2.1 Layout of the target area

In general, a high-energy proton beam bombards a carbon target to produce a large number of pions. A pion soon decays to a corresponding charge-state muon and a neutrino. Given that negative pions are mostly captured by target nuclei, the negative-muon yield is much lower than that of the positive muon yield. Therefore, the positive muons are used in multi-purpose studies based on the μSR (muon-Spin Resonance, Relaxation, and Rotation) method. Usually, positive muons are classified into three categories: decay muons, cloud muons, and surface muons. Decay muons are produced from pions that decay in flight. Low-energy pions soon decay to muons in the region of the target and form so-called cloud muons. Surface muons refer to the muons produced from the decay associated with stopping pions near the surface area of the target, as shown in Fig.2.

Fig.2.Full-screen View

Surface muons produced from the target

Apart from the commonly used graphite target, other materials such as beryllium, copper, and tungsten can also be used to produce pions and muons. For example, the muon sources at KEK and TRIUMF once employed a beryllium target [13]. According to previous research [12], the yield advantage of the high-Z target is not significant, but the radiation in the vicinity of the target is substantial. In our design, a carbon target is planned with a size of 10 cm × 5 cm × 1.5 cm. A 5-kW proton beam bombards the thick carbon target and the surviving proton beam is collimated and then delivered to a beam dump. The surface muons will then be collected at 0.6 m from the lateral side. Figure 3 shows the carbon target and geometry of the target area.

Fig.3.Full-screen View

Target area scheme for the simplified design of EMuS.

2.2 Muon distribution at entrance and parameters

To obtain more realistic initial beam parameters for the beamline design, FLUKA [14] is employed to simulate the particle interactions and transport processes in the target. The muon phase distributions at the lateral side of the target are scored. For surface muon applications, the beamline only needs to transport a muon beam with a narrow momentum spread. This is beneficial in reducing the proportion of cloud muons, which effectively promotes beam polarization. Fig.4 illustrates the initial phase map of the muon with momentum in the range between 26 ~ 29.8 MeV/c at 0.4 m from the center of the target. The center of the momentum range is 28 MeV/c, while the momentum dispersion is approximately 14%. The surface muon will be collected by the first doublet at 0.6 m from the target.

Fig.4.Full-screen View

Initial phase map and emittance ellipses (dashed line) at 0.4 m from the target

Similar to the beamlines at PSI and ISIS, a beamline with a large transverse acceptance is expected to generate the most intensive muon flux from the low proton beam power of 5 kW at EMuS. In general, with a muon-beamline of physical aperture of approximately 400 mm, the lattice based on room-temperature quadrupoles has an acceptable maximal divergence of approximately 120 mrad. Under this upper limit, phase-space ellipses are selected based on the density distribution of muons in the x-x’ or y-y’ phase planes. According to the previous research [12], considering an expected muon intensity at the sample, a beam emittance of 5200 (horizontal) × 3000 (vertical) π∙mm∙mrad at 0.4 meters from the right lateral side of the target would be reasonable as an acceptable muon flux. According to the ellipse emittances and maximal divergence, the Twiss parameters could be calculated using Formula 1 where βγ - α² = 1. In Fig.4, the muon density distribution and emittance ellipses are presented. The Twiss parameters are listed in Table 2.

3.1 Optics and multi-particle simulation

A simplified surface muon beamline with a length of approximately 10 m will facilitate the achievement of several goals with regards to surface muons: 1) collection from the target; 2) transmitting from the target to the experimental terminal; 3) focusing and collimating to a small beam spot; 4) Suppression of the in-beam background. Three doublets are used for the collection and transmission of surface muons. Two dipoles are used to obtain a small momentum acceptance and to sweep the neutral particles. A triplet is used to focus the spot focus. A Wien filter is used to sweep out in-beam positrons. The overall lattice structure of the beamline is shown in Fig.5.

Fig.5.Full-screen View

Layout of the baby-scheme muon beamline

The quadrupoles of the doublets are all at room-temperature with an aperture of 400 mm. In addition, the distance between the quadrupoles in a doublet is 0.15 m. Two rectangular bend magnets are used to bend the beam and define the beam momentum spread. Rectangular bend magnets with a deflection angle of 30 degrees are employed. A Wien Filter with a short length and strong field strength is placed before the triplet to sweep off in-beam positrons. The specific deflection angle can be calculated using Formula 2a and 2b [3] as follows:

where φ_E and φ_B are the deflection angles of the magnetic field and electric field respectively, and $l$ is the length of the field (hard boundary), B and E are the field strength, while p is the momentum of the particles. The units are in parentheses. Apparently, a strong magnetic field with a long length will reduce the beam polarization because it can facilitate both deflection and spin rotation. A compromise between the polarization reduction and positron background level is considered. Based on the initial beamline parameters that were previously mentioned, the optical calculation was started from 0.4 m from the carbon target. Fig.6 shows the horizontal and vertical linear optical envelopes of the beamline using graphic TRANSPORT [15].

