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ISSN: 1600-5767

Neutron Larmor diffraction on powder samples

aMax Planck Institute for Solid State Research, Stuttgart, Germany, bMax Planck Society Outstation at the Forschungsneutronenquelle Heinz Maier-Leibnitz (FRM-II), Garching, Germany, and cFaculty of Physics, University of Warsaw, Poland
*Correspondence e-mail: t.keller@fkf.mpg.de

Edited by G. J. McIntyre, Australian Nuclear Science and Technology Organisation, Lucas Heights, Australia (Received 2 August 2019; accepted 29 November 2019)

A hitherto unrecognized resolution effect in neutron Larmor diffraction (LD) is reported, resulting from small-angle neutron scattering (SANS) in the sample. Small distortions of the neutron trajectories by SANS give rise to a blurring of the Bragg angles of the order of a few hundredths of a degree, leading to a degradation of the momentum resolution. This effect is negligible for single crystals but may be significant for polycrystalline or powder samples. A procedure is presented to correct the LD data for the parasitic SANS. The latter is accurately determined by the SESANS technique (spin–echo small-angle neutron scattering), which is readily available on Larmor diffractometers. The analysis technique is demonstrated on LD and SESANS data from α-Fe2O3 powder samples. The resulting d-spacing range agrees with experimental data from high-resolution synchrotron radiation powder diffraction on the same sample.

1. Introduction

Neutron Larmor diffraction (LD) is a high-resolution technique which permits the measurement of lattice spacings dhkl and their range Δdhkl. The latter arises, for example, from microstrains, magnetostriction, or structural and magnetic domains, or from a small splitting of Bragg peaks resulting from distortions of the crystal lattice. The resolution of current Larmor diffractometers is ∼10−6 (relative) of the lattice spacing. Further improvement by one order of magnitude is possible by an optimized instrument design.

The present work is related to a review of the symmetry properties and subsequent revision of the crystal structures of numerous compounds. Synchrotron radiation (SR) powder diffraction studies of the multiferroic BiFeO3 (Sosnowska et al., 2012[Sosnowska, I., Przeniosło, R., Palewicz, A., Wardecki, D. & Fitch, A. (2012). J. Phys. Soc. Jpn, 81, 0446041.]), CaCO3, calcite (Przeniosło et al., 2016[Przeniosło, R., Fabrykiewicz, P. & Sosnowska, I. (2016). Physica B, 496, 49-56.]), the transition metal oxides V2O3, Al2O3 and Cr2O3 (Fabrykiewicz et al., 2018[Fabrykiewicz, P., Przeniosło, R., Sosnowska, I. & Fauth, F. (2018). Acta Cryst. B74, 660-672.]), and hematite (α-Fe2O3) (Przeniosło et al., 2014[Przeniosło, R., Sosnowska, I., Stękiel, M., Wardecki, D., Fitch, A. & Jasiński, J. B. (2014). Physica B, 449, 72-76.]) have shown mono­cli­nic symmetry instead of the previously established trigonal one. Similar studies have shown distortions of the cubic symmetry of chromium (Przeniosło et al., 2018[Przeniosło, R., Fabrykiewicz, P., Sosnowska, I., Wardecki, D., Sławiński, W., Playford, H., Hempelmann, R. & Bukowski, M. (2018). Physica B, 530, 183-190.]) and MnO (Fabrykiewicz & Przeniosło, 2016[Fabrykiewicz, P. & Przeniosło, R. (2016). Physica B, 489, 56-62.]). The monoclinic distortion observed in α-Fe2O3 is of the order of Δd/d ≃ 2 × 10−4, i.e. close to the detection limit of SR powder diffraction methods. The LD experiments in this work were conducted in order to confirm this small lattice distortion with a complementary technique and thus to validate the SR data.

LD is based upon the Larmor precession of the neutron spins in parallelogram-shaped magnetic fields along the incident and scattered beams. The geometry of these fields, sometimes labelled precession devices (PDs), was first proposed by Rekveldt (Rekveldt, 2000[Rekveldt, M. T. (2000). Physica B, 276-278, 55-58.]; Rekveldt et al., 2001[Rekveldt, M. T., Keller, T. & Golub, R. (2001). Europhys. Lett. 54, 342-346.]). Tuning of the field boundaries of the PDs parallel to the diffracting lattice planes yields a Larmor precession phase ϕLD ∝ dhkl, where ϕLD is decoupled from the spectral width and collimation of the neutron beam. The characteristic limitation of conventional diffractometers imposed by the inverse proportionality between intensity and resolution is not effective in LD, and so excellent momentum resolution without excessive loss of intensity is achieved. In this respect, LD is similar to neutron spin–echo, where the energy resolution is decoupled from the monochromaticity of the incident neutron beam.

Experimental studies taking advantage of LD include thermal expansion under hydrostatic and uniaxial pressure (Pfleiderer et al., 2007[Pfleiderer, C., Böni, P., Keller, T., Rössler, U. K. & Rosch, A. (2007). Science, 316, 1871-1874.]; Niklowitz et al., 2010[Niklowitz, P. G., Pfleiderer, C., Keller, T., Vojta, M., Huang, Y.-K. & Mydosh, J. A. (2010). Phys. Rev. Lett. 104, 106406.]; Bourdarot et al., 2011[Bourdarot, F., Martin, N., Raymond, S., Regnault, L.-P., Aoki, D., Taufour, V. & Flouquet, J. (2011). Phys. Rev. B, 84, 184430.]), the formation of structural and antiferromagnetic domains (Náfrádi et al., 2016[Náfrádi, B., Keller, T., Hardy, F., Meingast, C., Erb, A. & Keimer, B. (2016). Phys. Rev. Lett. 116, 047001.]), and the splitting of Bragg peaks by lattice distortions (Inosov et al., 2013[Inosov, D. S., Friemel, G., Park, J. T., Walters, A. C., Texier, Y., Laplace, Y., Bobroff, J., Hinkov, V., Sun, D. L., Liu, Y., Khasanov, R., Sedlak, K., Bourges, P., Sidis, Y., Ivanov, A., Lin, C. T., Keller, T. & Keimer, B. (2013). Phys. Rev. B, 87, 224425.]; Hu et al., 2015[Hu, D., Lu, X., Zhang, W., Luo, H., Li, S., Wang, P., Chen, G., Han, F., Banjara, S. R., Sapkota, A., Kreyssig, A., Goldman, A. I., Yamani, Z., Niedermayer, C., Skoulatos, M., Georgii, R., Keller, T., Wang, P., Yu, W. & Dai, P. (2015). Phys. Rev. Lett. 114, 157002.]; Wang et al., 2018[Wang, W., Song, Y., Cao, C., Tseng, K.-F., Keller, T., Li, Y., Harriger, L. W., Tian, W. C. S., Chi, S., Yu, R., Nevidomskyy, A. H. & Dai, P. (2018). Nat. Commun. 9, 3128.]). For thermal expansion only relative changes in the lattice constants are relevant, but LD can also provide absolute values of dhkl when the instrument is calibrated with a reference monocrystal, such as silicon or germanium. In this way, dhkl values for polycrystalline Inconel alloys were accurately measured to determine residual stresses (Repper et al., 2010[Repper, J., Keller, T., Hofmann, M., Krempaszky, C., Petry, W. & Werner, E. (2010). Acta Mater. 58, 3459-3467.]), and in spin–ice materials the exact values of the lattice constants allowed quantification of the emergent magnetic monopole charge (Ruminy et al., 2016[Ruminy, M., Groitl, F., Keller, T. & Fennell, T. (2016). Phys. Rev. B, 94, 174406.]).

