1763:
31:
1207:
293:
1827:
has a strong dependency on certain system parameters: dimensionality, strength of interatomic interactions and Fermi level. This law is not valid or the value of the Lorenz number can be reduced at least in the following cases: manipulating electronic density of states, varying doping density and
1874:
nanobeams. In the metallic phase, the electronic contribution to thermal conductivity was much smaller than what would be expected from the
Wiedemann–Franz law. The results can be explained in terms of independent propagation of charge and heat in a strongly correlated system.
442:(c. 1900) realized that the phenomenological description of conductivity can be formulated quite generally (electron-, ion-, heat- etc. conductivity). Although the phenomenological description is incorrect for conduction electrons, it can serve as a preliminary treatment.
1037:
1429:
920:
683:
1431:
which is very close to experimental values. This is in fact due to 3 mistakes that conspired to make his result more accurate than warranted: the factor of 2 mistake; the specific heat per electron is in fact about 100 times less than
1025:
387:
The mathematical expression of the law can be derived as following. Electrical conduction of metals is a well-known phenomenon and is attributed to the free conduction electrons, which can be measured as sketched in the figure. The
361:
150:
1951:
1797: = 3.2×10VK for tungsten at 100 °C. Rosenberg notes that the Wiedemann–Franz law is generally valid for high temperatures and for low (i.e., a few Kelvins) temperatures, but may not hold at intermediate temperatures.
2031:
519:
743:
575:
1828:
layer thickness in superlattices and materials with correlated carriers. In thermoelectric materials there are also corrections due to boundary conditions, specifically open circuit vs. closed circuit.
1321:
1711:
1202:{\displaystyle {\frac {\kappa }{\sigma }}={\frac {cm\,\langle {v}\rangle ^{2}}{3e^{2}}}={\frac {4}{\pi }}{\frac {k_{\rm {B}}^{2}T}{e^{2}}}=0.94\times 10^{-8}\;\mathrm {V} ^{2}\mathrm {K} ^{-2}}
1521:
1247:
131:
788:
1469:
824:
1316:
1283:
1613:
580:
This would lead, however, to a constant acceleration and, ultimately, to an infinite velocity. The further assumption therefore is that the electrons bump into obstacles (like
2355:
Paothep
Pichanusakorn, Prabhakar Bandaru. Nanostructured thermoelectrics, Materials Science and Engineering: R: Reports, Volume 67, Issues 2–4, 29 January 2010, pages 19–63,
1673:
1570:
608:
1852:
diverges with decreasing temperature, reaching a value five orders of magnitude larger than that found in conventional metals obeying the
Wiedemann–Franz law. This due to
2054:
1572:
K) the heat and charge currents are carried by the same quasi-particles: electrons or holes. At finite temperatures two mechanisms produce a deviation of the ratio
812:
2635:
943:
288:{\displaystyle L={\frac {\kappa }{\sigma T}}={\frac {\pi ^{2}}{3}}\left({\frac {k_{\rm {B}}}{e}}\right)^{2}=2.44\times 10^{-8}\;\mathrm {V^{2}{\cdot }K} ^{-2},}
1897:
1962:
1887:
derived a
Wiedemann-Franz law for molecular systems in which electronic conduction is dominated not by free electron motion as in metals, but instead by
2469:
Lee, Sangwook; Hippalgaonkar, Kedar; Yang, Fan; Hong, Jiawang; Ko, Changhyun; Suh, Joonki; Liu, Kai; Wang, Kevin; Urban, Jeffrey J. (2017-01-27).
1836:
In 2011, N. Wakeham et al. found that the ratio of the thermal and electrical Hall conductivities in the metallic phase of quasi-one-dimensional
459:
2339:
A. Putatunda and D.J. Singh. Lorenz number in relation to estimates based on the
Seebeck coefficient, Materials Today Physics, 2019, 8, 49-55,
2323:. Bulk nanostructured thermoelectric materials: current research and future prospects, Energy & Environmental Science, 2009, 2, 466–479,
2266:
2221:
2196:
2101:
1717:
scattering values are possible (trajectory b in the figure) and electrons can be transported without the transport of a thermal excitation
1800:
In many high purity metals both the electrical and thermal conductivities rise as temperature is decreased. In certain materials (such as
2261:
Thermal conductivity: theory, properties, and applications, edited by Terry Tritt, Kluwer
Academic / Plenum Publishers, New York (2004),
2278:
695:
2640:
1483:, the heat capacity, mean free path and average speed of electrons are modified and the proportionality constant is then corrected to
2246:
525:
2384:
Wakeham, Nicholas; Bangura, Alimamy F.; Xu, Xiaofeng; Mercure, Jean-Francois; Greenblatt, Martha; Hussey, Nigel E. (2011-07-19).
