1754:
2395:
3259:
1541:
1370:
231:
2773:
622:
348:
1697:
2248:
1361:
939:
740:
808:
1232:
680:
867:
547:
1612:
2659:
2621:
2044:
1895:
2307:
552:
1131:
1621:
1536:{\displaystyle \omega ^{2}=\omega _{\mathrm {pe} }^{2}+{\frac {3k_{\mathrm {B} }T_{\mathrm {e} }}{m_{\mathrm {e} }}}k^{2}=\omega _{\mathrm {pe} }^{2}+3k^{2}v_{\mathrm {e,th} }^{2},}
1099:
2138:
119:
2822:
2650:
2482:
994:
1263:
1164:
471:
385:
965:
2389:
2133:
2354:
1020:
242:
2793:
2510:
2327:
2564:
2537:
2453:
2426:
2098:
2071:
1988:
1961:
1934:
1839:
1812:
1785:
440:
3281:"Negative Effective Mass in Plasmonic Systems II: Elucidating the Optical and Acoustical Branches of Vibrations and the Possibility of Anti-Resonance Propagation"
1280:
685:
2488:
The negative effective mass (density) becomes also possible based on the electro-mechanical coupling exploiting plasma oscillations of a free electron gas (see
745:
409:
1169:
627:
1703:
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
1363:
are taken into account, the electron pressure acts as a restoring force as well as the electric field and the oscillations propagate with frequency and
1265:
is approximately 10 cm, which brings the plasma frequency into the ultraviolet region. This is why most metals reflect visible light and appear shiny.
876:
813:
1236:
Metals are only transparent to light with a frequency higher than the metal's plasma frequency. For typical metals such as aluminium or silver,
2824: from above. Metamaterials exploiting the effect of the negative mass in the vicinity of the plasma frequency were reported.
351:
3350:
2993:
2948:
491:
1738:
443:
1569:
1753:
485:
that the ion mass is infinite. This is generally a good approximation, as the electrons are so much lighter than ions.
2569:
1993:
1844:
1722:
plasma, fringing electric fields can result in propagation of plasma oscillations, even when the electrons are cold.
2856:, in particular for a discussion of the modification to the mode at propagation angles oblique to the magnetic field
110:
If the thermal motion of the electrons is ignored, it is possible to show that the charge density oscillates at the
2253:
2394:
2484: is the plasma frequency (the left sketch). The equivalent mechanical scheme of the system (right sketch).
2843:
2985:
1104:
1062:
226:{\displaystyle \omega _{\mathrm {pe} }={\sqrt {\frac {n_{\mathrm {e} }e^{2}}{m^{*}\varepsilon _{0}}}},\left}
1566:
term, but at small scales the pressure term dominates and the waves become dispersionless with a speed of
2768:{\displaystyle m_{\rm {eff}}=m_{1}+{m_{2}\omega _{\rm {p}}^{2} \over \omega _{\rm {p}}^{2}-\omega ^{2}},}
2798:
2626:
2458:
970:
474:
1239:
1140:
617:{\displaystyle \nabla \cdot \mathbf {j} =-{\frac {\partial \rho }{\partial t}}=i\omega \rho (\omega )}
449:
361:
98:. If one displaces by a tiny amount an electron or a group of electrons with respect to the ions, the
3368:
3112:
2848:
944:
2853:
2359:
2103:
90:
Consider an electrically neutral plasma in equilibrium, consisting of a gas of positively charged
2492:). The negative mass appears as a result of vibration of a metallic particle with a frequency of
2332:
999:
343:{\displaystyle \omega _{\mathrm {pe} }={\sqrt {\frac {4\pi n_{\mathrm {e} }e^{2}}{m^{*}}}},\left}
3012:
2652: is the plasma frequency. Thus, the metallic particle vibrated with the external frequency
2879:
1692:{\displaystyle v\sim v_{\mathrm {ph} }\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\omega }{k}},}
55:
1737:' periodic potential must be taken into account. This is usually done by using the electrons'
2778:
2495:
2312:
2243:{\displaystyle m_{\rm {eff}}=m_{1}+{m_{2}\omega _{0}^{2} \over \omega _{0}^{2}-\omega ^{2}},}
1356:{\textstyle v_{\mathrm {e,th} }={\sqrt {k_{\mathrm {B} }T_{\mathrm {e} }/m_{\mathrm {e} }}}}
3292:
3220:
3166:
3024:
2894:
2542:
2515:
2431:
2404:
2076:
2049:
1966:
1939:
1912:
1817:
1790:
1763:
418:
8:
3373:
2939:
1905:β. The mechanical model giving rise to the negative effective mass effect is depicted in
1712:
1551:
3296:
3224:
3179:
3170:
3154:
3028:
2898:
3315:
3280:
3243:
3208:
3090:
3040:
2915:
394:
3346:
3320:
3248:
3184:
3132:
3094:
3082:
3044:
2989:
2944:
1134:
79:
1022:
apply to make possible also the propagation of density waves in the charge density.
