Radial spin echo small-angle neutron scattering method: concept and performance

The concept of a radially symmetric spin echo small-angle neutron scattering (SESANS) setup is introduced and analyzed.


Introduction
In neutron scattering, the energy and momentum of the neutron beam are usually measured both before and after scattering in order to determine the energy or momentum transfer at the sample. Achieving a high accuracy of these measurement parameters requires high collimation and monochromatization which, in turn, leads to a dramatic loss in beam intensity and thus limits the practically achievable resolution. This limitation can be overcome by neutron spin echo techniques. These use the Larmor precession of polarized neutrons in a magnetic field to measure directly the transfer of energy ! or momentum Q [Q = (4/) sin(/2), where is the scattering angle and is the wavelength of the incident radiation] at the sample without the need to determine these quantities independently for the incoming and scattered neutron beams. In this way, the accuracy with which ! or Q are determined becomes independent of beam characteristics, and very high resolutions can be reached without the prohibiting low-intensity penalty of other neutron scattering methods.

Basics of neutron spin echo spectroscopy
Neutron spin echo (NSE) spectroscopy was first introduced by Mezei (1972) to measure the energy transfer in inelastic scattering. In an NSE spectrometer, a polarized neutron beam traverses two equal magnetic precession regions in which it experiences Larmor precession in opposite directions. A simplified NSE setup is shown in Fig. 1(a). The /2 spin flipper at the beginning of the first precession region turns the polarization of the neutron beam P perpendicular to the magnetic field B. This induces Larmor precession with the characteristic Larmor frequency ! L = B, where = 1.832 Â 10 8 s À1 T À1 is the neutron gyromagnetic ratio. For a neutron beam propagating with a speed v over a path of length ', where a constant magnetic field of strength B is applied, the accumulated precession phase ' is given by (Mezei, 1980c) with the wavelength, m the neutron mass, h the Planck constant and c = 4.636 Â 10 14 T À1 m À2 . The precession angle ' is thus proportional to both the neutron beam wavelength and the magnetic field integral (along the neutron trajectory), which in our simplified case (homogeneous magnetic field, highly collimated, i.e. non-divergent, neutron beam) reduces to the product of B and '. A flipper at the symmetry point of the setup, which is close to the sample position, reverses the precession direction in the second field area. Consequently, if the two precession regions are identical, the precession angles before and after the sample cancel each other. This leads to the spin echo condition Á' = ' incoming À ' scattered = 0 at the second /2 flipper, which is positioned at the end of the second precession region in front of the analyser and the detector. This flipper stops the precession and turns P in the direction of the magnetic field and thus of the analyser, so that ideally the maximum intensity is recovered at the detector (Mezei, 1980c;Mezei et al., 2003).
Inelastic scattering from the sample breaks the symmetry of the setup as the energy transfer ! modifies the speed, and thus the wavelength, of the scattered neutron beam. As a result, the precession phases accumulated before and after the sample are no longer equal to each other, leading to a non-zero total precession phase Á' and a reduction in the detected intensity. In the limit, where the energy transfer at the sample is much smaller than the energy of the incoming neutron beam, Á' is directly proportional to !, where the proportionality factor is the Fourier time (Mezei, 1980c). The component of the beam polarization P transmitted by the analyser is proportional to cosðÁ'Þ. Thus, the net precession angle Á' is reflected in the normalized spin echo signal determined by integrating over all possible energy transfers, which is equal to the intermediate scattering function (Mezei, 1980c): where S(Q, !) ¼ d 2 AE/dQ d!, with AE the total scattering cross section of the sample, and SðQ; ¼ 0Þ ¼ R SðQ; !Þ d! is the static structure factor. In other words, NSE measures the real (cosine) part of the ! Fourier transform of S(Q, !) and thus the dependence of the correlation function (Gä hler et al., 1996;Mezei et al., 2003).

Linear SESANS
The same principles of Larmor precession and neutron spin echo can be applied in experiments with elastic scattering [i.e. for ! = 0, in which case S(Q, !) reduces to S(Q) = dAE(Q)/dQ] to measure the scattering angle and thus the momentum transfer. This labelling of a neutron beam trajectory by the Larmor precession phase (Pynn, 1978(Pynn, , 1980 has been applied for small-angle neutron scattering (SANS), neutron reflectivity and neutron diffraction , leading to an increase in resolution by several orders of magnitude compared with conventional neutron scattering methods.
Encoding the neutron trajectory using the Larmor precession is possible by introducing specially shaped precession regions with inclined front and end faces, as shown in Fig. 1(b). This is achieved by adding linear field gradients that span the whole beam cross section. Analogously to NSE, the neutron polarization undergoes Larmor precession in opposite directions in two precession regions before and after the sample. Furthermore, the magnetic field configuration and arrangement of the gradients is such that the spin echo condition is satisfied in the absence of scattering.
