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Simulation of a rotating neutron moderator
Nuclear Inst. and Methods in Physics Research, A 959 (2020) 163562
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Nuclear Inst. and Methods in Physics Research, A journal homepage: www.elsevier.com/locate/nima
Simulation of a rotating neutron moderator G. Harrisson a ,∗, G. Li a , B. van der Ende a , R.B. Rogge a , Z. Tun b a b
Canadian Nuclear Laboratories, Chalk River, Ontario, K0J 1J0, Canada National Research Council Canada, Chalk River, Ontario, K0J 1J0, Canada
ARTICLE
INFO
ABSTRACT
Keywords: Neutron source Moving media High rotational speed Cryogenic GEANT4
This concept study demonstrates numerically the neutronic properties of a rotating neutron moderator, a device designed to moderate neutrons from a primary source and to emit the neutrons preferentially in a plane within a narrow energy band. The guiding principles are that a narrow energy band is achieved by cooling the moderator to cryogenic temperatures while the directionality and emitted neutron energy are attained by rotating the moderator at an appropriate angular velocity. Proof-of-principle simulations using GEANT4 demonstrate that the number of neutrons emitted at the desired energy and direction from the rotating moderator is about 60 times higher than that emitted by a static room temperature moderator of the same geometry.
1. Introduction
source emitting MeV-neutrons is sufficiently large, virtually all neutrons that are not lost to absorption will reach thermal equilibrium with the material before they escape to the surrounding space. The fully thermalized neutrons will have a speed (or energy) spectrum given by the Maxwell–Boltzmann distribution (MBD) of the characteristic temperature, T, which is the same as the temperature of the moderator. If the moderator is not sufficiently large, some neutrons will remain at higher energies. This condition is desired for some applications, for instance in the design of time-of-flight neutron spectrometers. In contrast, if the application requires a monochromatic beam, it is best to fully thermalize neutrons to ensure the peak flux is at the desired neutron energy. When neutrons of particular energy are selected out of the MBD, e.g. by a velocity selector or a crystal monochromator, those that fall outside the energy window are, of course, rejected. This loss of neutrons can be minimized by setting T to as close as possible to 0 K. A very cold moderator would be the best option if the only requirement was to concentrate neutrons into a sharply peaked distribution. However, for the reasons explained above, neutrons from a 0 K moderator may be too slow for a particular application. As a solution to this problem, we herein demonstrate the possibility of moderating neutrons to a very low temperature (as low as practically possible) and then accelerating them to a higher energy by Doppler shifting. Neutrons emitted in the forward direction by a moderator in motion will be Doppler shifted to higher energies in the stationary laboratory reference frame. Since an object cannot be in linear motion indefinitely in a finite space (the laboratory), the practical solution is a rotating moderator. Our literature search revealed that the concept of
Every application with neutrons involves neutron interaction with materials, be it a sample or a neutron detector. Since the interaction probability generally increases with decreasing neutron speed, neutrons with low energy (<1 eV) are preferred in most applications. However, the preferred neutron energy is not simply as low as reasonably achievable, due to several other factors. One such factor is how the neutron speed affects the beam flux. As neutrons propagate from a source to a sample, the volume-based flux may be obtained as the spatial density of the neutrons multiplied by their speed (dimensionally, neutrons/cm3 × cm/s = neutrons/cm2 /s). Hence, excessively slow neutrons can lead to unacceptably low neutron flux. Another factor, critically important in the case of neutron scattering, is the maximum amount of momentum transferable to the sample. If the incident neutron is too slow, it may not have enough momentum to transfer to the sample even in the case of back-scattering. Therefore, depending on the application, there is an optimal choice of neutron speed to achieve the most efficient use of neutrons. The initial energy of free neutrons obtained from nuclear reactions is in the order of 1 MeV. Slowing neutrons down to speeds corresponding to eV or meV energy range requires a moderator, a material made up of light nuclei with low neutron absorption and high probability for elastic scattering with large energy loss per scattering event. Hydrogenrich materials or liquid hydrogen are the most common moderator materials. The temperature, physical size and shape of the moderator, are important design parameters. The temperature is chosen in accordance with the desired neutron energy: room-temperature (∼300 K), for example, corresponds to ∼25 meV. If a moderator surrounding a
∗ Correspondence to: Canadian Nuclear Laboratories Ltd Station 68, 286 Plant Road, Chalk River, Ontario, K0J 1J0, Canada. E-mail address: [email protected] (G. Harrisson).
https://doi.org/10.1016/j.nima.2020.163562 Received 14 August 2019; Received in revised form 24 January 2020; Accepted 27 January 2020 Available online 31 January 2020 0168-9002/Crown Copyright © 2020 Published by Elsevier B.V. All rights reserved.
G. Harrisson, G. Li, B. van der Ende et al.
Nuclear Inst. and Methods in Physics Research, A 959 (2020) 163562
Doppler shifting neutrons to a higher energy by a room-temperature rotating disc was explored by Sjenitzer [1] with neutrons thermalized beforehand by an external medium. The device proposed in this work is a cryogenically cooled rotating and moderating ‘‘wheel’’ or tube containing an unaltered neutron source at its center. In addition to the Doppler energy shifting, we anticipate that the emissions viewed from our device in the laboratory frame will be highly anisotropic because, in the limit of infinitely high speed, every circumferential point of the tube will emit neutrons only in the forward direction and none (or very few) in the aft and axial directions. This anisotropy, combined with a highly peaked MBD within the cold moderator, is expected to yield a significantly higher and focused neutron flux compared to a stationary moderator optimized in temperature to emit the desired energy neutrons. In other approaches, Mayer et al. [2] demonstrate a system that shifts the energy of ultracold neutrons, UCN, (∼10−6 eV) to cold (10−3 eV). The system requires first producing a container of UCNs which relies on the total external reflection possible at those energies, and then, an energy shift with Bragg scattering from crystals on a high-speed rotation device to transfer some energy to the neutrons. Furthermore, Steyerl [3] demonstrates that cold neutrons emerging from a guide system fed by a cold source can be further reduced in energy by multiple momentum losses utilizing total external reflection upon receding turbine blades in order to produce UCNs. In contrast to these two examples, our concept does not require two distinct steps and facilities for each, or total external reflection of cold or ultracold neutrons, or Bragg diffraction. Rather, in a single device, the neutrons, through neutron–hydrogen nucleus interactions (scattering), are moderated to produce a narrow energy band width spectrum that shifts to higher energies through energy transfer from the hydrogen nuclei traveling at very high velocity. Our device can be viewed as a continuum of stages, that is, the thermalization primarily occurs at low radii and the shift to higher energies primarily occurs at large radii. Our concept is smaller than those of Mayer et al. and Steyerl as lengthy neutron guides/tubes are not required and the source of primary neutrons is closely surrounded by the moderator. It should also be noted that the conceptual device studied here produces thermal neutron energies (29 meV suitable for materials research) emitted preferentially in the rotation plane as a result of multiple neutron interactions with hydrogen nuclei traveling in a circle. In Section 2, we describe the computer simulation program and the hypothetical model moderator used to numerically provide a proofof-principle of the expectations outlined above. The simulation results along with discussions are in Section 3, followed by conclusions in Section 4.
Fig. 1. Rotating neutron moderator as simulated with GEANT4. The model moderator (blue hollow cylinder) and the arrangement of neutron detectors (yellow and green) around the moderator. See Table 1 for physical dimensions. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 1 Dimensions of the neutron moderator.
