Simulation of spin-echo SANS (SESANS) using McStas on monochromatic and time of flight instruments

Abstract

We conduct simulations of Spin Echo Small Angle Neutron Scattering (SESANS) by employing Monte Carlo methods to a setup using four magnetic Wollaston prisms. Our primary focus involves the validation of these models, encompassing monochromatic scenarios across various neutron wavelengths to ascertain the reliability of the simulations. Subsequently, we extend this validation to encompass simulations in time-of-flight mode. Our model consistently and precisely predicts the scattering patterns emanating from dilute spheres in both monochromatic and time-of-flight modes. Notably, it also accurately reproduces the intricate encoding associated with scattering occurring between the third and fourth magnetic Wollaston prism, which provides us with another approach to increase the solid angle coverage of a SESANS instrument. This validation process conclusively demonstrates the efficacy of our simulation methods. Importantly, it paves the way for simulating more intricate and realistic instrumental configurations, broadening the horizons for future research endeavours.

1. Introduction

Since the invention of neutron scattering techniques, there has been a concerted endeavour to enhance the resolution. One avenue for achieving this objective involves exploiting the Larmor precession of neutron spins, whereby slight perturbations in neutron energy or momentum transfer induce notable variations in the Larmor phase [8]. This precession phenomenon is dictated by the magnetic field environment through which neutrons traverse. Diverse magnetic field configurations have been devised to encode a myriad of scattering geometries. However, a fundamental challenge arises from the inherent inhomogeneity in magnetic field distributions, potentially leading to aberrations in the accumulated Larmor phase during neutron traversal.

Recent advancements in finite element-based simulations for magnetic fields afford the opportunity to comprehensively assess and characterise magnetic field performance. Such evaluations facilitate the investigation and subsequent minimisation of aberrations in the Larmor phase of neutron spins, guided by specific criteria. Despite the progress in magnetic field simulations, a commensurate level of sophistication in modelling neutron instrumentation has been lacking – a gap that now beckons attention through simulation methodologies.

This study specifically focuses on the Spin Echo Small Angle Neutron Scattering (SESANS) technique, which utilises a series of inclined magnetic fields to encode scattering into neutron polarisation. For a comprehensive understanding, readers are directed to pertinent references [2,26,27].

With an increasing number of instruments adopting SESANS [3,6,14,22,24–26], there is burgeoning interest in employing Monte Carlo methods to optimise these instruments and associated techniques [4,19]. Until recently, accurate simulation of Larmor labelling for scattering from such instruments was unattainable. However, recent enhancements in scattering models for Small Angle Neutron Scattering (SANS) have rendered this achievable [5]. These advanced models not only predict the scattered beam but also effectively consider the transmitted direct beam. Given the expanding presence of spin echo instruments and routine application of the technique at time-of-flight sources, it becomes pivotal to validate instrument behaviour using detailed Monte Carlo models. SESANS is increasingly finding applications across diverse fields such as colloids [23,32], food science [2,31], composites [9], gravitational studies [7,21] and as a prospective quantum probe [16,30]. This study aims to contribute to these advancements by validating various components and simple setups against established theoretical frameworks.

In this work we utilise the McStas Monte Carlo Ray Tracing software [17,33–36]. The remainder of the structure of this paper is as follows. We first define the established analytical expressions which predict the expected SESANS correlation function and degree of scattering expected. We then test these in various stages in our simulations presented here. We also present a description and parameterisation of the components used in the simulations. The validation steps are as follows:

1. Validate the sphere model against monochromatic SANS for different neutron wavelengths and both coherent and incoherent scattering.

2. Validate the sphere model in time of flight (TOF), similar to (1), for both coherent and incoherent scattering.

3. Validate the model for the Wollaston prism module by simulating this in monochromatic mode for several neutron wavelengths, considering simulations performed using the inclined foils [28].

4. Combine the TOF setup with the Wollaston prism model to simulate TOF SESANS.

5. Model a non-standard configuration with the sample between the third and fourth Wollaston prisms and compare against established theory.

Steps (1) and (2) are routine and included in Appendix B.1 for completeness.

