Characterization of Pr-Doped LaF3 Nanoparticles Synthesized by Different Variations of Coprecipitation Method

A set of Pr:LaF3 nanoparticles (NPs) were synthesized via coprecipitation method at three stoichiometric proportions of La(NO3)3, Pr(NO3)3, and NaF (1 : 0.8, 1 : 1, and 1 : 6, respectively). Two ways of mixing of the La(NO3)3, Pr(NO3)3, and NaF solutions (dropwise and swift addition) were used. One sample was subjected to microwave (MW) treatment for 30, 90, and 180min. All the samples were characterized by transmission electron microscopy (TEM) and X-ray diffraction (XRD). For all the samples, optical spectroscopy experiments were carried out. The XRD data were analyzed via the Debye-Scherrer and Williamson-Hall methods. It was revealed that the way of mixing of the La(NO3)3, Pr(NO3)3, and NaF solutions strongly affects the shape of the NPs. The slow dropwise addition of the NaF solution leads to the plate-like NP (PLNP) formation; otherwise, the swift addition of the NaF solution leads to the formation of more sphere-like NPs (SLNPs). The size and regularity in shape of the NP increase with the increasing stoichiometric proportion of La(NO3)3, Pr(NO3)3, and NaF from 1 : 0.8 to 1 : 6. The size and regularity in shape of the SLNPs increase with the increasing time of MW treatment. The Debye-Scherrer and Williamson-Hall methods confirmed the anisotropic shape of the PLNPs. The Williamson-Hall method showed that the values of strain are almost similar for all the samples (around 14∗10). Optical spectroscopy experiments revealed that although all the samples have an equal chemical composition, the luminescence lifetimes for different samples differ between each other. The luminescence lifetime of the PLNPs is less than that of the SLNPs having an equal stoichiometric proportion of La(NO3)3, Pr(NO3)3, and NaF. The luminescence lifetime of the 1 : 1 SLNPs increases with the increasing time of MW treatment.

During the last two decades, immense progress has been done toward facile and economy methods of synthesis of LnF 3 (Ln = La, Ce, Pr, and others) nanoparticles (NPs) doped by rare-earth ions [12]. Among these methods of synthesis, a coprecipitation method is considered one of the cheapest and easiest methods of synthesis of NPs [13]. On the one hand, it provides a synthesis of LnF 3 NPs with desired size, shape, and structure [14]. Usually, this method does not require toxic organic precursors as well as sophisticated and expensive laboratory equipment. On the other hand, this method has some disadvantages such as the presence of captured [15] and absorbed [8] water in NPs. These water molecules may significantly contribute into the luminescence-quenching processes [16]. Also, for some cases, NPs synthesized via such method can be characterized by relatively low crystallinity [17], presence of undesirable crystal phase, and broad size distribution and irregularity of NPs shape [18]. For this method, additional modifications of some parameters of synthesis such as the stoichiometric ratio of fluorinating agents and rare-earth salts or/and using microwave-assisted treatment can significantly improve the quality of nanomaterials.
Microwave-assisted treatment of fluoride NPs was developed in [15,17,[19][20][21]. This method has been widely applied in chemical reactions and material synthesis due to its unique reaction effects such as rapid volumetric heating and consequent dramatic increase in reaction rates [22]. In this case, the growth mechanism is likely a dissolutionrecrystallization process according to the conventional hydrothermal method for preparing rare-earth fluoride NPs and hydroxide nanorods/nanotubes [23]. However, in [17], it was shown that the recrystallization process for PrF 3 and DyF 3 NPs during the microwave-assisted treatment is different and depends on such factors as difference in symmetry and difference of lattice energies for lanthanide ions Pr 3+ and Dy 3+ . Also, in [17], it was reported that crystallinity of DyF 3 NPs obtained via coprecipitation method was significantly improved after microwave-assisted treatment for 7 hours without considerable changing of an average size of the DyF 3 NPs. In [8], it is reported that fullerene-like PrF 3 NPs were obtained after microwaveassisted treatment of a colloidal solution of irregularshaped PrF 3 NPs synthesized via coprecipitation method crystallinity [17], and remove captured water from the NPs core [15]. Hence, the luminescence lifetime and luminescence quantum yield will be improved without the significant complication of the synthesis procedure.
