Issue 76

Beam Dynamics Newsletter

3.8 Intense Muon Source with FFA Based Energy Recovery Internal Target (ERIT)1

Yoshiharu Mori, Hidefumi Okita, Tomonori Uesugi, Yasutoshi Kuriyama, Akihiro Tanigchi, Yoshihiro Ishi, Masayuki Muto,Yuka Ono,
Research Reactor Institute, Kyoto University, Japan.
Yujiro Yonemura, Hidehiko Arima, Nobuo Ikeda,
Dept. of Engineering, Kyushu University, Japan.
Akira Sato, Dept. of Physics, Osaka University, Japan.
Michikazu Kinsho, Yasuhiro Miyake, Masahiro Yoshimoto, Kota Okabe,
JPARC, JAEA-KEK, Japan.

1 This work was partially funded by ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan).

Introduction

Muons are of interest in various fields such as particle physics, nuclear physics and material science. Important among them is the transmutation of nuclear waste, especially high-level radioactive species such as long-lived fission products (LLFPs) and minor actinides (MAs), for which copious supplies of neutrons are needed. Transmutation can be triggered via muon nuclear transformations. A muonic atom can be formed by trapping a negative muon in the atomic nucleus, and this process has a probability of about 95 per cent success if the atomic number (\( Z \)) is more than 30. The muonic atom then transforms to a stable nucleus by beta decay and the emission of neutrons. For transmutation purposes a very high yield of negative muons of the order of 10\( ^{16} \)\( \mu \)\( ^- \)/s is necessary  [1].

Negative muons from the decay of negative pions are efficiently produced by nucleon-nucleon interactions with a high energy hadron beam colliding with a target nucleus containing neutrons. A hadron beam energy of 300 - 400 MeV/u or more is necessary to produce negative pions effectively  [2].

There are difficulties for negative pion production with this energy range. One is energy loss from the projectile proton by ionization in the target. The efficiency of negative pion production drops until the proton energy reaches the threshold energy of pion production at about 250 MeV/u. Another problem is the absorption of negative pions in the solid target. The absorption cross section of negative pions with the target nucleus is so large that a thinner target must be used to eliminate the effect. Thus, a high beam current and a thin target are essential to keep the efficiency large in negative muon production. In order to realize the required negative muon yield of 10\( ^{16} \)\( \mu \)\( ^- \)/s, a proton beam current of more than 3 A is necessary if a thin lithium target of 1 cm thickness is used.

Recovery of the beam energy loss by re-acceleration and use of a thinner target, the so called energy recovery internal target (ERIT) scheme  [3, 4], is a useful and convenient method for overcoming these difficulties. The ERIT scheme was first proposed for the efficient production of secondary particles such as neutrons and unstable nuclei, and the first ERIT ring was constructed at the research reactor institute of Kyoto University  [5]. This paper describes the possibility of applying the ERIT principle for intense negative pion/muon production of more than 10\( ^{16} \)\( \mu \)\( ^- \)/s with a wedge type of Li solid target  [1] or by creating a target by filling the ring with deuterium gas  [6].

3.8.1 Principle of ERIT

In the ERIT scheme, projectile particles circulate and pass through a thin target placed in the ring and generate secondary particles such as neutrons, pions, etc. Particles lose energy through ionization interactions (electronic stopping power) and are re-accelerated in rf cavities. Beam emittance growth caused by multiple scattering (Rutherford scattering and straggling) at the target can be counteracted by ionization cooling  [7, 8]. Re-acceleration imposes only longitudinal momentum transfer, while the energy losses occur in the directioin of particle travel; the overall effect is that both transverse and longitudinal momenta are reduced and the beam emittance does not increase.

Longitudinal cooling occurs if

\[ \frac {\partial }{\partial E}\left (\frac {\mathrm {d}E}{\mathrm {d}s}\right )>0. \]

Here, \( \mathrm {d}E/\mathrm {d}s \) is the stopping power. Therefore, if a wedge-shaped target is placed in the ring at a particular position causing orbit dispersion, the longitudinal emittance can be cooled.

