Optimizing moderator dimensions for neutron scattering at the spallation neutron source J. K. Zhao, J. L. Robertson, Kenneth W. Herwig, Franz X. Gallmeier, and Bernard W. Riemer Citation: Review of Scientific Instruments 84, 125104 (2013); doi: 10.1063/1.4841875 View online: http://dx.doi.org/10.1063/1.4841875 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/84/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in GIOVE, a shallow laboratory Ge-spectrometer with 100 μBq/kg sensitivity AIP Conf. Proc. 1549, 12 (2013); 10.1063/1.4818065 Boron neutron capture enhancement of fast neutron radiotherapy utilizing a moderated fast neutron beam Med. Phys. 32, 666 (2005); 10.1118/1.1861156 Special absorber neutron detector moderator assembly: A new detector system for flux measurements of collimated 2.5 MeV neutrons Rev. Sci. Instrum. 70, 1194 (1999); 10.1063/1.1149322 Neutron Scattering with Spallation Sources Phys. Today 38, 38 (1985); 10.1063/1.881009 Spallation sources for neutron scattering Phys. Today 35, 19 (1982); 10.1063/1.2914843
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REVIEW OF SCIENTIFIC INSTRUMENTS 84, 125104 (2013)
Optimizing moderator dimensions for neutron scattering at the spallation neutron source J. K. Zhao, J. L. Robertson, Kenneth W. Herwig, Franz X. Gallmeier, and Bernard W. Riemer Instrument and Source Division, Spallation Neutron Source, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
(Received 2 August 2013; accepted 16 November 2013; published online 11 December 2013) In this work, we investigate the effect of neutron moderator dimensions on the performance of neutron scattering instruments at the Spallation Neutron Source (SNS). In a recent study of the planned second target station at the SNS facility, we have found that the dimensions of a moderator play a significant role in determining its surface brightness. A smaller moderator may be significantly brighter over a smaller viewing area. One of the immediate implications of this finding is that for modern neutron scattering instrument designs, moderator dimensions and brightness have to be incorporated as an integrated optimization parameter. Here, we establish a strategy of matching neutron scattering instruments with moderators using analytical and Monte Carlo techniques. In order to simplify our treatment, we group the instruments into two broad categories: those with natural collimation and those that use neutron guide systems. For instruments using natural collimation, the optimal moderator selection depends on the size of the moderator, the sample, and the moderator brightness. The desired beam divergence only plays a role in determining the distance between sample and moderator. For instruments using neutron optical systems, the smallest moderator available that is larger than the entrance dimension of the closest optical element will perform the best (assuming, as is the case here that smaller moderators are brighter). © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4841875] I. INTRODUCTION
The upgrade of the Spallation Neutron Source (SNS) facility1 at Oak Ridge National Laboratory calls for increase of the accelerator power to 2 MW, eventually 3 MW,5 and a low repetition rate, long wavelength second target station.2, 3 Current studies point to a 10 Hz source running at one fifth the power of the first SNS target station. At this power level (333–500 kW), various source optimizations become possible such as a smaller proton beam profile on the target. At the same level of proton power input, next generation moderators can be significantly brighter than the current ones on the first SNS target station.4, 6 Recent studies on moderator optimization find that the dimensions of the neutron moderator play a critical role in thermal and cold neutron production.4, 7 Traditionally, some of the important parameters to be optimized in moderator designs include reflector coupling for coupled moderators and pulse shape for decoupled ones. Ideally, moderator optimization iterates with instrument designs in order to best meet the instruments’ requirements. Because the transverse dimension of the moderator has not been systematically studied as an optimization parameter in the past, generally it has not been incorporated into the design and optimization of current neutron instruments. Moderators on the first SNS target station have a viewing area of 10 cm × 12 cm, which was chosen from past experience where neutron instruments without neutron guides require a certain amount of beam angular divergence on the sample. At the time, the 10 cm × 12 cm viewing surface was a reasonable choice even when neutron guides are considered. With a 5 cm wide, 2.5m neutron guide (m = 1 being natural nickel 0034-6748/2013/84(12)/125104/8/$30.00
guide) starting at ∼2m from the moderator, for example, the moderator can comfortably fill the phase space at the guide entrance for thermal neutrons. Modern neutron instruments can incorporate advanced neutron optical design and components to take advantage of smaller, brighter moderators. Even some current instruments, such as the SNS EQ-SANS diffractometer,8 view only a small, brighter part of the 10 cm × 12 cm moderator. Viewing a smaller, overall significantly brighter moderator4 would directly improve the performance of this instrument. Thus, for modern instrument design, the instrument needs to be matched with an appropriately sized moderator in order to maximize its performance. In order to examine the effect of moderator dimensions on instrument performance, we chose to study two simplified instrument types, those with natural collimation and those that use neutron guide systems. The moderator surface brightness data from Ref. 4 are used which include four coupled and four decoupled moderators of different sizes (Fig. 1). Coupled moderators allow neutrons of all energies reflected by the surrounding materials to enter the moderator. For decoupled moderators, neutron absorbers, such as cadmium, are placed between the moderator and reflector. The absorber prevents thermal and cold neutrons (those with energy below ∼1 eV) from entering the moderator from the reflector. The four coupled moderators are cylinders and are viewed in the radial direction with viewing areas of 3 cm × 3 cm, 5 cm × 5 cm, 7 cm × 7 cm, and 10 cm × 10 cm, respectively. The four decoupled moderators are poisoned slabs with the same sized viewing areas as the coupled moderators. The poison is a neutron absorber in the middle of the moderator slab. It enables the moderator to be thicker for higher brightness while
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FIG. 1. Per-pulse surface brightness of the coupled (upper, thicker lines) and decoupled, poisoned (lower, thinner lines) cold hydrogen moderators on the short pulsed second SNS target station, reproduced from Ref. 4. The coupled moderators are optimized cylinders with different sized viewing areas as labeled in the legend. The decoupled moderators are optimized slabs 4 cm thick for the 10 cm × 10 cm moderator and 4.4 cm thick for the remainder. All decoupled moderators are poisoned with a neutron absorber placed in the center of the slab.
retaining a sharp pulse timing profile. The geometry of each of the eight moderators is optimized for its given viewing area.4 The range of moderator dimensions in this study is selected with the following considerations. On the smaller side, there is a practical limit on how small a moderator can be constructed, especially for cold moderators due to cryogenic requirements. A viewing area around 3 cm × 3 cm appears to be possible, though detailed engineering studies will be required to confirm its feasibility. It is conceivable that an even smaller and brighter moderator7 could be built. However, as the moderator size decreases, the viewpoint of the neutron guides on the moderator gets closer to the target. There is then a potential for increased instrument background. Detailed understanding of instrument background requires extensive studies and is beyond the current scope. In any case, the general conclusions of this work can be readily extended to even smaller and brighter moderators. On the larger end, the 10 cm × 10 cm moderator is about the same size as those used on the first SNS target station. There appears to be no demand for even larger moderators by the existing instruments at the SNS. In either the coupled or decoupled cases, the time structure of the initial neutron pulse does not vary significantly when the viewing area changes (Ref. 4). For that reason, we focus our study only on the neutron intensity at the sample position. II. INTENSITY ON THE SAMPLE WITH NATURAL COLLIMATION
Figure 2 shows the schematic diagram of a neutron scattering instrument with natural collimation and the neutron phase space acceptance diagram at the sample.9, 10 The sample is illuminated directly by the moderator. For simplicity, we consider beam divergence in one dimension
FIG. 2. (Top left) Illustration of a simple instrument with natural collimation. The sample is located at a distance L away from the moderator and is illuminated directly by the moderator. dm and ds are the linear dimensions of the moderator and the sample, respectively. The dashed lines indicate the most divergent neutrons accepted by the sample with the angle θ . (Top right) Illustration of neutron phase space acceptance diagram at the sample position for simple instruments with natural collimation.9, 10 (Bottom) Illustration of intensity integration at the sample position. Surface elements dAm = dxm dym and dAs = dxs dys are highlighted on the moderator and the sample, respectively. (xm ,ym ) and (xs ,ys ) are the coordinate systems on the surface of the moderator and sample, respectively. The distance between dAm and dAs is shown as l = [(xs − xm )2 + (ys − ym )2 + L2 ]1/2 . The projection of dAs onto the plane perpendicular to l is given by dAs cosα, where α is the angle between l and L, and cosα = L/l. The solid angle subtended by dAm at dAs is given by dAm cosα/l2 . Therefore, the neutron intensity received by dAs from dAm is given by dI = Bm dAm dAs cos2 α/l2 = Bm dAm dAs L2 /l4 , where Bm is the moderator brightness. Integration over dI results in Eq. (2).
only. The maximum beam divergence accepted by the sample is determined by 1 (1) (dm + ds )/L, 2 where L is the distance from the sample to the moderator and dm and ds are the maximum linear dimensions of the moderator viewing area and the sample cross-section (i.e., transverse areas), respectively. For circular cross-sections, dm and ds are the diameters of the moderator and the sample, respectively. For rectangular and square cross-sections, dm and ds are the diagonals. If the horizontal and vertical divergences are considered separately, then dm and ds can be considered as the horizontal and vertical dimensions, respectively. Within the limit of beam divergence angles ±θ , the intensity on the sample, I, in unit of neutrons Å−1 s−1 (or Å−1 pulse−1 ), is obtained by integrating the moderator brightness Bm (in unit of neutrons cm−2 sr−1 Å−1 s−1 or cm–2 sr−1 Å–1 pulse–1 ) over the solid angle m subtended by the moderator area Am at the sample position, and over the sample area As . Expressed in terms of the moderator and sample dimensions xm , ym , xs , and ys integrated over the surfaces of the moderator and sample (See Fig. 2 for details), I is given as tan θ =
I = L2 ∫ Bm dxs dys ∫[(xs − xm )2 + (ys − ym )2 + L2 ]−2 × dxm dym .
