Making Nd³⁺ a Sensitive Luminescent Thermometer for Physiological Temperatures—An Account of Pitfalls in Boltzmann Thermometry

Abstract

Ratiometric luminescence thermometry employing luminescence within the biological transparency windows provides high potential for biothermal imaging. Nd³⁺ is a promising candidate for that purpose due to its intense radiative transitions within biological windows (BWs) I and II and the simultaneous efficient excitability within BW I. This makes Nd³⁺ almost unique among all lanthanides. Typically, emission from the two ⁴F_3/2 crystal field levels is used for thermometry but the small ~100 cm⁻¹ energy separation limits the sensitivity. A higher sensitivity for physiological temperatures is possible using the luminescence intensity ratio (LIR) of the emissive transitions from the ⁴F_5/2 and ⁴F_3/2 excited spin-orbit levels. Herein, we demonstrate and discuss various pitfalls that can occur in Boltzmann thermometry if this particular LIR is used for physiological temperature sensing. Both microcrystalline, dilute (0.1%) Nd³⁺-doped LaPO₄ and LaPO₄: x% Nd³⁺ (x = 2, 5, 10, 25, 100) nanocrystals serve as an illustrative example. Besides structural and optical characterization of those luminescent thermometers, the impact and consequences of the Nd³⁺ concentration on their luminescence and performance as Boltzmann-based thermometers are analyzed. For low Nd³⁺ concentrations, Boltzmann equilibrium starts just around 300 K. At higher Nd³⁺ concentrations, cross-relaxation processes enhance the decay rates of the ⁴F_3/2 and ⁴F_5/2 levels making the decay faster than the equilibration rates between the levels. It is shown that the onset of the useful temperature sensing range shifts to higher temperatures, even above ~ 450 K for Nd concentrations over 5%. A microscopic explanation for pitfalls in Boltzmann thermometry with Nd³⁺ is finally given and guidelines for the usability of this lanthanide ion in the field of physiological temperature sensing are elaborated. Insight in competition between thermal coupling through non-radiative transitions and population decay through cross-relaxation of the ⁴F_5/2 and ⁴F_3/2 spin-orbit levels of Nd³⁺ makes it possible to tailor the thermometric performance of Nd³⁺ to enable physiological temperature sensing.

1. Introduction

Temperature is an important control parameter that governs, e.g., the rate of chemical reactions, but also the optimum working efficiency of electronic devices or dynamics and viability of biological systems. As such, non-invasive, sensitive and remote temperature measurement techniques with the capability to spatially resolve temperature variations down to the micrometer range are becoming increasingly relevant [1,2,3,4,5,6,7]. Physiological temperature sensing is especially demanding in that regard as it even requires to accurately distinguish temperature fluctuations below 1 K. Luminescence nanothermometry is an appealing and rapidly emerging technique that meets those requirements and constantly improves [8,9,10,11,12,13,14]. Among the various temperature-dependent optical parameters that can be used for temperature calibration, the luminescence intensity ratio (LIR) of two emissive transitions from thermally coupled excited states has emerged as an especially robust and yet easily measurable representative to extract information about the local temperature of a medium in contact with luminescent nanocrystals. Also, other methodological approaches such as lifetime thermometry [15,16,17,18,19,20,21], or the concept of excited state absorption (ESA) [22,23,24] have been developed for that purpose while organic framework-based thermometers increasingly attract attention [25,26].

Besides the requirement for a low temperature uncertainty (<±1 K) over a narrow temperature range (30 °C–60 °C), luminescence nanothermometry under physiological conditions has additional restrictions. Absorbance, light scattering or even autofluorescence of locally surrounding tissue pose challenges that have to be overcome to guarantee a reliable detection of the LIR outside the biological system of interest [27,28,29]. Thus, near infrared (NIR) luminescence within the biological transparency windows (BW I: 650–950 nm; BW II: 1000–1350 nm; BW III: 1550–1850 nm) is highly desirable for minimized attenuation of the emitted light to guarantee high temperature precision. While quantum dots rely on a sensitive thermal quenching behavior of their NIR luminescence within physiological temperature regimes and have successfully been employed in various biomedical applications [30,31,32,33,34,35,36], their usage in thermometry still requires careful calibration [37].

As an alternative, the lanthanide ion Nd³⁺(4f³) has become a promising candidate for in vivo nanothermometry [30,38,39,40,41,42,43]. This is related to the fact that its most intense radiative transitions all lie within BW I and BW II. Moreover, the large energy gap between the most dominantly emissive ⁴F_3/2 spin-orbit level and the next lower ⁴I_15/2 level typically allows for rather high brightness of Nd³⁺-activated luminescent nanocrystals. Finally, Nd³⁺ most efficiently absorbs within the first biological window. Altogether, those optical properties make Nd³⁺ almost unique among all lanthanides for NIR luminescence thermometry. Since the ⁴F_3/2 spin-orbit level of Nd³⁺ splits into exactly two energetically well isolated Kramers’ doublets (R₁ and R₂) in any lower symmetric crystal field with an energy gap of around 100 cm⁻¹, the vast majority of studies aiming at single ion in vivo nanothermometry with Nd³⁺ addressed the LIR stemming from emission from those two crystal field states [38,40,41,44,45]. Among the various possibilities of synthetically well accessible and size-controllable nanocrystals such as fluorides [46,47,48], nanosized garnets have become especially interesting in that sense due to the well resolved and intense ⁴F_3/2(R_1,2) → ⁴I_9/2(Z₅) emissive transitions at 935 nm and 945 nm. This allowed researchers to tune the relative thermal sensitivity S_r of this particular LIR towards the theoretically expected maximum value of around 0.15% K⁻¹ at 37 °C according to the given energy gap [49,50]. The value obtained with that concept of Nd³⁺-based nanothermometry could only be exceeded in lattices with higher crystal field splitting, for example LiLuF₄ nanocrystals, with a value of around 0.5% K⁻¹ at 37 °C [51].

Usage of the two Kramers’ doublets for in vivo nanothermometry is, however, connected to some inherent problems. Besides the challenges in size and morphology control of nanosized garnets [52] compared to, e.g., well-established synthesis routes to fluoride nanocrystals [46,47,48], the requirement for high spectral resolution and unambiguous assignment of the two closely lying emissive transitions to be attained under real in vivo conditions such as light scattering and absorption of surrounding tissue, dynamic flow, or movement of biological entities, is challenging. Even more severely, the theoretically achievable maximum relative sensitivity is well below 1% K⁻¹ and does not allow to achieve the required temperature measurement precision below 1 K even with use of laser excitation sources and sensitive light detection systems. Finally, the small energy gap between the two Kramers’ doublets inherently diminishes the absolute thermal response of the LIR because the two states have similar population densities at room temperature, as has been shown explicitly by thermodynamic modelling of a generalized excited two-level system to be reported in detail elsewhere [53].