The excess of fluorinating agents is commonly used in the synthesis of fluoride NPs in order to provide a single-phased composition of NPs and regularity of size and shape. For example, the nonstoichiometric proportions of rare-earth nitrates and fluorinating agent are often used for the synthesis of hexagonal structured Yb 3+ /Er 3+ :NaYF 4 NPs. Also, it is reported in [18] that the regularity if the size and shape of Eu 3+ :NaYF 4 was achieved by using the 5-fold stoichiometric proportion of NaF.
Usually, NPs synthesized via the methods mentioned above are crystalline particles. In order to determine the phase of the NPs and assess their crystallinity, the X-ray diffraction method is used. Moreover, the additional information can be extracted from X-ray data. For example, unlike the bulk crystals, the nanosized crystalline particles demonstrate broadened diffraction peaks. This peak broadening stems from crystallite size and different crystal imperfections such as lattice strains. Hence, the two main properties which can be extracted from the analysis of peak width are the crystallite size and lattice strain [24]. There are methods such as Debye-Scherrer and Williamson-Hall which enable to estimate for example lattice strains for additional characterization of different nanomaterials.
The Pr 3+ -doped LaF 3 (C Pr = 7%) NPs were chosen as an object of research because of their unique properties such as thermally coupled 3 P 1 to 3 P 0 electron states of Pr 3+ ions [25][26][27]. This property can be used in luminescent nanothermometry [12,28]. The emission spectrum of Pr 3+ in lanthanum fluoride host matrix overlaps with photosensitizers such as acridine (C 13 H 9 N) and cyanine which are highly relevant in hybrid radiotherapy-photodynamic therapy mentioned above [3]. Hence, this system Pr 3+ -doped LaF 3 nanoparticles can be used for different application including biology and medicine [29].
In this study, we focus on studying the physical properties of the Pr 3+ -doped LaF 3 of different size and shape synthesized via coprecipitation method by using different ways of mixing of rare-earth salts and fluorinating agents, varying stoichiometric proportion of rare-earth salts and fluorinating agents, and performing microwave-assisted treatment for chosen samples.
In order to assess the contribution of size and shape of the NPs and also their crystallinity into the optical properties of the NPs, we characterize the obtained NPs via transmission electron microscopy (TEM), X-ray analysis, and optical spectroscopy. We also analyze X-ray data via the Debye-Scherrer and Williamson-Hall methods in order to calculate crystallite size and strains of the NPs. For a chosen sample, we studied the influence of microwave irradiation on the physical properties of the NPs. Additionally, the information concerning microstrains into the NPs extracted from the X-ray data can be very useful for understanding some physical properties of NPs.

Materials and Methods
2.1. Classification of the Samples. All the Pr 3+ :LaF 3 (C Pr = 7% (atomic)) NPs were synthesized via coprecipitation method using a chemical reaction described in [8,15]. The obtained Pr 3+ :LaF 3 (C Pr = 7%) samples can be divided into two groups: sphere-like NPs (SLNPs) and plate-like NPs (PLNPs). Synthesis of both SLNPs and PLNPs was carried out at 3 different stoichiometric proportions of La(NO 3 ) 3 , Pr(NO 3 ) 3 , and NaF (1 : 0.8, 1 : 1, and 1 : 6, respectively). For the sake of simplicity, we will use abbreviations 1 : 0.8, 1 : 1, and 1 : 6 and SLNPs or PLNPs in order to name the samples. For example, 1 : 6 SLNPs means sphere-like Pr 3+ :LaF 3 (C Pr = 7%) NPs synthesized at 1 : 6 stoichiometric proportion of La(NO 3 ) 3 , Pr(NO 3 ) 3 , and NaF. Regardless of the stoichiometric proportion of La(NO 3 ) 3 , Pr(NO 3 ) 3 , and NaF, the same volumes of NaF and La(NO 3 ) 3 /Pr(NO 3 ) 3 solutions, time of reactions, temperature, and pH values were used intentionally for all the samples. Hence, the main strategy of synthesis of both SLNPs and PLNPs is exactly the same except for one parameter: for SLNPs, the NaF solution was poured very swiftly to the La(NO 3 ) 3 and Pr(NO 3 ) 3 solution, and for PLNPs, the NaF solution was added dropwise.

