We introduce a simple method to extract the nuclear coherent and isotope incoherent, spin incoherent, and magnetic neutron scattering cross section components from powder scattering data measured using a single neutron beam polarization direction and a position-sensitive detector with large out-of-plane coverage. The method draws inspiration from polarized small-angle neutron scattering and contrasts with conventional so-called “*xyz*” polarization analysis on wide-angle instruments, which requires measurements with three orthogonal polarization directions. The viability of the method is demonstrated on both simulated and experimental data for the classical “spin ice” system Ho_{2}Ti_{2}O_{7}, the latter from the LET direct geometry spectrometer at the ISIS facility. The cross section components can be reproduced with good fidelity by either fitting the out-of-plane angle dependence around a Debye–Scherrer cone or grouping the data by angle and performing a matrix inversion. The limitations of the method and its practical uses are discussed.

## I. INTRODUCTION

The combination of neutron scattering with polarization analysis grants the ability to separate both the scattering cross section components—nuclear coherent and isotope incoherent, nuclear spin incoherent, magnetic, nuclear–magnetic interference, and chiral magnetic—as well as the directional components of the sample magnetization.^{1} It is therefore applicable to a diverse range of problems, including nano-magnetism,^{2} liquid dynamics, and magnetic frustration, and, as such, has been implemented on nearly every type of neutron scattering instrument. The simplest form of polarization analysis—the so-called uniaxial or longitudinal polarization analysis—is sensitive only to scattering processes that either reverse the polarization along a particular direction or leave it unaltered. These processes and their corresponding cross sections are referred to as spin-flip (sf) and non-spin-flip (nsf), respectively. In the uniaxial case, it is possible to unambiguously separate the cross section components for a non-magnetic sample, as the nuclear coherent contribution is entirely nsf, while the spin incoherent is divided between the nsf (1/3) and sf (2/3) channels. This may be used to, for example, investigate collective vs single-particle dynamics in soft matter or liquid systems, as demonstrated by a recent study of diffusion and structural relaxation at intermediate length-scales in D_{2}O.^{3} For magnetic samples, on the other hand, the division of the scattering between channels depends on the relative orientation of the polarization **P**, the scattering vector **Q** = **k**_{i} − **k**_{f}, and the sample magnetization **M**. As such, measurements along several polarization directions are necessary to perform the cross section separation. These directions are conventionally chosen to form a Cartesian coordinate system with $x\u0302\Vert Q$. Due to the requirement that $x\u0302\Vert Q$, longitudinal polarization analysis is mainly employed on instruments with a single detector, such as triple axis spectrometers. The generalization of the method to a two-dimensional detector necessitates the assumption that the magnetic scattering is isotropic in space, i.e., that the sample scatters like a paramagnet, and that there is no nuclear–magnetic interference or chiral magnetic scattering. Then, applying the shorthand $\Sigma =\u22022\sigma \u2202\Omega \u2202E$ for the total double differential cross section, the magnetic scattering contribution in the nsf and sf channels may be written as

and

Using the previously defined coordinate system with **Q** now lying in the horizontal plane, six equations $\Sigma \xi nsf/sf$, where $\xi ={x\u0302,y\u0302,z\u0302}$, can be written down in terms of the cross section components and terms containing the in-plane angle *α* between $x\u0302$ and **Q**. The components of Σ are then separated by linear combinations that eliminate the *α* terms. This method, known as the *xyz* method,^{4,5} is routinely used to investigate polycrystalline magnetic samples ranging from spin glasses to frustrated magnets, as well as magnetic nanoparticles.^{6,7} It can also be applied to single crystals,^{8} where its use is less widespread due to the difficulty of the data analysis vs the single-detector case.

