How to design a focusing guide: The large moderator case 1

Abstract

As continuously shaped super-mirrors are becoming available, the conceptual design of focusing guides should explore a wider range of possibilities to accomplish an efficient neutron beam extraction. Starting from a desired phase-space volume at the sample position and using an upstream ray-tracing approach, the acceptance diagram of any focusing guide can be calculated at the moderator position. To ensure high brilliance transfer and homogeneous coverage, the acceptance diagram should be fully included in the neutron source emission phase-space volume. Following this idea, the guide system can be scaled into dimensionless geometric figures that convey performance limits for a desired cross-section reduction. Moreover, if we impose a monotonic increase of the reflection angle with divergence angle at the sample position, the shape of the mirror is analytically determined. This approach was applied in the design of a focusing guide for SNAP instrument at SNS, at ORNL, USA. The results of McStas simulations are presented with different options included. Our approach facilitates finding an optimal solution for connecting multiple guide pieces to avoid excessive losses and ensure a homogeneous phase space coverage.

Keywords

Neutron optics neutron guides neutron scattering supermirrors neutron instrumentation

1. Introduction

Maier-Leibnitz and Springer [15] introduced the neutron guide concept (neutron conducting tubes in their words) as “a well-polished, straight, rectangular tube”. The thermalized neutrons introduced in the tube experience specular reflections on walls if the incident reflection angle is not surpassing a critical value controlled by the refractive index of the bulk and the wavelength of the neutron. With the advent of supermirrors [16], the critical angle was pushed toward much higher values by employing neutron diffraction on a complex multi-layered structure. This ingenious advance in the making of neutron mirrors triggered a tremendous progress in neutron reflective optics, especially in the design and building of guides for neutron beam extraction. In the original concept, the divergence of the beam is limited by the critical angle, but a converging straight tapering can help to avoid this limitation and enhance the local flux [18]. In relation with this application, the name “focusing supermirror guide” appeared in the literature [20] and the idea of combining straight tapered pieces of guides to achieve even more ‘focusing’ became more ubiquitous [9,10]. Following this trend, some of the neutron guides built at the Spallation Neutron Source (SNS) were designed to incorporate multiple tapered guide elements with the goal of maximizing the neutron flux in a small area at the sample position [23]. This approach was implemented at the diffraction instruments VULCAN [23,26] and TOPAZ [22] and later at MANDI [6] and CORELLI. However, the straight tapering introduces certain discontinuities in the reflection angle at the junctions between pieces, which induce inhomogeneities in the angular distribution particularly in the short wavelength range. Thus, the next logical step in the neutron guides development was the fabrication of continuously shaped elements (a first example being the parabolically tapered guide [12]). With the opportunity afforded by nonlinearly tapered ‘trumpets’ the design of complex guide systems entered in a new era. It was found that parabolic [27] and elliptic [2] shapes must be carefully considered to ensure beam homogeneity in the phase space. New tools for neutron guide optimization were developed to address these problems [3,4].

In a general sense, the purpose of a ‘focusing’ neutron guide is to deliver a specified phase space volume at the sample position with a minimal decrease of neutron density in the phase space. Another requirement, which is not always strictly followed, is to ensure a homogenous density distribution inside the useful phase space volume without discontinuities (e.g. holes or stripes). Thus, the first step in the conceptual design of a neutron guide should include exploring the limits of the available phase space for a given moderator-sample arrangement with the assumption of a single reflection per guide element [8]. Here we will analyze only the source-in-view guide configurations, when a direct path is available for neutrons from moderator toward the sample. We also describe an optimal way to determine the shape of the mirrors, which minimizes reflection losses. Our first-principle approach was first summarized in [8]. This publication contains a detailed account on the basic design principles and on the use of a new general solution for the neutron guide profile.

Fig. 1.

Schematics of the back projection approach: a) Notations; the transverse positions are normalized to half of the moderator height and the positions along the beam are normalized to the sample-moderator distance. b) First condition; the entrance aperture position and size should permit only the phase-space needed to fully illuminate the sample. c) Second condition; the minimum surface angle at the exit is determined by reflection of a ray originating from the top of the entrance aperture to the top of the sample. d) Third condition; the maximum surface angle at the exit is determined by reflection of a ray originating from the bottom of the source to the bottom of the sample.

