The loss mechanisms in SFHe

For He-II, it is possible to formulate the expression for the ultracold neutron surviving time in terms of various loss mechanisms [37-39]:(1)Here, the reciprocal of τ represents the UCN total loss rate. is influenced by the UCN upscattering effect in He-II. arises from the substantial neutron absorption cross section of ³He. When a UCN encounters a wall coated with a material of high Fermi potential, losses may occur due to wall absorption and upscattering (). The parameter τβ = 880 s represents the neutron decay time. is the UCN leakage from gaps of UCN valve and storage volume wall. τ_impuritie is the loss caused by absorption and up-scattering of impurities which deposit on the walls of the production volume [39]. While MCUCN includes wall collisions, leakage and decay models, it lacks crucial additional loss mechanisms.

The upscattering rate of superfluid ⁴He τ_up is temperature dependent and can be divided into three parts [40]: one-phonon absorption, two-phonon scattering and roton-phonon scattering. H. Yoshiki summarized the formula for τ_up [41] as:(2)where the first term originates from one-phonon absorption, the second from two-phonon scattering, and the third from roton-phonon scattering. Thus, the upscattering time constant is related to the He-II temperature. According to measurements in Ref. [42], A=61.1 s^-1, B=7.6×10^-3 s^-1 K^-7, and C=5.22 s^-1 K^-7 for 1.6 K He-II, resulting in τ_up=3.4 s. For T = 0.7 K, A=130 s^-1 [40, 43], B=9.0×0-3 s^-1 K^-7, C=18 s^-1 K^-7, leading to τ_up=1247 s. The UCN upscattering loss rate in He-II has been experimentally confirmed in Refs. [40, 44] and plays a significant role at higher temperatures but a lesser role at lower temperatures.

According to Ref. [38], the wall loss time constant τ_wall is defined as:(3)where is the average loss probability per bounce, dependent on UCN energy. It’s noteworthy that higher energy UCNs are more likely to be lost compared to lower energy ones. Here, V represents the converter volume, S is its surface area, and v denotes the neutron velocity in vacuum.

The absorption time constant τ_abs is given by:(4)This term remains independent of UCN velocity due to the 1/v principle of the ³He thermal neutron absorption cross section σa. σa is 5333 b at a velocity of 2200 m/s. The notation denotes the number density of ³He atoms. The natural abundance ratio of to , at 1.37 × 10^-6, corresponds to 3×10¹⁶ cm^-3 at 1 K and an absorption time constant of 0.028 s. This parameter remains crucial in the total loss rate calculation until the abundance of ³He is reduced to below 10^-12.

The leakage time constant τ_leakage is defined as:(5)In comparison with τ_wall, this term corresponds to wall loss with μ = 1, where S_leakage represents the total area of gaps that can leak UCNs.

Initial Energy Distribution in SFHe

The MCUCN code incorporates uniform and Gaussian energy distributions but does not include other forms of energy spectrum distribution. To simulate the SFHe UCN source, we integrated a Monte Carlo sampling function to generate the energy distribution that accurately represents the superfluid helium UCN source.

The relationship between the real space UCN density n with energy E in the range (E, ) and the phase space density ρ is [38](6)where is the phase space volume and is the real space volume. N is the UCN number. When produced in SFHe, the UCNs fill the phase space with a constant density of ρ. Then(7)Thus the initial UCN spectrum in SFHe converter is(8)

The iMCUCN code

As depicted in Fig. 1, the iMCUCN code models UCN losses during transportation. The red sections delineate mechanisms for (a) generating the energy distribution of the UCN source in SFHe, (b) simulating the upscattering effect of ⁴He, (c) simulating absorption by residual ³He contamination, (d) simulating absorption and upscattering by impurities, (e) simulating UCN transmission through materials with negative optical potential. Polypropylene (PP) possesses a negative optical potential and serves as the foil confining SFHe [45]. The code MCUCN incorporates transmission physics through materials with positive optical potential but overlooks those with negative potential. Thus it is necessary to integrate the negative potential for PP foil in next section. The iMCUCN code also integrates geometry descriptions to simulate various volumes and includes a visualization component. UCNs are considered lost upon encountering the detector, the heat exchanger, or other components.

Fig. 1Full-screen View

Illustration of the flow of physical models in the iMCUCN code [36]. Diamonds indicate the physical models utilized for selecting a loss mechanism or going to exit window during transportation. Black characters represent original code sections, whereas our additions are highlighted in red

Figure 1 showcases the iMCUCN’s physical model, symbolized by diamonds [36]. Absorption may result in the loss of UCNs produced in the deuterium converter, yet the absorption rate within the superfluid ⁴He UCN source is zero. Neutrons undergo continuous decay during simulations, with ⁴He upscattering effects and ³He absorption occurring solely in He-II. Impurities are identified on the wall of converter. UCNs encountering surfaces of the detector, heat exchanger, and other components are either lost or detected. Upon collision with a wall surface, UCNs will pass through if the wall is virtual, or they may be reflected, absorbed, or transmit through the surface. Reflections manifest as either diffuse or specular, allowing reflected UCNs to persist and embark on subsequent collision. Surface absorption arises from both absorption and inelastic scattering by the wall. The simulation concludes for a UCN upon reaching the exit window, i.e., the detector, marking the end of its journey and prompting the initiation of a new cycle with the generation of another UCN in the converter.

