Suppressed-moment 2-k order in the canonical frustrated antiferromagnet Gd2Ti2O7

In partially ordered magnets, order and disorder coexist in the same magnetic phase, distinct from both spin liquids and spin solids. Here, we determine the nature of partial magnetic ordering in the canonical frustrated antiferromagnet Gd2Ti2O7, in which Gd3+ spins occupy a pyrochlore lattice. Using single-crystal neutron-diffraction measurements in applied magnetic field, magnetic symmetry analysis, inelastic neutron-scattering measurements, and spin-wave modeling, we show that its low-temperature magnetic structure involves two propagation vectors (2-k structure) with suppressed ordered magnetic moments and enhanced spin-wave fluctuations. Our experimental results are consistent with theoretical predictions of thermal fluctuation-driven order in Gd2Ti2O7, and reveal that inelastic neutron-scattering measurements on powder samples can solve the longstanding problem of distinguishing single-k and multi-k magnetic structures.


INTRODUCTION
Geometrical frustration is a central theme of condensed-matter physics because it can generate exotic magnetic states. These states can typically be divided into spin liquids, in which frustration inhibits long-range magnetic order, and spin solids, in which perturbations to the dominant frustrated interactions drive magnetic order 1 . Defying this classification, some frustrated magnets exhibit partial magnetic order 2-8 -the coexistence of order and disorder in the same magnetic phase. Magnetic partial order can be driven by fluctuations in an "order-by-disorder" scenario 9 , by interactions between emergent degrees of freedom in spin-fragmented states [10][11][12] , or by proximity to a quantum critical point 13 , while structural partial order can drive the behavior of materials such as fast-ion conductors 14,15 , Pb-based photovoltaics 16,17 , and high-pressure elemental phases 18 . To benchmark theories of partially ordered states 9,19,20 , experimental determination of the nature of partial magnetic ordering in real materials is crucial.
Materials in which magnetic ions occupy a pyrochlore lattice of corner-sharing tetrahedra provide opportunities for realizing exotic frustrated states 21 . The frustrated pyrochlore antiferromagnet Gd 2 Ti 2 O 7 is a canonical partially ordered system in which magnetic Gd 3+ ions (S = 7/2, L = 0) undergo two phase transitions at T 1 = 1.1 K and T 2 = 0.75 K [22][23][24][25][26] . Both low-temperature (T ≪ T 2 ) and intermediate (T 2 < T < T 1 ) phases have magnetic propagation vector k ¼ ð 1 2 1 2 1 2 Þ 27,28 , which is uncommon among rare-earth pyrochlores and suggests exchange interactions extend beyond nearest neighbors 20 . Both magnetic structures are also partially ordered, as shown by the coexistence of magnetic Bragg and diffuse scattering in polarized-neutron scattering measurements 28 . The presence of partial ordering in a magnetic ground state at millikelvin temperatures is highly unusual 21,28 . However, the low-temperature magnetic structure of Gd 2 Ti 2 O 7 has not yet been conclusively solved, for two reasons. First, the large neutron absorption cross-section of natural Gd makes neutron-scattering experiments on large crystals challenging. Second, most experimental probes are unable to distinguish a magnetic structure that orders with a single k ¼ ð 1 2 1 2 1 2 Þ wavevector (1-k structure) from structures that superpose symmetry-equivalent k 2 h 1 2 1 2 1 2 i (2-k, 3k, and 4-k structures). This phenomenon is known as the "multi-k problem" 29 . Developing methodologies to solve the multi-k problem is of broad relevance, because multi-k structures often correspond to noncollinear spin textures with nontrivial topological properties, such as skyrmions, hedgehogs, and vortex crystals (see, e.g., refs. [30][31][32][33][34]. Figure 1 shows the 1-k, 2-k, 3-k, and 4-k magnetic-structure candidates and their space groups. All are partially ordered, but each has a different modulation of the ordered magnetic moment μ ord : the 1-k and 4-k structures have 25% interstitial paramagnetic sites, whereas 2-k and 3-k structures have more complicated μ ord modulations. It was proposed in ref. 28 that magnetic diffusescattering measurements in the low-temperature phase support a 4-k structure with cubic magnetic symmetry. However, this result was called into question by the observation of transverse magnetization in small applied magnetic fields H∥〈112〉 and 〈100〉, which is inconsistent with cubic symmetry 35,36 . Theory proposes that the ordering is driven by a subtle interplay of energetic and entropic terms 9 . For interaction parameters relevant to Gd 2 Ti 2 O 7 , 1-k and 4-k structures are degenerate, and the 4-k structure is stabilized at T 1 by thermal fluctuations in an "order-bydisorder" mechanism. The same model predicts a second phase transition at T 2 into a 2-k ground state 9 . These striking predictions have awaited a conclusive experimental test. In this article, we experimentally determine the nature of partial magnetic order in Gd 2 Ti 2 O 7 using neutron-scattering measurements of isotopically enriched powder and single-crystal samples, combined with symmetry analysis and spin-wave calculations. We show that the low-temperature state of Gd 2 Ti 2 O 7 is actually 2-k, in agreement with theory 9 but in contradiction with the interpretation of previous experiments 28 . Remarkably, inelastic neutronscattering (INS) experiments on a powder sample unambiguously distinguish between n-k structures, demonstrating that neither single crystals nor symmetry-breaking perturbations such as applied field are prerequisite to solve the multi-k problem.
Our paper is structured as follows. We first present single-crystal neutron-diffraction measurements in applied magnetic field that suggest non-cubic magnetic symmetry. We then perform a comprehensive symmetry analysis of candidate magnetic structures. Finally, we show that only a partially ordered 2-k structure is consistent with low-temperature INS data. We conclude by discussing the general implications of our study for understanding partially ordered states and solving multi-k structures.