2. Back projection approach

To represent the beam extraction geometry, we introduce a generic half extension sample aperture $y_{S}$ and the moderator half extension $y_{M}$ located at a distance $L_{M}$ from sample position. We will use a ‘back projection’ approach to evaluate the neutron raytracing; the origin of the coordinate along the beam nominal direction, x, is placed at the sample position and the direction toward the moderator is positive (Fig. 1a). The rectangular coordinates x and y (along and across the nominal beam direction, respectively) will be scaled to $L_{M}$ and moderator extension, $y_{M}$ , respectively: $\begin{matrix} (1) & r = \frac{x}{L_{M}} and q = \frac{y}{y_{M}} \Rightarrow r \in (0, 1) \end{matrix}$ The main point of this exercise is that $y_{S}$ is smaller than $y_{M}$ ( $g = y_{S} / y_{M} < 1$ ) and the word ‘focusing’ seems to be appropriate, but the neutron beam at the sample position is not constrained inside the sample aperture. The value of $y_{S}$ should be considered an effective value, as the last slit in front of the sample cannot be placed in contact with the sample. However, the influence of the positioning of the last slit can be treated as a separate subject and will not be considered in the following analysis. For the moment we will ignore the shape of the guide and consider only the entrance aperture, $(r_{en}, q_{en})$ , and exit aperture, $(r_{ex}, q_{ex})$ , of the guide (with $r_{en} > r_{ex}$ as the entrance and exit are assigned along the neutron flow, contrary to the setting of coordinate system). In the small angle approximation, the maximum divergence angle for the direct view of the moderator from the center of the sample would be $α_{M} = y_{M} / L_{M}$ in units of milli-rad when $y_{M}$ is measured in millimeters and $L_{M}$ in meters. Likewise, the divergence angle on a specified point along the guide surface would be: $α = y / x$ and the extreme values $α_{en}$ and $α_{ex}$ define the range of divergence angles available in the center of the sample for the neutrons that are reflected from the guide. There is an obvious condition that $α_{ex} > α_{en}$ for the mirror to be visible from the sample position; moreover, $d α / d x ⩽ 0$ to ensure the local visibility in each point along the mirror. The α angle can be also normalized to $α_{M}$ by making $z = α / α_{M} = q / r$ and $d z / d r ⩽ 0$ . Given these scaling conventions, three conditions can be formulated to ensure a homogeneous phase space distribution at the sample position.

2.1. First condition: Continuous acceptance diagram

A first constraint should be considered, to ensure a compact angular distribution inside the sample aperture (a prerequisite for a smooth instrument resolution function), that will eliminate any gaps between the acceptance diagrams corresponding to the direct view of the moderator and the mirror. Thus, the mirror walls of the guide should intercept the extreme ray traces linking the sample aperture to the moderator (see Fig. 1b). A straightforward geometrical description gives a simple general condition for entrance aperture: $\begin{matrix} (2) & q_{en} ⩽ q_{en}^{max} = r_{en} - (1 - r_{en}) g or z_{en} = q_{en} / r_{en} ⩽ z_{en}^{max} = 1 + g - g / r_{en} \end{matrix}$ The guide entrance will always decrease the direct view of the moderator with a loss factor of: $\begin{matrix} (3) & R_{loss} = 1 - z_{en} ⩾ (1 / r_{en} - 1) g \end{matrix}$ Thus, the relations (2) should be considered an equality to minimize the loss in the field of view for the direct beam, but that ‘loss’ is not affecting the total intensity on the sample, which is mostly determined by the exit of the guide and the divergence it defines. The only neutron intensities ‘truly lost’ are the ones with wavelengths too short to be reflected by the guide.

