Defect clustering in an Eu-doped NaMgF₃ compound and its influence on luminescent properties

Abstract

Luminescent properties of compounds based on lanthanide ions are strongly influenced by defect clustering, and unfortunately, these defects are not fully understood in lanthanide-doped fluoroperovskite materials. In this context, we studied the structural properties and effects upon incorporation of divalent and trivalent Europium dopant ions in the orthorhombic phase of NaMgF₃, combining classic atomistic simulations and crystal field models. We developed a new set of interatomic potentials that reproduce the structural properties, as well as lattice parameters, interatomic distances and volumes, and elastic properties, with good accordance with experimental results. Analysis of the solution energy revealed that Eu³⁺ is most energetically favourable in the Mg site, while Eu²⁺ is most favourable in the Na site. The mechanism of charge compensation was investigated in both cases. We also analysed the local symmetry, charge transfer in Eu–F chemical bonding, crystal field parameters, and ⁷F₁ energy sub-levels of the Eu³⁺ ion in the host matrix based on crystal field and electronegativity models. In addition, we discussed the photoionization cross-section and optically stimulated luminescence (OSL) decay pattern for Eu²⁺-doped NaMgF₃. Thus, this work provides direction for new material design and opens up a framework to analyse structural and defect changes of fluroperovskite compounds upon lanthanide ion insertion.

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1 Introduction

Perovskite compounds are a class of special materials that have attracted wide attention in recent years for interesting applications, such as magnetoelectrics,^1,2 photovoltaics devices,^3–6 light-emitting diodes,^7,8 lasers,^9,10 photocatalysis,¹¹ memristors,¹² and ionizing radiation detectors.^13–15 Fluoroperovskite materials, ABF₃ (where A and B stand for alkali and alkaline earth metals, respectively), are a sub-class of perovskite compounds. In particular, NaMgF₃ is a material inserted in this family, with interesting properties related to optics and ionizing radiation dosimetry.^16,17 Rare earth-doped NaMgF₃ compounds have been considered promising materials for personal dosimetry because of the effective atomic number, similar to human tissue, and high sensitivity at low dosages.^18,19 Unlike the other materials of this class, such as AMgF₃ (A = Rb, K, and Cs), NaMgF₃ presents an orthorhombic perovskite structure with space group Pbnm at room temperature and standard pressure.²⁰

Luminescent properties of NaMgF₃ nanoparticles doped with lanthanide ions and Mn, synthesized using a reverse micro-emulsion method, have been reported.^21,22 Furthermore, lanthanide ions doped into NaMgF₃ polycrystalline samples have also been prepared by the conventional solid-state reaction method.^23,24 In fact, lanthanide incorporation in compounds has been largely used to enhance luminescent properties. In particular, Eu³⁺-doped materials are a well-known red emitting phosphors, widely used as spectroscopic probes because of their unique emission characteristics.^25–27 Valuable characterization information, such as local symmetry of the optically active ion, occupancy number, and Stark levels, can be obtained from emission characteristics of the Eu³⁺ ion. However, in many cases, the Eu ion is incorporated in a host matrix and aliovalent substitution occurs. The difference between the ionic radii of both ions (doped and host) is an important factor in evaluating the influence of aliovalent substitution, giving rise to material defects. Identifying these defects is crucial to accurately describing the spectroscopic properties and understanding specific mechanisms relevant to their application in optics and ionizing radiation detectors. Mechanisms of charge compensation are not yet established for Eu³⁺- and Eu²⁺-doped NaMgF₃. Some reports have suggested different types of charge compensation in materials of the same family (KMgF₃^28–30 and RbMgF₃³¹) doped with different lanthanide ions. However, these discussions are based on ion sizes (doped and host), ignoring discussions about lattice solution energy. In addition, the symmetry site and coordination number of the optically active ion are not clear, as well as the substitution site in the host matrix. The optically stimulated luminescence (OSL) decay pattern and high sensibility of the NaMgF₃:Eu²⁺ compound for low dose levels are not fully established. Therefore, a systematic study is necessary to make predictions about the incorporation of defects in the NaMgF₃ structure.

Classical atomistic simulation is a reliable tool for modelling a range of ionic materials and to help understand theoretical and experimental results. In this methodology, interactions between atoms are determined by interatomic potentials that are essential to studying physical properties of the simulated systems. Several studies have been widely used to examine structural, mechanical, elastic, and dielectric properties in solid-state materials.^32–36 Furthermore, atomistic simulation is able to perform studies on defect properties with low computational cost, compared with other methodologies, and has been successfully employed to study defects.^37–40 In addition, atomistic simulation procedures, combined with crystal field theory, is an practical method for describing spectroscopic properties of lanthanide ion-doped compounds and their dopant-related effects. Recently, Otsuka et al.⁴¹ performed a study from a spectroscopic point of view, combining atomistic simulation, the simple overlap model (SOM),⁴² and the method of nearest neighbours⁴³ (theoretical models of crystal field). The combination of both methodologies successfully described local symmetry and coordination number of the optically active ion, crystal field parameters, crystal field strength, ⁷F₁ stark sub-levels, and splitting.

Thus, in this work, we used a combination of classic atomistic simulation-based ionic models and crystal field models to study the orthorhombic phase of NaMgF₃. Firstly, atomistic simulation was used to describe structural properties and the defect formation process with the incorporation of Eu³⁺ and Eu²⁺ ions into NaMgF₃. For this, we developed a new set of interatomic potentials to describe the interactions between ions for the compound in the orthorhombic phase and performed a study of the structural and elastic properties. We carried out a defect study to obtain the most favourable charge compensation mechanism. Secondly, crystal field models were used to study spectroscopic properties of Eu²⁺ and Eu³⁺ ion-doped NaMgF₃. Detailed local geometry of the optically active ion in this host matrix was obtained. In addition, photoionization cross-section calculations, associated with the first-order kinetic model, gave us information about the OSL decay pattern and high sensibility of the Eu²⁺-doped NaMgF₃ compound.

