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Plasma oscillation

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From Wikipedia, the free encyclopedia

Rapid oscillations of electron density

This article is about a specific type of plasma wave. For plasma waves in general, seeWaves in plasmas.

Plasma oscillations, also known asLangmuir waves (eponymously afterIrving Langmuir), are rapidoscillations of theelectron density inconductive media, most notablyplasmas as well asmetals, at frequencies typically corresponding to theultraviolet band of theelectromagnetic spectrum. The oscillations can be described as an instability in thedielectric function of a free electron gas. The frequency depends only weakly on the wavelength of the oscillation. Thequasiparticle resulting from thequantization of these oscillations is theplasmon.

Langmuir waves were discovered by American physicistsIrving Langmuir andLewi Tonks in the 1920s.^[1] They are parallel in form toJeans instability waves, which are caused by gravitational instabilities in a static medium.

Mechanism

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Consider an electrically neutral plasma in equilibrium, consisting of a gas of positively chargedions and negatively chargedelectrons. If one displaces an electron or a group of electrons slightly with respect to the ions, theCoulomb force pulls the electrons back, acting as a restoring force.

Cold electrons

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

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

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

where $n_{\mathrm {e} }$ is the electron number density, $e {\displaystyle e}$ is the elementary charge, $m^{*}$ is the electron effective mass, and $\varepsilon _{0}$ 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 $\epsilon (\omega )=0$ . This is the condition for plasma transparency and wave propagation.

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

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

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

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

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

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

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

Plasma oscillations and negative effective mass

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**Figure 1.** Core with mass $m_{2}$ connected by a spring $k_{2}$ to a shell mass $m_{1}$ . The system experiences force $F(t)={\widehat {F}}\sin \omega t$ .

{\displaystyle m_{2}} — **Figure 1.** Core with mass $m_{2}$ connected by a spring $k_{2}$ to a shell mass $m_{1}$ . The system experiences force $F(t)={\widehat {F}}\sin \omega t$ .

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]

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

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

**Figure 2.** Electron gas $m_{2}$ inside an ionic lattice $m_{1}$ . Plasma frequency $\omega _{\rm {p}}$ defines spring constant $k_{2}=\omega _{\rm {p}}^{2}m_{2}$ .

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:

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

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

References

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^Tonks, Lewi; Langmuir, Irving (1929)."Oscillations in ionized gases"(PDF).Physical Review.33 (8):195–210.Bibcode:1929PhRv...33..195T.doi:10.1103/PhysRev.33.195.PMC 1085653.PMID 16587379.
^Ashcroft, Neil; Mermin, N. David (1976).Solid State Physics. New York: Holt, Rinehart and Winston. p. 19.ISBN 978-0-03-083993-1.
^Milton, Graeme W; Willis, John R (2007-03-08)."On modifications of Newton's second law and linear continuum elastodynamics".Proceedings of the Royal Society A.463 (2079):855–880.Bibcode:2007RSPSA.463..855M.doi:10.1098/rspa.2006.1795.
^Chan, C. T.; Li, Jensen; Fung, K. H. (2006). "On extending the concept of double negativity to acoustic waves".Journal of Zhejiang University Science A.7 (1):24–28.Bibcode:2006JZUSA...7...24C.doi:10.1631/jzus.2006.A0024.
^Bormashenko, Edward; Legchenkova, Irina (April 2020)."Negative Effective Mass in Plasmonic Systems".Materials.13 (8): 1890.Bibcode:2020Mate...13.1890B.doi:10.3390/ma13081890.PMC 7215794.PMID 32316640.
^Bormashenko, Edward; Legchenkova, Irina; Frenkel, Mark (August 2020)."Negative Effective Mass in Plasmonic Systems II".Materials.13 (16): 3512.doi:10.3390/ma13163512.PMC 7476018.PMID 32784869.

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Plasma oscillation

Mechanism

Cold electrons

Warm electrons

Plasma oscillations and negative effective mass

See also

References

Further reading