Introduction

Reciprocal expression has been traditionally used for the spin excitations as a magnon dispersion, which can be measured directly by inelastic neutron scattering (INS) with the energy and the momentum transfers in various magnets. The magnon (spin wave) dispersion has been a unique description of the elementary excitation. Meanwhile, recent developments of INS spectrometers at pulsed neutron sources provide extremely high signal to noise ratio above a few hundreds. This low background enables us to measure the weak magnetic diffuse scattering in a wide Q-range. So far, total scattering spectrometers in the facilities have been used for local structure analysis that utilizes a wide Q-range of the scattering pattern by the time-of-flight (TOF) method in various liquid and amorphous materials, and even crystalline solids1. Recently, the research has been extended to the inelastic scattering measurement of the dynamic pair-density function2. The time-dependence has also been discussed by Fourier transform on the energy axis as well. Meanwhile, the magnetic pair-density function analysis has also been applied to several materials such as a spin-glass system3. One of the pioneering works is the spin–spin correlation study on an amorphous alloy (Mn0.4Ni0.6)75P16B6Al3,4 where polarized neutron scattering has been carried out using the scattering pattern up to Qmax = 7 Å-1. The magnetic pair distribution function analysis is formulated in Ref.5. The spin density background is evaluated in Ref.6. As for the dynamics, the dynamic atomic pair-density function analysis has been performed in amorphous solids like SiO2 glass7,8. The method is also applied to a ferroelectric material9. After these pioneering works, the method has been widely used in liquid and glassy materials2.

Here, we measured the magnetic excitation of a powder sample of FeTiO3 by TOF INS. The crystal and magnetic structures of FeTiO3 are depicted in Fig. 1. FeTiO3 powder sample was selected because the magnon dispersions with optical modes are well studied10 and the Fe2+ magnetic moment is relatively large. The dynamic magnetic structure factor was analyzed by the dynamic magnetic pair-density function (DymPDF) analysis. The DymPDF analysis program is implemented in the ‘Utsusemi’ visualization software11.

Figure 1
figure 1

Magnetic structure of FeTiO3. Every other Fe atom (brown ball) slightly buckles along the c-axis. The spin (red arrow) directs along the c-axis with a small tilting of about 2°10. The nearest-neighbor bonds (brown rods) between Fe spins with r = 3.05 Å form a honeycomb lattice in the ab-plane.

Full size image

Dynamic magnetic pair-density function analysis

The dynamic pair-density function of the lattice DL(r, E) is calculated based on the following equation7,8.

$${D}_{L}\left(r,E\right)=\frac{2}{\pi }\int Q\left[\frac{{S}_{L}\left(Q,E\right)-{S}_{s0}\left(Q,E\right)}{B{Q}^{2}\mathrm{exp}\left(-{Q}^{2}\langle {u}^{2}\rangle /3\right)\frac{\langle n\left(E\right)+1\rangle }{E}}\right]\mathrm{sin}(Qr)dQ$$
(1)

where SL(Q, E) is the lattice part of the dynamic structure factor at the scattering vector Q and the energy E, and the self-dynamic structure factor Ss0(Q, E) can be written as follows.

$${S}_{s0}\left(Q,E\right)=B{Q}^{2}exp\left(-{Q}^{2}\langle {u}^{2}\rangle /3\right)\frac{\langle n\left(E\right)+1\rangle }{E}$$
(2)

where B is a constant; <u2> is the mean-square atomic displacement; n(E) is the Bose factor at an energy E.

In Eq. (1), DL(r, E) is multiplied by E based on the energy dependence of the phonon intensity. In the present case of magnetic excitation, we can omit the energy dependence in the following Eq. (3). In contrast to the phonon, the magnetic signals sharply decrease with increasing the scattering vector Q by the squared magnetic form factor fM(Q)2. Because of the effect, the magnetic signals become negligibly small above Q = 5 Å−1 for 3d transition metals such as Fe2+, as shown in Fig. 2. Therefore, the maximum Q is effectively limited to about 5 Å−1 for the DymPDF analysis, resulting in the limitation of the r-resolution. In other words, the outer orbital with magnetic moments such as 3d spins spread to some extent. Because of this reason, the maximum Q was set to be 4.3–5 Å−1 in the present DymPDF analysis. This small Qmax results in the suppression of the phonon contribution in the analysis.

