The photoluminescence of Gd³⁺-containing compound 3 has been used to assess the energies of the triplet states located on the terminal acac ligand since Gd³⁺ has a high energy-accepting electronic level, which prevents any population through the energy transfer from the electronic level of the antenna ligand. Supplementary Figure S3, ESI, shows the emission spectrum of compound 3 obtained upon excitation at 246 nm at 77 K. The energy of the triplet state is 25,316 cm⁻¹ (corresponding to an emission peak at 395 nm).

The excitation spectra of Tb³⁺-containing compound 1 monitored within the main ⁵D₄→⁷F₅ transition (543 nm) in the range 11–300 K exhibit large broadband, with a main component at ca. 330 nm attributed to the acac ligand excited state (Figure 3A). A series of low-intensity narrow lines attributed to transitions between ⁷F₆ and the ⁵G₅, ⁵G₆, and ⁵D₃ excited states indicates that Tb³⁺ is mainly populated through antenna-assisted sensitization rather than by a direct excitation into the intra-4f⁸ lines. The emission spectra of compound 1 recorded under excitation at 330 nm in the 11–297-K and 298–378-K intervals (Figures 3B,C) exhibit the classical Tb^{3+ 5}D₄→⁷F_6-0 characteristic emission lines. The profile of the emission spectra is similar under direct 4f⁸ excitation at 485 nm (Supplementary Figures S4A, B, ESI). The most intense emission, centered at 543 nm, corresponds to the ⁵D₄→⁷F₅ transition, displaying a gradual decrease with increasing temperature, as shown in Figure 4 and Supplementary Figures S4C, S5 (ESI). The emission decay curve of compound 1 was monitored at room temperature within ⁵D₄→⁷F₅. The curve is well-reproduced by a single exponential function, yielding a ⁵D₄ lifetime of 0.851 ± 0.002 ms, which is a rather typical value for Tb³⁺ complexes (Supplementary Figure S6). The presence of two maxima in the ⁵D₄→⁷F₅ transition, I ₁ and I ₂ in Figure 4, is associated with two distinct Stark components. Although the maximum number of ⁵D₄ and ⁷F₅ Stark levels can lead to 99 components in the ⁵D₄→⁷F₅ transition, once the majority is degenerate, only a few can be observed (Figure 4). A question can be raised concerning whether some of these transitions between Stark levels could be attributed to vibronic transitions (or sidebands). This can be answered with the emission spectrum recorded at 11 K (Figure 5A), where vibrational modes that could couple in this spectral region can be suppressed. As the emission intensity is directly proportional to the population of the emitting level, the model suggests that emission I ₁ originates from a Stark component ( 1 in Figure 5B) associated with the lower Stark level of ⁵D₄. In contrast, emission I ₂ comprises the emission from a Stark level with higher energy ( 2 in Figure 5B). It is important to note that the high-energy emission is not always attributed to a Stark level with higher energy as the energy of the transition also depends on the energy of the ending level (⁷F₅), as shown in Figure 5B.

The model assumes that the high-energy emission stems from a lower-energy Stark component of ⁵D₄. This assumption is relevant because, at 11 K, there would be no thermal population arising from 1 to 2 , and the emission I ₂ should vanish at low-temperature ranges because W 2 → 1 >> W 1 → 2 (Figure 5C), where W 2 → 1 and W 1 → 2 are the multiphonon decay rate (creation of phonons) and absorption (annihilation of phonons) rates, respectively, between 1 and 2 Stark components of the ⁵D₄ level. The decay rate W 2 → 1 was calculated from the energy gap law (Moos, 1970; Reisfeld et al., 1977; Riseberg et al., 1977; Malkin et al., 2005) in the Miyakawa–Dexter approach (Supplementary Eqs S1,S2) (Miyakawa and Dexter, 1970). The intramolecular energy transfer (IET) rates from T₁ to ⁵D₄ levels were calculated using Supplementary Eqs S5–S7 (Carneiro Neto et al., 2019) (details given in ESI) using the JOYSpectra web platform (Moura Jr. et al., 2021b).