2.2. Second condition: One reflection

A second condition imposes a single reflection event for each neutron reaching the sample aperture (but it does not prevent the neutron path to cross the mirror located on the other side of the guide: zig-zag reflections). This means that the limiting trajectory starting from the upper end of the sample aperture, $+ g$ , and reflected by the upper exit of the guide, $+ q_{ex}$ , should pass through the entrance aperture after $reflection ⩽ q_{en}$ (see Fig. 1c). This condition involves the slope of the guide at the exit (given by the derivative of the guide profile), $Δ θ_{ex}$ , and can be written as an inequality between angles: $\begin{matrix} (4) & 2 Δ θ_{ex} ⩽ α_{S}^{min} + α_{G}^{*}; α_{G}^{*} = [(q_{en} - q_{ex}) / (r_{en} - r_{ex})] α_{M} \end{matrix}$ where $α_{S}^{min}$ is the minimum divergence angle (starting from upper end of the sample range) and $α_{G}^{*}$ is the overall slope of the guide profile defined by the upper coordinates of exit and entrance apertures. Written in q and r coordinates the second condition becomes: $\begin{matrix} (5) & 2 Δ θ_{ex} ⩽ (\frac{q_{en} - q_{ex}}{r_{en} - r_{ex}} + \frac{q_{ex} - g}{r_{ex}}) α_{M} \end{matrix}$ In terms of normalized divergence the second condition becomes: $\begin{matrix} (6) & \frac{2 Δ θ_{ex}}{α_{M}} ⩽ \frac{ρ - 2}{ρ - 1} z_{ex} + \frac{ρ}{ρ - 1} z_{ex}^{0} - \frac{ρ}{ρ - 1} (z_{en} - z_{en}^{max}) \end{matrix}$ with $ρ = r_{en} / r_{ex}$ and $z_{ex}^{0} = (1 + g - g / r_{ex})$ . $z_{ex}^{0}$ represents the maximum opening at the guide exit location that allows full view of the moderator from the sample position. It is also necessary to consider the value of reflection angle, $θ_{ex} = z_{ex} α_{M} - Δ θ_{ex}$ , as compared with the maximum divergence angle ( $α_{ex} = z_{ex} α_{M}$ ) at the exit of the guide: $\begin{matrix} (7) & \frac{2 θ_{ex}}{α_{M}} ⩾ \frac{ρ}{ρ - 1} [(z_{ex} - z_{ex}^{0}) - (z_{en} - z_{en}^{max})] ⩾ \frac{ρ}{ρ - 1} (z_{ex} - z_{ex}^{0}) \end{matrix}$ Thus, fulfilling the second condition (7) as an equality, means reaching the minimum reflection angle compatible with a given guide set of parameters: ( $g, r_{ex}, r_{en}, z_{ex}, z_{en}$ ) or $(g, r_{ex}, r_{en}, z_{ex})$ if we consider $z_{en} = z_{en}^{max} (g, r_{en})$ . It must be emphasized that this condition will eliminate only the garland multiple reflections. By contrast, zig-zag multiple reflections may occur, but their rather large reflection angles will force their trajectory outside of the sample aperture, i.e. they must be considered a background contribution.

2.3. Third condition: Matching acceptance diagrams

Another restriction must be introduced concerning the acceptance diagram of the sample aperture that is back reflected to the moderator location. In other words, the back projected field of view through the guide entrance aperture from any points located inside the sample aperture should be within the active area of the moderator. This is the extreme trajectory starting from the lower end of the sample, $- g$ , and crossing the upper exit of the guide, $+ q_{ex}$ ; this trajectory should pass through the active area of the moderator, i.e. above the lower limit of the moderator, $- 1$ (see Fig. 1d). Written in angular terms: $\begin{matrix} (8) & 2 Δ θ_{ex} ⩾ α_{S}^{max} - α_{G}^{max} \end{matrix}$ where $α_{S}^{max}$ is the maximum divergence angle (starting from the lower end of the sample range) and $α_{G}^{max}$ is the slope of the trajectory defined by the upper edge of the guide exit and lower edge of the moderator. Now, a similar lower limit for $Δ θ_{ex}$ can be defined: $\begin{matrix} (9) & 2 Δ θ_{ex} ⩾ (- \frac{1 + q_{ex}}{1 - r_{ex}} + \frac{q_{ex} + g}{r_{ex}}) α_{M} = \frac{(1 - 2 r_{ex}) z_{ex} - z_{ex}^{0}}{1 - r_{ex}} α_{M} \end{matrix}$ This condition imposes a lower limit of the slope of the guide profile at its exit. The lower limit of the slope corresponds to an upper limit for the reflection angle: $\begin{matrix} (10) & \frac{θ_{ex}}{α_{M}} ⩽ \frac{1}{2} \frac{ρ}{ρ - r_{en}} [z_{ex} + 1] - \frac{1}{2} g / r_{ex} = \frac{1}{2} (z_{ex} + z_{ex}^{0}) / (1 - r_{ex}) \end{matrix}$ Thus, by fulfilling the third condition (10) as an equality one finds the maximum reflection angle compatible with a given reduced set of parameters: $(g, r_{ex}, z_{ex})$ . This condition is necessary, but not sufficient: a similar constrain has to be required for any point along the mirror, including the entrance. However, the right part of inequality (10) is increasing when moving from exit toward the entrance and the logical choice would be to reduce the reflection angle when the divergence angle decreases approaching the guide entrance. Thus, the equation (10) may be enough in most of practical cases, unless a peculiar mirror shape is considered.

