Exploring the structural and optical properties of lithium-chromium phosphate Li3Cr2(PO3)4

The Lithium-chromium phosphate Li3Cr2(PO3)4 sample was synthesized via the solid-state reaction method. The morphological integrity and chemical homogeneity were verified by energy dispersive X-ray analysis (EDX) and scanning electron microscopy (SEM). Infrared and Raman patterns were also analyzed. Optical absorption spectrum analysis, conducted within the range of 10000 cm−1 to 30000 cm−1 at room temperature, yielded some optical parameters (Eg, Eu, δ, k, n). The Neuhauser model is used to interpret the interference dip which was on the absorption spectrum of Li3Cr2(PO3)4. The Fourier transform of the autocorrelation function leads to the Zero Phonon Lines of the observed absorption energies. The electronic structure of Cr3+ (3d (Huang et al., 2009) 33) ions in Li3Cr2(PO3)4 was calculated using Racah method, which allowed for precise calculations of Racah and crystal-field parameters. The results showed good agreement between the theoretical and experimental energy levels.


Introduction
Lithium metal phosphate materials Li 3 M 2 (PO 4 ) 3 with M is a transition metal ion (V, Fe, Mn, Cr …) are the newest generation of active materials.Li 3 M 2 (PO 4 ) 3 are also an intriguing materials with potential applications in various fields, including energy storage and optoelectronics [1][2][3].Beyond their use in energy storage, knowing how structural and optical characteristics interact within Li 3 M 2 (PO 4 ) 3 creates opportunities to improve their efficiency, safety, and performance.The presence of chromium (Cr) within the structure offers opportunities to enhance the optical properties, and consequently particularly interesting for optical applications [4,5].In fact, chromium, can introduce unique optical characteristics to the material.One notable aspect is its ability to exhibit different oxidation states, leading to diverse optical properties [6,7].In the context of Li 3 Cr 2 (PO 4 ) 3 , chromium likely occupies octahedral coordination sites within the crystal lattice, influencing its electronic band structure and optical behavior [8,9].
Moreover, the octahedral environment surrounding chromium ions can induce interesting optical phenomena, such as crystal field effects and charge transfer transitions.These effects can impact the absorption and emission spectra of the material, making it potentially useful for applications such as phosphor materials, solid-state lighting, or optical sensors [10][11][12][13].However, it's important to note that chromium is typically present in smaller amounts compared to phosphorus in Li 3 Cr 2 (PO 4 ) 3. While this may limit the extent of its influence on the material's optical properties.
This paper explores the intriguing properties of Li 3 Cr 2 (PO 4 ) 3 , with a particular focus on the investigation of their optical and structural properties.First, we will concentrate on the Li 3 Cr 2 (PO 4 ) 3 compound's preparation and structural analysis.Next, we intend to present some important optical parameters: the band gap energy (E g ), the Urbach energy (E u ), the skin depth (δ), the extinction coefficient (k) and the refractive index (n).Furthermore, we analyze the feature observed in the absorption band of Cr 3+ ions using the Neuhauser model.The objective of this study is to identify the cause of interference dips.Finally, a crystal field theory study was conducted for the Cr 3+ (3d [3]) in an O h site symmetry using Racah tensor algebraic methods.It's notable that a few investigations have used crystal field theory to analyze the compound's absorption spectrum.