Based on the obtained IET and multiphonon rates (Figure 5C; Supplementary Table S3), together with the energy level diagram shown in Figures 5A,B, a three-level rate equation model can be considered as follows: d η 1 t d t = W 2 → 1 η 2 t − W 1 → 2 + 1 τ η 1 t (1) d η 2 t d t = W 1 → 2 η 1 t + ϕ η 3 t − W 2 → 1 + 1 τ η 2 t (2) d η 3 t d t = 1 τ η 1 t + η 2 t − ϕ η 3 t (3) where η 1 , η 2 , and η 3 are the populations of 1 , 2 , and ⁷F₅, respectively, with initial conditions (when t = 0) of η 1 0 = η 2 0 = 0 and η 3 0 = 1 . τ = 0.851 ms is the measured decay lifetime of the ⁵D₄ level, and ϕ is the feeding rate of the emitting level ⁵D₄ that comes mainly from the energy transfer rates involving the ⁷F₆ and ⁷F₅ levels as the starting level (e.g., IET rates from ligand states to Tb^{3+ 7}F₆→⁵D₄ and ⁷F₅→⁵D₄) (Kasprzycka et al., 2020; Carneiro Neto et al., 2022).

One premise of the present model is that the ϕ feeding rate is attributed to the direct energy transfer from the T₁ state to the ⁵D₄ level, following the pathways [T₁→S₀]→Tb³⁺[⁷F₆→⁵D₄] and [T₁→S₀]→Tb³⁺[⁷F₅→⁵D₄] (Supplementary Table S3). It is noteworthy that the last pathway dominates the direct energy transfer process with a rate of 1.9 × 10⁶ s⁻¹, and the exchange mechanism (Supplementary Eq, S7) has the most significant contribution (Supplementary Table S3).

Although the energy transfer from the S₁ state has been a recent topic of debate in the literature (Alaoui, 1995; Rodríguez-Cortiñas et al., 2002; Yang et al., 2004; Kasprzycka et al., 2017; Moura Jr. et al., 2021a; Aquino LE do et al., 2021; Manzur et al., 2023), as certain levels of Tb³⁺ may serve as better acceptors due to the high values of matrix elements involved in the IET process (e.g., ⁷F₆→⁵G₆ and ⁷F₅→⁵G₆) (Moura Jr. et al., 2021a), the population in the upper levels essentially decays not as quickly to the ⁵D₄ level compared to the rising of the population from the ⁷F₆ and ⁷F₅ to the ⁵D₄ level in the direct energy transfer process. Thus, this premise can be justified by the subsequent multiphonon decay between adjacent levels of Tb³⁺, forming a ladder-like decay process. These decays are generally slower than direct energy transfer, and consequently, decay steps like ⁵G₆→⁵D₃→⁵D₄ can be neglected in the present model.

For comparison, the decay from ⁵D₃→⁵D₄ involves a large energy gap of Δ(⁵D₃→⁵D₄) ≈ 5,792 cm⁻¹ (Carnall et al., 1978), leading to a decay rate of W m p = 7.4 × 10³ s⁻¹ if two optical phonons with ℏ ω ¯ = 2,896 cm⁻¹ each are considered to bridge the Δ(⁵D₃→⁵D₄) gap. This calculation is based on the application of Supplementary Eqs S1, S2 within the energy gap law framework. If a three-phonon process is required to bridge the gap, the multiphonon rate will be even lower ( W m p ∼ 60 s⁻¹), as expected when the number of phonons is increased. This underscores that the direct IET rates from the T₁ state are more than two orders of magnitudes higher (see Supplementary Table S3) than the multiphonon decay in the ⁵D₄ sensitization process.

The rate equation model (Supplementary Eqs S1–S3) was numerically propagated using the Radau method, a numerical approach belonging to the class of fully implicit Runge–Kutta methods (Hairer et al., 2015), over a time span of 0–10 ms with a step size of 10 ns. This implies that 1,000,000 points were calculated for each temperature, ranging from 11 to 297 K in steps of 2 K.

As the intensity is directly proportional to the population of the emitting level, the experimental intensity ratio ( I 1 / I 2 in Figure 4) and the calculated population ratio ( η 1 / η 2 ) in the steady-state regime can be compared. Figure 5D illustrates this comparison, and it can be concluded that the calculated trend is similar to the experimental trend. This observation suggests that the Stark components of the ⁵D₄ are governed by an equilibrium between non-radiative decay and rising among these components ( 1 and 2 ), driven by a Boltzmann distribution between W 2 → 1 and W 1 → 2 .