The accumulated total precession phase averaged over the neutron beam now also depends on the linear gradient, which leads to different phases for different trajectories through the instrument. The vertically inclined field boundaries lead to a linear relation between ' and the vertical transmission angle Cross-sectional views of the discussed setups showing the shape of the effective precession regions. (a) An NSE spectrometer with homogeneous magnetic field integrals shown as flat precession field area boundaries. (b) Linear SESANS with shifters generating a gradient in the magnetic field integral along the x direction, as illustrated by the inclined front and end faces of the precession field areas. This setup works with longitudinal or transverse precession fields. (c) A SESANS instrument with radial shifters, which generate a radial gradient of the magnetic field integral gradient with rotational symmetry about the optical axis of the instrument. This is illustrated by the conical front and end faces of the precession field area boundaries.
x of the neutron beam. If there is scattering at the sample, the transmission angle changes and, in the small-angle scattering approximation, the difference in transmission angles Á = incoming À scattered directly yields the momentum transfer for elastic scattering. Therefore, the net precession angle encodes the momentum transfer along the linear gradient Q x Bouwman et al., 2000), where 0 is the inclination angle of the field boundaries. The quantity x is the spin echo length, which is proportional to 2 and the magnetic field integral.
In the following we focus on isotropic elastic scattering and on the spin echo small-angle neutron scattering (SESANS) (Rekveldt, 1996) realization of this method. We designate the geometry of Fig. 1(b) as linear SESANS. As the measurements are performed in the direct beam it is convenient to consider the average spin echo polarization, In the case of scattering from a sample with total scattering cross section AE and thickness t, P SES is related to the SESANS correlation function G( x ), In the small-angle approximation, the total cross section is given by with k 0 = 2/ and the probed solid angle. This description includes the fraction of the neutron beam that does not scatter, as well as all scattering including multiple scattering . The normalized SESANS correlation function G( x ) is given by Equations (6) and (8) show that, similarly to NSE, SESANS measures a function which involves the Fourier transform of the scattering cross section dAE(Q)/d. However, the integration in this case is with respect to the momentum transfer, which is a two-dimensional vector in the detector plane. Consequently, for isotropic density distributions (r) the measured G( x ) and dAE(Q)/d are connected by a Hankel transform (Krouglov, de Schepper et al., 2003;Andersson et al., 2008;Kohlbrecher & Studer, 2017), as illustrated in Fig. 2. G( x ) is the projection of the normalized density autocorrelation function (r) along the neutron path (Andersson et al., 2008;Krouglov, de Schepper et al., 2003;Kohlbrecher & Studer, 2017), with z the coordinate parallel to the beam propagation, x the coordinate along the magnetic gradients [as shown in Fig. 1 and r = (x 2 + z 2 ) 1/2 . The described SESANS setup is not sensitive in the y direction and only measures correlations along x, the direction of the magnetic field integral gradient. In the second step in equation (9), a coordinate transformation from x, z to x, r is performed, and the resulting formula shows that G( x ) is an Abel transform of (r) (Krouglov, de Schepper et al., 2003).
The first dedicated SESANS instrument was built at TU Delft (Rekveldt, 1996) and uses transverse precession fields, i.e. perpendicular to the optical axis z, which are generated by permanent magnets. SESANS finds a broad range of applications such as in colloid science Li et al., 2011;Washington et al., 2014;van Gruijthuijsen et al., 2014van Gruijthuijsen et al., , 2018 or food science (Bouwman, 2021). Most SESANS instruments access spin echo lengths spanning three orders of magnitude from 10 nm up to 20 mm. The maximum length that can be probed is two orders of magnitude larger than that in conventional SANS. Besides the increased resolution, a significant advantage is the simple consideration of multiple scattering . An interesting feature, especially for the communication of the data to non-scattering experts, is the real-space character of the direct measurements .
A more recent variation in the technique is spin echo modulation small-angle neutron scattering (SEMSANS) (Bouwman et al., 2009(Bouwman et al., , 2011Sales et al., 2015;Strobl et al., 2015). A SEMSANS setup consists of only two precession devices with different gradients that create a spatial modulation of intensity at a detector. All spin manipulations can be performed before the sample, which gives more freedom to work with magnetic samples (Li et al., 2021) or sample environments. SEMSANS can also be combined with normal SANS to probe a much longer range of length scales (Schmitt et al., 2020).

Introducing radial SESANS
When it was first proposed to use the neutron spin echo principle for structural investigations, it was considered that the method would deliver the Fourier transform of dAE(Q)/d (Pynn, 1978(Pynn, , 1980Gä hler et al., 1996) closely following the mathematical concepts of NSE. This would imply that linear SESANS would measure the three-dimensional autocorrelation function directly (Gä hler et al., 1998). However, it turns out that only one direction, the one along the magnetic field gradient, is probed. This leads to a two-dimensional projection of the autocorrelation function and to the Hankel transform between G( x ) and dAE(Q)/d discussed above.
