287:
region around itself in which there are fewer electrons. This region can be treated as a positively charged "screening hole". Viewed from a large distance, this screening hole has the effect of an overlaid positive charge which cancels the electric field produced by the electron. Only at short distances, inside the hole region, can the electron's field be detected. For a plasma, this effect can be made explicit by an
1551:
1873:. It deals with a single realization of a one-component plasma whose electrons have a velocity dispersion (for a thermal plasma, there must be many particles in a Debye sphere, a volume whose radius is the Debye length). On using the linearized motion of the electrons in their own electric field, it yields an equation of the type
1556:
This result follows from the equations of a Fermi gas, which is a model of non-interacting electrons, whereas the fluid, which we are studying, contains the
Coulomb interaction. Therefore, the ThomasâFermi approximation is only valid when the electron density is low, so that the particle interactions
2230:
by the particle, and the other one is its screened potential, as classically obtained by a linearized
Vlasovian calculation involving a test particle. The screened potential is the above screened Coulomb potential for a thermal plasma and a thermal particle. For a faster particle, the potential is
1229:
372:
Consider a fluid of electrons in a background of heavy, positively charged ions. For simplicity, we ignore the motion and spatial distribution of the ions, approximating them as a uniform background charge. This simplification is permissible since the electrons are lighter and more mobile than the
286:
Consider a fluid composed of electrons moving in a uniform background of positive charge (one-component plasma). Each electron possesses a negative charge. According to
Coulomb's interaction, negative charges repel each other. Consequently, this electron will repel other electrons creating a small
210:
is the relative position between the charges. This interaction complicates the theoretical treatment of the fluid. For example, a naive quantum mechanical calculation of the ground-state energy density yields infinity, which is unreasonable. The difficulty lies in the fact that even though the
2276:
In real metals, the screening effect is more complex than described above in the ThomasâFermi theory. The assumption that the charge carriers (electrons) can respond at any wavevector is just an approximation. However, it is not energetically possible for an electron within or on a
1380:
598:. We consider two possible approximations, under which the two quantities are proportional: the DebyeâHĂŒckel approximation, valid at high temperatures (e.g. classical plasmas), and the ThomasâFermi approximation, valid at low temperatures (e.g. electrons in metals).
924:
202:
307:-body calculation. If the background is made up of positive ions, their attraction by the electron of interest reinforces the above screening mechanism. In atomic physics, a germane effect exists for atoms with more than one electron shell: the
2293:, and applies both to surface and bulk screening. In each case the net electric field does not fall off exponentially in space, but rather as an inverse power law multiplied by an oscillatory term. Theoretical calculations can be obtained from
2081:
1079:
1309:
1375:
819:
1758:
1668:
824:
108:
744:
574:
1834:
1546:{\displaystyle k_{0}\ {\stackrel {\mathrm {def} }{=}}\ {\sqrt {\frac {3e^{2}\rho }{2\varepsilon _{0}E_{\mathrm {F} }}}}={\sqrt {\frac {me^{2}k_{\mathrm {F} }}{\varepsilon _{0}\pi ^{2}\hbar ^{2}}}}}
1050:
2118:
2002:
2266:
2201:
1907:
1760:
which is called a screened
Coulomb potential. It is a Coulomb potential multiplied by an exponential damping term, with the strength of the damping factor given by the magnitude of
1241:
1975:
is the
Fourier-Laplace transform of the electrostatic potential. When substituting an integral over a smooth distribution function for the discrete sum over the particles in
2289:
for functions that vary rapidly in space are not good approximations unless a very large number of terms in the series are retained. In physics, this phenomenon is known as
1997:
1933:
2144:
2164:
2367:
Escande, D F; Elskens, Yves; Doveil, F (1 February 2015). "Direct path from microscopic mechanics to Debye shielding, Landau damping and wave-particle interaction".
1973:
2221:
1953:
1867:
305:
1320:
764:
1677:
239:
interaction between particles to a short-range "screened" Coulomb interaction. This system corresponds to the simplest example of a renormalized interaction.
