Fig. 4: Normalized onset temperatures for various Boltzmann thermometers.

All onset temperatures were calculated using Eq. 6 and normalized to \(\Delta E/k_{{{\mathrm{B}}}}\). The results of Figs. 2 and 3 were used to calculate the onset temperatures of the Eu³⁺-based thermometers. Decay rates were extracted from literature to calculate the onset temperatures of CsCdBr₃:Pr³⁺ (\(k_{{{{\mathrm{nr}}}}}\left( 0 \right) =\) 1.5 × 10² ms⁻¹, \(k_{1,{{{\mathrm{r}}}}} =\) 51 ms⁻¹)²⁷, LaCl₃:Pr³⁺ (\(k_{{{{\mathrm{nr}}}}}\left( 0 \right) =\) 3.1 × 10² ms⁻¹, \(k_{1,{{{\mathrm{r}}}}} =\) 68 ms⁻¹)^28,29, LaPO₄:Nd³⁺ (\(k_{{{{\mathrm{nr}}}}}\left( 0 \right) =\) 57 ms⁻¹, \(k_{1,{{{\mathrm{r}}}}} =\) 2.3 ms⁻¹)⁵, LaBO₃:Gd³⁺ (\(k_{{{{\mathrm{nr}}}}}\left( 0 \right) =\) 11 ms⁻¹, \(k_{1,{{{\mathrm{r}}}}} =\) 0.29 ms⁻¹)³⁰, Y₂(B₂SO₄)₆:Gd³⁺ (\(k_{{{{\mathrm{nr}}}}}\left( 0 \right) =\) 8.9 ms⁻¹, \(k_{1,{{{\mathrm{r}}}}} =\) 0.21 ms⁻¹)³⁰, YVO₄:Er³⁺ (\(k_{{{{\mathrm{nr}}}}}\left( 0 \right) =\) 3.0 × 10⁴ ms⁻¹, \(k_{1,{{{\mathrm{r}}}}} =\) 5.1 ms⁻¹)³¹. Decay rates were measured to calculate the onset temperatures of NaYF₄:Er³⁺ (\(k_{{{{\mathrm{nr}}}}}\left( 0 \right) =\) 1.9 × 10⁵ms⁻¹, \(k_{1,{{{\mathrm{r}}}}} =\) 1.4 ms⁻¹) and NaYF₄:Dy³⁺ (\(k_{{{{\mathrm{nr}}}}}\left( 0 \right) =\) 2.0 × 10² ms⁻¹, \(k_{1,{{{\mathrm{r}}}}} =\) 1.3 ms⁻¹). In principle, the energy gap of LaPO₄:Nd³⁺, LaBO₃:Gd³⁺, Y₂(B₂SO₄)₆:Gd³⁺, and YVO₄:Er³⁺ could be bridged by one phonon mode, but Fig. 2b suggested that more than one mode is typically necessary to realize resonance when the gap is small compared to the phonon energy. We, therefore, determine the onset temperature by inserting both 1 and 2 phonons in Eq. 6 and using the average \(p\) and \(T_{{{{\mathrm{onset}}}}}\)