The recent availability of polarization analysis on instruments with pixelated area detectors requires a further generalization of the *xyz* method to account for the variation of $(P\u0302\u22c5Q\u0302)2$ with the out-of-plane scattering angle. This has been provided by Ehlers *et al.*,^{9} who show that the conventional *xyz* separation remains straightforward provided that the angle *γ* between the horizontal plane and **Q** is known. If it is not, two additional sets of measurements in a coordinate frame rotated by *π*/4 about $z\u0302$ are required. On the other hand, the practical requirement of surrounding the sample with bulky electromagnetic coils to rotate the beam polarization means that it is difficult to avoid shadowing a significant fraction of the scattered beam, and hence to take full advantage of the full detector solid angle. This is further complicated by the strict field homogeneity requirements imposed by the use of wide-angle ^{3}He spin filters to analyze the scattered beam polarization. Compromises have been devised that either use highly permeable material to “smooth” the magnetic field in the region of the analyzer^{10} or concentric coils tilted away from $z\u0302$.^{11} In the former case, the vertical opening is ±20°, but with $\u223c3\xb0$ dead angles every 90° in the plane, while the latter places a restriction of ±6° on the vertical scattering angle.

The LET instrument at the International the ISIS Neutron and Muon Source, UK, is a cold direct geometry time-of-flight spectrometer.^{12} It features one of the largest position-sensitive detector arrays in the world, covering an angle of 180° in plane and ±30° out of plane, giving a total solid angle of ∼*π* st. The instrument was recently equipped with a uniaxial polarization analysis mode:^{13–15} the polarized incident beam is generated by a double-“V” supermirror cavity, flipped by a pulsed Mezei-type precession-coil flipper, and analyzed by a wide angle ^{3}He spin filter. In order to make full use of the LET detector, the scattered polarization is analyzed in only the vertical direction, which eliminates the horizontal field coils that would otherwise create dead angles or restrict the vertical opening of the analyzer system. Despite the loss of information this entails, it is still possible to separate the nuclear coherent and isotope incoherent and the spin incoherent scattering by a simple linear combination of the nsf and sf cross sections. This has already been exploited for several experiments on soft matter and liquids, as mentioned earlier. On the other hand, cross section separation for magnetic powders has yet to be attempted, mainly because it is no longer possible in the case where nuclear coherent, spin-incoherent, and magnetic scattering coexist.

In this paper, we will describe how the large vertical detector coverage on instruments like LET can be used to perform cross section separation in polycrystalline magnetic samples without chiral or nuclear–magnetic interference scattering. This is achieved by exploiting the smooth variation of $(P\u0302\u22c5Q\u0302)2$ around a Debye–Scherrer cone in a similar way to the analysis of polarized small angle scattering data from magnetic systems. The method, which we call *z*^{+}, is demonstrated first on simulated elastic diffuse scattering data from a so-called spin ice—a type of frustrated magnet where the magnetic anisotropy and interactions conspire to suppress the selection of a unique ground state. The cross section components are recovered with good fidelity through both fitting and direct matrix inversion of data grouped by vertical detector angle. Then, the method is demonstrated on real data on the spin-ice realization Ho_{2}Ti_{2}O_{7} from the LET spectrometer, where a good separation of the cross section components is once again obtained. Finally, the strengths, weaknesses, and foreseen applications of the method are discussed, particularly in the context of energy-resolved data from time-of-flight spectrometers, and relative to the established *xyz* method.

## II. METHOD

As a starting point, we will consider polarized small angle neutron scattering (PSANS) from a superparamagnet, a class of material that contains well-separated nanosized clusters of ferromagnetic spins. The scattering geometry for a typical SANS instrument with a position-sensitive detector is shown in Fig. 1(a). Since **Q** is approximately perpendicular to the incident wave-vector **k**_{i} and lies in the plane of the polarization, the magnetic parts of the nsf and sf cross sections at a given |*Q*| can be straightforwardly written down in terms of the azimuthal detector angle *ϕ* between **Q** and the polarization direction,^{2}

The modulation of the magnetic scattering amplitude with *ϕ* means that it is possible to separate the magnetic scattering from the coherent and spin incoherent components, which are both uniform in *ϕ*, and also to extract additional directional information about the sample magnetization in the case where the superparamagnet is placed in a large external magnetic field. When the magnetization of the clusters is random, however, $\Sigma mag,x\u0302=\Sigma mag,y\u0302=\Sigma mag,z\u0302=\Sigma mag$, and Eqs. (3) and (4) reduce to