Figure 2
figure 2

Squared magnetic form factor of Fe2+ as a function of Q (solid black line) and window function of Eq. (5) with Qmax = 5 Å−1 (broken red line).

Full size image

The dynamic magnetic structure factor SM(Q, E) of FeTiO3 was Fourier transformed to the dynamic magnetic pair-density function DM(r, E) based on the following Eq. (3), after being divided by Bose factor n(E) + 1 and squared magnetic form factor fM(Q)2 of Fe2+.

$${D}_{M}\left(r,E\right)=\frac{2}{\pi }{\int }_{{Q}_{min}}^{{Q}_{max}}Q\left[\frac{{S}_{M}\left(Q,E\right)}{\langle n\left(E\right)+1\rangle }-1\right]w\left(Q\right)\mathrm{sin}(Qr)dQ$$
(3)

where SM(Q, E) is obtained as follows.

$$\frac{{S}_{M}\left(Q, E\right)}{\langle n\left(E\right)+1\rangle }-1=\frac{S\left(Q, E\right)-{S}_{s}\left(Q, E\right)-{S}_{L}\left(Q, E\right)-{S}_{0}\left({Q}_{max}, E\right)}{{f}_{M}^{2}\left(Q\right)\langle n\left(E\right)+1\rangle }$$
(4)

The window function w(Q) is written as,

$$w\left(Q\right)=\frac{{Q}_{max}}{\pi Q}\mathrm{sin}\left(\frac{\pi Q}{{Q}_{max}}\right)$$
(5)

This window function reduces the error by suppressing the difference of SM(Q, E)/<n(E) + 1> from unity near Qmax. The window function effect can be found in Suppl. Note 1. Figure 2 shows an example of the window functions with Qmax = 5 Å−1.

Here, the incoherent dynamic structure factor SS(Q, E) and the phonon dynamic structure factor SL(Q, E) in Eq. (4) are approximated into one Q2 function as follows.

$${S}_{S}\left(Q, E\right)+{S}_{L} \left(Q, E\right) \sim A{\left(Q-{Q}_{max}\right)}^{2}$$
(6)

where A is a constant parameter obtained by a least-squares fitting to minimize the integral of [S(Q, E) − A(Q − Qmax)2 − S0(Qmax, E)] from Qmin to Qmax. This subtraction corrects the oscillation balance of SM(Q, E)/<n(E) + 1 > around unity, leading to DM(0, E)/<n(E) + 1 > = 0. The S0(Qmax, E) in Eq. (4) is determined from the following Eq. (7) to converge the DM(r, E).

$$S\left({Q}_{max}, E\right)-{S}_{s}\left({Q}_{max}, E\right)-{S}_{L}\left({Q}_{max}, E\right)-{S}_{0}\left({Q}_{max}, E\right)=0$$
(7)

The combination of Eqs. (5–7) protects the divergence of the dynamic magnetic pair-density function DM(r, E). Equation (6) corresponds to second-degree polynomial correction in the PDF analysis12. The detailed processes are described in Suppl. Note 2.

Local spin dynamics of FeTiO3 in real space

Bose-factor corrected dynamic structure factors S(Q, E)/<n(E) + 1 > of FeTiO3 measured at T = 8 and 200 K with Ei = 46 and 95 meV are shown in Fig. 3. The Q–E range of S(Q, E) with Ei = 46 meV was limited for the analysis up to 5 Å−1 and 22 meV in Fig. 3a,b, where only magnetic signals are visible. The spin excitation was clearly observed at 8 K below 3 Å−1 and 16 meV in Fig. 3a. A gap-like feature is observed at E = 11 meV, which can be attributed to the zone boundary gap between 9 and 12 meV at (0.5, 0, 0)10,13. The Q–E range of S(Q, E) in the case of Ei = 95 meV spreads up to 12 Å−1 and 80 meV (Fig. 3e,f). However, the magnon excitations are not well-defined due to the low resolution. Instead, phonon excitations become visible above 5 Å−1 and up to about 60 meV. The results support our setting of Qmax < 5 Å−1 for the DymPDF analysis.