The Larmor diffractometers TRISP at the Heinz Maier-Leibnitz Zentrum (Keller et al., 2002[Keller, T., Habicht, K., Klann, H., Ohl, M., Schneider, H. & Keimer, B. (2002). Appl. Phys. Mater. Sci. Process. 74, s332-s335.]; Keller & Keimer, 2015[Keller, T. & Keimer, B. (2015). J. Large-Scale Res. Facil. 1, A37.]), FLEXX at the Helmholtz-Zentrum Berlin (Groitl et al., 2015[Groitl, F., Keller, T., Quintero-Castro, D. L. & Habicht, K. (2015). Rev. Sci. Instrum. 86, 025110.]) and ZETA at the Institut Laue–Langevin (Martin et al., 2012[Martin, N., Regnault, L. & Klimko, S. (2012). J. Phys. Conf. Ser. 340, 012012.]) were originally designed as three-axis spin–echo spectrometers optimized for the spectroscopy of phonons and spin excitations. These instruments take advantage of the resonant neutron spin–echo technique (NRSE) to define the parallelogram-shaped field regions by radio frequency (RF) spin-flip coils (Golub & Gähler, 1987[Golub, R. & Gähler, R. (1987). Phys. Lett. A, 123, 43-48.]; Gähler & Golub, 1988[Gähler, R. & Golub, R. (1988). J. Phys. Fr. 49, 1195-1202.]). A recent development uses Wollaston prisms, triangular-shaped DC magnetic fields bounded by superconducting sheets (Li et al., 2017[Li, F., Feng, H., Thaler, A. N., Parnell, S. R., Hamilton, W. A., Crow, L., Yang, W., Jones, A. B., Bai, H., Matsuda, M., Baxter, D. V., Keller, T., Fernandez-Baca, J. A. & Pynn, R. (2017). Sci. Rep. 7, 865.]), as PDs. The advantages of the latter technique are the compact geometry and the increased flexibility achieved by tuning the inclination angles of the magnetic fields by electrical currents, instead of mechanical rotation as in the case of NRSE.

The resolution of LD depends only weakly on the sample mosaic, and for polycrystalline samples no significant reduction in the resolution compared with single crystals is expected. This was confirmed experimentally on polycrystalline aluminium (Rekveldt et al., 2001[Rekveldt, M. T., Keller, T. & Golub, R. (2001). Europhys. Lett. 54, 342-346.]) and Inconel alloy (Repper et al., 2010[Repper, J., Keller, T., Hofmann, M., Krempaszky, C., Petry, W. & Werner, E. (2010). Acta Mater. 58, 3459-3467.]). If the size of the crystallites is in the sub-micrometre range, small-angle neutron scattering (SANS) of the incident and diffracted beams might lead to a blurring of the Bragg angles. This effect is typically small, with SANS angles of the order of a few hundredths of a degree, and there will be no SANS intensity at the large Bragg angles θB ≃ 50° used in LD experiments. Initial LD measurements performed with α-Fe2O3 polycrystalline samples in cylindrical vanadium containers have shown d-spacing ranges considerably broader than those obtained with SR diffraction. This broadening results from small-angle scattering in the sample, which is negligible in the SR experiments, as it increases with sample diameter and neutron or X-ray wavelength. Both parameters are more favourable in the case of SR, with a sample diameter (capillary) of 0.5 mm and an X-ray wavelength of 0.4 Å, compared with a sample width of several millimetres and a neutron wavelength of ∼2 Å in LD.

In this paper we discuss the resolution of LD including SANS. The effect can be corrected for, if the SANS scattering probability S(θ) is known. Calculation of S(θ) is possible, but it requires detailed knowledge of the powder grain morphology and scattering length density profiles. In addition, strong multiple scattering is expected for the large sample diameters of several millimetres typically used in neutron diffraction experiments. Accurate modelling of this multiple scattering is difficult. Thus, a measurement of S(θ) is preferable. This can be efficiently performed on the Larmor diffractometer, which is readily converted into a spin–echo small-angle scattering (SESANS) instrument by changing the field polarities and inclination angles of the PDs (Rekveldt, 1996[Rekveldt, M. T. (1996). Nucl. Instrum. Methods Phys. Res. B, 114, 366-370.]; Rekveldt et al., 2003[Rekveldt, M. T., Bouwman, W. G., Kraan, W. H., Uca, O., Grigoriev, S. V., Habicht, K. & Keller, T. (2003). Neutron Spin Echo Spectroscopy, Lecture Notes in Physics, Vol. 601, edited by F. Mezei, C. Pappas & T. Gutberlet, pp. 87-99. Heidelberg: Springer.]; Keller et al., 1995[Keller, T., Gähler, R., Kunze, H. & Golub, R. (1995). Neutron News, 6(3), 16-17.]). Both LD and SESANS data are then consistently collected on the same sample in one experimental run.

In the following we first review LD including parasitic SANS. We discuss the properties of the SESANS technique and derive a method for correcting LD data for SANS effects. Finally, we work out an example of the analysis technique using experimental data from α-Fe2O3 powder samples.

2. Larmor diffraction including SANS

In this section we review the principles of LD and then include the effect of SANS.

Rekveldt proposed the field geometry shown in Fig. 1[link], a uniform DC magnetic field B0 with boundaries oriented parallel to the diffracting lattice planes (Rekveldt, 2000[Rekveldt, M. T. (2000). Physica B, 276-278, 55-58.]; Rekveldt et al., 2001[Rekveldt, M. T., Keller, T. & Golub, R. (2001). Europhys. Lett. 54, 342-346.]). The neutron spins are initially polarized perpendicular to B0 and cross the field twice, before and after being diffracted at the lattice planes of a crystal with inter-planar spacing dhkl and corresponding reciprocal-lattice vector G = 2π/dhkl. The Larmor phase ϕLD = ωLtt accumulated by a neutron spin after passing both fields is proportional to the Larmor frequency ωL = γB0, with γ = 2π × 2.916 kHz gauss−1 and the transition time tt = L/v that the neutron spends in the field. The parameter v is the velocity component perpendicular to the field boundary, with v = [(\hbar/m) k_\perp], m the neutron mass and k = G/2 = π/dhkl:

[\phi_{\rm LD} = \omega_{\rm L} {{2L} \over {v_\perp}} = {{2m} \over {\pi\hbar}} \, \omega_{\rm L} L d_{hkl} . \eqno(1)]

ϕLD only depends on the field integral J0 = 2ωLL and the lattice spacing dhkl, and is independent of the Bragg angle. The resonant spin–echo technique offers an easy way of defining the required flat-field boundaries by radio-frequency spin–flip coils (Fig. 1[link]).

[Figure 1]
Figure 1
Sketch of a Larmor diffractometer. A polarized neutron beam crosses the uniform field B0 twice. The boundaries of B0 are oriented parallel to the lattice planes dhkl, G = 2π/dhkl. In the resonant spin–echo configuration (NRSE), instead of a uniform field B0 four RF spin flippers C1–C4 define the boundaries of the precession regions, and there is no field along the flight path between the flippers and in the sample region. C1–C4 are double coils, each consisting of two single RF spin-flip coils with opposite field direction (known as the `bootstrap' technique; Gähler & Golub, 1988[Gähler, R. & Golub, R. (1988). J. Phys. Fr. 49, 1195-1202.]). The Larmor phase shift ΔϕLD and the polarization are obtained from the count rate I versus the position ΔL of the coil C4 (ΔL in the inset is in units of millimetres): I(ΔL) = I0{1 + Pcos[2π(ΔLΔL0)/Lper]}, where the period is Lper = 2πvn/ωL, vn is the neutron velocity and ωL the Larmor frequency. With the offset ΔL0, the phase is ΔϕLD = 2πΔL0/Lper.