2. Analytical correlation functions for dilute hard spheres

In any SESANS experiment, the measured quantity is the neutron polarisation as a function of the spin echo length. Typically, the polarisation resulting from scattering at the sample is normalised to the instrumental polarisation ( $\frac{\mathit{P}}{{\mathit{P}}_0}$ ). This is measured as a function of length termed the spin-echo length δ, defined as follows: $$\begin{array}{}\text{(1)}& \mathit{\delta }=\mathit{c}\mathit{B}\mathit{L}{\mathit{\lambda }}^2\mathit{cot}\mathit{\theta }\end{array}$$ where c is a constant $\mathit{c}=\frac{\mathit{\gamma }\mathit{m}}{\mathit{\pi }\mathit{h}}=1.476\times {10}^{14}{\mathit{T}}^{-1}{\mathit{m}}^{-2}$ , B is the strength of the magnetic field in the prisms, L is the separation between the prisms and θ is the angle between the hypotenuse of the prism and the beam direction as shown in Fig. 1.

Fig. 1.

Diagram illustrating a SESANS setup employing MWPs. The magnetic fields within the shaded and unshaded triangles are identical but antiparallel to each other. The solid black regions denote the $\mathit{\pi }\mathrm{/}2$ flipper, with magnetic fields perpendicular to the MWPs. The path of the neutron beam is represented by the dashed line. This depiction is adapted from reference [22], with adjustments made to the axes to align with the coordinate system in McStas.

In SESANS the measured quantity of the polarisation can be expressed as; $$\begin{array}{}\text{(2)}& \mathit{P}\mathrm{(}\mathit{\delta }\mathrm{)}=exp\mathrm{\left(}{\mathrm{\Sigma }}_{\mathit{t}}\right\{\mathit{G}\mathrm{(}\mathit{\delta }\mathrm{)}-1\left\}\mathrm{\right)}\end{array}$$ Here $\mathit{G}\mathrm{(}\mathit{\delta }\mathrm{)}$ is related to the Debye correlation function of the sample, for a dilute sphere sample with spheres of radius r, $\mathit{G}\mathrm{(}\mathit{\zeta }\mathrm{)}$ where $\mathit{\zeta }=\frac{\mathit{\delta }}{\mathit{r}}$ is given analytically by [15]; $$\begin{array}{}\text{(3)}& \mathit{G}\mathrm{(}\mathit{\zeta }\mathrm{)}=\mathrm{\Re }\left\{{[1-\mathrm{(}{\displaystyle \frac{\mathit{\zeta }}{2}}\mathrm{\left)}\right]}^{\frac{1}{2}}\mathrm{\right(}1+{\displaystyle \frac{1}{8}}{\mathit{\zeta }}^2\mathrm{)}+{\displaystyle \frac{1}{2}}{\mathit{\zeta }}^2[1-{\mathrm{\left(}{\displaystyle \frac{\mathit{\zeta }}{4}}\mathrm{\right)}}^2]ln[{\displaystyle \frac{\mathit{\zeta }}{2+{\mathrm{(}4-{\mathit{\zeta }}^2\mathrm{)}}^{\frac{1}{2}}}}\left]\right\}\end{array}$$ Here ℜ denotes taking the real part, evaluated for δ in the range 0 → 2r. ${\mathrm{\Sigma }}_{\mathit{t}}$ is given by [1] as; $$\begin{array}{}\text{(4)}& {\mathrm{\Sigma }}_{\mathit{t}}={\displaystyle \frac{3}{2}}\mathit{\varphi }\mathrm{(}1-\mathit{\varphi }\mathrm{)}{\mathrm{(}\mathrm{\Delta }\mathit{\rho }\mathrm{)}}^2{\mathit{\lambda }}^2\mathit{r}\mathit{d}\end{array}$$ Here d is the thickness of the sample and ϕ is the volume fraction. $\mathit{G}\mathrm{(}\mathit{\delta }\mathrm{)}$ is the correlation function and ${\mathrm{\Sigma }}_{\mathit{t}}$ a measure of the scattering power. For more details the reader is directed to references [1,23,32].

The form of $\mathit{G}\mathrm{(}\mathit{\delta }\mathrm{)}$ is depicted schematically in Fig. 2 (a) for two different sphere radii. The critical observation is that the correlation function approaches zero at the particle diameter. Figure 2 (b) illustrates the calculated polarisation for two monochromatic neutron wavelengths and a time-of-flight scan, highlighting the wavelength dependence on the scattering power. In Fig. 2 (c), the normalised scattering correlation function is presented, normalising out the neutron wavelength and sample thickness contributions. This results in a measured correlation function that is independent of the instrumental configurations and solely dependent upon the spin echo length and  ${\mathrm{\Sigma }}_{\mathit{t}}$ .

