# Analysis of SESANS data by numerical Hankel transform implementation in SasView

#### Abstract

SESANS data analysis has been implemented in the SasView software package, allowing SESANS experiments to be analyzed using a numerical Hankel transformation of isotropic small-angle scattering (SAS) models. The error of the numerical approximation is three orders of magnitude below typical experimental errors. All advanced data fitting features of SasView (multi-model fitting, batch fitting, and simultaneous/constrained fitting) are now also available for SESANS and this is demonstrated by examples of fitting SAS models to SESANS measurements.

## 1.Introduction

Spin-Echo Small-Angle Neutron Scattering (SESANS) is a small-angle scattering technique for analyzing properties of condensed matter. It was first experimentally demonstrated by Keller et al. [13] and subsequently developed and made useful for analysis of real samples at the Delft University of Technology [6], where it has since been used to characterize many microscopic soft matter structures, such as colloidal suspensions [29], granular materials [1], food structures [31,33], and kinetics of structural changes [34]. The pioneering Delft SESANS instrument extends SANS to length scales reaching 20 *μ*m and has inspired several other instrument development projects, such as a set-up using magnetic Wollaston prisms [19] and two time-of-flight instruments: Offspec [7,20,26] and Larmor [27], collaboratively developed by the Rutherford Appleton Laboratory and Delft University of Technology. Despite the successful design and implementation of these SESANS instruments, user-friendly data analysis tools have remained unavailable to a large extent. Here, we introduce the underlying theory of the numerical Hankel transform, and explain how it has been implemented into the SasView software package. We finally present several examples of successful data analysis, which highlight the capabilities of SESANS and the efficacy of the developed data analysis routines.

SESANS is a variant of the well-established Small-angle neutron scattering technique (SANS), which is itself a form of Small-angle scattering (SAS). The latter has been used over many decades to investigate properties of solids and liquids [10,11] and is a popular technique for material characterization. In a SANS experiment, a sample is placed in a beam of well-collimated neutrons and the small-angle scattering is detected by a position-sensitive detector. SESANS, on the other hand, uses the Larmor precession of polarized neutrons in carefully designed magnetic fields to label the scattering angle. The spin-echo principle can then be used to quantify this labelling to far smaller angles than a SANS instrument of comparable footprint and divergence could detect. Consequently, SESANS uses the available neutron flux far more efficiently than SANS at longer length scales. SESANS however, does not discriminate the scattered neutrons from the direct beam and thus works better in the presence of strong scattering, where multiple scattering corrections become important. Nonetheless, in contrast to SANS, these multiple scattering corrections can be easily accounted for in the SESANS data analysis [24]. Furthermore, the data interpretation is more accessible to non-experts in scattering techniques [4,5,12,22,24] for SESANS than it is for SANS. For the construction of an instrument with the combination of SANS and SEMSANS (Spin-Echo Modulated SANS) the reader is referred to the paper by Kusmin *et al.* [18].

SasView is an open-source software package for analyzing SAS experimental data written largely in the high-level programming language Python. It has a large, active community of developers from neutron and X-ray facilities and contains an extensive library of SAS model functions and fitting algorithms. As we will explain in the theory section below, for the analysis of the SESANS data the capabilities of SasView have been expanded to include the numerical Hankel transform, which is treated as a SAS resolution function.

## 2.Theory

In the following, we consider only small-angle scattering. In this case, the wave vector momentum transfer’s modulus *Q* is fully determined by the neutron wavelength *λ* and the scattering angle *θ*:

The measured quantity of a SESANS experiment is the polarization of the neutron beam *P* as a function of spin-echo length *δ* [1,13,17,22]. Spin-echo length is an experimental parameter that is determined by the neutron wavelength and the total magnetic field integral experienced by the neutrons.

*t*is the sample thickness and

*δ*is the 2D projection of a radial distance

*r*in 3D space. Alternatively, this can be re-written as the

*absolute scattering correlation function*[16,24,35]:

*x*can be one or several SAS model object parameters (e.g., the radius of a spherical colloid).