(2)
In practice, the condition L » dm is almost always true. Additionally, the moderator brightness Bm can often be approximated to be uniform across the moderator face, i.e., Bm is a constant. The intensity on the sample is
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dAs dm , and further to then simplified to I = Bm I = Bm Am As / L2 . Replacing L using Eq. (1) yields I = 4Bm As Am tan θ/(dm + ds ) . 2
2
(3)
Because we are interested in the relative performance between the moderators, we use the largest moderator in the current study, the 10 cm × 10 cm one, as our baseline. The intensity on the sample for other moderators is then expressed as the ratio to that obtained from the 10 cm × 10 cm case, rm = Im /I10 , where Im is the intensity on the sample when a moderator with an m × m viewing area is used. For the current discussion with square moderator viewed surfaces and square samples, the surface area is approximated by A ∼ d2 . The relative intensity on the sample is then given by (4) rm = (Bm /B10 )dm 2 (d10 + ds )2 / d10 2 (dm + ds )2 . In order for moderators of different dimensions to provide the same beam divergence, ±θ , on the same sized sample, the distance L between the moderator and the sample must be adjusted. We chose to make the comparison between moderators for fixed beam divergence at a given sample size as this is often a parameter that is critical to the performance of neutron scattering instruments. We note though, that the wavelength resolution will change for different L on instruments using time-of-flight techniques and this may place a constraint on freely adjusting L. According to Eq. (4), the intensity ratio rm is now only a function of the moderator and sample sizes but not the beam divergence. Obviously, with any given moderator, the on-sample neutron intensity increases as the maximum beam divergence increases (the sample moves closer to the moderator). However, as long as the sample is far enough from the moderator such that Eq. (3) is valid, such dependency is the same for all moderators. It is instructive to consider the limit as the sample dimension approaches zero in Eq. (4). In this case, only the moderator brightness is relevant and rm = (Bm /B10 ). The trend is that small samples will always be best illuminated by the brightest moderators, regardless of the moderator dimension. At the other limit, we can consider the case where the sample dimension is much greater than that of the moderators, ds » dm . In this case, rm = (Bm /B10 ) · (dm 2 /d10 2 ) and unless the moderator brightness increases faster than the viewed area, large samples will favor larger moderators. The relative moderator brightness (Bm /B10 ) in Eq. (4) is shown in Fig. 3. In principle, the wavelength dependent rm value for each individual instrument should be calculated in order to determine the best moderator for that instrument. On the second target station with a repetition rate of ∼10 Hz, most instruments will be using a broad neutron wavelength band of several Angstroms. Therefore, we believe that it is reasonable to use a (Bm /B10 ) value averaged over the neutron wavelength range of 1–5 Å (Table I). For instruments that can use an even broader wavelength band, the average (Bm /B10 ) values vary somewhat but not significantly (Table I). Using these average moderator brightness ratios, rm as a function of sample size is plotted in Fig. 4. As indicated by the limiting behavior discussed above, the small bright moderators significantly outperform larger ones for small samples. As the
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FIG. 3. Ratio of the coupled (thicker lines) and decoupled (thinner lines) moderator surface brightness versus that of the corresponding 10 cm × 10 cm moderator as a function of neutron wavelength. The average brightness ratios for the wavelength ranges of 1–5 Å and 0.5–10 Å are listed in Table I.
sample dimension increases, larger moderators start to become more favored. Generally speaking, when choosing coupled moderators, the moderator dimension should be as small as possible but still slightly larger than the desired sample size. For example, if the maximum sample size for an instrument is 2.5 cm or less, then the 3 cm × 3 cm moderator will deliver the best performance. For larger samples, the 5 cm × 5 cm, 7 cm × 7 cm, or the 10 cm × 10 cm moderator will become the better choice. For decoupled moderators, the moderator should be at least twice as large as the sample. The difference between the coupled and decoupled moderators is due to the fact that the brightness gains for small decoupled moderators are less than their coupled counterparts (see Fig. 3). III. MONTE CARLO SIMULATIONS
The fact that the on-sample intensity ratio, rm , from Eq. (4) is independent of the maximum beam divergence is readily confirmed by Monte Carlo simulations. In Fig. 4, the intensity gains simulated for the 3 cm × 3 cm, 5 cm × 5 cm, and 7 cm × 7 cm moderators over that of the 10 cm × 10 cm moderator are shown for maximum beam divergence angles of θ = 0.1◦ , 0.5◦ , and 0.9◦ , respectively. The simulations were carried out using the program ib.11 All the simulated data agree reasonably well with the calculated curves regardless of the maximum acceptable beam divergence values at the sample. Note that for any given moderator and sample size, the results at all beam divergences lie essentially on top of TABLE I. Ratios of the average moderator brightness for the 3 × 3, 5 × 5, and 7 × 7 moderators to the 10 × 10 moderator over the neutron wavelength ranges of 1–5 Å and 0.5–10 Å. Coupled moderators Wavelength range 1–5 Å 0.5–10 Å
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FIG. 4. On-sample intensity ratios for instruments with natural collimation when using the 3 cm × 3 cm, 5 cm × 5 cm, and 7 cm × 7 cm coupled (upper) and decoupled (lower) moderators with the corresponding 10 cm × 10 cm moderator. The curves are calculated using Eq. (4) using the average moderator brightness values given in Table I. The symbols indicate the result of Monte Carlo simulations with three different maximum beam divergence angles, 0.1◦ , 0.5◦ , and 0.9◦ . The simulated intensities are averaged over a neutron wavelength range of 1–5 Å. For any given moderator and sample size, the results for all three beam divergences lie essentially on top of one another.
one another. The small differences between the calculations and simulations can be attributed to various factors. In the calculations, for example, the moderator brightness Bm is assumed to be uniform across all emission angles. In the Monte Carlo simulations, the moderator is treated as a Lambert source which is at its brightest in the direction of the moderator surface normal. Also, Eq. (4) uses the average moderator brightness values whereas the simulations use the actual source files from Ref. 4. On the second SNS target station, the number of instruments that use natural collimation will likely be limited. For those that do, Fig. 4 offers a guideline for their moderator selection. In Sec. IV, we will examine the moderator selection criteria for instruments that use neutron transport optics in their design. IV. INSTRUMENTS WITH NEUTRON OPTICS
Most modern neutron scattering instruments use neutron optical systems to reduce the instrument background,
2
4 Sample size [cm]
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FIG. 6. Calculated (lines) and simulated (symbols) on-sample intensity ratios of the 3 cm × 3 cm, 5 cm × 5 cm, 7 cm × 7 cm coupled moderators with the 10 cm × 10 cm moderator for an instrument with a 50 m long neutron guide as shown in Fig. 5. The sample size is considered to be half that of neutron guide cross-section. A perfect neutron guide is assumed for the numerical calculations. Namely, the neutron intensities at the guide exit and entrance are the same. For the Monte Carlo simulations, the maximum beam divergence accepted by the neutron guide and the sample are both kept at ±0.2◦ . The neutron guide has a 3.5m supermirror coating with a 70% reflectivity at the critical angle. The intensity ratios are averaged over the neutron wavelength range of 1–5 Å.
manipulate the neutron phase space, or extend the total instrument length for increased time-of-flight timing resolution. For the current purpose of evaluating moderator performance, a simple straight neutron guide is used as a representative optical system (see Fig. 5). The maximum neutron beam divergence accepted by the sample is defined by the transverse size of the neutron guide or a slit at the end of the neutron guide, the transverse size of the sample, and the distance between them. A simple optimization is to have the neutron guide accept the same maximum beam divergence, ±θ , as that desired at the sample, namely by placing the neutron guide at L1 = 12 (dm + dg ) cotθ away from the moderator (Fig. 5). The cross-section of the neutron guide now becomes the equivalent of the sample area in Eq. (3) and for reasonable values of beam divergence, θ < 2◦ , the requirement that L1 » dm is valid. Because neutron guides commonly have cross-sections greater than the sample, the effectiveness of the small bright moderators is diminished. This effect is demonstrated in Fig. 6 where the sample cross-section is taken to be half that of the neutron guide. Obviously, such guide placement is not optimal and results in a severely under-illuminated guide.9, 12 The effect
FIG. 5. Schematic diagram of an instrument with a straight guide between the sample and the moderator. The neutron guide, represented by the solid-line rectangle, accepts the same amount of maximum beam divergence (θ ), represented by the oblique dashed lines, as the sample. dm and dg are the linear dimensions of the moderator and the neutron guide cross-sections. The red dashed lines represent optimal guide placement with the guide located L1 = 12 (dm − dg )cotθ away from moderator.