Fig.6.Full-screen View

Linear envelope calculated by TRANSPORT

Following the optimal optical design, G4beamline [16] is used to perform the multi-particle simulation for this beamline. In the case of no collimation, the optical parameters are used directly for the multi-particle simulation. The beam spot at the sample position is much larger in the vertical direction than that of the horizontal direction as shown in Fig.7a, because of nonlinear effects. Hence the triplet is optimally retuned to achieve a better spot both in terms of shape and size. Given that the distribution of the actual surface muons (26 ~ 29.8 MeV/c) is almost uniform, a uniform distribution of primary muons is employed in this simulation that is defined by the acceptance phase ellipses, for better statistics in G4beamline. In this case, the transmission efficiency is approximately 88 % based on a comparison of the particles in the beam spot with those at the initial position. A total of 12 % of the total muons are lost along the beamline, as shown in Fig.8. More than half of them are lost at the first doublet. The main component parameters of the beamline are given in Table 3.

Fig.7.Full-screen View

(a) Graph of beam spot when using the triplet given by TRANSPORT; (b) Graph of beam spot when using a retuned triplet

Fig.8.Full-screen View

Beam loss and remaining along the transport line

3.2 Beam collimation

Due to the large emittance of the secondary muon beam and the limitation of the number of components, it is difficult to implement an adjustable narrow momentum spread and small beam spot for different experiments. In our design, the beam collimation in the beamline is employed to reduce the background and to control the beam spot. Initially, we simulate the nuclear reaction process of proton bombardment on the carbon target by employing FLUKA. All the produced muons, positrons and other particles are collected before the entrance of beamline and then transported in the lattice of the beamline as a primary beam.

(1) Momentum collimation

Surface muons emitted from the target have the same momentum of approximately 29.8 MeV/c if they are not moderated. To obtain higher polarization of the muon beam, it is important to minimize the decay or cloud muons. Given that muons with momentum larger than 29.8 MeV/c must be cloud muons, we employ a momentum collimator to intercept them. The normalized dispersion function is considered to identify the suitable positions for momentum collimation,

where D_d is the horizontal normalized dispersion function, α and β are the Twiss parameters, and D_x is the horizontal dispersion function.

The normalized dispersion function along the beamline is shown in Fig.9a. After the first dipole, there is a maximum value which is the best collimating position. The μ⁺ with momentum larger than 29.8 MeV/c must be cloud muon and will lead to a decrease in polarization. At this position, a portion of the cloud muons can be removed using a collimator placed at approximately x=70 mm as shown in Fig.9b. In this case, there is still approximately 10 % of the cloud muons remaining in the beam. Considering the requirements of the μSR experiments, an accepted polarization of 92% is achieved.

Fig.9.Full-screen View

(a) Normalized dispersion function along this beamline; (b) x-x’ phase map of muons with different momentums at maximum normalized dispersion

2) Removal of positrons

The μSR technique gives the magnetic characteristics of the sample by measuring the information of the outgoing positrons that decay when muons are stopped in the material. Usually, positrons from the target are the dominant in-beam background particles for a muon source and must be minimized. In this case, we employ a Wien filter to deflect out the positrons that have the same momentum as the reference muons. Avoiding vital reduction of the muon beam polarization, the length of the Wien Filter is chosen as 0.6 m. Moreover, an additional collimator (collimation for x ≤ 50 mm) at the entrance of the down-streaming quadrupole Q7 helps to block the biased positions as shown in Fig.10.

Fig.10.Full-screen View

Positron and muon distributions at the entrance of Q7 and the collimator (shadowed part).

3) Small beam spot at the experimental sample

In general, given that the size of the μSR experiment sample is small, the size of the beam spot should be controlled within approximately Ø30 mm (FWHM). If no collimation is applied, the beam spot and profiles tend to be large and is given by Fig.7. Collimation using slits is used to reduce the beam spot size. The transfer matrix between two points $s_2$ and $s_1$ in the beamline could be expressed as Eq. (4) [17],

where α, β are the Twiss parameters and $\psi$ is the phase function. In addition, the transport equation between two points is expressed as follows:

Thus $x_2$ and $x_1$ will be multiples if the element ${\text{M}}_{12}$ is equal to zero. This means that $\sin \psi =0$, that is,

In this beamline, ψ = π would be a good choice for collimation. Fig.11 shows the phase shift in the x-direction and y-direction for the entire beamline. For a uniform distribution, the phase shift is 180 degrees from z = 6620 mm to the final spot in the x-direction and z = 3970 mm in the y-direction.

Fig.11.Full-screen View

Phase shift of the uniform-distribution beam within the nominal acceptance

Considering nonlinear effects and simulated distributions from the target, these positions will be slightly adjusted. Three groups of slits are placed at different positions along the z-axis as shown in Table 4. With collimation of three slits, beam spot and profiles are shown in Fig. 12. The shape of the beam spot is an approximate square and the FWHM values are approximately 30 mm. The flux within the FWHM (30 mm × 30 mm) is approximately 1.2×10⁵ μ⁺/s. The momentum is slightly larger than 10%.

Fig.12.Full-screen View

Resultantx beam spot and profiles with slits. (a) Profile at y = 0; (b) Profile at x = 0; (c) Beam spot; (d) Momentum spread.

Considering the different cases of collimation in the beamline, several backgrounds are shown in Table 5. Other particles, especially positrons, are successfully swept off. The low background meets the requirement of the μSR experiments.