2.1 Model description

Figure 1 shows the schematics of the μSR prototype spectrometer built in musrSim. In the upper part of Fig. 1, the two-ring structure comprises two detector rings with 64 detectors per ring, degraders, a cryostat, and a beam pipe. The new four-ring structure (bottom) changed the cryostat to a close cycle refrigeration (CCR) system with cruciform similar to that of the European Muon (EMU) spectrometer [19]. The four-ring spectrometer has 32 detectors per ring, resulting in a reduction of the radius of the detector ring from 150 mm to 90 mm. The arrangement of detectors in this four-ring structure is similar to that of the EMU spectrometer. Table 1 presents the main parameters of these two models. The optimal size of the degraders will be described in Sect. 3. Note that the detector ring near the sample is assigned as the inner ring. Five million muons were simulated in each run to obtain good statistical precision. The sample size used in the ISIS muon facility generally ranges from 10 mm to 50 mm (in both horizontal and vertical directions). Therefore, we set the size of the sample to Φ30 mm. The parameters of the EMuS beam are listed in Table 1. The size of the muon beam spot on the sample is optimized to 30×30 mm² in full width at half maximum (FWHM) accordingly. A small beam spot can be achieved by modifying slit settings if required in a specific experiment.

Fig. 1Full-screen View

Illustrations (cutting view) of a 128-channel μSR prototype spectrometer. (a): upper: two-ring structure (Φ300-mm detector ring), bottom: four-ring structure (Φ180-mm detector ring); (b): projection of detector rings along the z-axis.

Figure 2 shows the RMS envelope of the EMuS muon beam. Solid lines are provided by the accelerator group of EMuS at CSNS, which is simulated using g4bl (G4Beamline) [20]. The spatial and angular distributions of the beam are set up in musrSim consistent with g4bl simulations (dots in Fig. 2).

Fig. 2Full-screen View

RMS envelope of the EMuS muon beam with a spot size of 30 × 30 mm².

2.2 Asymmetry of a μSR spectrometer

For a bunch of surface muons, their decay positrons emit asymmetrically but preferentially along the polarization. The angular distribution [21] of positrons is

where A denotes the asymmetry, and x is the ratio of the positron’s kinetic energy to the maximum energy 52.8 MeV. The asymmetry A(x) is the asymmetry for a well-defined positron energy x. In an experiment, the observable asymmetry of a spectrometer is larger than 0 integrating over all the measured positron energies. A higher statistical precision can be achieved if a spectrometer has a high asymmetry. According to Eq. (1), high-energy positrons (x > 0.5) have a positive decay asymmetry, and their emission angles with respect to the muon spin become smaller. Degraders can be used to block low-energy positrons (x < 0.5) to increase the observable asymmetry of the spectrometer.

For a ZF / TF (zero field / transverse field) case, the asymmetry is determined by

where the "plus sign" indicates the muon polarization parallel to the forward detector ring, and the "negative sign" indicates the anti-parallel case; B and F denote the detector rings in the downstream or upstream direction along the beam direction, respectively. For a TF simulation, muons are precessing in a transverse field applied in the vertical direction in the sample area, and the asymmetry can be derived by directly fitting μSR "spectra" with

where N₀ is the initial muon count, τ_μ is the lifetime of the muon, ω is the angular frequency dependent on the gyromagnetic ratio of the muon and the strength of the field, ϕ₀ is the initial phase, and B is the background.

2.3 Energy threshold of a detector

As the scintillator is thin (5 mm), over 99% positrons penetrate the detector. The energy loss spectrum is given by the Landau distribution as shown in Fig. 3. The energy threshold can be set to 0.8 MeV to discriminate events with lower energy deposition. These events are due to low-energy positron events, gamma events, or PM tube noise.

Fig. 3Full-screen View

Energy deposition spectrum in a detector module

3.1 Validation of the simulation model

A detector module of the two-ring structure was tested on the EMU spectrometer at the ISIS Muon Source. During the test, an EMU detector module (consisting of three segments) was replaced by the Chinese detector as shown in Fig. 4 (a). Fig. 4 (b) is the result obtained from the experiment. The asymmetry is fitted by 0.2540±0.0017. The single Chinese detector was also modeled in musrSim. Here, 10⁷ muons were simulated for this specific experiment. The polarization of the ISIS muons is over 99%. Therefore, the polarization of the muons in the simulation is set to 100%. The simulated asymmetry is fitted by 0.2547±0.0073 (Eq. (4)), which is consistent with the experiment. Details of optimization of this two-ring spectrometer are described in [22].

