Introduction

High-temperature (high-Tc ) superconductivity in iron arsenides arises from electron or hole doping of their antiferromagnetic (AF) parent compounds1,2,3,4,5. The electron pairing, as well as the long range AF order, can arise from either quasiparticle excitations between the nested hole and electron Fermi surfaces6,7,8,9,10,11, or local magnetic moments12,13,14,15,16. In the itinerant picture, the superconducting pairing causes the opening of sign-reversed (s± wave) gaps in the respective hole and electron Fermi surfaces, as evidenced by a strong neutron spin resonance below Tc17,18,19,20,21,22,23,24,25. The observation of an in-plane momentum dependence of the resonance, with lengthened direction transverse to the AF wave vector QAFM (Figs. 1a and 1b), in single crystals of electron-doped BaFe2−x(Co,Ni) x As2 superconductors23,24,25 suggests a fully gapped s± state26, but the predicted momentum anisotropy of the spin excitations in optimally hole-doped materials has not been observed23. Here we report inelastic neutron scattering experiments on single crystals of superconducting Ba0.67Ka0.33Fe2As2 (Tc = 38 K). In addition to confirming the resonance19, we find that spin excitations in the normal state form longitudinally elongated ellipses that are rotated 90° to be along the QAFM direction in momentum space, consistent with the density functional theory (DFT) prediction23. On cooling below Tc , the resonance preserves its moment anisotropy as expected23,26, but the spin excitations for energies below the resonance unexpectedly become essentially isotropic in the in-plane momentum space and dramatically increase their correlation length. These results suggest that the superconducting gap structures in Ba0.67Ka0.33Fe2As2 are more complicated than those suggested from angle resolved photoemission experiments11.

Figure 1
figure 1

Schematic diagram of the reciprocal space probed, transport and neutron scattering data on Ba0.67K0.33Fe2As2.

(a) Real and reciprocal space of the FeAs plane. The light and dark As atoms indicate As positions below and above the Fe-planes, respectively. (b) Fermi surfaces at kz = 0 calculated using the tight-binding model of Graser et al.42 for electron and hole-doped BaFe2As2. The different colors indicate orbital weights around the Fermi surface with red = dxz , green = dyz and blue = dxy . (c) Temperature dependence of the in-plane resistivity ρ(T) shows the onset of superconductivity at Tc = 38 K. We find no resistivity anomaly that might be associated with structural or AF phase transitions above Tc . The inset shows the temperature dependence of the bulk susceptibility for a 1 mT in-plane magnetic field giving Tc = 38 K. (d) Elastic neutron scattering along the (H, 0, L) direction with fixed values of L = 0, 2 and 3 at 2 K, demonstrating that there is no static AF order in our samples. (e) Energy scans at the AF signal [Q = (1,0,0)] and background [Q = (1.3,0,0)] positions from 0.5 to 8.5 meV at 45 K and 2 K. (f) χ”(Q,ω), obtained by subtracting the background and removing the Bose population factor, clearly shows that a spin gap opens below E∼5.5 meV at 2 K. (g) Energy scans at Q = (1,0,1) and background Q = (1.3,0,1) positions from 0.5 to 8.5 meV at 45 K and 2 K. (h) χ”(Q,ω), obtained using the identical method as in (f), shows quite different behavior from the results in (f). Solid lines are guides to the eye. Data in (d) are from HB-3 and those in (e–h) are from cold triple-axis SPINS. The error bars indicate one sigma throughout the paper.

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Soon after the discovery of high-Tc superconductivity in iron arsenides, band structure calculations predicted the presence of two hole-type cylindrical Fermi surfaces around the zone center (Γ point) and electron-type Fermi surfaces near zone corners (the M points, Fig. 1b)6,7. The unconventional electron pairing in these materials can arise from either a repulsive magnetic interaction between the hole and electron Fermi surfaces6,7,8,9 or local AF moment exchange couplings10,12,14, both of which necessitate a sign change in their superconducting order parameters. In the simplest picture of this so-called “s± -symmetry” pairing state, nodeless superconducting gaps open everywhere on the hole and electron Fermi surfaces below Tc . One of the most dramatic consequences of such a state is the presence of a neutron spin resonance in the superconducting state, which occurs at the AF ordering wave vector QAFM with an energy at (or slightly less than) the addition of hole and electron superconducting gap energies () and a clean spin gap below the resonance17,18. The intensity gain of the resonance below Tc is compensated by the opening of the spin gap at energies below the resonance. The observed transverse momentum anisotropy of spin excitations in the electron-doped materials23,24,25 favors a fully gapped s± -symmetry superconductivity due to enhancement of the intraorbital, but interband, pair scattering process26 (see also Fig. 1b and supplementary information).

