ALetterofIntentto The J-PARC 50 GeV Proton Synchrotron ... · The J-PARC 50 GeV Proton Synchrotron Experiments The PRISM Project ... One of the main physics topics with PRISM-II is

A Letter of Intent toThe J-PARC 50 GeV Proton Synchrotron

Experiments

The PRISM Project

−A Muon Source of the World-Highest Brightnessby Phase Rotation −

PRISM Working Group

January 1st, 2003

2

The PRISM Working Group Members

Shinji Machida, Yoshiharu Mori∗, Chihiro Ohmori, and Takeichiro YokoiAccelerator Laboratory, High Energy Accelerator Research Organization (KEK),

Tatsushi Nakamoto, Toru Ogitsu, and Akira YamamotoCryogenics Science Center, High Energy Accelerator Research Organization (KEK),

Yoichi Igarashi, Shigeru Ishimoto and Koji Yoshimura∗

Institute of Particle and Nuclear Studies, High Energy Accelerator ResearchOrganization (KEK),

Noboru Sasao∗

Department of Physics, Kyoto University,

Yoshihisa IwashitaAccelerator Laboratory, Institute for Chemical Research, Kyoto University,

Hiroyuki OhnishiGraduate School of Engineering, Kyushu University,

Masaharu Aoki, Yoshitaka Kuno∗, Fumitaka Maeda, Kengo Nakahara,Norihito Nosaka, Akira Sato, and Yosuke Takubo

Department of Physics, Graduate School of Science, Osaka University.

The ∗ mark indicates the contact persons.

Abstract

We, the PRISM working group, would like to express our interest to construct a muonbeam facility of the highest beam brightness in the world, based on a novel techniqueof phase rotation. The facility is called PRISM. PRISM stands for Phase Rotated

Intense Slow Muon source. It would provide a muon beam having an intensityof about 1011 − 1012µ±/sec, a narrow momentum width of a few %, and no pioncontamination. The aimed beam intensity is about four orders of magnitude higherthan the highest presently available. The energy of the PRISM beam is relatively low(20 MeV in kinetic energy), since it is aimed to be used for stopped muon experimentsin general. For particle physics interest, in particular, an experimental search forlepton flavor violating µ− − e− conversion with PRISM is being planned.

The PRISM-II is another muon source with its momentum of around 500 MeV/cand high muon polarization. It would provide a beam for an experimental search forthe muon electric dipole moment (EDM). And it could serve, in future, as the firstring for a neutrino factory.

PRISM and PRISM-II will use a pulsed proton beam from the J-PARC 50 GeVproton synchrotron (50 GeV PS) by fast extraction. Experimental apparatus andPRISM/PRISM-II will be placed at a proposed pulsed proton beam facility, whichis not included in the Phase I plan of J-PARC. Some special arrangement to preventsoil activation at the potential beam extraction port to the facility is requested.

Contents

1 Executive Summary 5

2 PRISM Overview 72.1 Phase Rotated Intense Slow Muon Source . . . . . . . . . . . . . . . 72.2 PRISM Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 PRISM-II Specification . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Proton Beam 113.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 Proton Beam Power and Time Structure . . . . . . . . . . . . 113.1.2 Why is 50-GeV Proton Beam used ? . . . . . . . . . . . . . . 11

3.2 8 Bunch Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 90 Bunch Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Pion Capture System 144.1 Pion Production by 50-GeV Proton Incident . . . . . . . . . . . . . . 144.2 Pion Production Target . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 Pion Capture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.3.1 Optimization of the Pion Capture System . . . . . . . . . . . 164.3.2 Superconducting Solenoid Magnet . . . . . . . . . . . . . . . . 17

5 Pion Decay And Muon Transport System 185.1 Adiabatic Transition From a High to a Low Magnetic Field . . . . . . 185.2 Curved Solenoid Section . . . . . . . . . . . . . . . . . . . . . . . . . 195.3 Decay Solenoid Design . . . . . . . . . . . . . . . . . . . . . . . . . . 195.4 Dispersion Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 Phase Rotation System 226.1 Principle of Phase Rotation . . . . . . . . . . . . . . . . . . . . . . . 226.2 Phase Rotation by FFAG . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.2.1 Advantage of Circular Machines . . . . . . . . . . . . . . . . . 236.2.2 Advantage of FFAG . . . . . . . . . . . . . . . . . . . . . . . 24

6.3 PRISM-FFAG Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.4 PRISM-FFAG Simulations . . . . . . . . . . . . . . . . . . . . . . . . 24

2

CONTENTS 3

6.4.1 Spread of muon Arrival Time . . . . . . . . . . . . . . . . . . 266.4.2 RF Shape Dependence . . . . . . . . . . . . . . . . . . . . . . 266.4.3 PRISM-FFAG Acceptance Study . . . . . . . . . . . . . . . . 29

6.4.4 Muon Survival Rate in the PRISM-FFAG . . . . . . . . . . . 30

7 Estimation of Muon Yield and Beam Emittance 337.1 Muon Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7.2 Pion Contamination in a Beam . . . . . . . . . . . . . . . . . . . . . 34

8 PRISM-II 368.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.2 PRISM-II Muon Beam Line . . . . . . . . . . . . . . . . . . . . . . . 36

8.2.1 Pion Capture at Forward Take-off Angle . . . . . . . . . . . . 378.2.2 Pion Decay and Muon Transport Solenoid . . . . . . . . . . . 398.2.3 Selection of Pion Momentum to Improve Muon Polarization . 41

8.3 PRISM-II FFAG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.3.1 Estimation of Muon Yield and Muon Polarization . . . . . . . 42

9 A Pulsed Proton Beam Facility for PRISM 469.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

9.2 Proton Beam Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.3 Experimental Area Layout . . . . . . . . . . . . . . . . . . . . . . . . 479.4 Request at Phase-I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

10 Conclusion 51

A 90-Bunche Operation of the 50-GeV PS 53A.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

A.2 Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54A.3 Bunched Beam Extraction . . . . . . . . . . . . . . . . . . . . . . . . 55

B Simulation on Pion Production and Capture by using MARS 56

B.1 Pion Production by 50-GeV Proton Incident . . . . . . . . . . . . . . 56B.2 Solenoid Pion Capture . . . . . . . . . . . . . . . . . . . . . . . . . . 56B.3 Radiation Heating to Superconducting Magnets . . . . . . . . . . . . 59

C Extraction from the Pion Decay Section 64

D Introduction to the FFAG 67D.1 The Principle of a FFAG Accelerator . . . . . . . . . . . . . . . . . . 67D.2 Zero Chromaticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

D.3 Large Acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68D.4 Radial and Spiral Type . . . . . . . . . . . . . . . . . . . . . . . . . . 69D.5 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4 CONTENTS

E PRISM-FFAG Design 71E.1 Triplet Radial Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . 71E.2 Lattice Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71E.3 Magnet Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73E.4 RF Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

F Hardware Design of PRISM-FFAG 77F.1 RF System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77F.2 Magnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

F.2.1 Yoke Free Design of FFAG Magnet . . . . . . . . . . . . . . . 77F.2.2 Kicker Magnet Design . . . . . . . . . . . . . . . . . . . . . . 78

G R&D of FFAG machines in Japan 82G.1 R&D of the POP FFAG Machine . . . . . . . . . . . . . . . . . . . . 82

G.1.1 Results of the Accelerator Studies . . . . . . . . . . . . . . . . 84G.1.1.1 Beam Acceleration . . . . . . . . . . . . . . . . . . . 84G.1.1.2 Tune Survey . . . . . . . . . . . . . . . . . . . . . . 85G.1.1.3 Beam Position . . . . . . . . . . . . . . . . . . . . . 85

G.1.2 Aperture Survey . . . . . . . . . . . . . . . . . . . . . . . . . 87G.1.3 Summary of POP FFAG R&D . . . . . . . . . . . . . . . . . . 87

G.2 R&D of the 150-MeV Proton FFAG Machine . . . . . . . . . . . . . . 88G.2.1 Main Features of 150-MeV FFAG . . . . . . . . . . . . . . . . 89G.2.2 R&D Status and Construction Schedule . . . . . . . . . . . . 90

H Alternative Phase Rotator Scheme 92H.1 Phase Rotator Linac . . . . . . . . . . . . . . . . . . . . . . . . . . . 92H.2 System Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93H.3 Magnetic Field Distribution along the Axis . . . . . . . . . . . . . . . 94H.4 Phase Rotator with Linac . . . . . . . . . . . . . . . . . . . . . . . . 95H.5 Muon Yield Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Chapter 1

Executive Summary

PRISM : PRISM is a project to construct a dedicated source of a high intensitymuon beam with narrow energy spread and small beam contamination. PRISM standsfor “Phase Rotated Intense Slow Muon source”. The aimed muon beam intensity isabout 1011 − 1012 µ±/sec, four orders of magnitude higher than that available atpresent. The narrow beam energy spread can be achieved by a novel technique of“phase rotation”, which accelerates slow muons and decelerates fast muons by a radiofrequency (RF) field. In the present design scheme, by phase rotation, the originalmomentum spread (of about ±30 %) would be squeezed down to ±3 %. Because ofthis, the brightness within ±3 % momentum spread is increased by a factor of aboutten. Regarding small beam contamination, in particular, pions of the level of 10−18

in a beam can be achieved by a long flight path of the beam so that most of pionsdecay out.

Physics Motivation : One of the important particle physics topics achievable withPRISM is a search for lepton-flavor violating muon rare processes. Lepton flavorviolation attracts much attention, theoretically and experimentally, since it wouldhave a large discovery potential to new physics beyond the Standard Model, forinstance supersymmetric extension to the Standard Model. An example of proposedexperiments of such at PRISM is a search for µ−− e− conversion process in a muonicatom at a sensitivity of 10−18.

PRISM Specification: The energy of the PRISM muon beam is set to be relativelylow, such as 20 MeV in muon kinetic energy (68 MeV/c in momentum). The PRISMbeam specifications are listed in Table 1.1.

PRISM-II for the Muon EDM Measurement: In addition to PRISM, therearises an additional significant demand to have a highly intense muon beam of around0.5 GeV/c in momentum. It is a separate facility called PRISM-II, since a differentbeam line and a ring are needed. One of the main physics topics with PRISM-II isa search for the electric dipole moment (EDM) of the muon with a sensitivity of lessthan 10−24 e·cm. PRISM-II would provide muons with high intensity ( 1010/sec)

5

6 CHAPTER 1. EXECUTIVE SUMMARY

Parameters Design goal CommentsBeam Intensity 1011 − 1012µ±/sec 1014 protons/sec is assumed.Muon kinetic energy 20 MeV pµ = 68 MeV/cKinetic energy spread ±(0.5 − 1.0) MeVBeam Repetition 100 - 1000 Hz

Table 1.1: PRISM Beam specifications

and high polarization (Pµ = 0.6) after phase rotation.

Requested Arrangement at Phase I : From the budget profile presently plannedat J-PARC, the construction of the pulsed proton beam facility could be after PhaseI. Even under this condition, at Phase I, we would like to ask the management ofJ-PARC a special arrangement to avoid soil activation at the location of protonextraction port to a planned facility. Once soil is activated, future excavation wouldbecome almost impossible. To be more specific, we would like to request placingconcrete shielding blocks at the location of future extraction. The detailsare given in this note, in Section 9.4.

Finally, the physics programs with PRISM and PRISM-II have a large discoverypotential of new physics beyond the Standard Model. We would like to request strongsupport to realize PRISM and PRISM-II at the earliest date as possible.

Chapter 2

PRISM Overview

2.1 Phase Rotated Intense Slow Muon Source

PRISM is a project to provide a dedicated source of a high intensity muon beam withnarrow energy-spread and small beam contamination. PRISM stands for “PhaseRotated Intense Slow Muon source”. The aimed beam intensity is 1011 − 1012µ±/sec,four orders of magnitude higher than that available at present. It is achieved by a largesolid-angle pion capture with a high solenoid magnetic field. Narrow energy spreadcan be achieved by phase rotation, which accelerates slow muons and decelerates fastmuons by a radio frequency (RF) field. The pion contamination in a muon beam canbe removed by a long flight path in PRISM so that most of pions decay out.

PRISM consists of

• a pulsed proton beam (to produce a short pion pulsed beam),

• a pion capture system (with large-solid angle by a high solenoidal magneticfield),

• a pion decay and muon transport system (in a long solenoid magnet of about10 m long), and

• a phase rotation system (which accelerates slow muons and decelerates fastmuons by an RF field).

Some of the key components are explained in detail in the following sections. Theconceptual structure of PRISM is shown in Fig. 2.1.

2.2 PRISM Specification

The PRISM beam characteristics are summarized in Table 2.1.In the pion capture system, low-energy pions (and cloud muons) are trapped in a

high solenoidal magnetic field to achieve high intensity. The magnetic field and theinner bore radius of the surrounding superconducting magnet would determine an

7

8 CHAPTER 2. PRISM OVERVIEW

Proton BeamSuperconductingSolenoid Magnet rf Cavities

Solenoid PionCapture Section

Pion DecaySection

Phase Rotation Section

not in scale

PionProduction Target

Figure 2.1: Conceptual Components of PRISM. It has the pion capture system, thepion decay system, and the phase rotation system.

Table 2.1: Anticipated PRISM beam design characteristics

Parameters Design goal CommentsBeam Intensity 1011 − 1012µ±/sec 1014 protons/sec is assumed.Muon kinetic energy 20 MeV → pµ = 68 MeV/cKinetic energy spread ±(0.5 − 1.0) MeVBeam Repetition 100 - 1000 Hz

effective transverse momentum (pt) cut. Given an effective pt cut, a number of pionscaptured can be estimated. If pt is about 0.1 GeV/c, about 0.2 pions per proton canbe anticipated from Monte Carlo simulation. Since the proton beam intensity of theJ-PARC 50-GeV synchrotron is about 1014 /sec, about 3×1013 pions /sec is expected.

When a magnetic field is higher, the beam emittance of muons becomes betterand smaller. Therefore, the higher magnetic field is preferable. In the phase rotationsystem, slow muons are accelerated, and fast muons are decelerated by a strongradio-frequency (RF) electric field to yield a narrow longitudinal momentum spread.A schematic view of the basic concepts of solenoid capture and phase rotation isshown in Fig. 2.1, where the solenoid capture system followed by the phase rotationsystem is seen.

2.3. PRISM-II SPECIFICATION 9

Schematic layout of PRISM is shown in Fig.2.2. It combines high-field pion cap-ture, the π-µ decay system of a 10-m long superconducting solenoid magnet, andphase rotation system. One of the features of PRISM is to do phase rotation at aFixed-Field Alternating Gradient synchrotron (FFAG), which has several advantages,such as a large momentum acceptance.

2.3 PRISM-II Specification

In addition to PRISM that is for experiments with stopped muons, there arises addi-tional significant demand to have a highly intense muon beam of around 0.5 GeV/c inmomentum. One of the main physics topics is a search for the electric dipole moment(EDM) of the muon with a sensitivity of less than 10−24 e·cm. The current limit ofthe muon EDM is (3.7 ± 3.4) × 10−19 e·cm.

The search needs muons of about 0.5 GeV/c and injects them into a dedicatedring (EDM ring). The PRISM-II would provide muons with high intensity and highpolarization after phase rotation. The most important requirement is Nµ ·P 2

µ > 1016,where Nµ is a number of the muons available and Pµ is the muon polarization. Thespecifications of the PRISM-II muon beam are shown in Table 2.2

Table 2.2: Anticipated PRISM-II beam design characteristics

Parameters Design goal CommentsBeam Intensity 1010 − 1012µ±/sec 1014 protons/sec is assumed.Muon kinetic energy 597 MeV pµ = 500 MeV/cKinetic energy spread ±2 %Muon polarization Pµ > 0.6Beam Repetition 100 - 1000 Hz

In order to fulfill these requirements, the design of PRISM-II is slightly different tothat of the original PRISM in the following places; (1) longer pion decay system, (2)larger FFAG phase rotator. The pion capture system could be the same as PRISM.