Transmission Electron Microscopy and X-Ray Diffraction
Experiments. The structure of the material was characterized by X-ray diffraction method (XRD) with Shimadzu XRD-7000S X-ray diffractometer. Analysis of samples was carried out in a transmission electron microscope Hitachi HT7700 Exalens. Sample preparation: 10 microliters of the suspension were placed on a formvar/carbon lacey 3 mm copper grid; drying was performed at room temperature. After drying, the grid was placed in a transmission electron microscope using a special holder for microanalysis. The analysis was held at an accelerating voltage of 100 kV in TEM mode.
2.5. Optical Spectroscopy. The luminescence spectra were recorded using CCD spectrometer (StellarNet), which detects the emission in 200 -1100 nm spectral range with a spectral resolution of 0.5 nm. The optical parametric oscillator laser system (420-1200 nm) from JV LOTIS TII was used for excitation of the luminescence of the samples. The pulse width and the pulse repetition rate were 10 ns and 10 Hz, respectively. The spectral width of laser radiation was less than 0.15 nm. The luminescent lifetimes of Pr 3+ ions were detected using BORDO 211А (10 bit, 200 MHz bandwidth) digital oscillography and MDR-3 monochromator. The experiments were carried out at room temperature.
2.6. Elemental Analysis. Elemental analysis was carried out by using field-emission high-resolution scanning electron microscope Merlin Carl Zeiss with AZtec X-Max EDS system at accelerating voltage of incident electron of 20 kV and working distance of 10 mm. Excitation area is 1 μm. Technique preparation: sample on chuck move in the chamber of vacuum apparatus Quorum Q 150T ES. Conductive layer apply by technique cathode sputtering using alloy Au/Pd by quantity proportion 80/20. The thickness of the alloy is 15 nm.

Results and Discussion
3.1. Transmission Electron Microscopy of the Pr 3+ :LaF 3 (C Pr = 7%) Nanoparticles. According to the TEM data ( Figures 1-6), all the samples differ between each other by size and shape. The size distribution histograms (insets of Figures 1-6) are fitted by Gaussian function from which diameter and a width of the size distribution are extracted. In order to build size distribution for the PLNPs, the length of each plate was measured. The values of diameter and width of size distribution are listed in Table 1.
As can be seen from Figure 1, the 1 : 0.8 SLNPs have a relatively irregular shape. For the 1 : 1 SLNPs (Figure 2(a)), the shape is more regular. The 1 : 6 SLNPs ( Figure 3) demonstrate the most regular shape among all the SLNPs. The average diameter of the SLNPs increases from 10 4 ± 0 2 to 16 5 ± 1 2 nm for 1 : 0.8 SLNPs and 1 : 6 SLNPs, respectively, with the increasing stoichiometric proportion of La(NO 3 ) 3 , Pr(NO 3 ) 3 , and NaF. For the SLNPs, the values of the width of the size distribution do not differ between each other significantly.    [15,17,19,20].
The shape of the PLNPs demonstrates a remarkable difference from the SLNPs. According to           b)). Most of the 1 : 6 PLNPs are more similar to the SLNPs. It can be concluded that for the 1 : 0.8 PLNPs and the 1 : 1 PLNPs, the surface-to-volume ratio is higher than that of the rest of the NPs which can affect their optical properties [4] discussed in Optical Spectroscopy and Luminescence Lifetimes of the Pr 3+ :LaF 3 (C Pr = 7%) Nanoparticles.
For all the samples mentioned above, the calculations of the crystallite sizes and lattice strains via Debye-Scherrer and Williamson-Hall methods were carried out. These calculations will be discussed in the next part of the article. However, one of the main conclusions made from microscopy data is that SLNPs are more isotropic in shape and, as a consequence, more appropriate for Debye-Scherrer and Williamson-Hall methods. For this reason, 1 : 1 SLNPs were chosen for further microwave-(MW-) assisted treatment in order to study the contribution of MW irradiation into physical properties of the NPs. Although the regularity of the shape of the 1 : 6 SLNPs is slightly higher, we did not use the 1 : 6 SLNPs intentionally because in the majority of the articles, the 1 : 1 stoichiometric proportions of rare-earth nitrates and fluorinating agent are used; hence, this information is more valuable.