3310:
3300:
3238:
3228:
3174:
3128:
3124:
3074:
3032:
3017:
Proceedings of the Royal
Society A: Mathematical, Physical and Engineering Sciences
2910:
2902:
2859:
1719:
35:
31:
47:
2934:
1548:
412:
71:
27:
3263:
1704:
1615:
1053:
1049:
388:
934:{\displaystyle \epsilon (\omega )=1+{\frac {4\pi i\sigma (\omega )}{\omega }}}
50:. The frequency depends only weakly on the wavelength of the oscillation. The
3362:
3188:
3136:
3086:
3013:"On modifications of Newton's second law and linear continuum elastodynamics"
2930:
2833:
1902:
1730:
482:
99:
51:
3078:
2512:
which is close the frequency of the plasma oscillations of the electron gas
3324:
3252:
3036:
2906:
1614:. For such waves, however, the electron thermal speed is comparable to the
1555:
1030:
735:{\displaystyle \mathbf {j} (\omega )=\sigma (\omega )\mathbf {E} (\omega )}
82:
waves, which are caused by gravitational instabilities in a static medium.
3155:"Experimental study on negative effective mass in a 1D massβspring system"
3062:
1748:
803:{\displaystyle i\omega \rho (\omega )=4\pi \sigma (\omega )\rho (\omega )}
742:
taking the divergence on both sides and substituting the above relations:
1559:
1042:
870:
43:
1227:{\displaystyle f_{\text{pe}}={\frac {\omega _{\text{pe}}}{2\pi }}~\left}
3305:
3233:
1700:
1544:
1364:
1038:
75:
3279:
Bormashenko, Edward; Legchenkova, Irina; Frenkel, Mark (August 2020).
675:{\displaystyle \nabla \cdot \mathbf {E} (\omega )=4\pi \rho (\omega )}
1034:
95:
68:
488:
Proof using
Maxwell equations. Assuming charge density oscillations
1563:
1274:
1046:
1026:
234:
46:
region. The oscillations can be described as an instability in the
3113:"On the negative effective mass density in acoustic metamaterials"
2566:. The plasma oscillations are represented with the elastic spring
3063:"On extending the concept of double negativity to acoustic waves"
2838:
2100: and replace the entire system with a single effective mass
478:
60:
3278:
3262:
Text was copied from this source, which is available under a
3258:
1726:
862:{\displaystyle 1+{\frac {4\pi i\sigma (\omega )}{\omega }}=0}
39:
1990:. The system is subjected to the external sinusoidal force
3153:
Yao, Shanshan; Zhou, Xiaoming; Hu, Gengkai (2008-04-14).
1936:
is connected internally through the spring with constant
1734:
1166:. The numeric expression for angular plasma frequency is
1025:
This expression must be modified in the case of electron-
91:
1901:
Plasma oscillations may give rise to the effect of the β
1749:
Plasma oscillations and the effect of the negative mass
102:
pulls the electrons back, acting as a restoring force.
3264:
Creative
Commons Attribution 4.0 International License
3207:
Bormashenko, Edward; Legchenkova, Irina (April 2020).
3206:
2256:
1283:
542:{\displaystyle \rho (\omega )=\rho _{0}e^{-i\omega t}}
3111:
Huang, H. H.; Sun, C. T.; Huang, G. L. (2009-04-01).
2801:
2781:
2662:
2629:
2572:
2545:
2518:
2498:
2461:
2434:
2407:
2362:
2335:
2315:
2141:
2106:
2079:
2052:
2046:. If we solve the equations of motion for the masses
1996:
1969:
1942:
1915:
1847:
1820:
1793:
1766:
1624:
1572:
1373:
1242:
1172:
1143:
1107:
1065:
1002:
973:
947:
879:
816:
748:
688:
630:
555:
494:
452:
421:
397:
364:
245:
122:
2943:. New York: Holt, Rinehart and Winston. p. 19.
1607:{\displaystyle {\sqrt {3}}\cdot v_{\mathrm {e,th} }}
3061:Chan, C. T.; Li, Jensen; Fung, K. H. (2006-01-01).
1715:is difficult to observe and seldom of consequence.
2816:
2787:
2767:
2644:
2615:
2558:
2531:
2504:
2476:
2447:
2420:
2383:
2348:
2321:
2301:
2242:
2127:
2092:
2065:
2038:
1982:
1955:
1928:
1889:
1841:. The system is subjected to the sinusoidal force
1833:
1806:
1779:
1691:
1606:
1535:
1355:
1257:
1226:
1158:
1125:
1093:
1014:
988:
959:
933:
861:
802:
734:
674:
616:
541:
465:
434:
403:
379:
342:
225:
3360:
2616:{\displaystyle k_{2}=\omega _{\rm {p}}^{2}m_{2}}
2039:{\displaystyle F(t)={\widehat {F}}\sin \omega t}
1890:{\displaystyle F(t)={\widehat {F}}\sin \omega t}
1787:is connected internally through the spring with
1554:. If the spatial scale is large compared to the
3011:Milton, Graeme W; Willis, John R (2007-03-08).
