Effective number of oscillators and B...

Effective number of oscillators and Born effective charge

The operational definition of effective number of oscillators is:

n_{eff}=\frac{2}{\pi\omega_p^2}\int_{\omega_1}^{\omega_2}{\epsilon_2\omega d\omega

where 

\omega_p^2=\frac{e^2}{V_0m\epsilon_0}

here e is electronic charge, V_0 is the unit cell volume, m is the reduced mass of the oscillator and \epsilon_0 is the vacuum dielectric constant.

If we introduce (Born) effective charge q_{eff}, we also get

n_{eff} =\frac{Nq_{eff}^2V_0\mu\epsilon_0}{e^2V_0m\epsilon_0} = N(\frac{q_{eff}}{e})^2 \frac{m}{\mu}