A shift in dhkl, for example by thermal expansion, is measured by tracking the phase shift Δϕ(T) = ϕLDhkl versus the temperature T, where

[\epsilon_{hkl} = d_{hkl}/d_{hkl}^\circ - 1 . \eqno(2)]

[d_{hkl}^\circ] is the mean or unperturbed value of dhkl. dhkl is typically spread, for example by the effect of microstrains. Then the phase is also spread as

[\Delta \phi_{\rm LD} = \phi_{\rm LD}^\circ \, \epsilon_{hkl} . \eqno(3)]

[\phi_{\rm LD}^\circ] is related to [d_{hkl}^\circ] via equation (1)[link]. The final beam polarization is given by the average

[\eqalignno{P_{\rm LD} (\phi_{\rm LD}) = & \, \left \langle \cos (\Delta \phi_{\rm LD}) \right \rangle \cr = & \, \int D(\epsilon_{hkl}) \cos(\phi_{\rm LD} \epsilon_{hkl}) \, {\rm d} \epsilon_{hkl} . &(4)}]

D(hkl) is a normalized distribution function. P(ϕLD) is measured for a series of values ϕLD(B0), where for each ϕLD the position ΔL of coil C4 is scanned through one Larmor period as sketched in Fig. 1[link]. P is then obtained from the amplitude of the cosine-shaped I(ΔL). (The integration limits of all integrals in this section can be chosen as ∓∞.)

For the measurement of thermal expansion it is important to note that Δϕ(T) is affected by D(hkl), if the latter is asymmetric and if the asymmetry changes with T. This might result from an asymmetric splitting of a Bragg peak related to a structural phase transition. In cases where D(hkl) is symmetric, even if the width or shape changes with T, there will be no effect on Δϕ(T).

We now include the distortion of the neutron trajectories by SANS, and the misalignment between the lattice planes and the boundaries of B0, which necessarily happens for mosaic or polycrystalline samples. Both effects disturb the correlation between the lattice vector G and k, such that k [\neq] G/2. This distortion of the trajectories by SANS leads to an accumulation of additional Larmor phase, resulting in a reduction in P in equation (4)[link]. The calculation of the phase change depends on whether SANS happens before or after Bragg diffraction. SANS in the incident beam changes the Bragg angle, whereas the path length in the first precession field stays the same, so that the Larmor phase changes. SANS in the diffracted beam does not affect the Bragg angle, but the path length in the second precession field changes.

We first calculate ϕLD for single SANS events in the incident and diffracted beams, and then take the average assuming a range of SANS angles, which will be quantitatively determined in a separate SESANS experiment. The geometry is defined in Fig. 2[link], with the following relations:

[\theta_1 = \theta_0 - \alpha_1 + \eta , \eqno(5a)]

[\theta_2 = \theta_1 + \alpha_2 + \eta , \eqno(5b)]

[L_1 = L / \sin(\theta_0) , \eqno(5c)]

[L_2 = L / \sin(\theta_2) , \eqno(5d)]

[\lambda = 2d_{hkl} \sin(\theta_1) , \eqno(5e)]

[v = {{\pi\hbar} \over {m d_{hkl} \sin(\theta_1)}} , \eqno(5f)]

[d_{hkl} = d_{hkl}^\circ (1 + \epsilon_{hkl}) . \eqno(5g)]

[The last equation is identical to equation (2)[link] and is repeated here for better readability.] m and v = ℏk/m are the neutron mass and velocity, respectively. The Larmor phases ϕ1,2 accumulated before and after Bragg diffraction are

[\phi_{1,2} = {{\omega_{\rm L} L_{1,2}} \over {v}} . \eqno(6)]

The total phase is

[\eqalignno{\phi_{\rm LD} = & \, \phi_1 + \phi_2 \cr = & \, {{\omega_{\rm L} (L_1 + L_2)} \over {v}} \cr = & \, {{m} \over {\pi \hbar}} \, \omega_{\rm L} L d_{hkl}^\circ (1 + \epsilon_{hkl}) \left [ {{\sin(\theta_1)} \over {\sin(\theta_0)}} + {{\sin(\theta_1)} \over {\sin(\theta_2)}} \right ] . &(7)}]

Expanding equation (7)[link] with the definitions of equations (5)[link] to first order in , α1,2 and η gives

[\Delta\phi_{\rm LD} = \phi_{\rm LD}^\circ \left [ \epsilon_{hkl} - {{\alpha_1 + \alpha_2} \over {2}} \, \cot(\theta_0) \right ] , \eqno(8)]

where

[\phi_{\rm LD}^\circ = \phi_{\rm LD} (\epsilon_{hkl} = 0, \theta_1 = \theta_2 = \theta_0) , \eqno(9)]

[\Delta \phi_{\rm LD} = \phi_{\rm LD} - \phi_{\rm LD}^\circ. \eqno(10)]

The first-order term in η is zero. The second-order terms of the preceding expansion are summarized in Appendix A[link]. The leading term is ∼[ \cot^2(\theta_0)\eta^2], which is negligible for the experimental parameters in this work. The first term in equation (8)[link] is the usual LD phase related to variations in dhkl, while the second term describes the small-angle scattering.

[Figure 2]
Figure 2
(a) SANS in a powder sample before and after Bragg diffraction at a crystallite. (b) A single Bragg diffraction event at a crystallite. L1,2 are the path lengths in the precession field B. L is the width of the field area B0. α1,2 are the SANS angles. θ0 and θ2 are the angles of the trajectories with the field boundaries, and θ1 is the Bragg angle. η is the angle between the field boundary and the diffracting lattice plane. The signs of θ0, θ1 and θ2 are positive. α1, α2 and η have a negative sign for clockwise rotation, as shown in the sketch.

The polarization P(ϕLD) is given as a generalization of equation (4)[link] by the average

[\eqalignno{P(\phi_{\rm LD}^\circ) = & \, \left \langle \cos (\Delta \phi_{\rm LD}) \right \rangle \cr = & \, \int \!\!\! \int \!\!\! \int D(\epsilon_{hkl}) \, S(\alpha_1) \, T_1(\alpha_1) \, S(\alpha_2) \, T_2(\alpha_2) \cr & \, \times \cos \left \{ \phi_{\rm LD}^\circ \left [ \epsilon_{hkl} - {{\alpha_1 + \alpha_2} \over {2}} \, \cot(\theta_0) \right ] \right \} \, {\rm d}\alpha_1 \, {\rm d}\alpha_2 \, {\rm d}\epsilon_{hkl}. \cr &&(11)}]

S(α1) and S(α2) are the normalized ranges of the SANS angles before and after scattering, respectively, and T1,2(α1,2) are angular transmission functions of the diffractometer, given by the collimations of the incident and scattered beams, respectively. The cosine term in equation (11)[link] splits according to cos(a + b + c) = cos(a) cos(b) cos(c) + sin…, and the sine terms integrate to zero, as the S and T distribution functions are symmetric. The polarization is then