3209:"Negative Effective Mass in Plasmonic Systems"
2929:
2302:{\textstyle \omega _{0}={\sqrt {k_{2}/m_{2}}}}
869:But this is also the dielectric constant (see
3110:
2877:
3117:International Journal of Engineering Science
3010:
78:in the 1920s. They are parallel in form to
2982:An Introduction to Hot Laser Plasma Physics
1367:related by the longitudinal Langmuir wave:
67:Langmuir waves were discovered by American
3060:
48:dielectric function of a free electron gas
3314:
3304:
3242:
3232:
3178:
3152:
2914:
2428: is embedded into the ionic lattice
3067:Journal of Zhejiang University Science A
2393:
1752:
941:and the condition of transparency (i.e.
3340:
2979:
2923:
1126:{\displaystyle \omega _{\mathrm {pe} }}
3361:
2878:Tonks, Lewi; Langmuir, Irving (1929).
2539: relatively to the ionic lattice
1094:{\displaystyle m^{*}=m_{\mathrm {e} }}
16:Rapid oscillations of electron density
3274:
3272:
2775:which is negative when the frequency
3202:
3200:
3198:
3148:
3146:
3106:
3104:
3056:
3054:
3006:
3004:
2966:Optical properties of nanostructures
2356: from above the effective mass
996:and above), the same condition here
2656:is described by the effective mass
1268:
105:
13:
3334:
3269:
2963:
2808:
2735:
2716:
2675:
2672:
2669:
2636:
2592:
2468:
2375:
2372:
2369:
2154:
2151:
2148:
2119:
2116:
2113:
1665:
1662:
1659:
1640:
1637:
1598:
1595:
1589:
1519:
1516:
1510:
1477:
1474:
1447:
1434:
1422:
1396:
1393:
1345:
1328:
1316:
1299:
1296:
1290:
1249:
1150:
1117:
1114:
1085:
980:
631:
584:
576:
556:
371:
332:
324:
321:
318:
280:
255:
252:
215:
207:
204:
201:
151:
132:
129:
14:
3385:
3195:
3143:
3101:
3051:
3001:
2817:{\displaystyle \omega _{\rm {p}}}
2645:{\displaystyle \omega _{\rm {p}}}
2477:{\displaystyle \omega _{\rm {p}}}
989:{\displaystyle \omega _{\rm {p}}}
30:), are rapid oscillations of the
3257:
1562:are only weakly modified by the
1258:{\displaystyle n_{\mathrm {e} }}
1159:{\displaystyle n_{\mathrm {e} }}
967:from a certain plasma frequency
719:
690:
638:
563:
466:{\displaystyle \varepsilon _{0}}
380:{\displaystyle n_{\mathrm {e} }}
2880:"Oscillations in ionized gases"
960:{\displaystyle \epsilon \geq 0}
3129:10.1016/j.ijengsci.2008.12.007
2972:
2957:
2871:
2844:Relativistic quantum chemistry
2006:
2000:
1857:
1851:
1029:plasmas, often encountered in
922:
916:
889:
883:
844:
838:
797:
791:
785:
779:
764:
758:
729:
723:
715:
709:
700:
694:
669:
663:
648:
642:
611:
605:
504:
498:
1:
3180:10.1088/1367-2630/10/4/043020
2986:Nova Science Publishers, Inc.
2865:
2384:{\displaystyle m_{\rm {eff}}}
2128:{\displaystyle m_{\rm {eff}}}
810:which is always true only if
58:of these oscillations is the
3341:Longair, Malcolm S. (1998),
2968:. Pan Stanford. p. 201.
85:
34:in conducting media such as
7:
2827:
2349:{\displaystyle \omega _{0}}
1963: to a shell with mass
1814: to a shell with mass
1015:{\displaystyle \epsilon =0}
10:
3390:
1707:. Consequently, the large-
475:permittivity of free space
2849:Surface plasmon resonance
549:the continuity equation:
2984:, Huntington, New York:
2854:Upper hybrid oscillation
2391: will be negative.