Moderator Yellow detector Green detector Reflector platesa
Height (cm)
Inner radius (cm)
Outer radius (cm)
Material
25 25 5 25
24 30.0100 40.0000 n/a
30 30.0101 40.0001 30
HDPE-like Vacuum Vacuum 4 Be
a
Not part of the original model. Introduced for follow-up investigation as described in the text.
frame attached to the moderator at the site of the most recent neutron– moderator interaction. The latter is the frame in which existing physical models and data libraries can be readily applied since all nuclei in the neighbourhood are stationary in this frame. Both scattering and capture reactions of neutrons in the moderator are taken into account in the neutron transport calculations [8]. The user-defined physics list makes use of the high precision neutron package (NeutronHP) as well as the data sets stored in the G4NDL4.2 library and based on the ENDF/B.VII nuclear data evaluation [9]. All simulations were performed using neutron scattering cross sections derived from the free gas approximation. 2.2. Primary neutron source and model moderator The model, shown in Fig. 1, consists of a point neutron source placed at the center of a tube made of a material having the same composition of high-density polyethylene (HDPE). In this conceptual study, the material of HDPE was used as a convenient means to have a high hydrogen density. The neutron–moderator interactions do not take into account the fact that the hydrogen atoms are bound; neutrons and hydrogen nuclei are simulated as free gases. Nonetheless, the objective is to compare a room temperature moderator in which the hydrogen nuclei have no collective motion with the case where the nuclei are collectively moving at high speed in the same direction. In this context, the results are normalized to the same number of primary neutrons and the comparison is made on the basis of a figure of merit that relates to a narrow energy band and small divergence perpendicular to the plane of rotation. Indeed, the simulations do find that the fraction of primary neutrons meeting the conditions is small, but considerably higher in the case of the cold rotating moderator. The tube is oriented with its cylindrical axis vertical, is 25 cm high, has an inner radius of 𝑅𝑖𝑛 = 24 cm and an outer radius of 𝑅𝑜𝑢𝑡 = 30 cm. The thickness of the moderating material, 𝑅𝑜𝑢𝑡 − 𝑅𝑖𝑛 = 6 cm, is a compromise between maximizing neutron moderation and minimizing loss of neutrons by absorption. The tube rotates clockwise when viewed from above. Further dimensions are given in Table 1. The primary source at the center of the moderator tube is an isotropic emitter of neutrons in accordance with a prescribed energy
2. Simulation details The general concept we investigated is: is it possible to produce a practical neutron beam by shifting the intrinsically narrow energy band of a cold neutron spectrum to a target thermal energy regime by transferring some momentum from hydrogen nuclei moving in a preferred direction, and without significantly broadening the spectrum? In the absence of an analytical solution, Monte Carlo simulations were pursued. This study is based on GEANT4 (version 9.6.p04) simulation toolkit [4,5] and on ROOT (version 5.34.36) data analysis framework [6]. 2.1. Modified GEANT4 and input nuclear data The GEANT4 package used was a version modified to simulate neutron transport in high speed moving media [7]. As explained in our previous paper, Li et al. [7], the relative velocity between the neutron and medium is taken into account by doing multiple frame transformations in real space between the laboratory and the local 2
G. Harrisson, G. Li, B. van der Ende et al.
Nuclear Inst. and Methods in Physics Research, A 959 (2020) 163562
1. Lost through the top or bottom openings with no moderator interaction; 2. Enters the moderator and is absorbed; 3. Exits the moderator through the top or bottom surfaces as a fast neutron (E > 0.18 eV); 4. Exits the moderator through the top or bottom surfaces as a thermal (or low-E). neutron (E ≤ 0.18 eV); 5. Exits the moderator through the outer curved (side) surface as a fast neutron; or 6. Exits the moderator through the outer curved (side) surface as a thermal neutron. Columns A to F in Table 2 list the probability of these outcomes for our model moderator under three different conditions: stationary moderator at 300 K and cooled to 20 K, and 20 K moderator rotating at 75 000 rpm (revolutions per minute). The probability for neutrons that never enter the moderator material due to geometry is independent of the moderator condition. The 53.8% observed in all three simulations (column A in Table 2) is a simple model validation as the result matches the analytical solution; the combined solid angle subtended by the source at the openings expressed as a fraction of 4𝜋 steradian. In a practical device, this loss of neutrons can be minimized by placing the primary source close to the inner wall of the moderator and filling the central space as well as the top and bottom empty spaces with neutron reflectors. The off-centered arrangement would also allow employing multiple primary sources. In this demonstrative work, however, we maintain full cylindrical symmetry to reduce complexity of simulation interpretation. Column B in Table 2 illustrates that the loss of neutrons by absorption depends strongly on the condition of the moderator. The increase of absorption from 300 K to 20 K, 4.5% to 8.3%, is mainly due to the increase of absorption by 1 H at low neutron energies [13]. When the 20 K moderator is in rotation, the absorption decreases slightly because within the local reference frame attached to the moderator, neutrons remain at low energy with respect to the surrounding 1 H nuclei. The next two columns (C and D) of Table 2 show that the condition of the moderator has little to no effect on neutrons with E > 0.18 eV. This result is understandable since at 0.18 eV, the neutron velocity is 5.87 km/s whereas the circumferential speed of the moderator at 75 000 rpm is less, at 2.36 km/s. For neutrons with E > 0.18 eV, the rotating moderator therefore adds little to the neutron velocity. The last two columns of Table 2 show the impact of rotation on relatively low energy (low-E) neutrons. The increased number of neutrons lost to absorption by cooling comes at the expense of low-E neutrons exiting through either the top and bottom surfaces (column E) or the side surface (column F). Indeed, the fractional reduction is the same for the two possibilities: 3.1% / 4.9% = 0.63 for the top/bottom exit and 2.8% / 4.6% = 0.61 in the case of the side exit. This directional indifference is greatly impacted by the rotation which makes exiting through the side surface a far more probable outcome (5.6% vs. 1.2%).
Fig. 2. Primary neutron energy distribution as given by the 𝟐𝟓𝟐 Cf spontaneous fission spectrum. The 252 Cf spontaneous √ fission spectrum is approximated here by the Watt distribution: 𝜒 (𝐸) = 𝑎𝑒−𝐸∕𝑏 sinh 𝑐𝐸 where E is the neutron emission energy in MeV, 𝑎 = 0.6380, 𝑏 = 1.18 and 𝑐 = 1.03419.
Fig. 3. Primary neutron energy distribution as given by the 𝟕 Li(p, n)𝟕 Be reaction. The 7 Li(p, n)7 Be reaction being initiated here by 1.912 MeV incident protons [11,12].
distribution. Two sources of primary neutrons were investigated: (1) neutrons emitted by the spontaneous fission of 252 Cf [10] and, (2) neutrons generated from the 7 Li(𝑝, 𝑛)7 Be reaction using 1.912 MeV incident protons [11,12]. The energy spectrum of a 252 Cf source, approximated by the Watt distribution, is shown in Fig. 2 and the softer energy spectrum from the (𝑝, 𝑛) reaction is shown in Fig. 3. Except when specified otherwise, 108 primary neutrons were used per simulation with the spectrum of 252 Cf. These provide sufficient fidelity information to make general observations. Due to the large openings directly above and below the point source (see Fig. 1), more than 50% of the primary neutrons escape without ever entering the moderator material. The benefit of using beryllium to reflect some of these neutrons back to the moderator was also investigated. Apart from adding beryllium reflectors directly above and below the moderator tube, the simulation model otherwise remains exactly as in Fig. 1.