Results

In the initial neutron scattering experiments on powder samples of hole-doped Ba0.6K0.4Fe2As2 (Tc = 38 K), a neutron spin resonance near 14 meV was identified19. While the mode occurred near the magnitude of the AF wave vector QAFM as expected, the powder nature of the experiment meant one could not obtain detailed information on the energy and wave vector dependence of the excitations19 and therefore could not test the DFT prediction that the in-plane anisotropic momentum dependence of the spin excitations in hole-doped Ba0.6K0.4Fe2As2 should be rotated 90 degrees from that of the electron-doped materials (Fig. 1b)23. Since spin excitations can directly probe the nature the superconducting gap symmetry and phase information18,23,26, a determination of the electron-hole asymmetry in the spin dynamics of iron-arsenide superconductors is particularly crucial in view of the conflicting reports concerning the pairing symmetries by angle resolved photoemission spectroscopy (ARPES)11,27,28, penetration depth29 and thermo-conductivity meassurements30,31,32. Surprisingly, we find that the spin excitations at energies below the resonance in Ba0.67K0.33Fe2As2 (Tc = 38 K, Fig. 1c) have strong sinusoidal c-axis modulations around qz = 2πL/c (L = 1,3,…Fig. 1a and Fig. 2g), similar to their undoped parent compounds33 and display clean spin gaps in the superconducting state only for energies below ∼0.75 meV (Figs. 1e–1h and Fig. 2a–d). Furthermore, we discovered that the in-plane momentum dependence of the spin excitations in the normal state of Ba0.67K0.33Fe2As2 is elongated along the QAFM direction (Figs. 4b and 4d), thus confirming the DFT prediction23. Although superconductivity does not change the momentum anisotropy of the resonance as expected23,26 (Fig. 3 and Fig. 4c), the spin excitations at energies below the resonance dramatically increase the correlation length along the QAFM direction and become essentially isotropic below Tc (Fig. 2 and Fig. 4). Our work is not consistent with the large three-dimensional superconducting electronic gaps observed by ARPES11,27,28. On the other hand, given the strong gap variation over the Fermi surfaces27,28, one might imagine that the low-energy spin excitations arise from weak nodes or small gaps not directly seen by ARPES.

Figure 2
figure 2

Wave-vector and temperature dependence of the scattering at excitation energies below the neutron spin resonance energy in Ba0.67K0.33Fe2As2.

(a), (b) Q-scans at E = 5 meV along the (H,0,0) and (H,0,1) directions above and below Tc . While the scattering centered at (1,0,0) clearly vanishes below Tc , at (1,0,1) it persists and sharpens. (d) Q-scans at E = 1.5 meV along the (H,0,1) direction at 2 K and 45 K. The normal state peak at 45 K clearly survives superconductivity at 2 K. (c) Similar scans at E = 0.75 meV, where the normal state peak clearly disappears at 2 K, indicating the presence of a 0.75 meV spin gap. (e) Temperature dependence of the E = 3 meV scattering at the signal [Q = (1,0,0)] and background [Q = (1.3,0,0)] positions. The signal shows a clear suppression below Tc , while the background scattering goes smoothly across Tc . At low temperature, the signal merges into the background scattering, thus confirming the vanishing magnetic scattering. (f) Similar data at QAFM = (1,0,1) and Q = (1.4,0,1). While the signal also responds to Tc , the background scattering does not merge into the signal at low-temperature, thus indicating the continued presence of magnetic scattering. (g) Overall scattering in the (H,0,L) plane at E = 5 meV and 2 K. The data show a clear sinusoidal modulation along the L-direction. Data in (a,b,d) are from SPINS, (c) from MACS, (e) from BT-7, (f) from HB-3 and (g) from MACS. The horizontal bars indicate instrumental resolutions. The slight off-centering in peak positions for different experiments from the expected QAFM = (1,0,1) position is due to small sample mis-alignment problems. The scattering at (0.6,0,L) with L = 0,2.5 is of phonon or spurious origin.

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Figure 3
figure 3

Energy and wave vector scans and temperature dependence of the neutron spin resonance in Ba0.67K0.33Fe2As2.