RESULTS
Evidence for field-dependent magnetic domain population An example of the multi-k problem 29 is that measurements of magnetic Bragg intensities in zero applied magnetic field do not directly distinguish the structures shown in Fig. 1, due to spherical averaging in powder samples or averaging over degenerate magnetic domains in single crystals. To address this problem, we performed single-crystal neutron-diffraction measurements and applied a weak magnetic field Hk½110 to break the domain degeneracy at T = 0.07 K after zero-field cooling. Domains of the cubic 4-k structure are related only by translational and time-reversal symmetries and hence appear identical to neutrons, whereas domains of other n-k structures are related by rotational symmetries and hence have different diffraction patterns. A fieldinduced domain imbalance is therefore expected to leave the diffraction pattern unchanged only if the low-temperature structure is 4-k.
The magnetic field dependence of selected magnetic Bragg intensities is shown in Fig. 2. Magnetic Bragg peaks in the (hhl) plane disappear in small applied field 0.2 ≤ μ 0 H ≤ 0.5 T, while magnetic Bragg peaks outside the (hhl) plane become more intense. These observations are incompatible with the cubic 4-k structure, unless the applied field actually causes a magnetic phase transition rather than a domain imbalance. This scenario occurs in Er 2 Ti 2 O 7 37 , but appears unlikely in Gd 2 Ti 2 O 7 , in which no experimental signature of such a phase transition is observed in either specific heat 25 or torque magnetometry 35 measurements at base temperature and for μ 0 Hk 〈110〉 of less than 2 T. The fieldinduced uniform magnetization is also too small to suppress antiferromagnetic Bragg peaks significantly (M = 0.2 μ B for μ 0 H = 0.2 T 26,36 ) and so need not be considered. Moreover, our conclusion is unaffected by the significant neutron absorption of our single-crystal sample, because it is indicated by the field dependence of the peak intensities, and not by comparing the intensities of different peaks at the same field. These observations provide evidence against the cubic 4-k structure; however, they do not distinguish between non-cubic structures such as 1-k and 2-k candidates.