1837:
1732:. At higher temperatures, the contribution of phonons to thermal transport in a system becomes important. This can lead to
1678:
2536:
1486:
1638:
in the figure). For each electron transported, a thermal excitation is also carried and the Lorenz number is reached
1424:{\displaystyle L=3\left({\frac {k_{\rm {B}}}{e}}\right)^{2}=2.22\times 10^{-8}\;\mathrm {V} ^{2}\mathrm {K} ^{-2},}
1219:
100:
2470:
748:
325:
915:{\displaystyle \langle v\rangle ={\sqrt {\frac {8k_{\rm {B}}T}{\pi m}}}={\sqrt {\frac {8}{3\pi }}}v_{\rm {rms}}}
380:
Qualitatively, this relationship is based upon the fact that the heat and electrical transport both involve the
2630:
1820:
1435:
1214:
599:
927:
689:
581:
404:
75:
1288:
1255:
1853:
1575:
791:
1762:
678:{\displaystyle {\frac {d\mathbf {v} }{dt}}=-{\frac {e\mathbf {E} }{m}}-{\frac {1}{\tau }}\mathbf {v} }
340:
has approximately the same value for different metals at the same temperature. The proportionality of
2485:
2449:
2397:
2320:
2130:
1627:
416:
67:
1652:
1549:
2316:
1480:
381:
1812:
also may decrease with temperature. In the purest samples of silver and at very low temperatures,
2607:
2573:
2517:
1868:
408:
306:
2360:
2157:
1867:
also found a large violation of the
Wiedemann–Franz law near the insulator-metal transition in
2599:
2591:
2509:
2501:
2431:
2413:
2356:
2262:
2242:
2217:
2192:
2097:
2039:
1888:
314:
1649:. Note that in a perfect metal, inelastic scattering would be completely absent in the limit
2583:
2493:
2421:
2405:
2364:
2340:
2324:
2300:
2169:
2138:
1857:
424:
411:
Ohm's law is expressed here in bold face. The conductivity can in general be expressed as a
797:
1020:{\displaystyle \sigma ={\frac {ne^{2}\tau }{m}}={\frac {ne^{2}\ell }{m\langle v\rangle }}}
389:
17:
2489:
2401:
2134:
1630:. As the temperature tends to 0 K, inelastic scattering becomes weak and promotes large
30:
2426:
2385:
2090:
1884:
1782:
815:
589:
396:
1946:{\displaystyle {\frac {\kappa }{\sigma }}=L_{\text{M}}{\frac {\lambda }{k_{\rm {B}}}}}
2624:
2611:
2521:
2304:
2057:
349:
329:
2026:{\displaystyle L_{\text{M}}={\frac {1}{2}}\left({\frac {k_{\rm {B}}}{e}}\right)^{2}}
34:
Plot of the
Wiedemann–Franz law for copper. Left axis: specific electric resistance
450:
431:
is the inverse of the conductivity. Both parameters will be used in the following.
2299:
K. Gloos, C. Mitschka, F. Pobell and P. Smeibidl. Cryogenics, 30 (1990), p. 14,
400:
2587:
2386:"Gross violation of the Wiedemann–Franz law in a quasi-one-dimensional conductor"
2344:
588:) once in a while which limits their free flight. This establishes an average or
2069:
1748:
934:
428:
87:
2561:
2368:
1766:
Sketch of the various scattering process important for the
Wiedemann–Franz law.
2471:"Anomalously low electronic thermal conductivity in metallic vanadium dioxide"
1471:; the mean squared velocity of an electron is in fact about 100 times larger.
439:
2595:
2505:
2417:
2173:
2142:
2497:
2216:(Repr ed.). South Melbourne: Brooks/Cole Thomson Learning. p. 23.
446:
420:
321:
2603:
2513:
2435:
514:{\displaystyle \mathbf {F} =-e\mathbf {E} =m{\frac {\;d\mathbf {v} }{dt}}}
1891:
between molecular sites. The molecular
Wiedemann-Franz law is given by
1805:
445:
The assumption is that the electrons move freely in the solid like in an
2409:
1751:
the phonon contribution to thermal transport is constant and the ratio
585:
59:
1781:, while roughly constant, is not exactly the same for all materials.
449:. The force applied to the electron by the electric field leads to an
360:
2328:
1801:
1623:
1318:, and also accidentally used a factor of 2. This meant his result is
412:
2537:"For This Metal, Electricity Flows, But Not the Heat | Berkeley Lab"
2118:
2578:
1761:
738:{\displaystyle \kappa ={\frac {1}{3}}cn\,\ell \,\langle v\rangle }
359:
83:
29:
2187:
Simon, Steven H. (2013). "3: Electrons in Metals: Drude Theory".
