Update on Overleaf.

main.tex

@@ -526,8 +526,8 @@ license.
  distribution during the \textit{fs} pulse, we introduced another
  asymmetry factor
  $G_{N_e} = (N_e^{front}-N_e^{back})/(N_e^{front}+N_e^{back})$
- indicating the relationship between the average EHP densities in the front and in the back halves of the NP, defined as $N_e^{front}=\int_{(z>0)} {N_e}d{\mathrm{v}}$ and
- $N_e^{back}=\int_{(z<0)} {N_e}d{\mathrm{v}}$. In this way, $G_{N_e} = 0$ corresponds to the quasi-homogeneous case and the assumption of the
+ indicating the relationship between the average EHP densities in the front and in the back halves of the NP, defined as $N_e^{front}=2\int_{(z>0)} {N_e}d{\mathrm{v}}/V$ and
+ $N_e^{back}=2\int_{(z<0)} {N_e}d{\mathrm{v}}/V$, where $V = \frac{4}{3}\pi{R}^3$ is the volume of the nanosphere. In this way, $G_{N_e} = 0$ corresponds to the quasi-homogeneous case and the assumption of the
  NP homogeneous EHP distribution can be made to investigate the
  optical response of the excited Si NP. When $G_{N_e}$ significantly
  differs from $0$, this assumption, however, could not be