Frustrated magnetism of the triangular-lattice antiferromagnets α-CrOOH and α-CrOOD

Powder samples of α-CrOOH and α-CrOOD were synthesized using the high-temperature hydrothermal technique at an autogenic pressure of ⩽ 20 MPa [61, 62]. A 50 mL stainless steel liner was charged with 15 mmol Na₂CrO₄·4H₂O (> 99%, Adamas Reagent Co., Ltd), 22.5 mmol CHNaO₂ (> 98.5%, Shanghai Titan Scientific Co., Ltd), and 10 ml of high-purity H₂O (for α-CrOOH) or D₂O (⩾ 99.9%, Shanghai Titan Scientific Co., Ltd, for α-CrOOD). The liner was capped and mounted into a stainless steel pressure vessel. The vessel was heated to 350 °C at a rate of 1 °C min⁻¹, and the temperature was maintained for 20 hours and then cooled down to room temperature at a rate of −0.5 °C min⁻¹. A green-grey powder (∼ 1.1 g, the yield of ∼ 85%) of α-CrOOH(D) was obtained after rinsing, filtration, and drying. The phase purity was confirmed by both x-ray and neutron diffraction measurements (appendix A).

Magnetization (susceptibility) of α-CrOOH(D) was measured down to 1.8 K using powder samples of ∼ 40 mg, by a magnetic property measurement system (MPMS, Quantum Design, up to 7 T) and by a physical property measurement system (PPMS, Quantum Design, up to 14 T). The specific heat (2 ⩽ T ⩽ 200 K) was measured at 0, 2, and 4 T using a dry-pressed disk of powder (∼ 4 mg) in a PPMS. N grease was used to facilitate thermal contact between the sample and the puck, and the sample coupling was better than 99%. The specific-heat contributions of the grease and puck in different applied fields were measured at first and subtracted from the data.

Both steady-high-field (high-frequency) and x-band ESR measurements (∼ 30 mg of samples) were performed on the Steady High Magnetic Field Facilities, Chinese Academy of Sciences. The high-field ESR data down to 2 K were collected by sweeping the steady magnetic field from 0 up to 20 T at selected frequencies of f = 214, 230, and 331.44 GHz, respectively. The x-band (f ∼ 9.4 GHz) ESR measurements were carried out using a Bruker EMX plus 10/12 continuous wave spectrometer. The measured ESR signals/modes were fitted using the derivative Lorentzian function on a powder average,

$$\begin{equation}\frac{\mathrm{d}\hspace{1.3pt}{A}_{\text{bs}}}{{\mu }_{0}\mathrm{d}H}={B}_{\text{ck}}+\frac{16{A}_{0}\omega }{3\pi }\left\{\frac{2{\mu }_{0}\left({H}_{\text{res}}^{ab}-H\right)}{{\left[4{\mu }_{0}^{2}{\left({H}_{\text{res}}^{ab}-H\right)}^{2}+{\omega }^{2}\right]}^{2}}+\frac{{\mu }_{0}\left({H}_{\text{res}}^{c}-H\right)}{{\left[4{\mu }_{0}^{2}{\left({H}_{\text{res}}^{c}-H\right)}^{2}+{\omega }^{2}\right]}^{2}}\right\},\end{equation} \tag{ 1 }$$

where A₀ and B_ck are the integrated intensity and background, ω is the full width at half maximum (FWHM), ${\mu }_{0}{H}_{\text{res}}^{ab}=hf/\left({\mu }_{\mathrm{B}}{g}_{ab}\right)$ and ${\mu }_{0}{H}_{\text{res}}^{c}=hf/\left({\mu }_{\mathrm{B}}{g}_{c}\right)$ are the resonance fields along the ab plane and c axis, respectively.

The neutron diffraction experiments were conducted with the high-intensity multi-section neutron powder diffractometer, Fenghuang [65], using powder samples of ∼ 2.2 g, at China Mianyang Research Reactor. The incident neutron wavelength of 1.5925 Å was determined by the monochromator, Ge (5 1 1). The INS experiment was performed using α-CrOOD of ∼ 6 g on the BOYA multiplexing cold neutron spectrometer stationed at the China Advanced Research Reactor. Utilizing the CAMEA concept [66], BOYA simultaneously probes five fixed final energies (E_f) of 3.0 (energy resolution of σ = 0.11 meV), 3.5 (σ = 0.15 meV), 4.0 (σ = 0.18 meV), 4.5 (σ = 0.21 meV) and 5.0 meV (σ = 0.26 meV), and 17–34 scattering angles spanning 120° in the horizontal plane. Highly oriented pyrolytic graphite crystals were used as the monochromator and analyzers. Cryogenic Be filters were applied after the sample to cut off energies higher than 5 meV.