One way to overcome this problem and achieve an order of magnitude higher relative sensitivities has already been demonstrated by a phonon-assisted energy transfer between Nd³⁺ and Yb³⁺ [30,54,55] or recently, also Nd³⁺ and Er³⁺ [43]. However, the underlying temperature calibration model for the temperature-dependent LIR based on energy transfer is far from trivial and, strictly stated, Boltzmann’s law does not apply for those situations [53]. In addition, the NIR emission of Yb³⁺ lies in between the two transparency windows BW I and II and strongly overlaps with the ⁴F_3/2 → ⁴I_11/2-related emission of Nd³⁺ thus introducing additional error sources for accurate ratiometric luminescence thermometry.

A promising alternative that would allow for calibration with Boltzmann’s law and does not require the introduction of additional dopants is usage of the two spin-orbit levels ⁴F_5/2 and ⁴F_3/2 of Nd³⁺ instead. They have an effective energy separation of roughly 1000 cm⁻¹ and emission from those two levels can be easily spectrally resolved. In addition, the higher energy gap permits an order of magnitude higher relative sensitivity (S_r > 1% K⁻¹) at physiological temperatures, which meets the requirements for in vivo nanothermometry. Besides an original work by Haro-González et al. in a Nd³⁺-doped Sr-Ba-based niobate glass demonstrating a proof of principle [56], a more elaborate analysis of thermometry with that energy gap has been recently reported by Kolesnikov et al. in Y₂O₃:Nd³⁺ nanocrystals [57]. However, thermal coupling between the ⁴F_5/2 and ⁴F_3/2 spin-orbit levels is an issue because of the relatively large gap of around 1000 cm⁻¹. The non-radiative transition rates between the two spin-orbit levels have to be faster than decay rates of these levels within the relevant temperature regime since otherwise Boltzmann-based equilibrium is not sustained [58]. In addition, higher doping concentrations to help increase the absorption efficiency of Nd³⁺ may also induce faster decay as a result of energy migration and cross-relaxation effects, preventing Boltzmann equilibrium. Thus, dependent on the electronic structure of a lanthanide ion, the effect of energy migration or cross relaxation at higher doping concentrations on the Boltzmann equilibrium between two excited levels of interest has to be investigated in detail.

Here we present and discuss the ⁴F_5/2 and ⁴F_3/2 based alternative thermometric sensing by Nd³⁺ and evaluate its applicability for in vivo nanothermometry using LaPO₄:Nd³⁺ nanocrystals as an illustrative example. The choice of the host material arises from the fact that phosphates have a large maximum phonon energy of around 1050 cm⁻¹ that allows for resonant thermalization between the ⁴F_5/2 and ⁴F_3/2 levels [59]. Moreover, phosphate nanocrystals may also be at least theoretically expected to show biocompatibility given its presence to a large fraction in the body of mammals. Besides a structural and optical characterization, especially the impact of both temperature and Nd³⁺ concentration on the thermometric performance of those nanocrystals will be analyzed. We will carefully compare the found results to the luminescence decay kinetics of the Nd³⁺ ions and compare predictions with experimental results. Our general purpose is to draw attention to some potential pitfalls in conventional Boltzmann thermometers, discuss their origin and provide solutions.

2. Materials and Methods

2.1. Synthesis

La_1−xNd_xPO₄ nanocrystals (x = 0.02, 0.05, 0.10, 0.25, 1.00) were synthesized with a common hydrothermal co-precipitation approach. For that, stoichiometric amounts of La(NO₃)₃ · 6 H₂O (Alfa Aesar, 99.99%, Karlsruhe, Germany) and Nd(NO₃)₃ · 6 H₂O (Alfa Aesar, 99.9%, Karlsruhe, Germany) were dissolved in deionized water. A small volume of 1 M NaOH (Chemos, 99%, Altdorf, Germany) solution was added under stirring of the solution to attain an alkaline medium. Afterwards, an aqueous solution containing a stoichiometric amount of (NH₄)₂HPO₄ (Alfa Aesar, 98%, Karlsruhe, Germany) was added to allow for precipitation of the rare earth phosphates. The resulting solution was transferred to a Teflon-lined stainless-steel autoclave and kept at 140 °C for 30 h to induce crystallization of the nanoseeds of the rare earth phosphates. After natural cooling to room temperature, the colorless precipitate was collected, washed with deionized H₂O and ethanol (EtOH) several times and finally dried at 40 °C for 22 h.

Microcrystalline, dilute La_0.999Nd_0.001PO₄ powder was synthesized for control experiments and to elucidate concentration effects. The synthesis was similarly performed by means of a co-precipitation approach from dissolved La(NO₃)₃ · 6 H₂O (Sigma Aldrich, 99.99%, Schnelldorf, Germany) and Nd(NO₃)₃ · 6 H₂O (Sigma Aldrich, 99.9%, Schnelldorf, Germany) in alkaline aqueous solution and precipitation with a saturated (NH₄)₂HPO₄ (Sigma Aldrich, 99%, Schnelldorf, Germany) solution. The colorless precipitated phosphate was filtered off, washed with deionized H₂O three times and carefully dried at 150 °C on air for 3 h. After grinding of the dried residue to a fine powder, it was finally annealed at 1000 °C in air for 3 days.

2.2. Structural and Morphological Characterization

The La_1−xNd_xPO₄ (x = 0.02, 0.05, 0.10, 0.25, 1.00) nanocrystalline powders were characterized by means of X-ray powder diffraction (XRPD) and transmission electron microscopy (TEM). XRPD patterns were measured on a SmartLab (Rigaku, Tokyo, Japan) instrument using Cu K_α radiation (λ = 1.54056 Å). The average crystallite size and internal strain of the nanocrystals were determined by a PDXL2 built-in package software using the known crystal structures of LaPO₄ and NdPO₄ as a structural input. The XRPD pattern of La_0.999Nd_0.001PO₄ was measured on a Philips PW391 X-ray diffractometer (Eindhoven, The Netherlands) with Cu K_α radiation (λ = 1.54056 Å) in reflection mode with an Al sample holder.