##### Fig. 1.

For our numerical approach, integrating Eq. (4) from zero to infinity is not possible. Therefore we must re-write this equation with lower and upper finite integration boundaries

*d*is the minimum characteristic length scale of the SAS model. Since every SAS model has a different

*d*, this criterion is not practical. However, in practice,

*Z*(taken directly from the experimental data) and

*Q*is a 1D array of size

*N*,

*Q*are determined by

*N*is determined from the cut-off value

*Q*and δ with the 0th order Bessel function of the 1st kind

*N*scales with

*Hadamard*product,

*transposed Q*)

*Z*times to match the dimensions of

*Hadamard*product (or

*entry-wise*product) is defined for each element as:

*Z*columns.

*N*, because it is derived from

*Q*, with each element

*Z*times, as with

*Q*dimension to produce

*Q*must be pre-calculated for efficiency. However, this single expression was not implemented because it would have been difficult to debug for future developers of the SESANS package within SasView.

## 3.Validation and examples

### 3.1.Validating the numerical approach with an analytical sphere model

The accuracy of the numerically Hankel transformed SAS models was tested on several cases. Here we present a calculation of the point-wise difference between the numerically Hankel transformed SAS model and the analytical (exact) SESANS model of a uniform sphere. The numerical Hankel transform is applied to the following SAS model:

*φ*is the volume fraction,

*R*is the radius of the solid sphere. The analytical solution of the Hankel transform (Eq. (2)) of a solid sphere (Eq. (24)), is [17]:

##### Fig. 2.

The difference between the analytical equation and SasView’s numerical Hankel transform in Fig. 2 is a factor 1000 smaller than the magnitude of the absolute correlation values of the models themselves and a factor 100 smaller than typical experimental errors (see Figs 3 and 5 for examples). The oscillations in Fig. 2 at small *δ* are caused by too small *δ* are caused by too high *Q*-range. Although the

### 3.2.Examples of fitting a numerical model to a data-set

In the following we validate our approach on experimental data, which have been acquired at the SESANS instrument in Delft [23]. These measurements used neutrons of *δ*-range of 10 nm to 20 *μ*m. Fits were performed on several sets of experimental data. The solid sphere model (Eq. (24)) was applied to SESANS data of _{2}O and 41% D_{2}O (sample provided by M. Strobl [29]) (Fig. 3). The fit correctly assigns a value of approximately *R*, was applied to SESANS data of a dispersion of NILAC fat-free milk powder in D_{2}O (powder sample provided by NIZO Research, Netherlands) (Fig. 4). The powder consists of highly polydisperse casein micelles [31,33].

*x*the polydisperse quantity,

*R*(431 Å) is much lower than the one obtained when fitting without any size distribution (1200 Å, not shown).

##### Fig. 3.

##### Fig. 4.

The polymeric gel model (Eq. (27), from [28]) was applied to SESANS data of a gel (yogurt) of NILAC fat-free milk powder in D_{2}O (powder sample provided by NIZO Research, Netherlands) (Fig. 5). This sample is a dispersion of casein micelles in D_{2}O treated with glucono δ-lactone (GDL) to create a yogurt-like substance. The resulting structure has been previously described as a self-affine random medium [2,14,15]. Here we selected a SAS model which approximates the expected structure of the sample (casein micelles connected by a network of gel fibres).

*D*is the mass fractal dimension of the gel,

*ξ*is its persistence length, and

*R*of the milk (≈ 1200 Å), most likely due to aggregation of the casein micelles before becoming “frozen in” in the gel structure [3,8,9].