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FIG. 7. Neutron phase space acceptance diagram at the entrance of a 3.2 cm neutron guide viewing a 5 cm × 5 cm moderator in the horizontal direction. The intended maximum beam divergence angle at the sample is ±0.2◦ . For the left panel, the neutron guide is placed at L1 = 12 (dm + dg )cotθ away from the moderator (see text). For the right panel, it is placed at L1 = 12 (dm − dg )cotθ . The apparent difference in the color scale in the graphs is due to the difference in the minimum of neutron counts.
can be seen from the neutron phase space acceptance diagram simulated at the entrance of a 3.2 cm wide neutron guide and a 5 cm × 5 cm moderator in Fig. 7. The guide width is chosen arbitrarily to be less than the dimension of the moderator. The instrument setup for the simulation is intended to deliver 0.2◦ maximum beam divergence to the sample. In the left panel of the figure, the neutron guide positioned at L1 = 12 (dm + dg ) cotθ accepts maximum beam divergence of ±0.2◦ , which leaves a large part of the phase space unfilled. To take the neutron guide totally out of consideration in moderator evaluations, one would need to design a perfect guide with the same cross-section as the moderator, and fully saturate it by placing the guide entrance at the moderator face. This is of course impossible in practice. Once the desired maximum beam divergence angle θ and the guide dimension dg are determined, however, we can extend the neutron guide upstream by L1 = dg cotθ and completely fill, even if not saturating, the phase space within the divergence angles of ±θ ◦ (dashed lines in Fig. 5), as seen in the right panel of Fig. 7. That is, the neutron guide should be placed downstream from the moderator at a distance of L1 =
1 (dm − dg ) cot θ. 2
(5)
This condition can only be satisfied when dg < dm , namely when the neutron guide cross-section is smaller than the moderator. In practice, engineering constraints (heat load,
radiation damage, crowding by neighboring beam lines) impose limits on the closest possible guide location. For the first SNS target station, this limit is on the order of 1 m from the moderator.8, 13 Here, we assume that this 1 m limit also applies to the SNS second target station. This condition arises for guide dimensions dg ≥ dm − tanθ · 200 cm. Figure 8 compares the beam delivery performance for the two guide entrance positions, L1 = 12 (dm − dg ) cotθ or L = 12 (dm + dg ) cotθ away from the moderator, subject to a 1 m minimum distance limitation. This set of simulations modeled a sample size half that of the neutron guide for a beam divergence at the sample, θ = ±0.2◦ . Placing the guide entrance at L1 resulted in a maximum gain of intensity on the sample of somewhat greater than a factor of 2 when the dimension of the neutron guide approaches that of the moderator. When the guide cross-section is much smaller than the moderator, L1 ≈ L1 and therefore the intensity delivered to the sample is nearly identical. When the guide cross-section is larger than the moderator dimension, neutron phase space cannot be filled even if the guide entrance were located at the moderator surface. For this simulation study, optimum performance occurs when the guide dimension matches that of the moderator and its entrance can be placed at L1 . At the relatively modest maximum beam divergence angle, θ = ±0.2◦ , used to generate the data in Fig. 8, the lower limit of 1 m for L1 essentially takes effect when dm = dg . At larger values of θ , however, this limit will be reached
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FIG. 8. Monte Carlo simulations of the ratio of the neutron intensity on the sample with the neutron guide placed at L1 = 12 (dm − dg )cotθ compared to L = 12 (dm + dg )cotθ away from the moderator (Fig. 5). Data for the 3 cm × 3 cm, 5 cm × 5 cm, 7 cm × 7 cm, and 10 cm × 10 cm coupled moderators are shown. The neutron guide has a 3.5m supermirror coating with a 70% reflectivity at the critical angle. The sample size is half the linear dimension of the guide cross-section. The maximum accepted beam divergence angle at the sample is set to be ±0.2◦ . L1 is restricted to L1 ≥ 1 m. The intensity ratios are obtained by averaging the simulated intensity within the neutron wavelength range of 1–5 Å.
at smaller values of dg and Fig. 8 will exhibit some divergence dependence. Nonetheless, the maximum gain still occurs when the neutron guide dimension matches that of the moderator, but it decreases progressively as θ increases (data not shown). We now come back to the choice of moderators for neutron scattering instruments with neutron guides. When placed at L1 , the neutron guide accepts more beam divergence than the sample requires and Eq. (3) cannot be used to compare moderator performance since it does not properly account for these extra neutrons. In order to compare the performance of different moderators we can use Monte Carlo simulations to obtain the useful intensity at the sample position, or we can numerically integrate Eq. (2) up to the desired beam divergence. Numerical integration of Eq. (2) determines the useful intensity accepted at the guide entrance. If we assume that the guide is effective at delivering the desired beam divergence, the intensity at the sample position should be proportional to the intensity at the guide entrance. In either case, the ratio to the baseline moderator size of 10 cm × 10 cm provides a valid metric for moderator selection and is shown in Fig. 9. Note that the entire brightness gain of the smaller moderators is experienced for all guide dimensions up to the moderator size. Implicit in the observation that the maximum gain is set by the ratio of moderator brightness, is that a larger moderator of equivalent brightness will not outperform the smaller moderator until the desired guide dimension exceeds that of the small moderator. This observation is dependent on the maximum beam divergence to be transmitted to the sample, see discussion below. The best-match moderator for an instrument with a guide should be the smallest one that is still larger than or approximately the same size as the neutron guide cross-section at the guide entrance. For example, if an instrument has a neutron guide with a 3 cm or less cross section at the entrance, then the 3 cm × 3 cm moderator will be
Intensity Ratio vs 10x10 Moderator
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FIG. 9. Intensity ratios of the 3 cm × 3 cm, 5 cm × 5 cm, and 7 cm × 7 cm coupled (upper) and decoupled (lower) moderators to the corresponding 10 cm × 10 cm moderator. The lines are a numerical integration of Eq. (2) within the beam divergence of ±0.2◦ at the entrance of the neutron guide located at L1 = 12 (dm − dg )cotθ . The symbols are Monte Carlo simulations of the intensity on the sample averaged over the neutron wavelength range of 1–5 Å. The linear dimension of the sample is half that of the neutron guide cross-section. A 50 m long, 3.5m supermirror coated neutron guide with 70% reflectivity is used for the simulation. L1 is restricted to L1 ≥ 1 m for both the simulations and the numerical integrations.