We carried out FLD calculations [67, 68] for the susceptibility, specific heat, and ESR linewidth on the 12-site S = 3/2 TAF clusters with periodic boundary conditions (PBC) (appendix B) [69]. Thereupon we simultaneously fitted this model to the susceptibility and ESR linewidth measured above 60 K by minimizing the following function,

$$\begin{equation}{R}_{\text{p}}=\sqrt{\frac{1}{N}\sum\limits _{j}{\left(\frac{{X}_{j}^{\text{obs}}-{X}_{j}^{\text{cal}}}{{\sigma }_{j}^{\text{obs}}}\right)}^{2}},\end{equation} \tag{ 2 }$$

where N, ${X}_{j}^{\text{obs}}$ and ${\sigma }_{j}^{\text{obs}}$ are the number of the data points, the measured value and its standard deviation, respectively, whereas ${X}_{j}^{\text{cal}}$ is the calculated value.

The powder INS spectra were calculated for the S = 3/2 TAF model using the Spinw-Matlab code based on the linear spin-wave theory [70]. An average of the results calculated along 10 000 random orientations was performed at each $\left\vert \mathbf{Q}\right\vert$ for the dynamic structure factor ${S}^{\perp }\left(\left\vert \mathbf{Q}\right\vert ,E\right)$. The INS spectra were calculated by the convolution of the instrumental resolution (Gaussian) function (σ = 0.25 meV) and ${S}^{\perp }\left(\left\vert \mathbf{Q}\right\vert ,E\right)$ as ${I}^{\perp }\left(\left\vert \mathbf{Q}\right\vert ,E\right)={\left\vert f\left(\left\vert \mathbf{Q}\right\vert \right)\right\vert }^{2}\int {S}^{\perp }\left(\left\vert \mathbf{Q}\right\vert ,{E}^{\prime}\right)\frac{\mathrm{exp}\left[-4\enspace \mathrm{ln}\enspace 2{\left(E-{E}^{\prime}\right)}^{2}/{\sigma }^{2}\right]}{\sigma \sqrt{\pi /\left(4\enspace \mathrm{ln}\enspace 2\right)}}\enspace \mathrm{d}{E}^{\prime}$, where ${\left\vert f\left(\left\vert \mathbf{Q}\right\vert \right)\right\vert }^{2}$ is the magnetic form factor of Cr³⁺ in the dipole approximation. The classical magnetic ground states used in the above spin-wave simulation were calculated by Monte Carlo simulation at D/J₁ = 5% and −5%, respectively. At D/J₁ = 5%, the spin vectors of three ions forming an equilateral triangle are arranged in the ab plane with $\langle {\left\vert {S}_{j}^{z}\right\vert }^{2}\rangle$ = 0, and the angles between them are the same, θ_1,2 = θ_2,3 = θ_3,1 = 120°. At D/J₁ = −5%, while $\langle {\left\vert {S}_{j}^{z}\right\vert }^{2}\rangle$ = 1.137, θ_1,2 = 120.14°, θ_2,3 = 118.98°, and θ_3,1 = 360° − θ_1,2 − θ_2,3 are obtained. The international system of units is used throughout this paper.

Both α-CrOOH and α-CrOOD are good insulators with room-temperature resistance larger than 20 MΩ. Through Rietveld refinements, both x-ray and neutron diffraction measurements indicate that the delafossite structure (space group: R $\bar{3}$ m) maintains at least down to 8 K (appendix A) [61, 62]. The crystal structure of α-CrOOH(D) is shown in figure 1(a), and the stacking pattern of O-H(D)-O layers is straight, which is similar to that of CuCrO₂ or AgCrO₂ [56]. The triangular layers of magnetic Cr³⁺ ions are well spatially separated by nonmagnetic layers (figure 1(a)), rendering the interlayer magnetic coupling weak. The symmetry of the space group ensures the triangular lattice regular without spatial anisotropy (figure 1(b)). The inversion centers of the periodic crystalline lattice are located at halfway sites between neighboring Cr³⁺ ions, and thus the DM interactions are forbidden symmetrically [71].