It is easy to see that this corresponds to Eqs. (1) and (2) when **P** lies in the plane of **Q**. Including the remaining scattering contributions except for the chiral and nuclear–magnetic interference terms, it is possible to extract the cross section components by (simultaneously) fitting the nsf and sf cross section to the above expressions. This is routinely done for SANS from both superparamagnets and other magnetic systems.^{6,7} A similar method has also recently been implemented to extract local susceptibility tensors from diffraction data taken on preferentially oriented powders in a large applied magnetic field.^{16}

We now turn to the case of a conventional (atomic) paramagnet, where the scattering is distributed over a much larger scattering angle. In this case, the situation is complicated by the fact that the direction of **Q** varies with both horizontal and vertical scattering angles and, for instruments that resolve the neutron energy, energy transfer Δ*E* = *E*_{i} − *E*_{f}. In addition, the horizontal and vertical angular coverage on real wide-angle instruments is often asymmetric, meaning that the full Debye–Scherrer cone can only be measured at a small scattering angle. The coordinate system we use for the cylindrical detector geometry of LET is shown in Fig. 1(c); in this case, **P** is perpendicular to the base of the cylinder. We, thus, substitute $(P\u0302\u22c5Q\u0302)2$ in Eqs. (1) and (2) for the detector geometry-dependent function *Z*(*ϕ*; 2*θ*, 2*θ*_{p}, *β*, Δ*E*), the form of which is given for both spherical and cylindrical geometries in Appendix A. Considering only the nuclear coherent and isotope incoherent, the spin incoherent, and the magnetic scattering components, the full nsf and sf cross sections read

The most obvious way to extract the cross section components is (again) by simultaneously fitting the experimental nsf and sf cross sections to the above expressions. This can be challenging for data from wide-angle instruments, both because the range in *ϕ* is typically limited and because the data statistics—owing to the smaller cross sections in absolute terms—tend to be poorer than in SANS. Rather than conventional curve fitting, an alternative approach is to reduce the data to a set of three linear equations in the three cross section components and solve them. This can be done, for example, by grouping the spin-flip data by sectors in *ϕ* and arranging the resulting linear equations as a matrix equation that can be inverted to solve for the components

Here, the angle brackets represent the mean over the sector centered on the angle in the subscript. The numerical error of the extracted component values is dependent on how near the matrix is to singular, which can be quantified by its condition number, the ratio *C* = *s*_{max}/*s*_{min} of its largest to its smallest singular values. The matrix is best conditioned, corresponding to the lowest *C*, when the sectors are full quadrants. On the other hand, the limited range in *ϕ* means that this is not practicable for large |*Q*|, where only a small part of the Debye–Scherrer cone is measured. The angular width of a sector is therefore set to be half of the available *ϕ* range at a particular value of |*Q*|.

Since both approaches provide a means to separate the cross section components from measurements taken with a single (vertical) polarization direction, we collectively dub them *z*^{+}.

## III. SIMULATED DATA

We begin by demonstrating *z*^{+} on simulated data for the classical spin ice material Ho_{2}Ti_{2}O_{7}. Ho_{2}Ti_{2}O_{7} was chosen as it has been used in several previous studies that focus on the development of polarized neutron data analysis.^{9,16,17} The elastic magnetic cross section was generated by forward Monte Carlo simulations and combined with a nuclear coherent and small spin-incoherent contribution to produce a full data-set. The scattering from the virtual sample was performed using Monte Carlo ray-tracing in the RAMP software.^{18} The polarizer and analyzer were both assumed to be perfect, and the detector and coil geometry of the LET spectrometer were used, i.e., $P\Vert z\u0302$. The nsf and sf elastic scattering patterns were obtained from a simulation of 10^{10} neutron rays scattering into a cylindrical detector spanning in-plane and out-of-plane angles of −40° < 2*θ* < 140° and −30° < *β* < 30°, respectively. The results are depicted in Fig. 1. The data were integrated over Debye–Scherrer cones of a constant angular thickness of 1° for further analysis.

