One ion to catch them all: Targeted high-precision Boltzmann thermometry over a wide temperature range with Gd3

Abstract

Ratiometric luminescence thermometry with trivalent lanthanide ions and their 4fⁿ energy levels is an emerging technique for non-invasive remote temperature sensing with high spatial and temporal resolution. Conventional ratiometric luminescence thermometry often relies on thermal coupling between two closely lying energy levels governed by Boltzmann's law. Despite its simplicity, Boltzmann thermometry with two excited levels allows precise temperature sensing, but only within a limited temperature range. While low temperatures slow down the nonradiative transitions required to generate a measurable population in the higher excitation level, temperatures that are too high favour equalized populations of the two excited levels, at the expense of low relative thermal sensitivity. In this work, we extend the concept of Boltzmann thermometry to more than two excited levels and provide quantitative guidelines that link the choice of energy gaps between multiple excited states to the performance in different temperature windows. By this approach, it is possible to retain the high relative sensitivity and precision of the temperature measurement over a wide temperature range within the same system. We demonstrate this concept using YAl₃(BO₃)₄ (YAB):Pr³⁺, Gd³⁺ with an excited ⁶P_J crystal field and spin-orbit split levels of Gd³⁺ in the UV range to avoid a thermal black body background even at the highest temperatures. This phosphor is easily excitable with inexpensive and powerful blue LEDs at 450 nm. Zero-background luminescence thermometry is realized by using blue-to-UV energy transfer upconversion with the Pr³⁺-Gd³⁺ couple upon excitation in the visible range. This method allows us to cover a temperature window between 30 and 800 K.

Fig. 1. Thermodynamic foundation for the optimum performance of an excited three-level Boltzmann-based luminescent thermometer.

All functions are plotted in terms of the dimensionless variable $r=\frac{\mathrm{\Delta }{E}_{21}}{k_{\mathrm{B}}T}$, and the higher energy gap was exemplarily set to ΔE₃₁ = 1.71ΔE₂₁. a Plots of the thermal population probabilities per microstate, p_m/g_m, of an excited three-level system in thermodynamic equilibrium determining the response of a luminescent thermometer. g_m denotes the degeneracy of the electronic state |m〉. b Relative net population changes per microstate, ∆N_mn/N_tot, of an excited three-level system in thermodynamic equilibrium determining the sensitivity of a luminescent thermometer. Given known ∆E₂₁ and ∆E₃₁ values, it is possible to state for which temperature a change in the thermometric measure is advisable. c Resulting normalized thermal response functions ρ(r) (solid) and relative sensitivity functions σ(r) (dashed-dotted) for thermometry along the energy gaps ∆E₂₁ (blue) and ∆E₃₁ (red). d Resulting curves for the relative statistical measurement uncertainty, ${\sigma }_T/T$, of the designed Boltzmann thermometer with excited-state energy gaps ∆E₂₁ and ∆E₃₁

Fig. 2. Thermodynamically optimized Boltzmann cryothermometry with Gd³⁺ in YAB:Pr³⁺, Gd³⁺.

a Splitting of the lowest excited ⁶P_7/2 spin-orbit level of Gd³⁺ into the different Kramers’ doublets at the present D₃ site symmetry in YAB. The doped Gd³⁺ ions are sixfold coordinated in the form of a twisted trigonal prism (inset). b Indirect upconversion excitation scheme for Gd³⁺ by energy transfer from the 4f¹5d¹-related electronic states of Pr³⁺. Both the ground state absorption (GSA) and the excited state absorption (ESA) of two 450 nm photons within the pulse period (T = 50 ms) of the used pulsed laser source are indicated. c High-resolution upconversion photoluminescence spectra of YAB: 0.7% Pr³⁺, 20% Gd³⁺ upon excitation of Pr³⁺ at 448 nm showing the temperature dependence of the two radiative transitions from the lower and higher crystal field states of the ⁶P_7/2 level. The arrows mark the two thermometrically employed transitions. d Boltzmann plot of the temperature-dependent LIR normalized to its value at 300 K. The fitted energy gap, statistical figures of merit, and expected onset temperature for Boltzmann behaviour (see Eq. (5)) are indicated

Fig. 3. Boltzmann thermometry was thermodynamically optimized at room temperature and higher temperatures (T > 500 K) with Gd³⁺ in YAB:Pr³⁺, Gd³⁺.

a Spin-orbit level scheme and respective radiative transitions depicting the concept for high-precision Boltzmann thermometry with Gd³⁺. b High-resolution upconversion photoluminescence spectra of YAB: 0.7% Pr³⁺, 20% Gd³⁺ upon excitation of Pr³⁺ at 448 nm showing the radiative transitions from the different ⁶P_J levels (J = 7/2, 5/2, and 3/2) of Gd³⁺. The spectra were normalized with respect to the highest intensity peak (I₁₀ ~ 10⁶ counts) to demonstrate the thermally induced intensity increase of the higher energetic emission peaks. c Boltzmann plot of the temperature-dependent LIR employing the transitions from the ⁶P_5/2 and ⁶P_7/2 levels, respectively, normalized to its value at 873 K. The fitted energy gap, statistical figures of merit and expected onset temperature for Boltzmann behaviour (Eq. (5)) are indicated. d Boltzmann plot of the temperature-dependent LIR, which employs the transitions from the ⁶P_3/2 and ⁶P_7/2 levels, normalized to its value at 873 K. The fitted energy gap, statistical figures of merit, and expected onset temperature for Boltzmann behaviour (Eq. (5)) are indicated

Fig. 4. Assessment of the optimum performance of a wide-range multilevel luminescent Boltzmann thermometer based on Gd³⁺ in YAB:Pr³⁺, Gd³⁺.

a Plot of the relative sensitivities S_r of the different temperature measures and their improvement upon thermodynamically guided change of the respective energy gap of the excited levels of Gd³⁺ used for temperature sensing. b Overall theoretical relative temperature uncertainty of the different energy gaps assuming a constant integrated intensity measure of I₁₀ = 10⁷ counts of the selected lowest energetic emission. A relative temperature uncertainty of 0.1% is desirable because it allows the measurement of temperatures below 1000 K with a statistical error of ±1 K. The shaded areas indicate the statistical fluctuations in the relative temperature uncertainty