Experimental details
The Li 3 Cr 2 (PO 4 ) 3 compound has been prepared by the conventional solid-state reaction.The starting reagents Li 2 CO 3 , Cr 2 O 3 , and NH 4 H 2 PO 4 were calculated in the desired proportion according to the following reaction: The precursors are thoroughly mixed in equimolar proportions and ground in an agate mortar for 2h.The powder obtained is heated in a crucible at a temperature of 573K for 8h with intermediate regrinding to exhaust NH 3 , H 2 O, and CO 2 .Secondly, the calcined powders were uniaxial pressed into a cylindrical pellet using a hydraulic press and then the pellets were sintered at an optimized temperature 1100 K for 6h to acquire close packed grains.
A range of techniques was employed for the structural and optical characterization of phosphor.These techniques are included, scanning electron microscopy (SEM) coupled with energy dispersive X-ray (EDX) analysis, RAMAN, FTIR reflectance and absorption spectroscopy.Using a TESCAN VEGA3 SBH scanning electron microscope (SEM) equipped with an energy dispersive microscopy (EDS detector Bruker XFlagh 410 M), the morphology of the powder and its chemical composition were investigated.Using an image processing tool (Image J, version 1.50), at least 60 randomly chased grains were measured in order to collect size and size-distribution statistics from SEM pictures.The Fourier Transform Infrared Spectroscopy (FTIR) spectrum has covered a spectral range between 400 and 4000 cm⁻ 1 , with the sample in solid form.The number of scans performed was 32, and the resolution was set at 4 cms⁻ 1 , using a DTGS (Deuterated Triglycine Sulfate) detector.For the Raman spectrum, the laser wavelength used was 532 nm, covering a spectral range of 0-3600 cm⁻ 1 .The Raman analysis also involved a solid sample, with 100 scans at a resolution of 30 accumulations, 0.5 s, detected by a CCD (Charge-Coupled Device)-black illuminate.For the evaluation of optical properties, we employed both reflectance and absorption spectroscopy techniques using the PerkinElmer Lambda 365 UV/Vis Spectrophotometer.This instrument, covering a wide wavelength range from 190 nm to 1100 nm, offers high resolution and sensitivity, crucial for detecting subtle spectral features.

Morphological studies
As it is mentioned previously, Li 3 Cr 2 (PO 4 ) 3 's surface morphology, size distribution, and elemental composition were all thoroughly examined using an Energy Dispersive X-ray Spectroscopy (EDX) in conjunction with a Scanning Electron Microscope (SEM).To determine the size, shape, and morphological characteristics of the particles in the sample, SEM observations were used.The (SEM) micrograph of our compound is shown in Fig. 1  particles.The fact that the particle is so small suggests that the synthesis of microparticles was efficient [14].
The EDX spectrum is displayed in Fig. 2. We notice that oxygen (O) is the predominant component, with phosphorus (P) and chromium (Cr) present in smaller quantities.Lithium (Li) is not detected in the EDX spectrum because it is a volatile element that tends to evaporate or sublimate when exposed to the electron beam used in EDX analysis.A non-negligible intensity peak labeled C is displayed in the spectrum.This peak corresponds to carbon emissions possibly coming from a carbon tape used during the measurement prosses.Fig. 2 shows also the presence of Na and Al, though in small quantities.These elements are considered impurities and may have originated due to contamination during sample preparation.The quantitative analysis of the compound's elemental composition as determined by EDX spectrum is summarized in Table 1.

Vibrational studies: FTIR and Raman
As described on the literature [15][16][17] Li 3 Cr 2 (PO 4 ) 3 crystallizes in a monoclinic structure, specifically in the space group P2 1 /n.The structure of our sample indeed consists of CrO 6 octahedra that share corners with six PO 4 tetrahedra.This creates a three-dimensional framework where the CrO 6 octahedra are interconnected by the PO 4 tetrahedra, contributing to the stability of the structure.This structural arrangement is common in various phosphate compounds [15][16][17].The FTIR pattern for the compound under investigation is displayed in Fig. 3a.The vibrational features in the lower wavenumber region (400 -1000 cm − 1 ) are like that of PO 4 3− in Ref. [18].
This part of our study is very interesting, especially in relation to the CrO 6 octahedron.For complexes having an inversion symmetry center, vibronic origin is found to be a particularly useful method of increasing electron-dipole intensity during d-d transitions.Table 2 recapitulates the selection rules outlined for the O h point group.
Concerning the octahedron CrO 6 octahedron, by eliminating the translation and rotation modes we have 15 internal vibration modes [4]: As shown in Equation (1), in our context, the t 1u mode is detected at 1166 cm − 1 in Infrared spectroscopy, while Raman spectroscopy identifies activation of the a 1g , e g , and t 2g modes [25].Notably, the t 2u mode remains inactive in both spectroscopic techniques [25].The observed d-d transitions are elucidated by vibronic coupling with the t 1u symmetry mode, which exhibits activity in Infrared spectroscopy.Notably, the vibronic frequency of the Raman-active a 1g mode is determined to be 376 cm − 1 , consistent with observations reported in previous studies.