1568:
1224:{\displaystyle \rho =2{\frac {1}{(2\pi )^{3}}}\left({\frac {4}{3}}\pi k_{\mathrm {F} }^{3}\right),\quad E_{\mathrm {F} }={\frac {\hbar ^{2}k_{F}^{2}}{2m}},}
311:. In plasma physics, electric-field screening is also called Debye screening or shielding. It manifests itself on macroscopic scales by a sheath (
627:
463:
982:) and at low temperature. The former condition corresponds, in a real experiment, to keeping the metal/fluid in electrical contact with a fixed
1005:
2424:
617:
2599:
2544:
2344:
2226:
By inverse
Fourier-Laplace transform, the potential due to each particle is the sum of two parts One corresponds to the excitation of
334:
2594:
2570:
235:
In reality, these long-range effects are suppressed by the flow of particles in response to electric fields. This flow reduces the
2472:
1777:
1876:
994:
is, by definition, the energy of adding an extra electron to the fluid. This energy may be decomposed into a kinetic energy
1565:
Our results from the DebyeâHĂŒckel or ThomasâFermi approximation may now be inserted into
Poisson's equation. The result is
275:
919:{\displaystyle k_{0}\ {\stackrel {\mathrm {def} }{=}}\ {\sqrt {\frac {\rho e^{2}}{\varepsilon _{0}k_{\mathrm {B} }T}}}}
197:{\displaystyle \mathbf {F} ={\frac {q_{1}q_{2}}{4\pi \varepsilon \left|\mathbf {r} \right|^{2}}}{\hat {\mathbf {r} }},}
2086:
957:
607:
346:
2231:
modified. Substituting an integral over a smooth distribution function for the discrete sum over the particles in
440:. After the system has returned to equilibrium, let the change in the electron density and electric potential be Î
2604:
2432:
2298:
1671:
326:
31:
374:
2234:
2169:
405:. At first, the electrons are evenly distributed so that there is zero net charge at every point. Therefore,
1076:. The Fermi energy for a 3D system is related to the density of electrons (including spin degeneracy) by
612:
In the DebyeâHĂŒckel approximation, we maintain the system in thermodynamic equilibrium, at a temperature
2294:
457:
2494:
2386:
2076:{\displaystyle \epsilon (\mathbf {k} ,\omega )\,\Phi (\mathbf {k} ,\omega )=S(\mathbf {k} ,\omega ),}
987:
983:
437:
1978:
1914:
232:. As a result, a charge fluctuation at any one point has non-negligible effects at large distances.
2565:
2290:
1772:
584:
342:
322:
243:
2127:
2510:
2484:
2402:
2376:
2339:(Reprinted with corrections, Reprinted ed.). Oxford: Oxford University Press. §1.2.1, §3.2.
2227:
975:
754:
402:
2281:
to respond at wavevectors shorter than the Fermi wavevector. This constraint is related to the
83:, composed of electrically charged constituent particles, each pair of particles (with charges
2540:
2340:
1304:{\displaystyle \Delta \rho \simeq {\frac {3\rho }{2E_{\mathrm {F} }}}\Delta E_{\mathrm {F} }.}
1055:
2441:
2149:
2502:
2394:
2282:
1768:
1066:
by the kinetic energy of an additional electron in the Fermi gas model, which is simply the
967:
308:
271:
102:
57:
1958:
1767:, the Debye or ThomasâFermi wave vector. Note that this potential has the same form as the
2121:
961:
330:
49:
2120:
is the plasma permittivity, or dielectric function, classically obtained by a linearized
2506:
2498:
2398:
2390:
17:
2310:
2286:
2206:
1938:
1870:
1852:
417:
394:
290:
274:, the electric fields of ions in conducting solids are further reduced by the cloud of
267:
259:
65:
45:
1054:
If the temperature is extremely low, the behavior of the electrons comes close to the
2588:
2514:
2278:
366:
251:
69:
2406:
2315:
1067:
971:
945:
312:
77:
2425:"The theory of electrolytes. I. Lowering of freezing point and related phenomena"
979:
362:
338:
337:, which explains the predictive power of introductory models of solids like the
61:
373:
ions, provided we consider distances much larger than the ionic separation. In
2268:, yields the Vlasovian expression enabling the calculation of Landau damping.
