A homogeneous, spherical electron cloud describes an atom (radius $a_0$ and total charge $^-e$ and positive point charge $^+e$ as the nucleus. An external electric field stimulates the electron cloud to oscillate relatively to the nucleus. Let the achieved deflections be small compared ot the atom radius. Determine the resonance frequency for the case
(a) of a hydrogen atom: $a_0=0.53\cdot 10^{-10}m$.
(b) of a sodium atom: $a_0=4.53\cdot 10^{-10}m$.
(c) Show that even for electric fields of magnitude $10^8V/m$ the deflection of the electron cloud is small compared to the atom radius.
To be honest I'm not sure how to approach this. The exercises are usually based on the last lecture. And the last thing we did was talking about plasma oscillation, where the plasma frequency is $\omega_{pe}=\sqrt{\frac{n_ee^2}{m_e\epsilon_0}}$. If that's really what I have to apply here this is what I was thinking:
(a): Since $r=a_0$ then $n_e=\frac{1}{\frac{4}{3}\pi a_0^3}$, since there is only one electron in the electron cloud, right? Which then would give me a frequency of $\omega_{pe}=3786808$Hz. Is that correct?
Anyway, similarly that would give me $\omega_{pe}=1295275.407$Hz. Again, could very well be wrong for this problem since I don't know whether we are looking for the plasma frequency or not.
(c) Pretty much lost here. I don't remember an expression including the electric field in regards to the plasma frequency.
I would greatly appreciate any hints or tips.
The electric field of the electron cloud is zero at the center and increases linearly away from the center. You should have that result either for a uniform charged sphere or a uniform mass for gravity. This makes the atom a harmonic oscillator. Compute the spring constant, which is the force applied by the electric field divided by the distance the nucleus is offset from the center of the cloud. The mass is the mass of the electron. For the last part, find how much offset it takes to neutralize that electric field and note that it is small.