Figure 3
figure 3

(a) Bose-factor corrected dynamic structure factor S(Q, E)/<n(E) + 1 > of FeTiO3 measured with Ei = 46 meV at T = 8 K. (b) at T = 200 K. (c) Dynamic magnetic pair-density function DM(r, E) of FeTiO3 at T = 8 K. (d) at T = 200 K. (e) Bose-factor corrected dynamic structure factor S(Q, E)/<n(E) + 1 > of FeTiO3 measured with Ei = 95 meV at T = 8 K. (f) at T = 200 K. Low-energy spiky peaks in S(Q, E)/<n(E) + 1 > are originated from strong Bragg peaks, which extend with the energy-resolution.

Full size image

As shown in Fig. 1, Fe2+ (3d6, S = 2) spins in FeTiO3 order ferromagnetically within the honeycomb planes, which are then stacked antiferromagnetically along the c-axis. As all spins in a single honeycomb plane are oriented along the same direction, their fluctuations in phase with each other will appear as positive peaks in the DymPDF. We thus expect positive peaks at r = 3.0 Å and = 5.1 Å which are the nearest-neighbor and the next-nearest-neighbor Fe2+–Fe2+ bond distances within honeycomb planes, respectively. In the meanwhile, the second shortest Fe2+–Fe2+ bond in FeTiO3 is at r = 4.0 Å which is vertically connecting two adjacent honeycomb planes. As the interplanar spin coupling is antiferromagnetic, the low-energy fluctuations at this distance will appear as negative peaks in the DymPDF. The low-energy parts in Fig. 3c indeed show such peaks with expected sign changes between r = 3.0 Å and 5.1 Å. Figure 3d shows, however, that these dynamic spin–spin correlations of the ordered magnet at T = 8 K apparently are absent at T = 200 K well above the Néel temperature. Nevertheless, we notice weak pair-density correlations remaining in the paramagnetic phase at 200 K (Fig. 3d) at the same positions, suggesting dynamic atomic correlations via phonon vibrations. The positive phonon peak at r = 2.1–2.2 Å above 16 meV can be attributed to the Fe–O bonds, whereas the negative phonon peak at r = 3 Å above 16 meV may be the Fe–Ti bonds due to the scattering length sign change between Fe and Ti.

Figure 4 shows the comparison between the present DM(r, E) and the simulated DMcal(r, E)/<n(E) + 1 > based on the exchange parameters reported in Ref.10. Figure 4a shows a wide-energy DymPDF DM(r, E) pattern combined from two data sets measured with Ei = 18 (E < 12 meV) and 46 meV (E > 12 meV). The magnon mode transition at r = 3 Å is observed at about 10 meV. The powder averaged SMcal(r, E)/<n(E) + 1 > pattern calculated by SpinW14 is shown in Fig. 4c. The calculated original magnon dispersions are shown in Fig. 4d. As expected, the present DM(r, E) (Fig. 4a) is reasonably consistent with the simulated DMcal(r, E) (Fig. 4b). The energy-dependences of DM(r, E) at r = 3, 4, and 5 Å are shown in Fig. 4e, which roughly correspond to the first, second, and third nearest-neighbor bond lengths, respectively. Interestingly, they exhibit sign changes around 10, 8, and 7 meV, respectively, suggesting the reversal of the spin–spin correlations. These sign changes can be interpreted as the changes in relative phases as the magnon dispersions approach the Brillouin zone boundary. At r = 3 Å, for instance, the low-energy part accounts for the cooperatively oscillating acoustic mode without a phase difference between a pair of nearest-neighbor spins as illustrated in Fig. 4g. Meanwhile, the sign change around 10 meV can be interpreted as the magnon mode attaining π-phase difference illustrated in Fig. 4h. This sign change with increasing energy reminds us of the Ni phonon mode change from acoustic to optical modes observed by DyPDF9. Naively, the transition energy of 10 meV is expected to correspond to the dispersion change from acoustic to optical magnon dispersions in Fig. 4d. The highest energy of the acoustic mode reaches 12 meV, whereas the lowest energy of the optical mode extends to 7 meV. The 10 meV is near the center of the overlapping modes. Each nearest neighbor bond length may correspond to a typical energy in the energy range from 7 to 12 meV. However, the present DymPDF represents local magnon mode different from the magnon dispersions in Fig. 4d. In a real space image, even the acoustic mode can show π-phase-shift at the zone boundary in the acoustic magnon dispersion. Therefore, the magnon mode transition does not simply correspond to the dispersion change from acoustic to optical ones. Anyway, the real space profile was well reproduced by the spin-wave dispersions, suggesting a close correlation between them.

