Plasma Frequency

The plasma frequency,

$$\omega_p^{~2} = \frac{n\,e^2}{\epsilon_0\,m},$$ (5)

is the most fundamental time-scale in plasma physics. Clearly, there is a different plasma frequency for each species. However, the relatively fast electron frequency is, by far, the most important, and references to ``the plasma frequency'' in text-books invariably mean the electron plasma frequency.

It is easily seen that $\omega_p$ corresponds to the typical electrostatic oscillation frequency of a given species in response to a small charge separation. For instance, consider a one-dimensional situation in which a slab consisting entirely of one charge species is displaced from its quasi-neutral position by an infinitesimal distance $\delta x$. The resulting charge density which develops on the leading face of the slab is $\sigma=e\,n\,\delta x$. An equal and opposite charge density develops on the opposite face. The $x$-directed electric field generated inside the slab is of magnitude $E_x= -\sigma/\epsilon_0 =- e\,n\,\delta x/\epsilon_0$. Thus, Newton's law applied to an individual particle inside the slab yields

$$m\,\frac{d^2 \delta x}{dt^2} = e\,E_x = -m\,\omega_p^{~2}\,\delta x,$$ (6)

giving $\delta x = (\delta x)_0\,\cos\,(\omega_p\,t)$.

Note that plasma oscillations will only be observed if the plasma system is studied over time periods $\tau$ longer than the plasma period $\tau_p\equiv 1/\omega_p$, and if external actions change the system at a rate no faster than $\omega_p$. In the opposite case, one is clearly studying something other than plasma physics (e.g., nuclear reactions), and the system cannot not usefully be considered to be a plasma. Likewise, observations over length-scales $L$ shorter than the distance $v_t\,\tau_p$ traveled by a typical plasma particle during a plasma period will also not detect plasma behaviour. In this case, particles will exit the system before completing a plasma oscillation. This distance, which is the spatial equivalent to $\tau_p$, is called the Debye length, and takes the form

$$\lambda_D \equiv \sqrt{T/m}\,\,\omega_p^{-1}.$$ (7)

Note that

$$\lambda_D = \sqrt{\frac{\epsilon_0\,T}{n\,e^2}}$$ (8)

is independent of mass, and therefore generally comparable for different species.

Clearly, our idealized system can only usefully be considered to be a plasma provided that

$$\frac{\lambda_D}{L} \ll 1,$$ (9)

and

$$\frac{\tau_p}{\tau}\ll 1.$$ (10)

Here, $\tau$ and $L$ represent the typical time-scale and length-scale of the process under investigation.

It should be noted that, despite the conventional requirement (9), plasma physics is capable of considering structures on the Debye scale. The most important example of this is the Debye sheath: i.e., the boundary layer which surrounds a plasma confined by a material surface.