54:/ K in U/K, pink line. Lorenz number is more or less constant.
570:{\displaystyle \;d\mathbf {v} =-{\frac {e\mathbf {E} }{m}}dt}
2290:
Rosenberg, H. 2004. The Solid State. Oxford University Press
66:
states that the ratio of the electronic contribution of the
38:
in 10 Ω m, red line and specific thermal conductivity
1479:
After taking into account the quantum effects, as in the
368:
The arrows indicate the direction of the electric field
407:. Since the electric field and the current density are
2562:"Wiedemann–Franz Law for Molecular Hopping Transport"
2042:
1965:
1900:
1681:
1655:
1578:
1552:
1489:
1438:
1324:
1291:
1258:
1222:
1040:
946:
827:
800:
751:
698:
611:
528:
462:
153:
103:
1706:{\displaystyle \kappa \rightarrow 0;L\rightarrow 0}
1213:which is the Wiedemann–Franz law with an erroneous
2089:
2048:
2025:
1945:
1705:
1667:
1607:
1564:
1515:
1463:
1423:
1310:
1277:
1241:
1201:
1019:
914:
806:
782:
737:
677:
569:
513:
287:
125:
1516:{\displaystyle {\frac {\pi ^{2}}{3}}\approx 3.29}
602:as becomes evident from the following relations.
2560:Craven, Galen T.; Nitzan, Abraham (2020-02-12).
1622:: (i) other thermal carriers such as phonons or
1546:results from the fact that at low temperatures (
2158:"Bestimmung der Wärmegrade in absolutem Maasse"
2119:"Ueber die Wärme-Leitungsfähigkeit der Metalle"
2379:
2377:
1793: = 2.23×10VK for copper at 0 °C to
395:is observed to be proportional to the applied
2239:Introduction to the Electron Theory of Metals
1242:{\displaystyle {\frac {4}{\pi }}\approx 1.27}
126:{\displaystyle {\frac {\kappa }{\sigma }}=LT}
8:
2212:Ashcroft, Neil W.; Mermin, N. David (2012).
1675:K and the thermal conductivity would vanish
1299:
1292:
1272:
1259:
1073:
1064:
1011:
1005:
834:
828:
732:
726:
136:Theoretically, the proportionality constant
2450:"Bristol physicists break 150-year-old law"
783:{\displaystyle c={\frac {3}{2}}k_{\rm {B}}}
1390:
1171:
529:
491:
364:Electric circuit with metal and a battery
251:
2577:
2425:
2088:Jones, William; March, Norman H. (1985).
2041:
2017:
2001:
2000:
1994:
1979:
1970:
1964:
1934:
1933:
1924:
1918:
1901:
1899:
1777:Experiments have shown that the value of
1680:
1654:
1588:
1577:
1551:
1523:, which agrees with experimental values.
1496:
1490:
1488:
1464:{\displaystyle {\frac {3}{2}}k_{\rm {B}}}
1454:
1453:
1439:
1437:
1409:
1404:
1397:
1392:
1381:
1362:
1346:
1345:
1339:
1323:
1302:
1290:
1266:
1257:
1223:
1221:
1190:
1185:
1178:
1173:
1162:
1141:
1127:
1121:
1120:
1113:
1103:
1091:
1076:
1067:
1063:
1054:
1041:
1039:
991:
981:
963:
953:
945:
899:
898:
877:
852:
851:
840:
826:
799:
773:
772:
758:
750:
725:
721:
705:
697:
670:
660:
646:
640:
618:
612:
610:
550:
544:
533:
527:
495:
488:
477:
463:
461:
273:
264:
258:
253:
242:
223:
207:
206:
200:
184:
178:
160:
152:
104:
102:
50:in 100 U/K, blue line and Lorenz number
2080:
2036:is the Lorenz number for molecules and
1863:A Berkeley-led study in 2016 by S. Lee
1816:can drop by as much as a factor of 10.
598:. The drift velocity is related to the
419:). Here we restrict the discussion to
2636:Electrical resistance and conductance
1311:{\displaystyle \langle v\rangle ^{2}}
1278:{\displaystyle \langle v^{2}\rangle }
42:in W/(K m), green line. Right axis:
7:
1608:{\displaystyle L=\kappa /(\sigma T)}
403:where the prefactor is the specific
2279:Introduction to Solid State Physics
2191:. Oxford: Oxford university press.
348:with temperature was discovered by
2002:
1935:
1615:from the theoretical Lorenz value
1455:
1405:
1393:
1347:
1252:In Drude's original paper he used
1186:
1174:
1122:
906:
903:
900:
853:
774:
269:
255:
208:
25:
2117:Franz, R.; Wiedemann, G. (1853).