It should be noted that the design of the PRISM-II FFAG ring is the same as thefirst FFAG ring of the Japanese Neutrino Factory. The first ring would acceleratemuons from 0.3 GeV/c to 1.0 GeV/c. In future, therefore, the PRISM-II FFAG ringwould serve an important role toward the realization of a neutrino factory in Japan.

10 CHAPTER 2. PRISM OVERVIEW

PROTON BEAM DUMP

CAPTURE

SOLENOID

PRODUCTION TARGET

PRIMARY PROTON

MATCHING

SECTION SOLENOID

FFAG RING

RF CAVITY

5 M

INJECTION SYSTEM

EJECTION SYSTEM

Figure 2.2: Schematic layout of PRISM.

Chapter 3

Proton Beam

3.1 Overview

3.1.1 Proton Beam Power and Time Structure

A number of produced pions (and therefore their daughter, muons) is proportionalto beam power which is given by the product of beam energy and beam current.Roughly speaking, as long as the beam power is the same, the pion yield would bealmost the same. It is based on the fact that the pion cross section increases linearlyas proton beam energy. At the initial stage of PRISM, beam power of 0.75 MW (50GeV and 1.5 µA) is assumed at the initial stage1

The time structure of protons is critical for phase rotation. PRISM would requirea pulsed proton beam produced by fast extraction, since phase rotation needs muons(and therefore their parents, pions) to be within a narrow time spread before phaserotation2. If a time spread of the muon beam is larger, the energy spread after phaserotation becomes worse. A pulse width of much less than 10 nsec is needed.

Although a pulsed proton beam from fast extraction should be used for PRISM,since there is no room in the neutrino beam line to build a dedicated experimentalhall for PRISM, we would like to request the construction of a new experimental halland a beam line for fast-extraction. This issue will be presented later in Chapter 9.

3.1.2 Why is 50-GeV Proton Beam used ?

If PRISM is constructed at J-PARC, a proton beam from the J-PARC 50-GeV protonsynchrotron (PS) would be used. It should be noted that a proton beam from the3-GeV PS can not be used. The reasons are the following: (1) PRISM needs a narrowpulsed beam of protons, but in the present extraction scheme, a beam from the 3-GeV PS has a pulse width of more than 100 nsec, and (2) it is of technical difficulty

1It will be adopted to accommodate future proton upgrade of 4.4 MW, where the target issuehas to be such as a liquid mercury jet target being developed at a neutrino factory R&D.

2The details of phase rotation will be described in Chapter 6.

11

12 CHAPTER 3. PROTON BEAM

to make a narrow pulse of 3 GeV since a space-charge effect, which blows a beamaway, is more severe at lower energy, and (3) in the present muon-science facility,a full target (of about 1-2 interaction lengths) can not be placed since it is locatedupstream from the neutron scattering facility in the same proton beam line.

3.2 8 Bunch Operation

At the 50-GeV proton synchrotron, the beam intensity of 3.3× 1014 proton per cycleand a cycle time of 0.3 Hz are expected. The 50-GeV protons are extracted to thenuclear-particle (NP) experimental hall by slow extraction, and to the neutrino beamline by fast extraction. When operated in a slow extraction mode, the average beamcurrent and the duty factor are 15 µA and 0.20 respectively.

To have the best performance of phase rotation, it is necessary to have a narrowpulse width of the proton beam. The present pulse width in the J-PARC 50-GeVproton synchrotron is 6 nsec in sigma3.

The typical cycle structure of the J-PARC 50-GeV PS is illustrated in Fig. 3.1.Four batches (each of which contains 2 bunches) from the 3-GeV PS are injected intothe 50-GeV PS ring when it stays at a low field. When 8 buckets out of 10 are filledwith beams, the 50-GeV ring starts acceleration. The time period of every bunch isabout 535 nsec (1.67 MHz) and a gap separation is about 300 nsec (i.e. 50 % filling).

3.3 90 Bunch Operation

As mentioned before, PRISM would require a sharp pulse width to achieve a betterperformance of phase rotation. And the search for the µ−−e− conversion process withPRISM would require many beam bunches with relatively low beam instantaneousintensity. To accomplish this, a new novel idea of 90-bunches operation has come out.By handling 90 bunches in the ring, the above two requirements would be met. Thedetail parameters are shown in Table 3.1. It is worth to mention that muon (g − 2)experiment also needs 90 bunching as might be discussed in an independent Letterof Intent.

To realize 90 re-bunching, the proposed scheme is the following.

1. After the proton acceleration, a beam will be de-bunched by shutting RF off.It is the same as in slow beam-extraction.

2. A beam will be re-bunched by an RF of 90 bunches.

3. A bunch will be kicked out one by one (single-kicking mode). Since a bunchseparation is about less than 52 nsec, a fast kicker magnet is also needed. A rep-etition frequency of the single bunch depends on the experimental requirements

3The accelerator people generally use a ±3 sigma width as a full pulse width. Therefore, thiscase implies a full width of 36 nsec.

3.3. 90 BUNCH OPERATION 13

535 nsecat 50 GeV

1

2

3 4

Neu

trino

K-are

na

Abort

Booster ( 400MeV to 3GeV )

Main-Ring ( 3 to 50 GeV )

(1) Injection at 3 GeVRF: 1.82 MHz( at 3GeV )

549nsec

(at 3GeV

)

Figure 3.1: Typical machine cycle structure

Table 3.1: Main parameters of the 90-bunches operation.

Parameters valuesRevolution time 5.23 µsecRevolution frequency 0.191 MHzRF frequency of 90 bunches 19.1 MHza time period between bunches (central position) 52 nsec

and technical feasibility. In particular, for the search for µ−−e− conversionprocess, either 100 Hz or 1000 Hz is being considered.

An expected full pulse width (±3 sigma) in the 90 bunches scheme is 3.6 nsec (or 0.6nsec sigma of the width) if a higher RF voltage is used to keep the bucket height thesame as in the 10 harmonics operation.

Chapter 4

Pion Capture System

The highest beam intensity in the world, which would be provided by PRISM, couldbe achieved by large-solid angle capture of pions at their production.

4.1 Pion Production by 50-GeV Proton Incident

PRISM would provide a low energy muon beam for stopped muon experiments, ingeneral. Therefore, to capture, pions of low energy, which are the parent particle ofthe muons are of interest. At the same time, high energy pions could become potentialbackgrounds to eliminate. To study pion capture, simulation was performed by usingboth MARS and GEANT3 with FLUKA. The MARS code is a hadron productioncode developed at Fermilab.

Fig.4.1 shows the momentum spectra of π− produced from a graphite target. Itis seen that the maximum of transverse momentum (pT ) is around 100 MeV/c for0 < pL < 200 MeV/c for both the forward and backward pions. The maximum ofthe total momentum for the backward pions is located at about 120 MeV/c, whereasthat for the forward pions is about 200-400 MeV/c. Is is also seen that pions at highenergy are suppressed in the backward direction. From these reasons, we decided touse the backward pions for PRISM.

4.2 Pion Production Target

Monte Carlo simulations have been also done to study a pion production target. Someof the results on the pion production targets are summarized as follows. The detailedstudies can be presented in Appendix B.

• A heavy material for the pion production target is better than a light material.For example, the pion yield by a tungsten target is about 3 times larger thanthat of a graphite target.

• The yield of pions at low energy is almost saturated at a target length of morethan 2 interaction lengths. The increase of the yield from 2 to 3 interaction

14

4.2. PION PRODUCTION TARGET 15

p t (G

eV/c

)

pl (GeV/c)

pt (GeV/c)pt (GeV/c)

p (GeV/c)

Figure 4.1: Pion production in a graphite target. (top) correlation between pL andpT , (middle) Total momentum distributions for forward and backward π−s, (bottom)pT distributions for 0 < pL < 0.2 GeV/c, 0.2 < pL < 0.4 GeV/c, and 0.4 < pL < 0.6GeV/c.

lengths is very small, of about 10 %. Both a heavy and a light materials showthe same tendency.

• The yield of pions at low energy decreases as the radius of the target increases.This would be explained by the absorption of pions at low energy. The optimumradius is about 0.5cm.

One of the disadvantages of heavy metals is their low melting point. They mightmelt down when a proton beam of 1 MW beam power hits. On the other hand, ithas been known that graphite could work with up to beam power of about 1 MWlevel, with either radiation-cooled or water-cooled configuration, but with the cost ofa relatively low pion production yield (which is smaller by a factor of about 3 smallerthan in heavier materials). It is however needed to replace frequently due to theradiation damage on the specific heat of graphite.

R&D of the target system has just started. Several options, such as (1) a rotatingmetal band system, (2) a liquid mercury jet, (3) a tantalum fine particles packed inthe titanium casing, and (4) a graphite with radiation or water cooling (as the basicoption). All these are studied as the R&D works for the neutrino factory projects inthe world-wide. We would like to keep exchange information among various foreignstudies.

16 CHAPTER 4. PION CAPTURE SYSTEM

4.3 Pion Capture

To collect as many pions (and cloud muons) of low energy as possible, they arecaptured under a high solenoidal magnetic field with a large solid angle. In this case,pions emitting in a half hemisphere can be captured within the transverse momentumthreshold (pmax

t ). The pmaxt is given by a magnetic field strength (B) and the radius

of the inner bore of solenoid magnet (R) as

pmaxT = 0.3 × B(kG) × R(cm)

2. (4.1)

4.3.1 Optimization of the Pion Capture System

The muon yield may depend on several parameters of the pion capture system, suchas target material, target radius, target length, magnetic field strength and boreradius. In order to optimize those parameters, we performed several computer simu-lation studies. The study of the low energy pion yield, which was assumed to changeproportionally to the low energy muon yield, by using MARS was already described.The muon yield including the effect of the pion decay in the decay solenoid were doneby using GEANT3.21.

The muon yields are estimated at 10 m from the entrance of the pion decay system,where most of pions decay into muons. Fig.4.2 shows the number of muons withinthe emittance of 10000 π mm·mrad in horizontal and 2000 π mm·mrad in vertical atthe exit of the pion decay section1 as a function of a pion capture magnetic field (4T, 8 T, 12 T, and 16 T). It is seen clearly that the higher the pion capture magneticfield is, the better the muon yield at the exit of the pion decay system becomes2.Therefore, a higher magnetic field is preferable.

Once the magnetic field is chosen, the bore radius of the capture solenoid givesthe maximum pT of the accepted pions by the Eq.4.1. According to Fig.4.1, placingpmax

T at around 200 MeV/c would be sufficient. As a matter of fact, the net muonyield does not seem to increase beyond pmax

T = 90 MeV/c since we are interested inthe muon momentum being less than 84 MeV/c. That corresponds to R = 10 cm ifB = 6 T.

We should install radiation shield between the target and the superconductingsolenoid magnet. This will further increase the total radius of the superconductingsolenoid magnet. From the MARS simulation study on the radiation shielding shownin Appendix B, a thickness of the radiation shield would be about 30cm or more if itis made of tungsten.

Detailed optimization of the bore radius strongly depends on the available tech-nology of the superconducting solenoid magnet. For the moment, we employed theconservative basic design values, namely of B = 6 T, R = 10 cm and pmax

T = 90

1This is the basic acceptance of the phase rotation system at a later stage.2The other conditions assumed in this Monte Carlo simulation is 4 T for a pion decay system, a

tungsten target.

4.3. PION CAPTURE 17

Figure 4.2: Muon yields at 10 m from the entrance of the pion decay system as afunction of magnitudes of a pion capture field.

MeV/c. The final design will be decided by considering the R&D studies on super-conducting solenoid magnets (especially for the tolerance to the radiation heat load)as well as the available budget for the magnet procurement.

4.3.2 Superconducting Solenoid Magnet

The radiation heat load to superconducting coils placed behind the radiation shieldof tungsten of 30 cm in length is the level of 3 ×10−3 W/g. The radiation heat comemostly from neutrons. If copper is used as the stabilizer of the superconducting coils,a total thickness of the coil might be about 2 cm or more3, and a total impact on the4.5 K refrigeration load is over 1 kW. In order to overcome this difficulty, we havestarted R&D works and come with the following new ideas.

• The aluminum stabilizer, instead of the copper stabilizer, is employed in order toreduce the total coil mass. Since the radiation heat load to the coil increases pro-portionally to its total mass, the expected heat load to the aluminum-stabilizedsuperconducting coil might be greatly reduced than that to the copper-stabilizedcoil.

• The coil is cooled down by using the combination of both conduction coolingand forced-flow cooling.

With these ideas, we aim to develop the superconducting coil that can be operatedunder 500 W of the heat load.

3For example, see “MECO Superconducting Solenoid System Conceptual Design Report”[3]

Chapter 5

Pion Decay And Muon TransportSystem

5.1 Adiabatic Transition From a High to a Low

Magnetic Field

Since the pions captured at the pion capture system have a broad directional dis-tribution, it is intended to make them more parallel to the beam axis by changinga magnetic field adiabatically. From the Liouville theorem, a volume in the phasespace that beam particles occupy do not change. Under a solenoidal magnetic field,the relation between the radius of curvature (R) and the transverse momentum (pt)leads to the relation given by

pt × R ∝ p2t

B= constant, (5.1)

where B is a magnitude of the magnetic field. Suppose the magnetic field decreasesgradually, pt also decrease, yielding a more parallel beam. This is the principle of theadiabatic transition. Namely, when a magnetic field is reduced by a factor of two, pt

decreases to a half. On the other hand, since

pt × R ∝ B × R2 = constant′1, (5.2)

the radius of curvature increase by a factor of√

2. Therefore, the inner radius of amagnet in the pion decay section has to be

√2 times that of the pion capture. With

the cost of a beam brow up, a pion beam becomes more parallel. Furthermore, itis not effective in reality to have a long magnet with a high magnetic field, and amagnetic field has to be lowered at some point. Fig.5.1 illustrates the principle ofadiabatic transition.

1It implies that the magnetic flux is conserved.

18

5.2. CURVED SOLENOID SECTION 19

z

pl

pt

Figure 5.1: Adiabatic transition from a high magnetic field to a low magnetic field.This adiabatic transition reduces the magnitude of transverse magnetic field.

5.2 Curved Solenoid Section

To bring pions and muons into the experimental area, a beam has to be bent. Also itis necessary to make a hole in the solenoid magnets for initial protons to enter. Thesecould be achieved by introducing a curved solenoid magnets.

It is known that a center of the helical trajectory of charged particles is drifted ina curved solenoid field. The drift (D[m]) is given by

D =1

0.3 × B× s

R× p2

l + 12p2

t

pl

(5.3)

where B[T] is a magnetic field at the axis, s [m] and R[m] are a path length andthe radius of curvature of a bent solenoid. Namely, s/R is a bending angle. pl

and pt[GeV/c] are parallel and transverse momentum respectively. Charged particleswith opposite sign move in the opposite direction. This can be used for charge andmomentum selection if a suitable collimator is placed after the curved solenoid. Thiskind of curved solenoid magnets have been already adopted in the MECO (BNL-AGSE940) experiment. Unless two curved solenoid of opposite bent are installed, a dipolemagnetic field to compensate a drift of the central momentum is needed.

5.3 Decay Solenoid Design

To let pions decay into muons, we need a long flight path. Also to contain thosepions and decay muons in a limited space, a long solenoid magnet is required. At themomentum of about 100 MeV/c, a mean decay length is about 10 m, and therefore

20 CHAPTER 5. PION DECAY AND MUON TRANSPORT SYSTEM

posi

tron

elec

tron µ

+ π+µ- π-

neut

ron

prot

on

Frac

tion

Figure 5.2: Fractions of pions, muons and electrons at the end of a 10-m long solenoiddecay system.

a flight length of 10 m is needed. Fig.5.2 shows the relative fractions of the pion, themuon and the electron at the exit of a 10-m solenoid magnet.