Fig. 2.

(a), the forms of $\mathit{G}\mathrm{(}\mathit{\delta }\mathrm{)}$ for dilute spheres are displayed as given by equation (3) for two different radii (as indicated in the legend). (b), the calculated polarisation as a function of spin echo length is depicted for two monochromatic neutron wavelengths and a time-of-flight scan. It’s important to note that in the time-of-flight simulation, we assume a static magnetic field of 117G, corresponding to a spin-echo length given by $\frac{1}{5}{\mathit{\lambda }}^2$  [Ų]. As a result, the TOF curve intersects the 3 Å curve at $\mathit{\delta }=1.8\mathit{\mu }\text{ m}$ and the 5 Å curve at 5 μm. Lastly, in (c), the three different simulations from (b) are corrected for wavelength, illustrating that they all converge onto the same curve, expressed as a normalised correlation function.

3. Components used

We now introduce the various components employed in this work and their associated parameterisation. SANS $\text{\_}$ spheres2 is a McStas component that mimics the scattering from a system of hard monodisperse dilute perfect spheres. An advantage of this sample over earlier samples defined for McStas is the consideration of both scattered and transmitted neutrons. Previous sample components only accounted for scattered neutrons, resulting in unrealistic SESANS outcomes. Further improvements include the distribution of scattered neutrons along a logarithmic distribution in the momentum transfer Q and the extension of the Q range to encompass the SESANS regime and the inclusion of multiple scattering [5,13].

The sample is initially utilised to simulate the expected scattering from a Small Angle Neutron Scattering (SANS) instrument, as shown in Appendix B.1. The sample configuration comprises a volume fraction ϕ of 0.001, a difference in scattering length density between the spheres and the solvent, denoted as $\mathrm{\Delta }\mathit{\rho }$ , of $6·{10}^{10}{\text{ cm}}^{-2}$ , a radius r of 150 Å and a thickness of 1 mm. The outcomes of this simulation are detailed in Appendix B.1.

Neutrons possess the capability to scatter both coherently and incoherently from the sample. In the SANS $\text{\_}$ spheres2 sample, incoherent scattering is also considered, allowing the resultant simulation of an incoherent background, typical in more hydrogenous samples. Consequently, for dilute spheres, the scattering follows the conventional expression [10]. $$\begin{array}{}\text{(5)}& \mathit{I}\mathrm{(}\mathit{Q}\mathrm{)}=\mathit{\varphi }\mathit{V}{\mathrm{(}\mathrm{\Delta }\mathit{\rho }\mathrm{)}}^2{\left[3{\displaystyle \frac{sin\mathrm{(}\mathit{Q}\mathit{r}\mathrm{)}-\mathit{Q}\mathit{r}cos\mathrm{(}\mathit{Q}\mathit{r}\mathrm{)}}{{\mathrm{(}\mathit{Q}\mathit{r}\mathrm{)}}^3}}\right]}^2+{\mathit{I}}_{\mathrm{bg}}\end{array}$$ Here, V represents the volume of the sphere with radius r. A Q range from $1\times {10}^{-5}$  Å⁻¹ to 0.4 Å⁻¹ and incoherent backgrounds of 0 and 0.5 cm⁻¹ are employed. To validate this sample component, simulations were conducted for regular monochromatic and TOF SANS setups, summarised in Section B.1. These simulations demonstrate that the component behaves as expected for two different incident neutron wavelengths in monochromatic mode and also in time of flight. Furthermore, it accurately accommodates the inclusion of incoherent scattering. The data obtained were fitted using SasView, resulting in derived radii and volume fractions that agree to within 0.1% with the input parameters, including the incoherent background when utilised.

For simulating a monochromatic neutron source, the source $\text{\_}$ simple component is utilised, representing a simple, rectangular neutron source targeting a rectangular target. The utilised source measures 6 cm × 6 cm, generating 1 × 10⁸ neutrons. Two neutron wavelengths of 3 Å and 5 Å are simulated to verify the wavelength dependence. 3 Å is short for a SANS instrument, however it is used here as SESANS instruments can often utilise short wavelength neutrons, for example in the Delft setup [28].