It can be seen from Table 1 that the diameter of the 1 : 1 SLNPs increases with the increasing time of MW treatment. The value of size distribution becomes narrower from 10 5 ± 3 7 to 5 3 ± 0 7 nm for 0 and 180 min of MW treatment, respectively. The regularity of shape also improves with the increasing time of MW treatment.
The difference in size and shape of the PLNPs and the SLNPs can be explained by different synthesis conditions. In case of swift addition of NaF solution, the formation of the SLNPs is caused by homogeneous nucleation occurring when the concentration of rare earth and Fions becomes significantly higher than the equilibrium concentration immediately and the spontaneous growth of NPs takes place [30,31]. In the case of dropwise addition, the fast increase of Fions concentration does occur. In such conditions, the final shape of a nanocrystal can be determined by the growth competition of different crystal planes [32].  3 , and NaF which leads to a more swift increase of F-ions concentration, and the nature of the reaction looks similar to the reaction for the SLNPs. According to the XRD data, all the NPs are hexagonal-structured nanocrystals that correspond to the structure of matrixes of LaF 3 and PrF 3 . Sharp peaks and lack of peaks from impurities are observed, suggesting the high purity and good crystallinity of these samples. Also, the lack of amorphous phase was detected. For all the samples, the lattice parameters a and c are calculated using the formulas from [33,34]. For all the samples, a and c are 0.7164 and 0.7330 nm, respectively. The lattice parameters for LaF 3 (JCPDS-32-0483) are a = 0 7186 and c = 0 7352 nm. The reduction of Pr 3+ :LaF 3 lattice parameters apparently related to crystal lattice distortion. The radius of Pr 3+ (0.105 nm) is smaller than that of La 3+ (0.113 nm) due to the lanthanide contraction, so the cell volume of Pr 3+ :LaF 3 reduces with more Pr 3+ replacing La 3+ [14].

Elemental Analysis.
NaF is considered a very specific fluorinating agent. On the one hand, according to the literature data in some cases, the use of NaF leads to Sr 1-x Na x F 2-x , Ca 1-x Na x F 2-x [35,36], NaF-RF 3 , and/or NaRF 4 (where R = rare earth) [37] formation in water-based coprecipitation method. On the other hand, in spite of the possibility of NaF-RF 3 and/or NaRF 4 formation, the synthesis of NaF-LaF 3 and/or NaLaF 4 is considered as a very specific task. This is because La 3+ has the largest cationic radius among the lanthanide ions, and the ionic bond of La 3+ -Fis stronger than that of Na + -Fand other RE 3+ -F -, as reflected by the melting points of several representatives: LaF 3 1493°C > NdF 3 1410°C > SmF 3 1306°C > TbF 3 1231°C > NaF (993°C) [38]. As a result, in coprecipitation route, cations are difficult to settle into the lattice in compounds with large RE cations such as NaLaF 4 , which makes the synthesis of these compounds more difficult than that of NaYF 4 or NaLuF 4 [39].  Journal of Nanomaterials For these reasons, NaLaF 4 compound is considered thermodynamically nonpreferred [39]. This fact results in the shortage of efficient synthetic methods for the preparation of pure β-NaLaF 4 nanocrystals [40]. Indeed, we failed to find a work devoted to low-temperature coprecipitation method of synthesis of pure NaLaF 4 NPs. On the other hand, in papers devoted to the synthesis and investigation of undoped and doped NaLaF 4 NPs, the high-temperature hydrothermal routes or melting in a corundum crucible are utilized [41,42]. Moreover, it is noteworthy that the β-NaLaF 4 has a very distinguishable XRD pattern (diffraction peaks at 16.6, 28.  [43]. This pattern differs from the LaF 3 one. In the case of complex NaF-LaF 3 system, the notable amount of NaLaF 4 would be detected via double-phase XRD pattern as it is observed in [43].
