Polarized and Unpolarized Neutron Scattering for Magnetic Materials at the Triple-axis Spectrometer PONTA in JRR-3

We briefly recall the basic formulas of polarized neutron scattering from magnetic materials. The elastic scattering cross section of neutrons \(d\sigma/d\Omega|_{\textit{el}}\) is described as follows:⁸⁾ \begin{equation} \frac{d\sigma}{d\Omega}\bigg|_{\textit{el}} \propto |\langle \boldsymbol{k}' s'|V|\boldsymbol{k}s\rangle|^{2}. \end{equation} (1) \(\boldsymbol{k}\) and \(\boldsymbol{k}'\) are wave vectors of the incident and scattered neutrons. s and \(s'\) are the initial and final spin state of the neutrons. V represents the interaction potential for a neutron during the scattering process. For magnetic scattering, V is given by \begin{equation} V = - {\boldsymbol{\mu}}_{n}\cdot\boldsymbol{B}, \end{equation} (2) where \({\boldsymbol{\mu}}_{n}\) and \(\boldsymbol{B}\) show the magnetic dipole moment of a neutron and the magnetic field from the magnetic moments in the sample, respectively. By executing the integral with respect to the space coordinates, the cross section is written in the form of \begin{equation} \frac{d\sigma}{d\Omega}\bigg|_{\textit{el}} \propto |\langle s'|{\boldsymbol{\sigma}}\cdot\boldsymbol{M}^{\bot}({\boldsymbol{\kappa}})| s\rangle|^{2}. \end{equation} (3) \({\boldsymbol{\sigma}}\) is the Pauli spin operator for the neutron spin. \(\boldsymbol{M}^{\bot}({\boldsymbol{\kappa}})\) is a square of Fourier-transformed magnetic moments projected onto a plane perpendicular to the scattering vector \({\boldsymbol{\kappa}}\) (\(\equiv\boldsymbol{k}-\boldsymbol{k}'\)). To describe the polarization direction of the neutrons, we introduce a Cartesian coordinate system xyz in which the x axis is parallel to \({\boldsymbol{\kappa}}\), the z axis is perpendicular to the scattering plane, and the y axis completes the right-hand set (see the inset of Fig. 1). By using this coordinate system, \(\boldsymbol{M}^{\bot}({\boldsymbol{\kappa}})\) is written as \(\boldsymbol{M}^{\bot}({\boldsymbol{\kappa}})=(0,M^{\bot}_{y}({\boldsymbol{\kappa}}),M^{\bot}_{z}({\boldsymbol{\kappa}}))\). Note that the x component is always absent, because only the magnetic moments projected onto the yz plane contribute to \(\boldsymbol{M}^{\bot}({\boldsymbol{\kappa}})\). Taking the z axis as the quantization axis of the neutron spins, the right side of Eq. (3) is described as follows: \begin{align} |\langle{\uparrow}|\sigma_{y} M^{\bot}_{y}({\boldsymbol{\kappa}})+\sigma_{z} M^{\bot}_{z}({\boldsymbol{\kappa}})|{\uparrow}\rangle|^{2} &= |M^{\bot}_{z}({\boldsymbol{\kappa}})|^{2}, \end{align} (4) \begin{align} |\langle{\uparrow}|\sigma_{y} M^{\bot}_{y}({\boldsymbol{\kappa}})+\sigma_{z} M^{\bot}_{z}({\boldsymbol{\kappa}})|{\downarrow}\rangle|^{2} &= |M^{\bot}_{y}({\boldsymbol{\kappa}})|^{2}. \end{align} (5) This means that the magnetic moments perpendicular to the neutron spins induce spin-flip (SF) scattering, while those parallel to the neutron spin account for non-spin-flip (NSF) scattering. This experimental configuration is referred to as \(P_{zz}\) configuration. To realize this experimental configuration in the instrument, we apply weak guiding fields along the z axis throughout the scattering path. The incident neutrons obtained by the Heusler monochromator are in the ↑ state, which can be flipped to ↓ state by the spin flipper. Finally, only the neutrons in the ↑ state are reflected by the Heusler(111) analyzer and observed at the detector. Therefore, the intensities measured when the spin flipper is on (\(I_{\text{ON}}\)) and off (\(I_{\text{OFF}}\)) correspond to the SF and NSF scatterings, respectively.