[P(\phi_{\rm LD}^\circ) = P_{S1} P_{S2} P_{\rm LD} , \eqno(12)]

[P_{S1} = \int S_1(\alpha_1) \, T_1(\alpha_1) \, \cos \left [ \phi_{\rm LD}^\circ \cot(\theta_0) \, \alpha_1/2 \right ] \, {\rm d}\alpha_1 , \eqno(13)]

[P_{S2} = \int S_2(\alpha_2) \, T_2(\alpha_2) \, \cos \left [ \phi_{\rm LD}^\circ \cot(\theta_0) \, \alpha_2/2 \right ] \, {\rm d}\alpha_2 , \eqno(14)]

where PLD is given by equation (4)[link]. The first two factors, PS1 and PS2, describe the effect of SANS before and after diffraction, respectively. These factors differ in the beam collimation T1,2 and the width of the small-angle scattering S1,2. The widths of S1,2 will typically be much smaller than the widths of T1,2, such that the latter have only a small influence on the value of the integral. Thus we will simplify T1 = T2 = T. The widths of S1 and S2 are only identical for symmetric scattering configurations, such as cylindrical sample cross sections, or flat samples in a symmetric configuration with respect to k1,2, and might be quite different for asymmetric sample shapes.

One aim of an LD experiment is to measure the distribution width of the lattice spacing D(hkl). To extract PLD from the experimental data [P(\phi_{\rm LD}^\circ)] in equation (12)[link], the SANS integrals PS1, PS2 have to be known. In the following section we show that PS can be determined experimentally by taking advantage of the SESANS technique, and thus for the data analysis there is no need to use an analytical expression for PS. Nevertheless, an analytical formula is often helpful. As a simple model, we take Gaussians for the distributions D, S1,2 and T, with FWHMs of [\epsilon_{\rm FW}], [\alpha_{S1,2}] and [\alpha_{T}], respectively. These Gaussians have the form

[\exp \left [ {{-4 \ln(2) \, x^2} \over {x_{\rm FWHM}^2}} \right ] . \eqno(15)]

Using the tabulated integral

[\int\limits_{-\infty}^{+\infty} \exp(-a^2 x^2) \, \cos(b x) \, {\rm d}x = \left ( {{\pi} \over {a^2}} \right )^{1/2} \exp \left ( {{-b^2} \over {4a^2}} \right ) , \eqno(16)]

we obtain for the polarization [equation (12)[link]]

[P_{\rm LD} = \exp \left ( -C_1 \epsilon^2_{hkl \, {\rm FW}} \right ) , \eqno(17)]

[P_{S1} = \exp \left ( {{-C_2} \over {\alpha_{S1}^{-2} +\alpha_{T}^{-2}}} \right ) , \eqno(18)]

[P_{S2} = \exp \left ( {{-C_2} \over {\alpha_{S2}^{-2} +\alpha_{T}^{-2}}} \right ) , \eqno(19)]

[C_1 = {{\left ( \phi_{\rm LD}^\circ \right )^2} \over {16\ln(2)}} , \eqno(20)]

[C_2 = {{\left ( \phi_{\rm LD}^\circ \right )^2} \over {16\ln(2)}} \, {{\cot^2(\theta_0)} \over {4}} . \eqno(21)]

3. The SESANS technique

We briefly review the principles of SESANS and give scaling formulas for flat and cylindrical samples to convert SESANS data to the LD geometry. SESANS uses two identical field regions (or precession regions defined by RF spin flippers) with opposite signs of the magnetic field, with boundaries inclined by an angle Ω (Fig. 3[link]). Unscattered neutrons gain equal precession phases in both regions (ϕ1 = −ϕ2), even if the path is inclined with respect to the central axis. If the neutron is scattered by a small angle α, the path length in the second field changes (Rekveldt, 1996[Rekveldt, M. T. (1996). Nucl. Instrum. Methods Phys. Res. B, 114, 366-370.]; Rekveldt et al., 2003[Rekveldt, M. T., Bouwman, W. G., Kraan, W. H., Uca, O., Grigoriev, S. V., Habicht, K. & Keller, T. (2003). Neutron Spin Echo Spectroscopy, Lecture Notes in Physics, Vol. 601, edited by F. Mezei, C. Pappas & T. Gutberlet, pp. 87-99. Heidelberg: Springer.]; Keller et al., 1995[Keller, T., Gähler, R., Kunze, H. & Golub, R. (1995). Neutron News, 6(3), 16-17.]):

[\Delta L = L_0 \tan(\Omega) \, \alpha . \eqno(22)]

The phase difference Δϕ = ϕ1ϕ2 is

[\Delta \phi = \phi_1 \tan(\Omega) \, \alpha , \eqno(23)]

where ϕ1 is the Larmor phase accumulated in one precession region,

[\phi_1 = {{m \omega_{\rm L} L_0} \over {\hbar k}} . \eqno(24)]

The polarization P(ϕ1) is given by an integral similar to PS1, PS2 in equation (12)[link]:

[\eqalignno{P_{\rm SES} (\phi_1) = & \, \left \langle \cos(\Delta\phi) \right \rangle \cr = & \, \int S(\alpha) \, T(\alpha) \, \cos[\phi_1 \tan(\Omega) \,\alpha] \, {\rm d} \alpha . &(25)}]

It is not necessary to model S(α) for subsequent analysis of the LD data, but it is often convenient to have an analytical expression. As a simple model we again take Gaussian distributions [equation (15)[link]] for S(α) and T(α) with FWHMs of αS and αT, respectively. Using equation (16)[link] we get

[P_{\rm SES} (\phi_1) = \exp \left [ -{{\phi_1^2 \tan^2(\Omega)} \over {16\ln(2) \, (\alpha_S^{-2} + \alpha_T^{-2})}} \right ] . \eqno(26)]

Typically [\alpha_S \ll \alpha_T], so that the latter is negligible in equation (26)[link].

[Figure 3]
Figure 3
Spin–echo small-angle scattering configuration. The fields B of the two precession regions have opposite signs, as opposed to the LD configuration.

It is useful to express the SESANS phase [equation (23)[link]] in terms of the spin–echo length z and the momentum transfer Q:

[\Delta \phi = z\, Q, \eqno(27)]

where

[z = {{\phi_1 \tan(\Omega)} \over k} \eqno(28)]

or

[\eqalignno{ z \,\, ({ \AA}) & = 0.2531\,\,({\rm m}^{-1}\,\,{\rm kHz}^{-1}\,\,{\AA}{}^{-1})\tan(\Omega) \times L_0 \,\, ({\rm m}) \cr & \quad\times \nu_L \,\, ({\rm kHz}) \times \lambda^2 \,\, ({\rm \AA}^2) , &(29)}]

with νL = ωL/(2π).

We now show how PSES  cyl for a cylindrical sample can be calculated using SESANS data PSES  flat from a flat sample, and vice versa. We will take advantage of a scaling relation for PSES with respect to k and the sample width t derived by Rekveldt and Andersson (Rekveldt et al., 2003[Rekveldt, M. T., Bouwman, W. G., Kraan, W. H., Uca, O., Grigoriev, S. V., Habicht, K. & Keller, T. (2003). Neutron Spin Echo Spectroscopy, Lecture Notes in Physics, Vol. 601, edited by F. Mezei, C. Pappas & T. Gutberlet, pp. 87-99. Heidelberg: Springer.]; Andersson et al., 2008[Andersson, R., van Heijkamp, L. F., de Schepper, I. M. & Bouwman, W. G. (2008). J. Appl. Cryst. 41, 868-885.]), which is valid even in the presence of strong multiple scattering:

[P_{\rm SES} (z, k, t) = P_{\rm SES} (z, k_0, t_0)^{(k_0/k)^2 \, (t/t_0)} . \eqno(30)]

The transformation [P_{\rm SES \, flat} \rightarrow P_{\rm SES \, cyl}] is useful to verify equation (30)[link] experimentally. The transformation [P_{\rm SES \, cyl} \rightarrow P_{\rm SES \, flat}] is needed in the next section to analyse LD data collected on a cylindrical sample.

