22:
1267:
1088:
521:
The phenomenon of flux quantization was first discovered in superconductors experimentally by B. S. Deaver and W. M. Fairbank and, independently, by R. Doll and M. Näbauer, in 1961. The quantization of magnetic flux is closely related to the
1009:
1185:
1336:
300:
1140:
228:
1016:
691:
and, correspondingly, will not change any physical properties. However, in the superconductor of non-trivial topology, e.g. superconductor with the hole or superconducting loop/cylinder, the phase
896:
1496:. The normal core plays a role of a hole in the superconducting phase. The magnetic field lines pass along this normal core through the whole sample. The screening currents circulate in the
569:
of the irradiation. The
Josephson effect is very widely used to provide a standard for high-precision measurements of potential difference, which (from 1990 to 2019) were related to a fixed,
1180:
934:
2110:"The Feynman Lectures on Physics Vol. III Ch. 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity, Section 21-7: Flux quantization"
1416:
only when the path/trajectory around the hole described above can be chosen so that it lays in the superconducting region without screening currents, i.e. several
925:
516:
485:
To understand this definition in the context of the Dirac flux quantum one shall consider that the effective quasiparticles active in a superconductors are
2036:"Quantized Magnetic Flux in Superconductors: Experiments confirm Fritz London's early concept that superconductivity is a macroscopic quantum phenomenon"
1274:
1099:
1567:
is generally associated with the behaviour of microscopically small systems, whereas the quantization of magnetic flux in a superconductor and the
40:
2264:
2211:
2182:
1894:
1865:
1715:
236:
739:
172:
1460:. When such a superconductor (now without any holes) is placed in a magnetic field with the strength between the first critical field
2133:
1435:) superconducting wire or the cylinder with the similar wall thickness. In the latter case, the flux has a quantum different from
1838:
1262:{\displaystyle \Phi _{B}=\oint \mathbf {A} \cdot d\mathbf {l} ={\frac {\hbar }{q}}\oint \nabla {\theta }\cdot d\mathbf {l} .}
1829:
1505:-vicinity of the core and screen the rest of the superconductor from the magnetic field in the core. In total, each such
1964:
Deaver, Bascom; Fairbank, William (July 1961). "Experimental
Evidence for Quantized Flux in Superconducting Cylinders".
1271:
Now, because the order parameter must return to the same value when the integral goes back to the same point, we have:
1425:
away from the surface. There are geometries where this condition cannot be satisfied, e.g. a loop made of very thin (
58:
1493:
1486:
consists of a normal core—a cylinder of the normal (non-superconducting) phase with a diameter on the order of the
1083:{\displaystyle \mathbf {J} ={\frac {\hbar }{m}}\left(\nabla {\theta }-{\frac {q}{\hbar }}\mathbf {A} \right)\rho .}
531:
1151:
128:
can be as well but increments of flux can be quantized. The wave function can be multivalued as it happens in the
1999:
Doll, R.; Näbauer, M. (July 1961). "Experimental Proof of
Magnetic Flux Quantization in a Superconducting Ring".
1692:
have fixed values, and the
Josephson effect along with the von Klitzing quantum Hall effect becomes the primary
1740:
157:
570:
2310:
2300:
2295:
1730:
928:
1145:
2259:
1705:
1357:
717:
as one goes around the hole/loop and comes to the same starting point. If this is so, then one has
129:
148:
The first to realize the importance of the flux quantum was Dirac in his publication on monopoles
1735:
1720:
1583:
1525:
583:
523:
1457:
36:
1004:{\displaystyle \Psi (\mathbf {r} )={\sqrt {\rho (\mathbf {r} )}}\,e^{i\theta (\mathbf {r} )}.}
1765:
1533:
2035:
1789:
2008:
1973:
1938:
1801:
562:
313:
2232:
2148:
901:
492:
8:
1915:
Loder, F.; Kampf, A. P.; Kopp, T.; Mannhart, J.; Schneider, C. W.; Barash, Y. S. (2008).
1568:
2012:
1977:
1942:
1805:
2305:
2071:
1928:
1857:
1760:
1572:
2063:
2055:
1834:
1750:
1628:
1409:-layer) will always be quantized. However, the value of the flux quantum is equal to
2274:
2075:
2047:
2016:
1981:
1946:
1809:
1529:
1506:
1483:
1479:
725:
558:
2109:
2051:
1886:
2090:
1745:
1710:
1553:
1341:
729:
474:
465:
133:
1528:, the magnetic flux quantum was measured with great precision by exploiting the
2203:
2174:
1755:
1576:
628:
419:
415:
2289:
2059:
2020:
1985:
75:
2067:
1916:
1813:
1725:
1450:
527:
486:
369:
309:
1369:
and usually ≈ 100 nm). The screening currents also flow in this
1384:
inside the superconductor, which perfectly compensates the applied field
733:
618:
The following physical equations use SI units. In CGS units, a factor of
1785:
414:
If one deals with a superconducting ring (i.e. a closed loop path in a
1478:, the field partially penetrates into the superconductor in a form of
1950:
1696:
for the definition of the ampere and other electric units in the SI.