1059:
1869:-body approach provides together the derivation of screening effect and of
948:. The Debye length is the fundamental length scale of a classical plasma.
1370:{\displaystyle e\Delta \rho \simeq \varepsilon _{0}k_{0}^{2}\Delta \phi }
814:{\displaystyle e\Delta \rho \simeq \varepsilon _{0}k_{0}^{2}\Delta \phi }
456:) respectively. The charge density and electric potential are related by
229:
1753:{\displaystyle \phi (r)={\frac {Q}{4\pi \varepsilon _{0}r}}e^{-k_{0}r},}
1663:{\displaystyle \left\phi (r)=-{\frac {Q}{\varepsilon _{0}}}\delta (r),}
378:
37:
319:
52:
carriers. It is an important part of the behavior of charge-carrying
2489:
2381:
247:
73:
53:
1238:
is the Fermi wavevector. Perturbing to first order, we find that
318:
The screened potential determines the inter atomic force and the
590:
To proceed, we must find a second independent equation relating
620:. At each point in space, the density of electrons with energy
263:
1984:
1920:
1882:
369:, dealt with a stationary point charge embedded in a fluid.
2473:"N -body description of Debye shielding and Landau damping"
739:{\displaystyle \rho _{j}(r)=\rho _{j}^{(0)}(r)\;\exp \left}
569:{\displaystyle -\nabla ^{2}={\frac {1}{\varepsilon _{0}}},}
329:
of a large variety of materials, often in combination with
325:
in metals. The screened potential is used to calculate the
1829:{\displaystyle \varepsilon (r)=\varepsilon _{0}e^{k_{0}r}}
315:) next to a material with which the plasma is in contact.
761:
and expanding the exponential to first order, we obtain
2237:
2209:
2172:
2152:
2130:
2089:
2005:
1981:
1961:
1941:
1917:
1879:
1855:
1780:
1680:
1571:
1383:
1323:
1244:
1082:
1045:{\displaystyle \Delta \mu =\Delta T-e\Delta \phi =0.}
1008:
1002:
part. Since the chemical potential is kept constant,
827:
767:
630:
466:
293:
111:
218:, the average number of particles at each distance
2260:
2215:
2195:
2158:
2138:
2112:
2075:
1991:
1967:
1947:
1927:
1901:
1861:
1828:
1752:
1662:
1553:is called the ThomasâFermi screening wave vector.
1545:
1369:
1303:
1223:
1044:
974:, the system is maintained at a constant electron
918:
813:
738:
568:
299:
196:
266:inside the solid. Like the electric field of the
2471:Escande, D F; Doveil, F; Elskens, Yves (2016).
2113:{\displaystyle \epsilon (\mathbf {k} ,\omega )}
966:In the ThomasâFermi approximation, named after
2337:Renormalization methods: a guide for beginners
8:
1313:Inserting this into the above equation for Î
270:is reduced inside an atom or ion due to the
1955:is a source term due to the particles, and
683:
616:high enough that the fluid particles obey
333:models. The screening effect leads to the
211:Coulomb force diminishes with distance as
2488:
2380:
2244:
2236:
2208:
2179:
2171:
2151:
2131:
2129:
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2088:
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2012:
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1919:
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1916:
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1815:
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1736:
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1007:
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629:
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292:
180:
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137:
127:
120:
112:
110:
2327:
2261:{\displaystyle S(\mathbf {k} ,\omega )}
2196:{\displaystyle S(\mathbf {k} ,\omega )}
1530:
1183:
1902:{\displaystyle {\mathcal {E}}\Phi =S,}
412:We now introduce a fixed point charge
1845:Classical physics and linear response
7:
2530:
2528:
2526:
2524:
2477:Plasma Physics and Controlled Fusion
2418:
2416:
2369:Plasma Physics and Controlled Fusion
2362:
2360:
2358:
2356:
2564:Fitzpatrick, Richard (2011-03-31).