2.4. Overall limits

There is a range in the $(g, r_{ex}, r_{en}, z_{ex})$ domain where the $Δ θ_{ex}$ values are located between the limits set by the last two conditions. The end of the range is reached when the two limiting values for $Δ θ_{ex}$ coincide, which correspond to getting the maximum value of $z_{ex}$ for given $(g, r_{ex}, r_{en})$ : $\begin{matrix} (11) & z_{ex}^{max} = (\frac{2 (ρ - 1)}{1 - r_{en}} + 1) z_{ex}^{0} \end{matrix}$ As this value should be positive, $z_{ex}^{0}$ must be positive too. Thus, there is a hard limit for $r_{ex}$ : $\begin{matrix} (12) & r_{ex} > r_{ex}^{0} = g / (1 + g) \end{matrix}$ $r_{ex}^{0}$ is the intersection of the ray from the top of the moderator to the bottom of the sample with the beamline axis. Below this limit there is no solution for the reflection angle at the exit of the guide, i.e. the back projected sample aperture cannot be included in the acceptance of the moderator with only one reflection inside the guide. For an elliptical guide, $r_{ex}^{0}$ corresponds to the location where the demagnification factor coincides with the value g, i.e. the image of the moderator is exactly the sample aperture. Obviously, moving closer to the sample position causes the image of the moderator to become smaller than expected at the sample position. On the other hand, for $r_{ex} = r_{ex}^{0}$ , $z_{ex}^{0} = 0$ and, following equation (11), $z_{ex}$ can be different from 0 only if $r_{en} \to 1$ , i.e. the guide should start quite close to the moderator, which is not a convenient arrangement. Therefore, $g / (1 + g) < r_{ex} < r_{en}$ , with $z_{ex}$ taking negligible values when $r_{ex}$ is approaching these limits.

Equation (12) can be also understood as a limit in g if the location of the guide exit is chosen: $\begin{matrix} (13) & g < r_{ex} / (1 - r_{ex}) = g^{max} \end{matrix}$ This means that $g^{max}$ would be quite small when the exit of the guide is closer to the sample position. A focusing arrangement using single reflection guides will not work for small moderators or large sample apertures.

The $z_{ex}^{max}$ value, given by equation (11), can be further optimized along the $r_{ex}$ axis. The maximum $z_{ex}^{max}$ value for fixed $(g, r_{en})$ is reached when $r_{ex}$ satisfies the following equation: $\begin{matrix} (14) & r_{ex}^{op} = \frac{2 g}{1 + (g / 2) (3 + 1 / r_{en})} < \frac{2 g}{1 + 2 g} \end{matrix}$ Thus, the optimal location of the guide exit should be found in a relatively narrow interval determined by the g value: $g / (1 + g) < r_{ex}^{op} < 2 g / (1 + 2 g)$ . Additionally, the optimal value of $z_{ex}^{max}$ results from (11): $\begin{matrix} (15) & z_{ex}^{op} = \frac{r_{en}}{1 - r_{en}} \times \frac{{(1 + z_{en}^{max})}^{2}}{8 g} < \frac{r_{en}}{2 g (1 - r_{en})} \end{matrix}$ This value can become quite high, if g is small enough and guide entrance $r_{en}$ not very far from the moderator. However, the corresponding reflection angle may become too large for real applications: $\begin{matrix} (16) & {(\frac{θ_{ex}}{α_{M}})}^{op} = \frac{1 + 3 g / 2 + g / 2 r_{en}}{1 + g / 2 - g / 2 r_{en}} \times \frac{z_{ex}^{op}}{2} \end{matrix}$ If the value of $r_{ex}$ is acceptable, but the reflection angle too large, the value of $z_{ex}$ can be reduced and the minimum value of reflection angle can be calculated according to equation (7). In that case, a minimum value for exit reflection angle can be reached: ${(θ_{ex} / α_{M})}^{min} = \frac{1}{2} g r_{en} / r_{ex}^{2}$ , which corresponds to: $z_{ex} = (1 + g) [(ρ - 2) (r_{ex}^{0} / r_{ex}) + 1]$ . Another way to achieve this is to reduce $r_{ex}$ away from the optimal value and maintain the maximum allowed $z_{ex}$ (equation (11)). Under these conditions, the minimum value for exit reflection angle will be reached at $r_{ex} = 2 g / (1 + g)$ , with: ${(θ_{ex} / α_{M})}^{min} = \frac{1}{4} g r_{en} / {(r_{ex}^{0})}^{2} / (1 - r_{en})$ and $z_{ex} = (1 + g) [ρ - (1 + r_{en}) / 2] / (1 - r_{en})$ .

3. Guide profile: Analytic solution

The function describing the guide profile should be determined by imposing its values at the entrance and exit, as well as the derivative at the exit. We will try to get an analytical solution by considering a class of functions, which defines the reflection angle along the guide, $θ / α_{M} = q / r - d q / d r$ , dependent on $z = q / r$ values, $f (z)$ . With this notation, the mirror profile equation becomes: $\begin{matrix} (17) & \frac{d (z)}{d r} = - f (z) / r or \frac{d (q / r)}{f (q / r)} = d (ln (\frac{1}{r})) \end{matrix}$ The solution of this equation can be written in an integral form: $\begin{matrix} (18) & ln (\frac{r}{r_{ex}}) = \int_{z}^{z_{ex}} \frac{d z}{f (z)} \end{matrix}$ Two restrictions originating from boundary conditions must be imposed on $f (z)$ function: $\begin{matrix} (19) & f (z_{ex}) = \frac{θ_{ex}}{α_{M}} and ln (\frac{r_{en}}{r_{ex}}) = \int_{z_{en}}^{z_{ex}} \frac{d z}{f (z)} \end{matrix}$