Magnetic symmetry analysis
To constrain further the low-temperature magnetic structure, we reinterpret published powder neutron-diffraction data measured at 0.25 K (from ref. 28 ) using a comprehensive symmetry analysis.   38 . Importantly, this analysis makes no assumptions about the nature of the phase transition at T 2 (continuous or discontinuous), because we do not assume an intermediate-temperature parent structure, but instead consider the paramagnetic space group as the parent symmetry. Previous studies have shown that nearly all the features of the data can be modeled using a single irrep, L 1+ , which yields the fit shown in Fig. 3a 27,28 . The L 1+ model is the best currently available and generates the four basic n-k structures shown in Fig. 1. Crucially, however, the ð 1 2 1 2 1 2 Þ magnetic Bragg peak observed in the 0.25 K data is absent for the L 1+ model (inset to Fig. 3a). This peak appears at T 2 28 , and is accompanied by a large specific-heat anomaly observed in several different samples 24,25 , suggesting that it is intrinsic and sample-independent. Hence, while the L 1+ irrep is the main contributor to the low-temperature magnetic structure, at least one other irrep must also be present. We therefore investigated the four possible combinations of L 1+ with one other irrep. Only the (L 1+ , L 3− ) and (L 1+ , L 3+ ) irrep pairs allow nonzero intensity of the ð 1 2 1 2 1 2 Þ peak and so are candidates. For each irrep pair, a given set of magnetic distortion-mode amplitudes 39,40 yields several magnetic structures with identical Bragg profiles. Accordingly, we treated the magnetic distortion-mode amplitudes as free parameters that we optimized against our diffraction data. Figure 3 compares fits to diffraction data for the single-irrep L 1+ model with the two-irrep (L 1+ , L 3− ) and (L 1+ , L 3+ ) models. The twoirrep models yield nonzero intensity of the weak ð 1 2 1 2 1 2 Þ peak-and slightly improved overall fits to the data-by increasing the number of free parameters as indicated in Fig. 3. Unfortunately, the inclusion of an additional irrep also increases the number of candidate structures from four to 32, all of which are consistent with the diffraction data. The magnetic space groups, mode amplitudes, and ordered-moment lengths for these structures are given in Supplementary Tables 1-3. To compress this search space, we apply a physical criterion that no site should have μ ord > 7.0 μ B -the maximum value for S = 7/2 Gd 3+ ions-which reduces the number of candidate structures to eight. Importantly, this criterion rules out all 3-k structures, so we do not consider these further. Four of the eight remaining structures are monoclinic (L 1+ , L 3− ) variants of the 1-k structure, in which paramagnetic spins order with a small μ ord 28 . However, these structures are disfavored by symmetry because the L 3− irrep is not a symmetry-allowed secondary order parameter (SOP) of the 1-k structure 39,40 . The other four structures comprise three 2-k and one 4-k. Of these, the second irrep is a symmetry-allowed SOP in only one candidate-a (L 1+ , L 3+ ) 2-k structure. This structure satisfies the moment-length criterion, unlike the single-irrep 2-k structure that was therefore discounted in previous work 28,41 . Hence, while the (L 1+ , L 3+ ) 2-k Inelastic neutron scattering and spin-wave analysis While powder-averaged Bragg scattering cannot directly distinguish n-k structures, this limitation need not apply to the powderaveraged excitation spectrum. We therefore turn to powder INS experiments with high energy resolution (≈0.025 meV FWHM). Figures 4a, b show background-subtracted powder INS data in the intermediate phase (0.77 K) and the low-temperature phase (~0.05 K), respectively. The magnetic scattering at 0.77 K is broad in Q and E. By contrast, the low-temperature data show two relatively flat modes at energies of approximately 0.06 meV and 0.17 meV. Additional INS measurements on a thin piece of our single crystal show that the background-subtracted single-crystal scattering integrated over ðh; k; lÞ ¼ ð0 ± 1; 0 ± 1; 3 2 ± 1 2 Þ qualitatively resembles the powder data (Fig. 4c).
