Editing Plasma oscillation (section)

== Mechanism ==

Consider an electrically neutral plasma in equilibrium, consisting of a gas of positively charged [[ion]]s and negatively charged [[electrons]]. If one displaces by a tiny amount an electron or a group of electrons with respect to the ions, the [[Coulomb force]] pulls the electrons back, acting as a restoring force.

=== 'Cold' electrons ===

If the thermal motion of the electrons is ignored, it is possible to show that the charge density oscillates at the ''plasma frequency''
: <math>\omega_{\mathrm{pe}} = \sqrt{\frac{n_\mathrm{e} e^{2}}{m^*\varepsilon_0}}, \left[\mathrm{rad/s}\right]</math> ([[SI units]]),
:<math>\omega_{\mathrm{pe}} = \sqrt{\frac{4 \pi n_\mathrm{e} e^{2}}{m^*}}, \left[\mathrm{rad/s}\right]</math> ([[Centimetre gram second system of units|cgs units]]),

where <math>n_\mathrm{e}</math>  is the [[number density]] of electrons, <math>e</math> is the [[electric charge]], <math>m^*</math>  is the [[effective mass (solid-state physics)|effective mass]] of the electron, and <math>\varepsilon_0</math> is the [[permittivity of free space]]. Note that the above [[formula]] is derived under the [[approximation]] that the ion mass is infinite. This is generally a good approximation, since electrons are much lighter than ions.

Proof using Maxwell equations.<ref name="Ashcroft">{{cite book | last1=Ashcroft|first1=Neil|author-link=Neil Ashcroft | last2=Mermin|first2=N. David|author-link2=N. David Mermin | title=[[Ashcroft and Mermin|Solid State Physics]] | publisher=Holt, Rinehart and Winston|location=New York | year=1976|isbn=978-0-03-083993-1 | page = 19}}</ref> Assuming charge density oscillations <math>\rho(\omega)=\rho_0 e^{-i\omega t}</math> the continuity equation:
<math display="block">\nabla \cdot \mathbf{j} = - \frac{\partial \rho}{\partial t} = i \omega \rho(\omega) </math>
the Gauss law
<math display="block">\nabla \cdot \mathbf{E}(\omega) = 4 \pi \rho(\omega)</math>
and the conductivity
<math display="block">\mathbf{j}(\omega) = \sigma(\omega) \mathbf{E}(\omega)</math>
taking the divergence on both sides and substituting the above relations:
<math display="block">i \omega \rho(\omega) =  4 \pi \sigma(\omega) \rho(\omega)</math>
which is always true only if
<math display="block">1+ \frac {4 \pi i \sigma(\omega)}{\omega} = 0</math>
But this is also the dielectric constant (see [[Drude Model]])
<math>\epsilon(\omega) = 1+ \frac {4 \pi i \sigma(\omega)}{\omega} </math>
and the condition of transparency (i.e. <math>\epsilon \ge 0</math> from a certain plasma frequency <math>\omega_{\rm p}</math> and above), the same condition here <math>\epsilon = 0</math> apply to make possible also the propagation of density waves in the charge density.

This expression must be modified in the case of electron-[[positron]] plasmas, often encountered in [[astrophysics]].<ref>{{cite book | last=Fu | first=Ying | title=Optical properties of nanostructures | year=2011 | publisher=Pan Stanford | pages=201}}</ref> Since the [[frequency]] is independent of the [[wavelength]], these [[oscillation]]s have an [[Infinity|infinite]] [[phase velocity]] and zero [[group velocity]].

Note that, when <math>m^*=m_\mathrm{e}</math>, the plasma frequency, <math>\omega_{\mathrm{pe}}</math>, depends only on [[physical constant]]s and electron density <math>n_\mathrm{e}</math>. The numeric expression for angular plasma frequency is
<math display="block">f_\text{pe} = \frac{\omega_\text{pe}}{2\pi}~\left[\text{Hz}\right]</math>

Metals are only transparent to light with a frequency higher than the metal's plasma frequency. For typical metals such as aluminium or silver, <math>n_\mathrm{e}</math> is approximately 10<sup>23</sup> cm<sup>−3</sup>, which brings the plasma frequency into the ultraviolet region. This is why most metals reflect visible light and appear shiny.