372:and the electric current density
1838:lithium molybdenum purple bronze
671:
647:
619:
551:
534:
496:
478:
464:
2092:Theoretical Solid State Physics
2241:. CAMBRIDGE UNIVERSITY PRESS.
2096:. Courier Dover Publications.
1713:. At finite temperature small
1697:
1685:
1668:{\displaystyle T\rightarrow 0}
1659:
1634:scattering values (trajectory
1602:
1593:
1565:{\displaystyle T\rightarrow 0}
1556:
930:of the particles in the gas.
1:
2189:The Oxford solid state basics
2162:Annalen der Physik und Chemie
2588:10.1021/acs.nanolett.9b04070
2345:10.1016/j.mtphys.2019.01.001
2305:10.1016/0011-2275(90)90107-N
332:, who in 1853 reported that
1759:) is again found constant.
427:conductivity. The specific
2657:
2535:Yang, Sarah (2017-01-26).
2369:10.1016/j.mser.2009.10.001
2237:Mizutani, Uichiro (2003).
1883:In 2020, Galen Craven and
18:Wiedemann–Franz–Lorenz law
2641:Eponymous laws of physics
1821:degenerate semiconductors
1773:Limitations of the theory
2174:10.1002/andp.18722231107
2143:10.1002/andp.18531650802
2049:{\displaystyle \lambda }
1808:) however, the value of
1215:proportionality constant
415:of the second rank (3×3
2498:10.1126/science.aag0410
2060:for electron transfer.
818:of the electrons, and
690:kinetic theory of gases
600:average scattering time
405:electrical conductivity
86:is proportional to the
76:electrical conductivity
2050:
2027:
1947:
1854:spin-charge separation
1767:
1707:
1669:
1609:
1566:
1527:Temperature dependence
1517:
1465:
1425:
1312:
1279:
1243:
1203:
1021:
916:
808:
784:
739:
679:
571:
515:
435:Drude model derivation
377:
289:
127:
55:
2390:Nature Communications
2281:. John Wiley and Sons
2058:reorganization energy
2051:
2028:
1948:
1856:and it behaving as a
1785:gives some values of
1765:
1708:
1670:
1610:
1567:
1518:
1466:
1426:
1313:
1280:
1244:
1204:
1022:
917:
809:
807:{\displaystyle \ell }
785:
740:
680:
572:
516:
363:
290:
128:
33:
2040:
1963:
1898:
1823:, the Lorenz number
1679:
1653:
1628:Inelastic scattering
1576:
1550:
1487:
1436:
1322:
1289:
1256:
1220:
1038:
944:
825:
798:
749:
696:
609:
526:
460:
151:
101:
68:thermal conductivity
2490:2017Sci...355..371L
2402:2011NatCo...2..396W
2214:Solid state physics
2156:Lorenz, L. (1872).
2135:1853AnP...165..497F
1481:free electron model
1475:Free electron model
1132:
324:law is named after
64:Wiedemann–Franz law
2410:10.1038/ncomms1406
2277:Kittel, C., 2005.
2123:Annalen der Physik
2046:
2023:
1943:
1768:
1703:
1665:
1605:
1562:
1513:
1461:
1421:
1308:
1275:
1239:
1199:
1116:
1017:
912:
804:
780:
735:
675:
567:
511:
378:
307:Boltzmann constant
285:
123:
56:
2484:(6323): 371–374.