Fig. 4Full-screen View

(a): Location of the Chinese detector, (b): asymmetry of the TF test

3.2 Influence of degrader materials on the asymmetry

Four types of metals were chosen as degrader materials as shown in Fig. 5. The asymmetry has a peak value when the degrader thickness changes. According to Eq. (1), the asymmetry has the range [-0.33, 1] and positrons with kinetic energy = 26.4 MeV have zero asymmetries. When the thickness of the degrader is small, low-energy positrons can be blocked, resulting in an increase in asymmetry. Thereafter, the asymmetry drops. This is because multiple scattering changes the angular distribution of positrons if the degrader becomes thicker. The similarity of peak values in Fig. 5 indicates that the gain effects of different materials are almost the same. Lead is soft and toxic. Tungsten is more expensive than brass. Aluminum requires more material to obtain the same asymmetry as that of brass. From the simulation, brass has slightly higher asymmetry. Therefore, brass was set as the degrader material in subsequent simulations.

Fig. 5Full-screen View

Variation of simulated asymmetry with degrader thickness

3.3 Optimizing the two-ring structure

As mentioned in Sect. 2.1, a cryostat is used in the two-ring structure as the sample environment. The sample is fixed on a large sample plate (made of silver). As the beam spot is large, samples with sizes of Φ30 mm and Φ40 mm are simulated. As shown in Fig. 6(a), approximately 80% and 60% detected positrons are decayed from the Φ40-mm and Φ30-mm samples, respectively. Reduced positron events from the sample are transferred into the sample plate. Positrons decayed from outside the sample will be detected by the spectrometer as background. To eliminate the influences of these backgrounds, the cryostat can be updated to a fly-past structure, and a small silver sample plate can be used. The fly-past structure can allow muons outside the sample through the spectrometer. It is beneficial to small samples [19] depending on the beam divergence. Silver has no electronic moment and has very small nuclear moments. Background positrons from the silver plate can be separated from any relaxing signals from a sample as the background is non-relaxing. However, for very low sample depolarization rate comparable to that of silver, the two components cannot be separated. Hence, measurements at very low sample depolarization rate are limited. Increasing the percentage of muons decaying inside the sample is important for reducing the background. Percentages of decayed positrons have no significant differences in the ZF (a) and TF (b) conditions.

Fig. 6Full-screen View

Variation of percentages of detected positrons decayed from different parts (a, b) and asymmetries (c, d) with the brass degrader thickness. Panels (a, c) are for ZF, and (b, d) are for TF (100 Gauss).

The asymmetry with or without a cryostat in ZF / TF was calculated in Fig. 6 (c / d). The cryostat is a three-layered structure (each layer is a 1.5-mm-thick aluminum layer) [23]. Windows on the cryostat with an aperture of 55 mm are made of Mylar film (total thickness of 25 mm). For a certain sample size, the asymmetry with a cryostat is higher than that without one. This is because the cryostat can be considered a degrader, which can block low-energy positrons with low asymmetries. This is because, as the position of the sample does not change, asymmetries related to different sizes are approximately the same. However, if the beam emittance is large, muons passing the sample may hit a degrader, resulting in a reduction of the asymmetry. The asymmetry in the TF condition (d) is smaller than that in the ZF condition (c). The reason is that the spin rotates in the TF condition, making the average emission symmetrical axis tilt to the z-axis. According to the simulation, the highest asymmetry obtained by this two-ring structure is 0.28 with 10-mm-thick brass degraders.

For a μSR spectrometer with FPGA readout electronics, each detector detects approximately 10 events/pulse, which is tolerable to the distortion of positron time spectra [19]. Hence, the acceptance of the beam for the 128-channel spectrometer is 128 × 10 × Repetition rate / Detection efficiency / Beam intensity. Table 2 lists the detection efficiency and acceptance for two sample sizes. Less than one-fifth of the beam can be accepted by the spectrometer. If the two-ring spectrometer is operated under this beam, a pile-up of signals will lead to significant distortions of μSR spectra, which cannot be corrected. The two-ring spectrometer must be updated to a four-ring one to obtain a higher asymmetry and accept the full beam intensity.

3.4 Scanning the asymmetry along the z-axis

Fig. 7 shows the projection of the detector rings in the Y-Z plane. According to Eq. (1), the asymmetry of these two rings can be calculated by

Fig. 7Full-screen View

Projection of detector rings along the spin. The red circle is the angular distribution of positrons averaged over all the positron energies.

where $\overline{A}$ is the asymmetry for an infinitesimally small solid angle (ϴ₁≈ϴ₂≈0) averaged over all the positron energies accepted by the spectrometer. Without degraders, i.e., accepting positron energies from close to zero up to the maximum energy, the average asymmetry $\overline{A}$ = 0.33. With a degrader, the minimum accepted positron energy will be considerably larger than 0, resulting in $\overline{A}$ > 0.33. The formula indicates that, if the radius is small or the detector ring is far away from the sample, indicating that ϴ₁ or ϴ₂ is small, high asymmetry can be achieved. It also indicates that the asymmetry cannot be higher than $\overline{A}$.