(a) Energy scans from 2 to 19 meV at the signal [Q = (1,0,2)] and from 2 to 17 meV at the background [Q = (1.4,0,2)] positions above and below Tc . The background scattering has some temperature dependence below about 11 meV, probably due to the presence of phonons. This, however, does not affect the temperature dependence below 60 K. (b) χ”(Q,ω) at Q = (1,0,2) across Tc . (c) Constant-energy scans at E = 15 meV at 45 K and 2 K. The scattering shows a well centered peak at Q = (1,0,2) that increases dramatically below Tc , thus confirming that the mode is centered at commensurate positions. (d) The corresponding χ”(Q,ω) at Q = (1,0,2). (e) Comparison of the temperature difference (2 K minus 45 K) spectra for the neutron spin resonance at Q = (1,0,2) and (1,0,3). The resonance energy is weakly L-dependent. (f) Temperature dependence of the scattering at the resonance energy E = 15 meV and Q = (1,0,2). The scattering shows a clear order-parameter-like increase below Tc . Data in (a–f) are from HB-3. The horizontal bar indicates instrumental resolution.

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Figure 4
figure 4

The in-plane wave-vector profile of the spin excitations and temperature dependence of the scattering at energies below and near the resonance.

(a) The in-plane (H,K) magnetic scattering integrated from 5 meV to 10 meV at 5 K. The data were collected on the ARCS spectrometer with incident beam energy Ei = 25 meV, c-axis along the incident beam direction. (b) Identical scan at 45 K. In both cases, the L-integration range is from 1.2<L<2.1 which corresponds to an energy integration from 5 to 10 meV. (c) The in-plane (H,K) magnetic scattering profile covering the resonance energy integrated between 10 meV and 18 meV at 5 K on ARCS with Ei = 35 meV. (d) Identical scan at 45 K. Here the L-integration range is from 1.7<L<3 which corresponds to an energy integration from 10 to 18 meV. The color bars indicate intensity scale and dashed circle and ellipses indicate scattering profiles. (e) Constant-energy scans for E = 4 meV along the Q = (H,0,1) direction at 2 K and 45 K with linear background subtracted. The solid lines are Gaussian fits to the data. (f) Temperature dependence of the FWHM obtained by carrying out identical scans as in (e) at different temperatures. A clear reduction in the FHWM is seen below Tc . (g,f) Similar data obtained for E = 6 meV. Data in (a–d) are from ARCS, those in (e–h) are from MACS. See supplementary information for raw data at different temperatures.

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Figure 1c shows the transport and magnetic properties of our single crystals of Ba0.67K0.33Fe2As2 which indicate Tc = 38 K. Our samples were grown by using the self-flux method similar to an earlier report34. In previous neutron scattering and muon-spin-relaxation measurements on Sn flux grown Ba1−xK x Fe2As2 single crystals35,36, static AF order was found to phase separate from the superconducting phase due to K-chemical inhomogeneity. We have carried out systematic inductively coupled plasma atomic-emission spectroscopy analysis on our samples (size up to 15 mm*10 mm*1 mm) to confirm their chemical composition. Although our analysis also showed that K-concentrations vary slightly (up to 3%) for different batches, the superconducting properties near optimal K-doping are insensitive to such concentration variations and these samples have no static AF order coexisting with superconductivity at 2 K as shown by the neutron diffraction measurements in Fig. 1d. Independent nuclear magnetic resonance (NMR) measurements on these samples (W. P. Halperin, private communication) also confirmed the absence of the static AF order at 4 K and showed that the local magnetic field distribution as determined by the NMR linewidth is much narrower than that of the earlier K-doped BaFe2As2 samples37.

To determine the energy dependence of the imaginary part of the dynamic spin susceptibility χ”(Q,ω), we measured energy scans at the QAFM = (1,0,0) and (1,0,1) which correspond to spin excitations at AF wave vector transfers purely in the plane (L = 0) and L = 1, respectively (Fig. 1a), in the orthorhombic notation suitable for the parent compounds21,33. Figures 1e and1g show the raw data measured on the cold neutron triple-axis spectrometer above and below Tc . The corresponding dynamic spin susceptibilities, χ”(Q,ω), obtained by subtracting the background and correcting the Bose population factors (Figs. 1e and 1g), are shown in Figs. 1f and 1h for Q = (1,0,0) and (1,0,1), respectively. In the normal state (T = 45 K), χ”(Q,ω) at both wave vectors increases linearly with increasing energy. On cooling the system to T = 2 K (well below Tc ), a spin gap opens to E = 5 meV at Q = (1,0,0), while little change of the magnetic scattering occurs at Q = (1,0,1).