##### Fig. 5.

### 3.3.Example of fitting multiple models to a data-set

Two models were fitted to SESANS data of 70 nm radius spherical core-shell colloids with poly-styrene (PS) core and poly-ethylene oxide (PEO) corona in D_{2}O (data from K. v. Gruijthuijsen *et al.* [32]).The form factors used were those of solid spheres (Eq. (24)) and core-shell spheres (Eq. (28)),

##### Fig. 6.

##### Table 1

Fit model | Sphere | Core-shell |

703 ± 6 | 508 ± 24 | |

748 ± 35 | ||

2.46 ± 0.01 | 0.28 ± 0.38 | |

5.61 ± 0.22 | ||

6.4 | ||

ϕ | 0.377 ± 0.008 | 0.338 ± 0.007 |

##### Fig. 7.

### 3.4.Examples of simultaneous constrained fitting of SANS and SESANS data

A simultaneous constrained fit was performed on SANS and SESANS data of spray-dried calcium caseinate micelles in D_{2}O (data from B. Tian, [30]). The data-sets were fitted with a double power law scattering function (Fig. 7).

*A*and

*C*are the scale factors for the power laws,

*Q*value separating the two power laws. The fit parameters are listed in Table 2. The fit indicates a cross-over between characteristic length scales at

*A*, which corresponds to the

*Q*power law. These samples are known to aggregate over time, and since SANS was performed after SESANS (on the same sample) the higher value of

*A*could be due to aggregation of small particles into larger clusters.

USANS and SESANS data of poly-styrene spheres in D_{2}O (data from C. Rehm *et al.* [21]) were also analyzed using simultaneous/constrained fitting. The data-sets were fitted with a solid sphere form factor (Fig. 8) and the fit parameters are listed in Table 3. It should be noted that the USANS and SESANS measurements were not done on the same sample: 2 samples (1 for USANS, 1 for SESANS) were prepared using the same recipe, but were not taken from the same batch. Also, the SESANS sample was synthesized much longer in advance of the experiment than the USANS one. Due to the size of the particles, this led to significant sedimentation of the SESANS sample. For SESANS, multiple scattering correction is included in the data analysis [2,24]. This is not the case for USANS [25] and SasView does not take multiple scattering into account for SANS. These three factors (different samples, sedimentation, and multiple scattering correction) may explain the factor of 5 difference in *φ*. The fitted value for *R* (≈ 21300 Å) agrees well with the published data.

##### Table 2

Fit model | SANS | SESANS |

A | 23.0 | 18.6 |

C | 0.0009 | 0.0007 |

9 × 10 -4 | ||

1.19 | ||

2.95 | ||

bkg ( | 0.05 | 0 |

##### Fig. 8.

##### Table 3

Fit parameter | USANS | SESANS |

R (Å) | 21295 ± 22 | |

1.3 | ||

φ | 0.293a | |

bkg | 500a | 0 |

^{a}SasView was unable to determine errors for these fit parameter values.

## 4.Conclusion

The validation of the numerical Hankel transform of a solid sphere against its projected correlation function showed excellent agreement and demonstrates that the numerical Hankel transform is, by extension, reliable for other SAS models. Fitting using single and multiple data-sets, in batch mode or with constrained fitting parameters has been proven to work without errors and examples of data analyses have been provided. SasView can now be used to analyze SESANS experimental data.

## Acknowledgements

The authors would like to thank the SasView development team for their support in enabling user-friendly SESANS data analysis. Of the members of the SasView team, special thanks go to dr. J.R. (Jeffery) Krzywon for help with understanding the SasView code structure in the early stages of the project, dr. A.J. (Andrew) Jackson for keeping a broad perspective and the brief, yet extremely fruitful discussion leading to the understanding that resolution functions are coordinate transforms, dr. P.A. (Paul) Kienzle for extensive discussions and assistance integrating the SESANS routine into the SasView code-base, and P. Rozyczko and W. Potrzebowski for help with GUI integration and ensuring code stability. This work benefited from the use of the SasView application, originally developed under NSF award DMR-0520547. The USANS data presented in this paper was measured on the BT-5 instrument at NCNR (NIST) and the SANS data was measured on the Larmor instrument at ISIS (RAL).

SasView contains code developed with funding from the European Union’s Horizon 2020 research and innovation programme under the SINE2020 project, grant agreement No 654000. The authors acknowledge Nederlandse Organisatie voor Wetenschappelijk Onderzoek Groot grant no. LARMOR 721.012.102.

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