2317:M. S. Dresselhaus
2267:978-0-387-26017-4
2223:978-0-03-083993-1
2198:978-0-19-968077-1
2103:978-0-486-65016-6
2011:
1987:
1973:
1941:
1921:
1909:
1889:electron transfer
1879:Molecular systems
1749:Debye temperature
1740:) >
1725:) <
1505:
1447:
1356:
1231:
1147:
1111:
1098:
1049:
1015:
976:
892:
891:
872:
871:
766:
713:
668:
655:
632:
559:
509:
315:elementary charge
217:
193:
173:
112:
16:(Redirected from
2648:
2616:
2615:
2581:
2557:
2551:
2550:
2548:
2547:
2532:
2526:
2525:
2475:
2466:
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2459:
2457:
2456:
2446:
2440:
2439:
2429:
2381:
2372:
2353:
2347:
2337:
2331:
2329:10.1039/b822664b
2319:, Z. F. Ren and
2313:
2307:
2297:
2291:
2288:
2282:
2275:
2269:
2259:
2253:
2252:
2234:
2228:
2227:
2209:
2203:
2202:
2184:
2178:
2177:
2153:
2147:
2146:
2114:
2108:
2107:
2095:
2085:
2055:
2053:
2052:
2047:
2032:
2030:
2029:
2024:
2022:
2021:
2016:
2012:
2007:
2006:
2005:
1995:
1988:
1980:
1975:
1974:
1971:
1952:
1950:
1949:
1944:
1942:
1940:
1939:
1938:
1925:
1923:
1922:
1919:
1910:
1902:
1858:Luttinger liquid
1712:
1710:
1709:
1704:
1674:
1672:
1671:
1666:
1614:
1612:
1611:
1606:
1592:
1571:
1569:
1568:
1563:
1545:
1543:
1522:
1520:
1519:
1514:
1506:
1501:
1500:
1491:
1470:
1468:
1467:
1462:
1460:
1459:
1458:
1448:
1440:
1430:
1428:
1427:
1422:
1417:
1416:
1408:
1402:
1401:
1396:
1389:
1388:
1367:
1366:
1361:
1357:
1352:
1351:
1350:
1340:
1317:
1315:
1314:
1309:
1307:
1306:
1284:
1282:
1281:
1276:
1271:
1270:
1248:
1246:
1245:
1240:
1232:
1224:
1208:
1206:
1205:
1200:
1198:
1197:
1189:
1183:
1182:
1177:
1170:
1169:
1148:
1146:
1145:
1136:
1131:
1126:
1125:
1114:
1112:
1104:
1099:
1097:
1096:
1095:
1082:
1081:
1080:
1071:
1055:
1050:
1042:
1026:
1024:
1023:
1018:
1016:
1014:
1000:
996:
995:
982:
977:
972:
968:
967:
954:
921:
919:
918:
913:
911:
910:
909:
893:
890:
879:
878:
873:
870:
862:
858:
857:
856:
842:
841:
813:
811:
810:
805:
789:
787:
786:
781:
779:
778:
777:
767:
759:
744:
742:
741:
736:
714:
706:
684:
682:
681:
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674:
669:
661:
656:
651:
650:
641:
633:
631:
623:
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560:
555:
554:
545:
537:
520:
518:
517:
512:
510:
508:
500:
499:
489:
481:
467:
326:Gustav Wiedemann
294:
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286:
281:
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132:
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124:
113:
105:
21:
2656:
2655:
2651:
2650:
2649:
2647:
2646:
2645:
2631:Heat conduction
2621:
2620:
2619:
2559:
2558:
2554:
2545:
2543:
2534:
2533:
2529:
2473:
2468:
2467:
2463:
2454:
2452:
2448:
2447:
2443:
2383:
2382:
2375:
2354:
2350:
2338:
2334:
2315:A. J. Minnich,
2314:
2310:
2298:
2294:
2289:
2285:
2276:
2272:
2260:
2256:
2249:
2236:
2235:
2231:
2224:
2211:
2210:
2206:
2199:
2186:
2185:
2181:
2168:(11): 429–452.
2155:
2154:
2150:
2116:
2115:
2111:
2104:
2087:
2086:
2082:
2078:
2066:
2038:
2037:
1996:
1990:
1989:
1966:
1961:
1960:
1929:
1914:
1896:
1895:
1881:
1872:
1851:
1847:
1843:
1834:
1775:
1746:
1731:
1677:
1676:
1651:
1650:
1648:
1621:
1574:
1573:
1548:
1547:
1541:
1539:
1537:
1529:
1492:
1485:
1484:
1477:
1449:
1434:
1433:
1403:
1391:
1377:
1341:
1335:
1334:
1320:
1319:
1298:
1287:
1286:
1262:
1254:
1253:
1218:
1217:
1184:
1172:
1158:
1137:
1115:
1087:
1083:
1072:
1056:
1036:
1035:
1001:
987:
983:
959:
955:
942:
941:
894:
883:
863:
847:
843:
823:
822:
796:
795:
768:
747:
746:
694:
693:
642:
624:
614:
607:
606:
597:
546:
524:
523:
501:
490:
458:
457:
437:
390:current density
358:
304:
254:
252:
238:
202:
196:
195:
180:
165:
149:
148:
140:, known as the
99:
98:
28:
23:
22:
15:
12:
11:
5:
2654:
2652:
2644:
2643:
2638:
2633:
2623:
2622:
2618:
2617:
2572:(2): 989–993.
2552:
2527:
2461:
2441:
2373:
2348:
2332:
2308:
2292:
2283:
2270:
2254:
2247:
2229:
2222:
2204:
2197:
2179:
2148:
2129:(8): 497–531.