2 Methodologies

2.1 Computational simulation

The atomistic simulation technique was used to study the perfect structure and defective lattice of orthorhombic NaMgF₃, performed by GULP code.⁴⁴ Relaxation of the lattice parameters and atomic positions was completed to find the lowest energy. A description of the structural properties of the system depends on a set of potential parameters, adopted for a reliable description of fundamental interactions between the ions. Long-range interactions were calculated by Coulomb potential and short-range interactions by Buckingham potential. Eqn (1) shows the representation of repulsive (or Pauli repulsion) and attractive (or van der Waals interaction) terms of the Buckingham potential:where A, ρ, and C are parameters obtained by a fitting procedure, and r is interatomic distance between ions.

In addition, a model for efficient treatment of ionic polarization effects is necessary, and a simple model, known as the shell model,⁴⁵ was used. Ions in this model are represented by a core (massive, includes the nucleus plus core electrons) and shell (massless, includes valence electrons) connected by a harmonic constant. The formal charge of the ion is obtained by the sum of the core and shell charges.

The defect calculation was performed using a two-region strategy.⁴⁶ This method is very useful for calculating defects in atomistic simulations and has been used successfully.^47–51 The crystal lattice is divided into two spherical regions (I and II), where the defect (or defect cluster) is placed in the centre of these regions. The inner region I is the portion of the crystal located around the defect, allowing explicit relaxation of all the ion positions under the action of a force field. Region II is more distant from the defect and can be treated using an approximate continuous method, since ions in this region exhibit an interatomic displacement smaller than the ions in region I. To obtain reliable results, a convergence test, with an appropriate radius for these regions, is necessary. In this work, we used 12 and 18 Å for regions I and II, respectively. This corresponds to approximately 1000 ions in the region I and 2400 ions in region II. The total energy (E_T) can be calculated by the expression E_T = E₁(x) + E₁₂(x,u) + E₂(u), where E₁(x) is the energy of region I, E₂(u) is the energy of region II, and E₁₂(x,u) is the energy of the interaction region between them.

2.2 Crystal field parameters and Stark levels of the ⁷F₁ multiplet

Interaction between the lanthanide ion and its nearest neighbours (NNs) has been a discussion theme in research groups that work with lanthanide spectroscopy, for a long time. The point charge electrostatic model (PCEM)⁵² was the first non-parametric model to discuss crystal field parameters from a theoretical point of view. The PCEM considers that the bond between the lanthanide ion and its chemical surrounding is purely ionic, where the charge factor is equal to ligand valence and is located at the NN's position. Although some considerations of the PCEM have led to unsatisfactory results from a quantitative point of view, it has been the base model for the development of other theoretical models.

The simple overlap model⁴² used in our predictions is a theoretical model based on the PCEM, which has been largely used in lanthanide spectroscopy with satisfactory predictions.^53,54 The SOM introduces a small covalent character to describe Ln–NN chemical bonding. In this assumption, the effective interaction charge is defined as −ρ_jg_je and is located around the Ln–NN middle distance (R_j/2β). g_j is the charge factor devoted to Ln–NN chemical bonding, R_j is j-th NN distance from the Ln ion, e is the elementary charge, and β_j = 1/(1 ± ρ_j) is a factor that determines the position of the effective charge in the middle distance. The minus sign is applied when the charge is closer to the Ln ion, and the plus sign is applied when the charge is closer to the ligand. ρ_j = ρ₀(R₀/R_j)^3.5 describes the overlap of interacting wavefunctions, where R₀ is the smaller Ln–NN distance, and ρ₀ = 0.05 is the maximum overlap between the 4f and 2s (or 2p) orbitals.⁵⁵

Through these considerations, the crystal field parameters (B^k_q) of the SOM can be related to PCEM, as show eqn (2):

The ⁷F₁ energy sublevels of Eu³⁺ can be obtained through diagonalization of the crystal field matrix within the ⁷F₁ manifold.⁵⁶ Thus,

where E₀, E₋, and E₊ are energy sublevels for J = 1, measured in relation to the barycentre. U² is the reduced matrix element.⁵⁶

2.3 Photoionization cross-section of trap levels

The photoionization cross-section (σ) is an essential quantity to understand the interaction processes of electromagnetic radiation with matter. Recently, Lima–Batista–Couto⁵⁷ proposed a model to obtain σ of localized traps in the band gap with activation energy E_i with respect to the conduction band, based on time-dependent perturbation theory. The model describes the trap level by a three-dimensional isotropic harmonic oscillator wavefunction with angular frequency ω₀, and the electron in the conduction band is described by the plane wavefunction. Following the same steps reported previously⁵⁷ and using the Fermi's golden rule, we obtain:where γ(ω) = 2kℏω/m*ω₀c is an energy function, ω is the angular frequency of incident electromagnetic radiation, m* is the electron effective mass, α is the fine structure constant, ℏ is Planck's reduced constant, c is the speed of light, and k is the wavevector of the electron. This expression is obtained considering all multipole terms in the Hamiltonian that couple the linear moment of the electron with the radiation electromagnetic field. This model was applied successfully to predict σ in promising materials for personal dosimetry and explain the mechanism of electron de-trapping with light stimulation.^57,58

3 Results and discussion

3.1 Interatomic potentials of NaMgF₃

To analyse the structural properties and influence of defect clustering on the luminescent properties of the compound, describing the interactions between ions of the materials through a reliable set of interatomic potentials is necessary. We developed a new set of interatomic potentials for the orthorhombic phase of NaMgF₃ from an empirical fitting procedure, carried out with GULP code.⁴⁴ The empirical fitting was used to obtain Buckingham potential parameters for the Na–F interaction. The potential parameters used for Mg–F and F–F interactions were taken from a previous study³³ and have already been tested and validated for compounds of the same family, AMgF₃ (A = K, Cs, and Rb). Table 1 shows the interatomic potentials and shell model parameters used in all calculations of this work. A short range potential cutoff of 12 Å was used.