The ERIT scheme was demonstrated at the Institute for Integrated Radiation and Nuclear Science of Kyoto University (KURNS) as a low energy neutron source  [5]. Figure 1(a) shows a schematic layout and a photograph of the world’s first ERIT neutron source with ionization cooling at Kyoto University. The beam optics of the ERIT ring are based on fixed field alternating gradient (FFA) focusing to provide large transverse and longitudinal acceptances. The 11 MeV H\( ^- \) ion beam is injected from a linear accelerator (linac) into the FFA ring.

The Be(p,n)B reaction is used for neutron production in this scheme. The diameter of the ring at the central orbit is about 4.5 m. A beryllium foil, a few micrometres thick, is placed in the straight section of the ring, where H\( ^- \) ions from the linac are charge-stripped to protons and merged with the circulating proton beam, and the energy loss at the target is restored by the rf cavity. In this design, neutron production of more than 10\( ^{12} \) n/s is expected with an average beam current of 100 \( \mu \)A injected from the linac and beam circulation of more than 500 turns in the ring.

We have found that the design configurations and requirements can be well satisfied experimentally. The emittance growth rate in the vertical direction as a function of turn number has been measured with a beam scraper and an electrostatic bunch monitor. The experimental results, shown in Fig. 1(b), indicate that these design configurations and requirements can be satisfied.

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Figure 1: (a) Schematic layout of the ERIT neutron source (upper) and a picture of the apparatus built at KURRI. (b) Beam emittance growth as a function of number of circulating turns in the ERIT neutron ring, measured by the beam scraping method. The error bars in the figure show the systematic errors from the beam scraper positions, which were 0.5 mm in accuracy. The theoretical values estimated from the equations of ionization cooling rate are also shown in the figure.

3.8.2 Production of negative muons
Scheme 1: MERIT with proton beam and wedge shaped target

In order to produce negative pions/muons with the ERIT scheme, some additional features should be included. In contrast to the original ERIT ring for neutron production where the proton beam energy was very low, the transverse emittance growth caused by multiple scattering is rather modest because the proton beam energy is relatively higher in the case of pion production. The energy spread of the proton beam colliding with the target is mostly enlarged by electron straggling where the emittance growth rate is almost proportional to the proton beam energy at relativistic energies. A wedge-shaped target placed at the dispersive orbit should be used. Moreover, if the injection beam energy is lowered, the load on the injector can be relaxed.

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Figure 2: Schematic diagrams of ERIT and MERIT.

For negative pion/muon production, the ERIT ring provides beam acceleration and storage at the orbit where a wedge shaped target is placed, and seems most appropriate. A new type of ERIT named MERIT (Multiplex Energy Recovery Internal Target) for negative muons has been proposed  [1, 9, 10]. A schematic diagram of MERIT is shown in Fig. 2 showing a comparison with the ordinary ERIT.

In this proposal, the projectile particles are protons that are injected at a relatively low energy of 500 MeV and accelerated up to about 800 MeV by an FFA ring accelerator with fixed frequency rf acceleration. To make the use of fixed frequency rf feasible, either serpentine (semi-isochronous) acceleration or a static bucket acceleration scheme is applied.

In serpentine acceleration, the momentum compaction, \( \alpha \), should satisfy the following condition,

\[ \alpha \sim \frac 1{\gamma _s^2}\,. \]

Here, \( \gamma \)\( _s \) is the relativistic factor (total energy/rest mass energy) of the beam between its initial and final energies. Momentum compaction in a scaling FFA can be derived from the geometrical magnetic field index, \( k \), through

\[ \alpha =\frac 1{k+1}\,. \]

Figure 3 shows particle paths in longitudinal phase space in a serpentine scenario. Here, the field index \( k=2.43 \), \( \gamma _s=1.853 \) and the total rf voltage per ring is about 10 MV. The design of the MERIT ring for pion production has been based on these underlying conditions.

The optics of the ring are those of a scaling FFA with an eight-fold symmetry FDF triplet lattice. Table 1 presents the basic parameters of the ring.

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Figure 3: Particle paths in longitudinal phase space under serpentine acceleration.

Table 1: Basic parameters of MERIT ring for muon production.