Beside the positivity of the reflection angle, $f (z) > 0$ , a desirable requirement would be to maintain a positive derivative of the reflection angle, $d f / d z ⩾ 0$ , along the guide. That will ensure a monotonic decrease of the supermirror reflectivity when the divergence of the neutron path is increasing; i.e. smooth angular distributions will be delivered at the sample aperture. Another reasonable condition would be: $d^{2} f / d z^{2} ⩾ 0$ , which extends the angular reflection range for a fixed supermirror cutoff angle. The last two properties of f are not necessary but seem to be advantageous for a cautious conceptual design. The simplest choice for f would be a linear function: $\begin{matrix} (20) & f (z) = a + b z \end{matrix}$ In this case, $d f / d z = b ⩾ 0$ and $d^{2} f / d z^{2} = 0$ . Integrating (18), the two conditions (19) become: $\begin{matrix} (21) & a = \frac{θ_{ex}}{α_{M}} - b z_{ex} and (a + b z_{en}) r_{en}^{b} = (a + b z_{ex}) r_{ex}^{b} \end{matrix}$ Consequently, the slope b must obey the following equation: $\begin{matrix} (22) & \frac{1 - {(r_{en} / r_{ex})}^{- b}}{b} = (z_{ex} - z_{en}) {(\frac{θ_{ex}}{α_{M}})}^{- 1} \end{matrix}$ If the equation (22) has a solution, the shape of the mirror is uniquely determined: $\begin{matrix} (23) & \frac{q}{r} = z = z_{ex} - \frac{1}{b} \frac{θ_{ex}}{α_{M}} [1 - {(\frac{r}{r_{ex}})}^{- b}] \end{matrix}$

Two extreme cases must be considered. First, the $b = 0$ case. Here the reflection angle is constant along the mirror. In that case, the exit reflection angle is not a free parameter, but will be determined by the aperture and location of the guide exit: $\begin{matrix} (24) & \frac{θ_{ex}}{α_{M}} = \frac{z_{ex} - z_{en}}{ln (r_{en} / z_{ex})} \end{matrix}$ This corresponds to the logarithmic spiral shape and equation (23) becomes: $\begin{matrix} (25) & z = z_{ex} - \frac{θ_{ex}}{α_{M}} ln (\frac{r}{r_{ex}}) \end{matrix}$ The second case considers $a = 0$ , when the reflection angle is proportional to the divergence. In that case, the b value is determined by the geometry: $b = ln (z_{ex} / z_{en}) / ln (r_{en} / r_{ex})$ and the exit reflection angle is again fixed: $θ_{ex} / α_{M} = b z_{ex}$ . This corresponds to an extended parabolic shape (parabola: $b = \frac{1}{2}$ ) and equation (23) becomes: $\begin{matrix} (26) & z = z_{ex} {(\frac{r_{ex}}{r})}^{b} \end{matrix}$ The special shapes designated by equations (25) and (26) may demand values of $θ_{ex} / α_{M}$ that are not even contained in the admissible range defined by equations (7) and (10), whereas the general solution (23) allows for tuning the $θ_{ex} / α_{M}$ anywhere inside the allowable interval. A reasonable name for this kind of solutions would be steady rising slope spiral (SRS).

Many other analytical solutions can be found by using more complicated expressions for $f (z)$ and integration of equation (18). For example, by considering a second degree polynomial $f (z) = a + b z + c z^{2}$ integration of (18) results in: $\begin{matrix} (27) & \int \frac{d z}{f (z)} = \frac{1}{μ} ln (\frac{b - μ + 2 c z}{b + μ + 2 c z}); μ = \sqrt{b^{2} - 4 a c} \end{matrix}$ Here, the two boundary conditions (19) are insufficient to determine all three parameters $(a, b, c)$ and additional optimization procedures may be implemented.