TEM images were acquired on a Thermo Fisher Scientific (formerly Philips, Eindhoven, The Netherlands) FEI Tecnai 12 microscope with an electron acceleration voltage of U = 100 kV. Samples were prepared by scooping a Cu TEM grid with carbon-coated polymer support film into the nanopowders.

2.3. Diffuse Powder Reflectance Spectroscopy

Diffuse reflectance spectra were acquired on a Shimadzu (Kyoto, Japan) UV-Visible UV-2600 spectrophotometer equipped with an integrated sphere (ISR-2600 Plus). All diffuse reflectance spectra were corrected for background using a BaSO₄ standard in the regarded spectral range (200–1200 nm).

2.4. (Time-Resolved) Luminescence Spectroscopy and Thermometry

Luminescence spectra were measured on an Edinburgh FLS920 spectrofluorometer (Livingston, UK) equipped with 0.25 m single grating monochromators and detected with a Hamamatsu R5509-72 NIR photomultiplier tube (PMT, Hamamatsu, Shizuoka, Japan) that was cooled with liquid N₂. All emission spectra were acquired by excitation with continuous wave (CW) power-tunable (power limit: P_max = 800 mW, Livingston, UK) lasers operating at λ_ex = 690 nm. A high-resolution emission spectrum to evaluate the impact of non-radiative relaxation from the ⁴F_5/2 to the ⁴F_3/2 level was measured upon excitation with a CW power-tunable (power limit: P_max = 2 W, Livingston, UK) laser with λ_ex = 808 nm. The incident laser power in each experiment was set to a sufficiently low value in order to avoid laser-induced heating with yet high signal-to-noise ratio. Photoluminescence excitation spectra were acquired upon monitoring the most intense emission of Nd³⁺ at λ_em = 1057 nm and a 450 W Xe lamp as external excitation source. Luminescence decay curves in the microsecond domain were obtained by external excitation with a pulsed wavelength-tunable Opotek (Carlsbad, CA, USA) Opolette 355 LD optical parametric oscillator (OPO) with a repetition rate of 20 Hz and temporal pulse width of around 6 ns. The time-resolved signal was detected with a multichannel scaler (MCS) connected to the NIR PMT. Temperature-dependent emission spectra above room temperature were measured in a Linkam (Surrey, UK) THMS600 Microscope Stage (±0.1 °C temperature stability) that could be placed in the spectrometer.

3. Results and Discussion

3.1. Structural and Morphological Characterization of the Nd³⁺-Activated LaPO₄ Nanocrystals

Both LaPO₄ and NdPO₄ crystallize in a monazite structure type with monoclinic crystal system and space group P2₁/n (no. 14) [60,61]. Only one crystallographically independent La³⁺ or Nd³⁺ site on the Wyckoff position 4e are present in both unit cells, respectively. Given the only slightly smaller ionic radius of Nd³⁺ (1.163 Å for nine-fold coordination) compared to that of La³⁺ (1.216 Å for nine-fold coordination) [62], a full range of solubility of NdPO₄ within the LaPO₄ host is expected. Synthesis of the solid solutions La_1−xNd_xPO₄ by a co-precipitation approach instead conventional heterogeneous mixing and subsequent thermal annealing permits a more random distribution of the Nd³⁺ activators substituting for the La³⁺ ions, as has also been recently independently established by solid state magic angle spinning nuclear magnetic resonance (MAS-NMR) experiments on the ³¹P nuclei in lanthanide-activated LaPO₄, whose resonances sensitively react on paramagnetic impurities in their close environment [63,64,65].

Figure 1a depicts the XRPD patterns of the Nd³⁺-activated LaPO₄ nanocrystals as prepared by the hydrothermal co-precipitation approach. The broad background in the low 2θ regime as well as the low intensities and large widths of most Bragg reflections already indicate a small average crystallite size of the particles. Rietveld refinement of the X-ray diffraction patterns employing the monazite structure type as structural input [60,61], affords estimated average crystallite sizes between 6 nm and 10 nm (see

Table 1. Average crystallite size estimates $\langle d\rangle$, lattice parameters a, b, c, cell volume V and strain of nanocrystals according to Rietveld refinement on the XRPD patterns depicted in Figure 1. The numbers in brackets refer to the errors in the last digits, respectively. The quality parameters for the Rietveld refinement are given below: Profile factor R_p, weighted profile factor R_wp and optimum expected R_exp factor together with the final goodness of the fit (G.o.f.) to the structural input of the monazite structure type.

	La0.98Nd0.02PO4	La0.95Nd0.05PO4	La0.90Nd0.10PO4	La0.75Nd0.25PO4	NdPO4
$\langle d\rangle$/nm	9.239(17)	6.852(18)	6.652(6)	9.057(19)	6.199(18)
a/Å	6.8631(13)	6.8504(14)	6.8592(19)	6.8390(15)	6.8380(4)
b/Å	7.1043(13)	7.0901(13)	7.0924(19)	7.0799(15)	6.9890(4)
c/Å	6.5290(12)	6.5171(12)	6.5141(18)	6.5053(14)	6.4210(4)
V/Å3	309.84(21)	308.08(39)	308.44(23)	306.57(26)	298.67(11)
Strain %	0.37(2)	0.39(5)	0.34(2)	0.50(8)	0.42(6)
Rp/%	5.79	6.22	6.58	5.93	7.80
Rwp/%	4.36	4.61	4.85	4.47	5.78
Rexp/%	3.84	3.94	3.81	3.68	3.13
G.o.f.	1.50	1.58	1.72	1.61	2.49

). The small R factors below 10% and the goodness of fit (G.o.f.) parameter close to 1 indicate a very good agreement between the theoretically expected powder diffraction pattern according to the structural input of the monazite-type phases and the experimentally measured diffraction patterns (see Table 1). Moreover, the resulting strain percentage within the nanocrystals is close to 0%, which reflects the miscibility of the two constituents LaPO₄ and NdPO₄ in the solid solution.

Additional evidence for the homogeneous distribution of the Nd³⁺ ions substituting for the La³⁺ sites in LaPO₄ is given by observation of a gradual shift of the Bragg reflections towards higher values of 2θ with increasing Nd³⁺ content in the nanocrystals (see Figure 1b). The microcrystalline sample was separated from that analysis since the well-defined reflections in that particular case merge together upon particle-size induced broadening of the Bragg reflections in the nanocrystals.