PSES  cyl is obtained from PSES  flat by averaging and scaling the latter for the different path lengths t(y) of a neutron beam crossing a cylindrical sample [see Figs. 4[link](a) and 4[link](c)]. The polarization of individual trajectories is weighted by the transmission Tr = exp[−Σkt(x)], where Σk and Σth are the total absorption cross sections for neutrons with wavevectors k and kth (th for thermal), with kth = 2π/λth and λth = 1.8 Å. For most isotopes there is 1/k scaling if [k \leq k_{\rm th}]:

[\Sigma_k = {{\Sigma_{\rm th} \, k_{\rm th}} \over k} . \eqno(31)]

With the path length t = 2(R2y2)1/2, the width of the flat sample t0 and the cylinder radius R we get

[P_{\rm SES \, cyl} (z) = A^{-1} \, I \left [ P_{\rm SES \, flat}(z) \right ] , \eqno(32)]

[\eqalignno{& I \left [ P_{\rm SES \, flat} (z) \right ] \cr & = \int\limits_0^R \left \{ \left [ P_{\rm SES \, flat} (z) \right ]^{(k_0/k)^2 \, (1/t_0)} \, \exp (-\Sigma_k) \right \}^{2\left ( {R^2 - y^2} \right )^{1/2}} \, {\rm d}y , &(33)}]

[A = \int\limits_0^R \exp \left [ -2\Sigma_k \left ( R^2 - y^2 \right )^{1/2} \right ] \, {\rm d}y . \eqno(34)]

Calculating PSES  flat from PSES  cyl data corresponds to an inversion of equation (32)[link]. This is easily achieved numerically by searching for solutions for PSES  flat(z) that satisfy the equation

[A^{-1} \, I \left [ P_{\rm SES \, flat}(z) \right ] - P_{\rm SES \, cyl}(z) = 0 . \eqno(35)]

A and I are the same as in equation (32)[link]. Suitable algorithms for finding roots of nonlinear functions, such as fzero() in MATLAB (The MathWorks Inc., Natick, MA, USA), are available in many numerical libraries.

[Figure 4]
Figure 4
(a), (b) Rectangular sample. (a) In the SESANS configuration, the sample is oriented perpendicular to the incident neutron beam. A scattering event changes the direction of the neutron trajectory by an angle α. (b) In the LD configuration, a neutron is Bragg-diffracted at a crystallite located at a distance x from the sample face. In the symmetric reflection configuration, the angles of the incident and outgoing neutron trajectories with the sample face are equal to the Bragg angle (θ1 = θ2 = θB). A small misalignment δ of the sample face leads to θ1 = θBδ, θ2 = θB + δ. (c), (d) Cylindrical sample in (c) the SESANS and (d) the LD configuration.

4. Scaling of SESANS data to the LD geometry

In this section, we will determine the integrals PS1,2 in equation (12)[link] from the previously measured SESANS data, for both flat and cylindrical sample geometries. For the flat sample, there are two symmetric configurations, reflection and transmission, with the sample surface perpendicular and parallel to Q, respectively. As the effective path length in a flat sample of width t0 at the Bragg angle θB is Lr = t0/sin(θ) in reflection and Lt = t0/cos(θ) in transmission, Lt > Lr for θ > 45°. For the LD experiments we used θ = 55°, such that Lt/Lr ≃ 1.4, leading to increased SANS in the transmission configuration. This configuration is unfavourable in our case and we will not include it in the following discussion.

PS1, PS2 [equation (12)[link]] and PSES [equation (25)[link]] are very similar expressions, if the field inclination angles Ω in both SESANS and LD are set to Ω = 90° − θB, so that tan(Ω) = cot(θB). However, the ranges of SANS angles S(α) in SESANS and S1,2(α1,2) in LD are not equal, as the geometries for the two experimental configurations are different, and thus the effective neutron path lengths inside the sample are not the same.

In the SESANS configuration [Fig. 4[link](a)], the faces of the flat sample are aligned perpendicular to the beam. The effective neutron path length corresponds to the sample width t0. Small variations in the neutron path length resulting from finite beam collimation αT ≃ 1° and from small-angle scattering of the order of t0/t ≃ 1 − cos(αT/2) can be neglected in equation (30)[link].

In the LD configuration, the path lengths vary with the location of the diffraction event. For a symmetric setup in reflection geometry as shown in Fig. 4[link](b), the path lengths of the incident and diffracted neutron trajectories are equal, with t1,2(x) = x/sin(θ1,2). We obtain the polarization PS1, PS2 by averaging and scaling the SESANS polarization PSES  flat (z,k0,t0) measured for a flat sample of width t0 according to equation (30)[link]. The weight of an individual neutron trajectory is given by the transmission Tr1,2(x) = exp[−Σkt1,2(x)].

[\eqalignno{P_{S1,2} (k, z) = & \, {{1} \over {B}} \, \int\limits_{x=0}^{t_0} \left \{ P_{\rm SES \, flat}^{(k_0/k)^2 [1/t_0\sin(\theta_{1,2})]} \exp{ \left [ -{{\Sigma_k} \over {\sin(\theta_{1,2})}} \right ]} \right \}^x \, {\rm d}x \cr = & \, {{1} \over {B}} \, {{A^{t_0} - 1} \over {\ln(A)}} , &(36)}]

[A = P_{\rm SES \, flat}^{(k_0/k)^2 [1/t_0\sin(\theta_{1,2})]} \exp{ \left [ -{{\Sigma_k} \over {\sin(\theta_{1,2})}} \right ]} , \eqno(37)]

[B = {{\sin(\theta_{1,2})} \over {\Sigma_k}} \, \left \{ 1 - \exp{ \left [ -{{\Sigma_k t_0} \over {\sin(\theta_{1,2})}} \right ]} \right \} . \eqno(38)]

In symmetric reflection geometry with θ1 = θ2, we get PS1 = PS2. A detuning δ as shown in Fig. 4[link](b)[link] increases t1(x) and decreases t2(x), so that the product PS1 × PS2 changes in second order in δ, and a small detuning of the order of a few degrees is not significant.

The cylindrical sample geometry [Figs. 4[link](c) and 4[link](d)] is treated in a similar way by taking the average of the SESANS polarization PSES  flat(z,k0,t0) with the transmission as a weight factor. In the case of a cylindrical sample, the SESANS data should be measured on the same sample. Then PSES  cyl has to be converted to PSES  flat(z,k0,t0) [equation (35)[link]] with a specific but arbitrary t0. For a neutron diffracted at a crystallite located at (x, y), the length of the incident trajectory inside a sample of radius R is t1 = x + (R2y2)1/2.