2223:source terms due to the particles.
357:The first theoretical treatment of
56:, such as ionized gases (classical
2027:
1962:
1887:
1578:
1501:
1465:
1413:
1410:
1407:
1361:
1327:
1292:
1283:
1274:
1245:
1170:
1144:
1030:
1018:
1009:
903:
857:
854:
851:
805:
771:
720:
545:
483:
471:
335:independent electron approximation
25:
2571:The University of Texas at Austin
2461:(Thomson Learning, Toronto, 1976)
2457:N. W. Ashcroft and N. D. Mermin,
409:is initially a constant as well.
48:caused by the presence of mobile
2245:
2180:
2132:
2097:
2057:
2034:
2013:
181:
160:
113:
2423:P. Debye and E. HĂŒckel (1923).
1163:
998:part and the potential energy â
377:, this model is referred to as
228:, assuming the fluid is fairly
2255:
2241:
2190:
2176:
2107:
2093:
2067:
2053:
2044:
2030:
2023:
2009:
1992:{\displaystyle {\mathcal {E}}}
1928:{\displaystyle {\mathcal {E}}}
1790:
1784:
1690:
1684:
1654:
1648:
1619:
1613:
1108:
1098:
709:
703:
680:
674:
669:
663:
647:
641:
560:
557:
551:
536:
530:
521:
498:
495:
489:
480:
416:at the origin. The associated
185:
1:
2537:Introduction to Plasma Theory
2507:10.1088/0741-3335/58/1/014040
2399:10.1088/0741-3335/57/2/025017
385:Screened Coulomb interactions
2139:{\displaystyle \mathbf {k} }
618:MaxwellâBoltzmann statistics
262:and Coulomb potential of an
76:). In a fluid, with a given
2272:Quantum-mechanical approach
2621:
1771:. This screening yields a
1561:Result: Screened potential
955:
952:ThomasâFermi approximation
605:
602:DebyeâHĂŒckel approximation
347:nearly free electron model
68:in electronic conductors (
29:
27:Damping of electric fields
2600:Electromagnetism concepts
2535:Nicholson, D. R. (1983).
2440:: 185â206. Archived from
2433:Physikalische Zeitschrift
2299:density functional theory
1672:screened Poisson equation
990:. The chemical potential
327:electronic band structure
32:Electromagnetic shielding
2595:Condensed matter physics
2539:. New York: John Wiley.
375:condensed matter physics
359:electrostatic screening,
18:Electric field screening
2159:{\displaystyle \omega }
2122:Vlasov-Poisson equation
101:) interact through the
2262:
2217:
2197:
2166:is the frequency, and
2160:
2140:
2114:
2077:
1993:
1969:
1949:
1935:is a linear operator,
1929:
1903:
1863:
1830:
1754:
1670:which is known as the
1664:
1547:
1371:
1305:
1225:
1062:. We thus approximate
1046:
958:ThomasâFermi screening
928:The associated length
920:
815:
740:
570:
301:
198:
2335:McComb, W.D. (2007).
2295:quantum hydrodynamics
2263:
2218:
2198:
2161:
2141:
2115:
2078:
1994:
1970:
1968:{\displaystyle \Phi }
1950:
1930:
1904:
1864:
1831:
1755:
1665:
1557:are relatively weak.
