turbances. However, another possibility to decrease this
we will try to separate again our task: first we separated sensitivity exists: broadening of the HOMs bandwidths
optimization of the fundamental mode and HOMs; now we [19].
are trying to separate optimization of the HOMs extraction Further decrease of p is limited by behavior of deriva-
and their absorption by the load. tives for two modes: 2511 and 2513 MHz, see Fig. 22. The

C. Usage of derivatives @p=@q

We can find derivatives @p=@q of the BBU parameter
with respect to any size of the end half-cells’ half-axes: so
that q ¼ Aa ; Ba ; . . . ; ab , or bb . We can do this for any
HOM dipole mode. Having the matrix of @p=@q, we can
minimize the maximal value of p for a given frequency
range. We will limit the task by eight most dangerous
modes, i.e., the modes with biggest p. After the first run
of optimization with the end iris Rae ¼ 37 (as in Sec. II), it
was decided to increase it to the same value as at the other
side. The shape of the end cell was corrected for this
aperture. For the transition radius Rae ¼ Raea ¼ 39 mm
for the type a end half-cell and Rae ¼ Raeb ¼ 39 mm for
the type b end half-cell (Fig. 6), such a matrix is presented
in Fig. 22, for frequencies shown in the upper line and
values of p shown in the lower line. Values of q are shown
in the column on the right. FIG. 23. (Color) BBU parameter p vs frequency for the cavity
The values of p were decreased nearly 3 orders of with minimal fundamental losses before and after optimization
magnitude from the initial geometry when all the cells for minimal p.

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VALERY SHEMELIN Phys. Rev. ST Accel. Beams 12, 114701 (2009)

FIG. 24. Electric field of eight modes with biggest p.

biggest derivatives correspond to half-axes Aa and Ba but An attempt to redirect the lowest mode of the two with
they have different signs. Values of p for these two modes highest @p=@q (f ¼ 2511 and 2513) was made. This sepa-
are nearly equal: 2261 and 2259. So, further improvement ration was successful, Fig. 25; frequencies of the modes
of p can be done by changing other half-axes of the end somewhat changed: to 2514 and 2517 MHz. Unfortunately,
cells, but it will be insignificant. after this procedure several other modes substantially in-
From Fig. 22, one can see that lowest modes are more creased their BBU parameter and this attempt was left
sensitive to the change of the type b end cell (left lower aside.
quarter of the matrix) whereas the higher modes depend Further improvement of the geometry can be done using
strongly on the type a cell, with the smaller pipe (right- the same procedure of decreasing the maximal BBU pa-
hand upper quarter). This means that the lowest HOMs are rameter—now for the inner cells. This tuning for lower p
directed to the broader pipe and the higher modes propa- can be closely related to the broadening of the bandwidths
gate to the smaller pipe though they could be tuned for of the HOMs.
propagation into the broader pipe as well. The example
pictures of electric field of these modes confirm the afore- IV. CONCLUSIONS
said, Fig. 24.
An algorithm for calculation of inner and end cells of a
multicell cavity with minimal losses or Hpk =Eacc is
presented.
It is shown that optimization of the end cells can be done
independently of the inner cells and only a small tuning is
needed when the end cell is added to the uniform cells’
chain.
The analysis shows that a proper choice of geometry can
help to use end cells either for their highest possible
acceleration or for minimal losses. In both cases, limita-
tions of maximal surface electric fields and wall slope
angle were the same as for the inner cells.
Comparison of the proposed ERL cavity geometry with
the TESLA cavity geometry illustrates the traits of this
FIG. 25. Redirection of modes with maximal @p=@q into two-sided approach to optimization. It is also shown that
different pipes. the lower values of the wall slope angle are preferable not