This effect leads to artificial shifts to yet lower Bragg angles despite an increase of the Nd³⁺ content from x = 0.001 to x = 0.02. Accordingly, the resulting cell volume as obtained from the Rietveld refinement also gradually decreases with higher Nd³⁺ content (see Table 1).

The TEM images (see Figure 2) of the La_1−xNd_xPO₄ nanocrystals confirm the estimated crystallite sizes according to the XRPD patterns and diameters in the range of 10 nm to 15 nm are found. The nanocrystals have an anisotropic rod-like shape, which is also expected given the monoclinic crystal system. An average aspect ratio of around length/width = 1.5 is found for all nanocrystals. Moreover, the TEM images reveal a strong aggregation tendency of the nanocrystals in the powder, which was also found previously in higher condensed La-based phosphates such as the tetraphosphates ALa_1−yNd_y(PO₃)₄ (A = Li – Rb; 0 ≤ y ≤ 1) [66].

3.2. Diffuse Reflectance and Optical Absorption

Diffuse reflectance spectra can serve as an additional characterization tool for the successful incorporation of Nd³⁺ ions into the synthesized Nd³⁺-activated LaPO₄ nanocrystals. The very low Nd³⁺ concentration in microcrystalline La_0.999Nd_0.0001PO₄ did not give rise to a measurable reflectance signal and was thus excluded. A measure for the absorption coefficient of the respective powdered samples is accessible by the Kubelka-Munk function K/S under the assumption of a constant scattering part of the powders:

$$\frac{K}{S}=f\left(R_{\infty }\right)=\frac{{\left({1-R}_{\infty }\right)}^2}{{2R}_{\infty }},$$

(1)

where $R_{\infty }$ denotes the diffuse reflectance in the limit of a much higher scale of layer thickness of the powder compared to the average crystallite size, which is clearly fulfilled under the employed measurement conditions. The Kubelka-Munk spectra are depicted in Figure 3a. As expected, the narrow 4f³(⁴I_9/2) → 4f³(^2S+1L_J) transitions of Nd³⁺ show an increasing absorption strength with increasing Nd³⁺ content of the nanocrystals. Indeed, a linear correlation between the integrated Kubelka-Munk signal—if accordingly translated to an energy scale—and nominal Nd³⁺ concentration x is present over the whole regime between x = 0 and x = 1, in agreement with expectations from electromagnetic dispersion theory (see Figure 3b) [67]. On a statistical significance level of α = 0.05, the intercept of the linear calibration line does not differ from zero, as verified by a conventional t test. Altogether, the diffuse reflectance spectra confirm a homogeneous miscibility between the LaPO₄ and NdPO₄ phases within the nanocrystals.

In the context of suitability for in vivo nanothermometry, the Kubelka-Munk spectra (see Figure 3a) also reveal that the strongest absorption transitions of Nd³⁺ within the discussed nanocrystals are located in the first biological window between 650 nm and 950 nm. This demonstrates the general suitability of Nd³⁺ as a potent absorber for in vivo applications since its absorption remains negligibly affected by light attenuation due to surrounding tissue. The linear correlation of the integrated absorption strength with concentration in the La_1−xNd_xPO₄ nanocrystals does in principle allow for a simple strategy to improve the absorption strength in the Nd³⁺-activated nanocrystals using higher Nd³⁺ contents. As will be discussed below, however, this can induce both energy migration and cross-relaxation effects that limit the temperature window and thermometric performance of the nanocrystals.

3.3. Photoluminescence Properties and Luminescence Decay Dynamics—Predictions on Consequences for Thermometry with Nd³⁺

Figure 4a depicts the photoluminescence emission spectra of the La_1−xNd_xPO₄ nanocrystals at room temperature upon CW laser excitation at 690 nm into the ⁴F_9/2 spin-orbit level of the Nd³⁺ ions. Irrespective of the Nd³⁺ content, the room temperature emission spectra show radiative transitions from the ⁴F_3/2 level into the ground levels ⁴I_9/2 ($\langle {\lambda }_{\mathrm{em}}\rangle$ = 890 nm), ⁴I_11/2 ($\langle {\lambda }_{\mathrm{em}}\rangle$ = 1063 nm) and ⁴I_13/2 ($\langle {\lambda }_{\mathrm{em}}\rangle$ = 1341 nm). The ⁴F_3/2 → ⁴I_11/2 transition has the largest intensity, reflecting a large branching ratio of around β_11/2(⁴F_3/2) = 0.67 compared to the other observable radiative transitions (β_9/2(⁴F_3/2) = 0.17, β_13/2(⁴F_3/2) = 0.16). This is understandable as $\left|\mathsf{\Delta }L\right|$ = 3, $\left|\mathsf{\Delta }J\right|$ = 4 implies a Judd-Ofelt allowed forced electric dipole transition and moreover, Nd³⁺ is typically characterized by large Ω₄ and Ω₆ Judd-Ofelt intensity parameters in phosphates [68,69]. Similar findings have been reported in the previously mentioned higher condensed phosphates ALa_1-yNd_y(PO₃)₄ (A = Li – Rb; 0 ≤ y ≤ 1) [66]. As expected, the emission spectra become inhomogeneously broadened and the spectral resolution of the different crystal field components decreases with increasing Nd³⁺ content. The corresponding photoluminescence excitation spectra at room temperature acquired upon monitoring the most intense ⁴F_3/2 → ⁴I_11/2 emissive transition of Nd³⁺ at 1057 nm are depicted in Figure 4b). While the Kubelka-Munk spectra clearly suggest that the ⁴I_9/2 → ⁴F_5/2 transition at $\langle {\lambda }_{\mathrm{abs}}\rangle$ = 808 nm shows the maximum absorption coefficient, as expected for a Judd-Ofelt-allowed transition ($\left|\mathsf{\Delta }L\right|$ = 3, $\left|\mathsf{\Delta }J\right|$ = 2), the excitation spectra reveal that absorption into any of the ⁴F_J (J = 3/2…9/2) levels efficiently induces emission from the ⁴F_3/2 level at room temperature, in particular with increasing Nd³⁺ content. Like in the emission spectra, the respective excitation transitions also suffer inhomogeneous broadening. Moreover, several artefacts due to the Xe lamp lines are present in the excitation spectra (see peaks marked with asterisks in Figure 4b). These artefacts can serve as an internal intensity standard. The strong decrease in relative intensity of Nd³⁺ excitation lines relative to the Xe-lamp lines with x indicates the presence of concentration quenching of the photoluminescence in the nanocrystals.