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Above ∼200 K, the inverse susceptibility of α-CrOOH shows a linear temperature dependence (Curie–Weiss law, see figure 1(c)). We obtain the effective magnetic moment of $g{\mu }_{\text{B}}\sqrt{S\left(S+1\right)}=\sqrt{3{k}_{\text{B}}C/\left({N}_{\text{A}}{\mu }_{0}\right)}=3.861\left(2\right){\mu }_{\text{B}}$, from the fitted Curie constant C = 23.42(2) Kcm³ mol⁻¹. The g factor is nearly constant and isotropic at high temperatures, g = 1.97(1) ∼ 2, from fitting the ESR spectra (see figure 1(d)), and we further get the experimental value of S = 1.52(1), which is well consistent with the expected one S = 3/2. The 3d electrons of Cr³⁺ are localized spatially as the conductivity is poor, and the spin–orbit couplings are weak as evidenced by g ∼ 2. Therefore, one can expect a nearly Heisenberg Hamiltonian for α-CrOOH(D) as proposed in other Cr³⁺-based TAF compounds [53, 55],

$$\begin{equation}\mathcal{H}={J}_{1}\sum\limits _{\langle jl\rangle }{\boldsymbol{S}}_{j}\cdot {\boldsymbol{S}}_{l}+{\mathcal{H}}^{\prime }={J}_{1}\sum\limits _{\langle jl\rangle }{\boldsymbol{S}}_{j}\cdot {\boldsymbol{S}}_{l}+D\sum\limits _{j}{\left({S}_{j}^{z}\right)}^{2}+{J}_{2}\sum\limits _{\langle \langle j{l}^{\prime }\rangle \rangle }{\boldsymbol{S}}_{j}\cdot {\boldsymbol{S}}_{{l}^{\prime }},\end{equation} \tag{ 3 }$$

where the strengths of the single-ion anisotropy (D) and NNN exchange (J₂) terms are much weaker than that of the NN exchange interactions (J₁). The fitted Weiss temperature (see figure 1(c)), θ_cw(H) = −211.6(5) K, estimates the antiferromagnetic exchange energy as J₁+J₂ ∼ −3θ_cw(H)/[6S(S+1)] ∼ 28 K.

Owing to the absence of strong DM interactions [72] and coupling anisotropy [34], the high-temperature paramagnetic ESR linewidths are relatively narrow, ω ∼ 0.0616(2) T in α-CrOOH and ω ∼ 0.0778(2) T in α-CrOOD (figure 1(d)), that are an order of magnitude smaller than those reported in the QSL candidates, such as ZnCu₃(OH)₆Cl₂ [72] and YbMgGaO₄ [34]. As the frequency increases to 214 GHz, the ESR linewidth is only slightly increased, ω(214 GHz) ∼ 1.7ω(9.4 GHz) (see figure 2(b)), suggesting both g-factor randomness and anisotropy have marginal effect, ∆g ∼ μ_B g²[ω(f₂) − ω(f₁)]/(hf₂ − hf₁) ∼ 0.006g. The contribution of ∆g to the ESR linewidth (Zeeman contribution) is negligible at f ∼ 9.4 GHz, ω_∆ = ∆ghf/(μ_B g²) ∼ 2 mT ≪ ω, and thus we can safely use the high-temperature X-band ESR linewidth to determine the strength of the anisotropy interactions (see below).

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The spin system of α-CrOOH is almost linearly polarized in an applied magnetic field up to 14 T at 2 K, with M < 5%M_sa, confirming the strong AF interactions (figure 1(e)). Here, M_sa = gS μ_B/Cr is the saturation magnetization. A high magnetic field of μ₀ H_1/3 ∼ 3J₁ S/(gμ_B) ∼ 90 T is required to stabilize the 1/3 magnetization plateau, and the linear field dependence of magnetization is expected well below μ₀ H_1/3, according to the reported numerical result [73]. Through a linear fit to the magnetization above 3 T (figure 1(e)), the concentration of impurity spins, M_im ⩽ 0.2%M_sa, is obtained from the fitted intercept, and is much smaller than that of many other frustrated magnets, e.g. 7.7% in ZnCu₃(OH)₆Cl₂ [74]. Therefore, similar to other delafossite oxides (e.g. CuCrO₂ and AgCrO₂) the impurity effect is also negligible in α-CrOOH(D).

