Absorption/Extinction formula of nanoparticles I know the absorption/extinction equations in nanoparticle physics should be:
$$Q_{abs}=\frac{1}{2}\mathbf{Re}\int \mathbf{J}_{tot}\cdot\mathbf{E}_{tot}^\ast dV=\frac{\omega}{2}\mathbf{Im}(\epsilon)\int|\mathbf{E}_{tot}|^2dV$$
also, for the extinction, it reads:
$$Q_{ext}=\frac{1}{2}\mathbf{Re}\int \mathbf{J}_{tot}\cdot\mathbf{E}_0^\ast dV$$
But I see in some papers people use the following equations:
$$Q_{abs}=\frac{\omega}{2}\mathbf{Im}(\mathbf{d}\cdot\mathbf{E}_{inside}^\ast)$$
where $\mathbf{d}$ is the total dipole moment of nanoparticles. Also, for the extinction, it reads:
$$Q_{ext}=\frac{\omega}{2}\mathbf{Im}(\mathbf{d}\cdot\mathbf{E}_0^\ast)$$
I failed to derive the two equations. Can anyone give some help? Or, some reference papers would be also very helpful. Thanks a lot for the help.
 A: This mistery has an easy answear: in absence of external currents the total current is the current induced by the electric field, which is just the derivative of the nanoparticle polarization. This holds in general in absence of external current:
$$
\mathbf{J}_{pol} = \frac{\partial\mathbf{P}}{\partial t}
$$
Now in the case of a nanoparticle (or a dielectric/metallic sphere), it is known that the response to an external field is dipole-like. The total electric dipole $\mathbf{d}$ is given by:
$$
\mathbf{d} = \int \mathbf{P}\, d\mathbf{r}
$$
Now, since the total field inside the sphere $\mathbf{E}_{inside}$ and the exciting field $\mathbf{E}_0$ are both uniform (Attention: the total field outside the sphere is not uniform!), you can write:
\begin{align}
Q_{abs} &= \frac{1}{2}\mathbf{Re}\int \mathbf{J}_{pol}\cdot \mathbf{E}_{inside}^* =  \frac{1}{2}\mathbf{Re}\left\{ \mathbf{E}_{inside}^*\cdot \int \mathbf{J}_{pol}\, d\mathbf{r} \right\}\\
&=\frac{1}{2}\mathbf{Re}\left\{ \mathbf{E}_{inside}^*\cdot \left(\frac{\partial}{\partial t}\int \mathbf{P}_{pol}\, d\mathbf{r} \right)\right\} = \frac{1}{2}\mathbf{Re}\left\{\left(\frac{\partial}{\partial t}\mathbf{d}\right)\cdot\mathbf{E}_{inside}^*\right\}
\end{align}
If the external source is time harmonic, then all the time dependences can be assumed to be of the type $e^{-i\omega t}$, thus
$$
 \partial_t \mathbf{d} = -i\omega \mathbf{d}
$$
Substituting into the above equation, you get:
$$
Q_{abs} = \frac{1}{2}\mathbf{Re}\{-i\omega \mathbf{d}\cdot\mathbf{E}^*_{inside}\} = \frac{\omega}{2}\mathbf{Im}\{\mathbf{d}\cdot\mathbf{E}^*_{inside}\}
$$
and similarly for the $Q_{ext}$.