The special case of elliptic guides also deserve attention, as this type of guide is widely used in neutron extraction applications. In the small angle approximation, the ellipse with focal points located at the moderator and sample positions can be represented by the following equation: $\begin{matrix} (28) & q = z_{0} \sqrt{r (1 - r)} \end{matrix}$ The constant $z_{0}$ represents the divergence at the middle of the guide: $r = \frac{1}{2}$ . The f function corresponding to the above elliptic profile is: $\begin{matrix} (29) & f (z) = \frac{1}{2} z_{0} [1 + {(\frac{z_{0}}{z})}^{2}] \end{matrix}$ This is converging to a parabola for large z values but becomes unreasonably large for small z values. The derivative of f becomes negative for $z < z_{0}$ , which corresponds to $r > \frac{1}{2}$ , thus, the elliptic shape can be effectively used only inside the second half of the flight path, closer to the sample. Besides, the whole profile is determined only by the entrance: $\begin{matrix} (30) & z_{ex} = (1 + g - \frac{g}{r_{en}}) \sqrt{\frac{r_{en} (1 - r_{ex})}{r_{ex} (1 - r_{en})}} \end{matrix}$ which is a fixed arrangement with $r_{en} ⩽ \frac{1}{2}$ . In spite of the intuitive belief that ellipse shape works for any sample-moderator relationship, because of the first condition (2), even for the best case of $r_{en} = \frac{1}{2}$ , the loss factor (3): $R_{loss} = g$ shrinks the direct field of view for small moderators (i.e. relatively large g values). It should also be noted that this analysis only concerns the ideal case of a single reflection in an elliptical guide, which accounts for only part of the transported intensity [7].

4. An application case

The SRS profile was used to design a short focusing guide for the SNAP instrument at BL3 of SNS. In this case, the entrance and exit locations were fixed as the guide had to reside inside the instrument accessible enclosure. Considering the distance sample-moderator: $L_{m} = 15 m$ , the entrance and exit locations are: $r_{en} = 0.19$ and $r_{ex} = 0.032$ , corresponding to a relatively small guide length of 2.355 m and located with its exit at 0.48 m from the sample position. The dimensions of the moderator: 97 by 120 mm (horizontal and vertical) allow for small g values, as the instrument specific science requires very small samples (in high pressure cells) and a millimeter sized beam spot on sample is desired. Thus, the sample aperture width and height (not fixed) were constrained in the 2–3 mm range, slightly larger in height, with the corresponding g values of 0.025–0.028. Consequently, the dimensions of the entrance aperture, constrained by equation (2), were maintained to about 16 by 20 mm, which correspond to a $q_{en}$ value around 0.168 and $z_{en} = 0.88$ , respectively. The goal was to explore the possibility of attaining different levels of divergence at sample while minimizing the reflection angle inside the guide. Three levels of divergence were considered: 0.8 by 1.2 deg, 1 by 1.4 deg and 1.2 by 1.6 deg (low, middle and high respectively), with the corresponding guide exit apertures given in Table 1. The exit reflection angle was selected by using equation (7) and b value associated with the SRS solution was calculated using equation (22). The b values obtained by this optimization, as well as the corresponding reflection angle ranges are shown in Table 1.

Table 1
Summary of SNAP guide design parameters used in simulations

Type $2 α$ [deg] $Δ y_{en}$ [mm] $Δ y_{ex}$ [mm] b $θ_{en}$ [deg] $θ_{ex}$ [deg] Gain

Low divergence horizontal 0.8 15.7 7.0 0.59 0.08 0.22 2.23

vertical 1.2 20.1 10.5 0.47 0.15 0.36 2.62

Middle divergence horizontal 1.0 16.3 8.7 0.45 0.13 0.28 2.70

vertical 1.4 20.2 12.2 0.38 0.20 0.39 3.06

High divergence horizontal 1.2 15.9 10.5 0.34 0.19 0.36 3.35

vertical 1.6 20.4 14.0 0.32 0.26 0.45 3.50

Type	$2 α$ [deg]	$Δ y_{en}$ [mm]	$Δ y_{ex}$ [mm]	b	$θ_{en}$ [deg]	$θ_{ex}$ [deg]	Gain
Low divergence	horizontal	0.8	15.7	7.0	0.59	0.08	0.22	2.23
vertical	1.2	20.1	10.5	0.47	0.15	0.36	2.62
Middle divergence	horizontal	1.0	16.3	8.7	0.45	0.13	0.28	2.70
vertical	1.4	20.2	12.2	0.38	0.20	0.39	3.06
High divergence	horizontal	1.2	15.9	10.5	0.34	0.19	0.36	3.35
vertical	1.6	20.4	14.0	0.32	0.26	0.45	3.50

The expected gain in divergence relative to the direct view of the moderator was also estimated in the last column of Table 1. The product of horizontal and vertical gain represents the maximum gain in flux, if the mirror reflectivity approaches unity.