To confirm this conclusion, we carried out constant-energy scans at E = 5 meV and E = 1.5 meV. Figures 2a and 2b show the raw data at E = 5 meV across Q = (Η,0,0) and (Η,0,1), respectively. The normal state scattering shows broad peaks centered at Q = (1,0,L) with L = 0, 1. To estimate the in-plane spin-spin correlation lengths ξ, we fit the scattering profile with a Gaussian on a linear background using , where the full width at half maximum in Å−1. Fourier transforms of the Gaussian peak in reciprocal space give normal state in-plane spin-spin correlation lengths of and 23±4 Å for L = 0 and 1, respectively38. Upon entering into the superconducting state, the magnetic scattering vanishes for L = 0, while the spin correlation length for L = 1 increases to 52±5 Å (Fig. 2b). These results are consistent with Figs. 1e–1h and confirm that the low-temperature spin gaps are strongly L-dependent. Similar L-dependence of the spin gaps have also been found in electron-doped materials21,22,23. Therefore, while superconductivity suppresses the dynamic susceptibility at L = 0, it dramatically increases the in-plane spin correlation length and slightly enhances χ”(Q,ω) at L = 1. To see what happens at lower energies, we show in Figs. 2c and 2d constant-energy scans at E = 0.75 and 1.5 meV along the (Η,0,1) direction above and below Tc , respectively. While a clean spin gap is found at E = 0.75 meV in the superconducting state (Fig. 2c), there is clear magnetic scattering at E = 1.5 meV below Tc (Fig. 2d).

If the opening of a spin gap as shown in Figs. 1 and 2a–d is associated with superconductivity, one should expect a dramatic reduction in magnetic scattering below Tc . Figure 2e shows the temperature dependence of the E = 3 meV scattering at the L = 0 signal [Q = (1,0,0)] and background [Q = (1.3,0,0)] positions. While the signal scattering shows a clear suppression below Tc indicating the opening of a spin gap, the background scattering has no anomaly across Tc and merges into the signal below 15 K. The vanishing magnetic scattering at Q = (1,0,0) below Tc is confirmed by Q-scans along the (H,0,0) directions (see supplementary information). At Q = (1,0,1) and E = 3 meV, the temperature dependence of the scattering again shows a clear suppression below Tc (Fig. 2f), but in this case the background scattering at Q = (1.4,0,1) does not merge into the signal at 4 K. This is consistent with the constant-energy scans along the (H,0,1) directions (see supplementary information). While the scattering shows a clear peak centered at Q = (1,0,1) in the normal state, the identical scan in the superconducting state also has a peak that becomes narrower in width, indicating that the spin-spin correlation length at this energy nearly doubles from 23±3 Å at 45 K to 40±7 Å at 2 K.

Figure 2g shows the scattering profile in the [H,0,L] scattering plane at E = 5 meV and T = 2 K. The magnetic signal displays a clear sinusoidal modulation along the (1,0,L) direction with maximum intensity at odd L and no intensity at even L. At E = 3 meV, the normal state spin excitations also exhibit a sinusoidal modulation along the c-axis, while the effect of superconductivity is to open spin gaps near the even L positions.

Having established the behavior of the low energy spin dynamics across Tc , we now turn to the neutron spin resonance19 above the spin gap energy. In previous work on single crystals of electron-doped superconducting BaFe2–x(Co,Ni) x As2 (refs. 20–25), the neutron spin resonance was found to be dispersive along the c-axis and occurred at significantly different energies for L = 0 and 1 (ΔE ∼ 1–2 meV, refs. 21,23). Furthermore, the spin excitations display larger broadening along the transverse direction with respect to QAFM in momentum space without changing the spin-spin correlation lengths across Tc (refs. 23–25). To confirm the resonance in the previous powder measurements19 and determine its dispersion along the c-axis for hole-doped Ba0.67K0.33Fe2As2, we carried out systematic energy scans above and below Tc . Figure 3a shows the outcome at the signal [Q = (1, 0, 2)] and background [Q = (1.4, 0, 2)] positions for T = 45 K and 2 K. Figure 3b plots χ”(Q,ω) at Q = (1, 0, 2) across Tc . Inspection of Figs. 3a and 3b reveals that the effect of superconductivity is to suppress the low energy spin excitations and create a neutron spin resonance near 15 meV consistent with earlier work19. To test the dispersion of the resonance, we have also carried out similar measurements at Q = (1,0,3). Figure 3e compares the temperature difference plots for Q = (1,0,2) and Q = (1,0,3), which reveals little dispersion for the neutron spin resonance in Ba0.67K0.33Fe2As2. This is clearly different from that of electron-doped pnictides21,23. Figure 3f shows the temperature dependence of the scattering at Q = (1,0,2) and E = 15 meV. Consistent with earlier work19, we find that the intensity of the resonance increases below Tc like a superconducting order parameter. Figure 3c shows constant-energy scans at E = 15 meV along the (H,0,2) direction above and below Tc and Figure 3d plots the temperature dependence of χ”(Q,ω). These data indicate that the effect of superconductivity is to enhance the scattering at the AF wave vector without significantly changing the spin-spin correlation length.