The high-temperature ESR linewidth is highly sensitive to the magnetic anisotropy of a spin system, and thus has been widely used in determining microscopic Hamiltonian parameters [34, 72, 75]. At T ≫ gμ_B μ₀ H/k_B, the ESR linewidth is given as [75]

$$\begin{equation}\omega =\frac{\sqrt{2\pi }}{g{\mu }_{\text{B}}}\sqrt{\frac{{M}_{2}^{3}}{{M}_{4}}},\end{equation} \tag{ 4 }$$

where ${M}_{2}=\langle \left[{\mathcal{H}}^{\prime },{M}^{+}\right]\left[{M}^{-},{\mathcal{H}}^{\prime }\right]\rangle /\langle {M}^{+}{M}^{-}\rangle$ and ${M}_{4}=\langle \left[\mathcal{H},\left[{\mathcal{H}}^{\prime },{M}^{+}\right]\right]\left[\mathcal{H},\left[{\mathcal{H}}^{\prime },{M}^{-}\right]\right]\rangle /\langle {M}^{+}{M}^{-}\rangle$ are the second and fourth moments, respectively, ⟨⟩ represents a thermal average, and ${M}^{{\pm}}\equiv \sum _{j}{S}_{j}^{{\pm}}\equiv \sum _{j}{S}_{j}^{x}{\pm}i{S}_{j}^{y}$. Therefore, the perturbation terms in the Hamiltonian that do not commute with M^± will contribute to the ESR linewidth.

Here, we discuss various sources that broaden the ESR signal. First of all, the hyperfine interactions broaden the ESR linewidth with ${\omega }_{\text{h}}\sim {\left\vert {A}_{\text{h}}\right\vert }^{2}/\left(g{\mu }_{\text{B}}{J}_{1}\right)$ < 0.3 μT, where |A_h| < 60 MHz is the hyperfine coupling of ⁵³Cr³⁺ (abundance of 9.55%) [76]. The dipole–dipole interactions also contribute to the linewidth with ${\omega }_{\text{d}}\sim {\left\vert {E}_{\text{d}}\right\vert }^{2}/\left(g{\mu }_{\text{B}}{J}_{1}\right)\sim$ 0.2 mT, where $\left\vert {E}_{\text{d}}\right\vert \sim {\mu }_{0}{g}^{2}{\mu }_{\text{B}}^{2}/\left(4\pi {a}^{3}\right)\sim$ 1.9 GHz is the strength of dipole–dipole interactions and a = 2.98 Å is the lattice constant. There is only one Wyckoff position for Cr³⁺ ions, and thus the almost uniform Zeeman interactions cannot significantly broaden the x-band ESR linewidth (ω_Δ ∼ 2 mT, see above). All of the above contributions together account for a linewidth that is more than one order of magnitude smaller than the observed value of ω ⩾ 0.06 T.

The single-ion anisotropy (D) terms in equation (3) do not commute with M^±, and thus can give rise to the ESR broadening. At T ≫ J₁ S² k_B ⁻¹, ⟨⟩ → tr() is expected in equation (4), and we obtain the ESR linewidth as

$$\begin{equation}\omega \left(T\to \infty \right)=\frac{\sqrt{2\pi }}{g{\mu }_{\text{B}}}\sqrt{\frac{{2.4}^{3}{D}^{4}}{108{J}_{1}^{2}+9.6{D}^{2}}}\sim \frac{0.897{D}^{2}}{g{\mu }_{\text{B}}{J}_{1}},\end{equation} \tag{ 5 }$$

with |D|, |J₂| ≪ J₁. Taking J₁ ∼ 28 K and ω(300 K) ∼ 0.06 T (see above), we reach a rough estimation of |D| ∼ 1.6 K.