At the exit of the long solenoid magnet, a fringing field might blow up the muondistribution. This was studied by using GEANT3.21 with a fringing magnetic fieldcalculated with POISSON/SUPERFISH. It was concluded that although most of themuons will spread out at the fringing field region, the core parts in the good emittance,which can be accepted by the PRISM-FFAG ring, would be contained within ±50cm, where the next focusing elements should capture them. The detailed study isdescribed at Appendix C.

5.4 Dispersion Matching

When the muons are injected into the PRISM-FFAG ring, it is necessary to make amuon beam dispersive to match the acceptance of the PRISM-FFAG. This require-ment is placed by the FFAG characteristics. In order to make the muon beam disper-sive, a system consisting of double bending magnets is adopted (dispersion matchingsection), and it would be located between the pion decay and muon transport sys-tem and the phase rotation system. A schematic layout is shown in Fig.5.3. Thepreliminary lattice design of the dispersion matching has been done, and is shown inFig.5.4.

5.4. DISPERSION MATCHING 21

radius(mm)

54.4 61.2 68.0 74.8 81.6 (MeV/c)

high mom.

low mom.

particle orbit in FFAG (simulation)

Figure 5.3: Dispersion matching section consisting of double bending magnets. It doesmake a dispersive beam where muons of particular momentum come to a particularradial position.

Figure 5.4: Designed lattice of the dispersion matching section.

Chapter 6

Phase Rotation System

6.1 Principle of Phase Rotation

Phase rotation is a method to achieve a beam of narrow energy spread. The principleof phase rotation is to accelerate slow muons and decelerate fast muons by a strongradio-frequency (RF) electric field, in order to yield narrow longitudinal momentumspread. It corresponds to 90 rotation of the distribution of the muons in the beam inthe energy-time phase space, as schematically shown in Fig.6.1. After phase rotation,the projection of the phase space onto the energy axis becomes narrower and sharper.To identify fast and slow muons, a time of flight (TOF) from the time of production oftheir parent pions is used. It implies that slow muons come late and fast muons comeearly after a long drift of 10 m in the pion decay system. The muon distribution inptotal vs. a time of flight (TOF) space, which is calculated by Monte Carlo simulations,is shown in Fig.6.2. By phase rotation, the initial time spread is converted into thefinal energy spread. The narrow width of a pulsed proton beam is very critical for anet performance of phase rotation.

Since stopped-muon experiments are one of the main subjects for PRISM, thecentral muon momentum is set to be about 68 MeV/c (corresponding to a kinetic

Accelerate

Dec

eler

ate

Phase

Ene

rgy

High EnergyAdvanced Phase

Narrow Energy Spread

Low EnergyDelayed Phase

Phase

Ene

rgy

Figure 6.1: Principle of phase rotation shown in the energy-time phase space. Theinitial narrow time spread is converted into the final narrow energy spread.

22

6.2. PHASE ROTATION BY FFAG 23

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

20 30 40 50 60 70 80 90 100 110 120

ToF(ns)p

tota

l(GeV

/c)

Figure 6.2: Distribution of Ptotal versus TOF.

energy of 20 MeV). From Monte Carlo simulation studies, the original momentumspread of about ±30 % is reduced down to ±3 %, after phase rotation.

6.2 Phase Rotation by FFAG

One of the features of the phase rotation in PRISM is to adopt a circular (ring)machine based on a Fixed-Field Alternating Gradient synchrotron (FFAG).

6.2.1 Advantage of Circular Machines

A linear system (such as a linac) has been discussed for phase rotation1. Instead ofa linear system which is generally considered, a circular system has been adopted forPRISM. The reasons are described as follows.

Since the muons of interest for PRISM are relatively low in energy and are non-relativistic, their velocities (β) within ∆p = ±30% are quite different in magnitude.After a drift, thereby, they spread longitudinally in space. To capture them in anaccelerating electric field used for phase rotation, it must have a long wavelength. Apossible scheme is either an induction linac or a low-frequency RF with its frequencyof 1∼10 MHz. For the low frequency RF, the electric field gradient achievable is nothigh, at the best, less than 0.5-1 MV/m. Thus, a total length of the phase rotationsystem becomes usually very long. For instance, the central momentum of 68 MeV/cwith the initial momentum spread of ±30 %, a total length of about 150 meters isneeded. In addition to RF, to contain the muons in space, transverse focusing suchas a solenoid field is needed along the path. Therefore a long solenoid magnet of atotal length of 150 meters has to be built. From these reasons, a linear system is veryexpensive.

1See Appendix H for our study on the linear phase rotation system

24 CHAPTER 6. PHASE ROTATION SYSTEM

A circular machine, instead, could have smaller and compact in geometrical size.The number of RF cavities and the required electric power is greatly reduced bya factor of the turns necessary for phase rotation. Therefore, a total cost can besignificantly smaller.

6.2.2 Advantage of FFAG

Among various circular machines (such as a synchrotron and a cyclotron), a FFAGhas several advantages, which are listed below.

• Synchrotron oscillation for phase rotation : Since we need synchrotronoscillation to do phase rotation, we can not use a cyclotron which is isochronous.

• Large momentum acceptance : It is necessary to accept large momentumdistribution (for instance ± 20-30 %) at the beginning to do phase rotation.Unfortunately, a synchrotron has a limited momentum acceptance (of 1 %),and therefore it can not be used.

• Large transverse acceptance : Since a muon beam is broad in space, a cir-cular ring has to have a large transverse acceptance. Only FFAG (and probablya cyclotron which can not be used for the other reason above) have a largetransverse acceptance (both horizontal and vertical).

From these reasons, a FFAG is an only choice for a circular machine to do phaserotation.

6.3 PRISM-FFAG Design

A preliminary design of the PRISM FFAG ring and its lattice function are shown inFig.6.3. The PRISM-FFAG accelerator consists of a normal conducting magnet. Thediameter is about 10 meters. It has 8 straight sections. One straight section is for themuon injection, and another straight section is for the muon extraction. In the restof 6 straight sections, a total of 12 RF cavities are installed. From acceleration pointof view, the straight section should be as long as possible so as to install as manyRF cavities as possible. Fig.6.4 shows an example layout of the PRISM-FFAG phase-rotator ring. Table 6.1 summarizes the main specifications of the PRISM-FFAG ring.

6.4 PRISM-FFAG Simulations

Since the beam optics in FFAG is non-linear, Monte Carlo tracking is needed insteadof linear optics. Three-dimensional Monte Carlo simulations for the PRISM-FFAGhave been done based on GEANT 3.21. Multiple scattering of muons, interactions,

6.4. PRISM-FFAG SIMULATIONS 25

Figure 6.3: A lattice design of the PRISM-FFAG ring and the beam orbit parameters.

and pion and muon decays are taken into account. The magnetic field of the PRISM-FFAG magnets were calculated by a 3-dimensional magnetic field calculation code,TOSCA. A typical muon track of its injected momentum of 54.5 MeV/c is shown inFig.6.5. Since the PRISM-FFAG has momentum dispersion, the radius of the muonorbits becomes gradually larger.


6.4.1 Spread of muon Arrival Time

The muons of interest for PRISM (∆p = ±30%) come at a different timing to thePRISM-FFAG ring owing to their different velocities and path length. Fig.6.6 showsthe result of Monte Carlo simulations, in which it is shown that a net differenceof their arrival timing is within ±20 nsec. The RF frequency of the PRISM-FFAGshould accept all the muons in this time range. The RF frequency considered is 5MHz, and it would covers those.

For the muons of the same momentum, owing to their different pitch angle, theirarrival time is different. This spread affect the performance of phase rotation andtherefore it should be minimized. The Monte Carlo simulation result in Fig.6.6 showsthat the time spread for the same momentum is less than ±5 nsec. In the followingphase rotation simulation, this effect has been taken into account.

6.4.2 RF Shape Dependence

A shape of the RF field is critical to carry out phase rotation since the muon velocityis related in a non-linear way to the muon momentum in the momentum range of ourinterest. The relation of velocity and momentum is shown in Fig.6.2.

To optimize an RF shape, two different RF types were studied. One was a si-

VacuumPump

Figure 6.4: Layout of the PRISM-FFAG ring


Table 6.1: Main parameters of PRISM-FFAG.

Parameters valuesCentral Energy 20MeVnumber of sector 8k value 5transition gamma 2.45average radius 5 mB@F/D 0.435/0.0717 TF/2 angle 0.037 radD angle 0.031 radF/2 bend angle 26 degreepacking f 0.18tune(H/V) 3.404/2.641

µ−

Figure 6.5: An example of the muon trajectory in the PRISM-FFAG ring. The muonhas its initial momentum of 54.5 MeV/c and its curvature becomes larger as phaserotation is applied. This is calculated by the Monte Carlo tracking method.


10ns 10ns

81.6

71.2

68.0

61.2

54.4MeV/c

t=0,+-5ns

Figure 6.6: Difference of the arrival timing of muons with different momentum.

nusoidal shape and the other was a saw-tooth shape. They are shown in Fig.6.7.In the Monte Carlo study shown in Fig.6.7, the muons at the same phase (0) withsome assumed initial phase distribution (about ±5 nsec) are injected (as shown in redpoints). After each turn, the muon distributions are shown in points with differentcolors (green for the 1st turn, blue for the 2nd turn, yellow for the 3rd turn, pink forthe 4th turn, and sky-blue for the 5th turn). The spread of the same color pointsreflects the initial phase spread of the injected muons. From Fig.6.7, it is found thatin the case of simple sinusoidal RF shape (left), phase rotation is not perfect, yielding∆p/p = ±6%. However, in the case of the saw-tooth RF shape (right), it is morecomplete, yielding ∆p/p = ±2%.

To realize a saw-tooth RF shape, it is necessary to include higher-harmonics inan RF shape. Fig.6.8 shows the case when the fundamental frequency of 5 MHz ismodulated with higher harmonics, such as of its double and triple frequencies so asto create a saw-tooth-like RF shape. The performance of phase rotation by this RDshape is shown in the upper plot in Fig.6.8. From Fig.6.8, it can be concluded thatinclusion of the double and triple frequencies (with a ratio of fundamental : double :triple = 3 : 2 : 1) would be sufficient to simulate a saw-tooth RF wave form in termsof the performance of phase rotation.

There will be two solutions to include higher harmonics. One is to operate oneRF cavity with the fundamental frequency modulated with the double and triplefrequencies. In this case, the RF cavity should have a broad band width (low Q). Theother is to operate 12 RF cavities with different frequency, for example, 6 cavities for


Figure 6.7: (left) sinusoidal RF shape, and (right) saw-tooth RF shape. The upperfigures show muon distribution in the ∆p/p vs. their phases with respect to the RFphase (arbitrary), and the lower figures are an RF shape.

the fundamental frequency, 4 for the double, 2 for the triple frequency. In this case,an RF cavity with high Q can be used for each, and therefore a higher RF field canbe achieved. At this moment of writing, we are still under studies to determine whichis better.

6.4.3 PRISM-FFAG Acceptance Study

It is needed to demonstrate that FFAG has a large horizontal acceptance by MonteCarlo simulation. Five different momenta (54.4 MeV/c, 61.2 MeV/c, 68.0 MeV/c,74.8 MeV/c, 81.6 MeV/c) were selected. And, muons of these five momenta withlarge emittance are generated in Monte Carlo simulations and are injected into thecorresponding radius.

The simulation result for the horizontal and vertical acceptances are shown inFig.6.9 and Fig.6.10, respectively. In Fig.6.9, it is found that the PRISM-FFAG of thepresent design has more than about 10,000 πmm·mrad in the horizontal acceptancefor 5 turns. Also, it is seen in Fig.6.9 that the closed orbit of muons with five differentmomenta are located at five different radii, the smaller radius for the lower momentumunder a fixed magnetic field. When phase rotation is done after 5 turns, the orbitswith different radii come into the orbit of central momentum.

The vertical acceptance is shown in Fig.6.10, where the vertical acceptance of more


Figure 6.8: Convolution of higher harmonics

than 2000 πmm·mrad is expected. It is known that this acceptance is limited by thevertical physical aperture of the FFAG magnet. It is therefore being considered tohave a wider gap of the FFAG magnet to improve the vertical acceptance.

6.4.4 Muon Survival Rate in the PRISM-FFAG

Phase rotation is completed after 5 turns which corresponds to about 1 µsec. After 5turns, it is known that the survival rate of muons in the ring is about 60 % in average.The momentum dependence of survival rates is shown in Fig.6.11, where a differencewith and without a muon-decay can be seen. From Fig.6.11, at lower momenta, asurvival rate is smaller. It should be noted that smaller events at higher momenta isdue to the iron yoke structure of the PRISM-FFAG. It has a smaller magnet gap atthe larger radius (namely for muon tracks of higher momentum). To improve this, anew type of iron yoke, where the magnet gap is relatively flat at any radius is beingconsidered.


Initial Phase

After 1 turn

After 2turns

After 3turns

After 4 turns

After 5turns

54.4 61.2 68.0 74.8 81.6MeV/c

Figure 6.9: Horizontal acceptance of the PRISM-FFAG by Monte Carlo simulation.

54.4 61.2 68.0 74.8 81.6MeV/c

Initial Phase

After 1 turn

After 2 turns

After 3 turns

After 4 turns

After 5 turns

Figure 6.10: Vertical acceptance of the PRISM-FFAG by Monte Carlo simulation.


Figure 6.11: Surviving muon events (counts) in the PRISM-FFAG as a function ofmuon momentum. Two points at the same momentum value represent a number ofsurviving muons after 5 turns with and without muon decays.

The muons in the ring decay in their flight. When an electron from muon decaysurvive in the ring, it will form a beam contamination. From Monte Carlo simulation,it is found that no electrons from muon decays survive in the ring out of 1600 muons.It implies that the survival rate of electrons is less than 1/1600, which is limited bystatistics that was tried in the Monte Carlo simulation.

Chapter 7

Estimation of Muon Yield andBeam Emittance

7.1 Muon Yield

The muon yield (Yµ) at PRISM can be given by

Yµ = Np · Nπ · επ-decay · εmom · εtime · εemittance · εdispersion · εFFAG · εµ-decay, (7.1)

where Np is a number of protons per second, Nπ is a number of pions captured at thepion capture system, εmom, εtime, εemittance, εdispersion, εFFAG, επ-decay, and εµ-decay are,respectively, the momentum acceptance (50 MeV/c < pµ < 90 MeV/c), the timingacceptance (of about ±5 nsec), the beam emittance acceptance of the PRISM-FFAGring, the efficiency of the dispersion matching between the muon transport systemand the PRISM-FFAG, the survival rate during 5 turns in the PRISM-FFAG, thepion decay rate, and the muon survival rate due to muon decay.

They can be estimated by using Monte Carlo simulations with the assumptionof particular design based on some technologies employed. The technology choicehas a large impact on the muon yield. In particular, Nπ and εemittance have strongdependence on the target material and length, the magnitudes of the pion capturefield (capture field) and of the pion-decay and muon-transport system (transportfield). Although we have not fully optimized, typical sampling cases are listed inTables 7.1, where the numbers of muons per 1014 protons per second (J-PARC phaseI) and 4×1014 protons per second (J-PARC phase II) are shown1. In this estimation,the 3-interaction target length of 120 cm and of 28.8 cm for graphite and tungstenrespectively are used. The factor επ−decay was included in Monte Carlo simulationof the pion capture system. The factor εµ−decay was estimated with another MonteCarlo simulation dedicated to the FFAG phase rotation, and it is about 60%. At the

1At 4 MW beam power (4 × 1014 protons per second), it is known that the solid targets, likegraphite and tungsten, might not be able to be used. Several alternatives are discussed such as amercury liquid jet, a rotating band target, a tantalum fine-particle target.