Finally, to determine the in-plane wave-vector dependence of the spin excitations for Ba0.67K0.33Fe2As2, we carried out neutron time-of-flight measurements imaging the in-plane spin excitations at energies below (Figs. 4a and 4b) and at the resonance (Figs. 4c and 4d). In the normal state, the spin excitations exhibit anisotropy along the QAFM direction (Figs. 4b and 4d) precisely as predicted by the DFT calculation for hole-doped materials23. Our calculations suggest that the longitudinal elongation in spin excitations arises from intra-orbital, inter-band scattering from dyz orbitals between the hole- and electron- pockets (Fig. 1b and supplementary information). In the electron-doped case, the main contribution to the spin susceptibility for Q near QAFM = (1,0) comes from scattering processes between the blue dxy orbitals indicated by arrows, i.e., between the upper electron pocket and hole pocket around (1,1) in Fig. 1b. Since the (1,1) hole pocket is quite small in the electron doped case, the nesting wave vector is offset from QAFM = (1,0) by a finite ΔQy , which leads to the incommensurate peaks along the direction from (1,0) to (1,1), i.e., along the transverse direction. In the hole-doped case, dxy to dxy scattering is still strong, but now there is also a large contribution from scattering between the dyz orbitals which gives rise to the incommensurate peaks along the Qx direction and therefore the longitudinal anisotropy. The main scattering processes are again indicated by the arrows and occur between well-nested (green) regions on the outer hole pocket around Γ and the electron pocket around (1,0). Since the hole pocket is much larger than the electron pocket, the nesting wave-vector is offset by a finite ΔQx from QAFM = (1,0) which leads to the longitudinal anisotropy (see supplementary information for detailed calculations).

On cooling below Tc , the resonance exhibits an anisotropy profile that is the same as the spin excitations in the normal state (Figs. 4c and 4d), while the anisotropic scattering (Fig. 4b) found in the normal state for energies below the resonance becomes isotropic (Fig. 4a). To test if the changing scattering profile for energies below the resonance is indeed associated with superconductivity, we carried out detailed temperature-dependent wave-vector measurements along the Q = [H, 0, L] (L = 1) direction at E = 4, 6 meV, below and above, respectively, the spin gap of 5 meV (Fig. 1f). Figures 4e and 4g show constant-energy scans above and below Tc , which confirm earlier measurements at E = 1.5, 3, 5 meV (Fig. 2 and supplementary information). By fitting the profile with a Gaussian on a linear background, we can extract the FWHM of the scattering profile at different temperatures (see supplementary information). Figures 4f and 4h show the temperature dependence of the FWHM for E = 4 and 6 meV, respectively. In both cases, there is a dramatic drop in the FWHM and a corresponding increase in the spin-spin correlation length below Tc . Therefore, the effect of superconductivity is to induce a resonance and change the shape of the wave-vector dependent magnetic scattering from anisotropic to isotropic for excitation energies below the resonance.

Discussion

The novel spin excitations in hole-doped Ba0.67K0.33Fe2As2 we have discovered differ from the typical resonance behavior in electron-doped materials20,21,22,23,24,25 in two important ways. First, the development of superconductivity dramatically sharpens the spin excitations for energies below the resonance and changes their dispersion from anisotropic above Tc to isotropic in momentum space below Tc . Second, the normal spin excitations in hole-doped materials have a momentum anisotropy that is rotated 90 degrees from that of the electron-doped pnictides23,24,25. The observed elongated scattering along the longitudinal QAFM direction for hole-doped superconducting Ba0.67K0.33Fe2As2 is consistent with the fact that incommensurate spin excitations are observed in the longitudinal direction in pure KFe2As2 (ref. 39).