Important information on radiative and non-radiative decay rates can be obtained from luminescence decay measurements. Figure 5a depicts the luminescence decay curves obtained upon direct excitation into the ⁴F_3/2 level of Nd³⁺ at 870 nm and detection of the most intense ⁴F_3/2 → ⁴I_11/2 emissive transition around 1060 nm. The intensity of the ⁴F_3/2–excited emission at λ_em = 1058 nm in a very dilute, microcrystalline La_0.999Nd_0.001PO₄ control sample (λ_ex = 870 nm) decays monoexponentially with an (assumed purely radiative) decay rate of k_r(⁴F_3/2) = 2.25 ms⁻¹ at room temperature (see Figure 5a). With increasing Nd³⁺ content, a faster and non-exponential decay is observed. This behavior is a clear signature of an energy transfer processes involving quenching of the ⁴F_3/2 emission. In order to gain insight in the underlying quenching mechanism, the average decay rates $\langle k\rangle$ were determined with Equation (2) from the decay data depicted in Figure 5a.

$$\langle k\rangle =\frac{1}{\langle \tau \rangle }=\frac{{{\displaystyle \sum }}_{j}I\left(t_j\right)}{{{\displaystyle \sum }}_j{t}_j\times I\left(t_j\right)}$$

(2)

In Equation (2), I(t_j) denotes the normalized, background-corrected luminescence intensity at time t_j and j runs over all acquired data points. In the case of NdPO₄, the luminescence decay is already so fast that it was not feasible to reliably determine an average decay rate (see Figure 5a). Thus, it was excluded from further analysis. A plot of the average decay rates in La_1−xNd_xPO₄ versus the Nd³⁺ concentration x reveals an approximately linear relation between the two quantities (see Figure 5b). This is a clear signature of a two-ion process [70] and indicates that cross-relaxation of the ⁴F_3/2 level between the Nd³⁺ ions becomes active in the LaPO₄ host even at concentrations as low as 2 mol%. An explanation for the high cross-relaxation efficiency is the small nearest neighbor distance of Nd³⁺ ions in the orthophosphates (only 4.036 Å (NdPO₄), 4.104 Å (LaPO₄)) [60,61], which allows for an efficient electric dipole—electric dipole-type energy transfer. At lower concentrations and homogeneous doping, the average distances between Nd³⁺ ions will be larger and thus, the cross-relaxation efficiency is reduced. However, for electric dipole-electric dipole type energy transfer, cross-relaxation can be still effective over distances as large as 10 Å.

Inspection of the energy level diagram of Nd³⁺ shows that indeed the phonon-assisted cross-relaxation pathways [Nd1, Nd2]: [⁴F_3/2, ⁴I_9/2] → [⁴I_15/2, ⁴I_13/2] + $\hslash {\mathsf{\omega }}_{\mathrm{eff}}$ and [⁴F_3/2, ⁴I_9/2] → [⁴I_13/2, ⁴I_15/2] + $\hslash {\mathsf{\omega }}_{\mathrm{eff}}$ can take place if $\hslash {\mathsf{\omega }}_{\mathrm{eff}}\approx$ 1050–1100 cm⁻¹ (see Figure 5c). The required phonon energy agrees very well with the asymmetric O-P-O stretching vibration [59,66], whose energy also matches the energy gap between the ⁴F_5/2 and ⁴F_3/2 level. The ⁴F_5/2 level is even prone to resonant cross-relaxation, [Nd1, Nd2]: [⁴F_5/2, ⁴I_9/2] → [⁴I_15/2, ⁴I_15/2] (see Figure 5c). The possibility of both the ⁴F_3/2 and ⁴F_5/2 level to decay via cross-relaxation effectively increases the decay rate of both spin-orbit levels. This is expected to have immediate consequences for the thermometric performance of Nd³⁺ once the average decay rates of those levels become similar to the non-radiative transition rates governing the thermal coupling of the excited states. If the average decay rates supersede the non-radiative transition rates to bridge the ⁴F_5/2 – ⁴F_3/2 gap, thermodynamic Boltzmann equilibrium cannot be sustained anymore, and the thermometric performance is lost.

In order to obtain a semi-quantitative measure for the non-radiative transition rates mediating the thermal coupling between the ⁴F_5/2 and ⁴F_3/2 levels of Nd³⁺, luminescence decay curves upon selective excitation into each of those levels were recorded for the dilute microcrystalline La_0.999Nd_0.001PO₄ sample. For excitation into either the ⁴F_3/2 (λ_ex = 870 nm) or ⁴F_5/2 level (λ_ex = 790 nm) and monitoring the luminescence decay of the ⁴F_3/2 → ⁴I_11/2-related transition at 1058 nm at room temperature, a purely single exponential decay with a radiative decay rate of k_r(⁴F_3/2) = 2.25 ms⁻¹ is observed with errors well below 0.01 ms⁻¹ (see Figure 6a,b).

This observation indicates that the sum of radiative and non-radiative decay rate from the ⁴F_5/2 level, k_r(⁴F_5/2) + $k_{\mathrm{nr}}^{\mathrm{em}}\left(T\right)$, have to be much higher than the radiative decay rate k_r(⁴F_3/2) since otherwise a rise component is expected in the decay curve recorded for ⁴F_5/2 excitation. For selective excitation at 870 nm (resonant with excitation into the ⁴F_3/2 level) very weak anti-Stokes ⁴F_5/2 emission is observed with a much faster initial decay component besides the slower decay of the overlapping ⁴F_3/2 → ⁴I_9/2 – related luminescence. This fast ~4 μs initial decay is assigned to the total decay rate of the ⁴F_5/2 level. In order to obtain a more precise value, upon selective excitation into the ⁴F_5/2 level, the luminescence decay was recorded for the strongest ⁴F_5/2 → ⁴I_9/2 – related emission at 804 nm (see Figure 7a). A fast decay with an average (radiative and non-radiative) decay rate of k_r(⁴F_5/2) + $k_{\mathrm{nr}}^{\mathrm{em}}(T=298\mathrm{K})$ = 204 ms⁻¹ was found using Equation (2). The value agrees well with the result for the fast component from the double exponential fit (k_r(⁴F_5/2) + $k_{\mathrm{nr}}^{\mathrm{em}}\left(298\mathrm{K}\right)$ = (242 ± 12) ms⁻¹) as depicted in Figure 6c. For the following analysis, we will employ the average value k_r(⁴F_5/2) + $k_{\mathrm{nr}}^{\mathrm{em}}\left(298\mathrm{K}\right)$ = (223 ± 19) ms⁻¹.