In our work, as it is mentioned above, we do not observe any impurity peaks. It can be suggested that there is no second β-NaLaF 4 phase or this phase is negligible and cannot be detected via out X-ray diffractometer.
On the other hand, a small amount of Na can form NaF-LaF 3 system without forming the second phase. Just in order to check the presence of Na in the samples, we have performed an elemental analysis. The elemental analysis revealed that all the samples do not contain sodium as well as other elements. The presence of Pr, La, and F was proved. The elemental analysis spectra of 1 : 0.8 PLNPs, 1 : 1 PLNPs, 1 : 6 PLNPs, 1 : 0.8 SLNPs, 1 : 1 SLNPs, 1 : 6 SLNPs, and 1 : 1 SLNPs 180 MW are shown in Figures 8(a)-8(g), respectively.     Journal of Nanomaterials The unidentified peaks are the peaks of the conductive layer which was used for the sample preparation. The elemental analysis data are listed in Table 2.

Debye-Scherrer Calculations.
The instrumental corrected broadening β hkl corresponding to the diffraction peak of Pr 3+ :LaF 3 was estimated using equation (1) [33,34] as follows: In order to estimate the average size of the NPs, the Debye-Scherrer method is used [44]: where D is a diameter of a NP, K is a shape factor (we used K = 0 9), λ is the X-ray wavelength (0.15418 nm), β D is the line broadening at half the maximum intensity (FWHM) in radians, and θ is the Bragg angle (in degrees). The diffraction peaks having the lowest values of the signal-noise ratio are chosen. The values of D in different crystallography orientations (hkl) are listed in Tables 3 and 4. Also, Figures 9(a) and 9(b) summarize these data. We do not compare TEM data and the Debye-Scherrer calculations of the D intentionally for some reasons. More specifically, equation (2) assumes that the peak broadening is related to the nanoscale dimensionality of the crystalline particles only. It does not take into control the presence of strains and distortions in NPs; hence, the presence of these strains and distortions can seriously confound the D value. More than that the size calculated via Debye-Scherrer formula is the size of coherently diffracting domains which is not generally the same as the particle size [24,34]. Additionally, according to the microscopy data, all the samples have relatively broad size distribution (several nm) of NPs which require calculation of the additional constant for the Debye-Scherrer equation [45]. Also, according to the microscopy data, the shape of the NPs is far from perfect spherical or cubic; hence, the shape factor K is very sophisticated and actually should be calculated for each crystallographic orientation (hkl) [45]. The shape of both the PLNPs and the SLNPs is not perfect, and for all the samples, the shape factor K cannot be calculated precisely. The value of 0.9 is taken just in order to estimate the size of the particles. Summarizing all the above-mentioned information, it is very difficult to compare TEM data and Debye-Scherrer calculations. Hence, it is difficult to estimate the contribution of size into the peak broadening for all the samples in this study. For more precise results, the Williamson-Hall method described in the next part of the article is used.
On the one hand, the shape factor K depends on the crystallographic orientation (hkl) and the symmetry [45]. The symmetry is the same for all the samples; hence, it can be assumed that the K is a function of the crystallographic orientation (hkl) only. For irregular-shaped NPs, the K is very sophisticated. For perfectly spherical NPs, the K = 0 9 and it is the same for all the crystallographic ori- This irregularity in shape can also be confirmed by assessing the linearity of the Debye-Scherrer formula. For this purpose, the Debye-Scherrer formula is rearranged.
The plots were drawn with 1/β D on the x-axis and Cosθ along the y-axis. The K is assumed 0.9. The anisotropy in shape of the NPs leads to the phenomenon that the XRD peaks are broadened differently. As a consequence, the linearity of this plot can additionally confirm or disprove the isotropy of the shape of the NPs within the Debye-Scherrer model. If the shape of the NPs is far from spherical, the linearity of equation (3) should be low. The almost spherical NPs should demonstrate good linearity of equation (3). In turn, the linearity can be estimated using the Pearson coefficient. The values of the Pearson coefficient are listed in Tables 5 and 6. Taking into control that good linearity of data requires the values of Pearson coefficient more than 0.9, the comparison of the samples is performed. It is seen that the Debye-Scherrer data do not demonstrate perfect linearity especially for the 1 : 0.8 PLNPs and the 1 : 1 PLNPs   (Table 3). These facts indicate the anisotropy in shape of the 1 : 0.8 PLNPs and the 1 : 1 PLNPs which is also in agreement with microscopy data. However, the 1 : 1 SLNPs and the 1 : 6 SLNPs demonstrate relatively good linearity having the Pearson coefficient around 0.87.
