Monte-Carlo ray-tracing studies of multiplexed prismatic graphite analyzers for the cold-neutron triple-axis spectrometer at the High Flux Isotope Reactor

The neutron triple-axis spectrometer (TAS) is an important tool for studying collective excitations of condensed matter systems [2]. In a TAS experiment, the dynamical scattering function of the sample, $S(\mathbf{Q},E)$, is measured (in selected regions of momentum-energy space) according to the scattering relations for momentum $\mathbf{Q}=\bm{k}_{i}-\bm{k}_{f}$ and energy transfer $E=E_{i}-E_{f}=\frac{\hbar^{2}}{2m_{n}}(k_{i}^{2}-k_{f}^{2})$ [3]. In simplified terms, this requires knowledge of the neutrons’ initial and final wave-vectors $\bm{k}_{i,f}$, the scattering angle $2\theta$ of the detector, and the azimuthal orientation angle $\phi$ of the sample in the $xy$-scattering plane, which are related by

In the traditional TAS technique, $S(\mathbf{Q},E)$ is measured in a point-by-point fashion (within a resolution volume). Measuring the entire sample’s response in a given scattering plane typically requires a large number of combinations of $\bm{k}_{i}$ ($E_{i}$), $\bm{k}_{f}$ ($E_{f}$), $2\theta$, and $\phi$, resulting in long measurement times when a survey of $(\mathbf{Q},E)$-space is desirable. Multiplexing techniques can alleviate this bottleneck by using several analyzers to measure multiple $(\mathbf{Q},E)$-points simultaneously. Although this can complicate the scattering geometry, the spectrometer’s resolution function, and the available background mitigation techniques, this can be advantageous over neutron time-of-flight (TOF) spectroscopy for constrained sample environments, when combined with polarization analysis techniques, to maximize neutron flux using focusing optics, and/or when only a single scattering plane is needed.

Following the distinction made previously [1], multiplexing spectrometers can be broadly categorized into two families: wide-angle multiplexing and local multiplexing. In wide-angle multiplexing – also described as multi-channel detector systems – several analyzer channels are placed around the sample in a concentric arc, generally with fixed $\bm{k}_{f}$ ($E_{f}$). This technique increases the solid angle coverage of the analyzer system but restricts the geometry. One of the earliest examples of this concept is the MADBox spectrometer [4] at the Forschungsreaktor München (FRM), which made use of 61 in-plane analyzer channels. The FlatCone design at the Institut Laue-Langevin (ILL) [5] built upon this concept and uses 31 analyzer channels placed around the sample which scatter out of plane. A particularly efficient example of wide-angle multiplexing is the MACS spectrometer at the National Institute of Standards and Technology (NIST) [6], which uses analyzers with variable $\bm{k}_{f}$ and a combination of filters to suppress higher-order wavelengths. The local multiplexing technique places multiple analyzers on the same channel. An example is the IMPS [7] spectrometer, which places several analyzers at slightly offset angles from each other. The distinct analyzer crystals or blades direct neutrons toward a different spot on a position-sensitive detector (PSD). Similar concepts have been utilized in the SIKA, RITA-II, and PUMA spectrometers [8, 9, 10]. A particularly novel design is the HODACA concept, which uses a Rowland “anti-focusing” effect by placing analyzers on a circumferential arc, each of which focuses towards a different detector, resulting in a constant $E_{f}$ but large $Q$ coverage [11].

One of the more recent developments in multiplexing TAS design is the Continuous Angle Multiple Energy Analysis (CAMEA) spectrometer, which is installed and commissioned at the Paul Scherrer Institute (PSI) [1, 12, 13]. CAMEA uses the nearly perfect transmission of highly-oriented pyrolytic graphite (HOPG) crystals [14] to combine the local and wide-angle multiplexing techniques. While traditional local multiplexing (e.g. IMPS [7]) requires analyzers to be placed at slightly different scattering angles, CAMEA places analyzers directly behind each other, simplifying the geometry. Each of the eight (8) arrays of analyzers, or stations, focuses a band of different $\bm{k}_{f}$ ($E_{f}$) out of the scattering plane at specific locations on a position-sensitive detector, hence creating a quasi-energy-resolved detector. Eight (8) channels of this multi-analyzer system are placed at different $2\theta$ angles, resulting in wide-angle coverage. A similar concept has been successfully implemented at spectrometers like MultiFlexx [15] and BAMBUS [16] and it is planned for BIFROST at the European Spallation Source (ESS) [17].