Basic conditions

According to Ref. [45], an overview of the horizontal extraction geometry is shown in Fig. 2. 10,000 UCNs with energy below neV are produced isotropically and uniformly in a 40 L He cylinder converter. As mentioned above, τ_up = 3.4 s for 1.6 K He-II. When propagating through PP foil, the UCN transmission loss attributable to neutron elastic scattering off density inhomogeneities within the bulk is contingent upon both the macroscopic cross-section and the PP foil thickness [46, 47]. With an elastic scattering mean-free-path, μm, for inhomogeneities, and the employed foil’s thickness, d = 30 μm, the value of was utilized as an input for the iMCUCN code.

Fig. 2Full-screen View

(Color online) The horizontal near-foil geometry of Lujan center Mark3 UCN transportation system [45]

The detector will count the arrival UCNs that are lower than [45]. The input parameters in the simulation are listed in Table 1. The angular distribution for diffuse reflection follows the Lambertian model [38]. Gravity, loss due to wall absorption, upscattering effect in superfluid helium, and decay during UCN storage and transportation are included.

The energy distribution in SFHe

When UCNs are produced by inelastic scattering of 8.9 Å neutrons in superfluid helium, their angular distribution has a 6% asymmetry [48]. This real but short-lived physical process can be disregarded, so the Monte Carlo method uniformly and isotropically generates the UCNs in He-II. UCN energy distribution in He-II in iMCUCN correctly follows Eq. 8, as shown in Fig. 3. It also presents the uniform energy distribution in MCUCN.

Fig. 3Full-screen View

(Color online) The normalized source UCN spectrum at different positions in the transport system. The red line stands for the formal expression . The green line is the UCN spectrum distribution in the UCN converter. The blue line is the uniform UCN source distribution in MCUCN

The geometry of UCN transport system

Extending the geometry code in MCUCN to accommodate various experiments or designs is crucial. Utilizing the developed Python code, we can assess the accuracy of the constructed geometry and identify any potential gaps. Figure 4(a), (b), and (c) depict the results of the simulations, indicating the points where the UCNs collide with the wall of the transport system. We projected the results into XOY, XOZ, and YOZ planes to indicate whether the geometry is the one we aim to construct and to know if there are some gaps. Figure 4(d) to (f) provide the trajectories of UCNs on XOY, XOZ, and YOZ planes in the extraction process. The horizontal near-foil extraction geometry reveals no gaps, confirming the accuracy of the geometry model.

Fig. 4Full-screen View

(Color online) Figures (a) to (c) show the collision points when UCNs collide with the geometry during the transport process. Figures (d) to (f) show the trajectories of the first 100 UCNs

Comparison of the transmission rates simulated by MCUCN, iMCUCN, and UCNtransport

Tables 2 through 3 delineate the simulated outcomes by UCNtransport, MCUCN, and iMCUCN. Table 2 details the optimization process for the horizontal near-foil extraction system, changing simulation parameters based on the preceding step. A marked discrepancy in transmission rates is observed between MCUCN and UCNtransport simulations, whereas the variance between iMCUCN and UCNtransport remains under 10%. This underscores the critical role of the upscattering mechanism in iMCUCN for accurately simulating the SFHe UCN source at 1.6 K.

Furthermore, the absence of a transmission model for materials with negative optical potential in MCUCN hinders the accurate inclusion of PP foil in simulations. Per Table 2, transitioning from "switch foil from Al to ideal PP" to "Add PP elastic scattering" does not yield changes in MCUCN simulations.

In the optimization steps detailed in Table 2, incorporating diffuse reflection within the UCN converter assists in extricating UCNs from long-lived orbits in symmetric geometries. Adding the same diffusion rate in the heat exchanger volume limits the increase in the transmission rate, as UCNs have the potential to escape from the heat exchanger. Changing Al foil to PP foil increases the rate by 8% because low-energy UCN can pass the foil. Considering the elastic scattering in the PP foil, 72% of UCNs pass through it in iMCUCN simulations, which is close to the 68% obtained by UCNtransport. "Add PP foil support grid loss" refers to the necessity of providing a support for the foil confining superfluid ⁴He in practical applications, which results in a neutron transmission rate of 90%. The converter is located near the spallation target and requires a thick shield. Therefore, a 4 m long UCN transport guide should be installed at the end of the system in Fig. 2, accompanied by an extraction rate of 80%.