Table 1 Buckingham potential and shell parameters for NaMgF₃

Interaction	A (eV)	ρ (Å)	C (eV Å^6)	k (eV Å^−2)	Y (e)	Ref.
Na^+core–F^−shell	1223.35	0.2682	0	—	—	This work
Mg^2+core–F^−shell	904.7	0.2825	0	—	—	Ref. 33
F^−shell–F^−shell	5050.2	0.2189	4	15	−1.378	Ref. 33

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This set of interatomic potentials was validated, and the calculated lattice parameters of the NaMgF₃ compound are in excellent agreement with X-ray diffraction values, as well as mechanical properties. Elastic and dielectric constants are close to experimental values (see next section). The fluoride precursors NaF and MgF₂ are commonly used to synthesize NaMgF₃. In addition, the same set of potentials is also capable of modelling precursor fluorides (NaF and MgF₂). Even though the focus of this work was to analyse the orthorhombic NaMgF₃ phase, we were able to show that this set of interatomic potentials is transferable to the cubic phase of NaMgF₃, as well. Calculations of the various precursor fluoride properties and cubic NaMgF₃ are shown in the supporting information (see Tables S1–S7, ESI†). These facts are important validation characteristics for successfully calculating defect properties.

3.2 Structural properties of the orthorhombic NaMgF₃

Table 2 presents a comparison between calculated and experimental data from lattice parameters and cell volumes for orthorhombic NaMgF₃. A relative error of less than 0.82% was calculated for all lattice parameters and cell volumes with respect to X-ray diffraction data.⁵⁹Table 3 shows the most relevant interatomic distances for orthorhombic NaMgF₃ calculated in this work compared to experimental data.⁵⁹ The distances presented a relative error below 3% in all cases. These results show that our atomistic simulation has good acceptance in the reproduction of NaMgF₃ structural properties. In addition, a similar relative error was observed for all properties studied for cubic NaMgF₃ using the same set of interatomic potentials, as shown in the supporting information.

Table 2 Lattice parameters and cell volumes for orthorhombic NaMgF₃

Lattice parameters	Ref. 59	This work	%
a (Å)	5.360	5.404	0.82
b (Å)	5.488	5.473	−0.27
c (Å)	7.666	7.689	0.30
V (Å^3)	225.53	227.40	0.83

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Table 3 Interatomic distances (in Å) for orthorhombic NaMgF₃

Distance	Ref. 59	This work	%
Na–F1(x1)	2.322	2.322	0.00
Na–F2(x2)	2.303	2.337	1.48
Na–F1(x1)	2.416	2.474	2.40
Na–F2(x2)	2.563	2.640	3.00
Na–F2(x2)	2.710	2.710	0.00
Na–F1(x1)	3.097	3.079	−0.58
Na–F1(x1)	3.185	3.118	−2.09
Mg–F2(x2)	1.981	1.974	−0.35
Mg–F1(x2)	1.979	1.976	−0.15
Mg–F2(x2)	1.989	1.977	−0.60

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Fig. 1 presents the orthorhombic NaMgF₃ phase with space group Pbnm calculated for this work. The structure contains four non-equivalent (Na, Mg, F1, and F2) atoms. Mg²⁺ ions are coordinated by six F atoms, organized into three pairs of Mg–F bonds with approximately the same distances (two pairs of Mg–F2 bonds are equatorials and one pair of Mg–F1 bonds is apical). The Na atom is coordinated by eight F atoms, with only two Na–F1 bonds, and almost all bond distances are between Na–F2. This compound presents a different structural behaviour at room temperature and pressure compared to other materials of the same family (AMgF₃, A = Cs, Rb, and K).³³

Fig. 1 The crystalline structure of NaMgF₃ in an orthorhombic lattice with space group Pbnm. Na and Mg sites are shown in detail.

The elastic constants (C₁₁, C₂₂, C₁₂, C₁₃, C₂₃, C₃₃, C₄₄, C₅₅, and C₆₆) of NaMgF₃ in the orthorhombic phase are shown in Table 4, and the values calculated in this work are compared with experimental data.⁶⁰ The elastic constants satisfy Born's criteria and prove its mechanical stability. The reproducibility of these properties validates the potentials and transferability, which is crucial for modelling physical properties under conditions different from the initial fitting procedure. The bulk modulus, shear modulus, static dielectric constant (ε₀), and high-frequency dielectric constant (ε_∞) for orthorhombic NaMgF₃ are also shown in Table 4. In addition, our results show excellent transferability of these potentials for cubic NaMgF₃ (see ESI†). Having successfully completed this first step, we next analysed defect properties and their influence on spectroscopic properties of NaMgF₃.

Table 4 Elastic constants, bulk modulus, shear modulus and dielectric constants (static and high-frequency) for orthorhombic NaMgF₃

Elastic constants (GPa)	Ref. 60	This work	%
C11	125.7	132.9	5.7
C22	147.3	134.9	−8.4
C12	49.5	50.2	1.4
C13	45.1	46.2	2.4
C23	43.1	47.8	10.9
C33	142.5	135.1	−5.2
C44	46.7	45.3	−3.0
C55	44.8	43.4	−3.1
C66	50.4	43.8	−13.1
Dielectric constants
ε 0	—	7.75	—
ε ∞	—	2.20	—
Bulk modulus (GPa)	—	76.82	—
Shear modulus (GPa)	—	43.72	—