We use linear spin-wave theory to test candidate magnetic structures against the low-temperature excitation spectrum. The minimal spin Hamiltonian for Gd 2 Ti 2 O 7 is given by where J 1 and J 2 denote Heisenberg exchange interactions between nearest neighbor and next-nearest neighbor spin pairs, which are denoted by angle brackets 〈 〉 and 〈〈 〉〉, respectively; D is a single-ion anisotropy term that arises from mixture of the excited 6 P 7/2 atomic state into the 8 S 7/2 ground state 42 ; and is the long-ranged magnetic dipolar interaction including contributions from all spin pairs i, j, whose energy scale D dip S(S + 1) = 0.84 K is fixed by S and the nearest-neighbor distance r nn = 3.60 Å 22 . We assume the literature value of J 1 S(S + 1) = 4.8 K 22 and include a small ferromagnetic J 2 = − 0.04J 1 to stabilize k ¼ ð 1 2 1 2 1 2 Þ ordering 9,20,22 ; however, the spin-wave spectrum is not strongly affected by choosing different small values of J 2 . Electron-spin resonance (ESR) experiments find an easy-plane anisotropy that favors spin alignment perpendicular to local 〈111〉 axes 42 ; we take DS 2 = 1.5 K to match our INS data optimally. As an independent check on these interaction parameters, we performed measurements of the paramagnetic (T > T 1 ) diffuse scattering using the D4 diffractometer at the ILL 43,44 , and find these are consistent with Monte Carlo simulations 45 of the paramagnetic diffuse scattering for our parameter values (Supplementary Fig. 1).
A prerequisite for spin-wave modeling is that the magnetic structure is a local energy minimum of Eq. (1). We therefore tested which of the eight candidate structures are proximate to energy minima by iteratively aligning each spin with its mean field and checking for stability via the absence of imaginary spin-wave modes. Two candidate structures-one 2-k and one 4-k-are locally stable; both derive from the (L 1+ , L 3+ ) irrep pair that yields the best fit to diffraction data (Fig. 3c). By contrast, all candidate 1-k structures with nonzero interstitial μ ord are unstable. Figure 4d shows the optimized (L 1+ , L 3+ ) 2-k structure, which resembles the refined structure ( Supplementary Fig. 2 Table  4). Compared to the L 1+ 2-k structure shown in Fig. 1b, it has canted magnetic moments with more uniform magnitudes, μ ord / μ B ∈ {6.1(1), 4.6(1), 6.2(1)} in a 1:1:2 ratio. However, the suppression of μ ord compared to its theoretical value of 7.0 μ B indicates partial ordering. Figure 4e-g shows the calculated spin-wave spectra for the L 1+ 4-k, (L 1+ , L 3+ ) 2-k, and L 1+ 1-k structures, respectively. The (L 1+ , L 3+ ) 4-k structure involves weak ordering of the 25% of spins that are paramagnetic in the L 1+ 4-k structure; however, the spectra for both these structures are very similar and we show only the latter. The 1-k and 4-k calculations strongly disagree with the 0.05 K experimental data. By contrast, the 2-k calculation reproduces well the experimental data, most importantly the prominent low-energy (~0.06 meV) mode that is absent for the other candidates. This key qualitative observation is not affected by different choices of interaction parameters in the physically appropriate regime J > D ≫ J 2 . The Q dependences of the low-E (0.03 ≤ E < 0.12 meV) and high-E (0.12 ≤ E ≤ 0.21 meV) modes shown in Fig. 4h confirm agreement between the 2-k calculation and the experimental data. We therefore conclude that the (L 1+ , L 3+ ) 2-k structure is the correct low-temperature model.