566:
162:
2138:
for the definition of the ampere and other electric units in the SI"
1624:
1331:{\displaystyle \Phi _{B}={\frac {\hbar }{q}}2\pi ={\frac {h}{2e}}.}
295:{\displaystyle \Phi _{0}={\frac {2\pi \hbar c}{q}}={\frac {hc}{q}}}
1933:
84:, threading some contour or loop is defined as the magnetic field
1456:
Flux quantization also plays an important role in the physics of
1135:{\displaystyle \nabla {\theta }={\frac {q}{\hbar }}\mathbf {A} .}
613:
223:{\displaystyle \Phi _{0}={\frac {2\pi \hbar }{q}}={\frac {h}{q}}}
1350:
inside the superconductor is zero. More exactly, magnetic field
1655:
1616:
721:
magnetic flux quanta trapped in the hole/loop, as shown below:
647:— the superconducting order parameter. As any complex function
1356:
penetrates into a superconductor over a small distance called
422:, the magnetic flux threading such a hole/loop is quantized.
1620:
1446:
482:. Its value is, therefore, the same for any superconductor.
402:
2144:
1917:"Magnetic flux periodicity of h/E in superconducting loops"
1092:
Inside the body of the superconductor, the current density
927:
is the charge of the Cooper pair. The wave function is the
406:
308:
The phenomenon of flux quantization was predicted first by
2164:
R. Shankar, "Principles of
Quantum Mechanics", eq. 21.1.44
319:
1561:
obtained until 2019. This may be counterintuitive, since
1013:
Plugged into the expression of the current, one obtains:
464:
is a combination of fundamental physical constants: the
326:
316:
and later discovered experimentally in superconductors (
2208:
The NIST Reference on
Constants, Units, and Uncertainty
2179:
The NIST Reference on
Constants, Units, and Uncertainty
1891:
The NIST Reference on
Constants, Units, and Uncertainty
1862:
The NIST Reference on
Constants, Units, and Uncertainty
1957:
1914:
1790:"Quantised Singularities in the Electromagnetic Field"
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1188:
1154:
1102:
1019:
937:
904:
742:
495:
239:
175:
2092:
Superfluids: Macroscopic theory of superconductivity
1992:
627:
The superconducting properties in each point of the
31:
may be too technical for most readers to understand
1400:The magnetic flux frozen in a loop/hole (plus its
1330:
1261:
1174:
1134:
1082:
1003:
919:
891:{\displaystyle \mathbf {J} ={\frac {1}{2m}}\left.}
890:
510:
294:
222:
2095:. John Wiley & Sons. pp. 152 (footnote).
2287:
1852:
1850:
1552:, this provided the most accurate values of the
1378:-layer near the surface, creating magnetization
586:, the Josephson constant has an exact value of
557:. It is the constant of proportionality of the
136:. The unit of quantization is therefore called
1963:
1826:
1519:
614:Derivation of the superconducting flux quantum
143:
1847:
1445:The flux quantization is a key idea behind a
1716:Committee on Data for Science and Technology
1615:which, together with the definitions of the
1175:{\displaystyle \nabla \times \mathbf {A} =B}
2196:
2167:
1998:
1879:
1532:. When coupled with the measurement of the
1858:"2022 CODATA Value: magnetic flux quantum"
1623:, provides the official definition of the
2262:and flux quantization in superconductors
1932:
1658:. Therefore, both the Josephson constant
1358:London's magnetic field penetration depth
973:
59:Learn how and when to remove this message
43:, without removing the technical details.
695:may continuously change from some value
489:with an effective charge of 2 electrons
122:can be arbitrary, meaning that the flux
1887:"2022 CODATA Value: Josephson constant"
1144:Integrating around the hole/loop using
2288:
2204:"2022 CODATA Value: elementary charge"
2088:
1820:
2104:
2102:
2033:
1784:
1509:carries one quantum of magnetic flux
1449:, which is one of the most sensitive
327:Superconducting magnetic flux quantum
41:make it understandable to non-experts
2175:"2022 CODATA Value: Planck constant"
15:
1830:Introduction to Solid State Physics
573:of the Josephson constant, denoted
565:across a Josephson junction to the
13:
2253:
2099:
1794:Proceedings of the Royal Society A
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1103:
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14:
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1833:. Wiley & Sons. p. 281.