the best choice for delivering the most neutrons to the sample. Guides whose entrance can be located at L1 take even better advantage of the small, brighter moderators than the natural collimation case (compare Fig. 9 with Fig. 4). For the Monte Carlo simulations in Fig. 9, we used 3.5m supermirror coating for the neutron guides, where m = 1 corresponds to the critical reflection angle of natural nickel. Because of the rather limited maximum divergence angle of θ = ±0.2◦ used in the simulation, these guides have little effect on our conclusion. It is the transverse size of the neutron guide at its entrance that determines the best moderator dimensions for an instrument (Fig. 9). At larger divergence angles, guide properties other than its entrance dimensions will have some influence on our discussion. This is because neutron transport by a guide may be less efficient and this reduction in efficiency will be neutron wavelength and beam divergence dependent. However, these changes in neutron transport will occur in the same way regardless what moderator is used. More important is the effect of a practical limit placed on the minimum value of L1 . When L1 from Eq. (5) is less than this limit, the relative moderator performance becomes divergence dependent. For example, imposing the 1m limit used above, a neutron guide cross-section of 2 cm has threshold angles above which the moderator performance is divergence dependent of about 0.3◦ , 0.9◦ , 1.4◦ , and 2.3◦ for the 3 cm × 3 cm, 5 cm × 5 cm, 7 cm × 7 cm, and 10 cm × 10 cm moderators, respectively. At different minimum L1 limits, the cotangents of these angles scale in proportion to the new limit. At larger θ , the small bright moderators will lose their advantage over the larger 10 cm × 10 cm one, because the neutron guide cannot be extended close enough to be fully illuminated by the smaller moderator. However, the loss is not significant until
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2.5
0.1 ° 0.5 ° 1° 1.5°
2 1.5 1 0.5
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0 1.8 0
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0.6 0
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FIG. 10. Intensity ratios for the 3 cm × 3 cm (upper), 5 cm × 5 cm (middle), and 7 cm × 7 cm (lower) coupled moderators with the corresponding 10 cm × 10 cm moderator at different maximum divergence angles at the entrance to the neutron guide. The curves for decoupled moderators have the same shapes but different vertical scales (see Fig. 9 and Table I). Note that the divergence ranges calculated for each moderator size are different. Except for the difference in beam divergence and the corresponding different guide to moderator distance, the instrument setup is the same as that in Fig. 9. The intensities are integrated at the entrance of the neutron guide using Eq. (2) within the divergence angles as labeled in the legend. The minimum guide to moderator distance is subject to a 1 m limitation.
the angles are two or three times above the threshold. This effect is seen in Fig. 10, which shows divergence-dependent intensity ratios at the entrance of the neutron guides for different sized moderators calculated using Eq. (2). The data are plotted using Bm values (Eq. (2)) for coupled moderators. For decoupled moderators, the curves are the same but the plots have different vertical scales due the difference in Bm between the coupled and decoupled moderators (Table I and Fig. 9). In a recent study, it is found that elliptic and parabolically tapered ballistic guides are more efficient at transporting divergent neutrons than straight guides.18 One interesting fact from the study is that the source extraction spot of these optics is smaller than the transverse dimensions of the guide. This means that these novel neutron optics favor small bright moderators even at larger divergences. Thus, the situations presented in Fig. 10 can be seen as the worse-case scenario for small bright moderators. V. CONCLUSIONS
The results from Secs. II and IV can be summarized by the following general strategies for determining the appropriate sized moderator for a neutron scattering instrument. For instruments that employ neutron guide systems, the choice for both coupled and decoupled moderators should be the smallest moderator available that is larger than or approximately the same size as the guide cross-section at the guide entrance.
For instruments using natural collimation, the choice for coupled moderators should be the smallest one that is larger than the sample size. For the decoupled moderators, the choice should be the smallest moderator that is at least twice the size of the sample. These conclusions are valid as long as the smaller moderators have a higher brightness than the larger ones. Conversely, these same strategies can be used to decide what size moderators should be built at spallation neutron sources such as the second SNS target station given a suit of instrument requirements. In general, neutron guides on instruments have larger cross-sections than the samples to be studied. For small angle scattering instruments, such as the EQ-SANS instrument on the first SNS target station,8 the dimension of the collimation slit at the end of the neutron guide is usually twice as large as the sample. Most samples studied on SANS instruments are 1 cm or less in size. The maximum beam divergence angle on these instruments is almost always less than 1◦ . A typical experimental configuration for a SANS experiment is to use a 2 cm collimation slit, 1 cm sample, and a 4 m collimation distance, which gives a maximum beam divergence angle of 0.21◦ . Instruments like SANS diffractometers will thus clearly benefit from the smallest, 3 cm × 3 cm moderator. Other instruments that view a coupled moderator such as cold neutron chopper spectrometers14 and spin echo spectrometers,15 may study larger samples. For these instruments, either the 5 cm × 5 cm or the 7 cm × 7 cm moderators offer a good choice. Instruments that will benefit most from viewing the smallest 3 cm × 3 cm decoupled moderator are single crystal diffractometers, especially those that are optimized for studying macromolecules16 where samples under study are typically less than 1 mm in size. Similarly for a high resolution powder diffractometer, the strict resolution requirements on beam divergence and sample size means that this instrument will experience a significant performance boost on the 3 cm × 3 cm moderator. On the other hand, a high intensity powder diffractometer,17 or a backscattering spectrometer13 can be designed to have a large beam cross-section and use beam focusing to increase the intensity on the sample. For these instruments, a 7 cm × 7 cm or even larger moderator should be considered. These conclusions on moderator choices should be valid regardless of what neutron optical systems are used, because it is the dimension of the neutron beam cross-section at the guide entrance and the maximum beam divergence the optics is designed to transport that are the deciding factors. The optimal guide dimensions or the maximum beam divergence must be optimized for each neutron scattering instrument individually. Once these parameters are finalized, the optical system should then be placed according to Eq. (5) relative to the moderator. On most instruments, the collimation before the sample is fixed and the maximum divergence angle θ is readily obtained. On others, the collimation is variable between experiments. In the latter case, an appropriate θ value needs to be chosen before applying Eq. (5). As an example, on the asbuilt EQ-SANS instrument on the first SNS target station,8 the shortest possible collimation length is ∼2 m. Assuming a typical 1 cm sample and 2 cm collimation slit, the maximum
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beam divergence accepted by the sample at this collimation distance is θ ∼ 0.43◦ . The neutron guide system on the instrument is 4 cm × 4 cm in cross-section. It views the brightest spot at the lower left corner of the 10 cm × 12 cm coupled moderator, which is 2 cm displaced from the moderator center both horizontally and vertically. The shortest distance between the moderator edge and the neutron guide projection onto the moderator is thus 1 cm. Therefore, for the current discussion, the effective moderator dimension can be considered to be 6 cm. Applying these values to Eq. (5), the neutron guide should start at 1.33 m from the moderator, which is essentially the same as the as-built value of 1.28 m. Ideally, each class of neutron scattering instruments should have a dedicated moderator optimized to the requirements for that instrument class. However, in both the coupled and decoupled cases, two to three moderators with viewing areas in the range of 3 cm × 3 cm to 7 cm × 7 cm appear to be enough to serve the needs of most instruments. In any case, appropriately designed moderators will be crucial to the scientific productivity of the SNS facility for years to come. ACKNOWLEDGMENTS
This article has been authored by UT-Battelle, LLC, under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The work was sponsored by the Office of Basic Energy Sciences, US Department of Energy. The authors wish to thank Dr. Erik Iverson of SNS for reading and critiquing the article. 1 T. E. Mason, D. Abernathy, I. Anderson, J. Ankner, T. Egami, G. Ehlers, A.
Ekkebus, G. Granroth, M. Hagen, K. Herwig, J. Hodges, C. Hoffmann, C. Horak, L. Horton, F. Klose, J. Larese, A. Mesecar, D. Myles, J. Neuefeind,
Rev. Sci. Instrum. 84, 125104 (2013) M. Ohl, C. Tulk, X.-L. Wang, and J. Zhao, Physica B 385–386, 955–960 (2006). 2 J. M. Carpenter and T. E. Mason, “Technical concepts for a long wavelength target station for the spallation neutron source,” SNS Document: ANL-02/16, ORNL/SNS-TM-2001/163, Oak Ridge, TN, USA, 2002. 3 Oak Ridge National Laboratory, “White paper: A second target station for the spallation neutron source,” SNS Document: 100000000-TR0029-R00, Oak Ridge, TN, USA, 2007. 4 F. X. Gallmeier, “Moderator studies for a SNS short-pulse second target station,” SNS Document: STS04-41-TR00-R00, Oak Ridge, TN, USA, 2013. 5 Oak Ridge National Laboratory, “SNS power upgrade project closeout report,” SNS Document: PUP0 301 PI0001 R00, Oak Ridge, TN, USA, 2011. 6 F. X. Gallmeier, E. B. Iverson, W. Lu, P. D. Ferguson, and R. K. Crawford, “SNS second target station moderator performance update,” 19th Meeting on Collaboration of Advanced Neutron Sources, Grindelwald, Switzerland, 2010. 7 K. Batkov, A. Takibayev, L. Zanini, and F. Mezei, Nucl. Instrum. Methods Phys. Res. A 729, 500–505 (2013). 8 J. K. Zhao, C. Y. Gao, and D. Liu, J. Appl. Cryst. 43, 1068–1077 (2010). 9 J. M. Carpenter and D. F. R. Mildner, Nucl. Instrum. Methods Phys. Res. 196, 341–348 (1982). 10 J. R. D. Copley, Nucl. Instrum. Methods Phys. Res. A 287, 363–373 (1990). 11 J. K. Zhao, Nucl. Instrum. Methods Phys. Res. A 659(1), 434–441 (2011). 12 C. J. Carlile, M. W. Johnson, and W. G. Williams, “Neutron guides on pulse sources,” Rutherford Laboratory, RL-79-084, November 1979. 13 E. Mamontov and K. W. Herwig, Rev. Sci. Instrum. 82, 085109 (2011). 14 G. Ehlers, A. A. Podlesnyak, J. L. Niedziela, E. B. Iverson, and P. E. Sokol, Rev Sci. Instrum. 82, 085108 (2011). 15 M. Ohl, M. Monkenbusch, N. Arend, T. Kozielewski, G. Vehres, C. Tiemann, M. Butzek, H. Soltner, U. Giesen, R. Achten, B. Lindenau, A. Budwig, H. Kleines, M. Drochnerd, P. Kaemmerling, M. Wagener, R. Moller, E. B. Iverson, M. Sharp, and D. Richter, Nucl. Instrum. Methods Phys. Res. A 696, 85–99 (2012). 16 L. Coates, A. D. Stoica, C. Hoffmann, J. Richards, and R. Cooper, J. App. Cryst. 43, 570–577 (2010). 17 A. Huq, J. P. Hodges, O. Gourdon, and L. Heroux, Z. Kristallogr. Proc. 1, 127–135 (2011). 18 K. H. Klenø, K. Lieutenant, K. H. Andersen, and K. Lefmann, Nucl. Instrum. Methods Phys. Res. A 696, 75–84 (2012).