3.2. Impact of rotation on neutron trajectories
2.3. Neutron detectors
The lines in Fig. 4 are trajectories of individual neutrons as seen by an observer in the laboratory reference frame: for clarity, only a small number of trajectories are shown. Given that the rotation affects only low-E neutrons, the selected trajectories are from the subset of neutrons that have acquired a final energy ≤ 0.18 eV among the fifty first primary neutrons randomly generated by the source. The upper panels (a and b) of Fig. 4 are from a simulation with the moderator at rest (300 K) while the lower panels (c and d) are obtained with a cooled (20 K) rotating moderator. Each pair of panels depicts the same trajectories in two projections: looking from above (the cylindrical moderator appears as a circle) and side view (the outline of the moderator is a rectangle). Panels a and b show that neutrons travel randomly inside a static moderator, resulting in emissions
Simulation results are recorded by two neutron detectors, each a thin cylindrical shell filled with vacuum (G4_Galactic). Detector #1 (yellow in Fig. 1) is close to the moderator to count virtually all neutrons as they exit the moderator through the outer curved surface of the tube. Detector #2 (green in Fig. 1) counts neutrons emitted from the central band of the moderator with a small vertical divergence. 3. Results 3.1. Impact of rotation on neutron emission Neutrons emitted by the primary source eventually end up in one of the following outcomes: 3
G. Harrisson, G. Li, B. van der Ende et al.
Nuclear Inst. and Methods in Physics Research, A 959 (2020) 163562
Table 2 Probability (%) of neutron outcomes. Condition of the moderator
(A) Lost through openings
(B) Absorbed
(C) Exit via top/bottom E > 0.18 eV
(D) Exit via side E > 0.18 eV
(E) Exit via top/bottom E ≤ 0.18 eV
(F) Exit via side E ≤ 0.18 eV
300 K 0 rpm
53.8
4.5
13.9
18.3
4.9
4.6
20 K 0 rpm
53.8
8.3
13.8
18.2
3.1
2.8
20 K 75 000 rpm
53.8
7.3
13.9
18.2
1.2
5.6
Fig. 4. Trajectories of individual neutrons. The trajectories of neutrons, among the fifty first primary neutrons randomly generated by the source, that were thermalized to a final energy E ≤ 0.18 eV in a stationary room-temperature (300 K) moderator (upper panels) and those in the same moderator but rotating and cooled down to 20 K (lower panels). The outline of the cylindrical moderator appears as a circle when viewed from above (panels a and c) and as a rectangle in the side view (panels b and d). 19 neutrons in the static case and 22 neutrons in the rotating case are shown. Some of these neutrons are lost to absorption.
without a preferential direction. In contrast, the overall path of lowfinal-E neutrons in a rotating moderator looks like a spiral (panel c) and they exit preferentially in the same direction as the rotation, and closer to the horizontal plane. Those that remain at high energy do not manifest this preferred directionality, a fact that could be exploited to ‘‘filter out’’ fast neutrons from a thermal neutron beam.
energy cut-off is the same between Fig. 5 and Table 2, the sum of the neutrons under each curve is equal to the corresponding number appearing in the last column of Table 2 multiplied by 108 neutrons (i.e. percentage/100 × 108 neutrons). The cold static moderator (black curve) produces a sharp spectrum with a peak at 2.1 meV. Equivalent to 24 K, this peak position corresponds closely to the chosen temperature of the moderator material (20 K). Compared to the static moderator at room temperature (red curve), the peak neutron count has markedly increased by cooling, from ∼1300 to more than 8000 neutrons/0.02 meV/108 . However, due to the much reduced width of the distribution, the integrated flux, i.e. the area under the curve, has decreased from 4.57 × 106 to 2.82 × 106 neutrons. Physically, this reduction is explained by an increase in losses by absorption. The same moderator rotating at 75 000 rpm yields a spectrum with a peak shifted to 29 meV (green curve). The shift in energy is as expected; however, due to broadening, the peak
3.3. Energy distribution of neutrons from the rotating moderator The energy distribution of the low-E neutrons is of critical importance in the design of a device intended as a source of thermal neutron beams. Fig. 5 shows the distributions obtained from the three simulations listed in Table 2, each based on 108 primary neutrons. The counts plotted on the 𝑦-axis are the number of neutrons escaping through the yellow detector grouped in energy bins of 0.02 meV width (hence the unit chosen is neutrons/0.02 meV/108 ). Given that the 4
G. Harrisson, G. Li, B. van der Ende et al.
Nuclear Inst. and Methods in Physics Research, A 959 (2020) 163562
2. The neutron passes through both the yellow and green detectors; and 3. The trajectory is within 1◦ of the most probable direction of emission. Conditions 2 and 3 together mean that those neutrons counted in the enhancement factor come from the middle 5.3 cm high band of the yellow detector; this is equivalent to viewing the side of the moderator with a 50 mm high 1◦ collimator. To support quantitative analysis of good statistical value, simulations with 109 primary neutrons were undertaken for the evaluation of the enhancement factor. This figure of merit is shown in Table 3 summarizing the conditions and results of the simulations where in one case the moderator is static at 300 K, while in the other it is rotating and at 20 K. These two simulations, as a pair for comparison, were originally carried out with 252 Cf as the primary neutron source. For the 252 Cf source, 114 neutrons in the static case and 6978 neutrons in the rotating case met the selection criteria, resulting in an enhancement factor of 61 ± 6. The ∼10% relative uncertainty in the enhancement factor is the consequence of the static moderator being very inefficient in producing a monochromatic beam, the very reason a rotating moderator is proposed as an alternative. A particularity of the 252 Cf source is that 17.8% of the primary neutrons entering the moderator escape without ever interacting with the moderator material. It is an understandable result, given that the scattering cross section of 1 H becomes small at neutron energies beyond 1 MeV [14] and a large portion of the 252 Cf spectrum is above 1 MeV. Another pair of simulations were performed with 7 Li(𝑝, 𝑛)7 Be reaction as the primary neutron source. With this softer spectrum, only 2.1% of the primary neutrons entering the moderator escape without ever interacting with the moderator material. As shown in Table 3, the number of neutrons meeting the selection criteria increases approximately by a factor of 1.7 in both static and rotating moderators. The enhancement factor obtained, 66 ± 5 is the same within uncertainty as the 61 ± 6 obtained with the 252 Cf source. These results suggest that while the moderator performs more effectively in terms of flux relative to source intensity, the enhancement factor does not really improve if the primary neutron energy is mostly below 1 MeV. This is also expected as the moderator cylinder dimensions have been chosen for enhanced thermalization (i.e. production of thermal neutrons versus absorption losses) and the softer spectrum will not significantly change those dimensions. The chosen moderator thickness for the softer spectrum is 4 cm instead of 6 cm with the same outer diameter used for the 252 Cf spectrum. To evaluate potential further gains to performance, a pair of simulations were undertaken with neutron reflectors. Two reflectors, each a solid cylinder of beryllium (4 Be), 25 cm high and 30 cm in radius, were placed directly above and below the tube moderator. Thus, the 252 Cf source at the center was fully enclosed. The simulations showed that the physics inside the moderator remains unchanged as the cylindrical symmetry was maintained. Better neutron utilization led to about 2.5 times more neutron counts than the case without reflectors and yielded an enhancement factor (cold rotating versus static room temperature) of 58 ± 4.