To further examine the design characteristics of these three focusing variants, we performed McStas [14,28] simulations with a realistic moderator source and wavelength dependent supermirror reflectivity. The guide was simulated using the Guide_tapering component, approximating the curved shape in 800 steps. In order to limit the parameter space for the optimization, all simulations were performed with $m = 6$ and default reflectivity parameters. Since the beam line is relatively short and the desired wavelengths are comparably short, gravity was not included in the simulations. Figure 2a shows the wavelength dependence of the intensity in the working band for all three settings. Although the high divergence option shows a large increase in intensity toward the larger wavelength side of the spectrum, in the short wavelength side the decrease of intensity is significant, due to a reduced mirror reflectivity for those energies. The corresponding horizontal divergence dependence of wavelength is shown in Fig. 2b, as well as the acceptance diagram (divergence angle versus horizontal coordinate at the sample position) in horizontal plane. The direct beam contribution can be recognized in the intense central band of the acceptance diagram, whereas the mirror reflection contributions are in close contact with the central band inside about a 2.5 mm span around the center of the beam. The adjacent features, including the twice reflected spots visible for low and middle divergence versions, can be easily removed by inserting a small aperture in front of the sample. Figure 2c shows the 2-D spatial distribution in the neutron beam, integrated over whole wavelength band, and a cross section along the horizontal axis. Scattered spectra were simulated in McStas using an idealized theoretical sample that produces Bragg scattering at regular intervals over a broad d-spacing range. Scattered neutrons were detected in time-of-flight (TOF) event mode in realistically sized pixelated monitors placed at SNAP’s three most common detector positions (0.5 m from sample, rotated by 48°, 90°, 115° around the sample). The simulated data was then analyzed using Mantid by converting TOF to d-Spacing, histogramming the data, and summing up the resulting spectra over the entire instrument or only over individual detectors. Figure 3 shows a comparison of the 1Å peak shapes from the three different guide options. After careful consideration of the instrument’s needs, the mid divergence version was chosen as the best upgrade path.

Fig. 2.

McStas simulation results for the three SNAP guide options: a) left side: wavelength spectra, frame overlap chopper set to center wavelength of 2.1 Å (standard setting for SNAP). Right side: Brilliance Transfer as a function of wavelength, no chopper in simulation. b) Horizontal divergence as a function of wavelength (top) and position (bottom), both with frame overlap chopper set to 2.1 Å. The asymmetric structures at the start and end of the wavelength dependent divergence distribution are an effect of the chopper sweeping though the beam, opening and closing on one side before the other.

Fig. 3.

Diffraction peak shape of a powder sample (d-spacing: 1 Å) in SNAP instrument detectors calculated for three different guide options. SNAP has movable detector panels ( $0.5 m \times 0.5 m$ , distance sample-detector: 0.5 m) in three preferred positions: backscattering (top left), perpendicular to beam direction (top right), and forward scattering (bottom left). The bottom right panel shows the diffraction peak shape if one were to combine all three detector panel positions for a measurement utilizing the full instrument.

5. Combining guides

5.1. Parallel connection

Usually, the neutron beam extraction systems contain various elements with specific functionality and the neutron guide must accommodate certain gaps, some significantly larger than those specified by the mounting requirements. It is an essential question to work out a way to ensure that consecutive pieces of guide are not leaving gaps in the acceptance diagram at the sample position. In this respect, each piece of guide can be considered independently as feeding from a unique source, the moderator. This can be conceptualized as a parallel connection. In this case, the second and third conditions (see Sections 2.2 and 2.3) stay unchanged, but the first condition (Section 2.1) must be changed. This is because the previous piece of guide restricts the field of view, if there is a gap between consecutive pieces of guide. Thus, the equation (2) must be modified as follows: $\begin{matrix} (31) & q_{en}^{(n)} ⩽ \frac{r_{en}^{(n)}}{r_{ex}^{(n - 1)}} (q_{ex}^{(n - 1)} + g) - g or z_{en}^{(n)} ⩽ z_{en}^{(n - 1)} + g / r_{ex}^{(n - 1)} - g / r_{en}^{(n)} \end{matrix}$ The index n represents the number of the piece with $n = 1$ being the closest to the moderator; $n = 0$ refers to the moderator and the relation (31) reduces to equation (2) for the first piece of guide. In a true parallel connection, the contributions of different elements are independent, and the total flux must be a sum of all components. In our case the guide pieces are displaced in space, and there are some losses in the direct beam field of view due to the partial overlapping of consecutive guides contributions. As such, the equation (3) must be correspondingly changed: $\begin{matrix} (32) & R_{loss}^{(n)} ⩾ (1 / r_{en}^{(n)} - 1 / r_{ex}^{(n - 1)}) g \end{matrix}$ These field of view losses caused by gaps in the guide are more important if the gap is closer to the sample position. They affect only the fast neutrons which are unable to reach the sample by anything other than direct line of sight. For neutrons of longer wavelengths, the loss of direct field of view is compensated by the additional guide surface, and the brilliance transfer of the guide system will not suffer for those wavelengths. An example of parallel connection is featured in Fig. 4a. Two pieces of guide leave a 3 meters empty space in-between and both are looking directly to the moderator, with minimal interference between the reflected beams.

Fig. 4.