A method for direct measurement of the density autocorrelation function would also be of interest because it would facilitate the comparison between SANS and SESANS data sets. As a key feature, such a SESANS method should be able to measure Q, the modulus of the total momentum transfer, not just the one component along a certain direction. For this purpose one might consider introducing radially symmetric magnetic field precession regions, which would allow for measuring correlations independent of a specific direction. This concept is discussed in the following and is referred to as radial SESANS. In order to respect the radial symmetry of the setup, we consider longitudinal precession fields with radially symmetric gradients as shown schematically in Fig. 1(c). This radial symmetry is also reflected in the precession angle which, as discussed in the following section, depends on the radial distance of the neutron path from the optical axis.
Such a setup was first discussed by Zhao (2001) with the aim of developing a SESANS method where the density distribution function would be the simple Fourier transform of dAE(Q)/d, similar to NSE spectroscopy. The radial SESANS setup discussed by Zhao uses longitudinal magnetic fields created by solenoids similar to those of NSE spectrometers with in-beam gradient coils. Therefore, it can be considered as a possible add-on for existing NSE spectrometers. In NSE, Fresnel coils (Mezei, 1980a;Monkenbusch, 1990), shown schematically in Fig. 3(a), are used to correct the magnetic field integral inhomogeneities inherent to divergent beams with large cross sections. Similarly, SESANS could use shifter coils, linear or radial [see Figs. 3(b) and 3(c), respectively], not to correct for magnetic field inhomogeneities but to encode the neutron trajectory by adding a shift to the precession angle depending on a specific neutron trajectory. Schematic drawings of the resulting configurations are shown in Figs. 1(b) and 1(c) for linear and radial SESANS, respectively.
In this paper we investigate the realization conditions and the performance of a radial SESANS setup for small-angle neutron scattering. Our calculations are based on parameters which are compatible with state-of-the-art NSE spectrometers, such as the IN15 instrument at the Institute Laue-Langevin (Schleger et al., 1999). The focus is on the measured momentum transfer and the obtained correlation function.
Our findings show that in radial SESANS the labelling of the trajectories, and thus of the momentum transfer, is along the radial directions. However, it is not possible to obtain the total momentum transfer from the measured signal as radial SESANS is also sensitive to only one component of the momentum transfer vector. Thus, in this case we also obtain a projected correlation function along the radius, which is similar to linear SESANS. This implies that, regardless of the symmetry, radial or linear, SESANS probes correlation along only one direction, which is the direction of the field gradient. We discuss the implications of this result for other scattering geometries.

Radial SESANS
As discussed above, when considering the implementation of SESANS on NSE spectrometers, where the magnetic fields are generated by solenoids and their direction is along the optical axis of the instrument (i.e. longitudinal), there are two possibilities to trace the neutron trajectory through the device, either with linear shifters or with radial shifters, as shown in Figs. 3(b) and 3(c). SESANS with linear shifters, shown schematically in Fig. 1(b), satisfies the spin echo condition for the unscattered neutron beam, and the labelling of the scattering angle or momentum transfer is along the direction of the magnetic field integral gradient. Consequently, the combination of linear shifters and longitudinal precession fields reproduces the conventional linear SESANS technique and may be considered as an add-on option for NSE spectrometers. The SESANS method discussed here uses radial shifters, and the resulting magnetic field integral is therefore rotationally symmetric about the optical axis. In the following we evaluate the performance and capabilities of radial  SESANS assuming that the magnetic field configuration of the host NSE spectrometer is ideal, i.e. the magnetic field integral is the same for all possible neutron trajectories before and after the sample. This is technically achievable using correction elements, in particular Fresnel coils (Monkenbusch, 1990). Thus, the host NSE spectrometer ideally satisfies the spin echo condition ' = 0 for both the non-scattered and the elastically scattered (! = 0) neutron beams considered here. For this reason, in the following we will make an abstraction of the host instrument magnetic field configuration and will consider solely the effect of additional coils, such as linear or radial shifters, on ' for both the non-scattered and the elastically scattered neutron beams. Ideally, Fresnel coils can be considered as consisting of concentric current loops, as shown schematically in Fig. 3(a), generating a magnetic field integral that counterbalances the intrinsic r 2 dependence of the magnetic field integral produced by solenoids (Mezei, 1980a;Monkenbusch, 1990). In this way, NSE spectrometers can reach long Fourier times while keeping large beam cross sections and thus without sacrificing their data acquisition rate.
Shifters are similar to Fresnel coils but they generate a magnetic field integral proportional to a characteristic distance, e.g. along x as in Fig. 3(b) or along the radius as in Fig. 3(c). Linear shifters as in Fig. 3(b) were introduced in the early days of NSE spectroscopy to produce linear gradients in the magnetic field integral that can map the Q-dependent energy (dispersion) of collective excitations (Mezei, 1980b).
