THE FIRST TRIAL OF XY-COUPLED BEAM PHASE SPACE MATCHING
FOR THREE-DIMENSIONAL SPIRAL INJECTION∗
M. A. Rehman† , S. Ohsawa, H. Nakayama, K. Furukawa, T. Mibe, H. Hisamatsu, N. Saito, K. Sasaki,
High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
H. Iinuma, H. Hirayama, K. Oda, Ibaraki University, Mito, Ibaraki, 310-0801, Japan
R. Matsushita, The University of Tokyo, Bunkyo, Tokyo 113-0033, Japan
Abstract
The most recent measurement of muon g – 2 results in a
3.8𝜎 discrepancy with the equally precise theoretical prediction. The J-PARC muon g – 2/EDM experiment (E34) is under preparation to decipher this discrepancy and unravel thenew physics beyond the standard model. The precision goalfor g - 2 is 0.1 ppm. To achieve this precision goal a novel
three-dimensional spiral injection scheme has been devised
to inject and store the beam into a small diameter MRI–typestorage magnet for E34. The new injection scheme featuressmooth injection with high storage efficiency for the compact magnet. However, the spiral injection scheme is anunproven idea, therefore, a Spiral Injection Test Experiment(SITE) at KEK Tsukuba Campus is underway to establish
this injection scheme. Due to the axial symmetric field of thesolenoid magnet, a strongly XY-coupled beam is required.
To produce the required phase space for the solenoid-type
storage magnet, a beam transport line consisting of three rotatable quadrupole magnets has been designed and built for
SITE. The vertical beam size reduction by means of phasespace matching and other geometrical information has beensuccessfully measured by the wire scanners. This reportdescribe the first trial of phase space matching for the SITE.

INTRODUCTION
The most latest measurement of muon's g – 2 at BNL [1]results in 3.8 𝜎 [2] discrepancy with the Standard Model
prediction. This discrepancy could be a hint of new physics
beyond the standard model. The new J-PARC muon g –2/EDM (E34) is under construction to measure muon g –2/EDM to unprecedented level. The E34 experiment will
use a fully new approach in order to measure the muon's g –2/EDM. The final goal of E34 is to measure the muon's g –2 with a precision of 0.1 ppm and EDM down to the valueof 10−21 e.cm [3, 4].
The muon's g – 2/EDM will be measured by the means oflow emittance muon beam storage into a precise magnetic
field. In E34 a low emittance muon beam of momentum
300 MeV/c will be injected vertically into a compact 3–T
Magnetic Resonance Imaging (MRI) type solenoid magnet,
the muon beam will store with a 0.66 m diameter orbit. The
use of MRI-type storage magnet gives an unprecedented local field uniformity of 0.2 ppm. A novel three–dimensional
spiral injection scheme has been invented in order to inject∗ Work supported by “Grant in Aid” for Scientific Research, JSPS (KAK-ENHI#26287055, KAKENHI#19H00673)† rehman@post.kek.jp

Figure 1: The 3-D model of Spiral Injection test Experiment(SITE) setup.
the beam into the compact MRI–type magnet. This newinjection scheme provides high injection efficiency and overcome technical challenges related to the compact nature ofsolenoid-type storage magnet.
In the 3-D spiral injection scheme, the beam will be injected at the vertical angle into the storage magnet. The
radial field of the solenoid will decrease the vertical angleof the beam as it approaches the mid plane of the magnet.
Finally, a magnetic kicker will guide the beam to the storage
volume where the beam will be stored under a weak focusing
field. Due to the axial symmetric field of the solenoid magnet a so called "XY-coupled" beam is needed to avoid thevertical defocusing of beam inside the storage magnet [5].
The three-dimensional spiral injection scheme is an unprecedented injection idea, therefore, a demonstration experiment to establish the feasibility of this new injection scheme
is inevitable. A scale down Spiral Injection Test Experiment(SITE) with an electron beam is under development at KEK
Tsukuba campus. This paper will describe the first trial ofbeam injection with the XY-coupled beam into the solenoid
storage magnet.