2109:
2102:
2079:
2077:
2074:
2073:
2072:
2065:
2062:
2045:
2034:
2033:
2020:
2015:
2010:
2004:
1999:
1993:
1986:
1983:
1978:
1969:
1954:
1953:
1937:
1932:
1928:
1917:
1913:
1908:
1905:
1885:Abraham Nitzan
1880:
1877:
1870:
1849:
1845:
1841:
1833:
1830:
1774:
1771:
1744:
1729:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1664:
1661:
1658:
1646:
1619:
1604:
1601:
1598:
1595:
1591:
1587:
1584:
1581:
1561:
1558:
1555:
1535:
1528:
1525:
1512:
1509:
1504:
1499:
1495:
1476:
1473:
1457:
1452:
1446:
1443:
1420:
1415:
1412:
1407:
1400:
1395:
1387:
1384:
1380:
1376:
1373:
1370:
1365:
1360:
1355:
1349:
1344:
1338:
1333:
1330:
1327:
1305:
1301:
1297:
1294:
1274:
1269:
1265:
1261:
1238:
1235:
1230:
1227:
1211:
1210:
1196:
1193:
1188:
1181:
1176:
1168:
1165:
1161:
1157:
1154:
1151:
1144:
1140:
1135:
1130:
1124:
1119:
1110:
1107:
1102:
1094:
1090:
1086:
1079:
1075:
1070:
1066:
1062:
1059:
1053:
1048:
1045:
1029:
1028:
1013:
1010:
1007:
1004:
999:
994:
990:
986:
980:
975:
971:
966:
962:
958:
952:
949:
924:
923:
908:
905:
902:
897:
889:
886:
882:
876:
869:
866:
861:
855:
850:
846:
839:
836:
833:
830:
816:mean free path
803:
794:per electron,
776:
771:
765:
762:
757:
754:
734:
731:
728:
724:
720:
717:
712:
709:
704:
701:
686:
685:
673:
667:
664:
659:
654:
649:
645:
639:
636:
630:
627:
621:
617:
595:
590:drift velocity
578:
577:
566:
563:
558:
553:
549:
543:
540:
536:
532:
521:
507:
504:
498:
494:
487:
484:
480:
476:
473:
470:
466:
436:
433:
397:electric field
384:in the metal.
382:free electrons
357:
354:
302:
296:
295:
284:
279:
276:
271:
267:
261:
257:
248:
245:
241:
237:
234:
231:
226:
221:
216:
210:
205:
199:
192:
187:
183:
177:
171:
168:
164:
159:
156:
144:, is equal to
134:
133:
122:
119:
116:
111:
108:
27:Law of physics
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2653:
2642:
2639:
2637:
2634:
2632:
2629:
2628:
2626:
2613:
2609:
2605:
2601:
2597:
2593:
2589:
2585:
2580:
2575:
2571:
2567:
2563:
2556:
2553:
2542:
2538:
2531:
2528:
2523:
2519:
2515:
2511:
2507:
2503:
2499:
2495:
2491:
2487:
2483:
2479:
2472:
2465:
2462:
2451:
2445:
2442:
2437:
2433:
2428:
2423:
2419:
2415:
2411:
2407:
2403:
2399:
2395:
2391:
2387:
2380:
2378:
2374:
2370:
2366:
2362:
2358:
2352:
2349:
2346:
2342:
2336:
2333:
2330:
2326:
2322:
2318:
2312:
2309:
2306:
2302:
2296:
2293:
2287:
2284:
2280:
2274:
2271:
2268:
2264:
2258:
2255:
2250:
2248:9780511612626
2244:
2240:
2233:
2230:
2225:
2219:
2215:
2208:
2205:
2200:
2194:
2190:
2183:
2180:
2175:
2171:
2167:
2164:(in German).
2163:
2159:
2152:
2149:
2144:
2140:
2136:
2132:
2128:
2125:(in German).
