Fundamental Concerns of Optical Fluorescence Intensity Ratio-Based Thermometry

Abstract

This chapter provides a comprehensive exploration of optical fluorescence intensity ratio (FIR) temperature sensing, blending theoretical underpinnings with practical applications. It underscores the intrinsic sensitivity and non-invasiveness of FIR technology, spanning diverse scientific disciplines where its utility is paramount. Central to the discussion are the intricate energy transfer mechanisms within fluorescence emissions from temperature-sensitive materials, revealing their nuanced responses to thermal changes. Fundamental to FIR thermometry are the lanthanide (Ln³⁺) ions, which play pivotal roles due to their unique electronic configurations. These elements exhibit temperature-dependent variations in fluorescence properties, including intensity and lifetime, crucial for accurate temperature determination. Specifically, the chapter delves into the utilization of erbium (Er³⁺) and holmium (Ho³⁺) ions in the context of FIR thermometry, highlighting their distinct contributions to enhancing temperature sensitivity. The Er³⁺/Ho³⁺ co-doped nano-garnet emerges as a promising material in this field, effectively bridging theoretical frameworks with practical implementations. The narrative is enriched by the incorporation of the Boltzmann distribution equation, which provides a robust theoretical foundation for understanding temperature-dependent fluorescence phenomena exhibited by Ln³⁺ ions. This chapter serves as a valuable resource, offering a concise understanding on the forefront of optical FIR-based thermometry for researchers and professionals alike.

1. Introduction

The foundations of temperature measurement date back to the sixteenth and seventeenth centuries, a crucial period in the history of scientific instrumentation. During this era, Galileo made significant contributions by introducing the thermoscope [1]. This early device served as a forerunner to modern temperature-measuring instruments. Although these early instruments provided reasonable precision, they suffered from a lack of reproducibility [2]. Despite these limitations, the thermoscope paved the way for future advancements in thermometry. The term “thermometer” emerged in 1624, marking a major milestone in the development of temperature-measuring instruments [3].

Building on this legacy, temperature—a fundamental property of matter—can now be measured using various types of thermometers. These modern instruments, which are macroscopic systems that respond to changes in temperature, utilize different principles, such as thermal expansion, gas pressure, resistance, and infrared radiation, to provide accurate and reliable temperature measurements across a variety of applications. Indeed, one common approach relies on the thermal expansion of a liquid, leveraging the principle that liquids expand when heated and contract when cooled. Another method involves using the expansion of a gas, which can measure temperature accurately in one of the two keyways: either maintaining a constant pressure or maintaining a constant volume. These gas-based thermometers operate according to the ideal gas law, which offers a robust foundation for temperature measurement.

In addition to these methods, other thermometers utilize the temperature dependence of electrical resistance. As temperature changes, the resistance of certain materials also changes, providing a means to measure temperature. Furthermore, thermometers can also use black body radiation to measure temperatures accurately in specific scenarios. This approach involves analyzing the spectrum of radiation emitted by a body to determine its temperature. A physical measurement is achieved by comparing the physical magnitude of a body with a standard. To accurately measure something, it is important to understand the basic principle behind it, which is the physical phenomenon the measurement relies on. In luminescence thermometry, this principle relies on the change in any parameter of the luminescence emitted by a material when it is exposed to different temperatures.

Lord Kelvin, also known as William Thomson, recognized the importance of aligning temperature measurement with the kinetic theory of gases. In 1848, he proposed the Kelvin scale, establishing absolute zero as its starting point [4]. This scale is widely used, particularly in thermodynamics and physics, providing precise temperature measurements essential for accurate calculations. According to the International System of Units (SI), the Kelvin scale is the official temperature scale. Since 2019, the definition of the Kelvin has been based on the Boltzmann constant rather than the water triple point.

Boltzmann’s constant, a pivotal concept, establishes a crucial link between temperature and energy. It is integrated into the ideal gas law and serves as a conversion factor between Kelvin and Joules. This constant underscores the inherent relationship between temperature and energy, revealing their intimate connection. By understanding these concepts, one gains insights into the intricate nature of temperature measurement and its profound implications in the realm of thermodynamics. Nevertheless, the concept of temperature is closely tied to our subjective sensory perception of an object’s heat or coldness. In simple terms, temperature is the measure of how hot or cold a body is, expressed in various temperature scales, such as Fahrenheit, Celsius, or Kelvin.