Ring configuration H\( ^- \) FFA
Energy range (MeV) 500 - 800
Magnetic rigidity (T.m) 3.633 - 4.877
Lattice FDF
Average radius (m) 5.044 - 5.5
Magnetic field (T): F 1.96 - 2.41
Magnetic field (T): D 1.71 - 2.11
Number of cells 8
Geometrical field index 2.43
Cell tune: H 0.212
Cell tune: V 0.180
Beta function (m) @SS:H 2.5
Beta function (m) @SS:V 2.8
Dispersion function (m) 1.5

A schematic layout of the MERIT ring for muon production is shown in Figure 4. A wedge-shaped liquid lithium target whose thickness gradient is \( \rho '=0.27 \) g/cm\( ^3 \) is placed at the radial position \( R=5.5 \) m. The effective thickness of the target is 1.35 g/cm\( ^2 \) at \( R=5.55 \) m; therefore the energy loss of protons becomes approximately 2.5 MeV, which can be successfully recovered by rf re-acceleration.

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Figure 4: Schematic layout of the MERIT ring for negative muon production.

After acceleration to the top energy, the beam is stored with the help of a wedge-shaped, liquid lithium target placed on the maximum energy orbit. If the target becomes thicker towards the outside of the ring, then the beam stays around the maximum energy orbit where beam acceleration and energy loss are well balanced. Also, the ionization beam cooling helps to suppress emittance growth transversely and longitudinally, and the beam circulates for the requisite number of turns needed to generate enough negative pions/muons.

The negative pion yield is estimated from,

\[ Y=N_p\,f_0\,n_t\,l_t\,\sigma _\pi \,. \]

Here, \( Y \) is the negative pion yield, \( \sigma _\pi \) is the negative pion production cross section, \( N_p \) is the number of projectile particles (protons) per ring, \( f_0 \) revolution frequency, \( n_t \) is the target particle density and \( l_t \) is the target thickness. Since the negative pion production cross section for an 800 MeV proton is about 5 mb, a negative muon yield of \( 10^{16}\,\mu ^- \)/s can be accomplished when \( N_p=1.2\times 10^{12} \), which is modest compared with the space charge limited beam intensity at the given beam emittance  [11]. If the injection proton beam current is 2 mA, the beam needs to be accumulated and stored for at least 600 turns at the circulating, lithium target orbit.

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Figure 5: Beam emittance growth in horizontal (upper left), vertical (upper right) and longitudinal (lower left) directions respectively are shown as a function of number of turns. The figure (lower right) shows longitudinal beam behavior in phase space over 500 turns following injection.

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Figure 6: Dynamic apertures (left: horizontal, right: vertical ) estimated numeri- cally.

The beam is blown up transversely and longitudinally by Rutherford multiple scattering and energy straggling. The beam emittance growth, however, could be cured through the effect of ionization cooling in the energy recovery system. These longitudinal and transverse emittance behaviors can be estimated by simulation codes using particle tracking.

Figure 5 shows the longitudinal and transverse emittance growths as a function of beam turn numbers circulating around the ring, which are simulated by multi-particle beam tracking including the ionization cooling effect. As can be seen from this figure, the beam emittance in the longitudinal direction decreases gradually as a function of turn number. The growths of transverse beam emittance in both horizontal and vertical planes are also suppressed and reach equilibrium values of about 700 mm.mrad (rms) in horizontal and 600 mm.mrad (rms) in vertical directions, respectively. The acceptance of the ring must be large enough to cover these equilibrium emittances during beam acceleration and storage.

In the longitudinal direction, although the phase acceptance is limited to about \( \pm \pi \)/2, the longitudinal acceptance is large in serpentine beam acceleration, as shown in Figure 3. Since the scaling FFA magnets include non-linear field components, the transverse dynamic apertures for both horizontal and vertical directions must be large enough for the beam emittance. The dynamic apertures have been estimated numerically by particle tracking simulations and the results are shown in Figure 6. The dynamic apertures are about 0.1 m.rad horizontally and 0.07 m.rad vertically, and both are quite large compared with the beam emittance.

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Figure 7: Capture efficiency of \( \pi ^- \) for various positions of the lithium target placed in the straight section.

The momentum distribution of negative pions produced at the lithium target was estimated by the Geant4 beam line code, G4BL  [12]. Since the dipole magnetic field strength of the F magnet is about 2.4 T and the effective field length is about 1.1 m, almost all the pions could be swept away from the ring. The pions are emitted in various directions and the number of pions within the aperture of the capture solenoid is limited.