Investigation of spin-lattice coupling
The 2-k structure has orthorhombic symmetry (magnetic space group C a mcm), which is therefore expected to drive a crystallographic distortion via spin-lattice coupling. To test this hypothesis, we performed high-resolution powder neutron-diffraction measurements with ΔQ/Q = 5 × 10 −5 on the HRPD instrument at ISIS 46,47 . However, our data do not show visible peak splitting [ Supplementary Fig. 2], and while a statistically significant rhombohedral distortion could be refined, orthorhombic refinements were inconclusive due to their increased number of parameters. This suggests that spin-lattice coupling in Gd 2 Ti 2 O 7 is too weak to yield an observable orthorhombic distortion in our measurements. Further experiments, such as NMR or Mössbauer spectroscopy, may allow the expected distortion to be observed, but the requirement for <0.7 K temperatures presents experimental challenges.

DISCUSSION
Our experimental result that the low-temperature structure of Gd 2 Ti 2 O 7 is 2-k confirms state-of-the-art theoretical predictions 9 . However, it contradicts a previous experimental study 28 , which proposed a 4-k low-temperature structure based on analysis of low-temperature magnetic diffuse scattering. This study did not consider 2-k structures, because the single-irrep 2-k structure is . c Single-crystal INS data integrated over ðh; k; lÞ ¼ ð0 ± 1; 0 ± 1; 3 2 ± 1 2 Þ in the low-temperature phase (0.05 K, solid red diamonds) and intermediate phase (0.85 K, empty gray circles). Error bars represent one standard deviation. d Optimized 2-k magnetic structure. e Powder-averaged linear spin-wave theory (LSWT) calculation for the L 1+ 4-k structure. f Powder LSWT calculation for the optimized (L 1+ , L 3+ ) 2-k structure discussed in the text. g Powder LSWT calculation for the L 1+ 1-k structure. h Powder INS data and spin-wave calculations at 0.05 K. Experimental data are given in absolute intensity units by normalization to the nuclear Bragg profile. Gray crosses show the measured elastic line (|E| < 0.03 meV) with the incoherent scattering cross-section subtracted, empty blue diamonds show the measured low-E mode (0.03 ≤ E < 0.12 meV), and filled red circles show the measured high-E mode (0.12 ≤ E ≤ 0.21 meV). Spin-wave calculations for low-E and high-E modes are shown as blue and red lines, respectively, and have been vertically scaled by the same factor to match the experimental data.
unphysical and two-irrep 2-k structures were not identified. It also assumed that ordered sites contribute no diffuse scattering. This assumption is incorrect, however, because spin-wave scattering from ordered sites contributes to the energy-integrated diffuse intensity. In general, the magnetic Bragg intensity is proportional to ðμ ord =μ B Þ 2 and the total magnetic intensity is proportional to g 2 S(S + 1) for a spin-only ion. For a fully ordered system, μ ord /μ B = gS, so that spin-wave scattering would comprise 1/(S + 1) = 22% of the total intensity for S = 7/2. By contrast, partially ordered Gd 2 Ti 2 O 7 has μ ord /μ B < gS. Combining the refined values of μ ord with the total-moment sum rule, we estimate that lowtemperature diffuse scattering comprises 46(2)% of the total intensity. The inelastic scattering (E > 0.03 meV) shows clear Q dependence that is fully reproduced by the spin-wave calculation, whereas the elastic scattering (|E| < 0.03 meV) is weak and essentially flat away from Bragg peaks. Therefore, the lowtemperature magnetic diffuse scattering arises primarily from enhanced spin-wave fluctuations, and not from static spin disorder or quasielastic (paramagnetic) spin fluctuations.
Our neutron-scattering experiments, magnetic symmetry analysis, and spin-wave modeling reveal that the low-temperature magnetic structure of Gd 2 Ti 2 O 7 is 2-k, solving a longstanding problem in the field of frustrated pyrochlore oxides. The 2-k structure does not contain any entirely paramagnetic sites, and the increase in uniformity of the ordered-moment magnitudes likely stabilizes it over higher-symmetry 1-k and 4-k candidates. Nevertheless, the 2-k structure remains partially ordered, because magnetic-moment magnitudes are modulated and suppressed by up to 35% compared to their fully ordered value, while spin-wave fluctuations are enhanced. The magnetic ground state thus combines partial ordering with a multi-k structure-an exotic combination that, to the best of our knowledge, has not been observed in any other material with localized magnetic moments. Its existence in Gd 2 Ti 2 O 7 is especially remarkable because the large value of the spin quantum number and absence of orbital angular momentum for Gd 3+ ions suggests that quantum effects are weak. Instead, the enhancement of spin fluctuations may be related to the near-degeneracy of 2-k and 4-k states that results from the interplay of frustrated exchange interactions with magnetic anisotropy and long-ranged dipolar interactions.
While our study has focused on the low-temperature ground state, our results also shed light on the intermediate phase. Theory predicts that a 4-k structure is stabilized by thermal fluctuations at T 1 for our interaction parameters, before transitioning to a 2-k ground state at T 2 9 . This prediction is consistent with the experimental observation that the ð 1 2 1 2 1 2 Þ magnetic Bragg peak is absent in the intermediate phase 28 , because the irreps required to generate this peak are not symmetry-allowed SOPs for the 4-k structure. Moreover, our INS data in the intermediate phase show only broad inelastic features (Fig. 4a), consistent with the observation of quasielastic scattering in neutron-spin-echo experiments in the intermediate phase 48 . A reduction in spinwave scattering intensity is expected due to elevated temperature (T = 0.75T 1 ), which suppresses the refined value of the ordered magnetic moment to 2.65(3) μ B per ordered Gd at 0.77 K, for a 4-k model [ Supplementary Fig. 4]. However, the near-complete suppression of spin-wave intensity of the intermediate phase may hint at the presence of purely paramagnetic sites in a 4-k structure. Such sites would contribute a broad continuum of magnetic scattering, as observed in our INS data (Fig. 4a), and may also suppress propagating spin-wave excitations, similar to the effect of paramagnetic impurities 49 . Modeling these effects in the intermediate phase would provide a possible avenue for future work.
Perhaps the most general implication of our study is that singlek and multi-k structures can be unambiguously distinguished using inelastic neutron-scattering experiments on powder samples in zero applied field. In contrast to previous approaches [50][51][52] , this approach requires neither single crystal samples nor the application of external fields that explicitly break the symmetry. We anticipate that this approach will be widely applicable, and will help identify and understand multi-k structures where noncoplanar spin textures have nontrivial topologies [30][31][32][33][34] .

