Plasma oscillation: Difference between revisions

Latest revision as of 03:24, 28 June 2025

Template:Short description Script error: No such module "about".

Plasma oscillations, also known as Langmuir waves (after Irving Langmuir), are rapid oscillations of the electron density in conducting media such as plasmas or metals in the ultraviolet region. The oscillations can be described as an instability in the dielectric function of a free electron gas. The frequency depends only weakly on the wavelength of the oscillation. The quasiparticle resulting from the quantization of these oscillations is the plasmon.

Langmuir waves were discovered by American physicists Irving Langmuir and Lewi Tonks in the 1920s.^[1] They are parallel in form to Jeans instability waves, which are caused by gravitational instabilities in a static medium.

Mechanism

Consider an electrically neutral plasma in equilibrium, consisting of a gas of positively charged ions and negatively charged electrons. If one displaces an electron or a group of electrons slightly 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, the charge density oscillates at the plasma frequency:

ω_{p e} = \sqrt{\frac{n_{e} e^{2}}{m^{*} ε_{0}}}, [rad/s] (SI units)

ω_{p e} = \sqrt{\frac{4 π n_{e} e^{2}}{m^{*}}}, [rad/s] (cgs units)

where $n_{e}$ is the electron number density, $e$ is the elementary charge, $m^{*}$ is the electron effective mass, and $ε_{0}$ 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 $ϵ (ω) = 0$ . 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 $m^{*} = m_{e}$ , the frequency depends only on electron density and physical constants. The linear plasma frequency is:

$f_{pe} = \frac{ω_{pe}}{2 π} [Hz]$

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 $v_{e, t h} = \sqrt{k_{B} T_{e} / m_{e}}$ , the dispersion relation becomes:

$ω^{2} = ω_{p e}^{2} + 3 k^{2} v_{e, t h}^{2}$

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 $v_{p h} = ω / k$ becomes comparable to $v_{e, t h}$ , 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 $m^{*}$ .

Plasma oscillations and negative effective mass

File:A mechanical model giving rise to the negative effective mass effect..jpg

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) = \hat{F} \sin ω 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]

$m_{e f f} = m_{1} + \frac{m_{2} ω_{0}^{2}}{ω_{0}^{2} - ω^{2}}$

where $ω_{0} = \sqrt{k_{2} / m_{2}}$ . As $ω$ approaches $ω_{0}$ from above, $m_{e f f}$ becomes negative.

File:Equivalent mechanical scheme of electron gas in ionic lattice..jpg

Figure 2. Electron gas

m_{2}

inside an ionic lattice

m_{1}

. Plasma frequency

ω_{p}

defines spring constant

k_{2} = ω_{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:

$m_{e f f} = m_{1} + \frac{m_{2} ω_{p}^{2}}{ω_{p}^{2} - ω^{2}}$

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

References

Template:Reflist

@@ Line 1: / Line 1: @@
 {{Short description|Rapid oscillations of electron density}}
 {{about|a specific type of plasma wave|plasma waves in general|Waves in plasmas}}
 '''Plasma oscillations''', also known as '''Langmuir waves''' (after [[Irving Langmuir]]), are rapid oscillations of the [[electron density]] in conducting media such as [[Plasma (physics)|plasmas]] or [[metal]]s in the [[ultraviolet]] region. The oscillations can be described as an instability in the [[Drude model#AC field|dielectric function of a free electron gas]]. The frequency depends only weakly on the wavelength of the oscillation. The [[quasiparticle]] resulting from the [[Quantization (physics)|quantization]] of these oscillations is the ''[[plasmon]]''.
-Langmuir waves were discovered by American [[physicist]]s [[Irving Langmuir]] and [[Lewi Tonks]] in the 1920s.<ref>{{cite journal |url=http://www.columbia.edu/~mem4/ap6101/Tonks_Langmuir_PR29.pdf |journal=Physical Review |year=1929 |volume=33 |issue=8 |pages=195–210 |title=Oscillations in ionized gases |first1=Lewi |last1=Tonks |first2=Irving |last2=Langmuir|bibcode=1929PhRv...33..195T |doi=10.1103/PhysRev.33.195 |pmc=1085653 }}</ref>  They are parallel in form to [[Jeans instability]] waves, which are caused by gravitational instabilities in a static medium.
+Langmuir waves were discovered by American [[physicist]]s [[Irving Langmuir]] and [[Lewi Tonks]] in the 1920s.<ref>{{cite journal |url=http://www.columbia.edu/~mem4/ap6101/Tonks_Langmuir_PR29.pdf |journal=Physical Review |year=1929 |volume=33 |issue=8 |pages=195–210 |title=Oscillations in ionized gases |first1=Lewi |last1=Tonks |first2=Irving |last2=Langmuir|bibcode=1929PhRv...33..195T |doi=10.1103/PhysRev.33.195 |pmid=16587379 |pmc=1085653 }}</ref> They are parallel in form to [[Jeans instability]] waves, which are caused by gravitational instabilities in a static medium.
+==Mechanism==
+Consider an electrically neutral plasma in equilibrium, consisting of a gas of positively charged [[ion]]s and negatively charged [[electrons]]. If one displaces an electron or a group of electrons slightly with respect to the ions, the [[Coulomb force]] pulls the electrons back, acting as a restoring force.
-== Mechanism ==
+===Cold electrons===
-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.
+If the thermal motion of the electrons is ignored, the charge density oscillates at the ''plasma frequency'':
+:<math>\omega_{\mathrm{pe}} = \sqrt{\frac{n_\mathrm{e} e^{2}}{m^*\varepsilon_0}}, \quad \text{[rad/s]} \quad \text{(SI units)}</math>
+:<math>\omega_{\mathrm{pe}} = \sqrt{\frac{4 \pi n_\mathrm{e} e^{2}}{m^*}}, \quad \text{[rad/s]} \quad \text{(cgs units)}</math>
-=== 'Cold' electrons ===
+where <math>n_\mathrm{e}</math> is the electron number density, <math>e</math> is the elementary charge, <math>m^*</math> is the electron effective mass, and <math>\varepsilon_0</math> is the vacuum permittivity. This assumes infinite ion mass, a good approximation since electrons are much lighter.
-If the thermal motion of the electrons is ignored, it is possible to show that the charge density oscillates at the ''plasma frequency''
+A derivation using Maxwell’s equations<ref name="Ashcroft">{{cite book | last1=Ashcroft|first1=Neil | last2=Mermin|first2=N. David | 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> gives the same result via the dielectric condition <math>\epsilon(\omega) = 0</math>. This is the condition for plasma transparency and wave propagation.
-: <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.
+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.
-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:
+For <math>m^* = m_\mathrm{e}</math>, the frequency depends only on electron density and physical constants. The linear plasma frequency is:
-<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]].
+<math>f_\text{pe} = \frac{\omega_\text{pe}}{2\pi} \quad \text{[Hz]}</math>
-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
+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.
-<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===
-=== 'Warm' electrons ===