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1059:
822:
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674:is the phase. Changing the phase
635:quantum mechanical wave function
537:The inverse of the flux quantum,
262:
198:
1494:superconducting coherence length
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1217:
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1021:
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20:
2225:
2158:
2126:
929:Ginzburg–Landau order parameter
526:, but was predicted earlier by
71:Quantized unit of magnetic flux
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2027:
1908:
1778:
1675:and the von Klitzing constant
1469:and the second critical field
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1:
2052:10.1126/science.146.3650.1429
1771:
1741:Macroscopic quantum phenomena
1579:large numbers of particles.
152:Dirac magnetic flux quantum
90:multiplied by the loop area
78:, represented by the symbol
7:
2114:feynmanlectures.caltech.edu
2034:Parks, R. D. (1964-12-11).
1699:
1520:Measuring the magnetic flux
1397:inside the superconductor.
317:
144:Dirac magnetic flux quantum
10:
2327:
2233:"BIPM – mises en pratique"
1631:also has a fixed value of
736:in the superconductor is:
1344:, the magnetic induction
333:
161:
156:
2021:10.1103/PhysRevLett.7.51
1986:10.1103/PhysRevLett.7.43
2266:(physics stackexchange)
2001:Physical Review Letters
1966:Physical Review Letters
1827:C. Kittel (1953–1976).
1736:Husimi Q representation
1721:Domain wall (magnetism)
1584:2019 revision of the SI
1526:2019 revision of the SI
1458:type II superconductors
1096:is zero, and therefore
584:2019 revision of the SI
2089:London, Fritz (1950).
1814:10.1098/rspa.1931.0130
1731:Ginzburg–Landau theory
1586:, the Planck constant
1332:
1263:
1176:
1136:
1084:
1005:
921:
892:
532:phenomenological model
512:
425:The (superconducting)
418:) or a hole in a bulk
296:
224:
2272:David tong lectures:
1766:von Klitzing constant
1534:von Klitzing constant
1333:
1264:
1177:
1137:
1085:
1006:
922:
893:
668:is the amplitude and
631:are described by the
513:
427:magnetic flux quantum
297:
225:
138:magnetic flux quantum
2260:Aharonov–Bohm effect
1706:Aharonov–Bohm effect
1390:, thus resulting in
1275:
1186:
1152:
1100:
1017:
935:
920:{\displaystyle q=2e}
902:
740:
563:potential difference
511:{\displaystyle q=2e}
493:
314:Aharanov-Bohm effect
237:
173:
130:Aharonov–Bohm effect
2276:Quantum hall effect
2046:(3650): 1429–1435.
2013:1961PhRvL...7...51D
1978:1961PhRvL...7...43D
1943:2008NatPh...4..112L
1806:1931RSPSA.133...60D
1800:(821). London: 60.
1627:. Furthermore, the
1582:As a result of the
1569:quantum Hall effect
524:Little–Parks effect
153:
132:or quantized as in
2311:Physical constants
1761:Topological defect
1592:has a fixed value
1573:emergent phenomena
1480:Abrikosov vortices
1328:
1259:
1172:
1132:
1080:
1001:
917:
888:
651:can be written as
571:conventional value
546:Josephson constant
508:
292:
220:
151:
2301:Quantum magnetism
2296:Superconductivity
1840:978-0-471-49024-1
1751:Magnetic monopole
1629:elementary charge
1577:thermodynamically
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2147:. Archived from
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2136:Mise en pratique
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1575:associated with
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1530:Josephson effect
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1507:Abrikosov vortex
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703:
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559:Josephson effect
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544:, is called the
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530:in 1948 using a
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2254:Further reading
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1746:Magnetic domain
1711:Brian Josephson
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1554:Planck constant
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1146:Stokes' theorem
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475:electron charge
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466:Planck constant
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134:superconductors
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37:help improve it
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2154:on 2021-03-08.
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1991:
1956:
1927:(2): 112–115.
1921:Nature Physics
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1756:Quantum vortex
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788:
783:
779:
774:
769:
762:
759:
755:
750:
746:
709:
700:
664:
654:
629:superconductor
624:would appear.
