Cold electrons

If the thermal motion of the electrons is ignored, the charge density oscillates at the plasma frequency:

${\displaystyle \omega _{\mathrm {pe} }={\sqrt {\frac {n_{\mathrm {e} }e^{2}}{m^{*}\varepsilon _{0}}}},\quad {\text{[rad/s]}}\quad {\text{(SI units)}}}$

${\displaystyle \omega _{\mathrm {pe} }={\sqrt {\frac {4\pi n_{\mathrm {e} }e^{2}}{m^{*}}}},\quad {\text{[rad/s]}}\quad {\text{(cgs units)}}}$

where ${\displaystyle n_{\mathrm {e} }}$ is the electron number density, ${\displaystyle e}$ is the elementary charge, ${\displaystyle m^{*}}$ is the electron effective mass, and ${\displaystyle \varepsilon _{0}}$ is the vacuum permittivity. This assumes infinite ion mass, a good approximation since electrons are much lighter.

A derivation using Maxwell’s equations^[2] gives the same result via the dielectric condition ${\displaystyle \epsilon (\omega )=0}$. This is the condition for plasma transparency and wave propagation.

In electron–positron plasmas, relevant in astrophysics, the expression must be modified. As the plasma frequency is independent of wavelength, Langmuir waves have infinite phase velocity and zero group velocity.

For ${\displaystyle m^{*}=m_{\mathrm {e} }}$, the frequency depends only on electron density and physical constants. The linear plasma frequency is:

${\displaystyle f_{\text{pe}}={\frac {\omega _{\text{pe}}}{2\pi }}\quad {\text{[Hz]}}}$

Metals are reflective to light below their plasma frequency, which is in the UV range (~10²³ electrons/cm³). Hence they appear shiny in visible light.

Warm electrons

Including the effects of electron thermal velocity ${\displaystyle v_{\mathrm {e,th} }={\sqrt {k_{\mathrm {B} }T_{\mathrm {e} }/m_{\mathrm {e} }}}}$, the dispersion relation becomes:

${\displaystyle \omega ^{2}=\omega _{\mathrm {pe} }^{2}+3k^{2}v_{\mathrm {e,th} }^{2}}$

This is known as the Bohm–Gross dispersion relation. For long wavelengths, pressure effects are minimal; for short wavelengths, dispersion dominates. At these small scales, wave phase velocity ${\displaystyle v_{\mathrm {ph} }=\omega /k}$ becomes comparable to ${\displaystyle v_{\mathrm {e,th} }}$, leading to Landau damping.

In bounded plasmas, plasma oscillations can still propagate due to fringing fields, even for cold electrons.

In metals or semiconductors, the ions' periodic potential is accounted for using the effective mass ${\displaystyle m^{*}}$.

Plasma oscillations and negative effective mass

Plasma oscillations can result in an effective negative mass. Consider the mass–spring model in Figure 1. Solving the equations of motion and replacing the system with a single effective mass gives:^[3][4]

${\displaystyle m_{\rm {eff}}=m_{1}+{\frac {m_{2}\omega _{0}^{2}}{\omega _{0}^{2}-\omega ^{2}}}}$

where ${\displaystyle \omega _{0}={\sqrt {k_{2}/m_{2}}}}$. As ${\displaystyle \omega }$ approaches ${\displaystyle \omega _{0}}$ from above, ${\displaystyle m_{\rm {eff}}}$ becomes negative.

This analogy applies to plasmonic systems too (Figure 2). Plasma oscillations of electron gas in a lattice behave like a spring system, giving an effective mass:

${\displaystyle m_{\rm {eff}}=m_{1}+{\frac {m_{2}\omega _{\rm {p}}^{2}}{\omega _{\rm {p}}^{2}-\omega ^{2}}}}$

Near ${\displaystyle \omega _{\rm {p}}}$, this effective mass becomes negative. Metamaterials exploiting this behavior have been studied.^[5][6]