The cross section separation using both the fitting and matrix inversion methods described earlier is shown in Fig. 2. The error bars were estimated from the Hessian of the parameter matrix in the former case and directly from the experimental error bars in the latter. The fidelity of the separated cross section components to the original data (black lines) is excellent, and within 10% of at most |*Q*|. The method predictably performs less well as 2*θ* approaches 90°, where fewer of the Debye–Scherrer cones are intercepted by the detector. In addition, small deviations are seen at small angles $<10\xb0$ due to angle averaging and the data binning algorithm used. The error bars associated with the matrix inversion are, on average, slightly higher than those from fitting, aside from near nuclear Bragg positions, where the latter shows large anomalies.

In order to compare the quality of the separation achieved between *z*^{+} and the conventional *xyz* method, we simulated data for two additional directions of **P** in the horizontal plane. The same number of neutrons were traced as for $P\Vert z\u0302$, meaning that the full *xyz* dataset would effectively take three times as long to collect in a real experiment. The separation was performed by averaging the magnetic cross sections obtained independently from the spin-flip and non-spin-flip cross sections,^{4,9}

At a small scattering angle (|*Q*|), the fidelity of the *xyz* cross section components to the simulation input is very similar to that of both the fitting and inversion approaches for the $P\Vert z\u0302$ data. On the other hand, performing the linear combinations in Eq. (10) results in larger total error bars for the magnetic cross section, despite the much longer effective measurement time. Both the fidelity and error bars for the *xyz* method are as expected, considerably better at scattering angles near 2*θ* = 90°.

## IV. EXPERIMENTAL VERIFICATION

A powder sample of Ho_{2}Ti_{2}O_{7} was synthesized by the literature method.^{19} 6 g of powder were loaded into an aluminum can, which was cooled to 1.8 K in a standard “orange” cryostat. The neutron scattering experiments were carried out on the LET spectrometer at the ISIS facility, UK, using an incident energy of *E*_{i} = 3.84 meV. The polarization analysis and corrections were carried out as described in Ref. 15.

The scattering of Ho_{2}Ti_{2}O_{7} at low temperature is mostly elastic, and the detector signal integrated over the elastic line (−0.2 < Δ*E* < 0.2 meV) is shown in Fig. 3(a). Despite the large angular gap due to the beam-stop and direct beam monitor, the expected intensity variation in the detector with respect to *ϕ* in the detector is clearly visible. Applying the fitting procedure to the data grouped in Debye–Scherrer cones of angular thickness 3° produces the cross sections shown in Fig. 3(b). Here, it was necessary to constrain the variation of the incoherent scattering to be at most 2% from point to point beyond |*Q*| = 0.7 Å^{−1} to avoid the fitting algorithm finding local minima. This assumption is further justified by the fact that the spin-incoherent scattering is a smooth and flat function of |*Q*|, and hence that constraining the incoherent limits the fits to a physical subset of solutions. At |*Q*| < 0.7 Å^{−1}, some additional spin-flip background of unknown origin is transmitted by the polarization analyzer collimation, and the fitting constraint was removed. This illustrates that *z*^{+} is more susceptible to systematic errors than the *xyz* method since it relies on fitting rather than linear combinations to extract the components. This weakness notwithstanding, the quality of the cross section separation is excellent, even at a high scattering angle, and the resulting magnetic component is in very good agreement with the classical Monte Carlo simulation used to generate the simulated data discussed in Sec. III.

A separation using the matrix inversion method was also attempted but failed due to strong cross correlation between the magnetic and incoherent components, exacerbated by systematic errors like imperfect polarization corrections and sample absorption. This does not discount its use in the future when the latter effects are more accurately accounted for. Particularly, since it represents a parameter-free approach to separating the cross sections, it may be useful to initialize the fitting.