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3.3 Defect calculations

The process of incorporating Eu³⁺ and Eu²⁺ ions into the NaMgF₃ compound requires a charge compensation mechanism to stabilize the local structure and accommodate extra charge in the relaxed structure. Interatomic potential used to describe Eu–F interactions was taken from a previous study for modelling natural apatite crystals,⁶¹ and have already been tested and validated for Rare-Earth fluorides. The incorporation of defects into the crystalline structure, obtained by our atomistic simulation, can estimate the preferred doping site and mechanism of charge compensation most favourably. Firstly, we consider the various possible schemes of charge compensation for the incorporation of Eu³⁺ and Eu²⁺ in the NaMgF₃ compound. Tables 5 and 6 show the proposed chemical reaction schemes, expressed in Kröger–Vink notation,⁶² for the incorporation of Eu³⁺ and Eu²⁺ ions, respectively. After that, the next stage is the calculation of solution energy (E_sol) for each reaction, performed to predict the most energetically favourable scheme. The solution energy was obtained by a combination of defect energies, lattice energy of the fluoride precursor, and lattice energy of the dopant ion. In this work, the solution energies are calculated considering defects as isolated species (or unbound defects) and simulated in a cluster of defects (or bound defects). The motivation for calculating a cluster of defects is to account for the binding energy for different arrangements that frequently present the lowest energy. An example of how to calculate the solution energy for unbound (E^unbound_sol) or bound defects (E^bound_sol) is shown in eqn (7a) and (7b) for the first reaction (Scheme I) in Table 5. For the other reactions, a similar procedure is employed.where E_def is the defect formation energy, and E_latt is lattice energy. More details regarding the calculation of solution energies are found in ref. 48.

Table 5 Solid state reactions in the Kröger–Vink notation for Eu³⁺-doped NaMgF₃

Schemes	Incorporating the Eu^3+ into Na^+ site
(I)
(II)
(III)
(IV)
(V)
	Incorporating the Eu^3+ into Mg^2+ site
(VI)
(VII)
(VIII)
(IX)
(X)
	Incorporating the Eu^3+ into Na^+ and Mg^2+ sites
(XI)
(XII)

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Table 6 Solid state reactions in the Kröger–Vink notation for Eu²⁺-doped NaMgF₃

Schemes	Incorporating the Eu^2+ into Na^+ site
(XIII)
(XIV)
(XV)
(XVI)
(XVII)
(XVIII)
(XIX)
(XX)
	Incorporating the Eu^2+ into Mg^+2+ site
(XXI)

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Tables S8–S12 in the ESI† show defect and lattice energies required to perform the solution energy calculations.

Fig. 2 presents a solution energy diagram for bound and unbound defects for each reaction of Eu³⁺-doped NaMgF₃, represented by schemes shown in Table 5. Notably, Eu³⁺ prefers to be incorporated into the Mg²⁺ site (E^bound_sol = 2.129 eV), compensated by a sodium vacancy (Scheme VII). Other reactions proposed here lead to values close to the red line. Reactions IV (entering into a Na⁺ site) and VI (entering into a Mg²⁺ site), for instance, have approximate differences of 0.5 and 0.3 eV, respectively, with respect to the lowest point. Our predictions show that mechanisms involving two substitutional defects in Na⁺, Mg²⁺, or both sites are practically unlikely in this configuration. In all cases, the calculations carried out in clusters lead to a decreased solution energy because of interactions between defects. Some reactions show considerable binding energy, leading to lower solution energy in comparison with unbound defect calculations.

Fig. 2 Solution energies of bound and unbound defects for incorporation of Eu³⁺ into NaMgF₃.

In fact, the valence of ions is a determining factor for Eu³⁺ entering into the Mg²⁺ site instead of the Na⁺ site. The ionic radius varies with coordination number, charge states, and other parameters.⁶³ Even though Eu³⁺ (0.947 Å for a coordination number of six) has a larger ionic radius than Mg²⁺ (0.72 Å for a coordination number of six) and is smaller than Na⁺ (1.18 Å for a coordination number of eight), Eu³⁺ is preferable for replacing Mg²⁺ in the host matrix. In this case, the difference between the ionic radii of Eu³⁺ and Mg²⁺ ions are practically the same as Eu³⁺ and Na⁺ ions. Thus, according to our calculations, the difference in valence between the Eu and Na ions leads to less favourable charge compensations for the system, even though the Eu³⁺ ion could be better accommodated by replacing the Na site instead of the Mg site.

Fig. 3 shows solution energy versus proposed charge compensation schemes (see Table 6) for incorporation of Eu²⁺ into the Na site of NaMgF₃. The most favourable mechanism of charge compensation is through the sodium vacancy (scheme XV) and anti-site (scheme XVIII), both with energy solutions of approximately 1.5 eV (see red arrow). Analyses of scheme XXI show (see Table 6, but not shown in Fig. 3), evidently, that the Eu²⁺-doped Mg site requires no compensation mechanism because these ions have the same valence. The solution energy calculated for this scheme is approximately 2 eV. In other words, the calculations show that the Na site is energetically most favourable for the incorporation of the Eu²⁺ ion, rather than at the Mg site. In addition, our predictions show that the solution energy of bound defects is lower than unbound defects in all proposed schemes. These results show the importance of defect clustering in this system.

Fig. 3 Solution energies of bound and unbound for incorporation of Eu²⁺ into NaMgF₃.

The most probable mechanism of Eu³⁺ and Eu²⁺ ion incorporation into NaMgF₃, charge compensation is illustrated in Fig. 4. Analysing the proposed chemical reactions and calculation of solution energies, we see that when calculations are carried out considering defects and the respective mechanism of charge compensation, as a defect cluster, the solution energy is less than when calculated as isolated defects. The configuration of the local site is modified in terms of distances and distortions. Understanding these changes is of great importance for a better understanding of luminescent properties of optically active ions in the host matrix.

Fig. 4 Illustration of Eu³⁺ and Eu²⁺ incorporation into NaMgF₃ with the most favourable charge compensation mechanism: (a) pure NaMgF₃, (b) cluster, (c) cluster, and (d) cluster.