The focus here is on radial shifters, the magnetic field integral of which varies linearly with the radius. In this case the resulting precession phase for neutrons crossing a radial shifter at a radius r is given by with 0 the magnetic permeability, n W the winding density, i.e. the number of wires per unit length, R the outer radius of the radial shifter (r < R) and I the current in the coil. From now on we will ignore the term KR since it will in all configurations be compensated by the same term in another radial shifter with opposite sign. All tuning parameters are summarized in the constant K. For example, for a monochromatic neutron beam with = 10 Å and a shifter with n W = 2 mm À1 and I = 10 A, K = 11651 m À1 . In the following we assume that the neutron beam is perfectly monochromated. However, in the absence of scattering the spin echo condition is wavelength independent, and for this reason a wavelength distribution does not affect the performance of the setup but leads only to a distribution of spin echo lengths . In SEMSANS this distribution would decrease the amplitude for the higherorder modulations.
In our calculations the beam divergence is defined by the diaphragms. For an improved data acquisition rate one could envisage focusing the beam at the sample position. This would be possible by adequate design of the neutron delivery system, e.g. using a focusing mirror, as foreseen on IN15 (Schleger et al., 1999).
The performance of SESANS as given by equation (10) is independent of the main magnetic field from the host NSE instrument, which in principle could stay constant during the SESANS measurements. In this way it would be easy to disentangle the SESANS signal from any inelastic scattering. The latter would reduce the amplitude of the modulation and could be determined by NSE measurements with zero current in the shifter coils. Fig. 4 provides a schematic illustration of the linear and radial SESANS setups discussed here, which are conceptually equivalent to those of Figs. 1(b) and 1(c). The precession field areas are positioned on each spectrometer arm symmetrically around the flipper, which is in the centre of the setup and next to the pinhole at the sample position. The precession fields are created by coils and thus point towards the optical axis z. The labelling of the neutron beam trajectories is performed by two pairs of shifters, one at each arm of the setup, and the shifters of each pair are at a distance ' 1 apart from each other. Furthermore, the inner shifters, which are positioned next to the pinhole at the sample position, are at a distance ' 2 from it. Thus, the total distance between the outer shifters and the pinhole is L = ' 1 + ' 2 . The arrows indicate the direction of the electric currents in the shifters. Ideally, both the flipper and the pinhole should be positioned at the geometric centre of the setup. However, as the pinhole has finite dimensions its exact positioning will not be critical for the final performance of the setup.

Definition of neutron beam trajectories
In the following we focus on the performance of radial SESANS and we start by defining the neutron beam trajectory coordinates, expressing them in terms of the beam's crossing points with the four shifters (Sh) and the pinhole (S), as illustrated in Fig. 5. For this purpose we will use cylindrical coordinates, which are concordant with the symmetry of the setup. The coordinate system is defined such that the z axis coincides with the optical axis of the instrument and the   pinhole is located at the origin. Without any loss of generality, the crossing points are given by S ¼ Here r i is the radius of the crossing point at shifter Sh i or at the pinhole, and i are the corresponding angles in the xy plane, as shown in Fig. 6. In the absence of scattering, two points are sufficient to define the trajectory through the whole setup. Thus, r 1 , r 2 and r 3 can be expressed as a function of the coordinates at the sample S and the fourth shifter Sh 4 : with S4 = S À 4 and L = ' 1 + ' 2 . In the case where r S =r 4 ( 1, i.e. when the radius of the beam at the sample position is small compared with that at the shifters, we can simplify the above expressions using the Taylor expansion:

Neutron beam trajectory and Larmor precession
In general, the total precession angle Á' is equal to the difference between the precession angle before and after the flipper. In this SESANS setup, Á' depends only on the contributions of the four shifters (' i , i = 1, . . . , 4) that encode the neutron trajectory, leading to  (a) The definition of the crossing point of a neutron with shifter Sh i in polar coordinates (r i , i ). (b) A projection of a direct beam trajectory on the shifter plane to illustrate the relationship between the radial coordinates from equations (11)-(15). The coordinate r S at the sample is restricted to the pinhole area with radius p which is indicated in grey.