In order to separate the contribution of radiative decay rate k_r(⁴F_5/2) from the non-radiative rate $k_{\mathrm{nr}}^{\mathrm{em}}\left(T\right)$ at temperature T, an emission spectrum containing the emissive transitions from both the ⁴F_5/2 and ⁴F_3/2 spin-orbit levels into the same ground level ⁴I_13/2 was recorded for selective excitation into the ⁴F_5/2 level (λ_ex = 808 nm). The ratio between the integrated intensities (if measured in photon counts) of the emission, I_em, from the ⁴F_5/2 and the ⁴F_3/2 is equal to the ratio between k_r(⁴F_5/2) and $k_{\mathrm{nr}}^{\mathrm{em}}\left(T\right)$, respectively:

$$\frac{k_{\mathrm{r}}\left({}_{}{}^{4}F{}_{5/2}\right)}{k_{\mathrm{nr}}^{\mathrm{em}}\left(T\right)}=\frac{I_{\mathrm{em}}\left({}_{}{}^{4}F{}_{5/2}\right)}{I_{\mathrm{em}}\left({}_{}{}^{4}F{}_{3/2}\right)}=\frac{\beta ({}_{}{}^{4}F{}_{3/2}\to {}_{}{}^{4}I{}_{13/2}{)I}_{\mathrm{em}}({}_{}{}^{4}F{}_{5/2}\to {}_{}{}^{4}I{}_{13/2})}{\beta ({}_{}{}^{4}F{}_{5/2}\to {}_{}{}^{4}I{}_{13/2}{)I}_{\mathrm{em}}({}_{}{}^{4}F{}_{3/2}\to {}_{}{}^{4}I{}_{13/2})}$$

(3)

with $\beta ({}_{}{}^{4}F{}_{3/2}\to {}_{}{}^{4}I{}_{13/2})$ = 0.16 and $\beta ({}_{}{}^{4}F{}_{5/2}\to {}_{}{}^{4}I{}_{13/2})$ = 0.60 as the branching ratios of the respective transitions derived from the luminescence spectra (see Figure 4a and Figure 7b). Equation (3) is only valid if thermally excited non-radiative absorption from the ⁴F_3/2 level back to the ⁴F_5/2 is negligible and if there are no other non-radiative decay paths for the two levels.

The temperature dependence of non-radiative rates among 4fⁿ-related spin-orbit levels is governed by thermally excited multi-phonon transitions. The thermal average number $\langle n_{\mathrm{eff}}\rangle$ of effective phonon modes with energy $\hslash {\mathsf{\omega }}_{\mathrm{eff}}$ resonantly bridging the regarded energy gap between two electronic levels is given by the Planck formula:

$$\langle n_{\mathrm{eff}}\rangle =\frac{1}{\exp \left(\frac{\hslash {\mathsf{\omega }}_{\mathrm{eff}}}{k_{\mathrm{B}}T}\right)-1}$$

(4)

with k_B as the Boltzmann constant. With an effective phonon energy of $\hslash {\omega }_{\mathrm{eff}}\approx$ 1050 cm⁻¹, even at room temperature (T = 298 K), it is $\langle n_{\mathrm{eff}}\rangle \approx$ 6.34 ∙ 10⁻³, i.e., effectively, one phonon mode is only thermally excited with a very low probability. Based on Equation (3) and the emission spectrum depicted in Figure 7b, a non-radiative emission rate of $k_{\mathrm{nr}}^{\mathrm{em}}$(298 K) = g₁k_nr(0)$\left(1+\langle n_{\mathrm{eff}}\rangle \right)$ ≈ g₁k_nr(0) = (219 ± 19) ms⁻¹ can be derived, where g₁ = 4 is the (2J + 1)-fold degeneracy of the ⁴F_3/2 level. Thus, the intrinsic non-radiative rate is estimated to be k_nr(0) = (54.6 ± 4.7) ms⁻¹.

The validity of the employed approximation of a still negligible thermal multi-phonon absorption rate for the non-radiative transition ⁴F_3/2 → ⁴F_5/2 can now be verified. With the value for k_nr(0) and the degeneracy of the ⁴F_5/2 level, g₂ = 6, it is $k_{\mathrm{nr}}^{\mathrm{abs}}$(298 K) = g₂k_nr(0)$\langle n_{\mathrm{eff}}\rangle$ = (2.08 ± 0.18) ms⁻¹, which agrees with k_r(⁴F_3/2) within the statistical error. Thus, the approximation to neglect that non-radiative absorption pathway for Nd³⁺ ions in the dilute La_0.999Nd_0.001PO₄ microcrystals is just about to fail at room temperature and Boltzmann behavior should be expected to set in just above room temperature. In the case of the concentrated Nd³⁺-doped nanocrystals La_1−xNd_xPO₄, however, the additional cross-relaxation pathways and possible quenching of the ⁴F_3/2 and ⁴F_5/2 emission by high energy vibrations of any capping ligands such as -OH groups on the nanocrystal surface effectively increases the decay rate of the ⁴F_3/2 and ⁴F_5/2 level (see Figure 5a). Consequently, Boltzmann equilibrium is then expected to become active only at successively higher temperatures, once $k_{\mathrm{nr}}^{\mathrm{abs}}\left(T\right)$ supersedes these higher average decay rates.

Table 2. Derived radiative and non-radiative rates characterizing the thermal coupling of the ⁴F_3/2 and ⁴F_5/2 spin-orbit levels of Nd³⁺ in microcrystalline La_0.999Nd_0.001PO₄.

kr(4F3/2)/ms−1	kr(4F5/2)/ms−1	knr(0)/ms−1	${\mathit{k}}_{\mathbf{nr}}^{\mathbf{em}}$(298 K)/ms−1	${\mathit{k}}_{\mathbf{nr}}^{\mathbf{abs}}$(298 K)/ms−1
2.25	3.16 ± 0.27	54.6 ± 4.7	219 ± 19	2.08 ± 0.18

compiles the relevant decay rates derived from decay dynamics and emission spectra of the dilute microcrystalline La_0.999Nd_0.001PO₄ sample.