Table 7 shows the interatomic distances (d) from the atomistic simulation after doping Eu³⁺ and Eu²⁺ ions into NaMgF₃ for the most favourable schemes found in this work. The percentage difference between Eu–F and Na–F (or Mg–F) interatomic distances is represented by Δ (%). The Na–F and Mg–F distances are taken from the pure NaMgF₃ phase (see Table 3) for comparison. We note that some distances are reduced, while others are increased for both cases (schemes) involving Eu²⁺. For the cluster , the atoms drastically approach Eu²⁺ with distances of less than 3 Å. For Eu³⁺, in contrast with Eu²⁺, all distances increase after doping. In this case, Δ (%) is around 10% for all interactions.

Table 7 Comparison between interatomic distances (d in Å) of Eu²⁺ and Eu³⁺ ions doped in Na and Mg sites, considering the most favourable schemes

Eu^2+ into Na site,			Eu^2+ into Na site,			Eu^3+ into Mg site,
Eu–F	d (Å)	Δ (%)	Eu–F	d (Å)	Δ (%)	Eu–F	d (Å)	Δ (%)
Eu–F2	2.436	4.2	Eu–F2	2.467	5.6	Eu–F2	2.163	9.4
Eu–F2	2.436	4.2	Eu–F2	2.479	6.1	Eu–F2	2.197	11.1
Eu–F1	2.458	5.9	Eu–F1	2.463	6.1	Eu–F1	2.205	11.6
Eu–F1	2.592	4.8	Eu–F1	2.486	0.5	Eu–F1	2.161	9.4
Eu–F2	2.672	−1.4	Eu–F2	2.532	−6.6	Eu–F2	2.201	11.5
Eu–F2	2.672	−1.4	Eu–F2	2.727	0.6	Eu–F2	2.168	9.8
Eu–F2	2.633	−0.3	Eu–F2	2.533	−4.1
Eu–F2	2.633	−0.3	Eu–F2	2.632	−0.3
Eu–F1	2.715	−11.8	Eu–F1	2.911	−5.5
Eu–F1	2.936	−5.8	Eu–F1	3.121	0.1

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Experimental results based on photoluminescence spectra indicate that both Eu²⁺ and Eu³⁺ ions can co-exist in NaMgF₃, even although the emission spectrum from Eu³⁺ is quite different from the Eu²⁺ ion. In this case, when the NaMgF₃ is excited at 256 nm the emission from Eu²⁺ is observed at 366 nm, and when excited at 396 nm emission from Eu³⁺ at 590 nm is observed.²² Although the Eu²⁺/Eu³⁺ ratio depends on the chemical composition of the host,⁶⁴ the synthesis method⁶⁵ and exposure to ionizing radiation,^66,67 our calculations suggest that, from the perspective of solution energy, the Eu ion prefers to be incorporated into NaMgF₃ in its divalent state (see Fig. 2 and 3). Unlike Eu in the Na site, our calculations reveal that on the Mg²⁺ site, both Eu²⁺ and Eu³⁺ trivalent Europium are energetically favourable, once the energy difference is about 0.133 eV. Thus, Eu²⁺ emission can arise from Eu localized in the Na and Mg sites. In fact, experimental results show that Eu²⁺ is dominant at lower concentrations, but Eu³⁺ becomes important when concentration increases. This behaviour is similar to that reported for Eu³⁺ and Eu²⁺ in CaF₂:Eu crystals.⁶⁸

3.4 Spectroscopic properties of Eu ions in NaMgF₃

3.4.1 Local structure of the Eu³⁺ ion. In order to analyse the local symmetry and charges transferred in Eu–F chemical bonding and to calculate the crystal field parameters and Stark levels from the ⁷F₁ multiplet, relaxed positions of Eu³⁺ and its NNs are necessary. These positions are obtained from the most favourable mechanism of atomistic simulation presented in Section 3.3. In scheme VII, the most energetically favourable, Eu³⁺ has the same coordination number (six NNs) when incorporated into the Mg²⁺ site.

Spherical coordinates of the Eu³⁺ ion, obtained by atomistic simulations of defect clustering, are shown in Table 8. In dealing with a distorted structure, all Eu–F distances are slightly different and, consequently, symmetry discussions here are approximate.

Table 8 Spherical coordinates with respect to the main axis of symmetry, with the Eu³⁺ ion at the origin of the system

Spherical coordinates^a
NN	Distorted S6			Ideal S6
	R	θ	ϕ	R	θ	ϕ
a The ideal S6 point symmetry corresponds to one site of Eu^3+ in C-rare earth sesquioxides, taken from ref. 69, for comparison. The radial coordinates are given in angstroms, and the angular coordinates are given in degrees. The centroid (F1, F2, F3) coming out of the x′–y′ plane is taken as the z′-axis.
F1	2.168	63.36	0	2.345	62.81	0
F2	2.161	60.77	240.83	2.345	62.81	240
F3	2.163	62.61	121.50	2.345	62.81	120
F4	2.197	128.30	302.49	2.345	117.19	300
F5	2.201	127.65	180.74	2.345	117.19	180
F6	2.205	130.11	61.56	2.345	117.19	60

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We chose the principal axis of symmetry (z′-axis) by diagonalizing the tensor of the quadrupolar field, which is experienced by the optically active ion. In this case, the eigenvector takes the highest eigenvalues. The centroid (coming out the x′–y′ plane) between the F1, F2, and F3 atoms is taken as the z′-axis to measure the spherical coordinates.

F1, F2, and F3 have slightly different distances, as well as F4, F5, and F6 atoms. The C₃ symmetry operation about the z′-axis takes, approximately, F2 to F1 and F1 to F3. Similarly, the atoms in the lower plane bounce off each other. Then, F1, F2, and F3 are approximately equivalent to each other. Likewise, F4, F5, and F6 may be considered approximately equivalent. This procedure reduces the degree of freedom in crystal field calculations by employing the same charge factor to equivalent atoms. Thus, we have used the charge factor g₁ to yellow atoms and g₂ to orange atoms.