33

34 CHAPTER 7. ESTIMATION OF MUON YIELD AND BEAM EMITTANCE

time of writing, detailed studies on εdispersion and εFFAG have not been completedyet, and therefore they are assumed to be 100 %.

7.2 Pion Contamination in a Beam

At PRISM, a total flight length is about 150 meters. The survival rate Nπsurvival is

given by

Nπsurvival = exp(− L

cβγτµ) (7.2)

where β, γ, and τµ are the Lorentz factors and the mean lifetime of the pion, respec-tively. For the pions of about 68 MeV/c, it gives

Nπsurvival ∼ 10−17. (7.3)

Therefore, there is absolutely no pion contamination for the particles with 5 turnsin the PRISM-FFAG ring. One of the issues is that when late pions, which comebetween the proton pulse, enter into the PRISM-FFAG ring, there would becomepion contamination. To eliminate them, (1) the proton extinction between the pulsesshould be small (10−3) and (2) the kicker magnet at the entrance of the PRISM-FFAGhas to have additional extinction for late charged particles. Our naive estimationshows that the reduction of those late pions can be achievable at our desirable levelof 10−18.

7.2. PION CONTAMINATION IN A BEAM 35

Table 7.1: Negative Muon yields (Yµ) per second for various target materials, pioncapture magnetic fields, muon transport magnetic fields with 1014 protons/sec (J-PARC Phase-I) and 4 × 1014 protons/sec (J-PARC Phase-II). The top table isfor εemittance of 10,000 πmm·mrad in horizontal and 2,000 πmm·mrad in vertical,whereas the bottom table is for εemittance of 20,000 πmm·mrad in horizontal and 3,000πmm·mrad in vertical.

Target material Capture Transport Muon yield per Muon yield perfield field 1014 protons 4 × 1014 protons

Graphite 16 T 4 T 2.4 × 1010 9.6 × 1010

16 T 2 T 1.8 × 1010 7.2 × 1010

12 T 4 T 1.8 × 1010 7.2 × 1010

12 T 2 T 1.2 × 1010 4.8 × 1010

8 T 4 T 1.8 × 1010 7.2 × 1010

8 T 2 T 1.2 × 1010 4.8 × 1010

6 T 4 T 0.6 × 1010 2.4 × 1010

6 T 2 T 1.2 × 1010 4.8 × 1010

Tungsten 16 T 4 T 4.8 × 1010 19 × 1010

16 T 2 T 5.4 × 1010 22 × 1010

12 T 4 T 4.2 × 1010 17 × 1010

12 T 2 T 4.2 × 1010 17 × 1010

8 T 4 T 3.0 × 1010 12 × 1010

8 T 2 T 3.0 × 1010 12 × 1010

6 T 4 T 1.8 × 1010 7.2 × 1010

6 T 2 T 2.4 × 1010 9.6 × 1010

Target material Capture Transport Muon yield per Muon yield perfield field 1014 protons 4 × 1014 protons

Graphite 16 T 4 T 4.8 × 1010 19 × 1010

16 T 2 T 3.6 × 1010 14 × 1010

12 T 4 T 3.6 × 1010 14 × 1010

12 T 2 T 3.0 × 1010 12 × 1010

8 T 4 T 3.0 × 1010 12 × 1010

8 T 2 T 2.4 × 1010 9.6 × 1010

6 T 4 T 1.8 × 1010 7.2 × 1010

6 T 2 T 1.8 × 1010 7.2 × 1010

Tungsten 16 T 4 T 13 × 1010 50 × 1010

16 T 2 T 11 × 1010 46 × 1010

12 T 4 T 9.6 × 1010 38 × 1010

12 T 2 T 9.0 × 1010 36 × 1010

8 T 4 T 6.0 × 1010 24 × 1010

8 T 2 T 7.2 × 1010 29 × 1010

6 T 4 T 4.2 × 1010 17 × 1010

6 T 2 T 4.8 × 1010 19 × 1010

Chapter 8

PRISM-II

8.1 Overview

In addition to PRISM that is for experiments with stopped muons, there arises addi-tional significant demand to have a highly intense muon beam of around 0.5 GeV/c inmomentum. One of the main physics topics is a search for the electric dipole moment(EDM) of the muon with a sensitivity of less than 10−24 e·cm. The current limit ofthe muon EDM is (3.7 ± 3.4) × 10−19 e·cm.

The search needs muons of about 0.5 GeV/c and injects them into a dedicatedring (EDM ring). The PRISM-II would provide muons with high intensity and highpolarization after phase rotation. A separate Letter of Intent on the search for theelectric dipole moment of the muon is submitted. Details are available there.

The most important requirement is Nµ · P 2µ > 1016, where Nµ is a number of

the muons available and Pµ is the muon polarization. The other specification of thePRISM-II muon beam is as follows.

Table 8.1: Anticipated PRISM-II beam design characteristics

Parameters Design goal CommentsBeam Intensity 1010 − 1012µ±/sec 1014 protons/sec is assumed.Muon kinetic energy 597 MeV pµ = 500 MeV/cKinetic energy spread ±2 %Muon polarization Pµ > 0.6Beam Repetition 100 - 1000 Hz

8.2 PRISM-II Muon Beam Line

The search for the muon EDM at a sensitivity of 10−24e·cm requires a polarized muonbeam with high intensity, which has never existed so far. Such a highly intense muon

36

8.2. PRISM-II MUON BEAM LINE 37

beam can only be achieved by a novel scheme of PRISM.Since the momentum acceptance of the muon storage EDM ring would be only a

level of ∆p/p 2%, it is desirable to employ the phase rotation technique to improvethe energy spread of the polarized muon beam. The momentum spread of the muonbeam could be thus reduced by almost an order of magnitude by phase rotation,namely from ∆p/p = ±30% to ∆p/p ±2%. This would provide us a sufficientnumber of the polarized muons injecting into the muon storage EDM ring.

To accomplish the phase rotated muon beam for the EDM experiment, a new beamline has to be constructed. We call it PRISM-II. The major difference from PRISM(which is primarily for the µ− − e− conversion experiment) exists in the momentumrange of the muon beam; 68±20 MeV/c for PRISM and 500±150 MeV/c for PRISM-II. This difference results in the two major replacements; (1) construction of a newFFAG phase rotator for 500 MeV/c muons, (2) installation of torus solenoids to selectthe pion momentum to improve the muon polarization, (3) installation of a longerpion decay solenoid. Regarding the target and the pion capture solenoid, the samedesigns as those in PRISM can be used.

PRISM-II consists of, as similar to PRISM-I,

1. pion capture system

2. semi-adiabatic transfer from a high to a low magnetic field,

3. selection of the pion momentum,

4. pion decay and muon transportation system

5. selection of the muon momentum, and

6. phase rotation system.

8.2.1 Pion Capture at Forward Take-off Angle

Table 8.2 summarizes the current design parameters of the pion capture section andthe pion decay and muon transport section of PRISM-II. As discussed before, thecentral momentum of the polarized muons should be about 500 MeV/c. Thus, therequirement to the muons injected into the PRISM-II FFAG is to provide polarizedmuons with momentum range of 350 MeV/c < pµ < 650 MeV/c with NµP

2µ as large as

reasonably achievable, where Nµ is a number of the muons available to the experimentand Pµ is the muon polarization.

Figure 8.1 shows the Monte-Carlo simulated result on the dependence of the (lon-gitudinal) muon polarization as a function of initial pion momentum, where the pionin corresponding to the muons of momentum between 350 MeV/c < pµ < 650 MeV/care shown. It is seen that the pions, which produce the muons in this momentumrange, can distribute from 350 MeV/c to 1.1 GeV/c. In order to avoid pion contam-ination in the polarized muon beam, it is better to select the muons whose momentado not overlap with the initial pion momentum. Thereby, backward decay muons

38 CHAPTER 8. PRISM-II

Table 8.2: The design parameters of the pion capture and decay beam line componentof the muon EDM experiment.

Production TargetMaterial GraphiteLength 80 cmDiameter 2.0 cm

Superconducting Pion Capture SolenoidMagnetic Field Strength 6 TLength 120 cmInner Bore Radius 10 cm (inside the radiation shield)Transverse Momentum Acceptance < 100 MeV/cMaterial of the radiation shield TungstenThickness of the radiation shield 30 cmMaximum heat load to the cold coil less than 500 W

Superconducting Matching SolenoidMagnetic Field Strength 6 T – 1 TLength 450 cmInner Bore Radius 10 cm – 45 cmTransverse Momentum Reduction Factor 40%

Superconducting Torus SolenoidMagnetic Field Strength 1 T (at the center of bore)Compensation Field Strength 0.60 TMajor Radius 5 mInner Bore Radius 45 cmAngle of Arc 50

Transmission Efficiency 90% for 0.7 – 1.1 GeV/cSharpness of the Cut-off 80 MeV/c (10–90%)

Superconducting Decay SolenoidMagnetic Field Strength 1 TInner Bore Radius 45 cmLength 20 mFraction of Pion Decays 25%

are employed. In addition, Fig.8.1 shows that the backward decay muons provide arather higher polarization than that from the forward decay muons. By selecting thepions with pπ > 0.7 GeV/c, NµP 2

µ can be maximized.

A field strength of a superconducting solenoid magnets in the present technologywould be as high as 12T or more. However, if a potential heat load to the magnetby high neutron flux coming from the production target is considered, it would beconservative reasonable to start considering a lower field strength such as of 6 T. Thus,a 6T-superconducting solenoid magnet for the pion collection magnet is assumed in

8.2. PRISM-II MUON BEAM LINE 39

Initial π momentum (GeV/c)

Muo

n L

ongi

tudi

nal P

olar

izat

ion

Figure 8.1: Muon longitudinal polarization versus initial pion momentum. Muonsare those selected with horizontal and vertical acceptances 800π mm·mrad each andmomentum acceptance 350 MeV/c < pµ < 650 MeV/c.

the present design. Figure 8.2 shows pT distribution of the initial pions with thecondition of beam emittance of their daughter muons being 800π mm·mrad for bothhorizontal and vertical axises and 350 MeV/c < pµ < 650 MeV/c in momentum.From Fig.8.2, the muon yield may not drop significantly with the cut of pT < 100MeV/c to the initial pions. Therefore, the inner bore of the solenoid magnet couldbe 10cm at 6 T. It is the same design of the entry level of PRISM.

8.2.2 Pion Decay and Muon Transport Solenoid

The pions collected by the capture solenoid are transferred to the pion-decay andmuon-transport solenoid, in which the pions decay in flight into muons. Since themomentum range of the initial pions are around 1 GeV/c, and its τβγc is almost80m length, the decay solenoid should be desirably over 80 m. However it would bedifficult to accommodate a 80-m solenoid magnet in the experimental hall. Therefore,a solenoid magnet of 20 m long is considered in the present design. It ends up withabout 25 % pions decaying into muons. If it is possible to make a longer transportsolenoid is placed, the muon yield would be greater.

If the high magnetic fields at the pion capture section and the low magnetic fieldof the pion decay section are connected without any magnetic field leakage and thefield gradient being sufficiently small, the charged particles traversing through thechannel will take place a so called “adiabatic transfer”. In adiabatic transfer, B/p2

T

and BR2 are preserved, where B is the magnetic field strength, pT is the transverse


Transverse Momentum of Pion (GeV/c)

Cou

nts

per

2 M

eV/c

Figure 8.2: Transverse momentum of pions from a production target with a conditionof those daughter muons being selected with horizontal and vertical acceptance of themuon storage ring (800π mm·mrad) and momentum acceptance 350 MeV/c < pµ <650 MeV/c at the exit of the decay solenoid.

momentum and R is the radius of the helix of the charged particle. Thus, RpT isindependent of the field strength in the adiabatic transfer. On the other hand, themuon yield into a fixed acceptance at the exit of the decay solenoid takes linearscale to R2p2

T of the beam. As a result, the muon yield is almost independent of themagnetic field strength of the decay solenoid as long as the adiabatic transfer beingmaintained. This increases the freedom of design and the field strength of the decaysolenoid could be much less than that of the capture solenoid. We chose 1T as thefield strength of the decay solenoid for the current design.

However, the connection between the capture solenoid and the decay solenoid isactually not the perfect adiabatic transfer but rather imperfect way (semi-adiabatic)for 1 GeV/c pions since the pitch of helix of the pion trajectory becomes a levelof several meters or more for this momentum region. If the length of the matchingsection was about 20 m or more, the adiabatic transfer would be restored for 1 GeV/cpions at a cost of expensive price of the solenoid magnet system. In the current design,we performed Monte Carlo simulations and estimated the phase space of pions afterthe matching section, and found that the length of 4.5m of the matching solenoidwould result in similar pion distributions as the adiabatic transfer. Although it isnot the adiabatic transfer, but only the choice of the length of the matching solenoidhappens to match the pitch of helix for 1 GeV/c pions.

The expected size of pion beam after 4.5 m of the matching section would be

RD = 10 cm×√

6 T/1 T 25 cm. Muons from the pion decay would have influenced

8.3. PRISM-II FFAG 41

by the transverse momentum kick (maximum 30 MeV/c), and which would result inthe further blow-up of the beam size with 20 cm maximum. The radius of the decaysolenoid is chosen to be about 45 cm so that the muons from pion decay would besafely contained without hitting the wall of inner bore of the decay solenoid.

8.2.3 Selection of Pion Momentum to Improve Muon Polar-

ization

The pions momentum selection could be achieved by using a curved solenoid. It wasalso adopted in the MECO (BNL-E940) experiment. In the curved solenoid, namelytorus, a charged particle follows helical trajectories with a drift towards the directionperpendicular to the plane of the curved solenoid. The distance of the drift (D[m])after traversing a length s[m] in the torus is expressed as

D =1

0.3B× s

R× (p2

L +1

2p2

T )/pL ,

where R[m] is a curvature of the torus, and pT and pL[GeV/c] are the perpendicularand parallel momentum components, respectively. The amount of drift can be furthercontrolled by applying an additional magnetic field (called a compensation field) alongperpendicular to the plane of the curve; the net drift of the particle with the particularpT and pL could be thus adjusted to zero.

Figure 8.3 shows a momentum spectrum of the charged particles at the exit of50-curved solenoid, when particles are generated equally for 0 – 2.0 GeV/c at theentrance of the solenoid. The torus field is 1T, and the compensation magnetic fieldis 0.60 T, the inner bore radius of the torus is 45 cm, and the radius of curvatureis 5 m for the present calculation. It is clearly seen that the momentum selection ofthe pion being more than 0.7 GeV/c can be easily achieved with almost 90% of thetransmission efficiency. The momentum width of the transmitted particles and theposition of the transmission window can be easily adjusted by changing the length ofthe arc and the compensation field, respectively.

It is worth to mention here that the current design is not a unique solution to themuon beam line for the muon EDM experiment. The pion capture, semi-adiabatictransition, and curved solenoid could be replaced with hone magnet system. Furtheroptimization studies will be done.

8.3 PRISM-II FFAG

A preliminary design of the PRISM-II FFAG ring has been made. The PRISM-IIFFAG accelerator can be either a normal conducting magnet FFAG or a supercon-ducting one. The main difference is a bending strength and an average radius; 1.8 Tand 21 m, respectively, for the normal conducting, and 2.8 T and 10m for the super-conducting, respectively. Fig.8.4 and 8.5 show a footprint of the normal conducting


Pion Momentum (GeV/c)

Tra

nsm

issi

on E

ffic

ieny

Figure 8.3: The transmission probability of charged particles as a function of momen-tum for the curved solenoid, where the field strength at the center of solenoid being 1T, major radius of the curved solenoid being 5 m, minor radius of the curved solenoidbeing 45 cm, and the degree of curve being 50.

version and its lattice functions in one sector and those for superconducting one, re-spectively. Table 8.3 summarizes main parameters of the PRISM-II FFAG ring. Thenormal conducting one has to have more cell, twice as much, although the lengthof each cell is about the same for two versions, that is 4m, resulting in the similarmaximum value of beta functions. The dispersion function of the superconductingversion is 50% more. The superconducting one can be accommodated in the initialproposed area of the experimental hall.

