# Nonlocal and Size-Dependent Dielectric Function for Plasmonic Nanoparticles
## 3 Results and Discussion
When compared with the Nonlocal model, the outcomes of the NLSD show that the relative reductions in the main energy loss peaks, and the increases of the outcomes in the low-energy  
Figure 2: EELS (electron energy loss spectroscopy) spectrums for Au spherical dimers of different radii \(R\), and for different dielectric models (black solid curves, NLSD (nonlocal and size-dependent) model; blue dotted curves, SD model; red dash-dotted curves, experimental dielectric data from [PERSON] et al. [33]; green dashed curves, DFT (density functional theory) calculated data from [PERSON] et al. [34]; thinner gray solid curves, Nonlocal model), where the gap \(d\) fixed on 1 nm. (**a**) electron beam penetrating the centre in one of the nanospheres, and (**b**) passing by the middle of the gap, respectively, as indicated in Figure 1.  
region are more remarkable, as shown in Figure 2b. All of these are because the size-dependent effects due to the size-corrective terms are contained in the NLSD model. For further confirmation of the present findings, EELS calculated with different dielectric models for isolated Au spherical nanoparticles of different sizes are presented in Figure 3, where electron beam passing near the nanoparticles and the contribution of the surface vibrational mode is dominant. The difference of the total (and surface) energy loss probabilities between the NLSD model and Nonlocal model becomes more pronounced with the reduction of particle size, as shown in Figure 3 where the radius \(R=1\) nm, and the prediction of our NLSD model are much more close to the outcomes from the SD model and the DFT model. The discrepancy in predicted energy loss probabilities between the NLSD model and the Nonlocal model would gradually narrow and ultimately disappear when the radius is greater than 26 nm, as shown in Figure 3b, in addition to the discrepancy in predicted energy loss probabilities between the SD model and experimental dielectric data. These results for isolated nanoparticles are more compatible with the conclusions that have been verified both experimentally and theoretically in the previous studies of the SD model in which the significant size-dependent effects can only be expected to be observed below a certain limited size range of approximately 20 nm. Although, under what specific physical mechanism will the differences due to the size-dependent effects between the SD model and experimental dielectric data disappear in the outcomes between the NLSD and Nonlocal model for dimers of a large radius merit further investigation. However, according to all of the above results of EELS, one can see that the NLSD model presented in this paper has the advantage of clearly separating the size regions where local, nonlocal or size-dependent effects are dominant, for both simulation cases (isolated particle or dimer). First, the size-dependent effects somehow can be revealed in small isolated nanoparticles by using the SD model, whereas the outcomes of this model do not contain any appreciable impact of the nonlocal effects. In addition, at least, for dimers of a large radius, the outcomes of the SD model should be much more consistent with those by using the classic local experimental dielectric data--particularly for an electron beam penetrating the centre in one of the nanospheres and the contribution of the bulk vibrational mode being dominant, as shown in Figure 2a. By contrast, the size-dependent effects due to the same size-corrective terms in the NLSD model have been  
Figure 3: EELS spectrums for isolated Au spherical nanoparticles of different radii \(R\), and for different dielectric models as defined in the text, when electron beam passing near the nanoparticles as depicted in the inset. Distances \(d^{\prime}\) between electron beam and nanoparticles were fixed at 2 nm.