Determination of some optical parameters 3.3.1. Optical band gap energy Eg
In this section we are going to determine the optical band gap Eg.We start our calculation by the Marotti method [26].In this method the Eg parameter is determined directly from the maximum detected from the evolution of the spectrum dR/dλbeing R the reflectance and λ the wavelength.Fig. 4 shows the reflectance and the evolution dR/dλ spectra versus the wavelength λ.As it is clear we found the band gap E g at 3.14 eV.
Another tool to determine the optical band gap is the Tauc method [27].This method is a helpful technique for determining a material's optical bandgap (E g ) from its absorption spectrum.This technique is also used especially for materials with a distinct absorption edge that matches the bandgap, such as semiconductors and insulators.The Tauc method uses the absorption spectrum which is represented in Fig. 5 a.The band gap energy (Eg) of Li 3 Cr 2 (PO 4 ) 3 was calculated using the widely recognized quadratic Tauc equation [27]: Fig. 2. The quantitative analysis of the compositional elements presents in Li 3 Cr 2 (PO 4 ) 3 using EDX.

Table 1
The elements present in Li 3 Cr 2 (PO 4 ) 3 and their percentage of occupation.
Where α is the absorption coefficient, h is the Planck constant, ν is the photon's frequency, E g is the band gap energy and A is a constant.
The exponent value n depends on the nature of the band gap which is 1/2 for an allowed direct band gap and has the value 2 for an allowed indirect bandgap [28].Fig. 5 b displays graphs of (αhν) 2 and (αhν) 1/2 plotted against photon energy (hν).Based on this analysis, we determined the optical energy band gaps for direct transition E gd to be 2.95 eV and for indirect transition E gi to be 2.46 eV.We reorganize Equation ( 2) to confirm the type of optical band gap (whether direct or indirect), and we obtain the following relationship: as shown in Equation ( 3), the plot of Ln(αhv) against Ln(hν-2.95) for Li 3 Cr 2 (PO 4 ) 3 is shown in Fig. 6.The power factor (n = 0.51),     H. Souissi et al. which is near to 0.5, is represented by the slope of the line in this plot, which depicts a linear relationship.According to this value, our sample coincides with wide optical band gap semiconductors that exhibit a direct band gap transition.The obtained value (2.95 eV) is in close agreement with the Marotti method result.

Urbach energy Eu
The Urbach energy is a fundamental parameter in solid-state physics that characterizes the exponential tail of the absorption spectrum in a semiconductor or insulator.It is typically determined by the following relation [29]: The Urbach energy can be calculated based on Equation ( 4) by plotting (Ln(α)) against photon energy (hv) as illustrated in Fig. 7.The calculation of the Urbach energy is done by taking the reciprocal of the slope of the linear portion of this curve.Based on our analysis, we found that the band gap energy is around 18.3 % of the Urbach energy (Eu), which is 0.54 eV.This value indicates the density of localized states that exist within the compound, so verifying the existence of imperfections and structural disorder.

The skin depth δ, the extinction coefficient k and the refractive index n of Li 3 Cr 2 (PO 4 ) 3
Penetration depth or alternatively optical depth or optical skin depth is a measure of how deep light can penetrate into a material before being significantly absorbed or attenuated.This parameter depends of the frequency of the light and it is from absorption coefficient (α) according to the formula [30][31][32]: As shown in Equation ( 5), Fig. 8 shows the plot of skin depth δ versus (hν) for Li 3 Cr 2 (PO 4 ) 3 .The spectrum exhibits the detection of two important penetration depths at specific energy levels within the visible region: hν 1 = 2.27 eV and hν 2 = 3.08 eV, signifies significant characteristics of materials' interactions with light in this spectral range.These particular penetration depths impact their optical behavior.
The absorption and scattering of light by a substance at a particular wavelength are quantitatively represented by the extinction coefficient k which is crucial in the design and characterization of optical materials and devices.The evolution of k parameter is determined by the following equation [26]: According to Equation ( 6), Fig. 9 illustrates the evolution of k parameter versus hν.This figure demonstrates the presence of at almost three peaks in the visible region.These variations in the extinction coefficient are caused by variation in absorbance and the dark blue colour of the crystal.
Understanding the refractive index (n) of materials is important for designing and optimizing optical components and devices [33][34][35].For reflectance and extinction, the coefficient n is defined by the following Equation ( 7) [36]:   Fig. 10 shows the curve of refractive index against wavelength λ.This curve displays two important peaks at almost 402 nm (3.1 eV) and 541 nm (2.3 eV) which are very close to the maxima of the optical depth depicted in Fig. 8 and the minima of extinction displayed in Fig. 9.