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REVIEW OF SCIENTIFIC INSTRUMENTS 84, 125104 (2013)
Optimizing moderator dimensions for neutron scattering at the spallation neutron source J. K. Zhao, J. L. Robertson, Kenneth W. Herwig, Franz X. Gallmeier, and Bernard W. Riemer Instrument and Source Division, Spallation Neutron Source, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
(Received 2 August 2013; accepted 16 November 2013; published online 11 December 2013) In this work, we investigate the effect of neutron moderator dimensions on the performance of neutron scattering instruments at the Spallation Neutron Source (SNS). In a recent study of the planned second target station at the SNS facility, we have found that the dimensions of a moderator play a significant role in determining its surface brightness. A smaller moderator may be significantly brighter over a smaller viewing area. One of the immediate implications of this finding is that for modern neutron scattering instrument designs, moderator dimensions and brightness have to be incorporated as an integrated optimization parameter. Here, we establish a strategy of matching neutron scattering instruments with moderators using analytical and Monte Carlo techniques. In order to simplify our treatment, we group the instruments into two broad categories: those with natural collimation and those that use neutron guide systems. For instruments using natural collimation, the optimal moderator selection depends on the size of the moderator, the sample, and the moderator brightness. The desired beam divergence only plays a role in determining the distance between sample and moderator. For instruments using neutron optical systems, the smallest moderator available that is larger than the entrance dimension of the closest optical element will perform the best (assuming, as is the case here that smaller moderators are brighter). © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4841875] I. INTRODUCTION
The upgrade of the Spallation Neutron Source (SNS) facility1 at Oak Ridge National Laboratory calls for increase of the accelerator power to 2 MW, eventually 3 MW,5 and a low repetition rate, long wavelength second target station.2, 3 Current studies point to a 10 Hz source running at one fifth the power of the first SNS target station. At this power level (333–500 kW), various source optimizations become possible such as a smaller proton beam profile on the target. At the same level of proton power input, next generation moderators can be significantly brighter than the current ones on the first SNS target station.4, 6 Recent studies on moderator optimization find that the dimensions of the neutron moderator play a critical role in thermal and cold neutron production.4, 7 Traditionally, some of the important parameters to be optimized in moderator designs include reflector coupling for coupled moderators and pulse shape for decoupled ones. Ideally, moderator optimization iterates with instrument designs in order to best meet the instruments’ requirements. Because the transverse dimension of the moderator has not been systematically studied as an optimization parameter in the past, generally it has not been incorporated into the design and optimization of current neutron instruments. Moderators on the first SNS target station have a viewing area of 10 cm × 12 cm, which was chosen from past experience where neutron instruments without neutron guides require a certain amount of beam angular divergence on the sample. At the time, the 10 cm × 12 cm viewing surface was a reasonable choice even when neutron guides are considered. With a 5 cm wide, 2.5m neutron guide (m = 1 being natural nickel 0034-6748/2013/84(12)/125104/8/$30.00
guide) starting at ∼2m from the moderator, for example, the moderator can comfortably fill the phase space at the guide entrance for thermal neutrons. Modern neutron instruments can incorporate advanced neutron optical design and components to take advantage of smaller, brighter moderators. Even some current instruments, such as the SNS EQ-SANS diffractometer,8 view only a small, brighter part of the 10 cm × 12 cm moderator. Viewing a smaller, overall significantly brighter moderator4 would directly improve the performance of this instrument. Thus, for modern instrument design, the instrument needs to be matched with an appropriately sized moderator in order to maximize its performance. In order to examine the effect of moderator dimensions on instrument performance, we chose to study two simplified instrument types, those with natural collimation and those that use neutron guide systems. The moderator surface brightness data from Ref. 4 are used which include four coupled and four decoupled moderators of different sizes (Fig. 1). Coupled moderators allow neutrons of all energies reflected by the surrounding materials to enter the moderator. For decoupled moderators, neutron absorbers, such as cadmium, are placed between the moderator and reflector. The absorber prevents thermal and cold neutrons (those with energy below ∼1 eV) from entering the moderator from the reflector. The four coupled moderators are cylinders and are viewed in the radial direction with viewing areas of 3 cm × 3 cm, 5 cm × 5 cm, 7 cm × 7 cm, and 10 cm × 10 cm, respectively. The four decoupled moderators are poisoned slabs with the same sized viewing areas as the coupled moderators. The poison is a neutron absorber in the middle of the moderator slab. It enables the moderator to be thicker for higher brightness while
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1E+12
Surface Brightness [n/steradian/pulse/Å/cm2]
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ys
ym xm 3cm×3cm, Coupled 5cm×5cm, Coupled 7cm×7cm, Coupled 10cm×10cm, Coupled 3cm×3cm, Decoupled 5cm×5cm, Decoupled 7cm×7cm, Decoupled 10cm×10cm, Decoupled
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FIG. 1. Per-pulse surface brightness of the coupled (upper, thicker lines) and decoupled, poisoned (lower, thinner lines) cold hydrogen moderators on the short pulsed second SNS target station, reproduced from Ref. 4. The coupled moderators are optimized cylinders with different sized viewing areas as labeled in the legend. The decoupled moderators are optimized slabs 4 cm thick for the 10 cm × 10 cm moderator and 4.4 cm thick for the remainder. All decoupled moderators are poisoned with a neutron absorber placed in the center of the slab.
retaining a sharp pulse timing profile. The geometry of each of the eight moderators is optimized for its given viewing area.4 The range of moderator dimensions in this study is selected with the following considerations. On the smaller side, there is a practical limit on how small a moderator can be constructed, especially for cold moderators due to cryogenic requirements. A viewing area around 3 cm × 3 cm appears to be possible, though detailed engineering studies will be required to confirm its feasibility. It is conceivable that an even smaller and brighter moderator7 could be built. However, as the moderator size decreases, the viewpoint of the neutron guides on the moderator gets closer to the target. There is then a potential for increased instrument background. Detailed understanding of instrument background requires extensive studies and is beyond the current scope. In any case, the general conclusions of this work can be readily extended to even smaller and brighter moderators. On the larger end, the 10 cm × 10 cm moderator is about the same size as those used on the first SNS target station. There appears to be no demand for even larger moderators by the existing instruments at the SNS. In either the coupled or decoupled cases, the time structure of the initial neutron pulse does not vary significantly when the viewing area changes (Ref. 4). For that reason, we focus our study only on the neutron intensity at the sample position. II. INTENSITY ON THE SAMPLE WITH NATURAL COLLIMATION
Figure 2 shows the schematic diagram of a neutron scattering instrument with natural collimation and the neutron phase space acceptance diagram at the sample.9, 10 The sample is illuminated directly by the moderator. For simplicity, we consider beam divergence in one dimension
FIG. 2. (Top left) Illustration of a simple instrument with natural collimation. The sample is located at a distance L away from the moderator and is illuminated directly by the moderator. dm and ds are the linear dimensions of the moderator and the sample, respectively. The dashed lines indicate the most divergent neutrons accepted by the sample with the angle θ . (Top right) Illustration of neutron phase space acceptance diagram at the sample position for simple instruments with natural collimation.9, 10 (Bottom) Illustration of intensity integration at the sample position. Surface elements dAm = dxm dym and dAs = dxs dys are highlighted on the moderator and the sample, respectively. (xm ,ym ) and (xs ,ys ) are the coordinate systems on the surface of the moderator and sample, respectively. The distance between dAm and dAs is shown as l = [(xs − xm )2 + (ys − ym )2 + L2 ]1/2 . The projection of dAs onto the plane perpendicular to l is given by dAs cosα, where α is the angle between l and L, and cosα = L/l. The solid angle subtended by dAm at dAs is given by dAm cosα/l2 . Therefore, the neutron intensity received by dAs from dAm is given by dI = Bm dAm dAs cos2 α/l2 = Bm dAm dAs L2 /l4 , where Bm is the moderator brightness. Integration over dI results in Eq. (2).