3.4. Consequences of Cross-Relaxation on Luminescence Thermometry Employing the ⁴F_5/2 and ⁴F_3/2 Spin-Orbit Levels of Nd³⁺

For Boltzmann-based luminescence thermometry, it is beneficial to select a different auxiliary excited state that feeds the thermally coupled states giving rise to the temperature-dependent luminescence phenomena. Direct excitation into one of those levels provides an additional non-equilibrium component to the excited state population and although steady-state conditions might be retained, the single ion luminescence thermometer is driven out of thermodynamic equilibrium. Thus, despite the high absorption strength of the ⁴I_9/2 → ⁴F_5/2 transition at 808 nm, we decided to selectively excite into the ⁴F_9/2 spin-orbit level with a corresponding absorption wavelength of 690 nm for all luminescence thermometry experiments. The temperature-dependent luminescence spectra of the microcrystalline La_0.999Nd_0.001PO₄ upon excitation at 690 nm are depicted in Figure 8.

All samples were investigated over a wide temperature range, also above physiological temperatures, for the sake of better insight into the thermal coupling between the ⁴F_3/2 and ⁴F_5/2 levels and to investigate the predicted thermometric performance at higher temperatures. In the case of thermodynamic equilibrium conditions between the two excited levels, the LIR, R(T), should obey Boltzmann’s law:

$$R\left(T\right)=\frac{I_{20}}{I_{10}}=C\frac{g_2}{g_1}\exp \left(-\frac{{\mathsf{\Delta }E}_{21}}{k_{\mathrm{B}}T}\right),$$

(5)

where we denote the excited ⁴F_3/2 level as state $|1\rangle$ and the ⁴F_5/2 level as state $|2\rangle$ in the following. The ground level ⁴I_9/2 is referred to as state $|0\rangle$. Then, g₁ = 4 and g₂ = 6 represent the (2J + 1)-fold degeneracies of the two spin-orbit levels, while ΔE₂₁ = 1020 cm⁻¹ as derived from the Kubelka-Munk spectra is the energy gap between the two excited levels. This value is in good agreement with expectations according to the Dieke—Carnall diagram [71]. k_B is Boltzmann’s constant and C is a pre-exponential factor that basically relates the electronic line strengths of the regarded radiative transitions of interest. It can also be estimated from Judd-Ofelt theory [14,69], but will be considered as a fitting parameter in the present study. Figure 9 depicts the thermometric Boltzmann plots derived from the temperature-dependent luminescence spectra of the La_1−xNd_xPO₄ nanocrystals in the wavelength range between 775 nm and 950 nm (i.e., in BW I). All spectra were corrected for constant instrumental background of the NIR PMT and additional blackbody radiation background, B(λ, T), at a given temperature T to obtain meaningful LIR data,

$$B\left(\lambda ,T\right)=A+\frac{{2Dhc}^2}{{\lambda }^5\left[\exp \left(\frac{hc}{{\lambda k}_{\mathrm{B}}T}\right) -1\right]}$$

(6)

with A and D as free parameters, λ as the wavelength, h as Planck’s constant and c as the light velocity. Due to concentration quenching the luminescence in NdPO₄ was so weak, especially at temperatures above 100 °C, that no meaningful information could be extracted and thus, it was excluded from further analysis. The predictions according to the luminescence decay kinetics of Nd³⁺ can be directly compared to the experimental data. The LIR of the ⁴F_J → ⁴I_9/2 (J = 3/2, 5/2) transitions in La_0.999Nd_0.001PO₄ shows Boltzmann behavior over the full temperature range from 30 °C to 500 °C. This observation is consistent with the observation that the non-radiative absorption rate $k_{\mathrm{nr}}^{\mathrm{abs}}$(T) governing the non-radiative ⁴F_3/2 → ⁴F_5/2 transition is similar to the radiative decay rate of the ⁴F_3/2 level at room temperature and increases at higher temperatures. This will result in Boltzmann equilibrium starting just above room temperature. Overall, thermodynamic equilibrium between the two spin-orbit levels can be sustained over the full temperature range above ~300 K.

In the La_1−xNd_xPO₄ (x > 0.001) nanocrystals, the average decay rate of the ⁴F_3/2 level increases with Nd³⁺ content due to the additional cross-relaxation pathways between neighboring Nd³⁺ ions. This additional decay channel competes with the non-radiative absorption rate and can hamper Boltzmann equilibration. Given the known temperature dependence of $k_{\mathrm{nr}}^{\mathrm{abs}}$(T) (scaling with the Planck factor in Equation (4)) and the faster decay of the ⁴F_3/2 level (from the decay curves of the ⁴F_3/2 emission in Figure 5), it is possible to determine the threshold temperature T_on, above which Boltzmann behavior is expected in the more concentrated Nd³⁺-samples. T_on is taken as the temperature at which the non-radiative absorption $k_{\mathrm{nr}}^{\mathrm{abs}}$ becomes faster than the total decay rate of the ⁴F_3/2 level. In very good agreement with expectations, a shifted onset of the Boltzmann behavior is observed in all higher concentrated La_1−xNd_xPO₄ nanocrystals, both experimentally and theoretically. The predicted onset temperatures T_on derived from the requirement of equal non-radiative absorption and average decay rates from the ⁴F_3/2 level increases with Nd³⁺ content x and are indicated in Figure 9. For x = 0.02, Boltzmann equilibrium becomes problematic in the physiological temperature window and for x = 0.05, temperature sensing becomes possible only above 450 K. For the higher Nd³⁺ concentrations (x = 0.25) no Boltzmann behavior is observed even up to 500 °C. The predicted temperature at which the non-radiative absorption rate dominates is even higher, above 900 °C. Thus, although a higher Nd³⁺ content may increase the absorption efficiency of the nanocrystals (cf. Figure 3b), the efficient cross-relaxation of Nd³⁺ ions prevents sustainment of a Boltzmann equilibrium for the excited ⁴F_3/2 and ⁴F_5/2 levels for a larger temperature range. As a result, higher Nd³⁺ contents destroy the promising potential of the large ⁴F_5/2–⁴F_3/2 gap for Boltzmann thermometry with high sensitivity at physiological temperatures. In contrast, higher temperature thermometry (100 °C–500 °C) is still feasible even for Nd³⁺ concentrations as high as x = 0.10. Measurements of higher temperatures in BW I (and consequently, also in BW II) with those nanocrystals is however cumbersome since the blackbody background starts to dominate the emission spectrum and temperature can be measured more accurately from the background itself.