When considering a substance in a classical sense, with its constituent molecules continuously moving in a random and chaotic manner due to thermal energy, temperature is often defined as a measure of the translational kinetic energy stemming from this disorderly motion [5]. Higher temperatures correlate with more significant thermal movement. In simple terms, in accordance with the classical statistical physics, the average kinetic energy per particle is related to thermal motion. From a physics perspective, temperature represents the average kinetic energy of the atoms or molecules in a system. Essentially, temperature is the external indicator of the thermal energy present within a system. Absolute zero (0 K) is the lowest possible temperature, where matter contains no thermal energy. By measuring the temperature difference between two substances, we can identify the direction of heat flow, which always travels from a region of higher temperature to one of lower temperature. The equipartition theorem delineates the intricate relationship between temperature and the contributions of individual degrees of freedom within a system. Therefore, the total average kinetic energy of a molecule in an ideal gas would be

$$\frac{1}{2}\mathit{kT}+\frac{1}{2}\mathit{kT}+\frac{1}{2}\mathit{kT}=\frac{3}{2}\mathit{kT}$$E1

Each degree of freedom contributes $\frac{1}{2}\mathit{kT}$ to the total energy, with $k$ representing the Boltzmann constant and T is the temperature [5]. This connection reveals the profound influence of temperature on the microscopic behaviors exhibited by particles in a substance. When two systems share the same temperature, they exist in a state of thermal equilibrium, signifying an absence of net thermal energy transfer between them. Conversely, if one system is at a higher temperature than another, energy will flow spontaneously from the hotter system to the colder one when they come into thermal contact [5]. Broadening our perspective to the macroscopic scale, the second law of thermodynamics emerges as a guiding principle. This law dictates that the total entropy of an isolated system tends to increase over time during spontaneous processes. Entropy, often described as a measure of disorder or randomness within a system, becomes intimately connected to temperature through the relationship expressed by the equation:

$$∆S=\frac{∆Q}{T}$$E2

where ∆S represents the change in entropy, ∆Q signifies heat transfer, and T stands for the absolute temperature. Through the blending of these concepts, the temperature serves as the bridge connecting the statistical behaviors of particles to the overarching principles governing energy exchange in physical systems [5].

Recognized as a fundamental variable in science, temperature plays a pivotal role in various industrial and scientific processes [1, 6]. It is essential for controlling chemical reactions, manufacturing processes, and energy production, among other applications. Accurate temperature measurement and control are vital for ensuring the quality and safety of products and for optimizing efficiency in systems. The significance of temperature spans across multiple fields. In the medical field, temperature is a critical factor for diagnosing and monitoring patients. Body temperature is a key indicator of health, and fluctuations can signal the presence of infection, inflammation, or other medical conditions. Temperature also plays a role in preserving and transporting biological samples, vaccines, and other medical supplies. Additionally, it is essential in various medical procedures, such as thermotherapy and cryotherapy, where precise temperature control is necessary for effective treatment. Understanding and managing temperature in medical settings is crucial for patient care and overall health outcomes.

In scientific research, temperature plays a significant role in the study of various physical and chemical phenomena. It influences the behavior of materials at the atomic and molecular levels, as well as the properties of substances across different phases.

The evolution of temperature measurement, from Galileo’s thermoscope to modern methodologies, traces a rich historical journey underscored by continual innovation. Overcoming the limitations of early thermoscopes has spurred the development of diverse thermometer technologies, culminating in the introduction of non-contact measurement techniques. This progression exemplifies the dynamic nature of temperature measurement, driven by technological advancements.

As technology has advanced, novel techniques have emerged to enhance the accuracy and precision of temperature readings. Among these modern methods is luminescence thermometry, leveraging the luminescent properties of materials to gauge temperature. This approach marks a significant departure from traditional methods, offering non-contact and high-resolution temperature measurement capabilities, thus ushering in a new era in temperature sensing technology. By observing the changes in luminescence intensity or wavelength in response to temperature variations, researchers and engineers can determine the temperature of a system with great accuracy. This method builds on the legacy of temperature measurement, linking the past to the present, by incorporating principles of light and energy that Galileo and his contemporaries explored. Luminescence thermometry is particularly useful in challenging environments such as high-temperature or remote locations, where traditional methods may be less effective.

1.1 Luminescence thermometry

Traditional contact methods encounter difficulties in precisely measuring a material’s temperature, mainly because introduced thermometers may cause interference and disturb the accurate evaluation of the material’s temperature [7]. Conventional temperature measurement tools, such as bimetallic and liquid/gas-based thermometers, pyrometers, and thermocouples, among others, lack the precision needed for high spatial resolution, especially at scales on the order of micrometers, which are typical in cellular systems.

Optical sensors have noticeable advantages compared to conventional methods [8]. In recent years, researchers have become increasingly interested in optical thermometry, a sophisticated method that examines the relationship between temperature and various optical parameters. This modern approach offers key advantages, such as high spatial resolution, fast response, and non-invasive measurement. These benefits make it particularly useful in challenging environments like submicron scales, high-voltage areas, and caustic conditions [9]. Consequently, optical thermometers have become essential tools for precise temperature measurements in demanding settings.

Luminescence thermometry is a non-contact technique that measures temperature by observing the luminescent (light-emitting) properties of materials. This method is particularly beneficial when traditional temperature measurement techniques are impractical or impossible [10]. Some elements have optical properties that make them suitable for spectroscopic studies. Ln³⁺ ions, which display luminescence properties like fluorescence emissions, are commonly used as dopants to aid in examining host materials due to their high efficiency as emitters [11]. This has promoted the development of new concepts for sensors based on the luminescence properties of materials, leading to the creation of optical sensors [10]. Therefore, Ln³⁺ has attracted significant interest as optical temperature sensors because their fluorescence intensity can change in response to variations in their absorption and emission properties with temperature. These variations in fluorescence properties are examined using different techniques and are described by parameters, such as spectral position, bandwidth, intensity, decay lifetime, and fluorescence intensity ratio (FIR) [12].