It should be noted that the design of the PRISM-II FFAG ring is the same as thefirst FFAG ring of the Japanese Neutrino Factory. The first ring would acceleratemuons from 0.3 GeV/c to 1.0 GeV/c. The PRISM-II FFAG ring would servean important role towards the realization of a neutrino factory in Japan.

8.3.1 Estimation of Muon Yield and Muon Polarization

Monte Carlo simulations by using GEANT 3.21+FLUKA were performed to esti-mate the muon yield and muon polarization. The Monte Carlo simulation was madefrom pion production down to the exit of the muon transport solenoid. The originalGEANT3.21 code was modified so that it can handle transportation of the muonpolarization in a magnetic field.

RF acceleration of muons in the PRISM-II FFAG phase rotator would be a level of1 MeV/m. Thus, muons of 750 MeV/c should travel through the ring for almost 150


Figure 8.4: Normal conducting version of the PRISM-II FFAG ring.


Figure 8.5: Superconducting version of the PRISM-II FFAG ring.


Table 8.3: Main parameters of PRISM-II FFAG.

Parameters normal conducting super-conductingnumber of sector 32 16k value 50 15transition gamma 7.1 4orbit excursion 0.50 m 0.77 maverage radius 21 m 10 mB@F/D 1.8 T 2.8 TF/2 angle 0.026 rad 0.052 radD angle 0.018 rad 0.036 radF/2 bend angle 17 degree 26 degreepacking f 0.45 0.46phase advance(H/V) 120/61 deg. 131/103 deg.drift length 2.060 m 2.120 mBF length 1.104 m 1.065 mBD length 0.382 m 0.367 m

m to get decelerated down to 500 MeV/c. The length of 150 m corresponds to only2.5 turns in the superconducting FFAG phase rotator (and even if a packing factorof RF cavities it is not long). Muon depolarization due to the g − 2 precession wouldbe thus negligible. The decay loss of the muons in the FFAG phase rotator can beonly several percents. A tracking study in the PRISM-II FFAG ring with computeraided simulation is being undertaken.

Table 8.4 summarizes the expected muon yield and polarization. The estimatedNµP 2

µ is about 5 × 1016, and which is sufficient for our goal.

Table 8.4: The expected muon yield and NP 2 for the PRISM-II

Momentum Acceptance of the downstream FFAG 350 – 650 MeV/cHorizontal and Vertical Acceptance of the PRISM-II FFAG 800π mm·mradMuon Yield per 50 GeV/c proton 0.04%Muon Polarization (Longitudinal) 60%Expected NµP 2

µ per 1-year run at J-PARC 5 × 1016

Chapter 9

A Pulsed Proton Beam Facility forPRISM

9.1 Overview

As described in Chapters 3 and 6, an initial proton beam has to be pulsed withvery narrow width to carry out phase rotation of muons. The present nuclear andparticle experimental hall (NP hall) at the 50-GeV PS is planned to have only acontinuous proton beam from slow extraction, and does not have any capability havinga pulsed proton beam. The neutrino beam line which uses a fast-extracted protonbeam can not accommodate a room for other experiments. Therefore, we would liketo strongly request the construction of a dedicated pulsed proton beamfacility. A possible location will be north from the 50-GeV PS tunnel near the pacificocean. A preliminary layout of the pulsed proton beam facility is presented in Fig.9.1.A letter of intent on the pulsed proton beam facility is separately submitted. Here, abrief overview is presented.

9.2 Proton Beam Line

Proton bunches in the 50-GeV ring are kicked off outward with respect to the 50-GeVring toward the pulsed proton beam facility, in a single-kicking mode. It is noted thatprotons to the neutrino beam line are kicked inward. The presently-designed kickermagnet does already have a both-side kick since there is an abort beam line outsidethe 50-GeV tunnel. Our proton beam line is split from the proton abort line, and isextended to the experimental hall. Fig.9.2 shows a possible layout of such a beamline. When 8 (or 18) bunch running mode is used for PRISM, the same kicker magnetcan be used. When 90 bunches is available, a new fast kicker magnet, which will beinstalled near by, is used to kick 90 bunches into the proton beam line.

46

9.3. EXPERIMENTAL AREA LAYOUT 47

Far facility

Pulsed proton beamline

2nd pbar ring/experimental hall

g-2 ring

Fast extractionKicker

Near facility

Production TargetProduction TargetProduction TargetBeam DumpBeam DumpBeam Dump

PRISM FFAGPRISM FFAGPRISM FFAG

PRISM-II FFAGPRISM-II FFAGPRISM-II FFAG

EDM RingEDM RingEDM Ring

PRIME DetectorPRIME DetectorPRIME Detector

Figure 9.1: A possible layout of the pulsed proton beam facility, which consists of thenear facility and the far facility.

Neutrino beamline

50 GeV Ring

Bunched proton beamline

50 GeV abort dump

Fast extractionKicker

Figure 9.2: A possible layout of the pulsed (bunched) proton beam line.

9.3 Experimental Area Layout

Fig.9.1 shows a possible layout of the experimental facility, which consists of thenear facility (closer to the 50-GeV PS) and the far facility (beyond the public road).

48 CHAPTER 9. A PULSED PROTON BEAM FACILITY FOR PRISM

PRISM and PRISM-II will be installed in the near facility, whereas the muon g − 2ring and the anti-proton facility are located at the far facility.

A layout of the near facility is shown in Fig.9.3. The area is composed of twoareas. One is for PRISM (of low momentum muons with backward extraction) andPRISM-II (of muons of 500 MeV/c with forward extraction). They may be place atdifferent heights so that the injection and extraction beam lines do not interfere. Theother is the experimental area, in which detectors are installed. The target stationsmay be different between PRISM and PRISM-II. Thus, experiments using a beamfrom either PRISM or PRISM-II will be carried out in a time-sharing way1 Also, afull proton beam dump is needed.

Production TargetProduction TargetBeam DumpBeam Dump

PRISM FFAGPRISM FFAG

PRISM-II FFAGPRISM-II FFAG

EDM RingEDM Ring

PRIME DetectorPRIME Detector

Figure 9.3: A possible layout of the near facility where PRISM and PRISM-II areinstalled.

9.4 Request at Phase-I

The construction of the pulsed proton beam facility is not included in the Phase-Iplan of J-PARC according to the J-PARC schedule. However, it is important to keep

1Primarily a search for µ− − e− conversion process is planned at PRISM and a search for themuon electric dipole moment is planned at PRISM-II.

9.4. REQUEST AT PHASE-I 49

its possibility at the phase-I construction. One concern is activation of soils aroundthe area of the fast extraction port from the 50-GeV protons in the ring. If soilsat that area are activated, future excavation for the tunnel of the proton beam linewould become severely difficult. To avoid activation, we like to place some concreteshielding blocks at the location of proton extraction. Possible arrangement which weare considering is shown in Fig.9.4.

The thickness of the shielding blocks required was calculated based on the Moyerformula which assumes a linear radiation loss. From the calculation, with a concreteshielding of 2.2 meter thick in addition to the 0.8 meter thick of the 50-GeV tunnelwall, soil activation would be about 0.07 msv/h and 0.02 Bq/g. It is much less thanthe acceptable level of 0.3 msv/h and 0.1 Bq/g. From this, it is shown that theconcrete shielding blocks of 2.2 meter thickness would be sufficient.

50 CHAPTER 9. A PULSED PROTON BEAM FACILITY FOR PRISM

Shield Block

A-A Cross section

Shield Block

To Facility

Figure 9.4: Possible arrangement to avoid soil activation around the location of protonextraction.

Chapter 10

Conclusion

We, the PRISM working group, would like to express our interest to construct themuon beam facility of the highest beam brightness in the world, based on a noveltechnique of phase rotation (PRISM). PRISM provides a muon beam of its highintensity of about 1011 − 1012µ±/sec with narrow momentum width of a few % andalmost no pion contamination. The aimed muon intensity is about four orders ofmagnitude better than the highest muon beam intensity presently available. ThePRISM beam is intended to be used in general stopped-muon experiments. As anexample, for particle physics, we like to carry out an experimental search for thelepton flavor violating process of µ− + (Z, A) → e− + (Z, A) (µ− − e− conversion) byusing a high intensity muon beam from PRISM.

PRISM-II is another muon source with its momentum of around 500 MeV/c andhigher muon polarization. It is mostly designed for the muon electric dipole moment(EDM) measurement.

PRISM, PRISM-II and its experimental apparatus could be placed at the proposedpulsed proton beam facility (of fast proton beam extraction). We would like torequest strongly the construction of the pulsed proton beam facility. Sinceit is not included in the J-PARC Phase I budget, its construction will be in a later time.To be competitive to the other programs in the world, the pulsed proton beam facilitytogether with PRISM and PRISM-II should be constructed at the earliest possibletime. At the Phase I construction, we would like to ask a special arrangement to avoidsoil activation at the location of proton extraction port to the pulsed proton beamfacility. Once soil is activated, future excavation would become almost impossible.To be more specific, we would like to request placing concrete shielding blocksat the location of future extraction.

Finally, the physics programs considered with PRISM and PRISM-II would havea large discovery potential of new physics and phenomena. They are the most com-petitive, and the opportunity should not be missed for future success.

51

Bibliography

[1] Huhtinen M and Mokhov N V, FERMILAB-FN-697 FERMILAB preprint 2000“A Cross-Comparison of MARS and FLUKA simulation codes”

[2] Brun R, Caillat M et al., CERN-DD/85/1 CERN preprint 1985 “The GEANT3Electromagnetic Shower Program and a Comparison with the EGS3 code”

[3] the Massachusetts Institute of Technology Plasma Science and Fusion Center“MECO Superconductiong Solenoid System Conceptual Design Report”, 2002

[4] Fujieda M et al. 1998 in Proceedings of the European Particle Accelerator Con-ference (EPAC’98), Stockholm, p.1796

52

Appendix A

90-Bunche Operation of the50-GeV PS

As mentioned before, PRISM would require a sharp pulse width to achieve a betterperformance of phase rotation. And the search for the µ−−e− conversion process withPRISM would require many beam bunches with relatively low beam instantaneousintensity. To accomplish this, a new novel idea of 90 bunches operation has come out.By handling 90 bunches in the ring, the above both requirements would be met. Thedetail parameters are shown in Table A.1.

Table A.1: Main parameters of the 90 bunches operation.

Parameters valuesRevolution time 5.23 µsecRevolution frequency 0.191 MHzRF frequency of 90 bunches 19.1 MHza time period between bunches (central position) 52 nsec

It is worth to mention that muon g−2 experiment also needs 90 bunches operationas might be discussed in the independent letter-of-intent.

A.1 Procedure

To realize 90 re-bunching, the proposed scheme is the following.

1. After the proton acceleration, a beam will be de-bunched by shutting RF off.It is the same as in slow beam-extraction.

2. A beam will be re-bunched by an RF of 90 bunches.

53

54 APPENDIX A. 90-BUNCHE OPERATION OF THE 50-GEV PS

3. A bunch will be kicked out one by one (single-kicking mode). Since a bunchseparation is about less than 52 nsec, a fast kicker magnet is also needed. Arepetition frequency of the single bunch depends on the experimental require-ments and technical feasibility. In paricular, for the search for µ−−e− conversionprocess, 100 Hz or 1000 Hz are being considered.

An expected full pulse width (±3 sigma) in the 90 bunches sheme is 3.6 nsec (or 0.6nsec sigma of the width) if a higher RF voltage is used to keep the bucket height thesame as in the 10 harmonics operation. The main parameters of the RF cavities issummarized in Table A.2. A total of 5 RF cavities will be used to produce a total ofaccelerating field of 25000 kV and a total power of 1.25 MW.

Table A.2: The main parameters of the 90-bunches RF cavity (one unit).

Parameters 90-bunches RF Nufact-J RFFrequency 26 MHz 19.1 MHzAverage accelerating field 0.5 MV/m 1.0 MV/mCavity Voltage 0.5 MV 1.5 MVShunt Impedance about 1 M ohm about 1.5 M ohmPower 250 kW 500 kW

A.2 Issues

There are several issues to be considered. They are listed as follows.

• Estimation of a time required for de-bunching and re-bunching: It is guessed tobe from 0.1 second to 1.0 second,

• Estimation of a possible beam loss in the process of beam de-bunching and re-bunching: There is a possibility to change a lattice in such a way that themomentum compaction factor from −0.001 to −0.002 so that longitudinal mo-tion is accelerated.

• Estimation of a pulse width after re-bunching,

• Estimation of feasibility on an RF of 90 bunches, and

• Estimation of feasibility on a kicker magnet which has 40 nsec of both rise andfall time.

A.3. BUNCHED BEAM EXTRACTION 55

A.3 Bunched Beam Extraction

In the 90 bunching scheme, the bunch spacing is only about 50 nsec thus a kickermagnet should have both rise and fall time much less than 50 nsec. In order to seethe brief idea of the difficulty of the development of such a kicker magnet, we couldrefer to the kicker magnet design at TRIUMF KAON project. Table A.3 shows themajor parameters of the C-ring injection kicker magnet at TRIUMF KAON project.Flat top and magnet length could be much small in PRISM kicker magnet, and which

Table A.3: The major parameters of the C-ring injection kicker magnet of the TRI-UMF KAON project.

Parameters C-ring inj. our golax aperture 76 mmy aperture 68 mm 100 mmfrequency 50 Hz 100 - 200 Hzflat top 659 ns 25 nsrise and fall time 82 ns 40 nsV PFN per module 44 kVNo. of module 3V PFN 131 kVpeak current 874 Aimpedance 25Ωlength 3.535 m 1 m

would conpensate the difficulty of improving the rise and fall time by factor two.

Appendix B

Simulation on Pion Productionand Capture by using MARS

B.1 Pion Production by 50-GeV Proton Incident

The pion production is evaluated by the MARS simulation code developed at Fermilab[1], and also by the FLUKA hadron code in GEANT 3.21. Figs.B.1 and B.2 show themomentum spectrum of pions produced by 50-GeV protons, calculated by MARS forgraphite and tungsten targets respectively. In this simulation, the graphite target was40 mm in radius and 3 interaction lengths, whereas the tungsten target was 5 mm inradius and 3 interaction lengths. A negative value of the longitudinal momentum pL

implies the backward-direction. A number of the produced particles were recordedat the surface of the target. The momentum spectrum of negative particles is almostthe same as that of positive one for a 50-GeV proton beam.

Pions at low energy are of interest for PRISM, whereas those at high energycould become some backgrounds. From Fig.B.1 and B.2, it is clearly seen that lesspions in high energy are emitted backward. Therefore, the backward extraction hasbeen adopted for PRISM. Fig.B.3 shows the π− yields (of their momenta less than200 MeV/c) from backward extraction as a function of target length and radius fortungsten and graphite targets. It should be noted that this simulation was doneunder no magnetic field. Figs.B.4 and B.5 show the spectra of π− produced from agraphite and tungsten target respectively. From these, it is seen that the maximumof transverse momentum (pT ) is around 100 MeV/c for 0 < pL < 200 MeV/c for bothforward and backward pions. The maximum of the total momentum distributionfor the backward pions is about 120 MeV/c, whereas that of forward pions is about200-400 MeV/c.

B.2 Solenoid Pion Capture

To collect as many pions (and cloud muons) of low energy as possible, they arecaptured under a high solenoidal magnetic field with a large solid angle. In this case,

56

B.2. SOLENOID PION CAPTURE 57

10-4

10-3

10-2

10-1

1

10

10 2

10 3

10 4

10 5

0 5 10 15 20 25 30 35 40 45 50

p l (GeV/c)

even

t/0.

