DESIGN OPTIMIZATION OF FLUX CONCENTRATOR FOR SuperKEKB∗
L. Zang† , T. Kamitani, S. Fukuda and Y. Ogawa
High Energy Accelerator Research Organization (KEK)
1-1 Oho, Tsukuba, Ibaraki, 305-0801, Japan
Abstract
For high luminosity electron-positron colliders, intense
positron beam production is one of the key issues as wellas electron. Flux Concentrator (FC) is a pulsed solenoid
that can generate high magnetic field of several Tesla andis often used for focusing positrons emerged from a production target. It works as an optical matching device in apositron capture section. With this device, high capture efficiency is achieved. In this paper, we will discuss a design
optimization of a FC for the SuperKEKB positron source.
Geometrical parameters of the FC are optimized to achieve
high peak field using the CST EM Studio [1]. Magnetic
field distribution evaluated with the EM Studio is implemented into a particle tracking code to see a performanceof the positron capture section. The tracking simulation
includes a positron production at the target, focusing by
the FC and subsequent solenoids and acceleration by RFstructures till the end of the capture section. We report theresults of a FC design optimized for higher positron yieldwith the tracking simulation.

INTRODUCTION
The SuperKEKB positron source is based on the conventional scheme. A 3.5 GeV electron beam strikes on a tungsten target to generate cascade shower [2]. The generated
positrons are focused by a strong magnetic field provided
by a matching device before acceleration. Positron beam
after the target has a small size but large divergence angle.
In the accelerating section it can have a relatively large size
but should have a small divergence angle. A matching device such as flux concentrator could transform the phasespace distribution from the target so that it is appropriate
for the solenoid focusing field in the accelerating section:
this improves the capture efficiency. The adiabatic matching device is made of a slowly changing magnetic field plus
a long magnetic field. Between the maximum field B 0 andthe minimum B s , the magnetic field is characterized by thetaper parameter g as shown in Eq.1.
Bz (z) =B0
1 + gz(1)

Key parameters for the adiabatic matching device are theinitial field B0 , the taper parameter and the physical aperture. Because the positron beam has a very wide energy∗ Work supported by KEK† zang@post.kek.jp

Figure 1: CST Model of the spiral slit Flux Concentrator.
spread, it is not possible to achieve perfect matching between target and solenoid in the linac. Optimization of theparameters to achieve high capture efficiency is best doneby simulation. In the following sections, tracking simulation results of the flux concentrator and solenoid key parameters will be presented.

FLUX CONCENTRATOR MODELINGThe flux concentrator consists of primary coil and conductor core. A pulsed current in the primary coil induces
an eddy current in the conductor. Due to the skin effect,
induced current is directed into the inner surface through aslit to produce a high magnetic field in small area [3]. In
this section two proposed models are introduced: straightslit FC and spiral slit FC. These two models have been built
in CST Studio to evaluate the field distribution. The simulations do not only help us understand the field shape,
but also generate the field to implement into the tracking
simulation. The spiral slit FC model has been shown in
Fig.1. This design is initially developed by SLAC [4].Thecopper coil is 100 mm, 12 turns and assuming a currentof 16000 A. The copper core has a outer radius of 40 mm
and a conical inner radius growing from 3.5 mm to 26 mm.
There are certain advantages and disadvantages for each
design. First of all, spiral slit FC is a rather complicateddesign compared with the straight slit FC, which makes it
harder for machining. Secondly, the straight slit FC could
produce higher peak field as shown in Fig.2, although thespiral slit has a better adiabatic field distribution. Finally,straight slit FC has a larger transverse field with long tailwhich could defocus the positron beam, whereas the spiralslit FC’s transverse component is smaller.

Px [MeV/c]200-20-15

Figure 2: Spiral (red line) and straight (blue line) slit flux
concentrator longitudinal field distribution.

-10-5051015x [mm]Figure 4: Initial positron beam phase space distribution.

1.2Figure 3: Schematic view of the SuperKEKB positronsource configuration.

Positron Yield [e+/e-]

1.0

0.8

0.6

0.4

0.2

0.00

POSITRON SOURCE TRACKINGSIMULATIONThe motivation of FC optimization and tracking simulation is to improve the positron yield for SuperKEKBpositron source. The capture section layout is based on theSuperKEKB design as shown in Fig.3. Positrons generation simulation is carried out by GEANT4 [5] assuminga 3.5 GeV electron strike on a 14 mm thick tungsten target. The positron beam phase space distribution is shownin Fig.4. The generated positrons go through the adiabatic
matching device consisting of FC and bridge coil. Downstream of the matching section is the accelerating structures
include two 2 m L-band and four 2 m of large aperture Sband accelerating units. Both kinds of accelerating units
have an accelerating gradient of 10 MV/m [6]. The wholeaccelerating structure is surrounded by the solenoid. Moredetailed simulation results of positron yield in capture section by investigating the dependence upon some key parameters such as peak field, taper parameter and solenoid
field strength will be presented in following sections.