Fig. 5. Energy spectrum of neutrons escaping the moderator material through the side surface of the model moderator. The condition of each simulation using 108 primary neutrons is as labeled in the figure. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
neutron count decreases nearly by a factor of two (from ∼8000 to ∼4000 neutrons/0.02 meV/108 ). The total number of neutrons under the curve increases from the static to rotating case (from 2.82 × 106 to 5.61 × 106 neutrons) indicating that more neutrons exit through the side surface when the moderator is rotating. The same moderator at room temperature and not rotating will yield a spectrum peaked at ∼29 meV (red curve). This is, of course, how thermal neutrons are traditionally generated. The benefit of the rotation is seen here as the peak neutron count is ∼4 times higher than from a comparable traditional moderator. 3.4. Directional distribution of neutrons from the rotating moderator The neutron exit directions are reported in a coordinate system defined by three mutually orthogonal unit vectors relative to the outer curved surface of the moderator: an outward pointing normal ̂ a tangent vector 𝐭̂ pointing in the same direction as the vector 𝐧, rotation and a vertical vector 𝐯̂ parallel to the axis of rotation and pointing upward. The orientation of the NTV system relative to the laboratory frame depends on the exit point of each neutron. Fig. 6 shows the spatial distribution of exit directions for neutrons with 28 ≤ E ≤ 30 meV, with the projection plane as shown being divided in 100 × 100 pixels. In the static room temperature case (Fig. 6a), the distribution is widespread in space and the emissions are maximized at the center of the plot. In the cold rotating case (Fig. 6b), the distribution is undeniably concentrated in space and the ⃖⃖⃖⃖⃖⃖⃖⃖⃖⃖⃑28≤𝐸≤30 meV = 0.29 𝐧̂ + 0.96 𝐭̂ + 0 𝐯. ̂ most probable exit direction is Mped 3.5. Comparison of the rotating moderator to a conventional neutron source The rotating moderator is intended as a source of thermal neutron beams at a particular energy. Its performance can be compared to the one of a traditional source in producing a highly collimated and monochromatic neutron beam. Table 2 (also in Fig. 5) demonstrates that, in terms of integrated thermal flux, the rotating moderator is not much better than a traditional (i.e. static at 300 K) moderator: an increase from 4.6% to 5.6%, or a mere 22% of improvement, is insufficient to justify the complexity and engineering challenges of designing a practical rotating moderator: simply increasing the power of the primary neutron source by 22% would achieve the same goal. A more appropriate figure of merit suitable for both the narrow energy band and preferred direction of a rotating moderator, is an enhancement factor evaluated from neutrons that meet the following selection criteria: 1. Emitted with an energy within the range 28 ≤ E ≤ 30 meV;
4. Conclusion The simulations presented here demonstrate in principle how a cooled rotating moderator is superior to a comparable traditional (i.e. static) moderator if the goal is to produce a collimated and monochromatic (narrow energy spectrum) neutron beam in the thermal energy range. Neutrons with energy of a few MeV emitted isotropically from a point source can be focused in energy and space by a cold rotating moderator. Compared to a static moderator at room temperature, it is possible with a rotating moderator to direct toward a sample about 60 times more neutrons with energy of ∼29 ± 1 meV. The moderator used in this work for simulations is a hypothetical device, chosen for its simplicity in demonstrating numerically 5
G. Harrisson, G. Li, B. van der Ende et al.
Nuclear Inst. and Methods in Physics Research, A 959 (2020) 163562
Fig. 6. Stereographic projection of neutron exit directions. For neutrons with 28 ≤ E ≤ 30 meV. Zero is white. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 3 Simulation results.
Moderator temperature (K) Rotation speed (rpm) Desired neutron energy (meV) Energy range of interest (meV) Angle window from the most probable exit direction (◦ )
Static
Rotating
300 0 29 28–30 1
20 75 000 29 28–30 1
𝟐𝟓𝟐
Cf spontaneous fission spectrum
Most probable exit direction (NTV system) Number of neutrons meeting selection criteria Enhancement factor (no units) Primary neutrons enter moderator and escape without interacting (%)
1 𝐧̂ + 0 𝐭̂ + 0 𝐯̂ 114
Most probable exit direction (NTV system) Number of neutrons meeting selection criteria Enhancement factor (no units) Primary neutrons enter moderator and escape without interacting (%)
1 𝐧̂ + 0 𝐭̂ + 0 𝐯̂ 185
0.29 𝐧̂ + 0.96 𝐭̂ + 0 𝐯̂ 6978 61 ± 6a 17.8
17.8 𝟕 Li(p,
a
2.1
The ± uncertainty is estimated by assuming that one standard deviation of counting error is
n)𝟕 Be reaction 0.29 𝐧̂ + 0.96 𝐭̂ + 0 𝐯̂ 12 203 66 ± 5a 2.1
√ N, N being the number of
9
neutrons counted. 10 primary neutrons were used per simulation for the evaluation of the enhancement factors.
CRediT authorship contribution statement
how a cold rotating moderator can enhance the production of a lowdivergence, narrow energy band thermal neutron beam. Designing and building an actual device is a significant engineering challenge (moderator material, cryogenic temperatures and high rotational forces), for which substantial design optimization can be sought. However, the task is less daunting if it is viewed as finding an engineering solution integrating the following three different technologies: 1. A compact neutron generator as the primary source; 2. Cryogenics to cool a suitable moderator to ∼20 K or lower temperature; 3. High-speed rotation. While each of these technologies is quite mature on its own, their combination is a unique challenge. For example, commercial compact neutron generators are available, cooling to 20 K or even lower in temperature is readily achievable, and high-speed rotation is utilized in some commercial items (e.g., turbo-molecular pumps operating at 90 000 rpm, high-speed centrifuges, and high-speed flywheels). Therefore, it should be possible to build a rotating moderator with characteristics not much different from our hypothetical model. Any practical design will require in-depth study of potential materials and engineering concerns.
G. Harrisson: Software, Validation, Formal analysis, Investigation, Resources, Data curation, Writing - original draft, Writing - review & editing, Visualization. G. Li: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Writing - review & editing. B. van der Ende: Conceptualization, Methodology, Validation, Investigation, Writing - review & editing, Supervision, Project administration, Funding acquisition. R.B. Rogge: Conceptualization, Methodology, Validation, Investigation, Writing - review & editing, Supervision, Project administration, Funding acquisition. Z. Tun: Conceptualization, Methodology, Validation, Investigation, Writing - original draft, Writing - review & editing. Acknowledgment This work was performed under the auspices of Atomic Energy of Canada Limited’s Federal Nuclear Science and Technology program at the Canadian Nuclear Laboratories. References
Declaration of competing interest
[1] B. Sjenitzer, Neutron Moderation in a Rotating Disc, Delft University of Technology, 2008. [2] S. Mayer, H. Rauch, P. Geltenbort, P. Schmidt-Wellenburg, P. Allenspach, G. Zsigmond, New aspects for high-intensity neutron beam production, Nucl. Instrum. Methods Phys. Res. A 608 (2009) 434–439.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 6