In the TOF setup, the source $\text{\_}$ gen component is used, allowing the specification of a range of wavelengths with a certain distribution. A wavelength range from 2 Å to 5 Å with a uniform distribution is employed, generating 1 × 10⁹ neutrons. This increased number of neutrons aims to ensure sufficient statistics due to the higher number of spin-echo length bins when scanning over a range of $\mathit{G}\mathrm{(}\mathit{\delta }\mathrm{)}$ .

Two Monitor components in McStas are utilised, which return a simple count of neutrons. These monitors remain idealised and do not affect the passing neutrons. In the time-of-flight simulation, the monitors are replaced by the wavelength-sensitive monitor L $\text{\_}$ monitor in McStas, which counts the neutrons passing through, binned within the specified wavelength range. In both the monochromatic and TOF simulation these monitors are directly after the sample position in the direct beam.

For the final detector, a Position Sensitive Detector (PSD) of the PSD $\text{\_}$ monitor type is employed. This component consists of multiple pixels in both width and height, individually summing all neutrons passing through each pixel and outputting the total intensity for each pixel. The utilised PSD in the setup measures 1 m × 1 m and consists of 249 × 249 pixels. An example of a PSD measurement is depicted in Fig. 5 (b). In the time-of-flight setup, the PSD $\text{\_}$ TOF $\text{\_}$ monitor is used, binned within a specified time range.

Lastly, the setup includes source and sample apertures of the Slit component type in McStas. Circular apertures are employed with radii ${\mathit{R}}_1=5\text{ mm}$ and ${\mathit{R}}_2=5\text{ mm}$ for the source aperture and sample aperture, respectively.

4. Results of the SESANS simulation

The SESANS instrument in McStas is visualised in Fig. 9. Simulations were conducted for both the spin-up ( ${\mathit{I}}_{+}$ ) and spin-down ( ${\mathit{I}}_{-}$ ) configurations. This was achieved by altering the initial polarisation state and calculating $\mathit{P}=\frac{\mathrm{(}{\mathit{I}}_{+}-{\mathit{I}}_{-}\mathrm{)}}{\mathrm{(}{\mathit{I}}_{+}+{\mathit{I}}_{-}\mathrm{)}}$ . This could also be equally performed by adding in a flipper before the analyser, which would allow simulation of effects such as wavelength dependent flippers. In practical experiments, the measurement of empty beam polarisation is essential and the sample polarisation is usually corrected for this depolarisation. However, in this idealised scenario, such corrections are considered unnecessary as the empty beam polarisation is unity.

Starting with the monochromatic case, we conducted simulations where the magnetic field strength in the prisms was systematically varied to scan the spin-echo length (δ). The resulting polarisation is illustrated in Fig. 3 (a), showing good agreement with the anticipated theory discussed in Section 2. These simulations effectively capture both the expected shape of the correlation function $\mathit{G}\mathrm{(}\mathit{\delta }\mathrm{)}$ and the extent of depolarisation, which correlates with ${\mathrm{\Sigma }}_{\mathit{t}}$  – associated with the fraction of scattered neutrons within the transmitted beam [26]. Figure 3 (b) displays the successful representation of the two neutron wavelengths used, demonstrating consistent outcomes for 3 and 5 Å neutrons.

The time-of-flight simulation reproduces the green curve depicted in 3 (a), which is a fixed magnetic field with neutron wavelengths from 2–5 Å neutrons. When compared against the analytical function presented in 3 (b) (bottom green), a notable alignment is evident, signifying robust agreement. These straightforward simulations validate the efficacy of employing both the SANS $\text{\_}$ sphere2 scattering component and the Wollaston prism component. These results confirming our capability to accurately simulate instruments in both TOF and monochromatic modes.

Fig. 3.

In (a), the simulated polarisations for the two monochromatic neutron wavelengths, as indicated in the legend, are depicted alongside the results from the time-of-flight simulation. Meanwhile, (b) shows the normalised functions, accounting for wavelength and sample thickness effects. These corrected functions are compared with the calculated forms derived from equation (3). Note that the error bars are smaller than the points.

Fig. 4.

In (a), both the calculated and simulated normalised SESANS polarisations as a function of spin-echo length are depicted. The simulations are conducted with and without an incoherent background, as indicated in the legend, and also account for time-of-flight variations. Meanwhile, (b) demonstrates the corresponding alteration in the ‘apparent’ correlation length when the sample is positioned between magnets three and four. The inset visualises the modified spin echo setup with the different sample position and shows how ${\mathit{L}}^{\prime }$ is defined with respect to the sample and prisms. Note that the error bars are smaller than the points.