only. The maximum beam divergence accepted by the sample is determined by 1 (1) (dm + ds )/L, 2 where L is the distance from the sample to the moderator and dm and ds are the maximum linear dimensions of the moderator viewing area and the sample cross-section (i.e., transverse areas), respectively. For circular cross-sections, dm and ds are the diameters of the moderator and the sample, respectively. For rectangular and square cross-sections, dm and ds are the diagonals. If the horizontal and vertical divergences are considered separately, then dm and ds can be considered as the horizontal and vertical dimensions, respectively. Within the limit of beam divergence angles ±θ , the intensity on the sample, I, in unit of neutrons Å−1 s−1 (or Å−1 pulse−1 ), is obtained by integrating the moderator brightness Bm (in unit of neutrons cm−2 sr−1 Å−1 s−1 or cm–2 sr−1 Å–1 pulse–1 ) over the solid angle m subtended by the moderator area Am at the sample position, and over the sample area As . Expressed in terms of the moderator and sample dimensions xm , ym , xs , and ys integrated over the surfaces of the moderator and sample (See Fig. 2 for details), I is given as tan θ =
I = L2 ∫ Bm dxs dys ∫[(xs − xm )2 + (ys − ym )2 + L2 ]−2 × dxm dym .
(2)
In practice, the condition L » dm is almost always true. Additionally, the moderator brightness Bm can often be approximated to be uniform across the moderator face, i.e., Bm is a constant. The intensity on the sample is
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Rev. Sci. Instrum. 84, 125104 (2013)
dAs dm , and further to then simplified to I = Bm I = Bm Am As / L2 . Replacing L using Eq. (1) yields I = 4Bm As Am tan θ/(dm + ds ) . 2
2
(3)
Because we are interested in the relative performance between the moderators, we use the largest moderator in the current study, the 10 cm × 10 cm one, as our baseline. The intensity on the sample for other moderators is then expressed as the ratio to that obtained from the 10 cm × 10 cm case, rm = Im /I10 , where Im is the intensity on the sample when a moderator with an m × m viewing area is used. For the current discussion with square moderator viewed surfaces and square samples, the surface area is approximated by A ∼ d2 . The relative intensity on the sample is then given by (4) rm = (Bm /B10 )dm 2 (d10 + ds )2 / d10 2 (dm + ds )2 . In order for moderators of different dimensions to provide the same beam divergence, ±θ , on the same sized sample, the distance L between the moderator and the sample must be adjusted. We chose to make the comparison between moderators for fixed beam divergence at a given sample size as this is often a parameter that is critical to the performance of neutron scattering instruments. We note though, that the wavelength resolution will change for different L on instruments using time-of-flight techniques and this may place a constraint on freely adjusting L. According to Eq. (4), the intensity ratio rm is now only a function of the moderator and sample sizes but not the beam divergence. Obviously, with any given moderator, the on-sample neutron intensity increases as the maximum beam divergence increases (the sample moves closer to the moderator). However, as long as the sample is far enough from the moderator such that Eq. (3) is valid, such dependency is the same for all moderators. It is instructive to consider the limit as the sample dimension approaches zero in Eq. (4). In this case, only the moderator brightness is relevant and rm = (Bm /B10 ). The trend is that small samples will always be best illuminated by the brightest moderators, regardless of the moderator dimension. At the other limit, we can consider the case where the sample dimension is much greater than that of the moderators, ds » dm . In this case, rm = (Bm /B10 ) · (dm 2 /d10 2 ) and unless the moderator brightness increases faster than the viewed area, large samples will favor larger moderators. The relative moderator brightness (Bm /B10 ) in Eq. (4) is shown in Fig. 3. In principle, the wavelength dependent rm value for each individual instrument should be calculated in order to determine the best moderator for that instrument. On the second target station with a repetition rate of ∼10 Hz, most instruments will be using a broad neutron wavelength band of several Angstroms. Therefore, we believe that it is reasonable to use a (Bm /B10 ) value averaged over the neutron wavelength range of 1–5 Å (Table I). For instruments that can use an even broader wavelength band, the average (Bm /B10 ) values vary somewhat but not significantly (Table I). Using these average moderator brightness ratios, rm as a function of sample size is plotted in Fig. 4. As indicated by the limiting behavior discussed above, the small bright moderators significantly outperform larger ones for small samples. As the
3.5 Brightness Ratiovs 10×10 Moderator
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FIG. 3. Ratio of the coupled (thicker lines) and decoupled (thinner lines) moderator surface brightness versus that of the corresponding 10 cm × 10 cm moderator as a function of neutron wavelength. The average brightness ratios for the wavelength ranges of 1–5 Å and 0.5–10 Å are listed in Table I.
sample dimension increases, larger moderators start to become more favored. Generally speaking, when choosing coupled moderators, the moderator dimension should be as small as possible but still slightly larger than the desired sample size. For example, if the maximum sample size for an instrument is 2.5 cm or less, then the 3 cm × 3 cm moderator will deliver the best performance. For larger samples, the 5 cm × 5 cm, 7 cm × 7 cm, or the 10 cm × 10 cm moderator will become the better choice. For decoupled moderators, the moderator should be at least twice as large as the sample. The difference between the coupled and decoupled moderators is due to the fact that the brightness gains for small decoupled moderators are less than their coupled counterparts (see Fig. 3). III. MONTE CARLO SIMULATIONS
The fact that the on-sample intensity ratio, rm , from Eq. (4) is independent of the maximum beam divergence is readily confirmed by Monte Carlo simulations. In Fig. 4, the intensity gains simulated for the 3 cm × 3 cm, 5 cm × 5 cm, and 7 cm × 7 cm moderators over that of the 10 cm × 10 cm moderator are shown for maximum beam divergence angles of θ = 0.1◦ , 0.5◦ , and 0.9◦ , respectively. The simulations were carried out using the program ib.11 All the simulated data agree reasonably well with the calculated curves regardless of the maximum acceptable beam divergence values at the sample. Note that for any given moderator and sample size, the results at all beam divergences lie essentially on top of TABLE I. Ratios of the average moderator brightness for the 3 × 3, 5 × 5, and 7 × 7 moderators to the 10 × 10 moderator over the neutron wavelength ranges of 1–5 Å and 0.5–10 Å. Coupled moderators Wavelength range 1–5 Å 0.5–10 Å
Decoupled moderators
3 cm × 3 cm
5 cm × 5 cm
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5 cm × 5 cm
7 cm × 7cm
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1.8 1.7
1.4 1.4
1.7 1.6
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On-Sample Intensity Ratio vs 10×10 Moderator
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3cm×3cm 5cm×5cm 7cm×7cm
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On-Sample Intensity vs 10x10 Moderator
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3 4 Sample size [cm]
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7
FIG. 4. On-sample intensity ratios for instruments with natural collimation when using the 3 cm × 3 cm, 5 cm × 5 cm, and 7 cm × 7 cm coupled (upper) and decoupled (lower) moderators with the corresponding 10 cm × 10 cm moderator. The curves are calculated using Eq. (4) using the average moderator brightness values given in Table I. The symbols indicate the result of Monte Carlo simulations with three different maximum beam divergence angles, 0.1◦ , 0.5◦ , and 0.9◦ . The simulated intensities are averaged over a neutron wavelength range of 1–5 Å. For any given moderator and sample size, the results for all three beam divergences lie essentially on top of one another.
one another. The small differences between the calculations and simulations can be attributed to various factors. In the calculations, for example, the moderator brightness Bm is assumed to be uniform across all emission angles. In the Monte Carlo simulations, the moderator is treated as a Lambert source which is at its brightest in the direction of the moderator surface normal. Also, Eq. (4) uses the average moderator brightness values whereas the simulations use the actual source files from Ref. 4. On the second SNS target station, the number of instruments that use natural collimation will likely be limited. For those that do, Fig. 4 offers a guideline for their moderator selection. In Sec. IV, we will examine the moderator selection criteria for instruments that use neutron transport optics in their design. IV. INSTRUMENTS WITH NEUTRON OPTICS
Most modern neutron scattering instruments use neutron optical systems to reduce the instrument background,
2
4 Sample size [cm]
6
FIG. 6. Calculated (lines) and simulated (symbols) on-sample intensity ratios of the 3 cm × 3 cm, 5 cm × 5 cm, 7 cm × 7 cm coupled moderators with the 10 cm × 10 cm moderator for an instrument with a 50 m long neutron guide as shown in Fig. 5. The sample size is considered to be half that of neutron guide cross-section. A perfect neutron guide is assumed for the numerical calculations. Namely, the neutron intensities at the guide exit and entrance are the same. For the Monte Carlo simulations, the maximum beam divergence accepted by the neutron guide and the sample are both kept at ±0.2◦ . The neutron guide has a 3.5m supermirror coating with a 70% reflectivity at the critical angle. The intensity ratios are averaged over the neutron wavelength range of 1–5 Å.