Single-crystal sample growth
A single crystal was prepared by the floating-zone image furnace method 53,54 and was 99.4% enriched with 160 Gd to minimize neutron absorption by 155 Gd and 157 Gd.

Neutron diffraction experiments
Single-crystal neutron-diffraction measurements were performed using the WISH time-of-flight diffractometer at ISIS 55 on a = 10 mm 3 piece cut from our single crystal. This sample was aligned with the (hhl) scattering plane horizontal, and loaded within a dilution refrigerator and magnet, with the magnetic field along the vertical ½110 direction. Details of the previously published powder neutron-diffraction measurements are given in ref. 28 .

Inelastic neutron-scattering experiments
Powder inelastic neutron-scattering experiment measurements were performed using the DCS spectrometer at NIST 56 with an incident wavelength of 8.0 Å. An approximately 0.2 g portion of the same isotopically enriched powder sample studied in ref. 28 was loaded in a Cu foil and wrapped around the circumference of a cylindrical Cu container, to minimize neutron absorption. Single-crystal inelastic neutronscattering experiments measurements were performed on a thin piece of our crystal using the CNCS instrument at ORNL with an incident wavelength of 9.0 Å. The samples were cooled using dilution refrigerators.

Magnetic structure refinement
Magnetic refinements to the powder neutron-diffraction data shown in Fig. 3 were performed using the Topas Academic software 57 . Lattice and profile parameters (peak shape, zero offset, and background) were obtained by Pawley refinement to the experimental magnetic Bragg profile, and the intensity scale factor was obtained from refinement to the nuclear Bragg profile. Refinement of an overall Debye-Waller parameter yielded a positive value, indicating that absorption is not a major problem for the powder sample. For each irrep or pair of irreps, the magnetic distortion-mode amplitudes were then refined to the experimental magnetic Bragg profile, keeping lattice, scale, and profile parameters fixed. To identify different sets of mode amplitudes with identical fit quality (degenerate solutions), multiple refinements were performed with different initial values of the mode amplitudes. For the (L 1+ , L 3− ) irrep pair, we identified two degenerate sets of mode amplitudes, A and B, given in Supplementary Table 1. For the (L 1+ , L 3+ ) irrep pair, we identified the four degenerate sets of mode amplitudes C-F given in Supplementary  Table 2. By combining these sets of mode amplitudes with the n-k structures using the Isodistort software 39,40 , we identified 32 candidate magnetic structures. The magnetic space groups, mode amplitudes, and ordered-moment magnitudes for these structures are given in Supplementary Table 3.

Spin-wave calculations
Spin-wave calculations shown in Fig. 4e-h were performed using the SpinW software package 58 . For each candidate magnetic structure, we generated a magnetic unit cell containing 2 × 2 × 2 crystallographic unit cells (128 ordered magnetic moments); this "supercell" approach is necessary to calculate spin-wave spectra of multi-k structures. The candidate structures were then optimized by iteratively aligning the orientation of each ordered magnetic moment with its mean field. The magnetic dipolar interaction was evaluated by direct summation to a maximum interatomic distance of 40 Å. We checked that the optimized magnetic structure and excitation spectrum remained stable to summations over longer distances (up to 100 Å), indicating that satisfactory convergence was achieved. The optimized and refined (L 1+ , L 3+ ) 2-k structures are given in Supplementary Table 4, and their calculated magnetic neutron-diffraction patterns match closely as shown in Supplementary Fig. 3. The calculated spin-wave spectra were powder averaged numerically using Fibonnaci spherical integration with 987 points on the sphere, multiplied by the squared magnetic form factor of Gd 3+ , and convolved with a Gaussian with FWHM of 0.025 meV to match the energy resolution of the experimental data.

DATA AVAILABILITY
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.