2GeV

/100

k pr

oton

10

10 2

10 3

10 4

-0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

p l (GeV/c)

even

t/25

MeV

/100

k pr

oton

Figure B.1: Longitudinal momentum spectra of π± and µ± from 50-GeV protonsfrom a graphite target. Top plot is all pions (and muons), and the bottom plot is anextended one for their longitudinal momentum of pL < 2 GeV/c.

10-3

10-2

10-1

1

10

10 2

10 3

10 4

10 5

0 5 10 15 20 25 30 35 40 45 50

p l (GeV/c)

even

t/0.

2GeV

/100

k pr

oton

1

10

10 2

10 3

10 4

-0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

p l (GeV/c)

even

t/25

MeV

/100

k pr

oton

Figure B.2: Longitudinal momentum spectra of π± and µ± from 50-GeV protons froma tungsten target. Top plot is all pions/muons, and the bottom plot is an extendedone for their longitudinal momentum of pL < 2 GeV/c.

58APPENDIX B. SIMULATION ON PION PRODUCTION AND CAPTURE BY USING MARS

Figure B.3: Backward π− yields as a function of target length and radius for tungstenand graphite targets in the case of no magnetic field. Here, the pions with theirmomentum less than 200 MeV/c (p < 0.2 GeV/c) are plotted.

pions emitting in a half hemisphere can be captured within the transverse momentumthreshold (pmax

T ).The maximum transverse momentum of pions captured (pmax

T ) is given by a mag-netic field strength (B) and the radius of the inner bore of solenoid magnet (R). Itis given by

pmaxT (GeV/c) = 0.3 × B(T) × R(m)

2. (B.1)

The simulation of pion capture under high solenoidal magnetic field has been done.The simulation program was based on MARS code. The kinetic energy of incidentproton was 50 GeV with its point beam size. Various sizes of the targets have been

B.3. RADIATION HEATING TO SUPERCONDUCTING MAGNETS 59

studied for tungsten and graphite uder a magnetic field of 6 T, 12 T ant 16 T. Thenumber of backward extracted π−s, which have momentum of less than 200 MeV/c,were counted. The results is shown in Fig.B.6.

In comparison with the pion yield under no magnetic field shown in Fig.B.3, theπ− yield under a magnetic field is smaller for all the cases. This would be an effect ofre-hitting to the target owing to their helical motion under a magnetic field. As theradius of the target is larger, this reduction is larger. It is studied that an optimizedtarget radius of tungsten and graphite targets are 0.5 mm. Regarding a target length,for more than 2 interaction length the pion yield are almost saturated. And the pionyield for tungsten is larger by about three times than those of graphite.

The dependence of the π− emittance on a magnetic field has been studied. Fig-ure B.7 shows the pion radial beam size under a different magnetic field. As themagnetic field is higher, the π− beam size becomes better. The pion size would affectthe final muon yiled at PRISM.

B.3 Radiation Heating to Superconducting Mag-

nets

A heat load to the superconducting solenoid magnet for pion capture must be reduceddown to a level of 100 W in total from the practical point of view. However, a heat loadto materials around the pion production target exceeds 100 kW for the full intensity(namely, 2×1014 protons/3-seconds spill) of the J-PARC 50-GeV proton synchrotron.Therefore, to reduce a heat load, a thick radiation shield must be employed betweenthe pion production target and the superconducting solenoid magnet. We have doneMonte Carlo simulation to study a shield material and thickness. In this Monte Carlosimulation, the MARS13 code was used to generate hadrons including neutrons.

Fig.B.8 shows typical energy deposit distributions in the radiation shield made oftungsten, as a function of radial position for various region of z coordinates (along thebeam axis). Fig.B.9 shows those for the shield made of the combination of tungstenand paraffin, where the first 14 cm is tungsten followed by 20 cm paraffin.


p t (G

eV/c

)

pl (GeV/c)


p (GeV/c)

Figure B.4: Pion production in a graphite target. (top) correlation between pL andpT , (middle) Total momentum distributions for forward and backward π−s, (bottom)pT distributions for 0 < pL < 0.2 GeV/c, 0.2 < pL < 0.4 GeV/c, and 0.4 < pL < 0.6GeV/c.

p t (G

eV/c

)

pl (GeV/c)


p (GeV/c)

Figure B.5: Pion production in a tungsten target: (top) correlation between pL andpT , (middle) Total momentum distributions for forward and backward π−s, (bottom)pT distributions for 0 < pL < 0.2 GeV/c, 0.2 < pL < 0.4 GeV/c, and 0.4 < pL < 0.6GeV/c.


Figure B.6: Negative pion yields as a function of target length and radius for tungstenand graphite targets. Here, the pions with their momentum less than 200 MeV/c(p < 0.2 GeV/c) are used.


Figure B.7: Radial profile of negative pions with momentum less than 200 MeV/c,extracted from backward. Here, the target length is 3 interaction lengths and itsradius is 0.5 cm.


1

10

10 2

10 3

10 4

0 10 20 30

z :-20cm ~ -10cmz :-10cm ~ 0cmz : 0cm ~ 10cmz : 10cm ~ 20cmz : 20cm ~ 30cmz : 30cm ~ 40cm

Shield depth(cm)

Ene

rgy

depo

sit(

W)

1

10

10 2

0 5 10 15 20 25

Magnet depth(cm)

Ene

rgy

depo

sit(

W)

1020

3040

500

2040

60

10 2

10 3

10 4

Shield dep

th(cm)

Shield Postionz(cm)

Ene

rgy

depo

sit

(W)

510

1520

250

2040

60

10

Magnet dep

th(cm)

Magnet Positionz(cm)

Ene

rgy

depo

sit

(W)

Figure B.8: Energy deposit distribution in the shield and magnet. The shield consistsof a total of 34 cm tungsten. The superconducting coil is assumed to be made ofcopper as a stabilizer.

1

10

10 2

10 3

10 4

0 10 20 30

z :-20cm ~ -10cmz :-10cm ~ 0cmz : 0cm ~ 10cmz : 10cm ~ 20cmz : 20cm ~ 30cmz : 30cm ~ 40cm

Shield depth(cm)

Ene

rgy

depo

sit(

W)

1

10

10 2

0 5 10 15 20 25

Magnet depth(cm)

Ene

rgy

depo

sit(

W)

1020

3040

500

2040

60

10

10 2

10 3

10 4

Shield dep

th(cm)

Shield Postionz(cm)

Ene

rgy

depo

sit

(W)

510

1520

250

2040

60

10

10 2

Magnet dep

th(cm)

Magnet Positionz(cm)

Ene

rgy

depo

sit

(W)

Figure B.9: Energy deposit distribution in the shield and magnet. The shield consistsof 14-cm tungsten in front and 20-cm paraffin in rear. The superconducting coil isassumed to be made of copper as a stabilizer.

Appendix C

Extraction from the Pion DecaySection

At the exit of the pion decay and muon transport section consisting of a long solenoidmagnet, a fringing field might blow up the muon distribution. To study this effect, afringing magnetic field was calculated by the POISSON/SUPERFISH program. ThePOISSON/SUPERFISH code was developed at the Los Alamos Laboratory in US.In this calculation, a long solenoid magnet with an inner bore of 15 cm and an outerbore of 20 cm in radius was used. The central magnetic field of the solenoid magnetis 4 T. The fringing field distribution thus calculated is shown in Fig.C.1.

Using the distribution of fringing magnetic field thus calculated, the muon distri-butions at different locations at the exit of the transport solenoid magnet (such as(a) inside the transport solenoid, (b) at the edge (z = 0) of the solenoid, (c) z = 50cm from the edge of the solenoid, and (d) z = 100 cm from the edge of the solenoid)are shown in Figs.C.2 and C.3. Dark points represent the muons in the emittance en-velops of 10,000 πmm·mrad in horizontal and 2,000 πmm·mrad in vertical and 20,000πmm·mrad in horizontal and 3,000 πmm·mrad in vertical, for Figs.C.2 and C.3, re-spectively. From this, it is concluded that although most of the muons will spread

C

YC

LE =

850

Figure C.1: Fringing magnetic field flux lines at the exit of a solenoid magnet, calcu-lated by the POISSON program. (left) a magnetic field Bz on the axis as a functionof position along the axis, and (right) a flux distribution at the exit with an iron yoke.

64

65

out at the flinging field region, the core parts in the good emittance, which can beaccepted by the PRISM-FFAG ring, would not be broadened in their distribution atwithin 100 cm, where next focusing elements should be located.

Figure C.2: Phase space distributions (left:horizontal and right:vertical) of the muonsat the exit of the transport solenoid at various locations, (a) in the transport solenoid,(b) at the edge of the solenoid (z = 0), (c) z = 50 cm from the edge of the solenoid,and (d) z = 100 cm from the edge of the solenoid. Dark points represent the muonsin the emittance of 10,000 πmm·mrad in horizontal and 2,000 πmm·mrad in vertical,respectively.

66 APPENDIX C. EXTRACTION FROM THE PION DECAY SECTION

Figure C.3: Phase space distributions (left:horizontal and right:vertical) of the muonsat the exit of the transport solenoid at various locations, (a) in the transport solenoid,(b) at the edge of the solenoid (z = 0), (c) z = 50 cm from the edge of the solenoid,and (d) z = 100 cm from the edge of the solenoid. Dark points represent the muonsin the emittance of 20,000 πmm·mrad in horizontal and 3,000 πmm·mrad in vertical,respectively.

Appendix D

Introduction to the FFAG

D.1 The Principle of a FFAG Accelerator

A Fixed-Field Alternating Gradient (FFAG) synchrotron, as its name suggests, con-sists of DC operation of magnets. It is quite different from an ordinary alternatinggradient (AG) synchrotron (hereafter simply referred just as a synchrotron ), in whichmagnets are AC operated in accordance of a beam momentum in such a way that theaverage radius (and therefore the path length) is constant. As a consequence, whena particle of light rest-mass such as an electron is accelerated in a synchrotron, theparticle reaches the speed of light quickly so that the revolution frequency becomesconstant. However, a speed of a heavy particle such as a muon and a proton is slowlyincreased. Therefore, the revolution frequency varies with its momentum, althoughthe path length is constant.

In a FFAG, on the contrary, the average radius increases (or decreases in somepeculiar field configurations) , when a particle is accelerated. It is just like a cyclotron.However, unlike a cyclotron, particles of different momentum have different revolutionfrequencies. In other words, isochronous condition is violated on purpose. Thatmakes longitudinal focusing and synchrotron oscillations. Of course, the RF frequencyof an accelerating cavity has to follow the speed of a particle. But unlike protonsynchrotrons, in which the RF frequency is determined with the excitation of bendingmagnets, the RF frequency pattern and therefore the momentum pattern in time ismade solely by itself free from other references. Table D.1 shows the function ofmagnets and RF in various accelerators.

Table D.1: Operation of magnet and RF in different accelerators

constant frequency variable frequencyconstant bending field cyclotron FFAG synchrotronvariable bending field electron synchrotron proton synchrotron

67

68 APPENDIX D. INTRODUCTION TO THE FFAG

D.2 Zero Chromaticity

In transverse directions, it has alternating gradient (or strong) focusing. Free fromthe isochronous condition that is required in a cyclotron, a field index of each gradientmagnet can be large. It is determined only by the transverse stability of betatronoscillations. Careful optimization of transverse tunes to avoid strong resonances arepossible. The strong focusing is valid for a wide range, for example a factor of 3 asa ratio of the highest momentum to the lowest one. In principle, the transverse tuneis constant for that whole range. That is called, in terms of accelerator terminology,zero chromaticity. A type of FFAG with zero chromaticity satisfied is called ”ascaled machine”. Zero chromaticity is desirable to accelerate a beam from injectionto extraction with constant field magnets. It is also ideal to accumulate a beam ofwide momentum spread.

The first principle of FFAG with zero chromaticity can be described with thefollowing two conditions.

∂

∂p(K

K0) |ϑ=const. = 0, (D.1)

where K0 is an average curvature and K is a local curvature. θ is generalized azimuth.That imposes geometrical similarity. In addition

<∂n

∂p|ϑ=const. = 0, (D.2)

where n is a field index of magnet. The constancy of n at corresponding orbit pointsis assured. Those conditions are satisfied by the magnet which has the function givenby,

B = B0(r

r0

)k (D.3)

In fact, there is a FFAG with non zero chromaticity or a non scaled one, inwhich transverse tunes varies as a particle gets accelerated. That occurs because themagnetic field distribution in a real magnet is not perfect as described. In order tosimplify magnet design, sometimes non scaled machine is introduced purposely.

Those non scaling machines may be adequate for muon applications since there isonly a few turns in each acceleration process.

D.3 Large Acceptance

In a synchrotron, acceptable momentum spread is limited by two factors. One is abucket height in longitudinal phase space determined by RF voltage. Above that,a particle is not bounded and no synchrotron oscillations is expected. The otheris momentum acceptance. A higher momentum particle has larger bending radiusso that it circulates (usually) outer orbit. Dispersion function is the measure of

D.4. RADIAL AND SPIRAL TYPE 69

displacement due to momentum spread. Since the vacuum pine has a limited radiusin transverse plane, a higher or lower particle hits the wall of pipe even though abetatron amplitude is zero. The momentum acceptance is a few percent in terms ofdp/p in an ordinary synchrotron.

FFAG, on the other hand, has a potential of acceleration of a beam with muchhigher momentum spread. As we have mentioned already, the magnets are DC oper-ated and it covers focusing from injection to extraction. The momentum ratio througha cycle is 3 for our design. In other words, the momentum acceptance is +-50% if wetake the central momentum of a beam matched to the mean value of injection andextraction momentum. As for the bucket height, higher voltage makes higher bucketheight. The bucket height is proportional to the square root of the voltage.

D.4 Radial and Spiral Type

Depending on the magnet profile in θ direction, there are mainly two kinds of FFAG;one is radial sector type and the other is spiral sector one. A radial sector type hasa magnet whose entrance and exit faces are on radial line drawn from the machinecenter. There are normal and reverse bending magnets with nonlinear gradient whichprovide focus and defocus property in each transverse plane. Focusing and defocusingactions come also from an angle between entrance and exit orbit and the magnet face.

A spiral sector type has no reversed bending magnets. The alternating gradientfocusing action comes mainly from the edge angle. That edge focusing acts in alter-native way at the entrance and at the exit in addition to the main body field whichhas the same field variation in a radial direction as that of radial sector type. Highermomentum particle needs stronger focusing action so that the edge angle becomeslarger at outer radius resulting in spiral shape of magnets.

D.5 Historical Background

The FFAG idea was first introduced by several people independently in early 1950’sright after the AG principle was found. Electron models, both radial and spiral sectortypes, to prove the principle were made. They are a few meter in size and the outputenergy is less than 1 MeV. Extensive study was carried out at Midwestern UniversityResearch Associate (MURA) and a proposal to construct a few 10 GeV machine wasmade. However, an ordinary synchrotron was chosen for pursuing the energy frontierand the development as well as MURA itself died out in early 1960’s. By the way, alot of inventions which are nowadays common to the community were first emerged inthe MURA era. Those are RF stacking, adiabatic capture, and concept of separatrix,among others.

In 1980’s, demands of high intensity machine draw people’s attention to a FFAGaccelerator. It is true that FFAG was not the right choice to explore the energyfrontier because of relatively large magnets due to orbit excursion between injection

70 APPENDIX D. INTRODUCTION TO THE FFAG

and extraction momenta. However, large acceptance and possible high repetition cycleare best for high intensity with medium energy machine. There were several studymostly in Europe and Japan but no real machine was constructed. Some difficultiesstill exist to realize a real scale FFAG. One is the modeling of a magnet. The magnetfor FFAG requires nonlinear gradient as described in Eq. D.3. Shaping of magnetpole to make a design field needed iterative efforts. The other difficulty, that is mostessential, is the RF cavity which works in wide frequency range with large physicalaperture, especially in horizontal direction.