[\eqalignno{& P_{S1} = {{1} \over {C}} \cr & \times \int\limits_{x = -R}^R \int\limits_{y = -(R^2 - x^2)^{1/2}}^{(R^2 - x^2)^{1/2}} \left [ P_{\rm SES \, flat}^{(k_0/k)^2 (1/t_0)} \exp{\left ( -\Sigma_k \right )} \right ]^{[x + (R^2 - y^2)^{1/2}]} \, {\rm d}x \, {\rm d}y , \cr &&(39)}]

with

[C = \int \!\!\! \int \exp{ \left \{ -\Sigma_k \left [ x + \left ( R^2 - y^2 \right )^{1/2} \right ] \right \} } \, {\rm d}x \, {\rm d}y . \eqno(40)]

The integration boundaries are the same as in equation (39)[link]. For symmetry reasons, PS2 = PS1.

5. Experimental

The LD measurements were conducted on the TRISP spectrometer at the FRM II. TRISP was designed as a thermal three-axis spectrometer incorporating the resonant spin–echo technique for the high-resolution spectroscopy of phonons and spin excitations (Keller et al., 2002[Keller, T., Habicht, K., Klann, H., Ohl, M., Schneider, H. & Keimer, B. (2002). Appl. Phys. Mater. Sci. Process. 74, s332-s335.]), and it also operates efficiently as a Larmor diffractometer and SESANS instrument. The primary spectrometer includes a V-cavity polarizing neutron guide, a velocity selector and a pyrolytic graphite (002) monochromator. The spin–echo precession devices and the sample are housed in a mu-metal shield with a residual magnetic field of <5 mOe to avoid parasitic spin precession. For the LD measurements, the scattering angle 2θB was kept constant at 110° and the incident wavevector was varied in the range 2.08 < ki < 2.66 Å−1 to access various Bragg peaks. In LD mode, the RF coil rotation angles were set to Ω = 90° − θB = 35° to align the field boundaries parallel to the lattice planes. The same angles Ω = 35° with opposite sign in the second precession field were used in SESANS mode.

The sample was hematite, α-Fe2O3 (powder), as used in earlier studies (Stekiel et al., 2015[Stękiel, M., Przeniosło, R., Sosnowska, I., Fitch, A., Jasiński, J. B., Lussier, J. A. & Bieringer, M. (2015). Acta Cryst. B71, 203-208.]; Fabrykiewicz et al., 2017[Fabrykiewicz, P., Stękiel, M., Sosnowska, I. & Przeniosło, R. (2017). Acta Cryst. B73, 27-32.]). Hematite is a promising material for such studies because of the negligible incoherent neutron scattering of iron and oxygen. The powder was annealed at 1100 K for 5 h in air to reduce internal strain. This annealing step reduced hkl by ∼30%. We used two sample cells, a cylindrical vanadium container (labelled `cylinder'), inner diameter 12 mm and wall thickness 0.2 mm, and a flat quartz glass cell (labelled `flat'), inner dimensions 30 × 30 × 2 mm and 0.5 mm glass width. The flat cell produced an unstructured background (glass peak) in the LD measurement. LD was measured for (024), (116), (214) and (300) at the ki values given in Table 1[link] using both cylindrical and flat containers.

Table 1
Experimental data on α-Fe2O3 powder

LD experiments were performed using four Bragg peaks. The flat container was set to reflection geometry.

No. k−1) (hkl) Mode Container Contents
1 2.082 (024) LD Flat, reflection Filled
2 2.082 (024) LD Cylindrical Filled
3 2.263 (116) LD Flat, reflection Filled
4 2.263 (116) LD Cylindrical Filled
5 2.581 (214) LD Flat, reflection Filled
6 2.581 (214) LD Cylindrical Filled
7 2.639 (300) LD Flat, reflection Filled
8 2.639 (300) LD Cylindrical Filled
9 1.70   SESANS Flat Empty
10 1.70   SESANS Flat Filled
11 2.30   SESANS Flat Empty
12 2.30   SESANS Flat Filled
13 2.30   SESANS Cylindrical Empty
14 2.30   SESANS Cylindrical Filled

For the SESANS experiments, the scattering angle was set to zero and the neutron beam was shaped by a cadmium mask to the sample size. PSES was measured for both empty and filled sample cells at k = 2.30 Å−1, which is roughly in the centre of the k range used for LD, and in addition at k = 1.70 Å−1 for the flat sample to test the scaling with k according to equation (30)[link]. All measurements are shown in Fig. 5[link], where the data with sample were normalized to the empty cell measurements. (See also Table 1[link] for a summary of all data collected on α-Fe2O3).

[Figure 5]
Figure 5
SESANS data for α-Fe2O3 powder. (a) PSES  flat for k = 1.7 and 2.3 Å−1 for the flat quartz glass container without sample (empty), and with sample normalized by the empty container data. The maximum z for a given maximum precession field B scales ∼1/k2. The blue open circles show the k = 2.3 Å−1 data scaled to k = 1.7 Å−1 by equation (30)[link]. These scaled data coincide with the experimental data for k = 1.7 Å−1. (b) PSES for the flat and cylindrical containers at k = 2.3 Å−1. Pflat  calc is the polarization calculated for the flat sample geometry by taking the PSES  cyl data [equation (35)[link]], and vice versa for PSES  cyl  calc [equation (32)[link]]. The data for the cylindrical (flat) container are plotted as blue triangles (red circles). Solid symbols show measured data, while open symbols are calculations. The error bars in the measured values are smaller than the symbols.

In LD mode, first the instrumental resolution Pinstr(ϕLD) was determined with a Ge monocrystal as reference sample with a negligible range of dhkl, where for a perfect instrument we expect Pinstr = 1. The instrument configuration for this reference measurement was the same as for the subsequent LD experiment on the powder samples, with fixed 2θ = 110° and Ω = 35°. The Ge (220) and (004) Bragg reflections were accessible at k = 1.92 and 2.72 Å−1, respectively. Fig. 6[link] shows these data for two different operation modes, where one (RFM = 4) or both RF coils (RFM = 8) in the individual coil sets C1–C4 (Fig. 1[link]) were energized. These two modes differ in the effective Larmor frequency, which is two (four) times the RF for RFM = 4 (8), and they are used to cover the range of low (high) Larmor phase. The k dependence of Pinstr is a property of the polarizing neutron guide. The dependence on frequency and RFM, especially the scatter of the data points at low frequency, results from field inhomogeneities in the RF coils and from stray fields in between the coils.

[Figure 6]
Figure 6
Instrumental resolution in LD mode measured on a Ge monocrystal using the (220) and (004) reflections with k = 1.92 Å−1 and k = 2.72 Å−1, respectively. The scattering angle was 110° and the coil inclination angle Ω = 35°. RFM = 4 (8) is the number of active RF coils. The lines are obtained by smoothing the data. Error bars are smaller than the symbols.

Fig. 7[link] shows the LD data for the (024) reflection of α-Fe2O3 measured at k = 2.08 Å−1 for the flat container (solid lines) and the cylindrical container (dotted lines). Data for the other Bragg peaks are shown in the supporting information. The SANS contribution PS is much smaller for the flat container. Normalizing the raw data by PS and by the instrumental polarization Pinstr (identical for both containers) leads to very similar intrinsic PLD.

[Figure 7]
Figure 7
LD data for the (024) reflection of α-Fe2O3 for the flat quartz glass sample container (width 2 mm, solid symbols, solid lines) and the cylindrical vanadium container (diameter 12 mm, open symbols, dotted lines). The counting time per point was 20 min. Pinstr is the same for both containers. The fit (blue line) to Pexp is a product of the instrumental Pinstr, SANS PS2 and PLD [see equation (41)[link]]. The latter depends on the range of dhkl. A constant factor P0 takes unpolarized background into account.