Identification of interference dip in 4 T 2g ( 4 F) exited state
In this part we will focus on the absorption spectrum of Fig. 11 recorded at ambient temperature.So, by examining the absorption features in this spectrum, we can elucidate the energy levels involved in the transitions and gain a deeper understanding of the electronic structure of the material.In this spectrum, two broad absorption bands are observed, corresponding to spin-allowed transitions from the ground state 4 A 2g ( 4 F) to the excited states 4 T 2g ( 4 F) and 4 T 1g ( 4 F).Notably, an interference dip is observed within the broad band associated with the 4 T 2g ( 4 F) transition.This interference dip is caused by the overlap of spin-forbidden transitions from the ground state 4 A 2g ( 4 F) to the excited state 2 Eg( 2 G).Our group is interested in understanding the likely features observed in absorption spectra [37][38][39] (see Fig. 12).
The interaction between states with distinct spin multiplicities is the source of the interference dip seen in this situation.In this work, we utilize the Neuhauser method for a main objective which is to accurately determine the levels of 2 E g ( 2 G) and the position of 4 T 2g ( 4 F) state.All the details of the methodology outlined by Neuhauser et al. [40] and Bussière et al. [41] are presented in the following references [5,38,39].
In the case of Li 3 Cr 2 (PO 4 ) 3 the analytical expression describing the profile of the absorption spectrum is written as: As given by Equation ( 8), γ 1 is the coupling constant of the forbidden state αwith the allowed one.ω is the frequency, β is the spectrum without coupling.The α and β equations are: The width of individual vibronic lines is determined by Γ, a phenomenological damping factor, where Δ is the maximum of the absorption band without coupling and ε F is the energies of the forbidden state.The width of the allowed band in Equation ( 9) is denoted by ̅̅̅̅̅̅̅̅̅ ̅ , where λ e = Δ -ε A and ω 0 is the metal-ligand band's stretching frequency.
The absorption spectrum is fitted by: Fig. 11.Absorption spectrum of Li 3 Cr 2 (PO 4 ) 3 phosphate and the results of the Neuhauser fit (Red colour), the Gaussian fit (blue colour).

Re[σ(ω)]
] 2 (11) Equation ( 10), which is specifically programmed in Origin software, was used in the study to examine a single interference dip that was seen on the 4 T 2g ( 4 F) band, as depicted in Fig. 11.The location of the dip is determined by this equation by calculating the energies associated with the 4 T 2g ( 4 F) and 2 E g ( 2 G) states.After the software's calculation of these energies, the coupling (γ) and damping (Γ) parameters are important for the fitting process.The optimal values for γ and Γ, with other fitting parameters, are found in Table 3 and lead to the best fit.
By applying the analytical Equation ( 10), the analysis successfully reproduces the observed dip by showing a strong correspondence between the computed results and the absorption spectrum.It is noteworthy to emphasize that the energy corresponding to the doublet state (ε F = 2 E g ( 2 G)) is located at the depth of the dip instead of at the maximum of the peak.The parameter Δ = E( 4 T 2g ( 4 F)) represents the energy at the maximum peak of the wider band, roughly at the halfway point.The depth of these dips depends on the parameter Γ. γ = 258 cm⁻ 1 is the coupling constant between 2 E( 2 G) and 4 T 2 ( 4 F).Furthermore, γ and the spin-orbit (S-O) coupling constant are related as γ = − ξ ̅̅ 6 5