Below ∼150 K, the susceptibility deviates from the Curie–Weiss law (figure 2(a)) and the ESR linewidth shows obvious temperature dependence (figure 2(b)), indicating the formation of low-T AF correlations. To reproduce better the experimental data and to determine more precisely the Hamiltonian parameters, we carry out the FLD calculations [67, 68] above 60 K, where the numerical finite size effects are negligible (appendix B). We first simultaneously fit both susceptibility and x-band ESR linewidth by fixing J₂ = 0 (figures 2(a) and (b)), and obtain J₁ = 23.5 K and D = 1.08 K with the least-R_p = 9.7. In the easy-axis region (D < 0), we find another minimum-R_p point at J₁ = 23.6 K and D = −1.08 K with R_p = 9.9. Furthermore, we also try to determine J₂. Unfortunately, the quality of the fit is not significantly improved with a non-zero J₂, as both susceptibility and ESR linewidth (the J₂ terms commute with M^±) are insensitive to J₂ above 60 K. In the similar way, we obtain J₁ ∼ 22.4 K and |D| ∼ 1.19 K for α-CrOOD.

Toward the experimental realizing of a QSL in the geometrically-frustrated delafossite family, the ground-state magnetic order must be significantly suppressed at first. In α-CrOOH, the paramagnetic ESR signal with a relatively broad linewidth of ∼0.4 T (PM) is clearly observed down to 2 K (≪T_f(H), see figure 2(c)), implying the presence of persistent spin fluctuations. In sharp contrast, the x-band ESR spectra of CuCrO₂ and AgCrO₂ get undetectable on approaching their transition temperatures of 24 and 21 K, respectively [60]. Besides the paramagnetic signal, two additional ESR models (A and B) with sharp linewidths of ∼ 0.04 T are also detected below T_f(H) (figure 2(c)), and the additional signals (μ₀ H_res/f) show obvious frequency dependence at 2 K (figure 2(d)), thereby indicating that a magnetic freezing occurs for α-CrOOH. Despite the transition, the integrated intensities of A and B modes are nearly one order of magnitude smaller than that of the predominant PM mode, suggesting that only a small fraction of Cr³⁺ spins (⩽ 20%) get frozen. Below T_f ∼ 25 K, the ESR intensity of the paramagnetic signal that is proportional to the imaginary part of the dynamic susceptibility [72] obviously deviates from the bulk susceptibility, and quickly decreases to a small constant of 0.006(1) cm³ mol⁻¹ (figure 2(a)). Meanwhile, the line width decreases to ∼ 0.4 T as well (figure 2(b)). These observations might be attributed to the spin freezing, while the decrease of the paramagnetic ESR intensity from ∼ 0.09(2) cm³ mol⁻¹ at ∼ 30 K to ∼0.006(1) cm³ mol⁻¹ at ∼ 2 K (figure 2(a)) seems also consistent with the theoretical prediction of the ideal Dirac spin liquid for susceptibility (χ) among the dynamic spins, χ ∝ T [46]. The bulk susceptibility contributed from both dynamic and frozen spins is nearly constant below ∼ 30 K, χ_bulk ∼ 0.08 cm³ mol⁻¹ (figure 2(a)), in comparison with the local ESR one.

For delafossite CuCrO₂, a long-range 3D 120° Néel order with a moment of (3.1 ± 0.2)μ_B in the ac plane has been revealed by neutron powder diffraction measured at low temperatures [53]. Similarly, we carry out the neutron powder diffraction measurements for both α-CrOOH (figure 3(a)) and α-CrOOD (figure 3(b)). In contrast to other delafossite oxides [53, 58, 59], no obvious difference of the neutron patterns at between 8 K (well below T_f) and 50 K (well above T_f) can be detected, except the very ambiguous signal around |Q| = 4π/(3a) ∼ 1.41 Å⁻¹ that may relate to the small amount of frozen spins in both α-CrOOH and α-CrOOD. For α-CrOOH, the upper bound of the static structure factor at |Q| = 4π/(3a) is roughly estimated as ${M}_{\text{sub}}^{2}/2=\langle \left\vert I\left(8\;\mathrm{K}\right)-I\left(50\;\mathrm{K}\right)\right\vert \rangle /\left[6{\times}{\left(5.4\enspace \mathrm{f}\mathrm{m}\right)}^{2}{S}_{\text{ph}}{\left\vert f\left(\left\vert \mathbf{Q}\right\vert \right)\right\vert }^{2}\right]{\leqslant}0.2$ per Cr [69], that is much smaller than the recently reported value of ∼0.75 per site calculated using the ideal S = 3/2 NN THAF model [73]. Here, S_ph = I_n /(2|F_n |²) ∼ 11.9 fm⁻² Cr is the scale factor with I_n = 2160(140) and |F_n |² = 91 fm²/Cr presenting the observed intensity and calculated structure factor of the (0 0 3) nuclear reflection (see figure 3(a)), respectively, and $\langle \left\vert I\left(8\;\mathrm{K}\right)-I\left(50\;\mathrm{K}\right)\right\vert \rangle$ ⩽ 300. This static structure factor corresponds to an average moment of gμ_B M_sub ⩽ 1.2μ_B in α-CrOOH, much smaller than that reported in CuCrO₂ [53]. For α-CrOOD, the similar conclusion can be drawn.