=== 'Warm' electrons ===

When the effects of the [[electron]] thermal speed <math display="inline">v_{\mathrm{e,th}} = \sqrt{k_\mathrm{B} T_{\mathrm{e}} / m_\mathrm{e}}</math> are considered, the electron pressure acts as a restoring force, and the electric field and oscillations propagate with frequency and [[wavenumber]] related by the longitudinal Langmuir<ref>*{{Citation |last=Andreev |first=A. A. |title=An Introduction to Hot Laser Plasma Physics |year=2000 |publisher= [[Nova Science Publishers, Inc.]] |location=Huntington, New York |isbn=978-1-56072-803-0}}</ref> wave:
<math display="block">
\omega^2 =\omega_{\mathrm{pe}}^2 +\frac{3k_\mathrm{B}T_{\mathrm{e}}}{m_\mathrm{e}}k^2=\omega_{\mathrm{pe}}^2 + 3 k^2 v_{\mathrm{e,th}}^2,
</math>
called the [[David Bohm|Bohm]]–[[Eugene P. Gross|Gross]] [[dispersion relation]]. If the spatial scale is large compared to the [[Debye length]], the [[oscillation]]s are only weakly modified by the [[pressure]] term, but at small scales the pressure term dominates and the waves become dispersionless with a speed of <math>\sqrt{3} \cdot v_{\mathrm{e,th}}</math>. For such waves, however, the electron thermal speed is comparable to the [[phase velocity]], i.e., 
<math display="block">
v \sim v_{\mathrm{ph}} \ \stackrel{\mathrm{def}}{=}\  \frac{\omega}{k},
</math>
so the plasma waves can [[accelerate]] electrons that are moving with speed nearly equal to the phase velocity of the wave. This process often leads to a form of collisionless damping, called [[Landau damping]]. Consequently, the large-''k'' portion in the [[dispersion relation]] is difficult to observe and seldom of consequence.

In a [[bounded function|bounded]] plasma, fringing electric fields can result in propagation of plasma oscillations, even when the electrons are cold.

In a [[metal]] or [[semiconductor]], the effect of the [[ion]]s' periodic potential must be taken into account. This is usually done by using the electrons' [[effective mass (solid-state physics)|effective mass]] in place of ''m''.