Atomistic simulation was used to better understand the true nature of the defects in NaMgF₃, giving us information about spatial coordinates of the Eu³⁺ ion, which is not easily obtained by X-ray diffraction because of the low concentration of the Eu ion in the host matrix. Notably, a C₃ symmetry operation (Fig. 5) following a σ_h operation (reflection plane that contains the x′–y′ plane) takes the same structural pattern, approximately. The bond distances illustrated in Fig. 5 can be found in Table 8. A combination of C₃ and σ_h operations is termed S₆ symmetry in group theory.⁵⁶ In this case, a distorted S₆ point symmetry occurs because all distances are slightly different, and the angles differ from that of ideal S₆ symmetry. Once in S₆ point symmetry, the electric dipole 4f–4f transitions are forbidden, and the distorted S₆ symmetry explains the weak electric dipole 4f–4f transitions of Eu³⁺ observed in NaMgF₃.

Fig. 5 Local symmetry of the optically active ion, and the axis adopted to obtain spherical coordinates. The centroid coming out the x′–y′ plane is taken as the z′-axis.

Spectroscopic properties of Eu³⁺ in the Na sites must be quite different from Eu³⁺ substituting Mg²⁺, due to low symmetry of the Na site. In addition, our simulation shows that there is also distortion when Eu²⁺ substitutes Na. This suggests that Eu²⁺ must be in a site with very low symmetry, which induces a high intensity 4f–4f transition. In fact, 4f–4f emission from Eu²⁺ in NaMgF₃ has intensity comparable with broad 4f⁶5d¹–4f⁷ emission, which is an allowed transition and occurs with a high transition probability. On the other hand, experimental results show that Eu³⁺ in low symmetry sites are less unlikely, and this has also been predicted in our simulations, showing that Eu³⁺ in a Na site is less probable.

3.4.2 Crystal field parameters. Table 9 shows charge factors and the set of crystal field parameters, B^k_q, for the distorted S₆ site. The B^k_q were calculated using spherical coordinates of Eu³⁺ ion incorporation into the Mn site of NaMgF₃ (Table 8), a set of charge factors (g₁ and g₂) that describes the interaction Eu–F in this dielectric medium, and the maximum overlap, ρ₀, between 4f wavefunctions with ligand orbitals. In this case, we used the value (ρ₀ = 0.05) obtained by Axe and Burns.⁵⁵ Notably, all B^k_q are nonzero, as expected, because the local structure of the optically active ion is distorted. The B^k_q also allow for identification of Eu³⁺ ion symmetry, as well as predicting the ⁷F₁ state energy sublevels. In ideal S₆ symmetry,⁵⁶ only the following B^k_q values are nonzero: B²₀, B⁴₀, B⁴₃, B⁶₀, B⁶₃, and B⁶₆. We observe that the contribution of B^k_q values that represent S₆ symmetry are much higher than the others. B²₁ and B²₂ have approximately a 10% contribution in relation to B²₀; B⁴₁, B⁴₂, and B⁴₄ have a contribution around 4% in relation to B⁴₀ and B⁴₃. Likewise, B⁶₁, B⁶₂, B⁶₄, and B⁶₅ contribute approximately 10% in relation to B⁶₀, B⁶₃, and B⁶₆.

Table 9 Crystal field parameters (B^k_q in cm⁻¹) and charge factors^a using β⁻ and β⁺

B ^k q	β ^−	β ^+
a β ^− and β^+ define the charge factor position around the middle distance of Eu–F. The minus signal means that g is closer to Eu^3+, and the plus sign indicates that g is closer to the ligands. A rotation (30.5°) about the principal axis was carried out to eliminate the imaginary part of B^22.
B ^20	−299.6	−299.53
B ^21	−8.144 − 27.578i	−7.555 − 26.408i
B ^22	22.523	22.225
B ^40	−269.799	−211.583
B ^41	−0.98 − 8.615i	−1.118 − 6.937i
B ^42	−20.045 − 8.944i	−16.209 − 6.732i
B ^43	−50.207 + 522.044i	−28.773 + 416.689i
B ^44	2.264 + 1.095i	1.657 + 1.034i
B ^60	413.269	272.774
B ^61	8.86 + 34.918i	6.044 + 22.983i
B ^62	5.69i	0.049 + 3.418i
B ^63	11.311–134.588i	5.575 − 93.281i
B ^64	35.648–4.526i	22.963 − 3.444i
B ^65	0.292 + 22.377i	0.333 + 14.947i
B ^66	−282.947 − 53.841i	−187.929 − 25.489i
g 1	0.435	0.58
g 2	0.034	0.031

---

This leads us to conclude that Eu³⁺ occupies is a distorted S₆ point symmetry. Furthermore, we have calculated B^k_q using β⁻ and β⁺, which define the charge factor positions around the middle distance of Eu–F.⁴² The minus signal means that g is closer to the Eu³⁺ ion, and the plus signal means that g is closer to the ligands. This parameter is a way (according to the SOM) to include covalence effects on the chemical bond because the charge is localized in a middle distance (R/2β) instead of being located at the position of the ligand, as proposed by the PCEM. In this case, the use of β⁺ leads to a lower contribution from B^k_q that does not belong to ideal S₆ symmetry. In addition, the phenomenological charge factors, adjusted to reproduce the ⁷F₁ state energy sublevels, are higher.

We also use the model proposed by Lima et al.⁷⁰ to calculate charge transferred to the Eu–F chemical bond. This model is valid for high symmetry systems, in which only one charge factor is needed to describe the system. With ideal S₆ point symmetry, the model would be well applied, but for the sake of comparison, we have calculated one of the charges through this model using the following expression:

g = Δχ(D)/R_Eu−F (8)

where Δχ(D) is the Pauling electronegativity difference in Debye units (D = 3.33 × 10⁻³⁰ C m), and R_Eu–F is the distance between the positive and negative charge centres. R_Eu–F can be obtained by the difference between the atomic and crystalline radii from the cations and anions (refer to ref. 71 for more information). By using benchmark values available in the publication by Shannon,⁶³ we find R_Eu–F = 0.982 Å. Thus, we obtain g = 0.594 using Δχ(D) = 2.8. This value is closer to g₁ adjusted with β⁺ (see Table 9).