In contrast to the above-mentioned samples, the 1 : 1 SLNPs treated by MW demonstrate the Pearson coefficient around 0.9 and more which confirms the conclusions based on microscopy data. Summarizing all the above-mentioned information, the Debye-Scherrer model confirmed the anisotropy in shape of some samples. However, comparison of Debye-Scherrer diameters and the TEM sizes seems to be difficult especially for the PNLPs. Hence, the contribution of the nanoscale dimensionality into the peak broadening is not clear.
In order to take into account both the nanoscale dimensionality of the NPs and lattice strain contribution into the peak broadening, the Williamson-Hall method is used.

The Williamson-Hall Calculations of Size and Strain.
If the peak broadening is related to the presence of strains induced in powders due to crystal imperfection and distortion only, these strains are calculated using the equation as follows [34,46]: where d is a distance between crystallographic planes, and the values of d are into an interval from d-Δd to d + Δd.
Assuming that the particle size and strain contributions to line broadening are independent to each other, the observed line breadth is simply the sum of β D + β S from equations (2) and (4).
By rearranging the above equation, we get Equations (5) and (6) are the Williamson-Hall equations. A plot (equation (6)) is drawn with 4sinθ along the x-axis and β hkl cos θ along the y-axis. From the linear fit to the data, the crystalline size was estimated from the y-intercept, and the strain ε, from the slope of the fit. The Williamson-Hall plots for the SLNPs and the PLNPs are shown in Figure 10(a). The Williamson-Hall method assumes that the particles are isotropic in shape and the size of strains in different crystallographic directions are similar  10 Journal of Nanomaterials [47]. The TEM data, Debye-Scherrer, and Williamson-Hall calculations are listed in Table 5.
According to the microscopy data and the Debye-Scherrer calculations, the 1 : 0.8 PLNPs and the 1 : 1 PLNPs are highly anisotropic in shape. The SLNPs also are not perfectly isotropic. Hence, the accuracy of the Williamson-Hall calculations is not supposed to be high for at least 1 : 0.8 PLNPs and 1 : 1 PLNPs. However, the SLNPs treated by MW irradiation demonstrate relatively good shape isotropy; hence, the Williamson-Hall method is more applicable for them. Here, we analyze size and strain as well as estimate applicability of the Williamson-Hall method by assessing the linearity of the Williamson-Hall plots.
As it is mentioned above, we do not quantitatively compare the TEM data and the Debye-Scherrer calculations. However, both Debye-Scherrer and Williamson-Hall calculations do not reflect such an important tendency as increasing of the sizes of the SLNPs with increasing stoichiometric proportion of rare-earth salts and NaF qualitatively. Actually, all the SLNPs are around 14.2 nm in diameter. This fact brings into a question the applicability of the Williamson-Hall method toward the samples. Indeed, it was already mentioned that in the Williamson-Hall method, it is assumed that the particles are isotropic in shape and the strain is uniform in different directions leading to independent crystal properties [33]. Analyzing the microscopy data and the Pearson coefficients for the Williamson-Hall plots, it can be concluded that irregularity in the shape of the NPs correlates with the Pearson coefficients for the Williamson-Hall plots (Table 7). Moreover,

11
Journal of Nanomaterials the accuracy and quality of the Williamson-Hall calculation seem to be dependent on the linearity of the plots.
