Editing Plasma oscillation (section)

=== 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" />