Luminescence thermometry involves the emission of light when materials are electronically excited by an external energy source, such as optical radiation (in the case of photoluminescence) [10, 11]. This process is common in materials, such as dyes, semiconductors, and phosphors. The characteristics of the emitted light, including spectrum shape, bandwidth, and spectral shift, are influenced by the local temperature of the material. The technique leverages the complex interaction between temperature and luminescence, enabling thermal sensing through careful spatial and spectral analysis of emitted light as a material undergoes temperature changes. It provides a comprehensive solution, featuring high thermal resolution (<0.1°C), significant relative thermal sensitivity (>1%/°C), and precise temperature mapping with exceptional optical spatial resolution (<10 μm) [13]. Among the optical parameters used in temperature sensing, the fluorescence intensity ratio (FIR) stands out as the most reliable technique.

The key to developing an effective optical temperature sensor lies in selecting the right optically active ion and host matrix. Incorporating Ln³⁺ ions is crucial due to their photostability and spectroscopic advantages, such as distinct emission spectra, extended fluorescence lifetimes, and high quantum yields [11]. Known for their strong luminescence and intricate energy level structures, Ln³⁺ ions often exhibit narrow energy gaps between levels. However, not all of them are suitable for calibrating optical responses. Studies by Zhou et al. [14, 15] and Soler-Carracedo et al. [16] demonstrate how these features collectively enhance the precision and sensitivity of optical temperature measurements.

Considerations include the need for a delicate balance in the energy gap between thermalized levels, ensuring it is large enough to prevent emission overlap yet also allowing for a minimum upper-level population within the desired temperature range. Additionally, radiative probabilities associated with thermalized levels should be sufficiently high to produce significant emission intensities, and the relative intensities of emission peaks from different energy levels can be directly related to temperature. Praseodymium (Pr³⁺), neodymium (Nd³⁺), samarium (Sm³⁺), thulium (Tm³⁺), europium (Eu³⁺), holmium (Ho³⁺), and erbium (Er³⁺) ions have energy level pairs suitable for optical temperature sensors. Berthou et al. [17] examined Er³⁺ thermalized levels in fluoroindate fibers using the FIR technique across a large range of temperatures. Subsequent studies explored Ln³⁺ ions and different host materials, such as Er³⁺ or Er³⁺/Yb³⁺-doped various types of glasses, such as tellurite, fluorotellurite, oxyfluoride, fluoroindate, chalcogenide, and fluorophosphate glasses, for optical luminescent temperature sensing [18].

Garnets possess remarkable physicochemical properties, including hardness, high optical transparency, and strong mechanical and chemical stability. Their structure has been used as a host matrix for Ln³⁺ ions [16]. Optically, the combination of the luminescence properties of Ln³⁺ and the crystal stability of garnets makes them ideal for the development of new sensors in the context of nanothermometry.

Radiative decay processes allow for the precise measurement of these intensities. For example, in the Nd³⁺-doped Y₃Ga₅O₁₂ nano-garnets, changes in the relative intensities of emission peaks between Stark levels of the ⁴F₃/₂ → ⁴I₉/₂ transition are used to measure temperature accurately [19]. Radiative decay is preferred over non-radiative decay because non-radiative processes (such as phonon emission or energy transfer to surrounding molecules) result in energy loss as heat rather than useful light emission. High radiative decay rates minimize these losses, improving the performance of the material in its intended application.

1.2 Fluorescence intensity ratio (FIR)

The way different luminescent properties—such as lifetime, intensity, and spectral characteristics—respond to temperature changes, as depicted in Figure 1, enables researchers to measure temperature. They do this by analyzing the emitted light from a material when it is exposed to an external excitation source.

Figure 1.

Diagram illustrating the potential impacts of an increase in temperature on luminescence. Adapted from Ref. [12].

One key aspect of luminescent thermometry is the change in luminescence lifetime [11]. Typically, as the temperature of a material increases, its luminescence lifetime decreases, resulting in a faster emission of light. Another important characteristic is the spectral changes that occur with temperature shifts. For instance, it can be observed as a decrease in luminescence intensity, a shift in peak wavelengths, or changes in the emission band as the temperature varies. However, one of the most effective and commonly used techniques in luminescent thermometry is based on the luminescence intensity ratio between two emission peaks [10, 16]. This method is particularly sensitive and widely applied in practice. By observing the changes in the intensity ratio between two distinct peaks in the emission spectrum of luminescent ions, researchers can accurately determine temperature variations. When these ions are excited, they emit light from two energy levels separated by an energy difference (ΔE), designated as thermal coupling levels (TCLs).