manipulate the neutron phase space, or extend the total instrument length for increased time-of-flight timing resolution. For the current purpose of evaluating moderator performance, a simple straight neutron guide is used as a representative optical system (see Fig. 5). The maximum neutron beam divergence accepted by the sample is defined by the transverse size of the neutron guide or a slit at the end of the neutron guide, the transverse size of the sample, and the distance between them. A simple optimization is to have the neutron guide accept the same maximum beam divergence, ±θ , as that desired at the sample, namely by placing the neutron guide at L1 = 12 (dm + dg ) cotθ away from the moderator (Fig. 5). The cross-section of the neutron guide now becomes the equivalent of the sample area in Eq. (3) and for reasonable values of beam divergence, θ < 2◦ , the requirement that L1 » dm is valid. Because neutron guides commonly have cross-sections greater than the sample, the effectiveness of the small bright moderators is diminished. This effect is demonstrated in Fig. 6 where the sample cross-section is taken to be half that of the neutron guide. Obviously, such guide placement is not optimal and results in a severely under-illuminated guide.9, 12 The effect
FIG. 5. Schematic diagram of an instrument with a straight guide between the sample and the moderator. The neutron guide, represented by the solid-line rectangle, accepts the same amount of maximum beam divergence (θ ), represented by the oblique dashed lines, as the sample. dm and dg are the linear dimensions of the moderator and the neutron guide cross-sections. The red dashed lines represent optimal guide placement with the guide located L1 = 12 (dm − dg )cotθ away from moderator.
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FIG. 7. Neutron phase space acceptance diagram at the entrance of a 3.2 cm neutron guide viewing a 5 cm × 5 cm moderator in the horizontal direction. The intended maximum beam divergence angle at the sample is ±0.2◦ . For the left panel, the neutron guide is placed at L1 = 12 (dm + dg )cotθ away from the moderator (see text). For the right panel, it is placed at L1 = 12 (dm − dg )cotθ . The apparent difference in the color scale in the graphs is due to the difference in the minimum of neutron counts.
can be seen from the neutron phase space acceptance diagram simulated at the entrance of a 3.2 cm wide neutron guide and a 5 cm × 5 cm moderator in Fig. 7. The guide width is chosen arbitrarily to be less than the dimension of the moderator. The instrument setup for the simulation is intended to deliver 0.2◦ maximum beam divergence to the sample. In the left panel of the figure, the neutron guide positioned at L1 = 12 (dm + dg ) cotθ accepts maximum beam divergence of ±0.2◦ , which leaves a large part of the phase space unfilled. To take the neutron guide totally out of consideration in moderator evaluations, one would need to design a perfect guide with the same cross-section as the moderator, and fully saturate it by placing the guide entrance at the moderator face. This is of course impossible in practice. Once the desired maximum beam divergence angle θ and the guide dimension dg are determined, however, we can extend the neutron guide upstream by L1 = dg cotθ and completely fill, even if not saturating, the phase space within the divergence angles of ±θ ◦ (dashed lines in Fig. 5), as seen in the right panel of Fig. 7. That is, the neutron guide should be placed downstream from the moderator at a distance of L1 =
1 (dm − dg ) cot θ. 2
(5)
This condition can only be satisfied when dg < dm , namely when the neutron guide cross-section is smaller than the moderator. In practice, engineering constraints (heat load,
radiation damage, crowding by neighboring beam lines) impose limits on the closest possible guide location. For the first SNS target station, this limit is on the order of 1 m from the moderator.8, 13 Here, we assume that this 1 m limit also applies to the SNS second target station. This condition arises for guide dimensions dg ≥ dm − tanθ · 200 cm. Figure 8 compares the beam delivery performance for the two guide entrance positions, L1 = 12 (dm − dg ) cotθ or L = 12 (dm + dg ) cotθ away from the moderator, subject to a 1 m minimum distance limitation. This set of simulations modeled a sample size half that of the neutron guide for a beam divergence at the sample, θ = ±0.2◦ . Placing the guide entrance at L1 resulted in a maximum gain of intensity on the sample of somewhat greater than a factor of 2 when the dimension of the neutron guide approaches that of the moderator. When the guide cross-section is much smaller than the moderator, L1 ≈ L1 and therefore the intensity delivered to the sample is nearly identical. When the guide cross-section is larger than the moderator dimension, neutron phase space cannot be filled even if the guide entrance were located at the moderator surface. For this simulation study, optimum performance occurs when the guide dimension matches that of the moderator and its entrance can be placed at L1 . At the relatively modest maximum beam divergence angle, θ = ±0.2◦ , used to generate the data in Fig. 8, the lower limit of 1 m for L1 essentially takes effect when dm = dg . At larger values of θ , however, this limit will be reached
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3cm×3cm 5cm×5cm 7cm×7cm 10cm×10cm
2
2
1.5
1 0
5
10
15
3cm×3cm by M.C. 5cm×5cm by M.C. 7cm×7cm by M.C. 3cm×3cm by Integration 5cm×5cm by Integration 7cm×7cm by Integration
2.5
20
Linear Dimension of Guide Cross Section [cm]
FIG. 8. Monte Carlo simulations of the ratio of the neutron intensity on the sample with the neutron guide placed at L1 = 12 (dm − dg )cotθ compared to L = 12 (dm + dg )cotθ away from the moderator (Fig. 5). Data for the 3 cm × 3 cm, 5 cm × 5 cm, 7 cm × 7 cm, and 10 cm × 10 cm coupled moderators are shown. The neutron guide has a 3.5m supermirror coating with a 70% reflectivity at the critical angle. The sample size is half the linear dimension of the guide cross-section. The maximum accepted beam divergence angle at the sample is set to be ±0.2◦ . L1 is restricted to L1 ≥ 1 m. The intensity ratios are obtained by averaging the simulated intensity within the neutron wavelength range of 1–5 Å.
at smaller values of dg and Fig. 8 will exhibit some divergence dependence. Nonetheless, the maximum gain still occurs when the neutron guide dimension matches that of the moderator, but it decreases progressively as θ increases (data not shown). We now come back to the choice of moderators for neutron scattering instruments with neutron guides. When placed at L1 , the neutron guide accepts more beam divergence than the sample requires and Eq. (3) cannot be used to compare moderator performance since it does not properly account for these extra neutrons. In order to compare the performance of different moderators we can use Monte Carlo simulations to obtain the useful intensity at the sample position, or we can numerically integrate Eq. (2) up to the desired beam divergence. Numerical integration of Eq. (2) determines the useful intensity accepted at the guide entrance. If we assume that the guide is effective at delivering the desired beam divergence, the intensity at the sample position should be proportional to the intensity at the guide entrance. In either case, the ratio to the baseline moderator size of 10 cm × 10 cm provides a valid metric for moderator selection and is shown in Fig. 9. Note that the entire brightness gain of the smaller moderators is experienced for all guide dimensions up to the moderator size. Implicit in the observation that the maximum gain is set by the ratio of moderator brightness, is that a larger moderator of equivalent brightness will not outperform the smaller moderator until the desired guide dimension exceeds that of the small moderator. This observation is dependent on the maximum beam divergence to be transmitted to the sample, see discussion below. The best-match moderator for an instrument with a guide should be the smallest one that is still larger than or approximately the same size as the neutron guide cross-section at the guide entrance. For example, if an instrument has a neutron guide with a 3 cm or less cross section at the entrance, then the 3 cm × 3 cm moderator will be
Intensity Ratio vs 10x10 Moderator
On-Sample Intensity Ratio
2.5
1.5 1 Coupled moderators
0.5 0 0
2
4
6
8 Decoupled moderators
10
1.5
1
3cm×3cm by M.C. 5cm×5cm by M.C. 7cm×7cm by M.C. 3cm×3cm by Integration 5cm×5cm by Integration 7cm×7cm by Integration
0.5
0 0
2
4
6
8
10
Linear Dimension of Guide Cross-Section[cm]
FIG. 9. Intensity ratios of the 3 cm × 3 cm, 5 cm × 5 cm, and 7 cm × 7 cm coupled (upper) and decoupled (lower) moderators to the corresponding 10 cm × 10 cm moderator. The lines are a numerical integration of Eq. (2) within the beam divergence of ±0.2◦ at the entrance of the neutron guide located at L1 = 12 (dm − dg )cotθ . The symbols are Monte Carlo simulations of the intensity on the sample averaged over the neutron wavelength range of 1–5 Å. The linear dimension of the sample is half that of the neutron guide cross-section. A 50 m long, 3.5m supermirror coated neutron guide with 70% reflectivity is used for the simulation. L1 is restricted to L1 ≥ 1 m for both the simulations and the numerical integrations.