Suta and Meijerink (2020) reported that there is potential temperature dependence in the use of crystal field splitting (Stark levels) for trivalent lanthanides at low temperatures. However, at higher temperatures, where the product k_BT is significantly greater than the crystal splitting energy, a thermodynamic equilibrium among Stark levels could be established, allowing for their treatment as an effectively thermally averaged single level with an average radiative decay rate. In our model, this implies that the non-radiative decays and absorptions between two Stark levels are approximately equal, rendering the model inapplicable. In other words, Boltzmann statistics between Stark levels become ineffective at higher temperatures, resulting in a shift in the trend for temperatures exceeding 250 K, as shown in Supplementary Figure S5A (ESI), and consistent with the behavior shown in Supplementary Figure S5B (ESI).

To provide a self-calibrated ratiometric luminescent thermometer, the Eu³⁺ ion was introduced in the structure to obtain compound 2, as previously demonstrated in different MOF materials and Ln³⁺-based complexes (Brites et al., 2019). The excitation spectra at different temperatures were recorded by monitoring the main emissions of both Eu³⁺ at 615 nm (⁵D₀→⁷F₂) and Tb³⁺ at 543 nm (⁵D₄→⁷F₅) (Figures 6A,B). The spectra are relatively similar and, as in the case of compound 1, present main broadband (at 330 nm) attributed to the acac ligand excited states, confirming the excitation through antenna sensitization. Low-intensity intra-4f transitions of both Ln³⁺ ions can also be visible in the spectra, e.g., ⁷F_0,1→⁵D₂ (Figure 6A) and ⁷F₆→⁵D₄ (Figures 6A,B). The observation of this latter Tb³⁺ line when monitoring Eu³⁺ emission at 615 nm points out the occurrence of Tb³⁺-to-Eu³⁺ energy transfer.

The emission spectra were measured upon excitation at 330 nm, which is operational for both Ln³⁺ ions, in the ranges 11–297 K (Figure 6C) and 298–378 K (Figure 6D). Compared with the emission spectra of compound 1, the Eu³⁺-characteristic transitions from ⁵D₀ to ⁷F_J (J = 0–4) appear besides the Tb³⁺ transitions. Note that the emission spectra obtained under a direct excitation at 484 nm (Tb^{3+ 7}F₆→⁵D₄ transition) show, besides the main Tb³⁺ transition ⁵D₄→⁷F₅ (indicated in green), a series of the Eu³⁺-related transitions ⁵D₀→⁷F_1,2,3,4 (indicated in red, Supplementary Figure S7, ESI). This fact points out the presence of the Tb³⁺-to-Eu³⁺ energy transfer, as already inferred from the excitation spectra shown in Figure 6B. This is not surprising considering that the shortest Ln³⁺–Ln³⁺ distance in compound 2 is equal to 3.728 Å. As expected, the emission spectra performed with direct excitation in the Eu³⁺-related band at 464 nm display only the Eu³⁺ transitions (Supplementary Figure S8, ESI).

To estimate the energy transfer rates, in addition to the calculations of pairwise interactions (Malta, 2008; Carneiro Neto et al., 2020), the distribution of donor–acceptor distances, where Tb³⁺ is the donor and Eu³⁺ is the acceptor, was calculated from the crystallographic structure using a custom program written in C. The simulations of the Tb³⁺/Eu³⁺ ratio (3:1) in the structure of an expanded 20 × 20 × 20 crystal, consisting of 64,000 Ln³⁺ sites, were conducted. The sites could be occupied by either Tb³⁺ or Eu³⁺ ions. The program performed 100 simulations while maintaining the 3:1 ratio to provide statistically reliable donor–acceptor distance results. Thus, the occurrence of the formation of a Tb-Eu pair with a given distance is given by (Trannoy et al., 2021) O i = N i s ∙ x (4) where N i is the counting of a donor–acceptor pair with distance R i , s = 64,000 is the number of total host sites (i.e., Ln³⁺ sites) available for Eu³⁺ and Tb³⁺ substitution, and x is the fraction of Eu³⁺ ( x = 0.25 for forward energy transfer) or Tb³⁺ ( x = 0.75 for backward energy transfer).

Figure 1A and Supplementary Figure S1, ESI, show that the shortest distance between two {Ln₄} clusters of Ln³⁺ is in the order of 15 Å. This results in a weak interaction concerning Tb-Eu energy transfer between different {Ln₄} clusters. Thus, the Tb-Eu interaction is restricted to intra-cluster energy transfer, which can lead to highly effective energy transfer rates, as discussed in the literature (Wang et al., 2014; Calado et al., 2023; Gálico et al., 2023; Pelluau et al., 2023).