The FIR technique offers a significant advantage by using the intensities from two closely spaced energy levels, known as thermally coupled energy levels (TCLs), to monitor temperature. The intensity ratio changes with temperature are independent of the source power since any variations in excitation power affect both levels equally. This enhances the measurement sensitivity and sensor stability. TCLs are in a thermodynamically quasi-equilibrium state, offering several benefits over non-coupled levels [20, 21]. First, the population of individual TCLs is directly proportional to the total population. Second, the theory of relative changes in fluorescence intensity from the two TCLs is well understood, making their behavior easier to predict. For FIR techniques to be effective, the energy gap between TCLs should be within the range of about 200–2000 cm⁻¹, which is satisfied by many rare-earth ions, such as Tm³⁺, Pr³⁺, Nd³⁺, Sm³⁺, Eu³⁺, Ho³⁺, Er³⁺, and Yb³⁺ [12]. As the temperature increases, the relative intensity of light emitted from the higher energy level rises, forming the basis of the fluorescence intensity ratio (FIR) method (green box in Figure 1). By observing the ratio between two peaks in the emission spectrum, we can precisely determine temperature. This approach is straightforward, versatile, and less susceptible to experimental errors or misalignments, making it a preferred choice for accurate temperature sensing method. The only constraint is that energy levels they contain must be closely spaced [22].

The appeal of rare-earth-doped materials for temperature sensors has risen due to their economical fabrication and the ease of excitation using low-cost diode lasers. These materials contain multiple pairs of energy levels with small separations, comparable to thermal energy. In practical sensor applications, these energy levels are not only optically connected to the ground state but also have a high likelihood of non-radiative transitions between paired levels [23].

In the field of luminescence thermometry employing lanthanides, the pivotal component lies within the Er³⁺ ion, with particular emphasis on its ⁴S_3/2 and ²H_11/2 levels, which are prominently employed. As an example, Figure 2 exhibits a segment of the erbium emission spectrum in an Er³⁺-doped YGG nano-garnet (Er_0.1Ho_0.1:Y_2.8Ga₅O₁₂).

Figure 2.

Emission spectrum of a sample (an Er³⁺-doped YGG nano-garnet) at 300 K showcasing distinctive peaks corresponding to fluorescence emissions from erbium ions under the specified excitation condition.

The 523-nm and 550-nm emission bands are attributed to transitions within Er³⁺ ions, specifically from ²H_11/2 to ⁴I_15/2 and from ⁴S_3/2 to ⁴I_15/2 energy levels, respectively. A distinct advantage of FIR thermometry resides in its ability to easily discern the sharp line emissions originating from these two levels. This clarity significantly simplifies the identification process of the ²H_11/2 and ⁴S_3/2 levels, particularly when contrasted with scenarios where bands overlap, presenting challenges in their differentiation.

At typical room temperature, there exists an equilibrium state between the ⁴S_3/2 and ²H_11/2 levels, and both levels undergo light emission, with the ⁴S_3/2 level displaying a notably stronger emission compared to the ²H_11/2 level. These levels, referred to as thermal coupling levels (TCLs), demonstrate a discernible energy discrepancy (E₂₁) of approximately 900 cm⁻¹, as illustrated in Figure 3. Owing to the narrow gap separating these levels, their population distributions adhere closely to a Boltzmann distribution pattern as the temperature undergoes fluctuations.

Figure 3.

Diagrams illustrating the energy levels and transitions utilized in the FIR method, focusing on thermally coupled excited states of Er³⁺ focusing on TCLs ²H_11/2 and ⁴S_3/2.

As temperature rises, there is a notable increase in emissions from the ²H_11/2 level, as depicted in Figure 4 [24]. The positions of the two bands remain constant while the temperature increases. This rise in intensity is crucial for FIR thermometry analysis. By comparing the intensities of these two emissions, precise delineation of the areas corresponding to the ⁴H_3/2 and ²H_11/2 emissions is facilitated. These areas will be used to obtain the FIR by the expression:

Figure 4.

Diagram showing how emission bands in luminescent materials doped with erbium ions vary with temperature. Adapted from Ref. [24].

$$\mathit{FIR}=\frac{{\mathit{\text{Area}}}_{|2⟩\to |0⟩}}{{\mathit{\text{Area}}}_{|1⟩\to |0⟩}}$$E3

These delineated areas serve as the basis for determining the relative intensity, which in turn allows for the calculation of the temperature of the environment. By analyzing the ratio of their intensities, a calibration curve can be generated. The methodology for determining temperature relies on meticulously assessing the relative intensities of the ⁴S_3/2 and ²H_11/2 emissions. This comparative analysis allows us to precisely understand the temperature variations within the environment, aiding in practical applications, such as optimizing thermal management systems or ensuring proper operation of heating or cooling equipment. Therefore, the FIR technique provides an effective method to calibrate optical temperature sensors [15], improving measurement sensitivity and reducing the influence of varying measurement conditions [23].