Maintenance and upgrade plans for the High Flux Isotope Reactor (HFIR) at Oak Ridge National Laboratory [18] offer the opportunity to design and execute the construction of a state-of-the-art cold-neutron triple-axis spectrometer on a new cold guide-hall location, NB-6. In this paper, we employ Monte-Carlo ray tracing simulations to explore the concept and the performance of a secondary spectrometer relying on the CAMEA design and an optimized primary spectrometer. The design principles, optimization, and performance of the primary spectrometer (guide system, velocity selector, and monochromator) are discussed in Ref. [19]. The secondary spectrometer is currently known as the Multi-Analyzer Neutron Triple Axis (MANTA) spectrometer. This project offers the opportunity to explore and implement various forms of prismatic analysis strategies, which greatly enhances the resolving power of the spectrometer, as shown conceptually in Fig. 1.

This paper is organized as follows. In Sec. II, we outline the geometrical and technical parameters of our model instrument, as well as the multiplexing and prismatic analysis concepts it utilizes. In Sec. III, we detail the methods of our ray-tracing Monte-Carlo simulations on a realistic sample kernel to benchmark the performance of our model instrument. In Sec. IV we analyze the simulation data and present the principles of prismatic analysis along with the demonstration of a novel approach called Positionally-Calibrated Prismatic Analysis (PCPA). In Sec. V, we present the results of our ray-tracing simulations and data analysis approach on a realistic sample kernel and present how modifications to the original design impact data acquisition and quality. We conclude this work in Sec. VI.

The model of the primary spectrometer comprises an optimized neutron guide system with supermirror coatings that directs neutrons from the HFIR cold-neutron source (with the parameters of the existing HFIR cold-source) onto a 300 mm $\times$ 168 mm doubly focusing monochromator. The primary was optimized for incident energies of 2.4 to 20 meV, see Ref. [19] for details. The sample is located 160 cm from the monochromator. An aspirational engineering rendering of the spectrometer in the HFIR cold guide hall is presented in Fig. 2(a). Using a focusing monochromator increases the neutron flux on the sample. However, it yields a broad divergence of the incident beam at the sample position, affecting the spectrometer’s resolution. By incorporating the guide system and the doubly focusing monochromator in our simulations (see Section III), these resolution effects can be accounted for. In some instances in Section V, we utilized a simplified, “toy model” design of the full primary spectrometer. The “toy model” was designed to ease computational intensity and used an idealized virtual source and curved monochromator that produced a similar spot size and energy resolution at the sample location.

For the secondary spectrometer, we adapted the back-end design of CAMEA [1] to our virtual instrument, see Fig. 2(b) for an engineering diagram. As the planned use of a beryllium filter in the scattered beam requires analyzed energies to below 5.2 meV, and geometrical constraints require the analyzed energy to be above 2.4 meV, the CAMEA design is particularly well suited for our purposes. The design consists of $N_{c}=8$ angular channels, each of which have $N_{d}=13$ detectors that span $\Delta(2\theta)\!=\!7.5^{\circ}$ per channel, see Fig. 2(b). Each channel comprises $N_{s}$ analyzer stations in series, each built from $N_{b}$ blades of flat, highly-oriented pyrolytic graphite (HOPG) crystals. The analyzer blades of a given station have a mosaic of 1^∘, equivalent to 60 arc minutes (60’), and are rotated to fulfill the Bragg condition for a given $E_{f}$. We label the analyzer station with index $s=1\cdots N_{s}$. Furthermore, analyzer stations between different channels are focused using Rowland geometry [3, 20]. Analyzer stations placed further from the sample are larger to account for decreased solid angle coverage for a given crystal size with increasing distance. Additional details of the instrument model are provided in Tab. 1. In some instances in Section V, we modify the CAMEA design by changing the number of stations $N_{s}$ and the analyzer mosaic.

The analyzer blades reflect neutrons from the sample to an out-of-scattering plane direction and towards a series of horizontal position-sensitive detectors (PSDs), laid out in a radial pattern above the stations. Furthermore, the radial pattern has two vertical rows to minimize gaps between detectors. Each detector has an active length of $\approx 0.9$ m and a diameter of 12.6mm. By focusing neutrons on specific locations on the detectors, each analyzer station generates a “strip” of neutrons associated with a particular $E_{f}$. As illustrated in Fig. 3 for the parameters of Tab. 1, this approach achieves local multiplexing by avoiding overlap between strips generated by different stations. Thus, at minima, this allows the acquisition of $N_{s}=8$ $E_{f}$ values for each channel.