SPIRAL INJECTION TEST EXPERIMENT(SITE)The SITE composed of a 2 m long straight beamline, asolenoid storage magnet for the electron beam storage anda bend section to inject the electron beam into the storage
magnet. A triode type thermionic electron gun with 𝐿𝑎𝐵6cathode is used to generate the DC electron beam of 80 keVwith the beam current in the range of a few µA. After theelectron gun, a magnetic lens focuses the beam. A pair ofsteering coils also have been installed to control the trans-

verse position of the beam. An electric chopper system after
the electron gun produces a pulsed beam. Details of theelectric chopper can be found in [6]. A collimator of diameter 3 mm and depth 5 mm is placed after the electricchopper. The collimator was used as the beam dump forthe chopper system and also used to create the differential
pumping system for the gas monitor in the storage magnet.
After the collimator, three rotation quadrupole magnets havebeen installed for the beam phase space matching for thespiral injection. Due to the radial fringe field of the storage magnet, an XY-coupled beam at the injection point isrequired to avoid beam de-focusing.
After the rotation quadrupole magnets, a bending magnet is placed on the straight beamline to inject the beam at
40 degree towards the storage magnet. A second bending
magnet was installed near the injection point for the slightcontrol of the injection angle.
The storage magnet is basically a solenoid which consistsof a 540 mm long solenoid coil. The solenoid coil is enclosed
in the iron yoke. An auxiliary coil was placed at the centerof the solenoid in order to produce a weak focusing field at
the center of the storage magnet [7]. The pulsed magnetic
kicker will keep the beam to the center of the storage magnet[8]. The layout of the SITE experimental setup is shown inFig. 1. A new wire scanner monitor for the beam profile andposition measurement was developed for the solenoid storage
magnet [9]. The vertical beam blow has been successfully
suppressed by the phase space matching. The details will
be described in the forthcoming sections. A comparison ofparameters between E34 and SITE is given in Table 1.
Table 1: Comparison of Parameters between E34 and SITE.
Parameters
Magnetic field strength
Momentum
Cyclotron Period
Storage orbit diameter

E343T
300 MeV/c
7.4 ns0.66 m

SITE
0.0082 T
296 keV/c
5 ns
0.24 mOne of the key requirement of the 3-D spiral injection
scheme is to produce the required beam phase-space for theinjection to avoid the vertical beam blow–up. Here, we useda transfer matrix method to compute the coupling of thebeam at the matching point. Let’s assume a matrix 𝑀 which
consists of rotation and Twiss parameters matrix [5, 10]:(1)

where, 𝑈𝑖𝑛 and 𝑈𝑜𝑢𝑡 are input (initial) and output (matching point) rotation matrix, which satisfy the symplectic condition. In the case of, uncoupled input beam phase-space
𝑈𝑖𝑛 will be equal to identity matrix. In the case of SITE, theinput beam phase-space has no coupling, therefore, Eq. (1)can be written as follow:
𝑀 = 𝑈 −1 · 𝐷,

Figure 3: The beam phase space with and without XYcoupling.
The 4 × 4 𝑈 and 𝐷 matrices are given as follow:𝜇©
­ 0𝑈=­­ 𝑅1« 𝑅3

THE MATCHING BEAMLINE−1
𝑀 = 𝑈𝑜𝑢𝑡· 𝐷 · 𝑈𝑖𝑛 ,Figure 2: The layout of the matching beamline, the beamline composed of three rotatable quadrupole magnets, arectangular bending magnet and drift space. The rotatable
quadrupole magnets were placed in the downstream direction of the collimator.