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OPTIMAL CHOICE OF CELL GEOMETRY FOR A . . . Phys. Rev. ST Accel. Beams 12, 114701 (2009)

for the inner cells only but for the end cells as well. and H. Padamsee, Cornell University LNS Report
Removing the angle restriction will lead to the reentrant No. SRF 020128-01, 2002; TESLA Report No. 2002-1.
cavity having minimal losses or maximal acceleration for a [5] R. L. Geng, H. Padamsee, A. K. Seaman, and V. D.
given Epk =Eacc and aperture. Shemelin, in Proceedings of the 21st Particle
Accelerator Conference, Knoxville, 2005 (IEEE,
As the next optimization step, mutual compensation of Piscataway, NJ, 2005), pp. 653–655.
variable geometric parameters is used to find a geometry [6] F. Furuta et al., in Proceedings of the 10th European
with better HOMs extraction properties with a very small Particle Accelerator Conference, Edinburgh, Scotland,
deterioration of the losses and the magnetic peak field. 2006 (EPS-AG, Edinburgh, Scotland, 2006), pp. 750–753.
A possibility to control tuning of the HOMs propagation [7] Valery Shemelin, in Proceedings of the 2007 Particle
into the beam pipes was demonstrated. Usage of deriva- Accelerator Conference, Albuquerque, New Mexico,
tives of the BBU parameter with respect to cell dimensions 2007 (IEEE, Albuquerque, New Mexico, 2007),
is a powerful method of suppression of the HOMs. pp. 2352–2354; Cornell SRF Group Report No. SRF
Minimization of the BBU parameter of dipole HOMs 070614-02, 2007.
was done changing the shapes of the end half-cells of the [8] D. Myakishev, Cornell SRF Group Internal Report
No. SRF/D 051007-02, 2005.
cavity with increase of power losses of the fundamental
[9] D. G. Myakishev and V. P. Yakovlev, in Proceedings of the
mode by 0.8%. Decrease of the BBU parameter was nearly Particle Accelerator Conference, Dallas, TX, 1995 (IEEE,
3 orders of magnitude compared to the original shape tuned New Jersey, 1995), pp. 2348–2350.
for minimal losses. The BBU threshold current increased [10] S. Belomestnykh, Cornell University LNS Report
by the same amount, up to 10 A. No. SRF 942008-13, 1994.
[11] Igor Zagorodnov and Nikolay Solyak, Proceedings of the
ACKNOWLEDGMENTS 10th European Particle Accelerator Conference,
Edinburgh, Scotland, 2006 (Ref. [6]), pp. 2862–2864.
The author wishes to thank Georg Hoffstaetter for useful [12] V. Shemelin, Cornell University LNS Report No. SRF
recommendations. The author is especially thankful for the 051010-09, 2005.
important comments and corrections from Sergey [13] J. Sekutowicz et al., in Proceedings of the 20th Particle
Belomestnykh. This work has been supported by NSF Accelerator Conference, Portland, OR, 2003 (IEEE, New
Award No. PHY-0131508, Empire State Development Jersey, 2003), pp. 1395–1397.
Corporation (ESDC) Energy Recovery Linac Project [14] S. Belomestnykh and V. Shemelin, Nucl. Instrum.
No. U316, and Cornell University. Methods Phys. Res., Sect. A 595, 293 (2008).
[15] H. Padamsee et al., RF Superconductivity for Accelerators
(John Wiley & Sons, New York, 1998).
[16] R. Rimmer, H. Wang, G. Wu, and D. Li, Proceedings of
[1] P. Kneisel et al., Nucl. Instrum. Methods Phys. Res. 188, the 20th Particle Accelerator Conference, Portland, OR,
669 (1981). 2003 (Ref. [13]), pp. 1389–1391.
[2] E. Haebel, A. Mosnier, and J. Sekutowicz, in Proceedings [17] V. Shemelin and S. Belomestnykh, Cornell LEPP Report
of the XV International Conference on HEACC, No. SRF 020620-03, 2002.
Hamburg, 1992, Vol. 2; ‘‘TESLA Test Facility Linac [18] I. Bazarov and G. Hoffstaetter, in Proceedings of the 9th
Design report,’’ edited by D. A. Edwards, Version 1.0, European Particle Accelerator Conference, Lucerne, 2004
1995; B. Aune et al., Phys. Rev. ST Accel. Beams 3, (EPS-AG, Lucerne, 2004); Georg H. Hoffstaetter and Ivan
092001 (2000). V. Bazarov, Phys. Rev. ST Accel. Beams 7, 054401 (2004).
[3] K. Saito, SRF 2001, Tsukuba, Japan, 2001, pp. 583–587. [19] M. Liepe and N. Valles, SRF 2009 Workshop, Berlin,
[4] V. Shemelin, H. Padamsee, and R. L. Geng, Nucl. Instrum. Germany, 2009.
Methods Phys. Res., Sect. A 496, 1 (2003); V. Shemelin

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