In most Small Angle Neutron Scattering (SANS) experiments, incoherent scattering often presents challenges by generating a high background, particularly at high Q values. However, SESANS is less significantly affected by such scattering. This is evident in the simulations presented in Fig. 4 (a), showcasing both monochromatic and time-of-flight simulations with and without incoherent scattering. Previous tests in a standard SANS configuration confirmed the presence of incoherent background scattering, showing reasonable agreement. However, the simulations consistently predict a slightly lower scattering intensity than the calculated values, a trend more pronounced in the time-of-flight case.

In the current Delft SESANS setups, detection of scattering from smaller structures (less than 100 nm), is difficult, due to the low magnetic fields required. Consequently, if the sample is positioned in a secondary location between the third and fourth encoding devices the accessible spin-echo length is reduced and given by; $$\begin{array}{}\text{(6)}& \mathit{\delta }=\mathit{c}\mathit{B}{\mathit{L}}^{\prime }{\mathit{\lambda }}^2\mathit{cot}\mathit{\theta }\end{array}$$

Where ${\mathit{L}}^{\prime }$ is the distance from the sample to the centre of the hypotenuse of prism four, as indicated in the inset of Fig. 4 (b). Typically this halves the spin echo length scale range. It is quite possible that this arrangement enhances the solid angle coverage, thereby extending the accessible Q range, albeit changing the spin echo length range, however this is a subject of further study [18] outside the scope of this article.

For a sample placed between prisms three and four, a spin echo length ${\mathit{\delta }}^{\prime }$ is defined for a given ${\mathit{L}}^{\prime }$ (as shown in the inset of Fig. 4 (b)). Consequently, as the angle ϑ remains constant, $\frac{{\mathit{\delta }}^{\prime }}{\mathit{\delta }}=\frac{{\mathit{L}}^{\prime }}{\mathit{L}}$ . Therefore, we anticipate the probed spin echo length range to be altered from the traditional sample position by a factor of $\frac{{\mathit{L}}^{\prime }}{\mathit{L}}$ . SESANS was implemented in this configuration, and the resulting correlation function was fitted. It’s important to note that the spin-echo length $\mathrm{(}\mathit{\delta }\mathrm{)}$ is calculated using equation (6), and the ratio of the determined radius $\mathrm{(}{\mathit{R}}^{\prime }\mathrm{)}$ over the model radius $\mathrm{(}\mathit{R}\mathrm{)}$ is plotted in Fig. 4 (b) for various sample positions. This representation demonstrates good agreement between the calculated and simulated values.

5. Discussion and future directions

The comprehensive simulations conducted in this study elucidate various aspects of SESANS setups, affirming the efficacy of McStas in modeling this technique. This research complements prior demonstrations of monochromatic SESANS and SEMSANS [5]. Moreover, recent advancements have highlighted McStas’ capacity to model intricate setups involving a sequence of MWPs [20].

While certain sample characteristics such as absorption and multiple scattering are presently absent, their inclusion would be beneficial. Recent studies investigating multiple scattering effects suggest the feasibility of incorporating these phenomena into the model [13].

It’s important to note that our approach is purely classical, accounting only for the net Larmor phase accumulated by the neutron as it traverses through the instrument. Although quantum descriptions of polarized neutron state interactions are possible [12,16,30], these aspects are not replicated in this Monte Carlo model.

Another significant avenue involves integrating SEMSANS with SANS [29], which offers a comprehensive understanding of hierarchical structures. A straightforward strategy could entail stacking two samples of spheres with different sizes, facilitating simultaneous simulation of scattering at two distinct length scales. This presents an opportunity for more detailed optimization of instrument geometry, focusing on angular acceptance, and potentially yielding a more refined treatment of data compared to our current phenomenological method of transmission corrections [18].

6. Conclusions

This study successfully validates SESANS simulations across various neutron wavelengths and, notably, for the first time in a TOF setup. These simulations provide a framework for extracting SESANS results, allowing exploration of changes in sample positioning and incorporation of incoherent scattering effects. The anticipated outcome is the utilization of these tools to enhance the precision of the techniques and foster the development of novel and advanced instrumental setups.