In Table 3, the extraction rates of UCN increase with the height of the heat exchanger column when P_diffuse = 50% on its side walls. The results simulated by MCUCN are more than two times those simulated by iMCUCN. However, the differences between iMCUCN and UCNtransport at various column heights are less than 10%.

In conclusion, the simulation results from iMCUCN and UCNtransport are in agreement.

The experimental configuration and parameters

Figure 5 illustrates the experimental configuration. The ⁴He converter exhibits a rectangular geometry measuring 100 cm × 7 cm × 7 cm, with Be coated on the front and rear faces and BeO used for the sides. The vertically oriented extraction guide is positioned at a height of 10 cm, with the shutter situated midway along its length. Following this, there is a 7 cm horizontal bend leading to a second vertical guide extending for 15 cm, ultimately leading to the detector at its end. Constructed from stainless steel, the guides have a diameter of 23 mm. The gap area at the shutter is 1.73 mm². Refer to Table 4 for detailed information on various input parameters.

Fig. 5Full-screen View

The geometry of SUN1 [39, 49]

The initial UCN spectrum follows Eq. 8, and the maximum energy is 261 neV decided by the optical potential of BeO. The measured UCN count rate at the detector of the SUN1 system is depicted in Fig. 6. During a scenario where cold neutrons are incident while the shutter is closed, the detector detects UCNs, attributed to small gap between shutter and guide. The UCN count rates at the detector reached a saturation point after approximately 200 s. At around 800 s, the cold neutron beam was deactivated, coinciding with the shutter’s opening to the UCN guide. A peak in the UCN count rate at the detector was observed after about 3 s. Subsequently, the shutter was closed after 135 s.

Fig. 6Full-screen View

(Color online) SUN1 experimental results at detector. The red lines are fitted [39, 49]

Scenario before opening shutter

Before opening the shutter, the UCN number in the converter as a function of time is [38]:(9)where ρ(t) is the UCN density in the converter. V is the volume of the converter. P is the UCN production rate per unit volume. τ is the buildup time, as defined in Eq. (1), representing the UCN surviving time in the SFHe converter. The UCN number can reach a saturation point when the UCN generated time is sufficiently long, and the saturation UCN number in the UCN converter will be(10)where N_sat represents the saturation UCN number in the converter.

The UCN count rate at the detector in Fig. 6 can be expressed as(11)τ_gap is determined by the gap area between the shutter and the guide before the shutter is opened. It also follows Eq. (5).

In the converter, the β decay term remains constant, and the absorption of ³He is not disregarded under this condition. Both the He-II upscattering and the abundance of ³He are independent of UCN velocity. According to Ref. [39], it was reported that , , , , . Therefore, . Both the impurities time constant and leakage time constant remain ambiguous. To simplify the simulation, we assume that the losses caused by impurities and leakage are energy-dependent, similar to wall loss.

The leakage count rate at the detector is proportional to the area of the gap between the shutter and guide. The gap of the shutter, unspecified in Ref. [39, 49], can only be determined by comparing with simulated results through changing the size of gap. As illustrated in Fig. 7, we observed an increase in the UCN count rate before opening the shutter with increasing shutter gap size because the leakage rate is proportional to the gap area. The gap whose simulation results coincide with the experimental results before opening the shutter is determined to be 1.73 mm².

Fig. 7Full-screen View

(Color online) Fit the shutter gap

Scenario after opening of the shutter

After opening the shutter, the accumulated UCNs in the converter will be released. Neglecting gravity and the energy distribution of UCNs, and closing the shutter of cold neutron, the decay of UCNs in the converter after opening the shutter can be(12)where τ_guide is the guide leakage constant after opening the shutter. According to Eq. 11, the UCN count rate at the detector can be expressed as . In real scenarios, this process is dependent on the UCN energy in the converter. Thus the UCN number in the converter does not follow a strict one time constant decay. The UCN count rate at the detector before opening the shutter is nearly proportional to UCN total number in the converter and researchers [39, 49] utilize two time decay constants to fit the detected UCN count rate after opening the shutter. In the SUN1 experiment, the fitted two decay time constants are (13±1) s and (34±1) s.

The decay time after opening the shutter is primarily influenced by the area of the guide, which has been determined experimentally. It can also be affected by the unknown diffuse reflection probabilities of the guide (d_g) and converter (d_c). The diffuse reflection probability of polished stainless steel coated in the guide typically falls within the range of 0.5 m to 1 m. Therefore, we present the decay time constants for some results in Table 5.

