Investigation of magnetic fluctuations in ${\text{Tb}}_2{\text{Sn}}_2{\text{O}}_7$ ordered spin ice by high-resolution energy-resolved neutron scattering

We have studied magnetically frustrated ${\text{Tb}}_2{\text{Sn}}_2{\text{O}}_7$ by neutron diffraction and high-resolution energy-resolved neutron scattering. At 0.1 K, we observed short-range magnetic correlations with a typical scale of $4\text{Å}$, close to the near-neighbor distance between ${\text{Tb}}^{3+}$ ions. This short-range order coexists with ferromagnetic correlations and long-range spin ice order at the scales of 18 and $190\text{Å}$, respectively. Spin dynamics was investigated at a time scale down to ${10}^{-9}\text{s}$ by energy-resolved experiments on a backscattering spectrometer. We observed a freezing of the spin dynamics for all length scales with a strong slowing down of the spin fluctuations when long-range order settles in. We discuss the spin fluctuations remaining in the ground state, in comparison with previous data obtained by muon spectroscopy.

Figure 1

(Color online) (a) Magnetic intensity measured ${\text{Tb}}_2{\text{Sn}}_2{\text{O}}_7$ by neutron diffraction at small angles versus the moment transfer $Q$ for several temperatures. A spectrum above 1.2 K was subtracted. Solid lines are Lorentzian fits (see text). In the inset, the total diffraction pattern at 0.1 K, showing the small-angle intensity together with the magnetic Bragg peaks. (b) Medium correlation length $L_C^{\text{MRO}}=1/\kappa$ (full squares) versus temperature $T$. The solid line is a guide to the eye. (c) Norm A of the Lorentzian (open triangles) and intensity of the magnetic Bragg peaks (full squares) versus temperature.

Figure 2

(Color online) (a) Magnetic diffraction spectrum of ${\text{Tb}}_2{\text{Sn}}_2{\text{O}}_7$ versus the moment transfer $Q$ at 0.1 K. A spectrum at 1.2 K was subtracted. The solid line is a refinement taking into account both LRO and SRO contributions. A linear background is also added. The arrows correspond to the $Q$ bands selected in the time-resolved experiment. White arrows correspond to the Bragg $Q$ bands, black arrows to the low, medium, and high $Q$ bands. [(b) and (c)] Calculated LRO (SRO) contribution to the diffraction pattern.

Figure 3

(Color online) Inelastic neutron spectra of ${\text{Tb}}_2{\text{Sn}}_2{\text{O}}_7$ for several temperatures measured in the $Q$ range of the 111 and 200 Bragg peaks. The solid line is a fit as described in the text. Dotted, dashed-dotted, and dashed lines correspond to the quasielastic, elastic, and flat background terms, respectively.

Figure 4

(Color online) Temperature dependence of the parameters deduced from the fit of the inelastic spectra in the region of the 111 and 200 Bragg peaks. (a) Elastic intensity $C_1$ and quasielastic intensity $C_2$ deduced from the fit of the energy patterns. In the inset: the elastic intensity (full dots) is compared to the intensity from elastic scans (open squares) and to the integrated intensity of a magnetic Bragg peak measured in diffraction (diamonds). All quantities are scaled to vary between 1 at 0.1 K and 0 at 1.4 K. (b) Quasielastic width $2\Gamma$ versus temperature. In the inset: the background level $C_B$, involving fluctuations at a shorter time scale than probed in the energy window. (c) The fluctuation time $\tau =\hslash /\Gamma$. Solid lines are guides to the eye.

Figure 5

(Color online) Temperature dependence of the integrated elastic and quasielastic intensities. The background level was fixed to zero at all temperatures. Solid lines are guides to the eye. (a)–(c) correspond to low, medium, and high $Q$ bands, respectively. In the insets: the quasielastic width $2\Gamma$ versus temperature. The width at 0.09 K in the inset of (a) was fixed.

Figure 6

(Color online) Positive muon depolarization function in the case of a dipolar field fluctuating perpendicular to the initial direction of polarization, for different values of the ratio $\nu /\Delta$ (see text).