1548:
1372:
1306:
1226:
1047:
921:
816:
741:
571:
302:
199:
2291:Friedel oscillations
2235:
2207:
2170:
2150:
2146:is the wave vector,
2128:
2087:
2003:
1979:
1959:
1939:
1915:
1877:
1853:
1778:
1678:
1569:
1381:
1321:
1242:
1080:
1006:
984:potential difference
825:
765:
628:
464:
438:Dirac delta function
291:
276:conduction electrons
109:
2499:2016PPCF...58a4040E
2459:Solid State Physics
2391:2015PPCF...57b5017E
1773:dielectric function
1604:
1360:
1206:
1154:
804:
673:
608:DebyeâHĂŒckel theory
585:vacuum permittivity
343:free electron model
323:dispersion relation
260:electrostatic field
244:solid-state physics
222:is proportional to
2258:
2213:
2193:
2156:
2136:
2110:
2073:
1989:
1965:
1945:
1925:
1899:
1859:
1826:
1750:
1674:. The solution is
1660:
1590:
1543:
1367:
1346:
1301:
1221:
1192:
1138:
1056:quantum mechanical
1042:
976:chemical potential
916:
811:
790:
755:Boltzmann constant
736:
653:
566:
458:Poisson's equation
403:electric potential
397:of electrons, and
297:
194:
44:is the damping of
2566:"Debye Shielding"
2216:{\displaystyle N}
1948:{\displaystyle S}
1862:{\displaystyle N}
1722:
1643:
1541:
1540:
1473:
1472:
1423:
1418:
1396:
1281:
1216:
1133:
1118:
914:
913:
867:
862:
840:
730:
519:
353:Theory and models
300:{\displaystyle N}
246:, especially for
204:where the vector
188:
175:
16:(Redirected from
2612:
2605:Plasma phenomena
2581:
2579:
2578:
2551:
2550:
2532:
2519:
2518:
2492:
2468:
2462:
2455:
2449:
2448:
2446:
2429:
2420:
2411:
2410:
2384:
2364:
2351:
2350:
2332:
2283:Gibbs phenomenon
2267:
2265:
2264:
2259:
2248:
2222:
2220:
2219:
2214:
2202:
2200:
2199:
2194:
2183:
2165:
2163:
2162:
2157:
2145:
2143:
2142:
2137:
2135:
2119:
2117:
2116:
2111:
2100:
2082:
2080:
2079:
2074:
2060:
2037:
2016:
1998:
1996:
1995:
1990:
1988:
1987:
1974:
1972:
1971:
1966:
1954:
1952:
1951:
1946:
1934:
1932:
1931:
1926:
1924:
1923:
1908:
1906:
1905:
1900:
1886:
1885:
1868:
1866:
1865:
1860:
1840:Many-body theory
1835:
1833:
1832:
1827:
1825:
1824:
1820:
1819:
1805:
1804:
1769:Yukawa potential
1759:
1757:
1756:
1751:
1746:
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1723:
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1134:
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1119:
1117:
1116:
1115:
1093:
1051:
1049:
1048:
1043:
968:Llewellyn Thomas
943:
925:
923:
922:
917:
915:
912:
908:
907:
906:
896:
895:
885:
884:
883:
870:
869:
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861:
860:
848:
843:
838:
837:
836:
820:
818:
817:
812:
803:
798:
789:
788:
757:. Perturbing in
745:
743:
742:
737:
735:
731:
729:
725:
724:
723:
712:
695:
672:
661:
640:
639:
575:
573:
572:
567:
520:
518:
517:
505:
479:
478:
309:shielding effect
306:
304:
303:
298:
272:shielding effect
256:screening effect
227:
221:
217:
209:
203:
201:
200:
195:
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100:
91:
82:
21:
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2414:
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2205:
2204:
2168:
2167:
2148:
2147:
2126:
2125:
2085:
2084:
2001:
2000:
1977:
1976:
1957:
1956:
1937:
1936:
1913:
1912:
1875:
1874:
1851:
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1708:
1701:
1676:
1675:
1633:
1577:
1576:
1572:
1567:
1566:
1563:
1529:
1519:
1509:
1508:
1495:
1485:
1481:
1459:
1449:
1445:
1431:
1427:
1384:
1379:
1378:
1336:
1319:
1318:
1286:
1268:
1264:
1256:
1240:
1239:
1237:
1208:
1182:
1181:
1164:
1124:
1120:
1107:
1097:
1078:
1077:
1075:
1004:
1003:
964:
962:Lindhard theory
956:Main articles:
954:
942:
935:
929:
897:
887:
886:
875:
871:
828:
823:
822:
780:
763:
762:
752:
714:
713:
696:
690:
631:
626:
625:
610:
604:
582:
509:
470:
462:
461:
387:
355:
331:pseudopotential
289:
288:
284:
223:
219:
212:
205:
155:
154:
144:
133:
123:
122:
107:
106:
99:
93:
90:
84:
80:
66:charge carriers
46:electric fields
34:
28:
23:
22:
15:
12:
11:
5:
2618:
2616:
2608:
2607:
2602:
2597:
2587:
2586:
2583:
2582:
2559:
2558:External links
2556:
2553:
2552:
2546:978-0471090458
2545:
2520:
2463:
2450:
2447:on 2013-11-02.