=== Plasma oscillations and the effect of the negative mass ===
[[File:A mechanical model giving rise to the negative effective mass effect..jpg|alt=A mechanical model giving rise to the negative effective mass effect|thumb|'''Figure 1.''' Core with mass <math>m_2</math> is connected internally through the spring with <math>k_2</math> to a shell with mass <math>m_1</math>. The system is subjected to the sinusoidal force <math>F(t) = \widehat{F}\sin\omega t</math>.]]
Plasma oscillations may give rise to the effect of the “[[negative mass]]”. The mechanical model giving rise to the negative effective mass effect is depicted in '''Figure 1'''. A core with mass <math>m_2</math> is connected internally through the spring with constant <math>k_2</math> to a shell with mass <math>m_1</math>. The system is subjected to the external sinusoidal force <math>F(t)=\widehat{F}\sin\omega t</math>. If we solve the equations of motion for the masses <math>m_1</math> and <math>m_2</math> and replace the entire system with a single effective mass <math>m_{\rm eff}</math> we obtain:<ref name=":0">{{Cite journal|last1=Milton|first1=Graeme W| last2=Willis|first2=John R| date=2007-03-08|title=On modifications of Newton's second law and linear continuum elastodynamics | url=https://royalsocietypublishing.org/doi/10.1098/rspa.2006.1795|journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=463|issue=2079|pages=855–880| doi=10.1098/rspa.2006.1795|bibcode=2007RSPSA.463..855M|s2cid=122990527|url-access=subscription}}</ref><ref name=":1">{{Cite journal| last1=Chan|first1=C. T.|last2=Li|first2=Jensen|last3=Fung|first3=K. H.|date=2006-01-01|title=On extending the concept of double negativity to acoustic waves|url=https://doi.org/10.1631/jzus.2006.A0024|journal= Journal of Zhejiang University Science A|language=en|volume=7|issue=1|pages=24–28| doi=10.1631/jzus.2006.A0024|bibcode=2006JZUSA...7...24C | s2cid=120899746| issn=1862-1775|url-access=subscription}}</ref><ref name=":2">{{Cite journal|last1=Huang|first1=H. H.|last2=Sun|first2=C. T.| last3=Huang|first3=G. L.|date=2009-04-01|title=On the negative effective mass density in acoustic metamaterials |url=http://www.sciencedirect.com/science/article/pii/S0020722508002279|journal=International Journal of Engineering Science|language=en|volume=47|issue=4|pages=610–617 | doi=10.1016/j.ijengsci.2008.12.007 |issn=0020-7225|url-access=subscription}}</ref><ref name=":3">{{Cite journal| last1=Yao|first1=Shanshan |last2=Zhou|first2=Xiaoming |last3=Hu|first3=Gengkai |date=2008-04-14|title=Experimental study on negative effective mass in a 1D mass–spring system |journal=New Journal of Physics |volume=10|issue=4|pages=043020|doi=10.1088/1367-2630/10/4/043020 |bibcode=2008NJPh...10d3020Y|issn=1367-2630|doi-access=free}}</ref><ref name=":4"/>
<math display="block">m_{\rm eff}=m_1+{m_2\omega_0^2\over \omega_0^2-\omega^2},</math>
where <math display="inline">\omega_0=\sqrt{k_2 / m_2}</math>. When the frequency <math>\omega</math> approaches <math>\omega_0</math> from above the effective mass <math>m_{\rm eff}</math> will be negative.<ref name=":0" /><ref name=":1" /><ref name=":2" /><ref name=":3" />
[[File:Equivalent mechanical scheme of electron gas in ionic lattice..jpg|thumb|'''Figure 2.''' Free electrons gas <math>m_2</math> is embedded into the ionic lattice <math>m_1</math>; <math>\omega_{\rm p}</math>  is the plasma frequency (the left sketch). The equivalent mechanical scheme of the system (right sketch).]]
The negative effective mass (density) becomes also possible based on the electro-mechanical coupling exploiting plasma oscillations of a free electron gas (see '''Figure 2''').<ref name=":4">{{Cite journal| last1=Bormashenko|first1=Edward |last2=Legchenkova|first2=Irina |date=April 2020|title=Negative Effective Mass in Plasmonic Systems |journal=Materials |language=en |volume=13 |issue=8 |pages=1890 |doi=10.3390/ma13081890 |pmc=7215794 |pmid=32316640 |bibcode=2020Mate...13.1890B |doi-access=free}} [[File:CC-BY icon.svg|50px]]  Text was copied from this source, which is available under a [https://creativecommons.org/licenses/by/4.0/  Creative Commons Attribution 4.0 International License].</ref><ref name=":5">{{Cite journal |last1=Bormashenko|first1=Edward |last2=Legchenkova|first2=Irina |last3=Frenkel|first3=Mark |date=August 2020 | title=Negative Effective Mass in Plasmonic Systems II: Elucidating the Optical and Acoustical Branches of Vibrations and the Possibility of Anti-Resonance Propagation |journal=Materials |language=en |volume=13 |issue=16 |pages=3512 |doi=10.3390/ma13163512|pmc=7476018|pmid=32784869|bibcode=2020Mate...13.3512B|doi-access=free}}</ref> The negative mass appears as a result of vibration of a metallic particle with a frequency of <math>\omega</math> which is close the frequency of the plasma oscillations of the electron gas <math>m_2</math> relatively to the ionic lattice <math>m_1</math>. The plasma oscillations are represented with the elastic spring <math>k_2 = \omega_{\rm p}^2m_2</math>, where <math>\omega_{\rm p}</math> is the plasma frequency. Thus, the metallic particle vibrated with the external frequency ''ω'' is described by the effective mass
<math display="block">m_{\rm eff}=m_1+{m_2\omega_{\rm p}^2\over \omega_{\rm p}^2-\omega^2},</math>
which is negative when the frequency <math>\omega</math> approaches <math>\omega_{\rm p}</math> from above. Metamaterials exploiting the effect of the negative mass in the vicinity of the plasma frequency were reported.<ref name=":4" /><ref name=":5" />