2124:
2120:
2113:
2110:
2105:
2099:
2094:
2093:
2084:
2081:
2075:
2071:
2068:
2067:
2063:
2061:
2059:
2043:
2018:
2013:
2008:
1997:
1991:
1984:
1981:
1976:
1967:
1959:
1958:
1957:
1930:
1926:
1915:
1911:
1906:
1903:
1894:
1893:
1892:
1890:
1886:
1878:
1876:
1873:
1866:
1861:
1859:
1855:
1839:
1831:
1829:
1826:
1822:
1817:
1815:
1811:
1807:
1803:
1798:
1796:
1792:
1789:ranging from
1788:
1784:
1780:
1772:
1770:
1764:
1760:
1758:
1754:
1750:
1743:
1739:
1735:
1728:
1724:
1720:
1716:
1700:
1694:
1691:
1688:
1682:
1662:
1656:
1645:
1642: =
1641:
1637:
1633:
1629:
1625:
1618:
1599:
1596:
1589:
1585:
1582:
1579:
1559:
1553:
1534:
1526:
1524:
1510:
1507:
1502:
1497:
1493:
1482:
1474:
1472:
1450:
1444:
1441:
1418:
1413:
1410:
1398:
1385:
1382:
1378:
1374:
1371:
1368:
1363:
1358:
1353:
1342:
1336:
1331:
1328:
1325:
1303:
1295:
1267:
1263:
1250:
1236:
1233:
1228:
1225:
1216:
1194:
1191:
1179:
1166:
1163:
1159:
1155:
1152:
1149:
1142:
1138:
1133:
1128:
1117:
1108:
1105:
1100:
1092:
1088:
1084:
1077:
1068:
1060:
1057:
1051:
1046:
1043:
1034:
1033:
1032:
1008:
1002:
997:
992:
988:
984:
978:
973:
969:
964:
960:
956:
950:
947:
940:
939:
938:
936:
931:
929:
928:average speed
895:
887:
884:
880:
874:
867:
864:
859:
848:
844:
837:
831:
821:
820:
819:
817:
801:
793:
792:heat capacity
769:
763:
760:
755:
752:
729:
722:
718:
715:
710:
707:
702:
699:
691:
665:
662:
657:
652:
643:
637:
634:
628:
625:
615:
605:
604:
603:
601:
594:
591:
587:
583:
564:
561:
556:
547:
541:
538:
530:
522:
505:
502:
492:
485:
482:
474:
471:
468:
456:
455:
454:
453:according to
452:
448:
443:
441:
434:
432:
430:
426:
422:
418:
414:
410:
406:
402:
398:
394:
391:
385:
383:
375:
371:
367:
362:
355:
353:
351:
350:Ludvig Lorenz
347:
343:
339:
335:
331:
330:Rudolph Franz
327:
323:
318:
316:
312:
308:
301:
282:
277:
274:
265:
259:
246:
243:
239:
235:
232:
229:
224:
219:
214:
203:
197:
190:
185:
181:
175:
169:
166:
162:
157:
154:
147:
146:
145:
143:
142:Lorenz number
139:
120:
117:
114:
109:
106:
97:
96:
95:
93:
89:
85:
81:
77:
73:
69:
65:
61:
53:
52:ρ λ
49:
45:
41:
37:
32:
19:
2569:
2566:Nano Letters
2565:
2555:
2544:. Retrieved
2540:
2530:
2481:
2477:
2464:
2453:. Retrieved
2444:
2393:
2389:
2351:
2335:
2311:
2295:
2286:
2273:
2257:
2238:
2232:
2213:
2207:
2188:
2182:
2165:
2161:
2151:
2126:
2122:
2112:
2091:
2083:
2035:
1955:
1882:
1864:
1862:
1835:
1824:
1818:
1813:
1809:
1799:
1794:
1790:
1786:
1778:
1776:
1769:
1756:
1752:
1747:. Above the
1741:
1737:
1733:
1726:
1722:
1718:
1714:
1643:
1639:
1635:
1631:
1616:
1532:
1530:
1478:
1251:
1212:
1031:Therefore,
1030:
932:
925:
687:
592:
579:
451:acceleration
444:
438:
399:and follows
392:
386:
379:
373:
369:
365:
345:
341:
337:
333:
319:
310:
299:
297:
141:
137:
135:
91:
79:
71:
63:
57:
51:
47:
43:
39:
35:
2541:News Center
2070:Drude model
1544:10 V⋅K
1285:instead of
935:Drude model
429:resistivity
88:temperature
2625:Categories
2579:1909.06220
2546:2017-01-28
2455:2017-01-28
2076:References
1832:Violations
1531:The value
440:Paul Drude
356:Derivation
2612:202572812
2596:1530-6984
2522:206650639
2506:0036-8075
2418:2041-1723
2361:0927-796X
2044:λ
1927:λ
1907:σ
1904:κ
1698:→
1686:→
1683:κ
1660:→
1597:σ
1586:κ
1557:→
1508:≈
1494:π
1411:−
1383:−
1375:×
1300:⟩
1293:⟨
1273:⟩
1260:⟨
1234:≈
1229:π
1192:−
1164:−
1156:×
1109:π
1074:⟩
1065:⟨
1047:σ
1044:κ
1012:⟩
1006:⟨
998:ℓ
970:τ
948:σ
933:From the
888:π
865:π
835:⟩
829:⟨
802:ℓ
733:⟩
727:⟨
723:ℓ
700:κ
666:τ
658:−
638:−
542:−
472:−
447:ideal gas
421:isotropic
401:Ohm's law
352:in 1872.
322:empirical
275:−
266:⋅
244:−
236:×
182:π
167:σ
163:κ
110:σ
107:κ
74:) to the
2604:31951422
2514:28126811
2436:21772267
2064:See also
1806:aluminum
1538: =
745:, where
2486:Bibcode
2478:Science
2427:3144592
2398:Bibcode
2396:: 396.