Figure 7 shows the \( \pi ^- \) capture efficiencies estimated by the G4BL code at the entrance to the 1 m diameter solenoid magnet for various positions of the lithium target in the straight section. The overall \( \pi ^- \) capture efficiency is estimated to be about 50 per cent or less. Nevertheless, the production yield of negative muons in this system will still satisfy the production requirements.

Scheme 2: ERIT with gaseous deuterium target

One of the difficulties is how to collect and transport sufficiently many of the negative pions and muons produced at the lithium target to the nuclear waste treatment area. A deuterium gas target filling up the beam pipe of the ERIT ring instead of a liquid lithium thin target may improve the pion/muon capture efficiency and make muon nuclear transformation of long-lived nuclear wastes more manageable. Pions are generated everywhere in the beam pipe around the ring. Negative muons from negative pion decay are swept away to the outside of the beam pipe by the magnetic field and finally trapped as muonic atoms by LLFPs which cover the beam pipe, as shown in Fig. 8. In this scheme, almost all negative muons could be efficiently utilized to mitigate the long-lived fission products. Other energetic particles produced in the deuterium gas target, such as neutrons, protons, positive and neutral pions and gamma rays, could be thermalized by a water cell surrounding the beam pipe.

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Figure 8: Schematic layout of the ERIT ring for negative muon production (left) and cross sectional view of the ring pipe (right).

In this design, in contrast to Scheme 1, deuterons are chosen as the projectile particles instead of protons. The cross section of negative pion production is almost five times greater than for protons for a beam energy of about 600 MeV/u, although beam loss caused by the break-up reaction to a proton and neutron is large. The negative pion production cross section for a 600 MeV/u deuteron is about 20 mb  [13]. The negative pion yield \( Y \) attained with this system is

\[ Y=L\sigma _\pi \,, \]

where \( \sigma _{\pi } \) is the negative pion production cross section and \( L \) is the luminosity expressed as:

\[ L=N_d\,v_d\,n_T\,. \]

Here, \( N_d \) is the number of projectile particle(deuteron) per ring, \( v_d \) (cm/s) is the projectile particle velocity and \( n_T \) (cm\( ^- \)\( ^3 \)) is the target particle density. A negative muon yield of \( 1\times 10^{16}\,\mu ^- \)/s can be achieved when the luminosity is \( 5\times 10^{41} \) cm\( ^{-2} \) s\( ^{-1} \). When the pressure of deuterium gas target is 1 atm, \( N_d \) becomes \( 7.9\times 10^{11} \) deuterons per ring.

The total beam loss cross section, \( \sigma _l \), including deuteron break-up and pion generation estimated by Geant4:G4BL [12], is approximately 120 mb, which is six times larger than the negative pion production cross section. Thus, a deuteron beam current of about 10 mA should be continuously supplied to the system to attain the negative muon yield of \( 1\times 10^{16} \) s\( ^{-1} \). On the other hand, the mean free path of the projectile particle is about \( l_\mathrm {mfp}=1/\sigma _ln_T\sim 3000 \) m, which is equivalent to a path length of about 90 turns circumference of the ring whose diameter is about 11 m. Thus, the ring acceptance should be large enough to accept the beam whose emittance has grown by multiple scattering and straggling after 90 turns. The system also involves ionization cooling and through this the beam energy loss by ionization is recovered by rf re-acceleration, and emittance blow up is suppressed. The emittance growth as a function of beam path can be calculated from the ionization cooling rate equations.

Figure 9 shows the transverse beam emittance behavior estimated by the rate equation for the deuteron beam and deuterium target described above as a function of number of turns. Here, a beta function of 4 m averaged over the ring is assumed. As can be seen from the figure, since the normalized rms emittance is about \( 6\times 10^{-5} \) m.rad after 100 turns, a ring acceptance (unnormalized) of more than \( 5\times 10^{-4} \) m.mrad, which is almost ten times the rms emittance, seems to be enough. The stopping power of 600 MeV/u deuterons in deuterium gas is about 3 MeV/g.cm\( ^{-2} \) and the energy loss per turn in the ring filled with 1 atm deuterium gas becomes approximately 1 MeV. Therefore, a total rf voltage of more than 10 MV, when the rf harmonic is \( h=1 \), seems to be enough for energy loss compensation.

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Figure 9: Transverse emittance growth as a function of number of beam circulating turns estimated with the rate equations of ionization cooling.