It is seen that the most irregular-shaped 1 : 0.8 PLNPs and 1 : 1 PLNPs have very poor linearity (the Pearson coefficients are 0.57 and 0.66, respectively). The Pearson coefficient increases with improving the regularity of the shape of the NPs. Summarizing the above-mentioned information, the Williamson-Hall calculations of the diameter do not reflect tendencies of increasing the sizes of the SLNPs, and the Pearson coefficients are less than 0.9. It can be concluded that all the samples synthesized via coprecipitation method are not isotropic enough. On the other hand, in the specified accuracy, the values of strain ε seem to be equal for all the samples. Hence, the different conditions of the coprecipitation method do not lead to significant changing of the values of strain.
However, for the 1 : 1 SLNPs treated by MW for 30, 90, and 180 min (named 1 : 1 SLNPs 30 min, 1 : 1 SLNPs 90 min, and 1 : 1 SLNPs 180 min, respectively), both the Debye-Scherrer and Williamson-Hall methods reflect the tendency of increasing the sizes of the SLNPs with the increasing of MW treatment time ( Table 6). The Williamson-Hall plots of the samples are shown in Figure 10(b).
Moreover, the Pearson coefficients for both Debye-Scherrer and Williamson-Hall plots of NPs treated with MW are equal or more than 0.9 (Table 8). Here, we try to assess how anisotropy in shape assessed via TEM influences the applicability of the Debye-Scherrer and Williamson-Hall theories by comparing the Person coefficients of linear fitting (Table 8).
It means that the Williamson-Hall model is more appropriate for the analysis of the samples treated by MW unlike the rest of the samples. For MW-treated NPs, the values of ε seem to be similar for all the samples within the accuracy. These values do not depend on the time of MW treatment. The values of ε are around 14 * 10 -4 . These values are of the same magnitude to the results for ZnO NPs obtained in [24].
In the Williamson-Hall method, the value of ε seems to be similar for all the samples within the accuracy and does not depend on the time of MW treatment. It is reported in [15,17,48] that MW treatment improves the crystallinity of PrF 3 , DyF 3 , and LaF 3 NPs, respectively. As it is mentioned above, the growth mechanism of NPs during MW treatment is dissolution-recrystallization. Probably during 30, 90, and 180 min of MW treatment, the complete recrystallization does not occur. The average size of the NPs increases but  Finally, it is noteworthy to say that the Debye-Scherrer diameters are less than the Williamson-Hall ones which is in accordance with [47]. In this case, the Debye-Scherrer formula provides only a lower bound for the crystallite size. The Williamson-Hall diameter for all the particles is bigger than TEM diameter which can be explained by the wide variety of mechanisms leading to the peak broadening. These mechanisms are not taken into consideration within these models.

Optical Spectroscopy and Luminescence
Lifetimes of the Pr 3+ :LaF 3 (C Pr = 7%) Nanoparticles. Although the initial chemical composition of all the Pr 3+ :LaF 3 (C Pr = 7%) samples is equal, some optical properties of the NPs differ from each other. On the one hand, the luminescence spectra of all the Pr 3+ :LaF 3 NPs do not differ between each other. Figure 11(a) shows the spectra of the most distinguishable NPs. The transitions were determined according to [49]. The luminescent spectra have the emission bands at about 487, 523, 537, 580, 601, and 672 nm which are interpreted as a result of the transition from the 3 P j (j = 0, 1, 2) excited states to 3 H 4 , 3 H 5 , 3 H 6 , and 3 F 3 states of Pr 3+ ions, respectively. The emission from the 1 D 2 state was not found under the excitation condition and at the studied temperature range. Probably the emission from 1 D 2 is not observed because of the lack of nonradiative relaxation of 3 P j to 1 D 2 due to low cutoff phonon frequency in LaF 3 (350-400 cm −1 ).
On the other hand, the lifetime curves of 3 P 0 state of Pr 3+ ions for different samples differ from each other notably ( Figures 11(b)-11(d)). In addition, luminescence lifetime curves are not one or double exponential. The fitting curves are more sophisticated. Since the theoretical description of lifetime curves seems to be difficult and lays behind the scope of this article, we qualitatively compare the lifetime curves between each other. It is seen from Figures 11(b) and 11(c) that the SLNPs demonstrate longer lifetime comparing with the suitable PLNPs. Moreover, for both the SLNPs and the PLNPs, the lifetime increases with increasing of the NPs size. The SLNPs treated by MW demonstrate the same tendency. The increasing of MW treatment time leads to increasing in lifetime (Figure 11(d)).
