1510:
976:
1505:{\displaystyle {\frac {\alpha _{s}}{\alpha _{n}}}={\frac {2}{\hbar \omega }}\int _{\Delta }^{\infty }{{\frac {\left}{(E^{2}-\Delta ^{2})^{1/2}^{1/2}}}dE}{-}\Theta (\hbar \omega -2\Delta ){\frac {1}{\hbar \omega }}\int _{\Delta -\hbar \omega }^{-\Delta }{{\frac {\left}{(E^{2}-\Delta ^{2})^{1/2}^{1/2}}}dE}}
1556:
should be conserved through the transition. This result implies that the missing area of the spectral weight is concentrated in the zero frequency limit, corresponding to the dirac delta function (which covers the conduction of the superconducting condensate, i.e. the Cooper pairs). Many experimental
969:
In finite temperature condition, the response of electrons due to the incident electromagnetic wave can be regarded as two parts, the “superconducting” and “normal” electrons. The first one corresponds to the superconducting ground state and the next to the thermally excited electrons from the ground
466:
In the superconducting state, each term of the
Hamiltonian is dependent, because of the superconducting state consists of a phase-coherent superposition of occupied one-electron states, whereas it is independent in the normal state. Therefore, there appear interference terms in the absolute square of
461:
823:
227:
629:
27:
It was derived to explain the anomalous skin effect of superconductors. Originally, the anomalous skin effect indicates the non-classical response of metals to high frequency electromagnetic field in low temperature, which was solved by
964:
36:) fails because of the enhancement of the mean free path of the electrons in a good metal. Not only the normal metals, but superconductors also show the anomalous skin effect which has to be considered with the
1572:, microwave or far-infrared spectroscopy is suitable technique applying this theory. With the Mattis–Bardeen theory, we can derive fruitful properties of the superconducting gap, like gap symmetry.
640:
319:
890:
1544:
is the
Heaviside theta function. The first term of the upper equation is the contribution of "normal" electrons, and the second term is due to the superconducting electrons.
70:
853:
1542:
970:
state. This picture is the so-called "two-fluid" model. If we consider the “normal” electrons, the ratio of the optical conductivity to the one of the normal state is
2071:
492:
311:
284:
257:
532:
512:
898:
1804:
1741:
1557:
data supports the prediction. This story on electrodynamics of superconductivity is the starting point of optical study. Because any superconducting
1779:
2163:
1901:
2011:
2099:
1820:
818:{\displaystyle \alpha _{s}=\int {\left|M\right|^{2}F(\Delta ,E,E+\hbar \omega )N_{s}(E)N_{s}(E+\hbar \omega )\times {\rm {}}}dE}
2016:
1734:
1799:
1643:
1769:
2140:
1992:
1936:
1911:
2042:
1971:
1987:
1906:
1774:
2194:
2094:
2089:
1727:
456:{\displaystyle H_{1}=\sum \limits _{k\sigma ,k'\sigma '}{B_{k'\sigma ',k\sigma }c_{k'\sigma '}^{*}}c_{k'\sigma '}}
2047:
56:
arises, and the dispersion relation can be described like the one of a semiconductor with band gap 2Δ around the
1830:
1881:
1589:
D. C. Mattis; J. Bardeen (1958). "Theory of the
Anomalous Skin Effect in Normal and Superconducting Metals".
2173:
2032:
1964:
2084:
2057:
2037:
1959:
1954:
222:{\displaystyle \alpha _{s}=\int {\left|M_{s}\right|^{2}N_{s}(E)N_{s}(E+\hbar \omega )\times {\rm {}}}dE}
52:). After the transition to the superconducting state, the superconducting gap 2Δ in the single-particle
2158:
858:
2114:
1668:
1600:
2145:
1896:
1856:
1684:
831:
29:
1518:
32:. At sufficiently low temperatures and high frequencies, the classically predicted skin depth (
1921:
1750:
1639:
61:
53:
21:
2130:
2104:
1876:
1851:
1784:
1676:
1631:
1608:
1553:
2153:
470:
289:
262:
235:
24:. It is commonly applied in the research field of optical spectroscopy on superconductors.