Though this method of discrete angular and analyzer channels to form a multiplexing instrument has been accepted as a viable approach for decades, one of the advantages and innovations of the CAMEA concept is the utilization of the prismatic analyzer concept [21, 1, 12]. The foundations of prismatic analysis come from a first-order expansion of Bragg’s law from a finite mosaic analyzer crystal (with lattice spacing $d$). This yields a linear relationship between small deviations in incident angles $\Delta\theta$ and small deviations in reflected neutron wavelength $\Delta\lambda$ around a nominal Bragg reflection ($\sin\theta_{0}=\lambda_{0}/2d$), as illustrated in Fig. 4(a). As a result of analyzer mosaic, neutrons with proximate wavelengths from the nominal wavelength $\lambda_{0}$ are Bragg-scattered in different directions, Fig. 4(b). Using a position-sensitive detector, this effect can be advantageously used to encode small wavelength deviations into detector positions, Fig. 4(c). Since its inception, prismatic analysis has been featured in the design of the planned time-of-flight instruments BIFROST [17] and FARO [22].

To assess the performance and validate the concept of our spectrometer model, we employ neutron ray-tracing simulations using the McStas package [23]. McStas is a state-of-the-art tool used in designing modern neutron scattering instruments [17, 22, 16, 7, 6, 15, 1, 9, 24, 25, 26, 27, 28]. The program uses Monte Carlo techniques to simulate the behavior of neutron scattering instrument components with high accuracy. It also allows testing the instrument’s performance on various simple sample scattering kernels to mimic actual experiments. In this work, we employed McStas versions 2.7, 3.1, and 3.4.

The primary spectrometer, from the cold source to 15 cm before the sample, was simulated in McStas for various choices of incident energy $E_{i}$, using the design of Ref. [19]. These results were pre-compiled to form a set of simulated incoming beam profiles for the secondary spectrometer. The latter was simulated by adapting the McStas code of the PSI-CAMEA back-end, which was generously provided to us by its designers [1]. The virtual instrument comprises $N_{b}=5$ analyzer blades per station, $N_{s}=8$ stations per channel, $N_{c}=8$ channels with $N_{d}=13$ detectors per channel. This results in a total continuous scattering angle coverage of $8\times 7.5^{\circ}\!=\!60^{\circ}$ and a total of $N_{s}*N_{c}*N_{d}=832$ points in $S(\vec{Q},\omega)$ measured simultaneously. While the actual spectrometer uses filters and a radial collimator after the sample, we did not include these components in our simulations to maintain a continuous coverage; including collimators in our simulations is left for future work. A 10 $\times$ 10 cm rectangular slit simulates the opening in front of the analyzer stations, allowing full illumination of the analyzers and unrestricted angular coverage. Unless otherwise noted, we used an infinitely thin HOPG analyzer with a mosaic of 60’. Note that modeling the analyzer in this manner does not take into account the background signal that arises from thermal diffuse scattering and phonons in HOPG, nor the effect of the decrease in peak reflectivity as a function of crystal mosaic [14, 29, 3].

We employed three types of sample kernels. The first virtual sample was a full cylinder of polycrystalline vanadium with a 2 cm height and a 1.0 cm radius. We used this incoherent scattering kernel to obtain calibration runs (see Section IV) of the analyzer system and the position-sensitive detectors. Then, a virtual single crystal with a similar cylindrical geometry was prepared using the McStas phonon kernel component to mimic a simple bosonic excitation (acoustic magnon or phonon) dispersing in three dimensions with a bandwidth below 2 meV and uniform intensity across the Brillouin zone. The parameters of this inelastic scattering kernel are found in Tab. 2. Finally, we made use of a single crystal elastic Bragg scattering sample with a mosaic of 10’, lattice constant c = 6.71 $\,\mathrm{\AA}$, and the same geometry as the inelastic scattering kernel, which produced a single Bragg peak at $\vec{Q}=(0,0,2)$ ($|\vec{Q}|=1.85\,\mathrm{\AA^{-1}}$). This sample was used to calculate the elastic $\vec{Q}$ broadening due to the instrument.