2. Fundamentals

Lanthanide-doped materials hold a distinct position in photonics, thanks to the exceptional spectroscopic properties exhibited by Ln³⁺ ions in terms of luminescence and amplification. The pivotal discovery of the bright red-emitting phosphor Y₂O₃:Eu³⁺ at the dawn of the twentieth century marked a significant milestone in this field. Furthermore, the introduction of the YAG:Nd³⁺ laser in 1964 further emphasized the crucial role of lanthanides in shaping modern photonic technologies. This is attributed to their remarkable photoluminescence properties that stem from their abundant energy level structure [25]. This rich energy level structure, depicted in the Dicke diagram reprinted in Ref. [25], highlights the intricate electronic configurations of lanthanides, contributing to their diverse and versatile photoluminescent capabilities and distinctive characteristics that set them apart. Primarily, they showcase exceptional spectroscopic properties. These include emitting sharp lines, maintaining stable energy levels, and absorbing light across a broad spectrum, from infrared to ultraviolet. Their electronic structure grants them insulation from external influences within a host environment, ensuring that their transition energies remain independent. This shielding effect preserves the positions of their energy levels, minimizing any alterations when incorporated into a host matrix. Certain ions exhibit luminescence in the visible or near-infrared spectral (NIR) regions when subjected to UV irradiation. The emitted light’s color varies depending on the specific Ln³⁺ ion present. For example, Eu³⁺ emits red light, Tb³⁺ emits green light, Sm³⁺ emits orange light, and Tm³⁺ emits blue light [26]. Yb³⁺, Nd³⁺, and Er³⁺ ions are renowned for their NIR luminescence. Additionally, Pr³⁺, Sm³⁺, Dy³⁺, Ho³⁺, and Tm³⁺ ions also feature transitions in the NIR region, while Gd³⁺ emits exclusively in the ultraviolet (UV) region [25].

2.1 Energy levels in lanthanide ions

Lanthanides (Ln) encompass a series of elements in the periodic table spanning from cerium (Z = 58) to lutetium (Z = 71), succeeding lanthanum (Z = 57). They are characterized by the progressive filling of their 4f orbitals within their electronic configuration, ranging from [Ar]5s² 5p⁶ 4f¹ 6s² in cerium to [Ar]5s² 5p⁶ 4f¹³ 6s² in ytterbium. In their +3-oxidation state (commonly found in solids as Ln³⁺), lanthanides typically shed two electrons from the 6 s orbital and one from the 4f orbital, leaving the 4f orbital partially filled. The electronic configuration of the Ln³⁺ ions is represented by [Xe]4f^N [25, 27].

The intense shielding of the 4f electrons by the complete 5s², 5p⁶, and 6s² shells is a defining trait of lanthanide ions, rendering them behaviorally akin to atoms. While their energy levels exhibit stability transitioning from a free state to a solid matrix, their spectroscopic characteristics, such as absorption and emission band widths, are influenced by the host matrix’s nature. The arrangement of neighboring ions around the lanthanide ion holds significance in various matrices, particularly in crystalline versus disordered structures, where the former yields narrow bands due to uniform crystal field effects, while the latter leads to widened bands owing to varied ligand field energy levels. Heterogeneous broadening, characteristic of amorphous materials like glasses, arises from the lack of periodicity and variation in the bonding environment. This variability affects the Stark sublevels, resulting in broad absorption bands. This phenomenon is detailed in studies on glasses and luminescent materials doped with rare-earth ions, where it is observed that the absence of periodicity in glasses leads to variation in spectroscopic properties. [28].

The shielding provided by the filled 5s and 5d shells, along with the partially filled 6s² shell, insulates lanthanide ions against external fields from the host medium, thereby preserving the spectral characteristics of emission lines and minimizing distortions, akin to free ions.

The trivalent state of lanthanides holds several unique characteristics. These characteristics encompass distinct spectroscopic properties, exemplified by sharp lines observed in the Er³⁺ emission spectrum of Figure 2. Moreover, the independence of energy levels from the host environment is illustrated in Figure 5, where the absorption spectra of Ho³⁺ ions in two distinct hosts display identical configurations. Additional, lanthanide ions exhibit emissions ranging from infrared to ultraviolet, as shown in the Dicke diagram [25]. Trivalent lanthanides also exhibit long emission lifetimes and high quantum yields [25].

Figure 5.

Absorption spectra of an oxyfluoride glass-ceramic and yttrium orthovanadate (YVO₄) crystal doped with erbium ions highlight the independence of energy levels from the host environment.

Energy from an electronically excited state dissipates through both radiative and non-radiative means [20]. The emitted intensity correlates with the electron population density (N) in excited states, and its temporal changes are governed by

$$\frac{\mathit{dN}}{\mathit{dt}}=-\left(k_R+k_{\mathit{NR}}\right)N\left(t\right)$$E4

The rates of radiative (k_R) and non-radiative (k_NR) transitions determine the processes of photon emission. Radiative de-excitation pathways lead to photon emission, while non-radiative pathways release vibrational energy. Consequently, the electron population in the excited state and the luminescence intensity decay exponentially over time, with a characteristic time constant (τ) often referred to as the lifetime of the excited state.

$$\tau =\frac{1}{k_R+k_{\mathit{NR}}}$$E5

The reciprocal of the radiative transition rate is known as the radiative lifetime or natural lifetime, denoted as τ_R:

$${\tau }_R=\frac{1}{k_R}$$E6

This quantity can often be accurately calculated from absorption and emission spectra, as well as from the ratio of the measured lifetime to the internal quantum efficiency of emission, also known as the quantum yield (η):

$${\tau }_R=\frac{\tau }{\eta }$$E7

Transition metals are well known for their characteristic d-orbital electronic transitions, which result in complex and varied spectra. Similarly, lanthanides possess distinct f-orbital electronic configurations that lead to their own set of spectral features. These differences lie once more in their distinctive electronic configuration. The 4f electrons of lanthanides are shielded by the 5s and 5p electrons. As a result, when trivalent lanthanides are embedded within a host matrix, the influence of the host ions is typically minimal [25].