615:
612:
608:... GHz⋅V
592:
579:
554:
540:
507:
504:
501:
498:
432:
420:superconductor
416:superconductor
410:
409:
400:
384:
381:
373:
372:
367:
348:
345:
339:
338:
335:
334:CODATA values
328:
325:
304:
303:
289:
285:
282:
276:
271:
267:
264:
261:
258:
252:
247:
243:
231:
217:
214:
209:
204:
200:
197:
194:
188:
183:
179:
166:
165:
160:
145:
142:
70:
67:
66:
28:
26:
19:
9:
6:
4:
3:
2:
2323:
2312:
2309:
2307:
2304:
2302:
2299:
2297:
2294:
2293:
2291:
2278:
2277:
2271:
2268:
2267:
2261:
2258:
2257:
2238:
2234:
2228:
2213:
2209:
2205:
2199:
2184:
2180:
2176:
2170:
2161:
2150:
2146:
2139:
2137:
2129:
2115:
2111:
2105:
2103:
2094:
2093:
2085:
2077:
2073:
2069:
2065:
2061:
2057:
2053:
2049:
2045:
2041:
2037:
2030:
2022:
2018:
2014:
2010:
2006:
2002:
1995:
1987:
1983:
1979:
1975:
1971:
1967:
1960:
1952:
1948:
1944:
1940:
1935:
1930:
1926:
1922:
1918:
1911:
1896:
1892:
1888:
1882:
1867:
1863:
1859:
1853:
1851:
1842:
1836:
1832:
1831:
1823:
1815:
1811:
1807:
1803:
1799:
1795:
1791:
1787:
1781:
1777:
1767:
1764:
1762:
1759:
1757:
1754:
1752:
1749:
1747:
1744:
1742:
1739:
1737:
1734:
1732:
1729:
1727:
1724:
1722:
1719:
1717:
1714:
1712:
1709:
1707:
1704:
1703:
1697:
1695:
1690:
1686:
1679:
1673:
1669:
1662:
1657:
1635:
1630:
1626:
1622:
1618:
1597:
1590:
1585:
1580:
1578:
1574:
1570:
1565:
1559:
1555:
1550:
1546:
1539:
1535:
1531:
1527:
1524:Prior to the
1517:
1508:
1500:
1495:
1490:
1485:
1481:
1473:
1464:
1459:
1454:
1452:
1451:magnetometers
1448:
1443:
1430:
1420:
1404:
1398:
1394:
1388:
1382:
1373:
1364:
1359:
1354:
1348:
1343:
1338:
1325:
1319:
1316:
1312:
1307:
1304:
1301:
1296:
1293:
1288:
1283:
1269:
1256:
1248:
1245:
1241:
1234:
1229:
1226:
1221:
1213:
1210:
1202:
1199:
1194:
1169:
1166:
1158:
1147:
1142:
1129:
1116:
1111:
1107:
1095:
1090:
1077:
1074:
1070:
1056:
1051:
1047:
1039:
1033:
1030:
1025:
1011:
998:
982:
979:
975:
957:
952:
930:
914:
911:
908:
905:
885:
881:
875:
852:
849:
846:
842:
836:
819:
816:
807:
792:
789:
781:
772:
767:
760:
757:
753:
748:
735:
731:
727:
722:
715:
708:
704:to the value
699:
685:
678:
672:
659:
644:
640:
634:
630:
625:
622:
611:
590:
585:
577:
572:
568:
564:
560:
552:
547:
535:
533:
529:
525:
519:
505:
502:
499:
496:
488:
483:
480:
476:
471:
467:
442:
438:
428:
423:
421:
417:
408:
404:
401:
385:
379:
375:
374:
371:
368:
349:
341:
340:
336:
332:
324:
321:
315:
311:
302:
287:
283:
280:
274:
269:
265:
259:
256:
250:
245:
232:
230:
215:
212:
207:
202:
195:
192:
186:
181:
168:
167:
164:
159:
155:
149:
141:
139:
135:
131:
126:
120:
114:
108:
104:
100:
94:
88:
82:
77:
76:magnetic flux
63:
60:
52:
42:
38:
32:
29:This article
27:
18:
17:
2275:
2265:
2240:. Retrieved
2237:www.bipm.org
2236:
2227:
2216:. Retrieved
2207:
2198:
2187:. Retrieved
2178:
2169:
2160:
2149:the original
2135:
2128:
2117:. Retrieved
2113:
2091:
2084:
2043:
2039:
2029:
2007:(2): 51–52.
2004:
2000:
1994:
1972:(2): 43–46.