The simulations with the inelastic scattering crystal kernel were conducted to model a realistic scattering experiment covering energy transfers between $E=0$ meV and $E=2$ meV. As the detector system has an overall $2\theta$ coverage of $60^{\circ}$, the single crystal sample was simulated for several positions of the detector system. Using the lowest angle of the detector systems as a reference, we aimed to measure 5 $2\theta_{i}$ values centered around three reference points, $6^{\circ}$, $66^{\circ}$, and $116^{\circ}$. However to minimize scattering signal from non-interacting neutrons, simulations where the lowest detector angle was below $2.5^{\circ}$ or the largest detector angle was at or above $180^{\circ}$ were not simulated, leading to a net total of 12 positions. For each detector position, the single crystal sample was rotated in steps of $\Delta\phi=2.5^{\circ}$ and we ran the Monte-Carlo process with $>10^{8}$ neutrons for each sample orientation. This was repeated for two $E_{i}$ values, 5.1 meV and 5.27 meV, to produce the simulated results. The kernel was designed to measure the entire dispersion curve with $E_{i}=5.1$ meV, and the second incident energy of $E_{i}=5.27$ meV was selected to fill in the gaps in energy space. These results are shown in Section V.

Additional simulations with an elastic Bragg scattering sample were conducted to determine the elastic $\vec{Q}$ broadening as a function of $E_{i}$. To do this, the full primary and secondary spectrometer were used in a similar setup as the inelastic sample simulations. However, only one angular channel was used and a single slit with an opening of $1^{\circ}$ was placed in half the simulations. Using the known position of the Bragg peak as a reference, $2\theta$ was varied from a range of $\Delta 2\theta=\pm 12^{\circ}$ from the Bragg peak in steps of $1^{\circ}$ and sample was rotated in a range of $\Delta\phi=\pm 7$ in steps of $0.5^{\circ}$. This was repeated for three $E_{i}$ values, 3.21 meV, 4.05 meV, and 5.01 meV. The results are tabulated in Section V.

For a given analyzer channel and orientation of the sample, it is possible to use prismatic analysis to obtain the scattering function $S(\mathbf{Q},E)$ for significantly more values of energies than the number of stations $N_{s}$ as shown conceptually on Fig. 1. While prismatic analysis relies on post-measurement data analysis techniques that require careful calibration, it is possible to devise several such strategies.

CAMEA has successfully implemented an “energy-calibration” approach [30, 12, 13]. As shown in Fig. 3, the neutrons scattered by a given analyzer station – nominally associated with $E_{f}(s)$ – produce a well-separated strip (or analyzer-specific area) on the position-sensitive detector. Each strip in the detector space is broken into $N_{p}$ prismatic sub-regions (with $N_{p}$ ranging from $3$ to $7$). The integrated signal detected in a given sub-region is entirely associated with a particular $E_{f}(s,p)$, proximate, but not equal to the reference $E_{f}(s)$, which effectively yields a continuous energy coverage but requires careful calibration [13]. To determine the average $\bar{E}_{f}(s,p)$ analyzed by the $p$-th detector sub-region associated with analyzer station $s$, a calibration is performed by measuring the scattering from a vanadium sample as a function of small steps in $E_{i}$ ($\Delta E_{i}\approx 0.01$ meV). The incoherent vanadium signal integrated over a given sub-region yields a Gaussian profile as a function of energy, which can be fitted to obtain $\bar{E}_{f}(s,p)$, hence the name of Energy-Calibrated Prismatic Analysis (ECPA). Because the efficiency of the prismatic analysis across sub-regions is not uniform, this approach also conveniently yields an intensity normalization $A(s,p)$ to take analyzer transmission and prismatic/detector efficiency into account.

While the above approach works very well, it is interesting to compare it to an alternative method of data analysis which we call the Positionally-Calibrated Prismatic Analysis (PCPA) technique. As opposed to defining discrete sub-regions in detector space and calibrating the average energy being detected, the alternative approach defines discrete steps for the final neutron energies $E_{f}$ and reconstructs the spatial distribution of each $E_{f}$ on the detector system. This approach is demonstrated in Fig. 5. The rationale for this approach is that neutron detection along the detector is very well-defined by the detector electronics, which for the CAMEA implementation at PSI has 1024 pixels over an active length of $\approx 0.9$ m [12]. Just as with ECPA, calibration with a vanadium standard is essential.