The B²₀ sign defines the position of the ⁷F₁ state ground sublevel from the barycentre. We see in Table 8 that it is correctly predicted using β⁻ and β⁺ because B²₀ is negative, and the ⁷F₁ state ground sublevel is non-degenerate. We will discuss this point in more detail in the next section.

3.4.3 ⁷F₁ state energy sublevels of the Eu³⁺ ion. Table 10 shows the experimental²² and predicted ⁷F₁ state energy sublevels and splittings. The energy sublevels are measured with respect to the barycentre of the ⁷F₁ level. We use a set of phenomenological charge factors in the calculations of B^k_q to reproduce the energy sublevels and, consequently, the splitting. Our predictions were carried out using β⁻ and β⁺ for comparison.

Table 10 Experimental²² and predicted ⁷F₁ state energy sublevels and ΔE. The experimental energy sublevels with respect to the ⁷F₀ level are shown in parentheses. The other values are measured in relation to the barycentre

E (cm^−1)	E exp	E(β^−)	E(β^+)
E 0	−65.344 (315.689)	−58.674	−58.660
E −1	32.672 (413.705)	19.230	19.357
E +1	32.672 (413.705)	39.444	39.303
ΔE	98.016	98.118	97.963

---

The photoluminescence emission spectra of NaMgF₃ nanoparticles containing Eu, excited at 396 nm, for 1% Eu show ⁵D₀ → ⁷F_J transitions from Eu³⁺. The emission spectrum reported by Gaedtke and William²² at room temperature presents one peak corresponding to the ⁵D₀–⁷F₀ transition, two peaks from ⁵D₀–⁷F₁, two peaks from ⁵D₀–⁷F₂, and four peaks from ⁵D₀–⁷F₄. ⁵D₀–⁷F₁ is a magnetic dipole transition, which is not influenced by the crystalline environment. The number of lines and intensities in relation to the ⁵D₀–⁷F₂ transition indicates if the system is lower or higher in symmetry. The other transitions are electric dipole moment transitions which are strongly influenced by the crystalline environment.

The second peak of the ⁵D₀–⁷F₁ transition is doubly degenerate, and the ⁷F₁ splitting is less than 100 cm⁻¹. The emission spectrum reported by Gaedtke and William²² shows that the ⁵D₀–⁷F₁ transition is approximately 50% more intense than the ⁵D₀–⁷F₂ transition. This suggests that the Eu³⁺ ion occupies point symmetry with a distorted inversion centre, although the emission spectrum shows peaks corresponding to the ⁵D₀–⁷F₀ and ⁵D₀–⁷F₂ transitions.

Schuyt and William,²⁴ based on the Tanner diagram,⁷² suggested that the Eu³⁺ occupies sites with C_s, C_nv or C_n symmetry because the ⁵D₀–⁷F₀ transition is presented in the emission spectrum. C_s is part of low symmetry groups, which is not the case here because the ⁷F₁ splitting is less than 350 cm⁻¹ (ref. 73). Analysing the number of lines for each transition in the emission spectrum and comparing it with the Tanner diagram⁷² indicates C_3v or C_4v symmetry. Previous work carried out with Eu³⁺-doped KMgF₃ suggested the same symmetry.^28–30 However, C_3v, C_4v, and C_n are symmetry groups without inversion centres. The crystal field parameters related to the odd part of the crystal field potential is different from zero for this symmetry set (C_3v, C_4v, and C_n). In this case, the ⁵D₀–⁷F₂ transition, allowed by electric dipole and strongly influenced by the environment, would be more intense than the transition ⁵D₀–⁷F₁. This is not observed in the NaMgF₃:Eu emission spectrum. Thus, the most probable symmetry is distorted S₆ point symmetry. Due to distortion in the luminescent site, other transitions, beyond ⁵D₀–⁷F₁, are apparent in the emission spectrum beyond transition.

E ₋₁ and E₊₁ are slightly different because B²₂ is nonzero. The distortion in S₆ point symmetry leads to a small contribution of this parameter in relation to B²₀. This behaviour is not observed in the emission spectrum obtained experimentally. However, Seo et al.²⁸ showed a slight splitting of the second line around 0.7 nm in KMgF₃:Eu using site-selective laser-excitation spectroscopy. This agrees very well with our predictions because E₋₁ and E₊₁ are separated by approximately 20 cm⁻¹. In fact, the spectrometer used has a lower resolution, and a splitting of 0.7 nm cannot be identified in the emission spectrum.

Another point that deserves to be highlighted is the substitutional defect and charge compensation created by inserting Eu³⁺ into the host matrix. Reports for materials from the same family (Eu³⁺-doped KMgF₃^28–30 and RbMgF₃³¹) have diverged in relation to substitutional defects and charge compensation. The reports for Eu-doped KMgF₃ have suggested a Na substitutional defect,^28–30 while one report for Eu-doped RbMgF₃ suggested an Mg substitutional defect.³¹

Our results for the Eu³⁺ ion in NaMgF₃ show a Mg substitutional defect compensated by a Na vacancy in the calculations with defect clustering (see Fig. 2, scheme VII). The energy difference of the most favourable mechanism for Eu³⁺ incorporation in Na and Mg sites is around 0.5 eV. Reaction (IV) in the Na site is less favourable; the local geometry of the optically active ion has a lower symmetry (with eight NNs) and higher distortion compared with the most favourable reaction. We tested this geometry in crystal field parameter calculations, but the predictions do not reproduce the emission spectrum characteristics through a set of positive charge factors. Moreover, values of B^k_q do not lead to any conclusion about the local symmetry of the Eu³⁺ ion (doped in the Na site). Thus, our conclusions are based on atomistic simulation, group theory, crystal field calculations, and emission spectrum characteristics of NaMgF₃:Eu. These results lead us to strongly believe that the Eu³⁺ ion is incorporated in the Mg site.