The optimal condition corresponds to a converter diffuse reflection probability d_c = 3.0%, a guide diffuse reflection probability d_g = 1.5%. The final buildup time is 71.0 s, the two decay time constants are 13.7 s and 31.7 s. And the comparison between the simulated results and the measured results under this condition are shown in Fig. 8. The simulation results from iMCUCN under this condition closely align with the experimental data. Specifically, UCNs with lower energy contribute to the second decay time constant τ₂. The discrepancy ranging from 840 s to 937 s between SUN1 measurements and data simulated by iMCUCN may arise from several factors. Firstly, the UCN spectrum generated in simulation may differ from the actual conditions in the experiment. Secondly, errors may be presented in the data reading process from [39]. To clarify these discrepancies, further experiments need to be conducted.

Fig. 8Full-screen View

(Color online) The top figure compares the simulation results of the iMCUCN with the measured data under the selected conditions. The middle figure is identical to the top figure, but is presented in logarithmic coordinates. The figure below illustrates the ratio of the experimental data to the iMCUCN simulation data

Discussion

To further enhance comprehension of the underlying physics discussed in this section, additional details will be presented below.

The results obtained with d_c = 3.0% and d_g = 1.5% as simulated by MCUCN exhibit discrepancies with the experimental data depicted in the upper panel of Fig. 9. In the MCUCN simulation, the buildup time is 153.0 s, and the decay time constants are 15.3 s and 38.3 s. In the absence of losses due to impurities, leakage, absorption of ³He, and upscattering in ⁴He, the time required to reach the saturation UCN count rate is longer, allowing more UCNs to survive and be extracted. According to Eq. 3, wall loss depends on UCN energy, i.e., the lower the UCN energy, the smaller the chance of being lost after collision with the wall. The velocity of low-energy UCN is slower and takes more time to reach the detector. Therefore, the slope of the UCN count rate at detector after opening shutter simulated by MCUCN is relatively smaller than that simulated by iMCUCN in the upper panel of Fig. 9.

Fig. 9Full-screen View

(Color online) The figure compares the simulation results of the iMCUCN and MCUCN with the measured data under the optimal condition. The data simulated by the iMCUCN agrees with the measured data, while that of MCUCN shows discrepancies

Figure 10 depicts the UCN spectrum within the converter at various time points simulated by iMCUCN. The figure at the top shows the UCN spectrum in the converter simulated by iMCUCN before 800 s, depicting the change over time. Initially, at 20 s, the UCNs begin to accumulate within the converter. Approximately 300 s later, a saturation point is nearly reached, indicating that the rate of UCN accumulation has equalized with the rate of their loss. However, from 804 s, the UCN spectrum within the converter begins to decrease due to the opening of the shutter. It is noteworthy that the saturated UCN spectrum observed at 800 s does not strictly adhere to the expected distribution as described by Eq. 8. This deviation is attributed to the presence of energy-dependent loss mechanisms, including impurities, leakage, and interactions with the converter walls. The figure below displays the UCN spectrum in the converter simulated by iMCUCN after 800 s. At 810 s, a greater number of faster UCNs have been extracted compared to those with lower energy. By 900 s, the majority of the UCN have been extracted, with only a small number of low-energy UCN remaining in the converter. The optical potentials of BeO, Be, and stainless steel are 261 neV, 252 neV, and 183 neV, respectively. UCNs with energy lower than 183 neV can be more effectively constrained than those ranging from 183 neV to 252 neV, as some of them may escape from the stainless steel guide before the shutter. UCNs with energies between 252 neV and 261 neV will be less constrained due to the presence of Be surfaces.

Fig. 10Full-screen View

(Color online) The top figure illustrates the UCN spectrum in the converter simulated by iMCUCN before 800 s, showing the change over time. The bottom figure represents the spectrum after 800 s

The total UCN number within the converter can be illustrated through simulation, as visualized in Fig. 11. Notably, there exists a significant discrepancy between the MCUCN simulated total UCN number and those obtained from the iMCUCN simulation within the converter due to the lack of losses models of impurities, leakage, absorption of ³He, and upscattering in ⁴He in MCUCN.

Fig. 11Full-screen View

(Color online) The toal UCN number in the converter as a function of time simulated by iMCUCN and MCUCN

Figure 12 illustrates the UCN spectrum at the detector. Prior to the opening of the shutter, the spectrum at the detector remains low due to the small leakage rate from the gap and the low statistical UCN count rate. At about 807 s, the UCN spectrum is near to the peak, indicating a significant extraction of UCNs to the detector. Higher energy UCNs dominate the spectrum during this period. Subsequently, after 33 s, the extracted UCN number notably decreases, with lower energy UCNs becoming more prominent than higher energy ones. At 870 s, only lower energy UCNs remain. Finally, by 900 s, only a few lower energy UCNs can be extracted and detected at the detector.

Fig. 12Full-screen View

(Color online) The detected UCN spectrum at detector with a function of time