Here the relative orientation of the electric currents in the shifters, which we will designate in the following by their signs (+ and À), affects the sign of the corresponding ' i . In the absence of scattering, the linear SESANS (+À, +À) configuration shown in Fig. 4(a) satisfies the spin echo condition for all beam sizes and divergences. The case of radial SESANS, however, is more restrictive as the beam dimensions at the sample position have a significant impact on the performance of the setup. This is most prominent for the (++, ++) configuration [equivalent to Figs. 1(b) and 1(c) in the report by Zhao (2001)], the net precession angle of which can be calculated using the approximations of equations (19)-(21): This result can be understood from the geometry of the precession regions. Considering the case of S4 = /2 or 3/2 in Fig. 6 will result in a symmetric situation for the radii of the trajectory in the first arm as in the second arm. In that case we expect an echo, which is illustrated mathematically by the fact that in those cases cos S4 = 0. On the other hand, when S4 = 0 or , then the asymmetry will be maximal. Only for r S = 0 will there be an echo, as sketched in Fig. 7(a). This implies that this configuration satisfies the spin echo condition only for infinitesimally small pinholes, i.e. when the neutron trajectory passes exactly though r S = 0, and is very sensitive to deviations from it. Any realistic pinhole would therefore lead to significant deviations from the spin echo condition and a loss of the spin echo modulation. This is a severe drawback and for this reason, in the following, we will discard this configuration and focus on the performance of the (+À, À+) setup. In that case, and in the presence of a pinhole with radius p at the sample position, so that 0 r S p, following equations (10) and (22) the net precession angle can be written assuming the approximations of equations (19)-(21): This result can also be understood from the geometry of the setup as sketched in Fig. 7(b) by considering only the most asymmetric case when S4 = 0 or . For trajectories where the signs of the x components of the radii of the first two shifters are both opposite to those of the last two shifters this will always result in an echo situation. The path length through either of the two precession regions is only dependent on the angle of the trajectory and not on its position. This implies that, in the absence of scattering, the spin echo condition is satisfied in this configuration for any small pinhole of finite size.
The resulting NSE polarization in the most general case, beyond the small sample pinhole approximation, can be calculated by taking all possible neutron trajectories into account and is given by where p and a are the maximum pinhole and shifter radii, respectively, = S4 , and the net precession angle Á' is given by equations (24) and (16)-(18). Fig. 8(a) depicts the effect of the pinhole radius on P NSE , as obtained by numerical integration for a = 20 mm, = 10 Å and different spin echo lengths . A radius of 20 mm is a reasonable assumption; smaller values would of course be possible but would also lead to a dramatic decrease in the data acquisition rate.
Only an infinitesimally small pinhole (p = 0) leads to P NSE = 1, thus only this case satisfies the spin echo condition perfectly. By increasing the pinhole size, an increasing number of neutron trajectories do not satisfy the spin echo condition, leading to a reduction in P NSE . This reduction is substantial, but not obstructive, as a pinhole with a radius of 5 mm (thus a/ 4) leads to P NSE ' 0.6, which, depending on the spin echo length of the measurement, would be sufficient for performing SESANS measurements.
For a more quantitative evaluation we calculated the figure of merit (FOM) of the setup, derived from the relative error of P NSE : where the error ÁP NSE is determined from the derivative of P NSE and I n is the total intensity of the detected neutron beam, proportional to the area of the pinhole, I n / p 2 . As shown in Fig. 8(b), if the FOM is plotted against the pinhole size, it goes through maxima which mark the optimal operating conditions with the best compromise between resolution and neutron beam intensity.  Schematic drawings of the precession path through the two different configurations for two different neutron trajectories. The r S direction is taken along the y direction (perpendicular to the paper). We consider only the trajectory for S4 = 0 or , which means that all r i are in the x direction. (a) In the (++, ++) configuration the path lengths through the first and second precession regions are only equal for trajectories going exactly through the centre of the setup, as illustrated for the lower trajectory. In the higher trajectory the path length through the first precession region becomes shorter, while it becomes longer for the second region, as reflected in equation (23). (b) In the (+À, À+) configuration the exact value of r S has no effect on the echo. For a certain angle of the trajectory it does not matter if the neutron path is higher or lower, as long as the x components of the radii in the shifters change sign from the first precession region with respect to the second precession region, as reflected in equation (24).

Effect of scattering: relation between precession and momentum transfer
In the case of scattering, the trajectories before and after the sample are no longer correlated with each other. Here we start by assuming an infinitesimally small pinhole at the sample position such that r S = 0. In this case there is a clear relation between the radii of the points where the neutron beam impinges on the shifters: where in the absence of scattering r 1 = r 4 and r 2 = r 3 . Using equation (27), the net precession angle defined in equation (24) becomes Fig. 5 shows the trajectory of a possible scattering process and defines the crossing points at the shifters. The direct neutron beam travels from A 0 at the first shifter to point A at the fourth shifter, with r A 0 ¼ r A because of the pinhole. Scattered neutrons are deflected to point B.