By measuring the SF and NSF intensities, we can distinguish magnetic structures having different orientations of the magnetic moments. For instance, a magnetic Bragg reflection from a collinear antiferromagnetic order with magnetic moments parallel to the z axis produces only NSF scattering signals, which are detected when the spin flipper is off, as shown in Fig. 2(a). As an another example, a cycloidal magnetic structure with magnetic moments on the xy plane leads to only SF scattering signals, which are observed as \(I_{\text{ON}}\), as shown in Fig. 2(b). As for a screw-type magnetic order with a q-vector on the xy plane, the NSF scattering is always observed owing to the finite z component of the Fourier-transformed magnetic moments. On the other hand, the SF intensity changes with the angle between the scattering vector and the q-vector, as we see in the next section.

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Figure 2. (Color online) Schematics showing possible SF and NSF scattering processes for magnetic Bragg reflections from (a) a collinear antiferromagnetic order with magnetic moments parallel to the z axis and (b) a cycloidal magnetic order with the magnetic moments in the xy plane.

Besides the \(P_{zz}\) configuration, we can also set the polarization direction of the incident neutrons to be parallel to the scattering vector by applying a weak horizontal field by the Helmholtz coil. In this case, both the y and z components of \(\boldsymbol{M}^{\bot}({\boldsymbol{\kappa}})\) are perpendicular to the neutron spins, and thus observed as the SF scattering. On the other hand, the nuclear scattering is always observed in the NSF channel, because it does not change the spin state of the neutrons. This configuration is referred to as \(P_{xx}\), which is used for separating magnetic scattering from nuclear scattering. The \(P_{xx}\) configuration is also used for determining sense of spin rotation in helical magnetic orders.^9–14)

These selection rules regarding the spin rotation is generalized to include the inelastic scattering. Thus polarized neutron inelastic scattering can separate magnon excitations from the phonon excitations, and can detect the sense of the spin precession in magnon excitations.¹⁵⁾

We applied the \(P_{zz}\) polarization analysis to determine the magnetic structure in the ground state of the centrosymmetric magnetic skyrmion host Gd₂PdSi₃.¹⁶⁾ This compound has a hexagonal crystal structure, in which magnetic Gd³⁺ ions are arranged to form triangular lattice layers stacked along the c axis. The recent resonant x-ray magnetic scattering study by Kurumaji et al. revealed that the ground state has screw-type magnetic modulations with the q-vectors of \((q,0,0)\), \((0,q,0)\), and \((q,-q,0)\), which are equivalent to each other under the threefold rotational symmetry about the c axis.¹⁷⁾ The modulation wavenumber q is approximately 0.14 (r.l.u.). In general, resonant x-ray scattering is sensitive not only to magnetic modulations but also charge and orbital modulations. We thus performed polarized neutron scattering measurements, which can exclusively detect magnetic moments, to verify the magnetic structure. For this experiment, we grew an isotope-enriched ¹⁶⁰Gd₂PdSi₃ single crystal to reduce the strong neutron absorption effect of natural Gd. The sample was mounted in a closed-cycle ⁴He refrigerator with the \((H,K,0)\) scattering plane. Other details of the experimental conditions are described in Ref. 16.

Figure 3(a) shows the polarized neutron scattering profile of a magnetic Bragg reflection at \((0,q,0)\) measured in the \(P_{zz}\) configuration. Both the SF and NSF are observed, which is consistent with the screw-type magnetic structure. However, the presence or absence of the magnetic moments along the q-vector direction was not concluded from this data, because the scattering vector is parallel to the q-vector, as shown in Fig. 3(b). We thus measured another magnetic reflection at \((-1+q,2,0)\), where the scattering vector and the q-vector are nearly perpendicular to each other, as shown in Fig. 3(d). If the spin component parallel to the q-vector exists, it should be observed in the SF channel. Figure 3(c) shows the observed profile, revealing that the SF scattering is absent at this reflection. This unambiguously shows that the spin spiral plane is perpendicular to the q-vector. We also note that the NSF and SF intensities at \((0,q,0)\) are not equal to each other. This means that the screw structure has ellipticity, which reflects the existence of the magnetic anisotropy for the magnetic moments on the Gd sites.