G. Harrisson, G. Li, B. van der Ende et al.
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[3] A. Steyerl, A neutron turbine as an efficient source of ultracold neutrons, Nucl. Instrum. Methods 125 (1975) 461–469. [4] S. Agostinelli, J. Allison, K. Amako, J. Apostolakis, H. Araujo, P. Arcel, M. Asai, D. Axen, S. Banerjee, G. Barrand, F. Behner, L. Bellagamba, J. Boudreau, L. Broglia, A. Brunengo, H. Burkhardt, S. Chauvie, J. Chuma, R. Chytracek, G. Cooperman, G. Cosmo, P. Degtyarenko, A. Dell’ Acqua, G. Depaola, D. Dietrich, R. Enami, A. Feliciello, C. Ferguson, H. Fesefeldt, G. Folger, F. Foppiano, A. Forti, S. Garelli, S. Giani, R. Giannitrapani, D. Gibin, J.J. Gomez Cadenas, I. Gonzalez, G. Gracia Abril, G. Greeniaus, W. Greiner, V. Grichine, A. Grossheim, S. Guatelli, P. Gumplinger, R. Hamatsu, K. Hashimoto, H. Hasui, A. Heikkinen, A. Howard, V. Ivanchenko, A. Johnson, F.W. Jones, J. Kallenbach, N. Kanaya, M. Kawabata, Y. Kawabata, M. Kawaguti, S. Kelner, P. Kent, A. Kimura, T. Kodama, R. Kokoulin, M. Kossov, H. Kurashige, E. Lamanna, T. Lampen, V. Lara, V. Lefebure, F. Lei, M. Liendl, W. Lockman, F. Longo, S. Magni, M. Maire, E. Medernach, K. Minamimoto, P. Mora de Freitas, Y. Morita, K. Murakami, M. Nagamatu, R. Nartallo, P. Nieminen, T. Nishimura, K. Ohtsubo, M. Okamura, S. O’Neale, Y. Oohata, K. Paech, J. Perl, A. Pfeiffer, M.G. Pia, F. Ranjard, A. Rybin, S. Sadilov, E. Di Salvo, G. Santin, T. Sasaki, N. Savvas, Y. Sawada, S. Scherer, S. Sei, V. Sirotenko, D. Smith, N. Starkov, H. Stoecker, J. Sulkimo, M. Takahata, S. Tanaka, E. Tcherniaev, E. Safai Tehrani, M. Tropeano, P. Truscott, H. Uno, L. Urban, P. Urban, M. Verderi, A. Walkden, W. Wander, H. Weber, J.P. Wellisch, T. Wenaus, D.C. Williams, D. Wright, T. Yamada, H. Yoshida, D. Zschiesche, GEANT4 – A simulation toolkit, Nucl. Instrum. Methods Phys. Res. A 506 (2003) 250–303. [5] J. Allison, K. Amako, J. Apostolakis, H. Araujo, P. Arce Dubois, M. Asai, G. Barrand, R. Capra, S. Chauvie, R. Chytracek, G.A.P. Cirrone, G. Cooperman, G. Cosmo, G. Cuttone, G.G. Daquino, M. Donszelmann, M. Dressel, G. Folger, F. Foppiano, J. Generowicz, V. Grichine, S. Guatelli, P. Gumplinger, A. Heikkinen, I. Hrivnacova, A. Howard, S. Incerti, V. Ivanchenko, T. Johnson, F. Jones, T. Koi, R. Kokoulin, M. Kossov, H. Kurashige, V. Lara, S. Larsson, F. Lei, O. Link, F. Longo, M. Maire, A. Mantero, B. Mascialino, I. McLaren, P. Mendez Lorenzo, K. Minamimoto, K. Murakami, P. Nieminen, L. Pandola, S. Parlati, L. Peralta, J. Perl, A. Pfeiffer, M.G. Pia, A. Ribon, P. Rodrigues, G. Russo, S. Sadilov, G. Santin, T. Sasaki, D. Smith, N. Starkov, S. Tanaka, E. Tcherniaev, B. Tomé, A. Trindade, P. Truscott, L. Urban, M. Verderi, A. Walkden, J.P. Wellisch, D.C. Williams, D. Wright, H. Yoshida, GEANT4 – Developments and applications, IEEE Trans. Nucl. Sci. 53–1 (2006) 270–278. [6] R. Brun, F. Rademakers, ROOT – AN object oriented data analysis framework, Nucl. Instrum. Methods Phys. Res. A 389 (1997) 81–86. [7] G. Li, B. Ciungu, G. Harrisson, R.B. Rogge, Z. Tun, B.M. van der Ende, I. Zwiers, Neutron transport simulation in high speed moving media using GEANT4, Nucl. Instrum. Methods Phys. Res. A 874 (2017) 66–71.
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Nuclear Inst. and Methods in Physics Research, A journal homepage: www.elsevier.com/locate/nima
Simulation of a rotating neutron moderator G. Harrisson a ,∗, G. Li a , B. van der Ende a , R.B. Rogge a , Z. Tun b a b
Canadian Nuclear Laboratories, Chalk River, Ontario, K0J 1J0, Canada National Research Council Canada, Chalk River, Ontario, K0J 1J0, Canada
ARTICLE
INFO
ABSTRACT
Keywords: Neutron source Moving media High rotational speed Cryogenic GEANT4
This concept study demonstrates numerically the neutronic properties of a rotating neutron moderator, a device designed to moderate neutrons from a primary source and to emit the neutrons preferentially in a plane within a narrow energy band. The guiding principles are that a narrow energy band is achieved by cooling the moderator to cryogenic temperatures while the directionality and emitted neutron energy are attained by rotating the moderator at an appropriate angular velocity. Proof-of-principle simulations using GEANT4 demonstrate that the number of neutrons emitted at the desired energy and direction from the rotating moderator is about 60 times higher than that emitted by a static room temperature moderator of the same geometry.
1. Introduction
source emitting MeV-neutrons is sufficiently large, virtually all neutrons that are not lost to absorption will reach thermal equilibrium with the material before they escape to the surrounding space. The fully thermalized neutrons will have a speed (or energy) spectrum given by the Maxwell–Boltzmann distribution (MBD) of the characteristic temperature, T, which is the same as the temperature of the moderator. If the moderator is not sufficiently large, some neutrons will remain at higher energies. This condition is desired for some applications, for instance in the design of time-of-flight neutron spectrometers. In contrast, if the application requires a monochromatic beam, it is best to fully thermalize neutrons to ensure the peak flux is at the desired neutron energy. When neutrons of particular energy are selected out of the MBD, e.g. by a velocity selector or a crystal monochromator, those that fall outside the energy window are, of course, rejected. This loss of neutrons can be minimized by setting T to as close as possible to 0 K. A very cold moderator would be the best option if the only requirement was to concentrate neutrons into a sharply peaked distribution. However, for the reasons explained above, neutrons from a 0 K moderator may be too slow for a particular application. As a solution to this problem, we herein demonstrate the possibility of moderating neutrons to a very low temperature (as low as practically possible) and then accelerating them to a higher energy by Doppler shifting. Neutrons emitted in the forward direction by a moderator in motion will be Doppler shifted to higher energies in the stationary laboratory reference frame. Since an object cannot be in linear motion indefinitely in a finite space (the laboratory), the practical solution is a rotating moderator. Our literature search revealed that the concept of
Every application with neutrons involves neutron interaction with materials, be it a sample or a neutron detector. Since the interaction probability generally increases with decreasing neutron speed, neutrons with low energy (<1 eV) are preferred in most applications. However, the preferred neutron energy is not simply as low as reasonably achievable, due to several other factors. One such factor is how the neutron speed affects the beam flux. As neutrons propagate from a source to a sample, the volume-based flux may be obtained as the spatial density of the neutrons multiplied by their speed (dimensionally, neutrons/cm3 × cm/s = neutrons/cm2 /s). Hence, excessively slow neutrons can lead to unacceptably low neutron flux. Another factor, critically important in the case of neutron scattering, is the maximum amount of momentum transferable to the sample. If the incident neutron is too slow, it may not have enough momentum to transfer to the sample even in the case of back-scattering. Therefore, depending on the application, there is an optimal choice of neutron speed to achieve the most efficient use of neutrons. The initial energy of free neutrons obtained from nuclear reactions is in the order of 1 MeV. Slowing neutrons down to speeds corresponding to eV or meV energy range requires a moderator, a material made up of light nuclei with low neutron absorption and high probability for elastic scattering with large energy loss per scattering event. Hydrogenrich materials or liquid hydrogen are the most common moderator materials. The temperature, physical size and shape of the moderator, are important design parameters. The temperature is chosen in accordance with the desired neutron energy: room-temperature (∼300 K), for example, corresponds to ∼25 meV. If a moderator surrounding a
∗ Correspondence to: Canadian Nuclear Laboratories Ltd Station 68, 286 Plant Road, Chalk River, Ontario, K0J 1J0, Canada. E-mail address: [email protected] (G. Harrisson).
https://doi.org/10.1016/j.nima.2020.163562 Received 14 August 2019; Received in revised form 24 January 2020; Accepted 27 January 2020 Available online 31 January 2020 0168-9002/Crown Copyright © 2020 Published by Elsevier B.V. All rights reserved.