2412:
2352:
2346:978-0199236527
2345:
2326:
2325:
2323:
2320:
2319:
2318:
2313:
2311:Bjerrum length
2306:
2303:
2287:Fourier series
2273:
2270:
2257:
2254:
2251:
2247:
2243:
2240:
2228:Langmuir waves
2212:
2203:is the sum of
2192:
2189:
2186:
2182:
2178:
2175:
2155:
2134:
2109:
2106:
2103:
2099:
2095:
2092:
2072:
2069:
2066:
2063:
2059:
2055:
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2049:
2046:
2043:
2040:
2036:
2032:
2029:
2025:
2022:
2019:
2015:
2011:
2008:
1986:
1964:
1944:
1922:
1898:
1895:
1892:
1889:
1884:
1871:Landau damping
1858:
1846:
1843:
1841:
1838:
1823:
1818:
1814:
1809:
1803:
1799:
1795:
1792:
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1735:
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1602:
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1589:
1584:
1580:
1575:
1562:
1559:
1536:
1532:
1526:
1522:
1516:
1512:
1503:
1498:
1492:
1488:
1484:
1477:
1467:
1462:
1456:
1452:
1448:
1443:
1438:
1434:
1430:
1415:
1412:
1409:
1403:
1391:
1387:
1366:
1363:
1358:
1353:
1349:
1343:
1339:
1335:
1332:
1329:
1326:
1300:
1294:
1289:
1285:
1276:
1271:
1267:
1262:
1259:
1253:
1250:
1247:
1235:
1220:
1214:
1211:
1204:
1199:
1195:
1189:
1185:
1178:
1172:
1167:
1162:
1158:
1152:
1146:
1141:
1137:
1132:
1129:
1123:
1114:
1110:
1106:
1103:
1100:
1096:
1091:
1088:
1085:
1073:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
953:
950:
944:is called the
940:
933:
911:
905:
900:
894:
890:
882:
878:
874:
859:
856:
853:
847:
835:
831:
810:
807:
802:
797:
793:
787:
783:
779:
776:
773:
770:
750:
734:
728:
722:
717:
711:
708:
705:
702:
699:
693:
689:
686:
682:
679:
676:
671:
668:
665:
660:
656:
652:
649:
646:
643:
638:
634:
606:Main article:
603:
600:
580:
565:
562:
559:
556:
553:
550:
547:
544:
541:
538:
535:
532:
529:
526:
523:
516:
512:
508:
503:
500:
497:
494:
491:
488:
485:
482:
477:
473:
469:
460:, which gives
418:charge density
395:number density
386:
383:
354:
351:
296:
283:
280:
258:describes the
252:semiconductors
193:
187:
183:
171:
166:
162:
158:
153:
150:
147:
140:
136:
130:
126:
119:
115:
97:
88:
70:semiconductors
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2617:
2606:
2603:
2601:
2598:
2596:
2593:
2592:
2590:
2573:
2572:
2567:
2562:
2561:
2557:
2548:
2542:
2538:
2531:
2529:
2527:
2525:
2521:
2516:
2512:
2508:
2504:
2500:
2496:
2491:
2486:
2483:(1): 014040.