Both items are overcome, recently. As for the magnet shaping, design is nowefficiently done with a help of 3D magnetic field computation code like TOSCA.More fundamental breakthrough comes with a new type of a RF cavity equippedwith magnetic alloy (MA). Because of the very high permeability, a MA cavity hasboth high shunt impedance and wide band frequency range at the same time. Fromthe fabrication point of view, MA is a tape in its original form so that any kind ofcore shape is possible.

With those recent development, another proof of principle machine, this time theproof of proton acceleration with 1 kHz repetition rate, was constructed at KEK.The first beam is commissioned in June 2000 and the design procedure is established.About the status of the PoP machine and the larger machine followed will be describedin the next appendix.

Appendix E

PRISM-FFAG Design

E.1 Triplet Radial Sector

We adopted a scaled radial sector type FFAG with a triplet focusing. Compared witha spiral sector type FFAG, the radial one tends to have larger circumference ratio,which is the ratio of bending radius and average machine radius. However, easinessof magnet fabrication and better quality of fringing field of the triplet magnet isattractive enough to discard the spiral option. The triplet configuration has severaladvantages to other radial and spiral types. One is the field crump effects that isexpected between adjacent focusing and defocusing magnets. Secondly, the lengthof each straight section becomes almost doubled because one focusing and two halfdefocusing magnets are combined together to make one multi function magnet. Aminimum magnet unit has both focusing and defocusing. Therefore, the number ofmagnets is reduced. Finally, the lattice functions has mirror symmetry at the centerof a straight section even though that may not be essential. An injection optics designis eased and diagnostics at that position produce better signal to be interpreted.

There is another argument whether we need to make the machine scaled, in otherwords, zero chromaticity. In PRISM ring, the phase rotation is completed within 5turns and it seems to be not enough time to develop any resonance behavior. We tryto design a scaled machine in any case and later ease that requirement according tothe further beam dynamics study and tolerance of magnet fabrication.

E.2 Lattice Design

In order to simplify the FFAG model and quickly estimate optical property, thefollowing steps are taken. First, we assume that the FFAG consists of N identicalsector. Thus, the total bending angle in each sector is 2π/N . Each sector consistsof a normal bending magnet at the center and reversed bends on both sides. Thosehave nonlinear gradient. We name the normal bend at the center F magnet and thereversed bends on both sides D magnet. There are straight sections between adjacentsectors, not between F and D magnets.

71

72 APPENDIX E. PRISM-FFAG DESIGN

In each kind of bending magnet, we assume that the bending radius is constant,which is not the case in reality. If we take the bending angle in F magnet as θF

and that in D magnet as θD, the following relation is satisfied; θF − 2θD = 2π/N .Once a closed orbit at a certain energy is fixed in that way, the focusing propertyis determined according to the linear field gradient derived nearby the closed orbit.Fig. E.1 shows one half of a sector, and Table E.2 explains the definition of symbols.All the nonlinear components at the large amplitude area are ignored. Again, weassume that the linear field gradient is constant along the closed orbit as we assumedthe constant bending radius. In order to estimate the linear field gradient, we takeaverage radius and expand the field with respect to it.

βF βD

θF

θD

ρF

ρD

LF

LD

r0

r1

Figure E.1: One half of a triplet radial sector FFAG and its approximated orbit.

Table E.1: Definition of symbols.

symbol definitionN number of cellk field indexβF opening angle of F/2 with respect to machine centerβD opening angle of D with respect to machine centerθF bending angle of F/2θD bending angle of DρF bending radius of FρD bending radius of DLF path length of F/2LD path length of Dr0 orbit radius of F centerr1 orbit radius of F exit

E.3. MAGNET DESIGN 73

Now we know the path length in each magnet, bending angle, and gradient nearthe closed orbit, we can calculate linear lattice functions. That is done with anordinary synchrotron design code SAD.

A preliminary design of PRISM FFAG ring and its lattice function are shownin Fig.E.2. The PRISM-FFAG accelerator consists of a normal conducting magnet.The diameter is about 10 meters. It has 8 straight sections designed to be as long aspossible so as to install RF cavities as much as possible. Table E.2 summarizes mainparameters of the PRISM-FFAG ring.

Table E.2: Main parameters of PRISM-FFAG.

Parameters valuesCentral Energy 20MeVnumber of sector 8k value 5transition gamma 2.45average radius 5 mB@F/D 0.435/0.0717 TF/2 angle 0.037 radD angle 0.031 radF/2 bend angle 26 degreepacking f 0.18tune(H/V) 3.404/2.641

E.3 Magnet Design

The FFAG magnet produces a magnetic field of

B(r) = B0(r

r0

)k, (E.1)

where r is a radial position, and B0 is a magnitude of a magnetic field at the radiusof r0. k is a constant. In our present design, k = 5. And, the magnetic field of BZ

(T) at a radius of r (m) should be given for the D-coil and F-coil magnets in thePRISM-FFAG ring by

BZ = −0.0717 × (r

r0)5[T ] for the D coil, and

BZ = +0.435 × (r

r0)5[T ] for the F coil, (E.2)

where r0 is 5 m (for 68 MeV/c). In the present PRISM-FFAG magnet design, amagnetic field gradient is created by changing a magnet gap size. The cross section


Figure E.2: A lattice design of PRISM-FFAG ring and beam orbit parameters.

of the magnet gap is shown in Fig.E.4, where the half of the gap size changes by

a half gap = 10 × (500

r)5 (cm). (E.3)

Therefore, the shape of the iron yoke and gap is something like in Fig.E.4.

E.4. RF DESIGN 75

VacuumPump

Figure E.3: Layout of the PRISM-FFAG ring

E.4 RF Design

The RF frequency of the PRISM-FFAG ring is chosen to be about 5 MHz. ThePRISM-FFAG ring can have a total of 12 RF cavities. In each of the RF cavity has 4gaps. 2 of RF cavities can be placed in one straight section of the PRISM-FFAG ring.The distance of the RF gap is designed to be about 25 cm. Therefore, one straightsection of the PRISM-FFAG ring has to be more 2 m in length, as shown in Fig.E.5


r

rin=460

rout=550cm

Figure E.4: Shape of the PRISM-FFAG magnet gap as a function of radius. Theinnermost radius is 460 cm and the outermost radius is 550 cm.

25cm

m

Figure E.5: Two RF cavities in one straight section of the PRISM-FFAG ring. OneRF cavity has 4-gaps with 25 cm gap distance.

Appendix F

Hardware Design of PRISM-FFAG

F.1 RF System

A high-gradient and low-frequency RF system is very important to realize phaserotation in the PRISM-FFAG. In the PRISM-FFAG, an electric field gradient ofmore than 1 MV/m at a relatively low frequency (such as 5 MHz) is requested. Sincethe PRISM-FFAG ring has a large momentum acceptance, the accelerating time andacceptable momentum spread are solely limited by RF voltage. As possible candidatesfor high gradient and low frequency RF cavities, the following three types are beingdeveloped. They are (1) the inductive material loaded RF cavity using a new typeof ferrite, (2) the capacitive material loaded RF cavity using high grain ceramic, and(3) the air gap RF cavity.

F.2 Magnet

F.2.1 Yoke Free Design of FFAG Magnet

One of the problems of the conventional radial sector type of FFAG is that the sectormagnets occupy most of the ring circumference. It results in the lack of free spacefor beam injection and extraction.

Up to now, the FFAG triplet magnet developed in KEK employs the conventionalH-type magnet, such that the field flux returns through the side yoke attached to itspole. The triplet magnet used in KEK POP FFAG synchrotron is a complex of threeindependent H-type sector magnets(see Appendix G).

However, considering the field direction of the F-magnet and D-magnet of thetriplet magnet, it is reversed each other. Thus, if the return yokes of F and D-magnets are removed, the flux circulates through the F-pole and the D-pole. Suchflux generates the reversed field in F-pole and D-pole(see Fig. F.1). This is a roughexplanation of the idea of a new type FFAG magnet, so called “yoke-free magnet”.

In fact, if the return yoke is completely removed, the net flux of F-pole and D-poleshould be equal. It is unsuitable for FFAG ring, since the field of D-pole should be

77

78 APPENDIX F. HARDWARE DESIGN OF PRISM-FFAG

F-poleD-pole D-pole

Figure F.1: Schematic view of the field flux in the yoke-free magnet

weaker than that of F-pole. In order to reduce the flux in the D-pole, shunt yokesare installed at the side of D-pole. In addition, so as to apply fine tuning of the fieldstrength, tuning coils are installed around D-pole.

Schematic view of the typical yoke free magnet is shown in Fig. F.2. The returnyoke of F-pole is completely removed. This structure is favorable for the beam ex-traction, since the beam radius becomes the maximum at the center of the F-pole.The beam could be extracted at that point.

Up to now, a prototype yoke free magnet was made (see. Fig. F.3) by modifyingthe prototype magnet of POP FFAG, and the field measurement was carried out. Theresult shown in Fig. F.4 indicates good agreement with the numerical calculation byTOSCA.

It should be mentioned that a new project to demonstrate the yoke-free magnetis under development(see Appendix G).

F.2.2 Kicker Magnet Design

A preliminar design of the kicker magnets for beam injection and extraction fromthe PRISM-FFAG is shown. Regarding the PRISM-FFAG, the beam injection andextraction are one of the critical issues. To investigate possible schemes, a simpletracking simulation was carried out. It was assumed that about 80 % of the singlestraight section can be used for instruments needed for beam injection and extraction,and a drift space of 3m long is available.

Fig.F.5 shows a typical extraction orbit. From the simulation result, the requiredfield strength of the kicker magnet was found to be more than 500 gauss. In addition,

F.2. MAGNET 79

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X350.0

X400.0

X450.0

X500.0

X550.0

X600.0

Y-150.0

Y-100.0

Y-50.0

Y50.0

Y100.0

Y150.0

Z0.0

Z50.0

Z100.0

Z150.0

Z200.0

Z250.0

Figure F.2: View of a half pole of a typical yoke-free magnet

from the simulation of PRISM-FFAG phase rotator, the required rise time of themagnet was found to be 50ns.

Hardware requirements to realize such a kicker magnet has been investigated.The parameters are summarized in Table F.1. If one kicking is shared by fifteensmall magnets, the voltage needed to the kicker power supply is reduced to a feasiblevalue, typically below 100kV. It should be noted that due to a large muon beam sizefrom the PRISM-FFAG, once it is kicked out by the kicker magnet, the beam couldbe severely distorted as traversing the sector magnet. Therefore, the scheme whichinstalls plural kicker magnets can not be adopted.

Table F.1: One example of hardware parameters of the extraction kicker

beam emittance 10,000×2,000 πmm·mrad (H×V)magnet aperture 40×15cm (H×V)

field strength 500 gaussrise time 50 nano seconds

N·A 6000 A·TurnTotal length 3 meters

80 APPENDIX F. HARDWARE DESIGN OF PRISM-FFAG

Figure F.3: Prototype yoke free magnet

-400

-200

0

200

400

600

800

1000

1200

1400

0 2 4 6 8 10 12 14 16 18 20

θ(degree)

Bz(

gaus

s)

measurecalculation

r=100cm

r=85cm

r=70cm

Figure F.4: Result of field measurement of the prototype yoke-free magnet

F.2. MAGNET 81

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

X(m)

Y(m

)

Figure F.5: Simulated trajectories of the muons extracted out of the FFAG ring.

Appendix G

R&D of FFAG machines in Japan

In this appendix, research and development on the two FFAG machines developed atKEK is reported.

G.1 R&D of the POP FFAG Machine

Since the proposal of the FFAG principle proposed by Dr. Chihiro Owkawa in 1950’s,only an electron FFAG machine (the MURA project) was built. However, no protonFFAG accelerator has not been constructed. There were two major problems toconstruct a proton FFAG at the time. They are

• Lack of reliable numerical magnetic-field calculation codes,

• Lack of a high gradient RF cavity to make fast beam acceleration possible

For the former, the FFAG machine obtains its focusing force through the strongnon-linear field. In order to design the magnets with such a non-linear field, numericalcalculations are indispensable. However, in 1960’s, there were no reliable code tocalculate a magnetic field.

Now, those difficulties has overcome. The high-gradient cavity using a magneticalloy (MA) core has been developed at KEK, and it has solved the difficulty associatedwith RF acceleration. Also, we have now a developed 3D field calculation code, suchas TOSCA, which offers a tool of numerical field calculations. Thereby, a time tobuild a FFAG machine has come. We could re-establish the FFAG principle withadvanced accelerator techniques.

Under the above motivations, the proof-of-principle (POP) FFAG machine wasdeveloped at KEK,

• to establish the FFAG principle,

• to prove fast acceleration by a FFAG synchrotron, and

• to demonstrate the large acceptance of FFAG synchrotron.

82

G.1. R&D OF THE POP FFAG MACHINE 83

It should be noted that it is the first proton FFAG accelerator in the world. Theschematic layout and the photo of the POP FFAG are shown in Figs. G.1 and G.2,respectively.

Scale : 1m0 0.5 1.0

1 2

3 4-1

4-2 4-3

55 6

7-1

8

10

11

12

12

9

7-2

13

1415

16

18

17

9

1: ion source 2: chopper electrode 3: triplet-quadrupole magnet 4: steering magnet 5: solenoid magnet 6: beam slit 7: Faraday cup 8: septum electrode 9: bump electrode10: sector magnet11: F-magnet pole12: D-magnet pole13: beam position monitior14: RF cavity15: RF amplifier16: vacuum bellows17: turbo molecular pump18: cryopump

Figure G.1: Layout of the POP FFAG machine

Figure G.2: A top-view of the POP FFAG machine

Table G.1 summarizes the main design parameters.

84 APPENDIX G. R&D OF FFAG MACHINES IN JAPAN

Table G.1: FFAG POP model parameters

Type of magnet Radial sector type(Triplet)No. of sectors 8Field index(k-value) 2.5Energy 50keV(injection) - 500keVRepetition rate 1kHzMagnetic field Focus-mag. : 0.14∼0.32 Tesla

Defocus-mag. :0.04∼0.13 TeslaRadii of closed orbit 0.81 ∼ 1.14Betatron tune Horizontal : 2.17∼2.22

Vertical : 1.24∼1.26RF frequency 0.61 ∼ 1.38MHzRF voltage 1.3 ∼ 3.0 kVp

G.1.1 Results of the Accelerator Studies

When the construction of POP FFAG was complete, intensive accelerator studieswere carried out to understand the characteristics of FFAG accelerator. The majoritems of the accelerator study are as follows.

• Demonstration of fast acceleration,

• a study of betatron tune and synchrotron tune in various conditions,

• a study of beam positions in various energies, and

• a study of beam acceptance.

In the following, the results of accelerator studies are briefly reviewed.

G.1.1.1 Beam Acceleration

Compared to an ordinary synchrotron, the acceleration time of the FFAG synchrotronis not restricted by a ramping time of a pulsed magnet. Thus, the higher the accel-eration field is, the quicker the acceleration is completed. It is one of the prominentmerits of the FFAG synchrotron. To demonstrate this feature of FFAG accelerationis one of the strong motivation to develop the POP FFAG machine.

In the case that a synchronous phase is set to be 20 degree, the RF voltage shouldbe at least 1.3 kV during acceleration. We have developed a RF cavity using tworectangular FINEMET1 which has a core of 1.1 m (width) × 0.7 m (height). Thethickness of the core is 30mm. A 55kW RF amplifier which consists of two tetrodes(Eimac 4CW25,000) was used.

1For the details of FINEMET, see [4]


Fig.G.1.1.1 shows a typical beam signal observed by the inner and outer electrodesof the beam position monitor during acceleration of a proton beam. As seen inFig.G.1.1.1, it indicates that, as it is accelerated, the beam moves outward. In thecase of Fig.G.1.1.1, a flat-top energy is about 350 keV. The acceleration time is about600 µsec. At present, the acceleration up to 500 keV completes within 1 msec, and thebeam characteristics in different energies were measured. Some of them are presentedin the following sections.