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Nuclear Inst. and Methods in Physics Research, A 959 (2020) 163562
Doppler shifting neutrons to a higher energy by a room-temperature rotating disc was explored by Sjenitzer [1] with neutrons thermalized beforehand by an external medium. The device proposed in this work is a cryogenically cooled rotating and moderating ‘‘wheel’’ or tube containing an unaltered neutron source at its center. In addition to the Doppler energy shifting, we anticipate that the emissions viewed from our device in the laboratory frame will be highly anisotropic because, in the limit of infinitely high speed, every circumferential point of the tube will emit neutrons only in the forward direction and none (or very few) in the aft and axial directions. This anisotropy, combined with a highly peaked MBD within the cold moderator, is expected to yield a significantly higher and focused neutron flux compared to a stationary moderator optimized in temperature to emit the desired energy neutrons. In other approaches, Mayer et al. [2] demonstrate a system that shifts the energy of ultracold neutrons, UCN, (∼10−6 eV) to cold (10−3 eV). The system requires first producing a container of UCNs which relies on the total external reflection possible at those energies, and then, an energy shift with Bragg scattering from crystals on a high-speed rotation device to transfer some energy to the neutrons. Furthermore, Steyerl [3] demonstrates that cold neutrons emerging from a guide system fed by a cold source can be further reduced in energy by multiple momentum losses utilizing total external reflection upon receding turbine blades in order to produce UCNs. In contrast to these two examples, our concept does not require two distinct steps and facilities for each, or total external reflection of cold or ultracold neutrons, or Bragg diffraction. Rather, in a single device, the neutrons, through neutron–hydrogen nucleus interactions (scattering), are moderated to produce a narrow energy band width spectrum that shifts to higher energies through energy transfer from the hydrogen nuclei traveling at very high velocity. Our device can be viewed as a continuum of stages, that is, the thermalization primarily occurs at low radii and the shift to higher energies primarily occurs at large radii. Our concept is smaller than those of Mayer et al. and Steyerl as lengthy neutron guides/tubes are not required and the source of primary neutrons is closely surrounded by the moderator. It should also be noted that the conceptual device studied here produces thermal neutron energies (29 meV suitable for materials research) emitted preferentially in the rotation plane as a result of multiple neutron interactions with hydrogen nuclei traveling in a circle. In Section 2, we describe the computer simulation program and the hypothetical model moderator used to numerically provide a proofof-principle of the expectations outlined above. The simulation results along with discussions are in Section 3, followed by conclusions in Section 4.
Fig. 1. Rotating neutron moderator as simulated with GEANT4. The model moderator (blue hollow cylinder) and the arrangement of neutron detectors (yellow and green) around the moderator. See Table 1 for physical dimensions. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 1 Dimensions of the neutron moderator.
Moderator Yellow detector Green detector Reflector platesa
Height (cm)
Inner radius (cm)
Outer radius (cm)
Material
25 25 5 25
24 30.0100 40.0000 n/a
30 30.0101 40.0001 30
HDPE-like Vacuum Vacuum 4 Be
a
Not part of the original model. Introduced for follow-up investigation as described in the text.
frame attached to the moderator at the site of the most recent neutron– moderator interaction. The latter is the frame in which existing physical models and data libraries can be readily applied since all nuclei in the neighbourhood are stationary in this frame. Both scattering and capture reactions of neutrons in the moderator are taken into account in the neutron transport calculations [8]. The user-defined physics list makes use of the high precision neutron package (NeutronHP) as well as the data sets stored in the G4NDL4.2 library and based on the ENDF/B.VII nuclear data evaluation [9]. All simulations were performed using neutron scattering cross sections derived from the free gas approximation. 2.2. Primary neutron source and model moderator The model, shown in Fig. 1, consists of a point neutron source placed at the center of a tube made of a material having the same composition of high-density polyethylene (HDPE). In this conceptual study, the material of HDPE was used as a convenient means to have a high hydrogen density. The neutron–moderator interactions do not take into account the fact that the hydrogen atoms are bound; neutrons and hydrogen nuclei are simulated as free gases. Nonetheless, the objective is to compare a room temperature moderator in which the hydrogen nuclei have no collective motion with the case where the nuclei are collectively moving at high speed in the same direction. In this context, the results are normalized to the same number of primary neutrons and the comparison is made on the basis of a figure of merit that relates to a narrow energy band and small divergence perpendicular to the plane of rotation. Indeed, the simulations do find that the fraction of primary neutrons meeting the conditions is small, but considerably higher in the case of the cold rotating moderator. The tube is oriented with its cylindrical axis vertical, is 25 cm high, has an inner radius of 𝑅𝑖𝑛 = 24 cm and an outer radius of 𝑅𝑜𝑢𝑡 = 30 cm. The thickness of the moderating material, 𝑅𝑜𝑢𝑡 − 𝑅𝑖𝑛 = 6 cm, is a compromise between maximizing neutron moderation and minimizing loss of neutrons by absorption. The tube rotates clockwise when viewed from above. Further dimensions are given in Table 1. The primary source at the center of the moderator tube is an isotropic emitter of neutrons in accordance with a prescribed energy
2. Simulation details The general concept we investigated is: is it possible to produce a practical neutron beam by shifting the intrinsically narrow energy band of a cold neutron spectrum to a target thermal energy regime by transferring some momentum from hydrogen nuclei moving in a preferred direction, and without significantly broadening the spectrum? In the absence of an analytical solution, Monte Carlo simulations were pursued. This study is based on GEANT4 (version 9.6.p04) simulation toolkit [4,5] and on ROOT (version 5.34.36) data analysis framework [6]. 2.1. Modified GEANT4 and input nuclear data The GEANT4 package used was a version modified to simulate neutron transport in high speed moving media [7]. As explained in our previous paper, Li et al. [7], the relative velocity between the neutron and medium is taken into account by doing multiple frame transformations in real space between the laboratory and the local 2
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Nuclear Inst. and Methods in Physics Research, A 959 (2020) 163562
1. Lost through the top or bottom openings with no moderator interaction; 2. Enters the moderator and is absorbed; 3. Exits the moderator through the top or bottom surfaces as a fast neutron (E > 0.18 eV); 4. Exits the moderator through the top or bottom surfaces as a thermal (or low-E). neutron (E ≤ 0.18 eV); 5. Exits the moderator through the outer curved (side) surface as a fast neutron; or 6. Exits the moderator through the outer curved (side) surface as a thermal neutron. Columns A to F in Table 2 list the probability of these outcomes for our model moderator under three different conditions: stationary moderator at 300 K and cooled to 20 K, and 20 K moderator rotating at 75 000 rpm (revolutions per minute). The probability for neutrons that never enter the moderator material due to geometry is independent of the moderator condition. The 53.8% observed in all three simulations (column A in Table 2) is a simple model validation as the result matches the analytical solution; the combined solid angle subtended by the source at the openings expressed as a fraction of 4𝜋 steradian. In a practical device, this loss of neutrons can be minimized by placing the primary source close to the inner wall of the moderator and filling the central space as well as the top and bottom empty spaces with neutron reflectors. The off-centered arrangement would also allow employing multiple primary sources. In this demonstrative work, however, we maintain full cylindrical symmetry to reduce complexity of simulation interpretation. Column B in Table 2 illustrates that the loss of neutrons by absorption depends strongly on the condition of the moderator. The increase of absorption from 300 K to 20 K, 4.5% to 8.3%, is mainly due to the increase of absorption by 1 H at low neutron energies [13]. When the 20 K moderator is in rotation, the absorption decreases slightly because within the local reference frame attached to the moderator, neutrons remain at low energy with respect to the surrounding 1 H nuclei. The next two columns (C and D) of Table 2 show that the condition of the moderator has little to no effect on neutrons with E > 0.18 eV. This result is understandable since at 0.18 eV, the neutron velocity is 5.87 km/s whereas the circumferential speed of the moderator at 75 000 rpm is less, at 2.36 km/s. For neutrons with E > 0.18 eV, the rotating moderator therefore adds little to the neutron velocity. The last two columns of Table 2 show the impact of rotation on relatively low energy (low-E) neutrons. The increased number of neutrons lost to absorption by cooling comes at the expense of low-E neutrons exiting through either the top and bottom surfaces (column E) or the side surface (column F). Indeed, the fractional reduction is the same for the two possibilities: 3.1% / 4.9% = 0.63 for the top/bottom exit and 2.8% / 4.6% = 0.61 in the case of the side exit. This directional indifference is greatly impacted by the rotation which makes exiting through the side surface a far more probable outcome (5.6% vs. 1.2%).