To explore the complexities of the 4f configuration, Eu³⁺ serves as an illustrative example (Figure 6). The electronic configuration of Eu³⁺ is [Xe] 4f⁶. Initially, interactions between the core and the electrons yield a 4F⁶ configuration, with europium accommodating six electrons in its 4f shell. Subsequent introduction of electron repulsion results in terms displaying significant splitting into J-levels, further accentuated by spin-orbit interaction, which segregates these terms into distinct levels. These properties are indicated by term symbols S, L, and J in the notation ^2S + 1L_J. Here, 2S + 1 denotes the spin multiplicity, L represents the total orbital angular momentum, and J signifies the total angular momentum of the 4f electrons [25].

Figure 6.

Diagram illustrating the interactions causing the splitting of electronic energy levels of a Eu³⁺ ion. Energy increases as depicted upward in the diagram [29].

Because of the screening effect of 4f electrons, their interaction with the crystal field of ligands is significantly weaker than their spin-orbit interaction. The crystal field causes only a slight change in energy, resulting in the splitting of levels into multiple sublevels, known as the Stark effect. The maximum number of these Stark sublevels is 2J + 1 [20]. However, the splitting and the specific number of sublevels are influenced by the symmetry of the crystal field surrounding the Ln³⁺ ion and the corresponding selection rules [20].

The selection rules for electric dipole transitions are derived from the Judd-Ofelt (JO) theory. This theory provides a framework for understanding and predicting the intensity of spectral lines in lanthanide ions by considering the interaction between electronic states and the surrounding crystal field. It accounts for the otherwise forbidden transitions by introducing mixed parity states, which allow for electric dipole transitions under non-centrosymmetric conditions. Consequently, the Judd-Ofelt (JO) theory not only explains the occurrence of these transitions but also establishes the criteria for their probabilities based on the symmetry and environment of the lanthanide ions.

The trivalent lanthanides can be treated as free ions, where the Hamiltonian accounts for their behavior. The surrounding crystal field perturbs the system, contingent upon crystal field parameters, inducing additional splitting. The J-levels can be further split into sublevels because of the electric field of the matrix. However, due to shielding, energy level positions weakly hinge on the host matrix, rendering transition energies predominantly independent, as evidenced by the Dicke diagram [29].

Within the host matrix context, level splitting varies based on symmetry considerations, often yielding diverse peak numbers. However, comprehending intensity origins remains a challenge in lanthanide ion studies within the host matrix.

The metastable energy bands in Ln³⁺ ions, influenced by factors like electronic repulsion and spin-orbit coupling, contribute to the structured arrangement of energy levels within these ions. This structured arrangement is crucial for understanding the spectroscopic properties of lanthanide ions, particularly in materials like glasses and luminescent materials doped with rare-earth ions. The JO theory, which provides a framework for analyzing the spectral properties of lanthanide ions, builds upon our understanding of these structured energy levels. By elucidating the energies of various multiplets and predicting important properties related to light emission, the JO theory enhances our comprehension of optical materials, bridging the gap between fundamental energy level organization and practical spectroscopic analysis. Initially, in the early twentieth century, scientists struggled to comprehend the spectral properties of these ions. Although they recognized the sharp lines in their spectra, explaining why they appeared was challenging due to their violation of certain rules in quantum mechanics, like the Laporte rule.

In the 1940s, Raka developed his renowned algebra, laying the groundwork for complex calculations facilitated by the advent of computers. Soon thereafter, a breakthrough emerged: within non-centrosymmetric crystal fields, coupling between odd and even parity states forced the creation of mixed parity states, effectively mitigating the Laporte rule. So, in 1962, Judd and Ofelt made significant contributions by solving the Schrödinger equation for the steady state of a many-electron system. This breakthrough enabled them to successfully ascertain the energies of the respective ^2S + 1L_J multiplets, which are pivotal for comprehending the emission of light by Ln³⁺ ions. Additionally, their JO theory pioneered a fresh approach to forecast crucial properties relevant to light emission analysis, marking a noteworthy advancement in the field of spectroscopy [30]. Despite its inherent complexity, the JO theory enables the prediction of crucial derived quantities, such as transition probabilities, branching ratios, lifetimes, and quantum efficiencies, utilizing just three parameters (Ω₁, Ω₂, and Ω₃). These parameters provide relevant information on the local environment of the rare-earth ions [31], making JO theory a powerful tool in understanding and optimizing optical materials.