Using the calculated Tb-Eu pairwise energy transfer rates (see Supplementary Table S4 and the theoretical section in the ESI for further details) and the O i coefficients (Supplementary Table S5; Eq. 4) obtained from doping simulations of Tb³⁺ and Eu³⁺, the average (or effective) energy transfer rates from Tb-to-Eu can be estimated as (Trannoy et al., 2021) W = ∑ i = 1 4 W i = 1 − x x ∑ i = 1 4 O i ∙ W i (5) where x = 0.25 is the fraction of Eu³⁺ in the compound and W i represents the pairwise Tb-Eu energy transfer for the ith distance (Supplementary Table S5).

Supplementary Figure S9 illustrates the temperature behavior of W i and W . Although the temperature increase provides high Eu³⁺ sensitization, the emission of Eu³⁺ undergoes quenching (Figures 6C,D), probably due to high-energy phonons that may couple with the ⁵D₀ level. Thus, although intra-cluster Tb³⁺-to-Eu³⁺ energy transfer can provide high rates in contrast to non-clustering systems (Carneiro Neto et al., 2020; Trannoy et al., 2021), there are other factors in the chemical environment around Ln³⁺ that may act as a quenching channel, in this case, selectively affecting the Eu³⁺ ion, while the decrease in the intensity of the Tb³⁺ emissions may be related to the increase in the Tb³⁺-to-Eu³⁺ rates with temperature.

To demonstrate the possibility of using compound 2 as a self-referenced luminescent thermometer, temperature dependence of the normalized integrated intensity area related with the two main transitions Tb^{3+ 5}D₄→⁷F₅ (in green) and Eu^{3+ 5}D₀→⁷F₂ (in red) was extracted for both emission spectra obtained upon excitations at 330 nm (antenna) and 484 nm (intra-4f⁸). The temperature-dependent variation in the corresponding thermometric parameter (I_5D4→7F5/I_5D0→7F2, LIR) in the ranges 11–297 K (Figure 7A) and 298–378 K (Figure 7B) shows exponential correlations, which can be used for temperature measurements. The relative thermal sensitivity (S _r ) is the main parameter, allowing the comparison of the performance among different types of thermometers (Bednarkiewicz et al., 2020). The S _r value represents the variation in the experimental thermometric parameter (LIR in the present case) per degree of temperature, which is expressed as S r T = 1 L I R T ∂ L I R T ∂ T (6)

The temperature dependences of S _r are shown in the insets of Figures 7A,B. The maximum S _r value is equal to 1.5%·K⁻¹ at 297 K, for 11–297 K, and 2.0%·K⁻¹ at 373 K, for 298–378 K. Both values are close to a frequently considered high relative thermal sensitivity (∼1%·K⁻¹) (Brites et al., 2019) and are close to the best S _r values reported for mixed Eu³⁺/Tb³⁺ compounds (Trannoy et al., 2021). Temperature uncertainty (or thermal resolution, δT) is the smallest temperature change that can be detected (Brites et al., 2016). This value is related to S _r as follows: δ T = 1 S r T δ L I R T L I R T (7) where δLIR(T) is the standard deviation in the LIR(T) obtained upon several temperature cycles. According to this, the minimal thermal resolution is 0.2 K. Note that the use of emission spectra upon excitation at 484 nm for thermometry is also possible. Supplementary Figures S10A, B, ESI show the corresponding temperature dependences of LIR and the maximal value of S _r of 1.3%·K⁻¹ at 297 K. Therefore, the temperature sensor 2 proposed here works reliably in the operating range 11─378 K.

The emission decay curves of compound 2 were monitored at room temperature upon excitation at 330 nm within the ⁵D₄→⁷F₆ (Tb³⁺) and ⁵D₀→⁷F₄ (Eu³⁺) transitions (Supplementary Figure S11, ESI). The curves are well-reproduced by double-exponential functions as the best fits to the experimental data (r ² > 0.99), yielding lifetimes of 0.126 ± 0.002 and 0.668 ± 0.002 ms for the former and 0.110 ± 0.002 and 0.406 ± 0.002 ms for the latter transitions. The occurrence of two lifetimes is under investigation and will be addressed later.