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Figure 3. (Color online) (a) Polarized neutron scattering profile of the magnetic Bragg reflection at \((0,q,0)\) in ¹⁶⁰Gd₂PdSi₃. (b) Schematic showing the directions of \({\boldsymbol{\kappa}}\), q-vector and guide field \(\boldsymbol{H}\) with respect to the xyz coordinates when measuring the reflection at \((0, q, 0)\). (c, d) Those for the reflection at \((-1+q,2,0)\). (e) Polarized neutron scattering profiles of the reflection at \((0,q,0)\) measured with the \(P_{xx}\) configuration. (f) Schematic showing the directions of \({\boldsymbol{\kappa}}\), q-vector and guide field \(\boldsymbol{H}\) for the \(P_{xx}\) measurement at \((0,q,0)\). All the data were measured at 2.5 K. The above profiles and schematics are replotted and redrawn using the data presented in Ref. 16.

We also note here that the polarization analysis with the \(P_{zz}\) configuration is also suited for studying magnetic structures when the sample has relatively strong neutron absorption. The ratio between NSF and SF intensities does not depend on the absorption effect, because they are measured through the same scattering path. We applied this technique to investigate magnetic structures in another centrosymmetric skyrmion host EuAl₄.¹⁸⁾

In the above discussion, the NSF intensities observed at the magnetic Bragg peaks in the \(P_{zz}\) configuration were interpreted as the presence of the c component of the magnetic modulation. However, the NSF signal can arise from nonmagnetic modulations such as lattice modulations associated with charge density waves. To exclude this possibility, we also measured the magnetic Bragg reflection at \((0,q,0)\) with the \(P_{xx}\) configuration, in which a weak horizontal guide field was applied along the x axis [see Fig. 3(f)]. As shown in Fig. 3(e), the intensities were observed only in the SF channel, demonstrating that the data observed in the \(P_{zz}\) configuration does not contain nonmagnetic scattering.

The longitudinal polarization analysis is applicable not only to elastic scattering but also inelastic scattering. For instance, we used the \(P_{xx}\) polarization analysis to study magnetic excitation of NiO, which exhibits a commensurate antiferromagnetic order with the q-vectors of \((\frac{1}{2},\frac{1}{2},\frac{1}{2})\) and its equivalents.¹⁹⁾ In this experiment, we employed the PG(002) monochromator, which the has vertical focusing option, to increase the neutron flux at the sample position. To polarize the incident neutron beam, a supermirror spin polarizer was installed between the monochromator and the sample. The scattered neutrons were analyzed by the Heusler(111) analyzer. Three co-aligned single crystals of NiO with the total mass of approximately 10 g were mounted in an Al cell, which was attached onto the cold head of a closed-cycle ⁴He cryostat.

The measurement was performed with the \(E_{f}\)-fix mode, in which we fixed the energy of the scattered neutrons to be 14.7 meV. Accordingly, we changed the energy transfer by varying the energy of the incident neutrons. At each energy transfer, we measured the signals from the sample until the monitor count reached \(3.96\times 10^{6}\), which took approximately 9.3 and 7.7 min when the energy transfer was 0 and 17 meV, respectively. Figure 4 shows the energy-transfer dependence of the NSF and SF intensities at \((\frac{1}{2},\frac{1}{2},-\frac{1}{2})\), at which a magnetic elastic peak is observed at the elastic condition. We observed that the inelastic scattering intensity develops above 5 meV, and that the signals are mostly observed in the SF channel. This indicates the presence of the magnon excitation with the wave vector of \((\frac{1}{2},\frac{1}{2},-\frac{1}{2})\).

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Figure 4. (Color online) Polarized inelastic scattering profile of the constant-Q scan at \((\frac{1}{2},\frac{1}{2},-\frac{1}{2})\) in NiO at 35 K.