Fig. 2. Primary neutron energy distribution as given by the 𝟐𝟓𝟐 Cf spontaneous fission spectrum. The 252 Cf spontaneous √ fission spectrum is approximated here by the Watt distribution: 𝜒 (𝐸) = 𝑎𝑒−𝐸∕𝑏 sinh 𝑐𝐸 where E is the neutron emission energy in MeV, 𝑎 = 0.6380, 𝑏 = 1.18 and 𝑐 = 1.03419.
Fig. 3. Primary neutron energy distribution as given by the 𝟕 Li(p, n)𝟕 Be reaction. The 7 Li(p, n)7 Be reaction being initiated here by 1.912 MeV incident protons [11,12].
distribution. Two sources of primary neutrons were investigated: (1) neutrons emitted by the spontaneous fission of 252 Cf [10] and, (2) neutrons generated from the 7 Li(𝑝, 𝑛)7 Be reaction using 1.912 MeV incident protons [11,12]. The energy spectrum of a 252 Cf source, approximated by the Watt distribution, is shown in Fig. 2 and the softer energy spectrum from the (𝑝, 𝑛) reaction is shown in Fig. 3. Except when specified otherwise, 108 primary neutrons were used per simulation with the spectrum of 252 Cf. These provide sufficient fidelity information to make general observations. Due to the large openings directly above and below the point source (see Fig. 1), more than 50% of the primary neutrons escape without ever entering the moderator material. The benefit of using beryllium to reflect some of these neutrons back to the moderator was also investigated. Apart from adding beryllium reflectors directly above and below the moderator tube, the simulation model otherwise remains exactly as in Fig. 1.
3.2. Impact of rotation on neutron trajectories
2.3. Neutron detectors
The lines in Fig. 4 are trajectories of individual neutrons as seen by an observer in the laboratory reference frame: for clarity, only a small number of trajectories are shown. Given that the rotation affects only low-E neutrons, the selected trajectories are from the subset of neutrons that have acquired a final energy ≤ 0.18 eV among the fifty first primary neutrons randomly generated by the source. The upper panels (a and b) of Fig. 4 are from a simulation with the moderator at rest (300 K) while the lower panels (c and d) are obtained with a cooled (20 K) rotating moderator. Each pair of panels depicts the same trajectories in two projections: looking from above (the cylindrical moderator appears as a circle) and side view (the outline of the moderator is a rectangle). Panels a and b show that neutrons travel randomly inside a static moderator, resulting in emissions
Simulation results are recorded by two neutron detectors, each a thin cylindrical shell filled with vacuum (G4_Galactic). Detector #1 (yellow in Fig. 1) is close to the moderator to count virtually all neutrons as they exit the moderator through the outer curved surface of the tube. Detector #2 (green in Fig. 1) counts neutrons emitted from the central band of the moderator with a small vertical divergence. 3. Results 3.1. Impact of rotation on neutron emission Neutrons emitted by the primary source eventually end up in one of the following outcomes: 3
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Nuclear Inst. and Methods in Physics Research, A 959 (2020) 163562
Table 2 Probability (%) of neutron outcomes. Condition of the moderator
(A) Lost through openings
(B) Absorbed
(C) Exit via top/bottom E > 0.18 eV
(D) Exit via side E > 0.18 eV
(E) Exit via top/bottom E ≤ 0.18 eV
(F) Exit via side E ≤ 0.18 eV
300 K 0 rpm
53.8
4.5
13.9
18.3
4.9
4.6
20 K 0 rpm
53.8
8.3
13.8
18.2
3.1
2.8
20 K 75 000 rpm
53.8
7.3
13.9
18.2
1.2
5.6
Fig. 4. Trajectories of individual neutrons. The trajectories of neutrons, among the fifty first primary neutrons randomly generated by the source, that were thermalized to a final energy E ≤ 0.18 eV in a stationary room-temperature (300 K) moderator (upper panels) and those in the same moderator but rotating and cooled down to 20 K (lower panels). The outline of the cylindrical moderator appears as a circle when viewed from above (panels a and c) and as a rectangle in the side view (panels b and d). 19 neutrons in the static case and 22 neutrons in the rotating case are shown. Some of these neutrons are lost to absorption.
without a preferential direction. In contrast, the overall path of lowfinal-E neutrons in a rotating moderator looks like a spiral (panel c) and they exit preferentially in the same direction as the rotation, and closer to the horizontal plane. Those that remain at high energy do not manifest this preferred directionality, a fact that could be exploited to ‘‘filter out’’ fast neutrons from a thermal neutron beam.
energy cut-off is the same between Fig. 5 and Table 2, the sum of the neutrons under each curve is equal to the corresponding number appearing in the last column of Table 2 multiplied by 108 neutrons (i.e. percentage/100 × 108 neutrons). The cold static moderator (black curve) produces a sharp spectrum with a peak at 2.1 meV. Equivalent to 24 K, this peak position corresponds closely to the chosen temperature of the moderator material (20 K). Compared to the static moderator at room temperature (red curve), the peak neutron count has markedly increased by cooling, from ∼1300 to more than 8000 neutrons/0.02 meV/108 . However, due to the much reduced width of the distribution, the integrated flux, i.e. the area under the curve, has decreased from 4.57 × 106 to 2.82 × 106 neutrons. Physically, this reduction is explained by an increase in losses by absorption. The same moderator rotating at 75 000 rpm yields a spectrum with a peak shifted to 29 meV (green curve). The shift in energy is as expected; however, due to broadening, the peak
3.3. Energy distribution of neutrons from the rotating moderator The energy distribution of the low-E neutrons is of critical importance in the design of a device intended as a source of thermal neutron beams. Fig. 5 shows the distributions obtained from the three simulations listed in Table 2, each based on 108 primary neutrons. The counts plotted on the 𝑦-axis are the number of neutrons escaping through the yellow detector grouped in energy bins of 0.02 meV width (hence the unit chosen is neutrons/0.02 meV/108 ). Given that the 4
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Nuclear Inst. and Methods in Physics Research, A 959 (2020) 163562
2. The neutron passes through both the yellow and green detectors; and 3. The trajectory is within 1◦ of the most probable direction of emission. Conditions 2 and 3 together mean that those neutrons counted in the enhancement factor come from the middle 5.3 cm high band of the yellow detector; this is equivalent to viewing the side of the moderator with a 50 mm high 1◦ collimator. To support quantitative analysis of good statistical value, simulations with 109 primary neutrons were undertaken for the evaluation of the enhancement factor. This figure of merit is shown in Table 3 summarizing the conditions and results of the simulations where in one case the moderator is static at 300 K, while in the other it is rotating and at 20 K. These two simulations, as a pair for comparison, were originally carried out with 252 Cf as the primary neutron source. For the 252 Cf source, 114 neutrons in the static case and 6978 neutrons in the rotating case met the selection criteria, resulting in an enhancement factor of 61 ± 6. The ∼10% relative uncertainty in the enhancement factor is the consequence of the static moderator being very inefficient in producing a monochromatic beam, the very reason a rotating moderator is proposed as an alternative. A particularity of the 252 Cf source is that 17.8% of the primary neutrons entering the moderator escape without ever interacting with the moderator material. It is an understandable result, given that the scattering cross section of 1 H becomes small at neutron energies beyond 1 MeV [14] and a large portion of the 252 Cf spectrum is above 1 MeV. Another pair of simulations were performed with 7 Li(𝑝, 𝑛)7 Be reaction as the primary neutron source. With this softer spectrum, only 2.1% of the primary neutrons entering the moderator escape without ever interacting with the moderator material. As shown in Table 3, the number of neutrons meeting the selection criteria increases approximately by a factor of 1.7 in both static and rotating moderators. The enhancement factor obtained, 66 ± 5 is the same within uncertainty as the 61 ± 6 obtained with the 252 Cf source. These results suggest that while the moderator performs more effectively in terms of flux relative to source intensity, the enhancement factor does not really improve if the primary neutron energy is mostly below 1 MeV. This is also expected as the moderator cylinder dimensions have been chosen for enhanced thermalization (i.e. production of thermal neutrons versus absorption losses) and the softer spectrum will not significantly change those dimensions. The chosen moderator thickness for the softer spectrum is 4 cm instead of 6 cm with the same outer diameter used for the 252 Cf spectrum. To evaluate potential further gains to performance, a pair of simulations were undertaken with neutron reflectors. Two reflectors, each a solid cylinder of beryllium (4 Be), 25 cm high and 30 cm in radius, were placed directly above and below the tube moderator. Thus, the 252 Cf source at the center was fully enclosed. The simulations showed that the physics inside the moderator remains unchanged as the cylindrical symmetry was maintained. Better neutron utilization led to about 2.5 times more neutron counts than the case without reflectors and yielded an enhancement factor (cold rotating versus static room temperature) of 58 ± 4.