1886:
1591:
624:{\displaystyle F(\Delta ,E,E')={\frac {1}{2}}\left(1\pm {\frac {\Delta ^{2}}{EE'}}\right)}
48:
The most clear fact the BCS theory gives is the presence of the pairing of two electrons (
1672:
1604:
1891:
1835:
1825:
1789:
497:
959:{\displaystyle {\frac {\alpha _{s}}{\alpha _{n}}}={\frac {\sigma _{1s}}{\sigma _{n}}}}
828:
where the transition rate can be translated to real part of the complex conductivity,
2188:
1688:
1926:
1916:
1871:
1866:
57:
2052:
1861:
49:
33:
1764:
37:
1635:
1612:
467:
the matrix element. The result of the coherence changes the matrix element
2079:
1564:
never exceeds 200K and the superconducting gap value is about the 3.5
855:, because the electrodynamic energy absorption is proportional to the
1680:
1719:
1628:
Electrodynamics of Solids: Optical
Properties of Electrons in Matter
2135:
2109:
1552:
The calculated optical conductivity breaks the sum rule that the
2168:
1723:
20:
is a theory that describes the electrodynamic properties of
1659:
R. G. Chambers (1950). "Anomalous Skin Effect in Metals".
1714:
Electrodynamics of Solids and
Microwave Superconductivity
1521:
979:
901:
861:
834:
643:
535:
500:
473:
322:
292:
265:
238:
73:
286:
2123:
2070:
2025:
2001:
1980:
1944:
1935:
1844:
1813:
1757:
1536:
1504:
958:
884:
847:
817:
623:
506:
486:
455:
305:
278:
251:
221:
64:, the transition probabilities can be written as
1735:
514:of single electron and the coherence factors
8:
1941:
1742:
1728:
1720:
1520:
1482:
1478:
1468:
1455:
1423:
1419:
1409:
1396:
1340:
1328:
1317:
1316:
1309:
1308:
1299:
1285:
1266:
1237:
1217:
1213:
1203:
1190:
1158:
1154:
1144:
1131:
1069:
1057:
1046:
1045:
1038:
1037:
1031:
1026:
1007:
996:
986:
980:
978:
948:
935:
929:
918:
908:
902:
900:
876:
866:
860:
839:
833:
805:
804:
735:
716:
673:
660:
648:
642:
595:
589:
568:
534:
499:
478:
472:
434:
423:
405:
373:
368:
340:
327:
321:
297:
291:
270:
264:
243:
237:
209:
208:
139:
120:
110:
100:
90:
78:
72:
1626:Dressel, Martin; Grüner, George (2002).
38:theory of Bardeen, Cooper and Schrieffer
1581:
1445:
1375:
1322:
1292:
1272:
1248:
1180:
1110:
1051:
1013:
792:
750:
703:
196:
154:
7:
337:
44:Response to an electromagnetic wave
1522:
1465:
1406:
1337:
1303:
1286:
1260:
1242:
1200:
1141:
1066:
1032:
1027:
685:
592:
542:
14:
1707:Introduction to Superconductivity
885:{\displaystyle \sigma _{1}E^{2}}
1531:
1525:
1475:
1452:
1436:
1433:
1416:
1389:
1384:
1381:
1366:
1351:
1263:
1245:
1210:
1187:
1171:
1168:
1151:
1124:
1119:
1116:
1101:
1092:
1086:
1080:
801:
798:
783:
774:
768:
762:
756:
741:
728:
722:
709:
682:
562:
539:
259:is the density of states. And
205:
202:
187:
178:
172:
166:
160:
145:
132:
126:
1:
634:Then, the transition rate is
848:{\displaystyle \sigma _{1}}
2211:
2072:Technological applications
1537:{\displaystyle \Theta (x)}
1814:Characteristic parameters
1831:London penetration depth
1636:10.1017/CBO9780511606168
494:into the matrix element
2124:List of superconductors
2002:By critical temperature
1613:10.1103/PhysRev.111.412
1538:
1506:
960:
886:
849:
819:
625:
508:
488:
457:
307:
280:
253:
223:
1770:Bean's critical state
1539:
1507:
961:
887:
850:
820:
626:
509:
489:
487:{\displaystyle M_{s}}
458:
308:
306:{\displaystyle H_{1}}
281:
279:{\displaystyle M_{s}}
254:
252:{\displaystyle N_{s}}
224:
18:Mattis–Bardeen theory
1945:By magnetic response
1548:Use in optical study
1519:
977:
899:
859:
832:
641:
533:
498:
471:
320:
290:
263:
236:
71:
1897:persistent currents
1882:Little–Parks effect
1673:1950Natur.165..239C
1605:1958PhRv..111..412M
1307:
1036:
428:
1857:Andreev reflection
1852:Abrikosov vortices
1534:
1502:
1281:
1022:
956:
882:
845:
815:
621:
504:
484:
453:
401:
367:
303:
276:
249:
219:
34:normal skin effect
30:Robert G. Chambers
2195:Superconductivity
2182:
2181:
2100:quantum computing
2066:
2065:
1922:superdiamagnetism
1751:Superconductivity
1709:. Second edition.