It can be suggested that the optical properties of the samples are mainly affected by the size and shape of the NPs and, as consequent, the volume-to-surface ratio [50,51]. In this case, probably the main mechanism of luminescence quenching is related to energy multiphonon transfer from the exited ion to the high-vibrionic energy molecule such as OH group adsorbed on the NPs surface [16,52]. The thinnest 1 : 0.8 PLNPs having the biggest surface-to-volume ratio demonstrate the lowest luminescence lifetime which can be attributed to the proximity of the highest amount of Pr 3+ ions to the surface OH groups which can be regarded as the main quenching centers in this system [53]. In [54], it is shown that the shape of rare-earth-doped dielectric NPs can affect the luminescence lifetime. Moreover, it is shown in [55] that the use of water-based coprecipitation method leads to the presence of OH groups into the NP's core as well. Hence, in the case of the MW-treated samples, the lifetime increasing can be attributed by two processes. The first is the increasing in size of the NPs which leads to a reduction of the role of the surface. The second is the migration of the OH groups from the NP's core to specific water clusters [30,56,57] which leads to reducing the total amount of Pr 3+ ions contacting with OH groups.
The theoretical description of the decay curves is behind the scope of the article. In order to estimate and compare the specific lifetimes, we calculated the effective decay time, τ eff , via equation (7) which is commonly used for such complicated nonexponential decay curves [58]: where I t , intensity; t, time. Values of the effective lifetime are listed in Table 9.

Conclusions
The 1 : 0.   3 , and NaF solutions strongly affects the shape of the NPs. The slow dropwise addition of the NaF solution to the La(NO 3 ) 3 and Pr(NO 3 ) 3 solution leads to the PLNPs formation; otherwise, the swift addition of the NaF solution leads to the formation of more spherical NPs (SLNPs).
The stoichiometric proportion of La(NO 3 ) 3 , Pr(NO 3 ) 3 , and NaF also strongly affects the size and the shape of the NPs. In the case of SLNPs, the size and regularity in shape of the SLNP increase with the increasing stoichiometric proportion of La(NO 3 ) 3 , Pr(NO 3 ) 3 , and NaF from 1 : 0.8 to 1 : 6. The increasing stoichiometric proportion of La(NO 3 ) 3 , Pr(NO 3 ) 3 , and NaF also affect the PLNPs leading to an increase in thickness of the PLNPs. In the case of the PLNPs, the growth along the [100] and [010] planes occur more effectively than along the [001] plane which leads to PLNPs formation.
The size and regularity in shape of the SLNPs increase with the increasing time of MW treatment.
The Debye-Scherrer calculations have shown that the size of the most irregular-shaped 1 : 0.8 PLNPs and 1 : 1 PLNPs strongly depends on the crystallographic plane which additionally confirmed the shape irregularity of these NPs. The values of the diameter were calculated for several [hkl]  Optical spectroscopy experiments revealed that although all the samples have an equal chemical composition, the luminescence lifetimes for different samples differ between each other. The luminescence lifetime of the PLNPs is less than that of the SLNPs having an equal stoichiometric proportion of La(NO 3 ) 3 , Pr(NO 3 ) 3 , and NaF. The luminescence lifetime of the 1 : 1 SLNPs increases with the increasing time of MW treatment. Commonly, the lifetime increase with the increasing size of the NPs. Hence, the role of surface-quenching agents reduces. However, the luminescence lifetime curves cannot be fitted exponentially, and the additional theoretical interpretation is required. This last task is behind the scope of this work.

Data Availability
The TEM microscopy, XDR, spectra, and lifetime data used to support the findings of this study have been deposited in the Google disk repository (https://drive.google.com/file/d/1D 9RSIqwNOBqElv7tSfZ1EYJomr0cFbGk/view?usp=sharing). The size distribution, Debye-Scherrer, Williamson-Hall, and elemental analysis data used to support the findings of this study are included within the article. These data are available upon request from the corresponding author.

Conflicts of Interest
The authors declare that they have no conflicts of interest.