In principle, spin excitations in a paramagnetic superconducting material can stem from itinerant electrons, local spin moments, or a combination of both and can directly probe the superconducting gap symmetry17,18. Since there is heavy debate on whether the magnetism in iron pnictides arises from itinerant or localized electrons6,7,8,9,10,40,41, we will not address this issue here but instead focus on what the temperature dependence of the spin excitations tells us about the superconducting gap structures in hole-doped materials.

From the Fermi surface nesting picture, recent DFT calculations23 have successfully predicted that the oval shape of the normal state spin excitations in the electron-doped pnictides should rotate by 90 degrees in the hole-doped materials. This prediction can be understood from a detailed comparison of the Fermi surfaces in electron and hole doped materials (Fig. 1b) and is consistent with our observations (Figs. 4b and 4d). In the simplest s±-symmetry electron pairing model17,18, the opening of isotropic s-wave superconducting gaps in the hole and electron Fermi surfaces should suppress any low-energy spin excitations below the resonance. The observation of nearly zero-energy spin excitations at 2 K (E > 0.75 meV) and QAFM = (1, 0, L = odd) demonstrates that the superconducting gaps must be very small on some parts of the Fermi surfaces. These also must be linked by QAFM = (1, 0, 1) and simultaneously be present on the hole and electron Fermi surfaces with sufficient phase space to account for the observed low-energy spin excitations (Figs. 1–4). The dramatic increase in the spin-spin correlation length below Tc reveals that the low-energy spin excitations are also strongly affected by the opening of superconducting gaps on other parts of the Fermi surfaces. In principle, the opening of large superconducting gaps on these Fermi surfaces may reduce the scattering between the two different parts of the Fermi surfaces, which can increase the lifetime and hence the spin-spin correlation length as observed in our experiments.

However, such a pure itinerant picture is inconsistent with angle resolved photoemission experiments, where the three-dimensional superconducting gaps are large in all of the observed Fermi surfaces with a minimum gap energy of 4 meV (refs. 11,27,28), a value much larger than the observed spin excitations at L = 1. Therefore, the only way to understand the observed low-energy spin excitations near L = 1 in the itinerant picture is to assume that there are significant parts of the Fermi surface (Fermi arcs), yet to be observed by photoemission experiments11,27,28, that are essentially gapless. In this case, further calculations below Tc using a random phase approximation (RPA) and the three-dimensional five-orbital tight-binding model42 are needed to see what kind of gap structure is consistent with the observed change in the spin excitation linewidth. Therefore, while our results clearly indicate that the superconducting gap structures in Ba0.67K0.33Fe2As2 are more complicated than those suggested by the current ARPES measurements11,27,28, further theoretical work is necessary to understand the temperature dependence of the spin excitations.

Methods

Single crystals of Ba0.67K0.33Fe2As2 were grown by the self-flux method43. The resistivity and magnetic susceptibility were measured by PPMS and SQUID from Quantum design. A 6.12 mg crystal cut from a big piece used for neutron scattering shows 100% superconducting volume fraction, see the inset in Fig.1c. Many others from different batches show very similar properties. The position in reciprocal space at wave vector Q = (qx , qy , qz ) Å−1 is labeled as (H, K, L) = (qxa/2π, qyb/2π, qzc/2π) reciprocal lattice units (rlu), where the tetragonal unit cell of Ba0.67K0.33Fe2As2 has been labeled in orthorhombic notation with lattice parameters of a = b = 5.56 Å, c = 13.29 Å (ref. 33). Our neutron scattering experiments were carried out on the HB-3 thermal neutron three-axis spectrometer at High Flux Isotope Reactor and the ARCS time-of-flight chopper spectrometers at Spallation Neutron Source, Oak Ridge National Laboratory and on the BT-7 thermal, SPINS and MACS cold neutron triple-axis spectrometers, at the NIST Center for Neutron Research. For the HB-3, BT-7, SPINS, MACS neutron measurements, we fixed the final neutron energies at Ef = 14.7 meV, 13.5 meV, 5.0 meV and 5.0 meV, respectively. For triple-axis measurements, we co-aligned 4.5 grams of single crystals on aluminum plates. For time-of-flight measurements on ARCS, we co-aligned 60 pieces of single crystals with a total weight of 20 grams on several aluminum plates. In both cases, the in-plane and c-axis mosaics of aligned crystal assemblies are about 3° and 6.5°, respectively.