## V. DISCUSSION

Having shown that cross section separation is possible for the case of elastic scattering from a strong magnetic scatterer, we now turn to the implications for a broader range of problems, and particularly for inelastic scattering. First, like *xyz* polarization analysis, *z*^{+} is highly flux limited; the data shown in Fig. 3 was collected in around 8 h. Given that the inelastic intensity is typically orders of magnitude weaker than the elastic, it is thus difficult to foresee the method being used for a full separation of *S*(|*Q*|, Δ*E*), at least with currently available neutron instrumentation. On the other hand, the ability to resolve the scattering in both energy and momentum transfer obviates this need in most cases. This is because all three scattering components usually coexist only in a relatively small area of *S*(|*Q*|, Δ*E*); coherent and incoherent phonons are localized in (|*Q*|, Δ*E*) and most intense at large |*Q*|, while magnetic scattering predominantly occurs at low |*Q*|. In regions of (|*Q*|, Δ*E*) where two cross section components are dominant, these can be determined directly by averaging *Z*(*ϕ*) over the full Debye–Scherrer cone in Eq. (9), then inverting the appropriate 2 × 2 sub-matrix. A useful strategy to automate the analysis of inelastic datasets may therefore be to first group the data in (|*Q*|, Δ*E*), then perform a full *z*^{+} separation over each of these regions, before assessing the relative contributions of the cross section components. Depending on these, the appropriate approach to extract the components, should more than one be present, can be chosen. The grouping of the data could either be carried out manually, by rebinning the data onto a grid to improve statistics, or by using machine learning to identify features in the data.

For magnetic systems, the elastic line will nearly always require the application of the full *z*^{+} analysis, except in the rare cases where the spin-incoherent cross section is entirely absent. Other approaches that could be used to improve the quality and reliability of the separation when doing full *z*^{+} analysis include constraining the spin-incoherent component to, e.g., a Debye-Waller form or simultaneous fitting of data taken at different incident neutron energies.

Given the above, as well as the inherent sensitivity of *z*^{+} to systematic error, it is foreseen that it will be a complement to the conventional *xyz* (or ten-point) methods, rather than a replacement. Nonetheless, it should benefit a large variety of systems, including magnetic battery cathode materials, where weak spin-incoherent quasi-elastic neutron scattering (QENS) often coexists with magnetic and structural excitations, as well as rare-earth systems with low-lying crystal field excitations.

## VI. CONCLUSION

We have demonstrated the viability of extracting cross section components from powder neutron scattering data collected for a single polarization direction on a position-sensitive detector with a large vertical angular coverage using both simulated and experimental data. We call the method *z*^{+} since it grants more information about the cross section components than has previously been available from this type of measurement. The method is most likely to be useful for analyzing elastic magnetic scattering, as well as a limited range of inelastic scattering problems, including diffusion in magnetic energy materials and crystal field excitations. It is finally hoped that *z*^{+} will find broader applications once the next generation of spectrometers with wide-angle position-sensitive detectors—including CSPEC at ESS^{20} and CHESS at the SNS, Second Target Station^{21}—become available.

## ACKNOWLEDGMENTS

We thank Mr. Mark Devonport, Ms. Holly McPhillips, and Ms. Emily McFarlane (ISIS) for assistance during the experiments and Dr. Ross Stewart and Dr. Robert Bewley (ISIS) for useful discussions.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX A: GEOMETRIC FACTORS

Assuming the polarization $P\u0302\Vert z\u0302$, $(P\u0302\u22c5Q\u0302)2$ is expressed in terms of the angle *ϕ* around a Debye–Scherrer cone starting from

where *β* is the angle between the horizontal plane and the scattered neutron wave-vector **k**_{f} (Fig. 1). For a spherical detector geometry where the pixels are indexed by the scattering angle 2*θ* and *β*, the *β*-dependent term becomes

Thus, for elastic scattering,

The corresponding *Z*(*ϕ*) for a cylindrical detector with the cylindrical axis along $z\u0302$ [Fig. 1(c)] and in-plane scattering angle 2*θ*_{p} is found by inserting the following *β*-dependent term into Eq. (A2):

### APPENDIX B: FITS

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