3.5 Photoionization cross-section and OSL decay pattern of NaMgF₃:Eu²⁺

Polycrystalline NaMgF₃:Eu²⁺ has been shown to be a suitable material for application in personal dosimetry. The material has high sensitivity and is able to monitor small doses, having a linear dose–response behaviour between μGy dose levels up to approximately 100 Gy.¹⁸ However, this behaviour and the mechanism of electron de-trapping are not completely explained in the literature.

In this section, we discuss this point based on the photoionization cross-section (σ) of the trap level and associated it with the substitutional defect to understand the origin of the OSL signal from NaMgF₃:Eu²⁺. We employed the model developed by Lima–Batista–Couto⁵⁷ to predict σ, and ref. 57 can be consulted for more details on the method.

Fig. 6 shows the photoionization cross-section as a function of the electromagnetic radiation energy for NaMgF₃:Eu²⁺. The curve has a broad excitation interval that leads to electron de-trapping processes of localized traps in the band gap. OSL emission reported by Dotzler et al.,¹⁸ excited at 450 nm, shows a broad interval of the emission spectra from samples pre-irradiated with X-rays. This agrees with our predictions. We also observe that the maximum peak occurs for light stimulus at approximately 2.2 eV, which corresponds to the maximum probability of electrons to be de-trapped from this localized trap.

Fig. 6 Photoionization cross-section versus electromagnetic radiation energy for NaMgF₃:Eu²⁺. Inset shows experimental data from ref. 77 and theoretical OSL decay patterns for NaMgF₃:Eu²⁺. The experimental decay curve was obtained after irradiation with an X-ray dose of 219 mGy.⁷⁷

We calculate the magnitude of σ to the specific wavelength, λ = 470 nm, which was the same wavelength used in the most OSL measurements. The phonon frequency used here is 325 cm⁻¹, obtained from ref. 74 CaF₂:Yb²⁺. The electron effective mass used is 0.74m_e, reported for fluoroperovskites in ref. 75. The activation energy used is 1.2 eV.⁷² By using these values in eqn (6), we obtain σ = 0.244 × 10⁻²⁰ m² for the NaMgF₃:Eu²⁺ compound. This value is close to that obtained by Daniel et al.⁷⁶ for NaMgF₃:Eu²⁺, Ce³⁺ using the fitting method of the linearly modulated (LM)-OSL experimental curve. There, the fitting curve was carried out with four components, and one value of σ was obtained for each component. The dominant term presents a magnitude of 0.112 × 10⁻²⁰ m².

Using the value of σ calculated here, we estimate the OSL decay pattern of NaMgF₃:Eu³⁺ based on the first-order kinetic approximation (no re-trapping), which assumes the OSL signal decay with stimulation time is due to de-trapping of captured electrons and subsequent radiative recombination. Fig. 6 (inset) shows the experimental and theoretical OSL decay patterns. The experimental OSL decay curve was obtained after irradiation with an X-ray dose of 219 mGy.⁷⁷ The decay time is slower than that exhibited in the commercial material, Al₂O₃:C.⁵⁸ We note that the theoretical curve deviates slightly from the experimental curve because the model used here is the first-order kinetic model.

The rate at which electrons captured in the trap are optically excited to the conduction band is proportional to σ, and the OSL decay pattern is governed by σ. Our predictions show that σ of NaMgF₃ is on the same order of magnitude (10⁻²⁰ m²) as the calculated value for Al₂O₃:C.⁵⁸ This explains the high sensibility when stimulated with blue light.

4 Conclusion

In summary, we combined classical atomistic simulation and crystal field models to describe the origin of defects and their influence on luminescent properties of Eu-doped NaMgF₃ in the orthorhombic phase. We proposed a new set of interatomic potentials that reproduce the main properties of the orthorhombic phase. Defect calculations based on these interatomic potentials provide information regarding the energetic balance of dopant incorporation in this fluoroperovskite compound. In addition, using crystal field calculations, we explored, in detail, the type of defect and spectroscopic properties of the optically active ion. The main findings of this work are summarized below.

– The new set of interatomic potentials reproduced structural and elastic properties in the orthorhombic phase and precursor fluorides. In addition, the interatomic potential is transferable to the cubic phase, consistent with the literature.

– Defect calculations show that incorporation of Eu³⁺ ions into the Mg site, compensated by the Na vacancy, is the most energetically favourable. Further, the Eu²⁺ ion prefers to incorporate into the Na site, compensated by a Na vacancy or anti-site, in the host matrix. In addition, the solution energy with Eu²⁺ is lower than with Eu³⁺.

– We predict the local symmetry and ⁷F₁ energy sub-levels of the Eu³⁺ ion by using the simple overlap model and the local geometry obtained in defect calculations.

– The weak intensity of the ⁵D₀–⁷F₂ transition, as well as the small splitting of the second peak of the ⁵D₀–⁷F₁ transition (observed in emission spectrum as doubly degenerate), occurs due to the distortion in S₆ local symmetry occupied by Eu³⁺ ions.

– Our predictions of the photoionization cross-section and OSL decay pattern show that NaMgF₃:Eu²⁺ presents a high sensibility for stimulus over a large range of wavelengths.

The new insights presented in this work show the importance of defect calculations, combined with crystal field and photoionization cross-section models, to successfully describe the luminescent properties of lanthanide-doped compounds.

Author contribution

Afranio Sousa: data curation, formal analysis, investigation, methodology, validation, writing – original draft, writing – review and editing. Adelmo Souza: visualization, investigation, data curation, writing – original draft. Heveson Lima: conceptualization, formal analysis, data curation, investigation, project administration, supervision, validation, visualization, writing – original draft, writing – review and editing.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors acknowledge the CNPq (No. 441451/2019-5), CAPES and FAPESB/BA, Brazilian funding agencies for financial support.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0ma00953a

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