2.2 Boltzmann equilibrium

When analyzing Ln³⁺-doped materials, the FIR technique quantifies the emission intensities originating from closely spaced excited states (also known as the Boltzmann-type FIR method). It also evaluates intensities from electron transitions ending at various Stark sublevels. This method is versatile, suitable for both down-conversion and up-conversion emissions [20, 21]. By analyzing the fluorescence intensities of two adjacent energy levels that are in thermal equilibrium and optically coupled to a common lower level, temperature variations can be determined. These levels are thermally coupled due to their high probability of non-radiative transitions. The FIRs, which follow the Boltzmann distribution law, are calculated from the emission peaks of these thermally coupled energy levels at different temperatures. Two excited energy states of lanthanide ions are termed thermally coupled if the energy gap between them is 2000 cm⁻¹ or less. This proximity enables electrons to transition between these states due to thermal energy. Consequently, both states share the electronic population following Boltzmann’s distribution:

$$N_2=N_1\mathit{exp}\left(-\frac{E_{21}}{\mathit{kT}}\right)$$E8

N₂ and N₁ represent the numbers of electrons in the higher and lower excited states, respectively. E₂₁ denotes the energy difference between these states. The absolute temperature (T) determines how electrons distribute between these states according to Boltzmann’s distribution. This distribution explains the probability of electrons occupying different energy levels based on the thermal energy available at temperature T. Consequently, the fluorescence intensity ratio (FIR) emitted from the higher (I₂₀) and lower (I₁₀) excited states can be estimated as follows:

$$\mathit{FIR}=\frac{I_{20}}{I_{10}}=\frac{{\omega }_{20}^R{g}_{2}h{\nu }_2}{{\omega }_{10}^R{g}_{1}h{\nu }_1}\mathit{exp}\left(-\frac{E_{21}}{\mathit{kT}}\right)=B\mathit{exp}\left(-\frac{E_{21}}{\mathit{kT}}\right)$$E9

Here, h represents Planck’s constant, g signifies the degeneracy (2J + 1) of the energy states, A represents the rate of spontaneous emission, ν denotes the emission frequency, 2 and 1 refer to the “higher” and “lower” energy states, respectively, 0 denotes the ground state or an electronic level with lower energy than 2 and 1, and ${\omega }_{20}^R$ and ${\omega }_{10}^R$ are the spontaneous emission rates of the E₂ and E₁ levels to the E₀ level, respectively.

The log(FIR) exhibits a linear relationship with the reciprocal of temperature:

$$log\left(\mathit{FIR}\right)=log\left(B\right)-\frac{E_{21}}{k}\frac{1}{T}$$E10

Thus, the values of log(B) and E₂₁/k can be experimentally determined from the slope and intercept, respectively, of the linear fit of log (FIR) versus 1/T.

At colder temperatures, the higher energy state often remains unoccupied due to electrons lacking sufficient energy to bridge the energy gap. Moreover, when energy states are closely spaced, the non-radiative relaxation rate from the higher to the lower energy state becomes significantly pronounced. Consequently, the FIR read-out method has a temperature threshold proportional to E₂₁. This implies that lower values of E₂₁ correspond to colder temperatures suitable for FIR measurements. As temperature rises, the higher energy state sees an increasing population, leading to a gradual rise in emission intensity from this state at the expense of reducing the population in the lower energy state. As temperature rises, both emissions from the higher and lower energy states decrease in intensity due to a phenomenon known as temperature quenching. This reduction in intensity continues until one or both emissions become so weak that they are no longer detectable. The upper temperature limit for applying the FIR method depends largely on the phonon spectrum of the material hosting the luminescent material and the specific lanthanide ion being used. Essentially, the ability to accurately measure temperature diminishes as temperatures approach these upper limits, mainly due to the increased uncertainties associated with detecting weakened emissions. The FIR method allows for the use of any emissions if they originate from two excited levels that are thermally coupled. This flexibility enables the method to be applied to a wide range of materials and experimental conditions, thereby enhancing its versatility in temperature sensing applications.

3. The performance of a luminescence thermometer

To evaluate the effectiveness of thermometers, various parameters come into play [20]. One key parameter is absolute sensitivity (S_FIR), which quantifies the change in FIR per temperature increment. However, perhaps more commonly considered is relative sensitivity (S_R), which measures the change in FIR per temperature increment relative to that FIR itself.