2321:G. Chen
2131:Bibcode
2056:is the
1626:, (ii)
1624:magnons
926:is the
814:is the
790:is the
586:phonons
582:defects
423:, i.e.
409:vectors
313:is the
305:is the
82:) of a
60:physics
2610:
2602:
2594:
2520:
2512:
2504:
2434:
2424:
2416:
2359:
2265:
2245:
2220:
2195:
2100:
1956:where
1865:et al.
1802:silver
1783:Kittel
425:scalar
417:matrix
413:tensor
346:σ
342:κ
338:σ
334:κ
298:where
80:σ
72:κ
62:, the
48:λ
46:times
44:ρ
40:λ
36:ρ
2608:S2CID
2574:arXiv
2518:S2CID
2474:(PDF)
688:From
320:This
84:metal
2600:PMID
2592:ISSN
2510:PMID
2502:ISSN
2432:PMID
2414:ISSN
2357:ISSN
2263:ISBN
2243:ISBN
2218:ISBN
2193:ISBN
2098:ISBN
1540:2.44
1511:3.29
1372:2.22
1237:1.27
1153:0.94
937:,
328:and
309:and
233:2.44
2584:doi
2494:doi
2482:355
2422:PMC
2406:doi
2365:doi
2341:doi
2325:doi
2301:doi
2170:doi
2166:223
2139:doi
2127:165
1842:0.9
1819:In
1804:or
1249:.
584:or
94:).
58:In
2627::
2606:.
2598:.
2590:.
2582:.
2570:20
2568:.
2564:.
2539:.
2516:.
2508:.
2500:.
2492:.
2480:.
2476:.
2430:.
2420:.
2412:.
2404:.
2392:.
2388:.
2376:^
2363:,
2160:.
2137:.
2121:.
1869:VO
1860:.
1850:17
1844:Mo
1840:Li
1379:10
1160:10
692:,
366:U.
317:.
240:10
2614:.
2586::
2576::
2549:.
2524:.
2496::
2488::
2458:.
2438:.
2408::
2400::
2394:2
2371:.
2367::
2343::
2327::
2303::
2251:.
2226:.
2201:.
2176:.
2172::
2145:.
2141::
2133::
2106:.
2019:2
2014:)
2009:e
2003:B
1998:k
1992:(
1985:2
1982:1
1977:=
1972:M
1968:L
1936:B
1931:k
1920:M
1916:L
1912:=
1871:2
1848:O
1846:6
1825:L
1814:L
1810:L
1795:L
1791:L
1787:L
1779:L
1757:T
1755:(
1753:L
1745:0
1742:L
1738:T
1736:(
1734:L
1730:0
1727:L
1723:T
1721:(
1719:L
1715:q
1701:0
1695:L
1692:;
1689:0
1663:0
1657:T
1647:0
1644:L
1640:L
1636:a
1632:q
1620:0
1617:L
1603:)
1600:T
1594:(
1590:/
1583:=
1580:L
1560:0
1554:T
1542:×
1536:0
1533:L
1503:3
1498:2
1456:B
1451:k
1445:2
1442:3
1419:,
1414:2
1406:K
1399:2
1394:V
1386:8
1369:=
1364:2
1359:)
1354:e
1348:B
1343:k
1337:(
1332:3
1329:=
1326:L
1304:2
1296:v
1268:2
1264:v
1226:4
1209:,
1195:2
1187:K
1180:2
1175:V
1167:8
1150:=
1143:2
1139:e
1134:T
1129:2
1123:B
1118:k
1106:4
1101:=
1093:2
1089:e
1085:3
1078:2
1069:v
1061:m
1058:c
1052:=
1027:.
1009:v
1003:m
993:2
989:e
985:n
979:=
974:m
965:2
961:e
957:n
951:=
922:,
907:s
904:m
901:r
896:v
885:3
881:8
875:=
868:m
860:T
854:B
849:k
845:8
838:=
832:v
775:B
770:k
764:2
761:3
756:=
753:c
730:v
719:n
716:c
711:3
708:1
703:=
672:v
663:1
653:m
648:E
644:e
635:=
629:t
626:d
620:v
616:d
596:d
593:V
565:t
562:d
557:m
552:E
548:e
539:=
535:v
531:d
506:t
503:d
497:v
493:d
486:m
483:=
479:E
475:e
469:=
465:F
393:j
376:.
374:j
370:E
344:/
336:/
311:e
303:B
300:k
283:,
278:2
270:K
260:2
256:V
247:8
230:=
225:2
220:)
215:e
209:B
204:k
198:(
191:3
186:2
176:=
170:T
158:=
155:L
138:L
121:T
118:L
115:=
92:T
90:(
78:(
70:(
20:)
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