1699:so the plasma waves can
1273:When the effects of the
1101:, the plasma frequency,
3079:10.1631/jzus.2006.A0024
2980:Andreev, A. A. (2000),
2788:{\displaystyle \omega }
2505:{\displaystyle \omega }
2322:{\displaystyle \omega }
94:and negatively charged
3159:New Journal of Physics
3037:10.1098/rspa.2006.1795
2907:10.1103/PhysRev.33.195
2818:
2789:
2769:
2646:
2617:
2560:
2533:
2506:
2485:
2478:
2449:
2422:
2385:
2350:
2323:
2303:
2244:
2129:
2094:
2067:
2040:
1984:
1957:
1930:
1898:
1891:
1835:
1808:
1781:
1693:
1608:
1537:
1357:
1259:
1228:
1160:
1127:
1095:
1037:is independent of the
1016:
990:
961:
935:
863:
804:
736:
676:
618:
543:
477:. Note that the above
467:
436:
405:
381:
344:
227:
2819:
2790:
2770:
2647:
2618:
2561:
2559:{\displaystyle m_{1}}
2534:
2532:{\displaystyle m_{2}}
2507:
2479:
2450:
2448:{\displaystyle m_{1}}
2423:
2421:{\displaystyle m_{2}}
2397:
2386:
2351:
2324:
2309:. When the frequency
2304:
2245:
2130:
2095:
2093:{\displaystyle m_{2}}
2068:
2066:{\displaystyle m_{1}}
2041:
1985:
1983:{\displaystyle m_{1}}
1958:
1956:{\displaystyle k_{2}}
1931:
1929:{\displaystyle m_{2}}
1892:
1836:
1834:{\displaystyle m_{1}}
1809:
1807:{\displaystyle k_{2}}
1782:
1780:{\displaystyle m_{2}}
1756:
1694:
1609:
1538:
1358:
1260:
1229:
1161:
1137:and electron density
1128:
1096:
1017:
991:
962:
936:
864:
805:
737:
682:and the conductivity
677:
619:
544:
481:is derived under the
468:
446:of the electron, and
437:
435:{\displaystyle m^{*}}
406:
382:
345:
228:
3345:, Berlin: Springer,
2799:
2779:
2660:
2627:
2570:
2543:
2516:
2496:
2459:
2432:
2405:
2360:
2333:
2313:
2254:
2139:
2104:
2077:
2050:
1994:
1967:
1940:
1913:
1845:
1818:
1791:
1764:
1733:, the effect of the
1622:
1570:
1371:
1281:
1240:
1170:
1141:
1105:
1063:
1000:
971:
945:
877:
814:
746:
686:
628:
553:
492:
450:
419:
395:
362:
243:
120:
3297:2020Mate...13.3512B
3225:2020Mate...13.1890B
3171:2008NJPh...10d3020Y
3029:2007RSPSA.463..855M
2940:Solid State Physics
2899:1929PhRv...33..195T
2745:
2726:
2602:
2401:Free electrons gas
2220:
2203:
1909:. A core with mass
1713:dispersion relation
1552:dispersion relation
1529:
1487:
1406:
54:resulting from the
20:Plasma oscillations
3306:10.3390/ma13163512
3234:10.3390/ma13081890
2814:
2785:
2765:
2729:
2710:
2642:
2613:
2586:
2556:
2529:
2502:
2486:
2474:
2445:
2418:
2381:
2346:
2319:
2299:
2240:
2206:
2189:
2125:
2090:
2063:
2036:
1980:
1953:
1926:
1899:
1887:
1831:
1804:
1777:
1689:
1604:
1533:
1504:
1468:
1387:
1353:
1255:
1224:
1156:
1135:physical constants
1133:, depends only on
1123:
1091:
1012:
986:
957:
931:
859:
800:
732:
672:
614:
539:
463:
432:
401:
377:
340:
223:
3352:978-3-540-63785-1
3023:(2079): 855β880.
2995:978-1-56072-803-0
2964:Fu, Ying (2011).
2950:978-0-03-083993-1
2795: approaches
2760:
2329: approaches
2297:
2235:
2135: we obtain:
2021:
1872:
1684:
1675:
1670:
1648:
1578:
1453:
1351:
1218:
1210:
1206:
1195:
1180:
929:
851:
591:
404:{\displaystyle e}
308:
307:
191:
190:
80:Jeans instability
3381:
3369:Waves in plasmas
3355:
3343:Galaxy Formation
3329:
3328:
3318:
3308:
3276:
3267:
3261:
3256:
3246:
3236:
3204:
3193:
3192:
3182:
3150:
3141:
3140:
3108:
3099:
3098:
3058:
3049:
3048:
3008:
2999:
2998:
2976:
2970:
2969:
2961:
2955:
2954:
2935:Mermin, N. David
2927:
2921:
2920:
2918:
2884:
2875:
2860:Waves in plasmas
2823:
2821:
2820:
2815:
2813:
2812:
2811:
2794:
2792:
2791:
2786:
2774:
2772:
2771:
2766:
2761:
2759:
2758:
2757:
2744:
2739:
2738:
2727:
2725:
2720:
2719:
2709:
2708:
2698:
2693:
2692:
2680:
2679:
2678:
2651:
2649:
2648:
2643:
2641:
2640:
2639:
2622:
2620:
2619:
2614:
2612:
2611:
2601:
2596:
2595:
2582:
2581:
2565:
2563:
2562:
2557:
2555:
2554:
2538:
2536:
2535:
2530:
2528:
2527:
2511:
2509:
2508:
2503:
2483:
2481:
2480:
2475:
2473:
2472:
2471:
2454:
2452:
2451:
2446:
2444:
2443:
2427:
2425:
2424:
2419:
2417:
2416:
2390:
2388:
2387:
2382:
2380:
2379:
2378:
2355:
2353:
2352:
2347:
2345:
2344:
2328:
2326:
2325:
2320:
2308:
2306:
2305:
2300:
2298:
2296:
2295:
2286:
2281:
2280:
2271:
2266:
2265:
2249:
2247:
2246:
2241:
2236:
2234:
2233:
2232:
2219:
2214:
2204:
2202:
2197:
2188:
2187:
2177:
2172:
2171:
2159:
2158:
2157:
2134:
2132:
2131:
2126:
2124:
2123:
2122:
2099:
2097:
2096:
2091:
2089:
2088:
2072:
2070:
2069:
2064:
2062:
2061:
2045:
2043:
2042:
2037:
2023:
2022:
2014:
1989:
1987:
1986:
1981:
1979:
1978:
1962:
1960:
1959:
1954:
1952:
1951:
1935:
1933:
1932:
1927:
1925:
1924:
1896:
1894:
1893:
1888:
1874:
1873:
1865:
1840:
1838:
1837:
1832:
1830:
1829:
1813:
1811:
1810:
1805:
1803:
1802:
1786:
1784:
1783:
1778:
1776:
1775:
1698:
1696:
1695:
1690:
1685:
1677:
1673:
1672:
1671:
1669:
1668:
1656:
1651:
1646:
1645:
1644:
1643:
1613:
1611:
1610:
1605:
1603:
1602:
1601:
1579:
1574:
1542:
1540:
1539:
1534:
1528:
1523:
1522:
1503:
1502:
1486:
1481:
1480:
1464:
1463:
1454:
1452:
1451:
1450:
1440:
1439:
1438:
1437:
1427:
1426:
1425:
1411:
1405:
1400:
1399:
1383:
1382:
1362:
1360:
1359:
1354:
1352:
1350:
1349:
1348:
1338:
1333:
1332:
1331:
1321:
1320:
1319:
1309:
1304:
1303:
1302:
1269:'Warm' electrons
1264:
1262:
1261:
1256:
1254:
1253:
1252:
1233:
1231:
1230:
1225:
1223:
1219:
1216:
1208:
1207:
1205:
1197:
1196:
1193:
1187:
1182:
1181:
1178:
1165:
1163:
1162:
1157:
1155:
1154:
1153:
1132:
1130:
1129:
1124:
1122:
1121:
1120:
1100:
1098:
1097:
1092:
1090:
1089:
1088:
1075:
1074:
1059:Note that, when
1021:
1019:
1018:
1013:
995:
993:
992:
987:
985:
984:
983:
966:
964:
963:
958:
940:
938:
937:
932:
930:
925:
902:
868:
866:
865:
860:
852:
847:
824:
809:
807:
806:
801:
741:
739:
738:
733:
722:
693:
681:
679:
678:
673:
641:
623:
621:
620:
615:
592:
590:
582:
574:
566:
548:
546:
545:
540:
538:
537:
519:
518:
472:
470:
469:
464:
462:
461:
441:
439:
438:
433:
431:
430:
410:
408:
407:
402:
386:
384:
383:
378:
376:
375:
374:
349:
347:
346:
341:
339:
335:
331:
309:
306:
305:
296:
295:
294:
285:
284:
283:
266:
265:
260:
259:
258:
232:
230:
229:
224:
222:
218:
214:
192:
189:
188:
187:
178:
177:
167:
166:
165:
156:
155:
154:
143:
142:
137:
136:
135:
112:plasma frequency
106:'Cold' electrons
32:electron density
22:, also known as
3389:
3388:
3384:
3383:
3382:
3380:
3379:
3378:
3359:
3358:
3353:
3337:
3335:Further reading
3332:
3277:
3270:
3205:
3196:
3151:
3144:
3109:
3102:
3059:
3052:
3009:
3002:
2996:
2977:
2973:
2962:
2958:
2951:
2928:
2924:
2887:Physical Review
2882:
2876:
2872:
2868:
2830:
2807:
2806:
2802:
2800:
2797:
2796:
2780:
2777:
2776:
2753:
2749:
2740:
2734:
2733:
2728:
2721:
2715:
2714:
2704:
2700:
2699:
2697:
2688:
2684:
2668:
2667:
2663:
2661:
2658:
2657:
2635:
2634:
2630:
2628:
2625:
2624:
2607:
2603:
2597:
2591:
2590:
2577:
2573:
2571:
2568:
2567:
2550:
2546:
2544:
2541:
2540:
2523:
2519:
2517:
2514:
2513:
2497:
2494:
2493:
2467:
2466:
2462:
2460:
2457:
2456:
2439:
2435:
2433:
2430:
2429:
2412:
2408:
2406:
2403:
2402:
2368:
2367:
2363:
2361:
2358:
2357:
2340:
2336:
2334:
2331:
2330:
2314:
2311:
2310:
2291:
2287:
2282:
2276:
2272:
2270:
2261:
2257:
2255:
2252:
2251:
2228:
2224:
2215:
2210:
2205:
2198:
2193:
2183:
2179:
2178:
2176:
2167:
2163:
2147:
2146:
2142:
2140:
2137:
2136:
2112:
2111:
2107:
2105:
2102:
2101:
2084:
2080:
2078:
2075:
2074:
2057:
2053:
2051:
2048:
2047:
2013:
2012:
1995:
1992:
1991:
1974:
1970:
1968:
1965:
1964:
1947:
1943:
1941:
1938:
1937:
1920:
1916:
1914:
1911:
1910:
1864:
1863:
1846:
1843:
1842:
1825:
1821:
1819:
1816:
1815:
1798:
1794:
1792:
1789:
1788:
1771:
1767:
1765:
1762:
1761:
1760:Core with mass
1751:
1711:portion in the
1676:
1658:
1657:
1652:
1650:
1649:
1636:
1635:
1631:
1623:
1620:
1619:
1588:
1587:
1583:
1573:
1571:
1568:
1567:
1524:
1509:
1508:
1498:
1494:
1482:
1473:
1472:
1459:
1455:
1446:
1445:
1441:
1433:
1432:
1428:
1421:
1420:
1416:
1412:
1410:
1401:
1392:
1391:
1378:
1374:
1372:
1369:
1368:
1344:
1343:
1339:
1334:
1327:
1326:
1322:
1315:
1314:
1310:
1308:
1289:
1288:
1284:
1282:
1279:
1278:
1271:
1248:
1247:
1243:
1241:
1238:
1237:
1215:
1211:
1198:
1192:
1188:
1186:
1177:
1173:
1171:
1168:
1167:
1149:
1148:
1144:
1142:
1139:
1138:
1113:
1112:
1108:
1106:
1103:
1102:
1084:
1083:
1079:
1070:
1066:
1064:
1061:
1060:
1001:
998:
997:
979:
978:
974:
972:
969:
968:
946:
943:
942:
903:
901:
878:
875:
874:
825:
823:
815:
812:
811:
747:
744:
743:
718:
689:
687:
684:
683:
637:
629:
626:
625:
583:
575:
573:
562:
554:
551:
550:
524:
520:
514:
510:
493:
490:
489:
457:
453:
451:
448:
447:
426:
422:
420:
417:
416:
413:electric charge
396:
393:
392:
370:
369:
365:
363:
360:
359:
327:
317:
313:
301:
297:
290:
286:
279:
278:
274:
267:
264:
251:
250:
246:
244:
241:
240:
210:
200:
196:
183:
179:
173:
169:
168:
161:
157:
150:
149:
145:
144:
141:
128:
127:
123:
121:
118:
117:
108:
88:
72:Irving Langmuir
28:Irving Langmuir
17:
12:
11:
5:
3387:
3377:
3376:
3371:
3357:
3356:
3351:
3336:
3333:
3331:
3330:
3268:
3194:
3142:
3123:(4): 610β617.
3100:
3050:
3000:
2994:
2971:
2956:
2949:
2931:Ashcroft, Neil
2922:
2893:(8): 195β210.
2869:
2867:
2864:
2863:
2862:
2857:
2851:
2846:
2841:
2836:
2829:
2826:
2810:
2805:
2784:
2764:
2756:
2752:
2748:
2743:
2737:
2732:
2724:
2718:
2713:
2707:
2703:
2696:
2691:
2687:
2683:
2677:
2674:
2671:
2666:
2638:
2633:
2610:
2606:
2600:
2594:
2589:
2585:
2580:
2576:
2553:
2549:
2526:
2522:
2501:
2470:
2465:
2442:
2438:
2415:
2411:
2377:
2374:
2371:
2366:
2343:
2339:
2318:
2294:
2290:
2285:
2279:
2275:
2269:
2264:
2260:
2239:
2231:
2227:
2223:
2218:
2213:
2209:
2201:
2196:
2192:
2186:
2182:
2175:
2170:
2166:
2162:
2156:
2153:
2150:
2145:
2121:
2118:
2115:
2110:
2087:
2083:
2060:
2056:
2035:
2032:
2029:
2026:
2020:
2017:
2011:
2008:
2005:
2002:
1999:
1977:
1973:
1950:
1946:
1923:
1919:
1886:
1883:
1880:
1877:
1871:
1868:
1862:
1859:
1856:
1853:
1850:
1828:
1824:
1801:
1797:
1774:
1770:
1750:
1747:
1739:effective mass
1705:Landau damping
1688:
1683:
1680:
1667:
1664:
1661:
1655:
1642:
1639:
1634:
1630:
1627:
1616:phase velocity
1600:
1597:
1594:
1591:
1586:
1582:
1577:
1532:
1527:
1521:
1518:
1515:
1512:
1507:
1501:
1497:
1493:
1490:
1485:
1479:
1476:
1471:
1467:
1462:
1458:
1449:
1444:
1436:
1431:
1424:
1419:
1415:
1409:
1404:
1398:
1395:
1390:
1386:
1381:
1377:
1347:
1342:
1337:
1330:
1325:
1318:
1313:
1307:
1301:
1298:
1295:
1292:
1287:
1277:thermal speed
1270:
1267:
1251:
1246:
1222:
1214:
1204:
1201:
1191:
1185:
1176:
1152:
1147:
1119:
1116:
1111:
1087:
1082:
1078:
1073:
1069:
1054:group velocity
1050:phase velocity
1011:
1008:
1005:
982:
977:
956:
953:
950:
928:
924:
921:
918:
915:
912:
909:
906:
900:
897:
894:
891:
888:
885:
882:
858:
855:
850:
846:
843:
840:
837:
834:
831:
828:
822:
819:
799:
796:
793:
790:
787:
784:
781:
778:
775:
772:
769:
766:
763:
760:
757:
754:
751:
731:
728:
725:
721:
717:
714:
711:
708:
705:
702:
699:
696:
692:
671:
668:
665:
662:
659:
656:
653:
650:
647:
644:
640:
636:
633:
624:the Gauss law
613:
610:
607:
604:
601:
598:
595:
589:
586:
581:
578:
572:
569:
565:
561:
558:
536:
533:
530:
527:
523:
517:
513:
509:
506:
503:
500:
497:
460:
456:
444:effective mass
429:
425:
400:
391:of electrons,
389:number density
373:
368:
356:
355:
338:
334:
330:
326:
323:
320:
316:
312:
304:
300:
293:
289:
282:
277:
273:
270:
263:
257:
254:
249:
238:
221:
217:
213:
209:
206:
203:
199:
195:
186:
182:
176:
172:
164:
160:
153:
148:
140:
134:
131:
126:
107:
104:
87:
84:
24:Langmuir waves
15:
9:
6:
4:
3:
2:
3386:
3375:
3372:
3370:
3367:
3366:
3364:
3354:
3348:
3344:
3339:
3338:
3326:
3322:
3317:
3312:
3307:
3302:
3298:
3294:
3290:
3286:
3282:
3275:
3273:
3265:
3260:
3254:
3250:
3245:
3240:
3235:
3230:
3226:
3222:
3218:
3214:
3210:
3203:
3201:
3199:
3190:
3186:
3181:
3176:
3172:
3168:
3165:(4): 043020.
3164:
3160:
3156:
3149:
3147:
3138:
3134:
3130:
3126:
3122:
3118:
3114:
3107:
3105:
3096:
3092:
3088:
3084:
3080:
3076:
3072:
3068:
3064:
3057:
3055:
3046:
3042:
3038:
3034:
3030:
3026:
3022:
3018:
3014:
3007:
3005:
2997:
2991:
2987:
2983:
2975:
2967:
2960:
2952:
2946:
2942:
2941:
2936:
2932:
2926:
2917:
2912:
2908:
2904:
2900:
2896:
2892:
2888:
2881:
2874:
2870:
2861:
2858:
2855:
2852:
2850:
2847:
2845:
2842:
2840:
2837:
2835:
2834:Electron wake
2832:
2831:
2825:
2803:
2782:
2762:
2754:
2750:
2746:
2741:
2730:
2722:
2711:
2705:
2701:
2694:
2689:
2685:
2681:
2664:
2655:
2631:
2608:
2604:
2598:
2587:
2583:
2578:
2574:
2551:
2547:
2524:
2520:
2499:
2491:
2463:
2440:
2436:
2413:
2409:
2400:
2396:
2392:
2364:
2341:
2337:
2316:
2292:
2288:
2283:
2277:
2273:
2267:
2262:
2258:
2237:
2229:
2225:
2221:
2216:
2211:
2207:
2199:
2194:
2190:
2184:
2180:
2173:
2168:
2164:
2160:
2143:
2108:
2085:
2081:
2058:
2054:
2033:
2030:
2027:
2024:
2018:
2015:
2009:
2003:
1997:
1975:
1971:
1948:
1944:
1921:
1917:
1908:
1904:
1903:negative mass
1884:
1881:
1878:
1875:
1869:
1866:
1860:
1854:
1848:
1826:
1822:
1799:
1795:
1772:
1768:
1759:
1755:
1746:
1744:
1740:
1736:
1732:
1731:semiconductor
1728:
1723:
1721:
1716:
1714:
1710:
1706:
1702:
1686:
1681:
1678:
1653:
1632:
1628:
1625:
1617:
1592:
1584:
1580:
1575:
1565:
1561:
1557:
1553:
1550:
1546:
1530:
1525:
1513:
1505:
1499:
1495:
1491:
1488:
1483:
1469:
1465:
1460:
1456:
1442:
1429:
1417:
1413:
1407:
1402:
1388:
1384:
1379:
1375:
1366:
1340:
1335:
1323:
1311:
1305:
1293:
1285:
1276:
1266:
1244:
1234:
1220:
1212:
1202:
1199:
1189:
1183:
1174:
1145:
1136:
1109:
1080:
1076:
1071:
1067:
1057:
1055:
1051:
1048:
1044:
1040:
1036:
1032:
1028:
1023:
1009:
1006:
1003:
975:
954:
951:
948:
926:
919:
913:
910:
907:
904:
898:
895:
892:
886:
880:
872:
856:
853:
848:
841:
835:
832:
829:
826:
820:
817:
794:
788:
782:
776:
773:
770:
767:
761:
755:
752:
749:
726:
712:
706:
703:
697:
666:
660:
657:
654:
651:
645:
634:
608:
602:
599:
596:
593:
587:
579:
570:
567:
559:
534:
531:
528:
525:
521:
515:
511:
507:
501:
495:
486:
484:
483:approximation
480:
476:
458:
454:
445:
427:
423:
414:
398:
390:
366:
353:
336:
328:
314:
310:
302:
298:
291:
287:
275:
271:
268:
261:
247:
239:
236:
219:
211:
197:
193:
184:
180:
174:
170:
162:
158:
146:
138:
124:
116:
115:
114:
113:
103:
101:
100:Coulomb force
97:
93:
83:
81:
77:
73:
70:
65:
63:
62:
57:
53:
52:quasiparticle
49:
45:
41:
37:
33:
29:
25:
21:
3342:
3291:(16): 3512.
3288:
3284:
3216:
3212:
3162:
3158:
3120:
3116:
3073:(1): 24β28.
3070:
3066:
3020:
3016:
2981:
2974:
2965:
2959:
2938:
2925:
2890:
2886:
2873:
2653:
2489:
2487:
2398:
1906:
1900:
1757:
1742:
1741:in place of
1724:
1717:
1708:
1560:oscillations
1556:Debye length
1272:
1235:
1058:
1043:oscillations
1033:. Since the
1031:astrophysics
1024:
487:
357:
111:
109:
89:
66:
59:
56:quantization
23:
19:
18:
3219:(8): 1890.
1543:called the
871:Drude Model
44:ultraviolet
3374:Plasmonics
3363:Categories
2866:References
2073: and
1701:accelerate
1365:wavenumber
1039:wavelength
76:Lewi Tonks
69:physicists
3285:Materials
3213:Materials
3189:1367-2630
3137:0020-7225
3095:120899746
3087:1862-1775
3045:122990527
2804:ω
2783:ω
2751:ω
2747:−
2731:ω
2712:ω
2632:ω
2588:ω
2500:ω
2464:ω
2399:Figure 2.
2338:ω
2317:ω
2259:ω
2226:ω
2222:−
2208:ω
2191:ω
2031:ω
2028:
2019:^
1882:ω
1879:
1870:^
1758:Figure 1.
1679:ω
1629:∼
1618:, i.e.,
1581:⋅
1470:ω
1389:ω
1376:ω
1203:π
1190:ω
1110:ω
1072:∗
1052:and zero
1035:frequency
1004:ϵ
976:ω
952:≥
949:ϵ
927:ω
920:ω
914:σ
908:π
887:ω
881:ϵ
849:ω
842:ω
836:σ
830:π
795:ω
789:ρ
783:ω
777:σ
774:π
762:ω
756:ρ
753:ω
727:ω
713:ω
707:σ
698:ω
667:ω
661:ρ
658:π
646:ω
635:⋅
632:∇
609:ω
603:ρ
600:ω
585:∂
580:ρ
577:∂
571:−
560:⋅
557:∇
532:ω
526:−
512:ρ
502:ω
496:ρ
455:ε
428:∗
352:cgs units
303:∗
272:π
248:ω
181:ε
175:∗
125:ω
96:electrons
86:Mechanism
3325:32784869
3253:32316640
2937:(1976).
2828:See also
2623:, where
2490:Figure 2
1907:Figure 1
1564:pressure
1275:electron
1047:infinite
1045:have an
1041:, these
1027:positron
235:SI units
3316:7476018
3293:Bibcode
3244:7215794
3221:Bibcode
3167:Bibcode
3025:Bibcode
2916:1085653
2895:Bibcode
2839:Plasmon
1720:bounded
479:formula
473:is the
442:is the
411:is the
387:is the
61:plasmon
42:in the
36:plasmas
26:(after
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2250:where
1674:
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1209:
358:where
40:metals
3091:S2CID
3041:S2CID
2883:(PDF)
1727:metal
1725:In a
1718:In a
1549:Gross
3347:ISBN
3321:PMID
3249:PMID
3185:ISSN
3133:ISSN
3083:ISSN
2990:ISBN
2945:ISBN
1735:ions
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92:ions
74:and
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3301:doi
3239:PMC
3229:doi
3175:doi
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