The momentum transfer is defined as Q = k i À k f , where k i and k f are the wavevectors of the incident and scattered neutrons, respectively. Only elastic scattering is considered here and therefore |k i | = |k f | = k 0 = 2/. The vector k i points along the trajectory of the incident or direct beam, along the vector SA from the crossing point with the sample S to the point A (equal to the vector between A 0 at the first shifter and S), while the direction of k f is given by the vector SB from S to B: Using the coordinate notation from above for the crossing points A and B, and the condition for a pinhole (p = 0), the wavevectors can be written as Consequently, the total momentum transfer vector becomes where the square roots are approximated assuming r A ; r B ( L, such that (r 2 + L 2 ) 1/2 ' L, and the modulus of the momentum transfer is given by Both the equation for Q and Fig. 5 show that the momentum transfer takes place in the xy plane. This is the detector plane, and looking at the scattering process in this plane involves the projected wavevectors derived from equations (30) and (31): The moduli of these vectors depend on the radial coordinates r A and r B : The total momentum transfer Q is now decomposed into a radial Q r and a complementary Q C component, as shown in Fig. 9. Q r points in a radial direction parallel to k xy i and its modulus is determined by the lengths of the wavevector projections given in equation (36). The difference between k xy i and k xy f labels the momentum transfer in the radial direction:   (25) and (26), respectively.
In contrast, the complementary component cannot be expressed clearly in terms of the wavevectors without knowing the polar coordinates A and B . By combining the derived expressions for the precession angle Á' [equation (28)] and Q r [equation (37)] while considering that r A = r 1 and r B = r 4 , we find that the net precession angle is proportional only to the radial momentum transfer. This is an important result, indicating that only one component of the momentum transfer vector is included in the measured signal. Since there is no unique relationship between Q r and Q C , it is not possible to calculate the magnitude or direction of Q from the vector equation Q r + Q C = Q without knowledge of Q C . In other words, radial SESANS probes only the momentum transfer along the radial direction without the possibility of obtaining the total momentum transfer, This equation has been derived in the limit of an infinitely small sample pinhole. A satisfactorily high P NSE can be obtained for a sample pinhole with a diameter p = a/4. Therefore, equation (38) will be less strict for scattering events, especially those where the incoming and outgoing vectors are close to the optical axis. However, these events occur less frequently than those where they are closer to the edge of the diaphragms, since there is more area. This small deviation from equation (38) gives rise to a spread in precession for a certain Q r , leading to a resolution effect for the SESANS measurement as a function of spin echo length. As in linear SESANS, the proportionality factor in equation (38) is the spin echo length, which can be varied by changing the current I in the shifter coils while keeping all other parameters, including the precession magnetic field, unchanged.
The NSE polarization and FOM in Fig. 8 were calculated for different values that can be achieved with the described setup. For I varying between 2 and 10 A in shifter coils with a winding density of n W = 2 mm À1 , ' 1 = 1 m and a neutron wavelength of = 10 Å , ranges from 0.37 to 1.85 mm.
In general, the defined momentum transfer components Q r and Q C are not perpendicular to each other. For scattering with a small total momentum transfer, however, this can be assumed since Q r is small and a linear approximation, as shown in Fig. 10, is justified. If so, Q ¼ ðQ 2 r þ Q 2 C Þ 1=2 holds for the measured component Q r and the complementary component Q C . This situation resembles the momentum encoding in linear SESANS, where the total momentum transfer is always decomposed into perpendicular components. In other words, radial SESANS is similar to the conventional linear method, as it does not allow for accessing more than just a component of the momentum transfer vector.
A consequence of the fact that Á' depends only on Q r is that radial SESANS probes only correlations along the radial direction. Similar to linear SESANS, this leads to a projected correlation function, defined as the Hankel transform of the sample's scattering function. For radially symmetric functions the Hankel transform can be derived from a two-dimensional Fourier transform (Bouwman et al., 2000;Andersson et al., 2008;Uca et al., 2003), The integral is over all Q values reaching the detector. In the given case, the integral translates to an integral over the radii of the first and last shifter and over the difference in polar angles (cf. Fig. 9). The normalized correlation function becomes where a is the shifter radius determining the maximum detectable Q value and thus the minimum size measurable.  The scattering event shown in Fig. 5 as seen from the sample, projected onto the xy plane. The total momentum transfer Q is decomposed into a radial component Q r and a complementary component Q C . Only the magnitude of Q r is included in the measured precession angle. The angle denotes the difference between the polar angle at the first and last shifter, = 1 À 4 .

Scattering from spherical particles
In the following we evaluate the performance of the radial SESANS setup and compare it with linear SESANS. For this purpose we consider the simple case of isolated spherical particles of radius R, whose scattering function is given by (Guinier & Fournet, 1955) The results for different particle radii are drawn in Fig. 11 for both radial and linear SESANS. For the latter, G() has been calculated analytically (Krouglov, de Schepper et al., 2003) and thus corresponds to an ideal setup where instrument parameters, such as angular acceptance, are not taken into account. Comparison of the two data sets shows that G() of radial SESANS also goes to zero at the values corresponding to the maximal distance of the scattering particle, which in the case of spheres corresponds to their diameter. On the other hand, for small R, and therefore large values of Q, the radial SESANS results deviate from those of the analytical calculation. This is due to the angular acceptance of the setup and thus the shifter size, which determines the maximal detectable Q value in radial SESANS. The calculation of equation (41) was performed with a shifter radius of a = 20 mm, which gives a minimum measurable particle radius of less than R = 50 nm for the given setup (' 1 = ' 2 = 1 m, = 10 Å ). Below this radius the SESANS correlation function deviates from the analytical result, and this holds not only for radial but also for linear SESANS (Rekveldt, 1996).