Figure G.3: Typical BPM signal during acceleration in POP FFAG, Upper signal:inside, Lower signal: outside

G.1.1.2 Tune Survey

Firstly, a betatron tune was measured in injection orbit in various field condition. Inthe FFAG machine, the betatron tune can be changed by varying the relative fieldstrength between the focusing magnet and the defocusing magnet, which is called“F/D ratio”. Fig.G.1.1.2 shows the betatron tune in various different F/D ratio. Theresults were consistent with the expectation from the computer simulation.

The synchrotron tune was measured in the energy range from 50 keV to 500 keV.The result in Fig.G.1.1.2 shows good agreement with the expected tune.

G.1.1.3 Beam Position

In FFAG, the orbit position changes as the beam energy increases. The consistencyof the beam position with the calculation is one of the measure to verify the currentdesign strategy.


00.05

0.10.15

0.20.25

0.30.35

0.40.45

0.5

3.4 3.6 3.8 4 4.2 4.4

tuneH(measured)tuneH(calculated)

(a)

F/D ratio in B*LF

ract

iona

l par

t of

tun

e

00.05

0.10.15

0.20.25

0.30.35

0.40.45

0.5

3.4 3.6 3.8 4 4.2 4.4

tuneV(measured)tuneV(calculated)

(b)

F/D ratio in B*L

Fra

ctio

nal p

art

of t

une

Figure G.4: Betatron tune in various field setting, (a) Horizontal tune, (b)Verticaltune

:1/01/27 10.42

0

5

10

15

20

25

30

0 100 200 300 400 500 600

: measure

: calculation

Beam energy(keV)

Sync

hrot

ron

tune

(kH

z)

Figure G.5: Synchrotron tune in various energies

The beam position was measured with energy from 100 keV to 400 keV. The resultswere summarized in Fig.G.1.1.3, and it is consistent with the simulation within thesystematics error.


0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

0 50 100 150 200 250 300 350 400 450 500

calculationmeasure

Ebeam(keV)

Rbe

am(m

)

Figure G.6: Radial position of the circulating beam observed by the Faraday cup inthe ring in various energies

G.1.2 Aperture Survey

It is known that one of the outstanding advantages of the FFAG accelerator is itslarge transverse acceptance. It would be a key issue to realize a FFAG-based neutrinofactory.

POP FFAG adopted the multi-turn injection scheme by using a DC septum anda pair of bump electrodes. Thus, by tuning the injection condition, it can inject asingle bunch with a large injection error. Making use of this feature, the capabilityof storage and acceleration of a large emittance beam in the FFAG ring has beensuccessfully demonstrated. Fig.G.1.2 shows a typical change of the horizontal beamposition observed by the beam position monitor, changing the amplitude of injectionerrors. It indicates that the beam with more than 3 cm in amplitude was circulated inthe FFAG ring. The beam with such a large amplitude was successfully acceleratedto the final energy. It means that the horizontal acceptance is at least more than 5000πmm·mrad.. At present, the acceptance presented here is no more than a lower limit,since the maximum injection error is restricted by the distance between the injectionseptum and the injection orbit. Thus, at present, a trial to move the injection orbitoutward is undertaken.

G.1.3 Summary of POP FFAG R&D

From the accelerator study of POP FFAG, the following results have been demon-strated.

• Fast beam acceleration, i.e. from 50 keV to 500 keV within 1 msec.


-9-339

0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18x 10

-3

time(s)

pos(

cm) Vbump=3kV

-9-339

0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18x 10

-3

time(s)

pos(

cm) Vbump=6kV

-9-339

0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18x 10

-3

time(s)

pos(

cm) Vbump=9kV

-9-339

0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18x 10

-3

time(s)

pos(

cm) Vbump=12kV

-9-339

0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18x 10

-3

time(s)

pos(

cm) Vbump=15kV

Figure G.7: Beam position observed by the beam position monitor in various injectionerror

• Large horizontal acceptance, typically more than 5000 πmm·mradian.

• Good agreement between the betatron tune, the synchrotron tune and the orbitposition with the numerical calculations.

They give strong supports for the application of FFAG to accelerate a beam of short-lived particles with large emittance. At present, further accelerator studies are goingon. In addition, a plan of beam extraction from the POP FFAG ring is under theinvestigation.

G.2 R&D of the 150-MeV Proton FFAG Machine

After the success of the POP FFAG commissioning, a new proposal to constructa larger-size FFAG accelerator was approved in Japanese fiscal year (JFY) 2000.

G.2. R&D OF THE 150-MEV PROTON FFAG MACHINE 89

In this project, a FFAG synchrotron to accelerate protons up to 150 MeV is beingconstructed. The main parameters are summarized in Table G.2. The schematic viewof this 150-MeV FFAG accelerator is shown in Fig. G.2

0 1m 5m

Figure G.8: Schematic view of the 150MeV FFAG

Table G.2: 150MeV FFAG main parameters

Type of magnet Radial sector type(Triplet)No. of sectors 12Field index(k-value) 7.5Energy 10MeV - 125MeV or 12MeV -150MeVRepetition rate 250HzMagnetic field Focus-mag. : 0.34∼1.63 Tesla

Defocus-mag. :0.15∼0.78 TeslaRadii of closed orbit 4.4m ∼ 5.3mBetatron tune Horizontal : ∼3.8

Vertical : ∼2.2RF frequency 1.5 ∼ 4.6MHz

G.2.1 Main Features of 150-MeV FFAG

Compared to POP FFAG, 150-MeV FFAG has three new features for R&D works.They are (1) adoption of yoke-free magnets, (2) beam extraction from the FFAG ring,and (3) operation with high repetition.


As the first item, yoke-free magnets are one of the key issues for future develop-ments of the FFAG-based neutrino factory. If it is successful, it could allows easyaccess of injection and extraction of the beam and also could offer large degree offreedom for possible configuration of beam apparatus. Thus, the demonstration ofyoke-free magnet is useful.

As the second item, it is now being considered to employ the following scheme.The beam in the FFAG ring is bend by a kicker magnet installed in the middle ofthe straight section. A typical field strength for 150-MeV FFAG is about 600 gauss.A decay time of the magnetic field is less than 150 ns. In the next straight section,a DC septum magnet is installed to give further horizontal kick to the beam. Therequired field strength is about 2 kgauss. Finally, the beam is extracted from thering. Fig.G.9 shows a typical beam extraction orbit.

Figure G.9: Typical extraction orbit of 150MeV FFAG

As the third item, in some sense it could be said that the capability of high-repetition operation has been already demonstrated at POP FFAG and, as shownbefore, it has proved that it can accelerate a beam within 1 msec. However, in POPFFAG, the actual repetition rate is 1 Hz due to limitation of the power consumption.Thus, in 150-MeV FFAG, a real high repetition operation has to be demonstrated.A repetition rate of 250 Hz is planned. A fast kicker magnet will also be needed fora neutrino factory.

G.2.2 R&D Status and Construction Schedule

Preliminary designs of the ring and the beam-extraction scheme have been almostfixed. The fabrication of the sector magnets and the extraction elements is under-

G.2. R&D OF THE 150-MEV PROTON FFAG MACHINE 91

taken. Until the end of JFY 2001, most of the components were ready, and theconstruction has already started in the beginning of JFY 2002 and the commission-ing will start at the end of JFY 2002.

Appendix H

Alternative Phase Rotator Scheme

H.1 Phase Rotator Linac

Although the base line of the phase rotator is FFAG, another phase rotator schemethat uses a simple linac is also being studied : namely PRISM-Linac. The main reasonfor this study is as follows. While the muons captured by the solenoid magnetic fieldcan be transported sufficiently as long as the magnetic field continues, we need aspecial care to transfer them to a FODO transport system composed of bendingmagnets, Q-magnets and so on. When we use a linac instead of the ring as a phaserotator, the muons are captured and transported by a continuous solenoid field tillthe end (stopping target), and thus the transfer between the two different transportsystem, solenoid channel and FODO channel, vanishes. In addition, this system doesnot need fast kickers for injection to and extraction from the ring, which requiresspecial care to be taken. The drawbacks of this linac scheme may be: 1) althoughunwanted particles such as oppositely charged muons diffuse in the phase rotator,they are also transported by the solenoid channel, 2) the cost may be high. The firstissue can be solved by use of a curved solenoid, which is already described in section5.2. The resulted system is shown in Fig.H.1 schematically.

The following sections describe such an alternative scheme and a rough simulationresult of a muon yield.

SpiralSolenoid

ø12m

~27m0.5MV/m x 25m x ø4.4m

SpiralSolenoidø12m

80m

100m

110m

130m

PRISM/L phase rotator

Primary Proton

Production TargetProtonBeamDump

Figure H.1: Schematic layout of the PRISM-Linac

92

H.2. SYSTEM OVERVIEW 93

10

15

20

25

30

35

0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3

MeV

µs

31m

6MHz+12MHz

5ns

20-10(sin(2 *6*m1*(m0-m2))+sin(2 *12*m1*(m0-m3))/m4)

χ

Figure H.2: Muon energy as a function of ToF at 31m downstream from the pro-duction target. The time spread corresponding to 10-30 MeV range is 100 ns. Therequired acceleration and deceleration voltage can be synthesized with 6 MHz and 12MHz RF component.

H.2 System Overview

There is not much difference between the two schemes before the long drift section;pion production target, pion capture-decay system and muon transport system arethe same as the base line design. The muon transport system, however, is longer thanthe base line design, which develops the ToF before the phase rotator. This featuremay be preferable to relax a restriction on the system layout: the production targetis supposed to be located underground, while the main system will be on the ground.

Because the energy spread is reduced 1/10 ( ±50% to ±5% ) by the phase rotationscheme, the bunch length should be elongated 10 times. Suppose that the initialproton bunch width is 10 ns, the bunch length should be elongated to 100 ns, whichis achieved by 30m drift from the production target(See Fig.H.2). The relationshipbetween the muon energy and its ToF is almost quadratic in the assumed energyregion, which is well fitted by 6 MHz and 12 MHz sine wave with proper phase andamplitude. Thus the required voltage for the phase rotation can be synthesized bythese RF voltages.

The total voltage of the acceleration and deceleration required for the phase rota-tion is 10 MV for 20MeV ± 50% bunch. This amount of voltage has to be applied tothe muon bunch within 30m; the bunch length develops further than the RF systemcan handle, otherwise. Fig.H.3 shows how the phase rotation goes on.

After the phase rotator, curved solenoids with dipole field are situated for particleselection. This makes the time delay of the muon arrival, and thus the survived pionsare much reduced.

94 APPENDIX H. ALTERNATIVE PHASE ROTATOR SCHEME

6MHz1.5MV

3m

6MHz1.5MV

3m

6MHz1.5MV

3m

6MHz1.5MV

3m

6MHz1.5MV

3m

6MHz1.5MV

3m

12MHz1.2MV

3m

1m spacing between cavities

10

15

20

25

30

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

MeV

28m before kick

32m (6MHz)

36m (6MHz)

40m (6MHz)

28m (6MHz)

44m(6MHz)

48m(6MHz) 52m(12MHz,1.2MV)

t [µs]

Figure H.3: Rough simulation shows how the phase rotation goes. Six 3m-cavitieslined up with 1m spacing generate 9 MV in total. 12-MHz one is used to fit thewaveform.

2m 8m

1T

20~30m

4T

Z

B

Production target

6 or 12T

Capture&Decay sectionAdiabatic expansion

Drift sectionPhase Rotator

Stopping target

Figure H.4: The magnet field distribution along the axis.

H.3 Magnetic Field Distribution along the Axis

The magnet field distribution along the axis is shown in Fig.H.4. The solenoid mag-netic filed starting from the production target area continues till the end. The baseline design adopts 6T field at the capture section for a practical reason on the fab-rication of the super conducting solenoid. The field level should be high to keep thepion beam size, the source size of the muons, small. Because more than half of thepions decay to muons at 3m downstream from the target, the magnetic field startsto decrease down to 1T at 3 m. This converts the transverse momentum to the lon-gitudinal component with increase of the beam size. Because the ToF develops asa function of the longitudinal velocity, transverse components has to be as small aspossible during the drift section. The excessive transverse momentum decreases thecorrelation between the ToF and the total energy which is not only a function of thelongitudinal momentum but also that of the transverse one. The field may have to beincreased in order to reduce the beam size in the phase rotator. Although this maydrop some amount of muons, such muons with big emittance are not usable afterward.The magnetic filed beyond the phase rotator should be designed under considerationof the curved solenoid. It should be noted again that the magnetic field continues tillthe end.

H.4. PHASE ROTATOR WITH LINAC 95

f=6MHzQ=17000Z=~300k /m

Emax=5.5MV/mW=500J

@Eave=0.5MV/m(0.75MV in total)

CLBeam Hole ø30cm16cm gap

Superconducting solenoid coil

L=1.5m

ø4.4m

ø80cm

Figure H.5: 6MHz air core cavity with φ4.4m in diameter.

H.4 Phase Rotator with Linac

Low frequency cavities have been usually realized by inductive load such as ferrite.The high gradient field, however, makes a use of such material difficult, because ofthe magnetic saturation. This may be overcome with use of special material or noveldesign. Among the options, air core cavities are described here. Fig. H.5 shows across-sectional schematic view of such a cavity with main parameters.

In order to reduce the resonant frequency, it has heavy capacitive load in thecavity with narrow electrode gap, where the peak surface field is 5.5 MV/m at 0.5MV/m. Because the cavity length is 1.5 m, this single cavity generates 0.75 MV intotal, and thirteen of these cavities should be lined up. RF power of 2 MW in peakwould be sufficient to generate 0.75 MV in a cavity. From the filling time value, theminimum pulse length will be 1 ms. This cavity have a space for the superconductingsolenoid in the drift tube sticking out from the inner wall. The magnetic field strengthof the solenoid is assumed to be 4T in this case. This value has to be optimized underconsideration of the bore size: the larger outer diameter of the drift tube decreasesthe total inductance in the cavity and thus the cavity size tends to increase.

The Kilpatrick’s criterion for maximum surface field Ekp is 4.6 MV/m at 6 MHz(seeFig.H.6). It should be noted that this value only suggests a criterion and the practicalfield gradient of 1.2 times higher than this value is easy to achieve.

96 APPENDIX H. ALTERNATIVE PHASE ROTATOR SCHEME

0

2

4

6

8

10

12

1 10 100

E [M

V/m

]

Kilpatrick's Sparking Criterion

Figure H.6: Kilpatrick’s criterion for maximum surface electric field in RF.

Figure H.7: Phase space distribution of muons at 27m downstream from the produc-tion target.

H.5 Muon Yield Simulation

Fig.H.7 shows the muon energy as a function of ToF from the production. Because ofthe transverse momentum component and disturbance at pion decay, the distributionsmears slightly. The phase space distribution after the phase rotator is shown inFig.H.8. About 400 muons are obtained out of 105 50GeV protons. Fig.H.9 showsthe simulation result starting from 12 T magnetic field at the capture section forfuture upgrade. About 103 muons are collected in the case. The muons with oppositesign also come to the end in both cases, which has to be removed by the curvedsolenoid.

H.5. MUON YIELD SIMULATION 97

µ+

µ-

0 50 100 150 200 25010

15

20

25

30MeV

Figure H.8: Left: Phase space distribution of muons after the phase rotator.Right:projected histogram to the energy axis: bin width 1 MeV. The capture field is6T.

0 100 200 300 400 50010

15

20

25

30

µ+

µ-

MeV

Figure H.9: Top: particle trajectories till the end of drift section. Left: phase spacedistribution of muons after the phase rotator. Right: projected histogram to theenergy axis: bin width 1 MeV. The capture field is 12T.

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