A preliminary ERIT ring design for negative muon production with a deuteron beam has been carried out. Table 2 is a list of basic parameters of the ring. The ring is composed of an eight-fold symmetric FDF scaling FFA lattice and superconducting (SC) magnets are assumed to be used for lower electricity consumption and a compact ring size. An average ring radius of 5.5 m is chosen at the central orbit so that the maximum magnetic field strength in the SC magnets can be modest.

Table 2: Parameters of the ERIT ring for muon production.

Energy 600 MeV/u
Magnetic rigidity 8.126 TM
Lattice FDF
Average radius 5.5 m
Magnetic field (F) 4.016 T
Magnetic field (D) 3.509
Opening angle (F) 0.2032 rad
Opening angle (D) 0.1432 rad
Geometrical field index 2.4
F/D ratio 1.1
Cell tune: V 0.180
Betatron tune (H) 0.2188/cell
Betatron tune (V) 0.1797/cell
Curvature (F) 2.023 m
Curvature (D) 2.316 m

The betatron tunes per cell are 0.22 and 0.18 in horizontal and vertical directions, respectively. The averaged beta functions over the ring are about 3.1 m and 3.8 m in horizontal and vertical directions, respectively. The dynamic aperture of the ring is greater than \( 2\times 10^{-2} \) m.rad for both horizontal and vertical directions, which is large enough for the requirement described above.

The multi rf cavities for recovering the energy lost in the deuterium gas target are placed in each straight section and the total rf voltage should be more than 20 MV when the rf harmonic number is two. The frequency of the rf cavity is 13.8 MHz.

The maximum beam intensity limited by space charge effects can be estimated by the following Laslett tune shift formula  [11]:

\[ N=-\frac {2\pi \beta ^2 \gamma ^3 B_f\Delta Q}{r_dF}\,. \]

Here, \( \Delta Q \) is the betatron tune shift caused by the space charge effect, \( N \) is the number of particles per ring, \( \epsilon \) is the beam emittance (m.rad), \( r_d \) is the classical deuteron radius (m), \( F \) is the form factor and \( B_f \) is the bunching factor, \( \beta \) is the relativistic velocity and \( \gamma \) is the Lorentz factor. Assuming \( \Delta Q \)=-0.25, \( \epsilon =6\times 10^{-5} \) m.rad, \( F=1 \), \( B_f=0.3 \), \( r_d=7.7\times 10^{-19} \) m, \( \beta =0.79 \) and \( \gamma =1.64 \), then

\[ N=1.0\times 10^{14}\,. \]

This is well above the beam intensity of \( 7.9\times 10^{11} \) deuterons per ring requested for the production of negative muons, with \( 1\times 10^{16}\,\mu ^- \)/s in this ring.

Beam injection is one of the issues in this scheme. It is favorable to have an injection scheme with continuous beam to avoid sacrificing beam duty factor. Continuous beam injection could be possible with a charge exchange injection scheme using a thin foil. In this case, however, the injected particle should not be a negative deuterium ion but rather a neutral deuterium atom. This is because the ring is fully occupied by a high pressure deuterium gas that would strip the electrons from negative ions. Neutral deuterium atoms can be stripped to deuterons with relatively higher deuterium gas pressure. However, the beam injection area allocated to the stripping foil should be evacuated to 0.1 atm or less to avoid electron stripping before the foil. An efficient differential gas pumping system at the injection region is essential to prevent the injected neutral deuterium beam from gas stripping and scattering. Obviously, a more detailed design for the beam injection is definitely necessary including new ideas.

Summary

New schemes with energy recovery internal target (ERIT) have been proposed for efficient production of negative muons for various applications including reduction of long lived nuclear waste through muon nuclear transformations.

One of the schemes consists of the MERIT (Multiplex Energy Recovery Internal Target) ring with a hadron beam and wedge-shaped lithium target, which could accelerate and store beam in the ring. In this scheme, it is expected that a muon yield of \( 10^{16}\,\mu ^- \)/s can be achieved.

A second scheme is to use an ERIT ring with gas target filling the beam pipe, which could also produce a muon yield of \( 10^{16}\,\mu ^- \)/s with a 600 MeV/u and 10 mA deuteron beam with a 1 atm deuterium gas target. Both schemes seem to be feasible using a scaling type of FFA which could provide the large beam acceptance needed in transverse and longitudinal directions.

References
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