2482:
2478:
2474:
2467:
2464:
2460:
2454:
2451:
2443:
2439:
2435:
2434:
2426:
2419:
2417:
2413:
2408:
2404:
2400:
2396:
2392:
2388:
2383:
2378:
2375:(2): 025017.
2374:
2370:
2363:
2361:
2359:
2357:
2353:
2348:
2342:
2338:
2331:
2328:
2321:
2317:
2314:
2312:
2309:
2308:
2304:
2302:
2300:
2296:
2292:
2288:
2284:
2280:
2279:Fermi surface
2271:
2269:
2252:
2249:
2238:
2229:
2224:
2210:
2187:
2184:
2173:
2153:
2123:
2104:
2101:
2090:
2070:
2064:
2061:
2050:
2047:
2041:
2038:
2020:
2017:
2006:
1942:
1909:
1896:
1893:
1890:
1872:
1856:
1849:A mechanical
1844:
1839:
1837:
1821:
1816:
1812:
1807:
1801:
1797:
1793:
1787:
1781:
1774:
1770:
1763:
1747:
1742:
1737:
1733:
1729:
1725:
1718:
1713:
1709:
1705:
1702:
1698:
1693:
1687:
1681:
1673:
1657:
1651:
1645:
1638:
1634:
1630:
1625:
1622:
1616:
1610:
1606:
1600:
1595:
1591:
1587:
1582:
1573:
1560:
1558:
1554:
1534:
1524:
1520:
1514:
1510:
1496:
1490:
1486:
1482:
1475:
1460:
1454:
1450:
1446:
1441:
1436:
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1428:
1401:
1389:
1385:
1364:
1356:
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1347:
1341:
1337:
1333:
1330:
1324:
1316:
1311:
1298:
1287:
1269:
1265:
1260:
1257:
1251:
1248:
1234:
1218:
1212:
1209:
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1197:
1193:
1187:
1176:
1165:
1160:
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1150:
1139:
1135:
1130:
1127:
1121:
1112:
1104:
1101:
1094:
1089:
1086:
1083:
1072:
1069:
1065:
1061:
1057:
1052:
1039:
1036:
1033:
1027:
1024:
1021:
1015:
1012:
1001:
997:
993:
989:
985:
981:
977:
973:
969:
963:
959:
951:
949:
947:
939:
932:
926:
909:
898:
892:
888:
880:
876:
872:
845:
833:
829:
808:
800:
795:
791:
785:
781:
777:
774:
768:
760:
756:
749:
732:
726:
715:
706:
700:
697:
691:
687:
684:
677:
666:
658:
654:
650:
644:
636:
632:
624:has the form
623:
619:
615:
609:
601:
599:
597:
593:
588:
586:
579:
563:
554:
548:
542:
539:
533:
527:
524:
514:
510:
506:
501:
492:
486:
475:
467:
459:
455:
451:
447:
443:
439:
435:
431:
427:
423:
419:
415:
410:
408:
404:
400:
396:
392:
384:
382:
380:
376:
370:
368:
364:
360:
352:
350:
348:
344:
340:
336:
332:
328:
324:
321:
316:
314:
310:
294:
281:
279:
277:
273:
269:
265:
261:
257:
253:
249:
245:
240:
238:
233:
231:
226:
216:
208:
191:
169:
164:
156:
151:
148:
145:
138:
134:
128:
124:
117:
104:
103:Coulomb force
96:
87:
79:
75:
71:
67:
63:
59:
55:
51:
47:
43:
39:
33:
19:
2575:. Retrieved
2569:
2536:
2480:
2476:
2466:
2458:
2453:
2442:the original
2437:
2431:
2372:
2368:
2336:
2330:
2316:Debye length
2275:
2225:
1910:
1848:
1761:
1564:
1555:
1314:
1312:
1232:
1070:
1068:Fermi energy
1063:
1053:
999:
995:
991:
972:Enrico Fermi
965:
946:Debye length
937:
930:
927:
758:
747:
621:
613:
611:
595:
591:
589:
577:
453:
449:
445:
441:
433:
429:
425:
421:
413:
411:
406:
398:
390:
388:
371:
367:Erich HĂŒckel
358:
356:
317:
313:Debye sheath
285:
255:
241:
236:
234:
224:
214:
206:
94:
85:
78:permittivity
62:electrolytes
41:
35:
1999:, one gets
1058:model of a
980:Fermi level
393:denote the
363:Peter Debye
339:Drude model
282:Description
2589:Categories
2577:2018-07-12
2490:1506.06468
2322:References
30:See also:
2515:118576116
2382:1409.4323
2253:ω
2188:ω
2154:ω
2105:ω
2091:ϵ
2065:ω
2042:ω
2028:Φ
2021:ω
2007:ϵ
1963:Φ
1888:Φ
1798:ε
1782:ε
1730:−
1710:ε
1706:π
1682:ϕ
1646:δ
1635:ε
1626:−
1611:ϕ
1588:−
1579:∇
1531:ℏ
1521:π
1511:ε
1451:ε
1442:ρ
1365:ϕ
1362:Δ
1338:ε
1334:≃
1331:ρ
1328:Δ
1284:Δ
1261:ρ
1252:≃
1249:ρ
1246:Δ
1184:ℏ
1136:π
1105:π
1084:ρ
1060:Fermi gas
1034:ϕ
1031:Δ
1025:−
1019:Δ
1013:μ
1010:Δ
889:ε
873:ρ
809:ϕ
806:Δ
782:ε
778:≃
775:ρ
772:Δ
701:ϕ
688:
655:ρ
633:ρ
549:ρ
546:Δ
540:−
528:δ
511:ε
487:ϕ
484:Δ
472:∇
468:−
436:) is the
428:), where
237:effective
230:isotropic
186:^
152:ε
149:π
42:screening
2305:See also
2285:, where
345:and the
2495:Bibcode
2407:8246103
2387:Bibcode
2301:(DFT).
1317:yields
753:is the
583:is the
448:) and Î
379:jellium
361:due to
268:nucleus
58:plasmas
38:physics
2543:
2513:
2405:
2343:
2083:where
1911:where
1422:
1395:
1377:where
1231:where
988:ground
866:
839:
821:where
746:where
576:where
341:, the
320:phonon
254:, the
248:metals
74:metals
64:, and
54:fluids
50:charge
2511:S2CID
2485:arXiv
2445:(PDF)
2428:(PDF)
2403:S2CID
2377:arXiv
986:with
2541:ISBN
2341:ISBN
2297:and
970:and
960:and
936:⥠1/
594:and
401:the
389:Let
365:and
250:and
92:and
2503:doi
2395:doi
685:exp
420:is
264:ion
242:In
105:as
60:),
36:In
2591::
2568:.
2523:^
2509:.
2501:.
2493:.
2481:58
2479:.
2475:.
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2401:.
2393:.
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2256:)
2250:,
2246:k
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2181:k
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2094:(
2071:,
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2045:)
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2035:k
2031:(
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2018:,
2014:k
2010:(
1985:E
1943:S
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1894:S
1891:=
1883:E
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1794:=
1791:)
1788:r
1785:(
1765:0
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1694:=
1691:)
1688:r
1685:(
1658:,
1655:)
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1617:r
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1466:F
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1315:Ό
1299:.
1293:F
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1177:=
1171:F
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1151:3
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1140:k
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1016:=
996:T
992:Ό
978:(
941:0
938:k
934:D
931:λ
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846:=
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114:F
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81:Δ
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