√
[39], which results in ξ = 235 cm⁻ 1 .For the band 4 T 1g ( 4 F), the maximum of the peak is obtained by Gaussian fit and is represented in Table 3.The Neuhauser method primarily allows for a precise reading of the spin-forbidden transition 4 A 2g ( 4 F)→ 2 E g ( 2 G) due to the overlap between the broad band 4 T 2g ( 4 F) and the narrow peak 2 E g ( 2 G) wich lead to interference deep.It's important to note that for the broad bands 4 T 2g ( 4 F) and 4 T1g( 4 F), the Neuhauser method fits a Gaussian band, and the obtained energies corresponds to the maximum of these bands.Additionally, it is worth noting that when considering the zero-phonon line, the 4 T 2g ( 4 F) level appears before the 2 E g ( 2 G) level, which is the opposite of what is obtained with the Neuhauser method.For this reason we determine the zero-phonon line ZPL energies (E( 4 T 2g ( 4 F)) ZPL and E( 4 T 1g ( 4 F)) ZPL energies) of the Cr 3+ ions from the absorption spectrum (Fig. 11).The E( 2 E g ( 2 G)) ZPL energy is already determined from the Neuhauser method (Table 3).For determining the ZPL energies, we model the absorption spectrum (Fig. 11) by using the autocorrelation function.This function 〈φ|φ(t)〉 illustrates the dynamic evolution of the wave function from its initial state to its final state within a potential energy well.The Fourier transform of the autocorrelation function, which is the overlap of φ(t) with φ(t = 0), results in the absorption spectrum [4]: The evaluation of the autocorrelation function is simplified by assuming (i) the potential energy wells are harmonic with identical  force constants for the initial and final states, (ii) a constant transition dipole moment, and (iii) identical normal coordinates for both states.Under these assumptions, the autocorrelation function can be expressed analytically as follows [4]: E ZPL is the energy at the origin of the broad band (Zero Phonon Line), the frequency of the absorption spectrum is ω, and the breadth of each line in the spectrum is determined by the phenomenological damping factor Γ. For all amounts, wavenumber units are used.The terms k j (frequency of 376 cm − 1 is used) and ΔQ j represent the vibrational frequencies of each mode and the displacement along the normal coordination Q j of the potential surfaces, respectively.The experimental absorption spectrum is fitted by adjusting the parameters E( 4 T 2g ( 4 F)) ZPL and E( 4 T 1g ( 4 F)) ZPL , Γ, and ΔQ.Table 4 lists the values that were obtained.

Crystal field study of the Cr 3+ in Li 3 Cr 2 (PO 4 ) 3
Understanding the electronic structure and behavior of transition metal complexes, especially coordination compounds, is possible through the application of Crystal Field Theory (CFT) framework.These complexes have ligands, or atoms or molecules that coordinate with the metal, all around a central metal ion.According to CFT, there is only electrostatic interaction between the metal ion and ligands.Ligands, which are usually negatively charged, surround the metal ion with a crystal field that affects the distribution and energy levels of its d-electrons.The following general Hamiltonian (Equation ( 12)) is utilized to ascertain the energy states of Cr 3+ in Li 3 Cr 2 (PO 4 ) 3 compound [42][43][44]: • H 0 the Hamiltonian configuration • H ee the electron-electron repulsion term given as function of the Racah parameters (B and C) [45][46][47].In our case Cr 3+ (3d [3]),the Russell-Saunders terms, representing the 2S+1 L states, comprise 4 F, 4 P, 2 H, 2 G, 2 F, and 2 P terms.
• The H Trees Hamiltonian relies on the fine structure parameter α Tree and incorporates the Trees correction [48,49].
• H CF is the crystal field Hamiltonian.In fact, in the case of Li 3 Cr 2 (PO 4 ) 3 , The Cr 3+ (3 d 3 ) occupy an octahedral site symmetry and is given by the following equation [50,51]: The ligand field splitting in the crystal field Hamiltonian H CF is denoted by the parameter Dq, which is frequently used to describe the structure of transition metal ions.The matrix elements of the C k q operators are numerically calculated using the Racah tensor algebraic methods [52].The crystal field Hamiltonian H CF is developed as a function of the Racah tensor operators C (k)  q defined as follows: k are the spherical harmonics.
For transition ions of the first series with electronic configuration 3d N , the crystal-field is of the intermediate strength [5,8,9], the basis functions in the LS-coupling scheme have been adopted for the theoretical calculation.By using the Racah tensor algebraic methods, the matrix elements of C q (k) Racah tensor operators are calculated numerically [27]:

Table 4
Parameters used to calculate the absorption spectrum shown in Fig. 11.
parameters values According to Equation ( 14), the 3j-symbols ) carry all the dependence on the labels M L , q and M L ' while the reduced matrix elements 〈LS ⃦ ⃦ C k ⃦ ⃦ L ʹ S〉 is independent of these labels, but depends on the normalization of the tensor operators.
H SO isthe spin-orbit coupling Hamiltonian and is function of the ζ parameter which is the strength of the interaction between the orbital motion of an electron and its spin.The α Tree and ζ parameters are calculated from the equations [26]: where the parameter N describes the average reduction factor due to covalency [26]is defined as follows: The free ion Cr 3+ parameters B 0, C 0 , α 0 and ζ 0 are illustrated in Table 5 [8,9].
The theoretical computation of energy levels for Cr 3+ ions in Li 3 Cr 2 (PO 4 ) 3 involves diagonalizing the Hamiltonian Equation ( 13), which consists of a (120x120) matrix representing free ion eigenstates within quartet ( 4 F ground state and 4 P excited state) and doublet excited terms ( 2 H, 2 G, 2 F, and 2 P).A specialized code developed by our team, based on Maple software, is employed for diagonalization.The computed energies align closely with Yeung et al.'s findings [53], validating our computational approach.These energies, expressed as analytical equations, depend on parameters B, C, Dq, α, and ζ.Dq is determined from quartet excited state absorption energy, while B and C are derived from observed quartet and doublet excited states, respectively.Parameters α and ζ are calculated from Equation 15-a and 15-b.Inputting these parameters into our software package allows simulation of Cr 3+ ion electronic structure and energy levels in Li 3 Cr 2 (PO 4 ) 3 .The resulting parameters, detailed in Table 5, enable deduction of theoretical energy levels for Cr 3+ (3 d 3 ) in O h site symmetry (Table 6).Our computational approach, simplified by the developed software package, shows remarkable consistency between predicted Stark energy levels and actual experimental results.
An intriguing approach for comprehending the absorption spectrum is the coupled potential energy surface model.This approach allows for the examination of electronic transitions, particularly intra-configuration d-d excitations.A single normal coordinate, Q, is employed in the model of coupled potential energy surfaces.Notably, the doublet 2 E g ( 2 G) state and the ground 4 A 2g ( 4 F) state share identical vibrational frequencies and positions of the potential energy minimum at Q = 0 Å [43].However, a shift ΔQ causes the 4 T 2g ( 4 F) state transitioning to longer metal-ligand bond distances.This change occurs because of the way certain molecular orbitals, specifically metal-ligand antibonding ones, are populated through a process called d-d excitation [43].The potentials for the excited states 2 E g ( 2 G), 4 T 2g ( 4 F) and 4 T 1g ( 4 F) are as follows: V V in the absorption spectrum, the breadth of the 4 T 2g ( 4 F) and 4 T 1g ( 4 F) bands corresponds to the magnitude of the displacements ΔQ.The E ZPL ( 2 E g ( 2 G)), E ZPL ( 4 T 2g ( 4 F)) and E ZPL ( 4 T 1g ( 4 F)) are the energies of the potential minimum for the 2 E g ( 2 G), 4 T 2g ( 4 F) and 4 T 1g ( 4 F) states, respectively.The autocorrelation function's modelling of the absorption spectra yielded the values of the quantities ΔQ, E ZPL ( 2 E g ( 2 G)), E ZPL ( 4 T 2g ( 4 F)) and E ZPL ( 4 T 1g ( 4 F)) (Table 4).In equation (15), k is the frequency observed on the Raman spectrum of Li 3 Cr 2 (PO 4 ) 3 assigned to Cr-O symmetric stretching mode (a 1g ) denoted as ω 0 (376 cm -1) in section 4-1.The potentials state surfaces, obtained from Equation ( 15) and from the spectroscopic parameters shown in Table 4, are depicted in Fig. 13.This diagram describes the electronic transitions at the zero-phonon line (ZPL) between the ground state 4 A 2g ( 4 F) and the excited states 2 E g ( 2 G), 4 T 2g ( 4 F) and 4 T 1g ( 4 F).
Assuming that for Cr 3+ , the value of k is 0.21 [54], the bonding nature of Cr 3+ is evaluated using the following expression [55]: Based on Equation (17), the calculation of h gives a value of 0.975.This value indicates a d-electron delocalization, indicating predominant ionic bonding between Cr 3+ and its ligands.Utilizing the Racah parameters B and C from Table 3 (with C/B = 4.37), we construct the Tanabe-Sugano diagram for Cr 3+ in an octahedral site symmetry (Fig. 14).This diagram illustrates the energy levels of Cr 3+ ions concerning Dq/B as influenced by the local field strength.The specific case of Cr 3+ in Li 3 Cr 2 (PO 4 ) 3 phosphate is represented by a vertical line corresponding to the Dq/B value obtained from our theoretical calculations.
An additional method to elucidate the origin of interference dips involves examining the exchange of characteristics between the 2 E g ( 2 G) and 4 T 2g ( 4 F) states, which exhibit proximal behavior.These two states may combine and exchange some of their respective characters.As illustrated in Fig. 15 through the process of spin-orbit coupling, the excited states 2 E g ( 2 G) and 4 T 2g ( 4 F) can be divides into distinct E 1 , E 2 , and G states.Remarkably, these closely positioned states maintain an identical G character, facilitating a confluence of their attributes.
Consequently, the typically forbidden transition 4 A 2g ( 4 F)→ 2 E g ( 2 G) assumes some features of the permitted transition 4 A 2g ( 4 F)→ 4 T 2g ( 4 F), resulting a gain in intensity which is lower than that of the allowed transition.