the best choice for delivering the most neutrons to the sample. Guides whose entrance can be located at L1 take even better advantage of the small, brighter moderators than the natural collimation case (compare Fig. 9 with Fig. 4). For the Monte Carlo simulations in Fig. 9, we used 3.5m supermirror coating for the neutron guides, where m = 1 corresponds to the critical reflection angle of natural nickel. Because of the rather limited maximum divergence angle of θ = ±0.2◦ used in the simulation, these guides have little effect on our conclusion. It is the transverse size of the neutron guide at its entrance that determines the best moderator dimensions for an instrument (Fig. 9). At larger divergence angles, guide properties other than its entrance dimensions will have some influence on our discussion. This is because neutron transport by a guide may be less efficient and this reduction in efficiency will be neutron wavelength and beam divergence dependent. However, these changes in neutron transport will occur in the same way regardless what moderator is used. More important is the effect of a practical limit placed on the minimum value of L1 . When L1 from Eq. (5) is less than this limit, the relative moderator performance becomes divergence dependent. For example, imposing the 1m limit used above, a neutron guide cross-section of 2 cm has threshold angles above which the moderator performance is divergence dependent of about 0.3◦ , 0.9◦ , 1.4◦ , and 2.3◦ for the 3 cm × 3 cm, 5 cm × 5 cm, 7 cm × 7 cm, and 10 cm × 10 cm moderators, respectively. At different minimum L1 limits, the cotangents of these angles scale in proportion to the new limit. At larger θ , the small bright moderators will lose their advantage over the larger 10 cm × 10 cm one, because the neutron guide cannot be extended close enough to be fully illuminated by the smaller moderator. However, the loss is not significant until
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2.5
0.1 ° 0.5 ° 1° 1.5°
2 1.5 1 0.5
3cm×3cm moderator
Intensity Ratio
0 1.8 0
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8
0.1 ° 10 1° 1.5° 2°
6
8
0.1 ° 10 1° 1.5° 2.5°
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8
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1.4 1 0.6 5cm×5cm moderator 0.2 1.4 0
2
4
1.2 1 0.8
7cm×7cm moderator
0.6 0
2
4
Linear Dimension of Guide Cross-Section[cm]
FIG. 10. Intensity ratios for the 3 cm × 3 cm (upper), 5 cm × 5 cm (middle), and 7 cm × 7 cm (lower) coupled moderators with the corresponding 10 cm × 10 cm moderator at different maximum divergence angles at the entrance to the neutron guide. The curves for decoupled moderators have the same shapes but different vertical scales (see Fig. 9 and Table I). Note that the divergence ranges calculated for each moderator size are different. Except for the difference in beam divergence and the corresponding different guide to moderator distance, the instrument setup is the same as that in Fig. 9. The intensities are integrated at the entrance of the neutron guide using Eq. (2) within the divergence angles as labeled in the legend. The minimum guide to moderator distance is subject to a 1 m limitation.
the angles are two or three times above the threshold. This effect is seen in Fig. 10, which shows divergence-dependent intensity ratios at the entrance of the neutron guides for different sized moderators calculated using Eq. (2). The data are plotted using Bm values (Eq. (2)) for coupled moderators. For decoupled moderators, the curves are the same but the plots have different vertical scales due the difference in Bm between the coupled and decoupled moderators (Table I and Fig. 9). In a recent study, it is found that elliptic and parabolically tapered ballistic guides are more efficient at transporting divergent neutrons than straight guides.18 One interesting fact from the study is that the source extraction spot of these optics is smaller than the transverse dimensions of the guide. This means that these novel neutron optics favor small bright moderators even at larger divergences. Thus, the situations presented in Fig. 10 can be seen as the worse-case scenario for small bright moderators. V. CONCLUSIONS
The results from Secs. II and IV can be summarized by the following general strategies for determining the appropriate sized moderator for a neutron scattering instrument. For instruments that employ neutron guide systems, the choice for both coupled and decoupled moderators should be the smallest moderator available that is larger than or approximately the same size as the guide cross-section at the guide entrance.
For instruments using natural collimation, the choice for coupled moderators should be the smallest one that is larger than the sample size. For the decoupled moderators, the choice should be the smallest moderator that is at least twice the size of the sample. These conclusions are valid as long as the smaller moderators have a higher brightness than the larger ones. Conversely, these same strategies can be used to decide what size moderators should be built at spallation neutron sources such as the second SNS target station given a suit of instrument requirements. In general, neutron guides on instruments have larger cross-sections than the samples to be studied. For small angle scattering instruments, such as the EQ-SANS instrument on the first SNS target station,8 the dimension of the collimation slit at the end of the neutron guide is usually twice as large as the sample. Most samples studied on SANS instruments are 1 cm or less in size. The maximum beam divergence angle on these instruments is almost always less than 1◦ . A typical experimental configuration for a SANS experiment is to use a 2 cm collimation slit, 1 cm sample, and a 4 m collimation distance, which gives a maximum beam divergence angle of 0.21◦ . Instruments like SANS diffractometers will thus clearly benefit from the smallest, 3 cm × 3 cm moderator. Other instruments that view a coupled moderator such as cold neutron chopper spectrometers14 and spin echo spectrometers,15 may study larger samples. For these instruments, either the 5 cm × 5 cm or the 7 cm × 7 cm moderators offer a good choice. Instruments that will benefit most from viewing the smallest 3 cm × 3 cm decoupled moderator are single crystal diffractometers, especially those that are optimized for studying macromolecules16 where samples under study are typically less than 1 mm in size. Similarly for a high resolution powder diffractometer, the strict resolution requirements on beam divergence and sample size means that this instrument will experience a significant performance boost on the 3 cm × 3 cm moderator. On the other hand, a high intensity powder diffractometer,17 or a backscattering spectrometer13 can be designed to have a large beam cross-section and use beam focusing to increase the intensity on the sample. For these instruments, a 7 cm × 7 cm or even larger moderator should be considered. These conclusions on moderator choices should be valid regardless of what neutron optical systems are used, because it is the dimension of the neutron beam cross-section at the guide entrance and the maximum beam divergence the optics is designed to transport that are the deciding factors. The optimal guide dimensions or the maximum beam divergence must be optimized for each neutron scattering instrument individually. Once these parameters are finalized, the optical system should then be placed according to Eq. (5) relative to the moderator. On most instruments, the collimation before the sample is fixed and the maximum divergence angle θ is readily obtained. On others, the collimation is variable between experiments. In the latter case, an appropriate θ value needs to be chosen before applying Eq. (5). As an example, on the asbuilt EQ-SANS instrument on the first SNS target station,8 the shortest possible collimation length is ∼2 m. Assuming a typical 1 cm sample and 2 cm collimation slit, the maximum
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beam divergence accepted by the sample at this collimation distance is θ ∼ 0.43◦ . The neutron guide system on the instrument is 4 cm × 4 cm in cross-section. It views the brightest spot at the lower left corner of the 10 cm × 12 cm coupled moderator, which is 2 cm displaced from the moderator center both horizontally and vertically. The shortest distance between the moderator edge and the neutron guide projection onto the moderator is thus 1 cm. Therefore, for the current discussion, the effective moderator dimension can be considered to be 6 cm. Applying these values to Eq. (5), the neutron guide should start at 1.33 m from the moderator, which is essentially the same as the as-built value of 1.28 m. Ideally, each class of neutron scattering instruments should have a dedicated moderator optimized to the requirements for that instrument class. However, in both the coupled and decoupled cases, two to three moderators with viewing areas in the range of 3 cm × 3 cm to 7 cm × 7 cm appear to be enough to serve the needs of most instruments. In any case, appropriately designed moderators will be crucial to the scientific productivity of the SNS facility for years to come. ACKNOWLEDGMENTS
This article has been authored by UT-Battelle, LLC, under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The work was sponsored by the Office of Basic Energy Sciences, US Department of Energy. The authors wish to thank Dr. Erik Iverson of SNS for reading and critiquing the article. 1 T. E. Mason, D. Abernathy, I. Anderson, J. Ankner, T. Egami, G. Ehlers, A.
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