1705:Michael Tinkham,
1667:(4189): 239–240.
1493:
1331:
1320:
1279:
1228:
1060:
1049:
1020:
1002:
954:
924:
614:
576:
507:{\displaystyle M}
336:
62:Fermi golden rule
54:density of states
22:superconductivity
2202:
2131:bilayer graphene
2105:Rutherford cable
2017:room temperature
2012:high temperature
1942:
1902:proximity effect
1877:Josephson effect
1821:coherence length
1744:
1737:
1730:
1721:
1693:
1692:
1681:10.1038/165239b0
1656:
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1501:
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1492:
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1490:
1486:
1473:
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1432:
1431:
1427:
1414:
1413:
1401:
1400:
1387:
1350:
1346:
1345:
1344:
1332:
1329:
1321:
1318:
1310:
1306:
1298:
1280:
1278:
1267:
1241:
1236:
1229:
1227:
1226:
1225:
1221:
1208:
1207:
1195:
1194:
1167:
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1162:
1149:
1148:
1136:
1135:
1122:
1079:
1075:
1074:
1073:
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1058:
1050:
1047:
1039:
1035:
1030:
1021:
1019:
1008:
1003:
1001:
1000:
991:
990:
981:
965:
963:
962:
957:
955:
953:
952:
943:
942:
930:
925:
923:
922:
913:
912:
903:
891:
889:
888:
883:
881:
880:
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854:
852:
851:
846:
844:
843:
824:
822:
821:
816:
808:
807:
806:
740:
739:
721:
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678:
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672:
653:
652:
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628:
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622:
620:
616:
615:
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612:
600:
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577:
569:
561:
513:
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493:
491:
490:
485:
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482:
462:
460:
459:
454:
452:
451:
450:
442:
429:
427:
422:
421:
413:
400:
399:
389:
381:
366:
365:
357:
332:
331:
312:
310:
309:
304:
302:
301:
285:
283:
282:
277:
275:
274:
258:
256:
255:
250:
248:
247:
228:
226:
225:
220:
212:
211:
210:
144:
143:
125:
124:
115:
114:
109:
105:
104:
83:
82:
2210:
2209:
2205:
2204:
2203:
2201:
2200:
2199:
2185:
2184:
2183:
2178:
2149:
2119:
2062:
2021:
2008:low temperature
1997:
1976:
1931:
1887:Meissner effect
1840:
1836:Silsbee current
1809:
1775:Ginzburg–Landau
1753:
1748:
1702:
1700:Further reading
1697:
1696:
1658:
1657:
1653:
1646:
1625:
1624:
1620:
1592:Physical Review
1588:
1587:
1583:
1578:
1569:
1562:
1554:spectral weight
1550:
1517:
1516:
1474:
1464:
1451:
1415:
1405:
1392:
1388:
1336:
1312:
1311:
1271:
1209:
1199:
1186:
1150:
1140:
1127:
1123:
1065:
1041:
1040:
1012:
992:
982:
975:
974:
944:
931:
914:
904:
897:
896:
872:
862:
857:
856:
835:
830:
829:
731:
712:
662:
661:
644:
639:
638:
605:
601:
591:
582:
578:
554:
531:
530:
496:
495:
474:
469:
468:
443:
435:
430:
414:
406:
382:
374:
369:
358:
350:
323:
318:
317:
293:
288:
287:
266:
261:
260:
239:
234:
233:
135:
116:
96:
92:
91:
74:
69:
68:
46:
12:
11:
5:
2208:
2206:
2198:
2197:
2187:
2186:
2180:
2179:
2177:
2176:
2171:
2166:
2161:
2156:
2151:
2147:
2143:
2138:
2133:
2127:
2125:
2121:
2120:
2118:
2117:
2112:
2107:
2102:
2097:
2092:
2087:
2085:electromagnets
2082:
2076:
2074:
2068:
2067:
2064:
2063:
2061:
2060:
2055:
2050:
2045:
2040:
2035:
2029:
2027:
2026:By composition
2023:
2022:
2020:
2019:
2014:
2009:
2005:
2003:
1999:
1998:
1996:
1995:
1993:unconventional
1990:
1984:
1982:
1981:By explanation
1978:
1977:
1975:
1974:
1969:
1968:
1967:
1962:
1957:
1948:
1946:
1939:
1937:Classification
1933:
1932:
1930:
1929:
1924:
1919:
1914:
1909:
1904:
1899:
1894:
1889:
1884:
1879:
1874:
1869:
1864:
1859:
1854:
1848:
1846:
1842:
1841:
1839:
1838:
1833:
1828:
1826:critical field
1823:
1817:
1815:
1811:
1810:
1808:
1807:
1802:
1797:
1795:Mattis–Bardeen
1792:
1787:
1782:
1780:Kohn–Luttinger
1777:
1772:
1767:
1761:
1759:
1755:
1754:
1749:
1747:
1746:
1739:
1732:
1724:
1718:
1717:
1712:Shu-Ang-Zhou,
1710:
1701:
1698:
1695:
1694:
1651:
1644:
1618:
1599:(2): 412–417.
1580:
1579:
1577:
1574:
1567:
1560:
1549:
1546:
1533:
1530:
1527:
1524:
1513:
1512:
1500:
1497:
1489:
1485:
1481:
1477:
1471:
1467:
1463:
1458:
1454:
1450:
1447:
1444:
1441:
1438:
1435:
1430:
1426:
1422:
1418:
1412:
1408:
1404:
1399:
1395:
1391:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1349:
1343:
1339:
1335:
1327:
1324:
1315:
1305:
1302:
1297:
1294:
1291:
1288:
1284:
1277:
1274:
1270:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1240:
1235:
1232:
1224:
1220:
1216:
1212:
1206:
1202:
1198:
1193:
1189:
1185:
1182:
1179:
1176:
1173:
1170:
1165:
1161:
1157:
1153:
1147:
1143:
1139:
1134:
1130:
1126:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1078:
1072:
1068:
1064:
1056:
1053:
1044:
1034:
1029:
1025:
1018:
1015:
1011:
1006:
999:
995:
989:
985:
967:
966:
951:
947:
941:
938:
934:
928:
921:
917:
911:
907:
879:
875:
869:
865:
842:
838:
826:
825:
814:
811:
803:
800:
797:
794:
791:
788:
785:
782:
779:
776:
773:
770:
767:
764:
761:
758:
755:
752:
749:
746:
743:
738:
734:
730:
727:
724:
719:
715:
711:
708:
705:
702:
699:
696:
693:
690:
687:
684:
681:
676:
671:
668:
665:
659:
656:
651:
647:
632:
631:
619:
611:
608:
604:
598:
594:
588:
585:
581:
575:
572:
567:
564:
560:
557:
553:
550:
547:
544:
541:
538:
503:
481:
477:
464:
463:
449:
446:
441:
438:
433:
426:
420:
417:
412:
409:
404:
398:
395:
392:
388:
385:
380:
377:
372:
364:
361:
356:
353:
349:
346:
343:
339:
335:
330:
326:
300:
296:
273:
269:
246:
242:
230:
229:
218:
215:
207:
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201:
198:
195:
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189:
186:
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177:
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168:
165:
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159:
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138:
134:
131:
128:
123:
119:
113:
108:
103:
99:
95:
89:
86:
81:
77:
45:
42:
13:
10:
9:
6:
4:
3:
2:
2207:
2196:
2193:
2192:
2190:
2175:
2172:
2170:
2167:
2165:
2162:
2160:
2157:
2155:
2152:
2150:
2144:
2142:
2139:
2137:
2134:
2132:
2129:
2128:
2126:
2122:
2116:
2113:
2111:
2108:
2106:
2103:
2101:
2098:
2096:
2093:
2091:
2088:
2086:
2083:
2081:
2078:
2077:
2075:
2073:
2069:
2059:
2056:
2054:
2051:
2049:
2046:
2044:
2043:heavy fermion
2041:
2039:
2036:
2034:
2031:
2030:
2028:
2024:
2018:
2015:
2013:
2010:
2007:
2006:
2004:
2000:
1994:
1991:
1989:
1986:
1985:
1983:
1979:
1973:
1972:ferromagnetic
1970:
1966:
1963:
1961:
1958:
1956:
1953:
1952:
1950:
1949:
1947:
1943:
1940:
1938:
1934:
1928:
1925:
1923:
1920:
1918:
1917:supercurrents
1915:
1913:
1910:
1908:
1905:
1903:
1900:
1898:
1895:
1893:
1890:
1888:
1885:
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1878:
1875:
1873:
1870:
1868:
1865:
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1837:
1834:
1832:
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1819:
1818:
1816:
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1798:
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1760:
1756:
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19:
2053:oxypnictides
1988:conventional
1927:superstripes
1872:flux pumping
1867:flux pinning
1862:Cooper pairs
1794:
1713:
1706:
1664:
1660:
1654:
1627:
1621:
1596:
1590:
1584:
1565:
1558:
1551:
1514:
968:
827:
633:
523:
519:
515:
465:
231:
58:Fermi energy
47:
26:
17:
15:
1912:SU(2) color
1892:Homes's law
1319:E(E +
1048:E(E +
60:. From the
50:Cooper pair
2048:iron-based
1907:reentrance
1645:0521592534
1576:References
1845:Phenomena
1523:Θ
1466:Δ
1462:−
1449:ω
1446:ℏ
1407:Δ
1403:−
1379:ω
1376:ℏ
1358:−
1338:Δ
1326:ω
1323:ℏ
1304:Δ
1301:−
1296:ω
1293:ℏ
1290:−
1287:Δ
1283:∫
1276:ω
1273:ℏ
1261:Δ
1255:−
1252:ω
1249:ℏ
1243:Θ
1239:−
1201:Δ
1197:−
1184:ω
1181:ℏ
1142:Δ
1138:−
1114:ω
1111:ℏ
1096:−
1067:Δ
1055:ω
1052:ℏ
1033:∞
1028:Δ
1024:∫
1017:ω
1014:ℏ
994:α
984:α
946:σ
933:σ
916:α
906:α
864:σ
837:σ
796:ω
793:ℏ
778:−
760:×
754:ω
751:ℏ
707:ω
704:ℏ
686:Δ
658:∫
646:α
593:Δ
587:±
543:Δ
445:σ
425:∗
416:σ
397:σ
384:σ
360:σ
345:σ
338:∑
200:ω
197:ℏ
182:−
164:×
158:ω
155:ℏ
88:∫
76:α
2189:Category
2080:cryotron
2038:cuprates
2033:covalent
1790:Matthias
1758:Theories
1689:33268593
610:′
559:′
448:′
440:′
419:′
411:′
387:′
379:′
363:′
355:′
2174:more...
2058:organic
1669:Bibcode
1601:Bibcode
40:(BCS).
1951:Types
1785:London
1687:
1661:Nature
1642:
1515:where
313:where
232:where
2164:TBCCO
2136:BSCCO
2115:wires
2110:SQUID
1685:S2CID
2169:YBCO
2159:NbTi
2154:NbSn
2141:LBCO
1640:ISBN
16:The
2146:MgB
2095:NMR
2090:MRI
1965:1.5
1805:WHH
1800:RVB
1765:BCS
1677:doi
1665:165
1632:doi
1609:doi
1597:111
526:).
518:(Δ,
2191::
1960:II
1683:.
1675:.
1663:.
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1630:.
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524:E'
2148:2
1955:I
1743:e
1736:t
1729:v
1716:.
1691:.
1679::
1671::
1648:.
1634::
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1611::
1603::
1570:T
1568:B
1566:k
1561:c
1559:T
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1529:x
1526:(
1499:E
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1043:[
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1005:=
998:n
988:s
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927:=
920:n
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683:(
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664:|
655:=
650:s
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607:E
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580:(
574:2
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437:k
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408:k
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376:k
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352:k
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342:k
334:=
329:1
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245:s
241:N
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98:M
94:|
85:=
80:s
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