The diabatic/adiabatic approach for the interpretation of the forbidden transition
As we have mentioned in previous, the existence of a dip in the 4 T 2g ( 4 F) band, which is a forbidden transition.This feature can be 0.772 B 0 (cm − 1 ) [8][9] 918 C 0 (cm − 1 ) [8][9] 4133 [8][9] 275 caused by the interaction between states that are close to one another but have different spin multiplicities.As shown by Fano in atomic spectroscopy, these bands are not just additive but also closely connected, leading to interference phenomena [37][38][39].To investigate the coupling between states of different spin multiplicities, we use again the model of coupled potential energy surfaces with a single normal coordinate Q.The potentials for the excited states 2 E g ( 2 G) and 4 T 2g ( 4 F) in the absence of spin-orbit coupling are given by equation 15a and b.In this diabatic approach, the two potential wells of the excited states are considered independently of each other.So, Rapid nuclear motion prevents transitions between these wells, effectively sealing off any exchange of characteristics.The uncoupled (diabatic) potentials, resulting in the absence of spin-orbit coupling are illustrated in Fig. 16a.For the sake of simplification, only the derivative states that have G irreducible representations are displayed in Fig. 16b.The curves obtained from Equation (15a) and b cannot describe the spin forbidden transition 4 A 2g ( 4 F) → 2 E g ( 2 G) detected in the absorption spectrum (Fig. 11).Doublet excited state G( 2 E g ) can interact with the G( 4 T 2g ) states.According to this analysis, the doublet states receive most of their intensities from the allowed quartet transitions and the spin forbidden transition 4 T 2g ( 4 F)→ 2 E g ( 2 G) can arise from the interaction of the Fig. 13.The diabatic potentials energy curves for 2 E g ( 2 G), 4 T 2g ( 4 F) and 4 T 1g ( 4 F).3d [3] electrons of Cr 3+ with the Li 3 Cr 2 (PO 4 ) 3 host matrix.The observed spin-forbidden transitions between the 4 A 2g ( 4 F) ground state and the 2 E g ( 2 G) excited state are caused by the interaction between the same symmetry levels (G) derived from electronic states of different multiplicities [4].Effectively, a constant with a magnitude proportional to the spin-orbit coupling λ( − 3λ ̅̅̅̅̅̅̅̅ 6/5

√
) [55] are considered to mix the characters of the 4 T 2g ( 4 F), and 2 E g ( 2 G) diabatic potentials.In this situation, the coupled potentials' matrix (adiabatic potentials) 2x2 is expressed in the following: The diagonals Terms in Equation ( 18) represent the equations of the diabatic potential energy wells and are therefore identical to those of Equation ( 16).The diabatic/adiabatic approach will considerably modify the appearance of the theoretical absorption spectrum and the appearance of observed forbidden transitions experimentally, such as the band corresponding to states 4 T 2g ( 4 F) and 2 E g of Fig. 11, will appear in the calculation.The adiabatic potentials state surfaces, obtained from Equation ( 18) and from the spectroscopic parameters shown in Table 4, are depicted in Fig. 16c.The spin-forbidden 4 A 2g ( 4 F) → 2 E g ( 2 G) transitions are caused by the adiabatic excited state potential energy curves V 1 and V 2 , which are produced when the spin-orbit coupling between 4 T 2g ( 4 F) and 2 E g ( 2 G) is taken into account.