+Including the effects of electron thermal velocity <math>v_{\mathrm{e,th}} = \sqrt{k_\mathrm{B} T_\mathrm{e} / m_\mathrm{e}}</math>, the dispersion relation becomes:
-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>
-<math display="block">
+\omega^2 = \omega_{\mathrm{pe}}^2 + 3 k^2 v_{\mathrm{e,th}}^2
-\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">
+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 <math>v_\mathrm{ph} = \omega / k</math> becomes comparable to <math>v_{\mathrm{e,th}}</math>, leading to [[Landau damping]].
-v \sim v_{\mathrm{ph}} \ \stackrel{\mathrm{def}}{=}\  \frac{\omega}{k},
+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 <math>m^*</math>.
+===Plasma oscillations and negative effective mass===
+[[File:A mechanical model giving rise to the negative effective mass effect..jpg|thumb|'''Figure 1.''' Core with mass <math>m_2</math> connected by a spring <math>k_2</math> to a shell mass <math>m_1</math>. The system experiences force <math>F(t) = \widehat{F} \sin\omega t</math>.]]
+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:<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|volume=463|issue=2079|pages=855–880|doi=10.1098/rspa.2006.1795|bibcode=2007RSPSA.463..855M }}</ref><ref name=":1">{{Cite journal|last1=Chan|first1=C. T.|last2=Li|first2=Jensen|last3=Fung|first3=K. H.|date=2006|title=On extending the concept of double negativity to acoustic waves|journal=Journal of Zhejiang University Science A|volume=7|issue=1|pages=24–28|doi=10.1631/jzus.2006.A0024|bibcode=2006JZUSA...7...24C }}</ref>
+<math>
+m_{\rm eff} = m_1 + \frac{m_2 \omega_0^2}{\omega_0^2 - \omega^2}
 </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.
+where <math>\omega_0 = \sqrt{k_2 / m_2}</math>. As <math>\omega</math> approaches <math>\omega_0</math> from above, <math>m_{\rm eff}</math> becomes negative.
+[[File:Equivalent mechanical scheme of electron gas in ionic lattice..jpg|thumb|'''Figure 2.''' Electron gas <math>m_2</math> inside an ionic lattice <math>m_1</math>. Plasma frequency <math>\omega_{\rm p}</math> defines spring constant <math>k_2 = \omega_{\rm p}^2 m_2</math>.]]
-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''.
+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:
-=== Plasma oscillations and the effect of the negative mass ===
+<math>
-[[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>.]]
+m_{\rm eff} = m_1 + \frac{m_2 \omega_{\rm p}^2}{\omega_{\rm p}^2 - \omega^2}
-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>
-<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" />
-== See also ==
+Near <math>\omega_{\rm p}</math>, this effective mass becomes negative. Metamaterials exploiting this behavior have been studied.<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|volume=13|issue=8|pages=1890|doi=10.3390/ma13081890|doi-access=free |pmc=7215794|pmid=32316640|bibcode=2020Mate...13.1890B }}</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|journal=Materials|volume=13|issue=16|pages=3512|doi=10.3390/ma13163512|doi-access=free |pmc=7476018|pmid=32784869}}</ref>
+==See also==
 * [[Electron wake]]
 * [[Plasmon]]
 * [[Relativistic quantum chemistry]]
 * [[Surface plasmon resonance]]
-* [[Upper hybrid oscillation]], in particular for a discussion of the modification to the mode at propagation angles oblique to the magnetic field
+* [[Upper hybrid oscillation]]
 * [[Waves in plasmas]]
-== References ==
+==References==
-{{reflist | 30em}}
+{{reflist |30em}}
-{{reflist|group=note}}
 ==Further reading==
-*{{Citation |last=Longair |first=Malcolm S. |title=Galaxy Formation |year=1998 |publisher=Springer |location=Berlin |isbn=978-3-540-63785-1 }}
+*{{Citation |last=Longair |first=Malcolm S. |title=Galaxy Formation |year=1998 |publisher=Springer |isbn=978-3-540-63785-1 }}
 [[Category:Waves in plasmas]]
 [[Category:Plasmonics]]
 [[pt:Oscilação plasmática]]

Plasma oscillation: Difference between revisions

Latest revision as of 03:24, 28 June 2025

Contents

Mechanism

Cold electrons

Warm electrons

Plasma oscillations and negative effective mass

See also

References

Further reading

Navigation menu

Plasma oscillation: Difference between revisions

Latest revision as of 03:24, 28 June 2025

Mechanism

Cold electrons

Warm electrons

Plasma oscillations and negative effective mass

See also

References

Further reading

Navigation menu

Search