Radial SEMSANS
The rotationally symmetric magnetic field geometry, consisting of radial shifters and longitudinal precession fields, can also be implemented for SEMSANS. In this technique, all spin manipulations take place in two precession regions before the sample (see Fig. 12).
Existing SEMSANS instruments are built with transverse precession fields that have inclined front and end faces, as for linear SESANS. In this case, the total Larmor precession angle accumulated by the neutrons depends linearly on the height of their trajectory at the detector but is independent of their divergence angle. When the focus condition B 1 ' 1 = B 2 ' 2 is satisfied (Bouwman et al., 2009(Bouwman et al., , 2011, where B 1 and B 2 are the precession magnetic fields and ' 1 and ' 2 are the distances of the respective precession areas from the detector, this leads to a spatial modulation of the neutron beam intensity at the detector. Any scattering by a sample decreases the amplitude of that modulation, also called visibility (Bouwman et al., 2011;Li et al., 2016).
The SEMSANS normalized visibility is analogous to the normalized NSE polarization in SESANS, and therefore both methods probe the same projected correlation function (Bouwman et al., 2011;Strobl et al., 2015). The labelled direction in the precession angle ', and thus the orientation of the modulation, are determined by the direction of the magnetic fields. While transverse fields create a modulation along the detector height (transverse direction), the modulation in radial SEMSANS would be along the radial coordinate, starting at the beam centre. Fig. 12 shows the setup of a radial SEMSANS instrument. Two shifters encode the radial coordinate of the neutron trajectory in '. The magnetic fields and positions are set to B 2 = 2B 1 and ' 1 = 2' 2 to satisfy the focus condition. As before, a neutron trajectory can be defined by its crossing points at the shifters Sh i , the sample S and the detector D using cylindrical coordinates:

Precession angle and momentum transfer
S ¼ Comparisons of the correlation function G of radial and linear SESANS, calculated for dilute spherical particles of different radii R. The solid lines represent G for radial SESANS as a function of the spin echo length , while the dashed lines are the corresponding G() for linear SESANS.
The minimum measurable length scale in radial SESANS is determined by the maximum Q value detected, which depends on the scattering angles and the size of the detector.

Figure 12
A schematic drawing of the radial SEMSANS instrument. The second shifter generates a magnetic gradient twice as strong as the first one, resulting in a modulation of the beam intensity at the detector in a radial direction.
Using equation (10), the total precession angle Á', defined as the difference between the precession accumulated before and after the flipper, becomes in which K is defined as in equation (10) for the first shifter. In SEMSANS it is crucial to have a direct proportionality between Á' and the radial position of a neutron at the detector r D . In the absence of scattering, the radial coordinate of the crossing point at the second shifter r 2 can be first expressed as a function of r 1 and r D . With ' 1 = 2' 2 and 1D = 1 À D this gives As a second step we introduce a small pinhole in front of the first shifter, so that, r 1 =r D ( 1 and we can use the Taylor expansion, This simplifies the expression for the total precession angle, which in the absence of scattering becomes This indeed provides a linear relation between Á' and r D , which shows that this setup creates a radial modulation. However, this result is not valid close to the detector centre (r D ! 0) where our approximation using the Taylor expansion breaks down.
In the presence of scattering, we can express Á' as a function of the radial momentum transfer Q r and a spin echo length in a way very similar to radial SESANS. For this derivation, we introduce an infinitesimally small pinhole in front of the first shifter so that r 1 ! 0. The radius r 2 can then be derived from the geometry of the setup: The precession angle from equation (47) becomes Considering the wavevectors for the incident and scattered neutron beams, we derive the moduli of their projections onto the xy plane as where SD = S À D , and r S and r D are small compared with ' 1 or ' S . The radial component of the momentum transfer vector thus becomes Using equations (52) and equation (54) we obtain where the factor in front of Q r denotes a spin echo length , with The spin echo length depends on the distance between the shifters, which is as expected and is also the case for radial SESANS.
