Crystal field states of ${\mathrm{Tb}}^{3+}$ in the pyrochlore spin liquid ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ from neutron spectroscopy

We report time-of-flight neutron scattering measurements of the magnetic spectrum of ${\mathrm{Tb}}^{3+}$ in ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. The data, which extend up to 120 meV and have calibrated intensity, enable us to consolidate and extend previous studies of the single-ion crystal field spectrum. We successfully refine a model for the crystal field potential in ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ without relying on data from other rare-earth titanate pyrochlores, and we confirm that the ground state is a non-Kramers doublet with predominantly $|\pm 4\rangle$ components. We compare the model critically with earlier models.

Figure 1

Neutron powder diffraction pattern of polycrystalline ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ together with Rietveld refinement fit. The data were measured at room temperature in bank 6 on GEM. The line below the data is the difference between the data and fit, and the tick marks show the peak positions.

Figure 2

Neutron scattering spectrum of polycrystalline ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ at $T=7$ K, measured on MERLIN with $E_i=150$ meV. The colors represent the value of $E\times S(Q,E)$ in units of mb ${\mathrm{sr}}^{-1}$ f.u.${}^{-1}$

Figure 3

Energy cuts through the spectrum of polycrystalline ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ measured with $E_i=36$ meV. The open and closed circles represent data averaged over $Q$ ranges 1.0–$1.8 {\AA }^{-1}$ and 6.0–$6.6 {\AA }^{-1}$, respectively. (a) $T=7$ K. The strong magnetic peaks at $E=10$ meV and 15.5 meV (the latter split) are about a factor of 6 smaller at $\langle Q\rangle =6.3 {\AA }^{-1}$ relative to $\langle Q\rangle =1.4 {\AA }^{-1}$, consistent with the variation in intensity due to the magnetic form factor of ${\mathrm{Tb}}^{3+}$. (b) $T=300$ K. The two peaks at 7 and 15 meV in the $\langle Q\rangle =6.3 {\AA }^{-1}$ spectrum are caused by phonons since they are not present at $\langle Q\rangle =1.4 {\AA }^{-1}$.

Figure 4

Energy cuts through the spectrum of polycrystalline ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. (a) Spectrum at four different $\langle Q\rangle$, recorded with $E_i=150$ meV at $T=7$ K, showing magnetic peaks at 49 and 61 meV and phonon peaks at 72, 91, and 99 meV. (b) Comparison of the low-angle $\left(\langle \varphi \rangle ={10}^{\circ }\right)$ spectrum at $T=7$ and 90 K, recorded with $E_i=65$ meV. The corresponding $Q$ is given by ${\hslash }^2{Q}^2/2m_n=E_i+E_f-2{\left(E_i{E}_f\right)}^{1/2}cos\varphi$, where $E_f=E_i-E$ and $m_n$ is the neutron rest mass. In (b) the error bars are smaller than the size of the symbols.

Figure 5

Comparison of the observed and calculated spectra of polycrystalline ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. The heavy solid (red) lines are calculated from Eq. (2) with eigenstates derived from the crystal field parameters given in the caption of Table 2, and the individual transitions are shown as thin (blue) lines. The broken (blue) lines are the estimated nonmagnetic background, and the fitted elastic peaks centered on $E=0$ are shown as pale (green) lines.