Peak Field
The lateral acceptances r max and the transverse momentum acceptace Pxmax could be roughly estimated by following expressions:Bsmax=a(2)rB02468101214

B0 [T]

Figure 5: Positron yield as a function of maximum adiabatic matching deive field B 0 .
Pxmax = e B0 Bs a(3)where B0 and Bs represents the maximum and minimum
magnetic field of adiabatic matching device respectively.
In the simulation, the field length is fixed (various taperparameters), and the peak field will be scanned from 1.5 T
to 11.7 T to investigate the correlation between the maximum field B0 and positron yield, where the positron yieldis defined as the number of positrons per primary electron
at the end of capture section. The results have been presented in Fig.5 which shows the positron yield as a functionof maximum adiabatic matching device field B 0 . We candivide the plot into two regions to analyze it: region 1 from
1.5 T to 6 T and region 2 from 6 T to 12.7 T. In region 1,B0 changes from 1.5 T to 6 T. In this process, theoreticallythe transverse momentum acceptance is increased by a factor of 2, whereas the lateral acceptance is reduced by thesame factor. Because the positron beam after the target has
a fairly small lateral dimension, the reduction of lateral acceptance will not cause serious positron losses. As we haveseen in Fig.4, the transverse momentum of positron beam
is rather large, so it is efficient to improve the positron yield
by increasing the transverse momentum acceptance. As
we can see in Fig.5 the positron yield could be improved

1.6
2.0

1.2

Positron Yield [e+/e-]

Positron Yield [e+/e-]

1.4

1.0
0.8
0.6
0.4

1.5

1.00.5Bz=6.2T
Bz=4.2T

6.2T
4.2T

0.2
0.0

0.0
0100200300

400500Taper Field Length [mm]

0.0

0.2

0.4

0.6

0.8

Solenoid Field [T]Figure 6: Positron yield as a function of taper field length.
The black dots and red dots represent the 6.2 T case and
4.2 T case respectively.

Figure 7: Positron yield as a function of solenoid fieldstrength. The black dots and red dots represent the 6.2 Tcase and 4.2 T case respectively.

from 0.6 to 1.2. However, in region 2, instead of capturing more positrons, the capture efficiency starts to saturate.
For instance, the extreme case of using 12.7 T field, andthe positron yield is even lower than the 6 T case. Whena much higher field is applied, the reduction of lateral acceptance starts to cut out more off-axis positrons. And thelarge transverse momentum acceptance is less effective because most of positrons have been already in the acceptancelevel. Therefore, positron capture efficiency is saturated fora higher field than 6 T.

20% gain of positron yield. For the SuperKEKB solenoid,
0.4 T is the nominal design value.

Taper Parameter
Ideally, an adiabatic field distribution could provide thedemanded phase space transformation. A smaller taper parameter defines a smoother adiabatic field distribution, butthat could lead to a very long FC. For a fixed peak fieldstrength, the larger the taper parameter is, the smaller theenergy acceptance. Fig.6 shows the positron yield as afunction of the taper field length for fixed peak field of 6.2 Tand 4.2 T.
Field of 6.2 T and 4.2 T has the similar tendency. If
we assume the 200 mm field length as a default choice,to increase the taper field length to 500 mm could effectively improve the positron yield about 18.5% and 13.6%
for 6.2 T and 4.2 T respectively. Whereas shortening thetaper field length could lead to the lower positron yield.

Solenoid
After passing through the matching device, positrons go
into a long solenoid field till the end of the capture section. In this section, we will look into the influence ofthe solenoid field to the positron yield. Fig.7 shows thepositron yield as a function of solenoid field. The blackdots and red dots represent the peak field of 6.2 T and 4.2 T.
When the solenoid field increase from 0.1 T to 0.8 T, thepositron number increase linearly. For example, solenoid
field increase from 0.4 T to 0.5 T would lead to an extra

CONCLUSIONS
Between the positron production target and the first acceleration section, the FC plays an important role as a partof matching device. We have studied the positron yield asfunctions of the peak field, taper parameter and solenoidfield. The initial field about 6 T is an optimal choice. Whenthe peak field is fixed, the taper parameter should be small.
However, small taper parameter could end with a very long
flux concentrator, so that this value is restricted by thespace that reserved for flux concentrator. Furthermore, thesolenoid field could help to improve the positron yield. So
far the simulation is finished at the end of capture section.
After the capture section, the focusing system and damping
ring has a small transverse phase space acceptance and energy acceptance respectively, which could reduce the final
positron yield. Hence, in the future, the tracking simulation
will be extended to the downstream including the quadruple and the damping ring.

REFERENCES[1] CST EM STUDIO, “Low Frequency Electromagnetic Designand Simulation”.
[2] SuperKEKB Task Force, “Letter of Intent for KEK Super B
Factory”, KEK Report 2004-4, June 2004.
[3] L. Hoffmann and V. Scheuing, “Transformer Coils for High
Magnetic Fields”, CERN 63-36, Nuclear Physics Division,
22nd October, 1963.
[4] A. V. Kulikov, S. D. Ecklund and E. M. Reuter, “SLC
Positron Source Pulsed Flux Concentrator”, SLAC-PUB5473, June, 1991.
[5] Geant4 Collaboration, “Geant4 User’s Guide for Application
Developers”, 2nd December, 2011;
http://geant4.cern.ch/[6] T. Kamitani, “Positron Source Upgrade”, KEKB review.