The effect of cross-relaxation on the useable temperature window for LIR temperature sensing can differ among lanthanide ions. Here we show for the ⁴F_3/2 and ⁴F_5/2 levels of Nd³⁺ that high dopant concentrations are clearly detrimental because cross-relaxation shortens the lifetime of the emitting levels, thus limiting the time available for Boltzmann equilibration. However, cross-relaxation can also be beneficial, if it provides an additional pathway for thermalization between the emitting levels, thus establishing Boltzmann equilibrium. This has been shown to be the case for the ⁵D₀ and ⁵D₁ levels of Eu³⁺ where cross-relaxation between neighboring Eu³⁺ ions provides an alternative path for relaxation and thus sustains Boltzmann behavior over a wider temperature range at elevated Eu³⁺ concentrations, as was demonstrated in the case of β-NaYF₄:Eu³⁺ [58].

It is noteworthy that the fitted effective energy gaps, ΔE₂₁, gradually increase with higher Nd³⁺ content x. This can be explained by the fact that the fitted effective energy gap for the nanocrystals with higher Nd³⁺ contents are obtained from higher temperature data. At higher temperatures, the probability for non-radiative absorption into the higher crystal field states of the ⁴F_5/2 spin-orbit level increases, which effectively increases the energy gap ΔE₂₁. Thus, only in the very dilute La_0.999Nd_0.001PO₄ compound does the energy gap agree with the spectroscopically deduced energy gap according to the Kubelka-Munk spectra (ΔE₂₁ = 1020 cm⁻¹, see Figure 3a). Interestingly, also the pre-exponential constant C systematically increases with increasing Nd³⁺ content x. As it is fundamentally related to both the branching ratios and the radiative decay rates from the two considered excited states of Nd³⁺, its increase may also be related to the temperature-induced population of higher energetic crystal field states of the ⁴F_3/2 and ⁴F_5/2 levels. In conjunction with those observations, the predicted onset temperatures for Boltzmann behavior for the nanocrystals activated with 5 mol% and 10 mol% Nd³⁺ are significantly higher than the experimentally observed onsets (see Figure 9c,d). A possible explanation is a decreasing cross-relaxation efficiency of the ⁴F_3/2 level at higher temperatures. Since cross-relaxation of the ⁴F_3/2 level requires one high energy phonon mode of the phosphate host, an increasing temperature and population of the higher energy crystal field level may lead to an energy mismatch of the necessary energy transfer resonance condition and could thus reduce the cross-relaxation efficiency. Temperature-dependent luminescence decay analyses and modelling of the energy migration processes are necessary to confirm this hypothesis.

Since thermodynamic equilibrium between the ⁴F_5/2 and ⁴F_3/2 level of Nd³⁺ is sustained over the full temperature range investigated (30 °C–500 °C) in La_0.999Nd_0.001PO₄, physiological temperatures are measurable by means of luminescence thermometry with that compound. As Boltzmann behavior is realized, the relative sensitivity S_r (in % K⁻¹) of the luminescence thermometer is given by

$$S_{\mathrm{r}}=\left|\frac{1}{R\left(T\right)}\frac{\mathrm{d}R}{\mathrm{d}T}\right|=\frac{{\mathsf{\Delta }E}_{21}}{k_{\mathrm{B}}{T}^2}$$

(7)

Figure 10 depicts the evolution of the relative sensitivity for La_0.999Nd_0.001PO₄ as obtained from Equation (7). In particular, it is higher than 1% K⁻¹ for the full physiological temperature regime (30 °C–75 °C), which is practically difficult to achieve around room temperature with any single ion Boltzmann thermometer, especially in the NIR regime [49,50,51,57]. Typically, energy transfer-based thermometers are used in those cases [30,43,54,55] for which the underlying thermometric mechanisms are often not well established. The present results show promising potential of Nd³⁺ for physiological temperature sensing by means of luminescence thermometry, if the boundary conditions for the validity of a Boltzmann equilibrium are met which requires low Nd³⁺ concentrations. The ⁴F_5/2–⁴F_3/2 spin-orbit gap in Nd³⁺ gives rise to relative sensitivities that are an order of magnitude higher than the typically found ones in the range of 0.25% K⁻¹ [50]. A disadvantage of the presented thermometric concept of Nd³⁺ is the rather low intensity of the ⁴F_5/2 → ⁴I_9/2 emission that can give rise to a higher relative intensity uncertainty, depending on the sensitivity of the detection system. Since it is the temperature uncertainty that matters in a well performing luminescence thermometer, both the relative sensitivity and emission intensities have to be optimized.

4. Conclusions

Nd³⁺ is a promising candidate for in vivo luminescence thermometry due to its intense radiative transitions in the biological transparency windows BW I and BW II. Most Nd-based single ion thermometers utilize the two crystal field states arising from the excited ⁴F_3/2 spin-orbit level. These thermometers are fundamentally limited in their relative thermal sensitivity S_r due to the small energy gap of ~100 cm⁻¹ and the high demands on spectral resolution which are difficult to meet under in vivo conditions. An alternative promising probe for physiological temperature sensing is the ⁴F_5/2–⁴F_3/2 spin-orbit gap of Nd³⁺ which is ~1000 cm⁻¹. In order to investigate the feasibility of temperature sensing based on the temperature dependent emission intensity ratio of ⁴F_3/2- and ⁴F_5/2-related emission, both microcrystalline, dilute La_0.999Nd_0.001PO₄ and nanocrystalline La_1−xNd_xPO₄ (x = 0.02, 0.05, 0.10, 0.25, 1.00) were synthesized by means of a co-precipitation approach and structurally, morphologically and optically characterized. Analysis of the decay kinetics reveals that the ⁴F_3/2 and ⁴F_5/2 excited states of Nd³⁺ are prone to efficient cross-relaxation quenching at higher Nd-concentrations. Cross-relaxation competes with the non-radiative transition rates governing the thermalization of the ⁴F_5/2 and ⁴F_3/2 spin-orbit levels. This additional decay pathway leads to a gradual breakdown of Boltzmann equilibrium at physiological temperatures with increasing Nd³⁺ concentration. Only for low Nd³⁺ concentrations (x < 0.02) is it possible to sustain Boltzmann equilibrium between the ⁴F_5/2 and ⁴F_3/2 levels of Nd³⁺ in the physiological temperature regime. The relative sensitivity in this important temperature regime exceeds 1% K⁻¹. Overall, the present study demonstrates that a careful analysis of the excited state dynamics allows for an assessment of the performance of a luminescence thermometer, but also demonstrates the potential pitfalls of Boltzmann thermometry that can be encountered. A mechanistic understanding of the competition between radiative and non-radiative decay processes can lead to a tailored design of novel luminescence thermometers with optimal performance in a specific temperature window.