To calibrate the analyzer-detector system, the primary spectrometer is tuned to a particular incident energy ($E_{i}$) to analyze the elastic incoherent scattering from the vanadium kernel. It is important to note that the vanadium calibration procedure must use a vanadium sample of the same height as the sample to be measured to properly account for the correct vertical divergence in the beam after scattering. We have verified that the $2\theta$-position of the secondary spectrometer does not affect the results; thus, the calibration procedure is performed at a single position of the detector system. After scattering from the vanadium and a given analyzer blade, neutrons with energy $E_{f}=E_{i}$ are detected at particular locations on the PSD. This yields a Gaussian distribution for each $E_{f}$, as seen in Fig. 5. This process is repeated by sweeping $E_{i}$ across various bands of energy centered around the nominal $E_{f}(s)$ of the analyzers, that is $E_{i}=E_{f}(s)\pm 0.12$ meV is scanned in steps of $0.04$ meV for $s=1\cdots N_{s}$ (total of $\approx 7$ $E_{i}$’s per station $s$).

This process provides the distribution of neutron counts versus detector position for each discrete $E_{f}=E_{i}$ within the calibration set. When comparing the different $E_{f}$’s, it is clear that the total detector-integrated intensities $I(E_{f})$ vary significantly due to the radial arrangement of the detectors and the fact that the additional energies measured from the prismatic concept are intrinsically less intense, see Fig. 5. To correct for this effect, we introduce a scale factor, $G(E_{f})=I(E_{f}=3.21\,\mathrm{meV})/I(E_{f})$, where $I(E_{f})$ is the integrated intensity of a given $E_{f}$ over the entire detector, therefore summing over all Gaussian profiles ascribed to a specific $E_{f}$. The integrated intensity of the $E_{f}=3.21$ meV energy was arbitrarily chosen as a reference as it is the most intense. The scale factor is then applied as a prefactor to each Gaussian profile in the detector space. The last, and central step is to transpose the ensemble of these scaled distributions (of neutron counts versus position at fixed $E_{f}$) into a new set of distributions representing neutron counts versus $E_{f}$ at fixed position. In other words, we create normalized histograms of final energies for each pixel, representing the likelihood that a neutron detected on a given pixel had energy $E_{f}$. A representative result of such a histogram is shown in Fig. 6. Note that the scaling factor $G(E_{f})$ only affects the relative distribution of energies within a histogram; in the end, the scaling does not affect the net flux at the detector. Additionally, to mimic the effect of cross-talk baffles, neutrons that land in between strips of neutrons (as seen in Figure 3) are discarded. In the next section, we employ the PCPA technique on a realistic sample kernel for a virtual spectrometer to assess its feasibility.

In this work, we have performed Monte-Carlo ray-tracing simulations of several multiplexed triple-axis spectrometer models relying on prismatic analysis to obtain a quasi-continuous energy-transfer coverage with only one or two incident energies $E_{i}$. Building on the concept and the design of the CAMEA spectrometer at PSI, we have introduced a distinct data analysis approach relying on the statistical likelihood that a neutron detected on a given pixel corresponds to a particular final energy $E_{f}$. This approach is dubbed Positionally-Calibrated Prismatic Analysis (PCPA). Simulations of the spectrometer model for a simple inelastic scattering signal show that PCPA data analysis maximizes the number of final energies measured simultaneously. Departing from the CAMEA design with eight analyzer stations, we explored further design evolutions, including changing the number of analyzer stations and the crystal mosaic of the graphite analyzers. These conceptual studies highlight the enormous flexibility offered by prismatic analysis. In principle, any existing TAS with a PSD located far enough from the analyzer to allow for distance collimation effects to occur can make use of the prismatic analyzer concept and the PCPA technique. Our conceptual work serves as a foundation for performing more comprehensive studies of multiplexed prismatic spectrometers, taking collimation and background mitigation into account. This includes testing a more complex sample kernel, considering thermal diffuse scattering from the analyzers, and dealing with complex sample environments. Our work utilized a realistic primary spectrometer developed in Ref. [19], including a fully optimized neutron guide system for the cold source of the High Flux Isotope Reactor. Although no formal concept has been chosen for the secondary spectrometer, when combined with a refined, CAMEA-style multiplexed prismatic system, a concept known as MANTA, our work shows that this modern beam-line would deliver world-class performance to study excitations in quantum condensed matter physics.