Both S_FIR and S_R can be derived from Eq. (9) as follows:

$$S_{\mathit{FIR}}=\frac{\partial \left(\mathit{FIR}\right)}{\partial T}=\frac{E_{21}}{k{T}^2}\left(\mathit{FIR}\right)$$E11

$$S_R=\frac{\partial \left(\mathit{FIR}\right)}{\partial T}\frac{1}{\left(\mathit{FIR}\right)}=\frac{E_{21}}{k{T}^2}$$E12

Maximizing these parameters is crucial for optimal thermometer performance. Even a slight temperature fluctuation can enhance sensitivity, indicative of a superior thermometer’s ability to accurately detect temperature changes. Another important metric in assessing thermometer performance is the limit of detection, denoted as δT_min. This metric is closely linked to the minimum temperature change that the thermometer is capable of registering. In practical terms, δT_min represents the smallest temperature variation that the thermometer can reliably detect and measure within its operational range. It’s mathematically defined as follows:

$${\mathit{\delta T}}_{\mathit{min}}=\frac{\delta \left(\mathit{FIR}\right)}{\left(\mathit{FIR}\right)}\frac{1}{S_R}$$E13

The term δ(FIR)/(FIR) represents the resolution limit or relative uncertainty of the FIR, indicating the smallest detectable change in the fluorescence intensity ratio. Enhancing this limit may require improving equipment and refining measurement techniques. For example, increasing integration time and averaging measurements can reduce experimental noise, thus improving resolution [16]. The uncertainty of the FIR is expressed as follows:

$$\frac{\delta \left(\mathit{FIR}\right)}{\left(\mathit{FIR}\right)}=\sqrt{{\left(\frac{\delta {\mathit{\text{Area}}}_{|1⟩\to |0⟩}}{{\mathit{\text{Area}}}_{|1⟩\to |0⟩}}\right)}^2+{\left(\frac{\delta {\mathit{\text{Area}}}_{|2⟩\to |0⟩}}{{\mathit{\text{Area}}}_{|2⟩\to |0⟩}}\right)}^2}$$E14

where ${\mathit{\text{Area}}}_{|1⟩\to |0⟩}$ and ${\mathit{\text{Area}}}_{|2⟩\to |0⟩}$ represent the areas associated with the emissions used to calibrate the thermometer and $\delta {\mathit{\text{Area}}}_{|1⟩\to |0⟩}$ and $\delta {\mathit{\text{Area}}}_{|2⟩\to |0⟩}$ are their uncertainties, measured as the standard deviations of these areas from multiple measurements taken under uniform conditions [16].

In a recent investigation [16], the sensitivity of the sensor to temperature variations was explored using both Er³⁺ and Ho³⁺ ions co-doped in nano-garnet as the emissive species. The FIR technique was employed for temperature detection across a broad range, from 30 K to 540 K. The thermally coupled levels ²H_11/2 and ⁴S_3/2​ for Er³⁺ ions and ⁵S₂​ and ⁵F₄​ for Ho³⁺ ions were utilized in the temperature sensing mechanism. This approach is significant, as few optical sensors are capable of effectively covering both high and low temperature ranges.

Upon excitation of the sample with lasers, distinct emission bands corresponding to both Er³⁺ and Ho³⁺ ions were observed, as shown in Figure 7. The emission bands facilitated accurate temperature measurements through the FIR technique, demonstrating the sensor’s efficacy in detecting temperature variations over a wide range, thereby offering potential advancements in optical temperature sensing technologies.

Figure 7.

Spectrum showing emissions from the sample when simultaneously excited at 385 nm and independently excited at 406 nm for Er³⁺ ions, and at 473 nm for Ho³⁺ ions. Adapted from Ref. [16].

Allowing for independent excitation of Er³⁺ or Ho³⁺ ions facilitates the calculation of S_R for emissions from both ions (Figure 8).

Figure 8.

Relative sensitivity of the sensor for YGG nano-garnet, measured independently for erbium (red, dashed line) and holmium (blue, solid line) transitions [16].

For Er³⁺ ions, the S_R was observed to increase notably from approximately 0.2% K⁻¹ at 540 K to 1.3% K⁻¹ at 200 K [16]. This dynamic underscores the sensor’s heightened responsiveness at lower temperatures, where even minor temperature changes prompt significant variations in the fluorescence intensity ratio. Conversely, at higher temperatures such as 540 K, the sensor exhibits a comparatively muted response to temperature fluctuations, reflecting reduced sensitivity. Meanwhile, the evaluation of S_R​ for Ho³⁺ ions across temperatures ranging from 30 K to 300 K revealed a peak sensitivity of about 0.4% K⁻¹ at 200 K [16]. This finding indicates effective temperature sensing capabilities at lower temperature regimes for Ho³⁺ ions, albeit with sensitivity characteristics distinct from those observed for Er³⁺ ions.

4. Conclusion

Optical FIR-based thermometry emerges as a cutting-edge technology poised to revolutionize temperature measurement across various disciplines. By leveraging the fluorescence properties of lanthanide ions, particularly erbium and holmium in doped nano-garnets, researchers have advanced our understanding of temperature-sensitive materials. The technology offers non-invasive and highly sensitive methods capable of operating from cryogenic to high temperatures, making it invaluable in diverse applications from materials science to biomedical research. Challenges remain in enhancing resolution and reducing experimental noise, yet ongoing advancements in material design and measurement techniques promise continued innovation. Optical FIR-based thermometry stands at the forefront of temperature sensing technologies, offering profound implications for future scientific and industrial applications.

Acknowledgments

The author wishes to thank the Direção Regional da Ciência, Inovação e Desenvolvimento, Governo Regional dos Açores, for their generous support through application M3.3.C/EDIÇÕES/022/2024 – “Fundamental Concerns of Optical FIR-Based Thermometry”.