Simulations of scanning the asymmetry along the z-axis were performed with the detector length of 10 mm (radius of the detector ring = 150 mm or 90 mm) to verify the relation between the asymmetry and location. For these specific simulations, 10⁷ muons were simulated to suppress statistical fluctuations. Ideally, a virtual detector with zero length can exactly detect the asymmetry in each point along the z-axis according to Eq. (5). However, we cannot determine the energy deposition used to confirm whether the deposited event is a muon-decay event. A detector with a length of 10 mm is sufficiently small to detect the asymmetry in its location. For the two-ring structure, each ring has 64 scintillators (see Table 1). The width of each segment is 13.5 mm, resulting in the detector ring with a radius of 150 mm. For the four-ring structure, each ring has 32 detectors. Similar to the fly-past device in EMU, the radius of the cruciform (see Fig. 1) is approximately 80 mm. Detectors should be placed outside the cruciform or degraders. Therefore, the minimum radius of the detector ring that can be chosen is 90 mm. Fig. 8 shows scanning of the asymmetry using degraders with different thicknesses along the z-axis in the ZF. Theoretical results were calculated using Eq. (5). The simulation results in Fig. 8 are consistent with their corresponding theoretical ones. The space occupied by the spectrometer is in the region (-350, 350) mm along the z-axis. In this region, detectors that use thicker degraders can achieve higher asymmetries. When detectors are further away from the sample, higher asymmetries can be achieved but the asymmetry increases slowly after z = 150 mm. Comparing detector rings with different radii but the same degrader thickness, asymmetries increase quickly with a smaller radius (90 mm). Furthermore, detectors with smaller radius can achieve higher asymmetries in the interested region along the z-axis. To achieve high asymmetries, the two-ring spectrometer should be updated to a four-ring one. According to the analysis, a detector ring can be placed after 150 mm to achieve the highest asymmetry and another detector ring can be mounted before 150 mm to cover more solid angle for the four-ring spectrometer. Therefore, the outer detector ring with a length of 100 mm is set up after 150 mm. To ensure that the inner and outer rings have the same covered solid angle, the length of the inner ring is set to 40 mm.

Fig. 8Full-screen View

Asymmetries of 10-mm-long detector rings (radius = 90 mm or 150 mm) using brass with different thicknesses in varied locations. Curves are calculated using Eq. (5).

3.5 Optimizing the four-ring structure

As discussed in Sect. 3.3, the sample environment should use a fly-past structure. Hence, the diameter of the sample plate is the same as that of the sample. Muons outside the sample can pass through the cruciform without any obstructions. Samples with four sizes were simulated to determine the influences of the sample size and the beam emittance on the asymmetry. Fig. 9 shows the percentages of positrons decayed from the sample with different sizes. If the diameter of a sample is not smaller than Φ20 mm, over 93% positrons are decayed from the sample. Accordingly, the largest sample (Φ30 mm) has the highest asymmetry (approximately 0.42) and the asymmetry does not change significantly for a reduction in the sample size from Φ30 mm to Φ20 mm as shown in Fig. 10(a). If a sample is small (Φ15 mm or Φ10 mm), the percentage of positrons from the sample and the observable asymmetry reduce significantly. This is because the angular dispersion of the real beam is relatively large for small samples. Muons passing through the sample could hit the inner wall of the cruciform or degrader in the right side of the sample as shown in the bottom part of Fig. 1. However, asymmetries for samples of different sizes are almost the same in the TF condition (Fig. 10(b)). Asymmetry of the inner or outer ring is also plotted in Fig. 10(c, d). The outer ring has higher asymmetry than the inner ring (up to ~13% in ZF and ~18% in TF). This is because high-energy positrons emit preferentially along the z-axis, indicating that the emission angle of these high-asymmetry positrons is small. The angle of the outer ring pointing to the sample is smaller than that of the inner ring. Therefore, the outer detectors can achieve higher asymmetries. From these simulations, the optimal degrader thickness is 10 mm.

Fig. 9Full-screen View

Percentage of positrons decayed from samples with four sizes in ZF

Fig. 10Full-screen View

Observable asymmetry as a function of the degrader thickness in ZF (a, c) and TF (b, d). Curves are eye-guide lines.

Table 3 shows the detection efficiency and mean count rate per pulse per detector. From Fig. 10 and Table 3, it can be observed that the four-ring structure can achieve the highest asymmetry (~0.42) and also accept a relatively high muon beam intensity.