Figure 4
figure 4

(a) Combined DymPDF DM(r, E) pattern at T = 8 K measured with Ei = 18 (E < 12 meV) and 46 meV (E > 12 meV). The magnon mode transition at r = 3 Å is observed at about 10 meV. (b) Simulated DMcal(r, E) pattern based on the SpinW powder averaged magnon dispersions14. (c) Powder averaged SMcal(Q, E)/<n(E) + 1 > calculated by SpinW based on the exchange parameters in Ref.10. (d) Magnon dispersions calculated by SpinW. (e) Energy dependence of DM(r, E) at r = 3, 4, and 5 Å integrated with r-width of 1 Å (Ei = 18 meV). The energy-resolution ΔE is shown on the left-side bottom. f Energy dependence of DMcal(r, E) calculated by SpinW at r = 3, 4, and 5 Å. (g) Schematic magnon mode with no-phase-shift between the nearest-neighbor spins. (h) Schematic magnon mode with π-phase-shift between the nearest-neighbor spins.

Full size image

In addition, there are some noticeable anomalies in Fig. 4e. The first kink appears as a peak or a bottom at 5.5–6.5 meV for these bond lengths. The second kink is at about 9 meV. The third kink is at about 10.5 meV clearly observed for all the bonds. The first kink energy agrees with the gap of one acoustic magnon mode. The second kink energy corresponds to the bottom energy of the optical magnon mode. The third kink energy is the crossing point of the acoustic and optical modes. The SpinW simulation roughly reproduced these kinks as shown in Fig. 4f. The reason why these curves do not match exactly with the simulation requires further study. However, this method provides a novel opportunity, for example, to observe the local spin dynamics of magnetic nanoclusters, which, so far, had been difficult to study.

In summary, we successfully observed the local magnon mode change on FeTiO3 in real space by the dynamic magnetic pair-density function (DymPDF) analysis. This real space image provides a novel possibility to study local spin dynamics in addition to the nearly static magnetic pair-density of various magnetic systems at low energy. The latter is particularly important because nuclear scattering components can be naturally removed by limiting Q-range. It means that a tedious process to subtract nuclear Bragg peaks from the measured pattern can be avoided. Possible suitable research by DymPDF would be molecular cluster magnets, nanomagnets, local magnetic clusters, and magnetic short-range orders in addition to magnetic amorphous alloys and magnetic quasicrystals. The local magnetic clusters are known to appear in spin frustration systems as a spin molecule in MgCr2O415. In a superconducting state, spin resonance mode appears in an unconventional superconductor such as CeCoIn516. By DymPDF analysis, it becomes possible to check whether spin-singlet or spin-triplet state is realized in the superconducting state, in addition to the correlation length that can be compared with the superconducting coherence length. Magnetic percolation network such as fracton17 or Mott transition by anion exchange18 may also be a suitable system to study a decay of the magnetic pair-density as a function of a bond length in the network. DymPDF analysis can also be used to study the spin dynamics of a crystalline magnet that can be synthesized only in powder form. The application is not limited to these systems. There can be vast other applications by the DymPDF analysis.

Methods

A FeTiO3 powder sample of 99.5% purity was purchased from Kojundo Chemical Lab. Co., Ltd. The X-ray diffraction powder pattern did not include any impurity phase. The magnetic structure is shown in Fig. 1. The space group is R-3 (#148) with lattice parameters of a = 5.087 Å and c = 14.092 Å (Hexagonal)19. FeTiO3 exhibits an antiferromagnetic phase transition at TN = 58.0 K19. The powder with a weight of 5.58 g was put into an aluminum cell. Inelastic neutron scattering measurements were carried out at the chopper spectrometer 4SEASONS with a multi-Ei option in J-PARC with a proton beam power of 600 kW20. The used incident energies were 17.8, 46.0, and 94.7 meV under a Fermi chopper frequency of 300 Hz. The energy resolutions at E = 0 for Ei = 17.8, 46.0, and 94.7 meV are 0.67, 2.48, and 7.43 meV, respectively. The dynamic structure factor S(Q, E) was subtracted by empty aluminum cell data measured under the same condition. The detector efficiency depending on Ef was corrected in the ‘Utsusemi’ software11. Crystal and magnetic structures of FeTiO3 were drawn by the software ‘VESTA’21. The magnon dispersions in Fig. 4c,d are calculated by ‘SpinW’ software based on the linear spin wave theory with the Holstein–Primakoff approximation14.