Examples of parallel (a) and serial (b) connections using the SRS guide surface calculation method. The case where the serial connection of guides is used requires understanding of the intensity that reflects off of both sides of the guide system, and should be accounted for in the design of surfaces.

Another question regarding the parallel connection of guides is the continuity of the reflection angle at the transition between two adjacent pieces. When considering a sequence of guide pieces (with or without gaps), keeping the continuity of the $f (z)$ function will provide a piecewise approximation of any global reflection angle change along the guide. This is because the value of b can be adjusted accordingly. This way the local approximation with SRS can become an efficient method to optimize the global shape of a long guide by means of maximizing different figures of merit.

5.2. Serial connection

In the parallel connection case, any of the neutron path projections in one plane (horizontal or vertical) contains only one specular reflection. Unlike the parallel connection, the serial connection supposes that for two pieces of guide, a reflection in the first piece will be followed by another one in the second piece. A famous example is the ballistic guide [1,17,21], when a parabolic guide transforms a divergent beam into a parallel beam, which is focused back with a second parabolic guide close to the sample position. To generalize such an approach and still preserve the continuity of the acceptance diagram, an intermediary aperture (real or imaginary) must be introduced between the exit from the first guide and the entrance into the second. This can be considered a ‘secondary source’ and its opening should satisfy an equivalent of equation (2): $\begin{matrix} (33) & g_{ss} ⩽ r_{ss} - (1 - r_{ss}) g \end{matrix}$ Here $g_{ss}$ is the ‘sample’ versus moderator ratio related to the first guide, whereas $r_{ss}$ and g are defined in the entire sample-moderator scheme. An example of serial connection is featured in Fig. 4b, with a secondary source aperture placed at two thirds of the distance from moderator to the sample position.

For the first piece of guide, all three conditions will refer to the moderator and the secondary source, whereas for the second piece the three conditions will refer to the secondary source and the sample aperture. By following the three conditions, the first guide will provide a homogenous coverage of the secondary source opening. However, following the corresponding three conditions of the second guide would not guarantee that the acceptance diagram of the sample aperture that is back reflected to the secondary source location falls inside the acceptance diagram produced by the first guide. A detailed check on the mutual setting of these two acceptance diagrams is necessary to maximize their intersection, otherwise the brilliance transfer will be affected. If using SRS functions for the guide design, the condition $b ⩾ 0$ can be relaxed for the first guide, as the neutron trajectories with large exit divergence in the first guide are linked to the small divergence angles at the sample positions and vice versa. Interestingly, as in the first guide $g_{ss}$ can become large and $b < 0$ , the limitation of the dimensions of the moderator (large moderators) can be removed in this case. A serial connection of two guides may be a good solution for the neutron beam extraction from small moderators. However, a careful optimization of the b values in the two guides will be necessary, aside from the acceptance diagrams matching. In general, it is expected that the serial connection will lower significantly the brilliance transfer due to the double reflections inside the assembly, but it is sometimes unavoidable when a small cross section element (as a Fermi chopper for example) must be inserted into the neutron flight path away from sample position [5,13]. The serial connection also allows for real focusing in the terminal device, which may image the secondary source aperture at the sample position or detector (in case of SANS). These are one sided (Kirkpatrick-Baez or Montel) arrangements of mirrors, and their serial connection problem [11] goes beyond the scope of this publication.

6. Discussion and conclusion

Restrictions in the dimensions of entrance and exit apertures of neutron guides have been established previously. For example, the conditions seen in (2) or (31) have been applied successfully when guides were made from straight tapered pieces [22,23]. For elliptic guides, the authors of [19] report limits for garland and zig-zag paths, but they refer to a specific guide shape and lack the generality of our deduction from Section 2.2. Other authors tried to deduce a new shape for the mirrors, which ensures a homogenous transformation of phase space volume [24,25]. However, their approach focuses on monochromatic beams and does not address issues of multiple reflections arising from multispectral beams. The last developments in neutron guides optimization [3,4] include phase space considerations, but the profile of each guide piece is fixed from the beginning.

Our three general conditions, discussed in Section 2, are independent of guide shape. For a fixed type of guide (parabolic, elliptic or logarithmic) the existence of a solution fulfilling all three conditions is not certain. Our general solution (SRS), is always available and, by assembling pieces of SRS tapered guides, it is possible to construct any complex guide system and optimize its shape as an ensemble. We demonstrated that this approach works well for large moderators ( $g < 1 / 2$ ) and parallel connections, but serial connection, as a general solution, was not investigated completely. To this point, our efforts concentrated on source-in-view guide design. Future work should address extending our approach to off-source-view deflecting guides, however, in this case, the serial connection will likely be required, making its analytical treatment more challenging.

A portion of this research used resources at the Spallation Neutron Source, a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory.

This research used resources of the Compute and Data Environment for Science (CADES) at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725.

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