(2)

0𝜇𝑅2𝑅4−𝑅4𝑅3𝜇0𝑅2ª−𝑅1 ®
®,
0 ®
𝜇 ¬
𝐷=𝐷𝑋
00𝐷𝑌


,(3)
where, 𝑅1 , 𝑅2 , 𝑅3 , 𝑅4 are known as coupling parameters and non-zero value of these parameters shows thestrength of the coupling. The quantity 𝜇 is given as 𝜇 =p
1 − (𝑅1 𝑅4 − 𝑅2 𝑅3 ). The matrix 𝐷 is a diagonal matrix
consist of beam input and output Twiss parameters.
An iterative procedure has been adopted to calculate thecoupling parameters. The emittance of the beam for SITE atthe collimator location was measured by the quadrupole scanmethod. The values of horizontal (𝜖 𝑥 ) and vertical emittance(𝜖 𝑦 ) are 0.61 ± 0.05 mm–mrad and 0.41 ± 0.04 mm–mrad,respectively [11].
The matching beamline consists of three rotatable
quadrupole magnets, a bending magnet, and drift spaces.
Figure 2 presents the photo matching beamline. These threerotatable quadrupole magnets can be rotated at an arbitrary
angle around the beam. The transfer matrix of the beamtransport line can be written as follow:
𝑀 = 𝐿5 · 𝐵 · 𝐿4 · 𝑄3 · 𝐿3 · 𝑄2 · 𝐿2 · 𝑄1 · 𝐿1

(4)

Figure 4: (a) The gas monitor view of the beam injection into the storage magnet without XY-coupling. (b) The verticalbeam size and position are measured by the wire scanner in the case of no coupling. The horizontal axis is the vertical
position of the wire scanner in the storage magnet and vertical axis is beam current deposited at the wire scanner. It isvery evident that beam blow-up drastically without appropriate phase space matching. (c) The gas monitor results withXY-coupling. (d) The vertical beam size was measured by the wire scanner with XY-coupling.
where, 𝑀 is a transport line matrix, 𝐵 is bending magnet,𝐷 represents the drift space, 𝑄 1 , 𝑄 2 , and 𝑄 3 denotes therotation quadrupole magnets. A custom program in Mathematica was developed to find the appropriate rotation angle
and strength of the rotation quadrupole magnets. The black
dots in Fig. 3 shows the beam phase space without XY–coupling and red dots are corresponds to the coupled thebeam phase. In the case of coupled beam phase space therotation angles of 𝑄 1 , 𝑄 2 , and 𝑄 3 were −20 deg, 25 deg and−45 deg respectively.
Figure 4 (a) and (b) shows gas monitor and wire scanner results of the beam injection without any phase space
matching, respectively, the beam blow–up in the verticaldirection is very clear from gas monitor and wire scanner
measurement. Whereas, Fig. 4 (c) and (d) gas monitor andwire scanner results of the beam injection with the appropriate XY-coupling as shown in Fig. 3 (red dots), respectively.
The reduction in the vertical beam size is visible by the gas
monitor and quantitative information can be seen by the wire
scanner. Without XY-coupling only 2–turns can be seen bythe gas monitor, whereas, when the beam was injected withappropriate XY-coupling 4–turns can be seen very clearly.
Figure 5 shows the comparison of vertical beam size withand without XY-coupling.

CONCLUSION

Figure 5: The comparison of vertical beam size with andwithout XY-coupling. The vertical beam size at the kickpoint (Y ≈-100 mm) was 8.18±0.03 mm (1𝜎) without XY–coupling, which has been controlled to 2.56±0.005 mm (1𝜎)with appropriate phase space matching.

In order to overcome the vertical beam blow–up in thesolenoid-type storage magnet, a beam transport line consisting of three rotatable quadrupole magnets has been designed
and built for SITE. The dramatic reduction in the verticalbeam size was observed with the appropriate setting of threerotatable quadrupole magnets. Without any beam phase–space matching, the beam size grows to 8.18±0.03 mm (1𝜎)at the kick point (Y ≈-100 mm) of the storage magnet. Thevertical beam blow–up was reduced to 2.56±0.005 mm (1𝜎)at the kick point with the appropriate combination of thethree rotation quadrupole magnets.

ACKNOWLEDGMENTS
We thank Prof. Tohru Honda, Prof. Toshiyasu Higo,Tsuyoshi Suwada and Dr. Izumi Mochizuki for their generous support for this project.

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