The results of LD measurements for α-Fe2O3 are compared with the results obtained by Stekiel et al. (2015[Stękiel, M., Przeniosło, R., Sosnowska, I., Fitch, A., Jasiński, J. B., Lussier, J. A. & Bieringer, M. (2015). Acta Cryst. B71, 203-208.]) on the high-resolution SR powder diffractometer ID22 at the ESRF (Fitch, 2004[Fitch, A. N. (2004). J. Res. Natl Inst. Stand. Technol. 109, 133-142.]). The observed SR diffraction peak shape was described using a pseudo-Voigt function. The instrumental contribution of ID22 was estimated by measuring a reference LaB6 sample.

6. Results and discussion

In this section we discuss the analysis of LD data from α-Fe2O3 powder. The analysis steps include (i) determination of the instrumental resolution by means of a monocrystal reference sample, (ii) measurement of the small-angle scattering by the SESANS technique, (iii) scaling these data to the LD geometry including absorption and (iv) determination of D(hkl).

In LD, SESANS and other neutron spin–echo techniques, the measured signal, that is the polarization versus the Larmor precession phase, is a product of intrinsic and instrumental effects. This is in contrast with conventional diffraction and spectroscopy, where the measured signals are convolutions of the instrumental resolution and sample-related effects (Mezei, 1980[Mezei, F. (1980). Editor. Neutron Spin Echo, Lecture Notes in Physics, Vol. 128. Heidelberg: Springer-Verlag.]; Mezei et al., 2003[Mezei, F., Pappas, C. & Gutberlet, T. (2003). Editors. Neutron Spin Echo Spectroscopy, Lecture Notes in Physics, Vol. 601. Heidelberg: Springer.]). This is explained, in brief, by the fact that the polarization is a probability, namely the expectation value of the neutron spinor. The probabilities of instrumental effects, such as non-perfect RF coils, and sample intrinsic effects, such as the range of D(hkl), are statistically independent, so that the probabilities (polarizations) are multiplied. A second qualitative argument is that LD and spin–echo techniques provide cosine-Fourier transforms of distribution functions [equations (11)[link] and (25)[link]], such that convolutions are converted to products. In the present case of an LD experiment, the measured polarization Pexp is

[P_{\rm exp} = P_0 P_{\rm inst} P_S^2 P_{\rm LD} , \eqno(41)]

where PS and PLD are defined in equations (12)[link] and (17)[link]. A constant factor P0 accounts for additional effects leading to a loss of polarization, independent of the Larmor phase, including non-polarized background from the sample and spin precession resulting from small randomly distributed ferromagnetic moments in the sample. Pinst is taken from Fig. 6[link] and is independent of the sample. The same argument holds for the SESANS experiment, where

[P_{\rm exp \, SES} = P_0 P_{\rm inst \, SES} P_{\rm SES} . \eqno(42)]

The factor PS in equation (41)[link] is calculated from Pexp  SES in equation (42)[link] by applying equations (36)[link] and (39)[link] for the flat and cylindrical samples, respectively. The instrumental resolution Pinst is measured by means of a `perfect' sample with no intrinsic effects, such as a monocrystal with no range of dhkl in the case of LD, an empty container in the case of SESANS, and a purely elastic scatterer in the case of energy-resolved spin–echo spectroscopy. In the current experiment, we used as reference samples a Ge single crystal for LD or an empty cell for SESANS. Pinst(freq, RFM, k) from the Ge crystal is shown in Fig. 6[link]. The polarization depends on the frequency, the number RFM of energized RF coils, and the wavevector k. We use the smoothed curves in Fig. 6[link] to average the small parasitic oscillations, and a linear interpolation in k between the two data sets taken at k = 1.92 and 2.72 Å−1. For the following analysis, Pinst(freq, RFM, k) has to be converted to Pinst(ϕLD), shown as a line in Fig. 7[link]. This conversion is ambiguous, as there are two data sets with different RFMs for Pinst. Thus, we always use the data with the same RFM as used for the LD measurement Pexp of the powder sample.

Fig. 5[link] shows SESANS data normalized by the corresponding empty cell scans to eliminate instrumental effects [equation (42)[link]], such as SANS in the coils close to the sample and non-ideal performance of the RF coils. The width αS (FWHM) for the small-angle distributions obtained by fitting the Gaussian model equation (26)[link] to the normalized data gives 0.0650 (5), 0.1138 (5) and 0.1427 (5)° for the flat cell at k = 2.3 and 1.7 Å−1 and for the cylindrical cell at 2.3 Å−1, respectively. The parameter αS depends strongly on k and on the sample width, but is still much smaller than the angular transmission of the instrument αT ≃ 1°, so that the latter has no effect in equation (26)[link].

The LD experiment is usually performed at many Bragg reflections, each with a different k. The SESANS measurement should at least be done at the lowest of these k values. Scaling of the SESANS data to other k is possible using equation (30)[link], as is demonstrated in Fig. 5[link](a) by scaling the k = 2.3 Å data to k = 1.7 Å. These scaled data coincide with the measured data. Scaling of SESANS data between flat and cylindrical sample shapes by equations (32)[link] and (35)[link] also works, as is shown in Fig. 5[link](b), where again the scaled and measured data coincide. The absorption of the α-Fe2O3 powder was included in these plots, although it is small and has no visible effect (σth = 0.10 cm−1 for bulk α-Fe2O3; the powder density is roughly 40% less). The scaling relation (30)[link] might not hold for unconventional SANS processes, such as multiple Bragg scattering (MBS), which might contribute significantly to the SANS cross section (Barker & Mildner, 2015[Barker, J. G. & Mildner, D. F. R. (2015). J. Appl. Cryst. 48, 1055-1071.]). Our analysis technique is also valid for such unconventional SANS processes, if the scaling equation (30)[link] holds. The latter can be experimentally tested by taking SESANS data at several different k values.

The LD data for a (024) reflection (k = 2.08 Å−1) of the α-Fe2O3 powder are shown in Fig. 7[link]. This peak was measured at the smallest k = 2.08 Å−1 and thus has the largest SANS contribution. PS shows a more pronounced decay for the cylinder due to the increased path length and increased SANS of the cylinder compared with the flat sample. The raw data without any corrections correspond to [\epsilon_{\rm 024 \, raw}] values of 13.5 (5) × 10−4 for the cylinder and 8.80 (11) × 10−4 for the flat container. Including all corrections, PLD is obtained from a fit to equation (17)[link], corresponding to 024 of 7.1 (2) × 10−4 and 7.2 (4) × 10−4 for the flat and cylindrical samples, respectively. (The errors are statistical errors from the fit.) This means that, after correction, the values for flat and cylindrical samples agree, where for the flat sample the SANS correction is much smaller. The polarization offsets P0 are 0.89 (1) for the cylinder and 0.74 (1) for the flat cell, indicating the increased background from the flat quartz glass container. Data for the other Bragg peaks are shown in the supporting information.