1969:
1965:
1959:
1924:
1920:
1910:
1899:. Retrieved
1890:
1881:
1870:. Retrieved
1861:
1828:
1822:
1797:
1793:
1780:
1726:Flux pinning
1693:
1688:
1684:
1677:
1671:
1667:
1660:
1633:
1611:10 J⋅Hz
1595:
1588:
1581:
1563:
1557:
1548:
1544:
1537:
1523:
1498:
1488:
1471:
1462:
1455:
1444:
1428:
1418:
1402:
1399:
1392:
1386:
1380:
1371:
1362:
1352:
1346:
1339:
1270:
1143:
1093:
1091:
1012:
734:Cooper pairs
723:
713:
706:
697:
683:
676:
670:
657:
642:
638:
632:
626:
620:
617:
588:
582:. With the
575:
550:
545:
536:
528:Fritz London
520:
487:Cooper pairs
484:
478:
469:
440:
436:
426:
424:
413:
377:
310:Fritz London
307:
233:
169:
147:
137:
124:
118:
112:
106:
102:
98:
92:
86:
80:
73:
55:
46:
30:
1786:Dirac, Paul
1453:available.
1340:Due to the
2290:Categories
2242:2020-01-21
2218:2024-05-18
2214:. May 2024
2189:2024-05-18
2185:. May 2024
2119:2020-01-21
1901:2024-05-18
1897:. May 2024
1872:2024-05-18
1868:. May 2024
1772:References
458:10 Wb
49:April 2024
2306:Metrology
2060:0036-8075
1934:0709.4111
1649:10 C
1571:are both
1360:(denoted
1305:π
1294:ℏ
1280:Φ
1246:⋅
1242:θ
1238:∇
1235:∮
1227:ℏ
1211:⋅
1203:∮
1191:Φ
1159:×
1156:∇
1120:ℏ
1108:θ
1104:∇
1075:ρ
1060:ℏ
1052:−
1048:θ
1044:∇
1031:ℏ
983:θ
958:ρ
939:Ψ
866:Ψ
847:−
837:∗
833:Ψ
826:∇
823:ℏ
817:−
811:Ψ
808:−
805:Ψ
799:∇
796:ℏ
790:−
782:∗
778:Ψ
567:frequency
263:ℏ
260:π
242:Φ
199:ℏ
196:π
178:Φ
163:CGS units
2076:30913579
2068:17753357
1788:(1931).
1700:See also
1625:kilogram
1619:and the
661:, where
473:and the
391:.8484...
158:SI units
2040:Science
2009:Bibcode
1974:Bibcode
1939:Bibcode
1802:Bibcode
1652:
1182:gives:
633:complex
461:
398:
365:
110:. Both
96:, i.e.
35:Please
2074:
2066:
2058:
1837:
1656:ampere
1617:second
1492:, the
1482:. The
898:where
728:, the
337:Units
2280:(PDF)
2152:(PDF)
2141:(PDF)
2072:S2CID
1929:arXiv
1639:1.602
1621:metre
1601:6.626
1447:SQUID
653:Ψ = Ψ
447:2.067
351:2.067
320:below
2212:NIST
2183:NIST
2145:BIPM
2064:PMID
2056:ISSN
1895:NIST
1866:NIST
1835:ISBN
1148:and
724:Per
601:.848
580:J-90
318:see
116:and
74:The
2048:doi
2044:146
2017:doi
1982:doi
1947:doi
1810:doi
1798:133
1666:= 2
1644:634
1641:176
1603:070
1395:= 0
732:of
712:+ 2
680:by
603:416
599:597
597:483
539:1/Φ
454:...
452:848
449:833
439:/(2
389:597
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356:848
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2292::
2235:.
2210:.
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2181:.
2177:.
2143:.
2112:.
2101:^
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2054:.
2042:.
2038:.
2015:.
2003:.
1980:.
1968:.
1945:.
1937:.
1923:.
1919:.
1893:.
1889:.
1864:.
1860:.
1849:^
1808:.
1796:.
1792:.
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1516:.
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1442:.
1427:≤
931::
714:πn
684:πn
641:,
637:Ψ(
610:.
606:98
595:=
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518:.
445:≈
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403:Hz
395:10
370:Wb
362:10
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105:⋅
101:=
2245:.
2221:.
2192:.
2134:"
2122:.
2078:.
2050::
2023:.
2019::
2011::
2005:7
1988:.
1984::
1976::
1970:7
1953:.
1949::
1941::
1931::
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1816:.
1812::
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1326:.
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1253:l
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1222:=
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1200:=
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1167:=
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1130:.
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693:θ
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643:t
639:r
621:c
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343:Φ
288:q
284:c
281:h
275:=
270:q
266:c
257:2
251:=
246:0
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213:h
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187:=
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125:Φ
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113:B
107:S
103:B
99:Φ
93:S
87:B
81:Φ
62:)
56:(
51:)
47:(
33:.
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