# NaYF4 Microstructure, beyond Their Well-Shaped Morphology
## 3 Results and Discussion
### Microstructural Analysis  
For all the 13 studied samples, all XRPD patterns exhibit peaks broader than the instrumental resolution (typically the FWHM for the samples was above 0.050\({}^{\circ}\)\(2\theta\) to be compared with 0.029\({}^{\circ}\)\(2\theta\) for the LaB\({}_{6}\) standard pattern). This broadening is known to be due to size effect and/or microconstrain in the system. From our previous fit, the shape factors (\(\phi\)) can be defined by \(\phi=IB_{vo}/FWHM_{vo}\) with \(IB_{vo}\) the observed integral breath (IB) and the observed FWHM. The Lorentzian and Gaussian limits for the shape factors are respectively \(\phi=2/\pi\approx 0.6366\) and \(\phi=2(\frac{log_{2}(2)}{pi^{2}})^{1/2}\approx 0.93949\)[32]. Shape factor with \(\phi<0.6366\) and \(\phi>0.9394\) can be referred as super Lorentzian and super Gaussian respectively [32]. Otherwise, it implies that the profile is a convolution of both shapes and justifies the use of a pseudo-Voigt. It is well admitted now that, for small geometrical effects of a single line instrument, the instrumental profile is approximately Lorentzian, whereas the profile arising from lattice strain is more Gaussian [33; 34]. The broadening due to defaults or small crystallite size depends respectively on their nature of the shape and the size distribution of the crystallites, and is assumed to be Lorenztian. These components can be taken as Voigtian, the Lorentzian and Gaussian parts are the limiting cases. If two or more reflections are available, size and strain effects can be determined from the variation of Lorentzian and Gaussian contributions in the IB depending on the hkl Miller indices [35]. We followed the procedure recommended by [PERSON] [32], computing the FWHM and IB of a pseudo-Voigt profile from the broadened profile to give the shape factor, calculating the breadths of the constituent profiles corrected by the instrumental broadening and analyzing the Williamson-Hall [36] and Halder-Wagner [37; 38] plots. The empirical procedure derived by [PERSON] et al. [35] for calculating the Lorentzian and Gaussian components (betaL, betaG) and the integral breadth (IB) has been used in the Fullprof software. The Figure 5a presents the dependence of the shape factor for four typical samples (NR 04-07-09-10) and the most different one (NR13), all the peaks show a shape factor between Gaussian and Lorentzian limits. The shape factor may not correlated with the Gd concentration (Figure 5b) To understand the origin of the broadening of our peaks, possibly due to both size and microstrain contribution, the Halder-Wagner method [37; 38] is recommended with such profile parameters instead of a Williamson-Hall approach [36], as the peaks have not a pure Lorentzian shape. Note that the fit is not so dependent on the \(2\theta\) range, even if Halder-Wagner plot insists on the small \(2\theta\) values (however the most intense peaks). To distinguish between size and microstrain respective contributions, the Halder-Wagner equation is written as \((IB^{*}/d^{*})^{2}=IB^{*}/(d^{*})^{2}\). By plotting this equation a linear behavior is observed for all our samples (Figure 6). The intercept of the linear regression in the Halder-Wagner plot is related to the importance of microstrain in the material and, on the other hand, the slope is related to a mean apparent size (in volume) of the NPs. In Figure 6, here for NR08, a clear linear regression is observed, with an origin close to zero. This behavior is common to all our synthesized NPs, as summarized in Table 4. Whatever the sample the microstrains are close to zero. For the rest of the study, the broadening of the Bragg peaks is then considered to be the broadening of a size effect only, and the microstrain effect can be neglected. One can also notice that the characteristic size of the NPs obtained from the Halder-Wagner plot is smaller that the one obtained by TEM imaging.