The flat sample shape is superior to the cylinder, since for a given beam diameter (typically 20 mm) a larger sample volume at a smaller width can be used. As a rule of thumb, the SANS should be limited to PSES [\gtrsim] 0.5 for the maximum z and minimum k, as the relative error usually increases with decreasing P. This condition is roughly met in the present experiment for the flat sample, but not for the cylinder, although the sample mass is the same in both cases. The only disadvantage of the flat sample is an additional decay of the polarization in the case of an offset δ [Fig. 4[link](b)] from the symmetric reflection geometry. For the present data, we numerically calculated that for δ < 10° the variation δ024 < 0.07, i.e. only 1/3 of the statistical error. These errors can be avoided by a precise alignment of the sample cell.

In Fig. 8[link] the hkl obtained by LD for four Bragg peaks (flat sample) are compared with the d-spacing distribution widths observed on ID22. The latter were determined by numerical deconvolution of the instrumental resolution function (LaB6 reference) from the observed raw-data Bragg peaks. The agreement of SR diffraction with LD confirms our earlier findings about the monoclinic deformation of the α-Fe2O3 lattice (Przeniosło et al., 2014[Przeniosło, R., Sosnowska, I., Stękiel, M., Wardecki, D., Fitch, A. & Jasiński, J. B. (2014). Physica B, 449, 72-76.]). The widths calculated for the monoclinic structure model (Przeniosło et al., 2014[Przeniosło, R., Sosnowska, I., Stękiel, M., Wardecki, D., Fitch, A. & Jasiński, J. B. (2014). Physica B, 449, 72-76.]) are 024 = 2.22 × 10−4, 116 = 1.67 × 10−4, 214 = 0.95 × 10−4 and 300 = 1.12 × 10−4. The measured values are much larger, but their variation follows the variation of the calculated values. The increased width is probably due to microstrains in the powder grains.

[Figure 8]
Figure 8
Δd/d spacing distribution width hkl (FWHM) versus Q = 1/dhkl for four Bragg peaks of α-Fe2O3 obtained from LD (flat container) and SR diffraction. The data correspond from left to right to the reflections (024), (116), (214) and (300).

The measured hkl are close to the resolution limit of SR diffraction. For LD, it is not straightforward to define the resolution. Variations in hkl are detected as a change in the polarization PLD. In typical experiments with monocrystalline samples without SANS, the error in hkl is close to 10−5. The situation is more complicated for powder samples, where in the case of the flat container at the (024) reflection SANS contributes a broadening of 1 × 104 to hkl. Thus, the statistical errors of the order of ±1.5 × 10−4 shown in Fig. 8[link] as a result of the fit are also a good estimate for the resolution. The good agreement between LD and SR diffraction validates the related data analysis techniques, i.e. the measurement of the instrumental resolution with the LaB6 reference sample in SR diffraction and the determination of the instrumental resolution by a monocrystal and the correction for SANS in the case of LD.

7. Summary

In this work we have shown that small-angle scattering makes a significant contribution to the range of dhkl measured by neutron Larmor diffraction. We have described a quantitative procedure to measure the SANS by means of the spin–echo small-angle neutron scattering technique, and derived scaling expressions to include the SESANS data in the analysis of the LD data collected on flat or cylindrical samples. We have shown that using a flat sample in reflection geometry is preferable to using cylindrical samples, as for a given sample mass the effective neutron path length, and thus the effect of SANS, is minimized for the flat sample. Finally, we have applied the new analysis method to LD data from hematite and calcite powder (for the latter see the supporting information). These LD data confirm the hkl-dependent d-spacing range previously observed by high-resolution SR diffraction.

APPENDIX A

Second-order terms of ϕLD

The preceding discussion was based on the first-order expansion [equation (8)[link]] of ϕLD in equation (7)[link]. Here, we show that the second-order terms of this expansion are negligible for the analysis of the experimental data in the present work.

The expansion of equation (7)[link] to second order in α1, α2, η and hkl gives

[\eqalignno{\phi_{\rm LD} = & \, \phi_{\rm LD}^\circ \left ( 1 + \epsilon_{hkl} \right ) - {{\phi_{\rm LD}^\circ} \over {2}} (\alpha_1 + \alpha_2) \cot (\theta_0) \cr & \, + {{\phi_{\rm LD}^\circ} \over {2}} \sum_{ij} M_{ij} X_i X_j , &(43)}]

with the parameter vector X = [(\alpha_1, \alpha_2, \eta, \epsilon_{hkl})] and the coefficients

[{\bf M} = \left ( \matrix{ -{{1} \over {2}} & -1-\cot^2(\theta_0) & -\cot^2(\theta_0) & -\cot(\theta_0) \cr 0 & {{1} \over {2}} + \cot^2(\theta_0) & 2 + 3\cot^2(\theta_0) & -\cot(\theta_0) \cr 0 & 0 & 1 + 2\cot^2(\theta_0) & 0 \cr 0 & 0 & 0 & 0 } \right ) . \eqno(44)]

The polarization in equation (41)[link] gains an additional factor PSO from the second-order terms:

[P_{\rm SO} = \textstyle\prod\limits_{ij} P_{ij \, {\rm SO}} , \eqno(45)]

with

[P_{ij \, {\rm SO}} = \int \!\!\! \int F_i (X_i) \, F_j(X_j) \, \cos \left ( {{\phi_{\rm LD}^\circ} \over {2}} M_{ij} \, X_{ij} \right ) \, {\rm d}X_i \, {\rm d}X_j . \eqno(46)]

Fi(Xi) are Gaussian distribution functions for the parameters Xi, with FWHM αS1 = αS2 = 0.09°, FW = 7 × 10−4 and ηFW = 0.33°. This last parameter was obtained from the width of the rocking scan of a single perfect crystal with the same spectrometer configuration as used for LD. ηFW is smaller than the collimation (∼1°), as the wavelength bands of the incident and diffracted beams are limited by the monochromator and analyser, and thus the range of Bragg angles for a given dhkl is limited. Calculation of ηFW is in principle possible, but a suitable resolution formalism for Larmor diffractometers taking all spectrometer parameters into account, similar to the work of Habicht et al. (2003[Habicht, K., Keller, T. & Golub, R. (2003). J. Appl. Cryst. 36, 1307-1318.]) and Groitl et al. (2018[Groitl, F., Keller, T. & Habicht, K. (2018). J. Appl. Cryst. 51, 818-830.]) for three-axis spin–echo spectrometers, is not yet available.

The second-order terms in equation (43)[link] are very small, except the η2 term (∼M33η2). Fig. 9[link] shows the polarization equation (46)[link] including all terms, and the η2 term P33 SO separately, for the experimental parameters of this work. This dominant term changes the resulting range of hkl = 7.1 (4) × 10−4 by 1.7 × 10−6, more than one order of magnitude less than the statistical error in hkl. Thus we neglected this term in the present analysis. If no monochromator or analyser is used, P33 SO is large and should be included in the analysis as an additional factor in equation (41)[link], such that

[P_{\rm exp} = P_0 P_{\rm instr} P_S^2 P_{\rm 33 \, SO} P_{\rm LD} . \eqno(47)]

[Figure 9]
Figure 9
Second-order terms of ϕLD [equation (46)[link]] and the dominant η2 term P33 SO.

Supporting information


Acknowledgements

We thank O. Holderer and A. Senyshyn for the sample containers, and F. Tralmer for technical support.

Funding information

We acknowledge funding from the Deutsche Forschungs­gemeinschaft (DFG, German Research Foundation), Projektnummer 107745057 – TRR 80. PF acknowledges financial support from `Kartezjusz' project co-financed from the European Social Fund under operational programme Knowledge Education Development, agreement No. POWR.03.02.00-00-I001/16-00.

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