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Our specific heat of α-CrOOH (α-CrOOD) measured at zero field shows the magnetic transition at T_f = 25 K (22 K), which is well consistent with the previously reported results [63, 64] (figure 3(c)). At the transition temperature, the magnetic specific heat (see appendix C for technical details) shows a small anomaly (kink) above the very broad hump (figure 3(c)). The small anomaly should correspond to the frozen spins, whereas the broad hump indicates the formation of short-range AF corrections, as reported in various QSL candidates [33]. In contrast, the magnetic specific heat of CuCrO₂ only shows a conventional sharp λ peak at T_f(Cu) = 24 K [57, 58].

Below the transition temperature, the magnetic specific heat behaves a power-law temperature dependence, C_m ∝ T^γ , with γ(H) = 2.28(1) in α-CrOOH and with γ(D) = 2.151(2) in α-CrOOD, suggesting the formation of either long-range quasi two-dimensional AF order (spin-wave magnons) or unconventional magnetic state (e.g. Dirac spin-liquid states [16, 77]). The absence of sharp strong magnetic Bragg peaks in the neutron diffraction patterns measured at low temperatures (figures 3(a) and (b)) favors the later scenario, and thereby suggests that the system of α-CrOOH(D) is in the vicinity of a spin-liquid phase (similar to the case of Cu_0.85Ag_0.15CrO₂ [58, 59]). In contrast, CuCrO₂ exhibits C_m ∝ T³, thus confirming the long-range 3D order there [57, 58].

The ESR, neutron diffraction, and specific heat measurements consistently indicate that the long-range 120° Néel order is significantly (at least partially) suppressed at low temperatures in both α-CrOOH and α-CrOOD (figures 2 and 3). This can be further confirmed by the INS spectra, as no sharp spin-wave dispersion is clearly observed down to 3.5 K in α-CrOOD (figure 4(a)). In contrast, the calculated spectra of the spin-wave excitations from the classical three-sublattice ordered ground states show sharp dispersions even on the powder average at the same temperature (figures 4(b) and (c)). The structural diffraction is very sharp with FWHMs of ∼ 0.1 Å⁻¹ in the wave-vector space (figure 3(b)), and the energies of phonon excitations, E ⩾ θ_E2 ∼ 60 meV (appendix C), are more than one order of magnitude larger than J₁ at |Q| = 4π/(3a). Therefore, the structural signals cannot account for the observed INS spectra with broad FWHMs ⩾ 0.5 Å⁻¹ (correlation length of ξ_cor ⩽ 10 Å). The broad magnetic INS spectrum measured at 3.5 K (figure 4(a)) is reminiscent of the continuous magnetic excitations observed in QSL candidates [36, 37, 39, 40], and thus indicates the presence of significant spin fluctuations in α-CrOOD. At 3.5 K (≪ −θ_cw), the residual magnetic entropy is almost zero, S_m ∼ 0.4%Rln4 (see figure C1(d)), and thus these fluctuations should be quantum in nature. Unfortunately, both ESR (figure 2) and specific heat (figure 3) measurements indeed have detected the signs of a magnetic transition at T_f, and thus the system of α-CrOOH(D) is in the vicinity of but not deeply inside a spin-liquid phase.

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