Conclusion
This study focused on the synthesis and analysis of Li 3 Cr 2 (PO 4 ) 3 phosphate.Scanning electron microscopy (SEM) imaging exhibited particle dimensions spanning from 0.2 to 1.2 μm, with an average size of 0.574 μm, indicating the emergence of micrometer-scale particles.UV/Vis absorption spectra, assessed through Tauc's law and Urbach energy calculations, revealed direct transition properties, with increased disorder and defect density attributed to the presence of Cr 3+ ions.Significantly, absorption spectra at room temperature displayed an interference dip within the extensive 4 T 2g ( 4 F) band, ascribed to the overlapping of spin-forbidden transitions 4 A 2g ( 4 F) → 2 E g ( 2 G).This phenomenon was clarified using the Neuhauser model, which relies on coupled potential energy surfaces.This model gives the ZPL energy of the 2 E g ( 2 G) level.The Fourier transform of the autocorrelation function leads to the Zero Phonon Lines of the observed absorption energies 4 T 2g ( 4 F) and 4 T 1g ( 4 F).From these ZPL energies, we determine a deeper understanding of the electronic configuration of Cr 3+ (3 d 3 ) ions within Li 3 Cr 2 (PO 4 ) 3 phosphate by using the tensoriel Racah theory.It aided in establishing Racah and crystal-field parameters, ensuring a good agreement between theoretical predictions and observed energy levels.The diabatic/adiabatic approach was an important model to demonstrate the origin of a dip in the 4 T 2g ( 4 F).The obtained results hold significance for numerous optical applications.

Declaration of competing interest
This work includes an experimental study Lithium-chromium phosphate Li 3 Cr 2 (PO 3 ) 4 using the solid-state reaction method.The morphological integrity and chemical homogeneity were verified by energy dispersive X-ray analysis (EDX) and scanning electron microscopy (SEM).Infrared and Raman patterns were also analyzed.The optical properties have been study using the absorption characterization.An interference dip was found on the 4 T 2g ( 4 F) band in the absorption spectrum of Li 3 Cr 2 (PO 3 ) 4 , which was explained by spin-orbit coupling between the 2 E g ( 2 G) and 4 T 2g ( 4 F) states.The electronic structure of Cr 3+ (3d [3]) ions in Li 3 Cr 2 (PO 3 ) 4 was calculated using the Neuhauser model.The theoretical part is also carried out using software that we have developed ourselves in our laboratory and it is rarely that we find this type of calculation because this software is based on Racah method (the most powerful method that describe the electronic structure of transition ions) and it is not easy and requires specialists.The programming lasted about 6 years and each time we add more precision.This software uses Maple 20 as software is as well tested and gives the same result as another software using the Fortran software and we can prove that.We have published several articles and in well-indexed newspapers.
Fig.1bshows a particle size that ranges from 0.2 to 1.2 μm, with an average size of 0.574 μm signifying the formation of micrometric

Fig. 4 .
Fig. 4. The spectrum of reflectance (dark solid line) and the evolution of dR/dλ(red circle) versus the wavelengthλ.

Fig. 14 .
Fig. 14.Tanabe-Sugano diagram for Cr 3+ in an octahedral site symmetry.The vertical red line corresponds to the Dq/B = 1.75 obtained from our theoretical calculations.

Fig. 15 .
Fig. 15.Splitting of Stark levels 4 T 2g ( 4 F) and 2 E g ( 2 G) under spin-orbit coupling of Cr 3+ in Li 3 Cr 2 (PO 4 ) 3 .Spin-allowed transition is indicated by solid arrows.Under spin-orbit coupling, pairs of interacting levels are connected by dot-point arrows.

Fig. 16 .
Fig. 16.a) The uncoupled (diabatic) potentials for 4 T 2g ( 4 F) and 2 E 2g ( 2 G) subdivided into E 1 , E 2 , and G states b) we represent only the derivative states that have G irreducible representations.c) the coupled (adiabatic) excited state potential energy curves V 1 and V 2 , (spin-orbit coupling between 4 T 2g ( 4 F) and 2 E g ( 2 G) is taken into account).

Table 2
Selection rules for Cr 3+ d-d electronic transitions in O h site symmetry for in Li 3 Cr 2 (PO 4 ) 3 .

Table 3
Fitting parameters in cm − 1 for the calculation of the absorption spectra of Cr 3+ in Li 3 Cr 2 (PO 4 ) 3 .