Plasma Frequency

The plasma frequency,

$${\mit\Pi} = \left(\frac{n\,e^2}{\epsilon_0\,m}\right)^{1/2},$$ (1.5)

is the most fundamental timescale in plasma physics. There is a different plasma frequency for each species. However, the relatively large electron frequency is, by far, the most important of these, and references to “the plasma frequency” in textbooks invariably mean the electron plasma frequency.

It is easily seen that ${\mit\Pi}$ 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 (whose bounding planes are normal to the $x$-axis) consisting entirely of particles of one species (with charge $e$ and mass $m$) is displaced from its quasi-neutral position by an infinitesimal distance $\delta x$ (parallel to the $x$-axis). The resulting charge density that 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 $E_x= -\sigma/\epsilon_0 =- e\,n\,\delta x/\epsilon_0$ (Fitzpatrick 2008). Thus, Newton's second law of motion applied to an individual particle inside the slab yields

$$m\,\frac{d^2 \delta x}{dt^2} = e\,E_x = -m\,{\mit\Pi}^{2}\,\delta x,$$ (1.6)

giving $\delta x = (\delta x)_0\,\cos\,({\mit\Pi}\,t)$.

Plasma oscillations are observed only when the plasma system is studied over time periods, $\tau$, longer than the plasma period, $\tau_p\equiv 2\pi/{\mit\Pi}$, and when external influences modify the system at a rate no faster than ${\mit\Pi}$. In the opposite case, one is obviously studying something other than plasma physics (for instance, nuclear reactions), and the system cannot usefully be considered to be a plasma. Similarly, observations over lengthscales $L$ shorter than the distance $v_t\,\tau_p$ traveled by a typical plasma particle during a plasma period will also not detect plasma behavior. 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 is defined

$$\lambda_D \equiv \frac{1}{{\mit\Pi}}\left(\frac{T}{m}\right)^{1/2}.$$ (1.7)

It follows that

$$\lambda_D = \left(\frac{\epsilon_0\,T}{n\,e^2}\right)^{1/2}$$ (1.8)

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

According to the preceding discussion, our idealized system can usefully be considered to be a plasma only if

$$\frac{\lambda_D}{L} \ll 1,$$ (1.9)

and

$$\frac{\tau_p}{\tau}\ll 1.$$ (1.10)

Here, $\tau$ and $L$ represent the typical timescale and lengthscale of the process under investigation.

It should be noted that, despite the conventional requirement given in Equation (1.9), plasma physics is actually capable of describing structures on the Debye scale (Hazeltine and Waelbroeck 2004). The most important example of this ability is the theory of the Langmuir sheath, which is the boundary layer that surrounds a plasma confined by a material surface (Lieberman and Lichtenberg 2005). (See Section 4.17.)