Fig. 5. Energy spectrum of neutrons escaping the moderator material through the side surface of the model moderator. The condition of each simulation using 108 primary neutrons is as labeled in the figure. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
neutron count decreases nearly by a factor of two (from ∼8000 to ∼4000 neutrons/0.02 meV/108 ). The total number of neutrons under the curve increases from the static to rotating case (from 2.82 × 106 to 5.61 × 106 neutrons) indicating that more neutrons exit through the side surface when the moderator is rotating. The same moderator at room temperature and not rotating will yield a spectrum peaked at ∼29 meV (red curve). This is, of course, how thermal neutrons are traditionally generated. The benefit of the rotation is seen here as the peak neutron count is ∼4 times higher than from a comparable traditional moderator. 3.4. Directional distribution of neutrons from the rotating moderator The neutron exit directions are reported in a coordinate system defined by three mutually orthogonal unit vectors relative to the outer curved surface of the moderator: an outward pointing normal ̂ a tangent vector 𝐭̂ pointing in the same direction as the vector 𝐧, rotation and a vertical vector 𝐯̂ parallel to the axis of rotation and pointing upward. The orientation of the NTV system relative to the laboratory frame depends on the exit point of each neutron. Fig. 6 shows the spatial distribution of exit directions for neutrons with 28 ≤ E ≤ 30 meV, with the projection plane as shown being divided in 100 × 100 pixels. In the static room temperature case (Fig. 6a), the distribution is widespread in space and the emissions are maximized at the center of the plot. In the cold rotating case (Fig. 6b), the distribution is undeniably concentrated in space and the ⃖⃖⃖⃖⃖⃖⃖⃖⃖⃖⃑28≤𝐸≤30 meV = 0.29 𝐧̂ + 0.96 𝐭̂ + 0 𝐯. ̂ most probable exit direction is Mped 3.5. Comparison of the rotating moderator to a conventional neutron source The rotating moderator is intended as a source of thermal neutron beams at a particular energy. Its performance can be compared to the one of a traditional source in producing a highly collimated and monochromatic neutron beam. Table 2 (also in Fig. 5) demonstrates that, in terms of integrated thermal flux, the rotating moderator is not much better than a traditional (i.e. static at 300 K) moderator: an increase from 4.6% to 5.6%, or a mere 22% of improvement, is insufficient to justify the complexity and engineering challenges of designing a practical rotating moderator: simply increasing the power of the primary neutron source by 22% would achieve the same goal. A more appropriate figure of merit suitable for both the narrow energy band and preferred direction of a rotating moderator, is an enhancement factor evaluated from neutrons that meet the following selection criteria: 1. Emitted with an energy within the range 28 ≤ E ≤ 30 meV;
4. Conclusion The simulations presented here demonstrate in principle how a cooled rotating moderator is superior to a comparable traditional (i.e. static) moderator if the goal is to produce a collimated and monochromatic (narrow energy spectrum) neutron beam in the thermal energy range. Neutrons with energy of a few MeV emitted isotropically from a point source can be focused in energy and space by a cold rotating moderator. Compared to a static moderator at room temperature, it is possible with a rotating moderator to direct toward a sample about 60 times more neutrons with energy of ∼29 ± 1 meV. The moderator used in this work for simulations is a hypothetical device, chosen for its simplicity in demonstrating numerically 5
G. Harrisson, G. Li, B. van der Ende et al.
Nuclear Inst. and Methods in Physics Research, A 959 (2020) 163562
Fig. 6. Stereographic projection of neutron exit directions. For neutrons with 28 ≤ E ≤ 30 meV. Zero is white. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 3 Simulation results.
Moderator temperature (K) Rotation speed (rpm) Desired neutron energy (meV) Energy range of interest (meV) Angle window from the most probable exit direction (◦ )
Static
Rotating
300 0 29 28–30 1
20 75 000 29 28–30 1
𝟐𝟓𝟐
Cf spontaneous fission spectrum
Most probable exit direction (NTV system) Number of neutrons meeting selection criteria Enhancement factor (no units) Primary neutrons enter moderator and escape without interacting (%)
1 𝐧̂ + 0 𝐭̂ + 0 𝐯̂ 114
Most probable exit direction (NTV system) Number of neutrons meeting selection criteria Enhancement factor (no units) Primary neutrons enter moderator and escape without interacting (%)
1 𝐧̂ + 0 𝐭̂ + 0 𝐯̂ 185
0.29 𝐧̂ + 0.96 𝐭̂ + 0 𝐯̂ 6978 61 ± 6a 17.8
17.8 𝟕 Li(p,
a
2.1
The ± uncertainty is estimated by assuming that one standard deviation of counting error is
n)𝟕 Be reaction 0.29 𝐧̂ + 0.96 𝐭̂ + 0 𝐯̂ 12 203 66 ± 5a 2.1
√ N, N being the number of
9
neutrons counted. 10 primary neutrons were used per simulation for the evaluation of the enhancement factors.
CRediT authorship contribution statement
how a cold rotating moderator can enhance the production of a lowdivergence, narrow energy band thermal neutron beam. Designing and building an actual device is a significant engineering challenge (moderator material, cryogenic temperatures and high rotational forces), for which substantial design optimization can be sought. However, the task is less daunting if it is viewed as finding an engineering solution integrating the following three different technologies: 1. A compact neutron generator as the primary source; 2. Cryogenics to cool a suitable moderator to ∼20 K or lower temperature; 3. High-speed rotation. While each of these technologies is quite mature on its own, their combination is a unique challenge. For example, commercial compact neutron generators are available, cooling to 20 K or even lower in temperature is readily achievable, and high-speed rotation is utilized in some commercial items (e.g., turbo-molecular pumps operating at 90 000 rpm, high-speed centrifuges, and high-speed flywheels). Therefore, it should be possible to build a rotating moderator with characteristics not much different from our hypothetical model. Any practical design will require in-depth study of potential materials and engineering concerns.
G. Harrisson: Software, Validation, Formal analysis, Investigation, Resources, Data curation, Writing - original draft, Writing - review & editing, Visualization. G. Li: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Writing - review & editing. B. van der Ende: Conceptualization, Methodology, Validation, Investigation, Writing - review & editing, Supervision, Project administration, Funding acquisition. R.B. Rogge: Conceptualization, Methodology, Validation, Investigation, Writing - review & editing, Supervision, Project administration, Funding acquisition. Z. Tun: Conceptualization, Methodology, Validation, Investigation, Writing - original draft, Writing - review & editing. Acknowledgment This work was performed under the auspices of Atomic Energy of Canada Limited’s Federal Nuclear Science and Technology program at the Canadian Nuclear Laboratories. References
Declaration of competing interest
[1] B. Sjenitzer, Neutron Moderation in a Rotating Disc, Delft University of Technology, 2008. [2] S. Mayer, H. Rauch, P. Geltenbort, P. Schmidt-Wellenburg, P. Allenspach, G. Zsigmond, New aspects for high-intensity neutron beam production, Nucl. Instrum. Methods Phys. Res. A 608 (2009) 434–439.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 6
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