The variation in the visibility of the modulation in the beam intensity at the detector can be calculated in the most general The normalized visibility V of the intensity modulation as a function of the radial distance from the detector centre r D . V was calculated with equation (57)  case, beyond the small pinhole approximation. For a pinhole of radius R in front of the first shifter, so that 0 r 1 R, we obtain In the following, Á' was determined using equations (47) and (48) in equation (57). The integration was performed numerically for different spin echo lengths between 0.1 and 1.8 mm. The results obtained in the absence of scattering, for a pinhole with R = 0.5 mm and as a function of r D , are illustrated in Fig. 13. This shows that it is also possible to perform SEMSANS measurements with radial shifters, even though the visibility is lower than for linear SEMSANS. As in the previous section for SESANS, we consider here shifters with a winding density of n W = 2 mm À1 . The generated magnetic field gradient is therefore not continuous, as assumed in the mathematical model, but a stepped function of the windings. This approach thus breaks down when the radius of the pinhole becomes comparable to or smaller than the radius of the innermost shifter coil loop. In this case the effect of the first shifter drops out, leading to another expression for the total precession angle: Á' = À2Kr 2 . The effect on the modulation can be seen in the recalculated visibility shown in Fig. 13(a). This effect will further reduce the visibility for radial SEMSANS.

Concluding remarks
Since the early days of NSE it has been acknowledged that shifter coils combined with longitudinal precession fields, as found in NSE spectrometers, may be used to encode the neutron beam trajectories for SESANS-like applications (Pynn, 1978(Pynn, , 1980. In this article we have discussed how linear shifter coils can lead to a configuration that is conceptually equivalent to the SESANS setup, which has been developed in Delft (Rekveldt, 1996;Rekveldt et al., 2003;Andersson et al., 2008) and uses transverse (vertical) magnetic fields with inclined boundaries. This configuration is only sensitive to the projection of the momentum transfer vector along the (vertical) direction, which is defined by the inclination of the magnetic field boundaries.
Here we have considered radial, instead of linear, magnetic field integral gradients and have evaluated the performance of a SESANS setup resulting from the combination of radial shifter coils with longitudinal precession fields, in concordance with the cylindrical symmetry of the setup. Our results show that this configuration is also sensitive to the projection of the momentum transfer vector along one direction, in this case the radial one. Thus, in this case SESANS also yields a projected correlation function. We argue that this should be the case for all SESANS realizations, independent of their specific magnetic field geometry. In fact, as the magnetic field integral varies in a continuous fashion across the area of the neutron beam it is always possible to identify continuous 'equipotential' lines along which the magnetic field (integral) gradient vanishes [e.g. along x in Fig. 4(a) or along all circles centred at y = 0 and z = 0 in Fig. 4(b)]. Consequently, the SESANS signal would be insensitive to the correlation function along these lines and would yield the projected correlation function along those direction(s) along which the magnetic field (integral) gradient is maximum: y in Fig. 4(a) or the radial direction in Fig. 4(b). This is an important difference between SESANS and NSE spectroscopy and explains why the former yields a projected correlation function, whereas the latter, which probes a scalar quantity (the energy transfer at the sample), delivers the intermediate scattering function directly.
The performance of linear SESANS does not depend on neutron beam characteristics such as beam dimensions and divergence. In contrast, our results show that radial SESANS performs best for p ( a, i.e. when the radius of the neutron beam at the sample is small in comparison with that at the outer shifter coils. This requirement ensures that the spin echo condition is satisfied for a majority of trajectories but also leads to substantial intensity losses. Nonetheless, the results of Fig. 8 show that for a realistic configuration, in the absence of scattering, it is possible to increase the beam size at the sample, e.g. p/a ' 1/4, while keeping a reasonable spin echo polarization. In addition, scattering from a model sample consisting of simple spherical particles provides results consistent with what would be expected for linear SESANS.
The simple setup considered here should reach spin echo lengths ranging from about 100 nm to a few micrometres. Higher resolutions could be obtained by using stacked shifter coils, as is commonly the case with Fresnel coils in NSE spectrometers.
Longitudinal magnetic fields combined with radial shifter coils could also be used for radial SEMSANS, although with a substantially reduced performance with respect to its linear SEMSANS counterpart. Radial SEMSANS requires the smallest possible beam aperture at the point (first shifter) where the Larmor labelling begins. This of course dramatically reduces the intensity of the neutron beam and at the same time makes the first shifter redundant when the dimensions of the aperture become comparable to the distance between the wires of this shifter coil.
To conclude, linear and radial SE(M)SANS are conceptually similar to each other but the linear realization clearly outperforms the radial one. In addition, both setups can be realized by combining shifter coils (radial or linear) with longitudinal magnetic fields, as found at NSE spectrometers, and could be implemented as add-ons to existing instruments. The spin echo length that can be reached would be similar to those obtainable on present SESANS instruments (some 10-20 mm). The advantage of such a realization would lie in the combination of the long wavelengths accessible to existing NSE instruments and their high neutron flux. In this way it would be possible to study weakly scattering systems and perform kinetic measurements on the timescale of minutes instead of hours, which is presently the limit.
Compared with existing SESANS instruments, such a realization would allow for using custom-made in-beam coils, with current densities adapted to specific requirements that research papers could enable new types of experiments with unexplored capabilities.