1433:
contain a blackbody-distributed photon gas. Unlike a massive gas, this gas will exist without the photons being introduced from the outside – the walls will provide the photons for the gas. Suppose the piston is pushed all the way into the cylinder so that there is an extremely small volume. The photon gas inside the volume will press against the piston, moving it outward, and in order for the transformation to be isothermic, a counter force of almost the same value will have to be applied to the piston so that the motion of the piston is very slow. This force will be equal to the pressure times the cross sectional area (
119:(gas of massive bosons) and a photon gas with a black-body distribution is that the number of photons in the photon gas is not conserved. A photon can be created upon thermal excitation of an atom in the wall into an upper electronic state, followed by the emission of a photon when the atom falls back to a lower energetic state. This type of photon generation is called thermal emission. The reverse process can also take place, resulting in a photon being destroyed and removed from the gas. It can be shown that, as a result of such processes there is no constraint on the number of photons in the system, and the
108:), so the equilibrium distribution must be established by other means. The most common way that an equilibrium distribution is established is by the interaction of the photons with matter. If the photons are absorbed and emitted by the walls of the system containing the photon gas, and the walls are at a particular temperature, then the equilibrium distribution for the photons will be a
1851:
In low-dimensional systems, for example in dye-solution filled optical microcavities with a distance between the resonator mirrors in the wavelength range where the situation becomes two-dimensional, also photon gases with tunable chemical potential can be realized. Such a photon gas in many respects
1432:
As an example of a thermodynamic process involving a photon gas, consider a cylinder with a movable piston. The interior walls of the cylinder are "black" in order that the temperature of the photons can be maintained at a particular temperature. This means that the space inside the cylinder will
104:. This distribution is established as the particles collide with each other, exchanging energy (and momentum) in the process. In a photon gas, there will also be an equilibrium distribution, but photons do not collide with each other (except under very extreme conditions, see
1162:
1292:
251:
1002:
886:
500:
390:
1777:
672:
82:
of the black-body photon gas is zero at thermodynamic equilibrium. The number of state variables needed to describe a black-body state is thus reduced from three to two (e.g. temperature and volume).
1609:
1521:
1671:
2041:
1050:
1385:
1337:
765:
1830:
78:
distribution is established by the interaction of the photons with matter, usually the walls of the container, and the number of photons is not conserved. As a result, the
1175:
533:
1037:
712:
1420:
582:
140:
899:
2266:
795:
406:
678:
The following table summarizes the thermodynamic state functions for a black-body photon gas. Notice that the pressure can be written in the form
1881:
298:
1987:
1683:
594:
1843:
is the enthalpy at the end of the transformation. It is seen that the enthalpy is the amount of energy needed to create the photon gas.
1852:
behaves like a gas of material particles. One consequence of the tunable chemical potential is that at high phase space densities then
101:
2156:
1965:
2054:
1537:
130:, with the radiation field being in equilibrium with the atoms in the wall. The derivation yields the spectral energy density
2334:
2172:
J. Klaers; J. Schmitt; F. Vewinger & M. Weitz (2010). "Bose–Einstein condensation of photons in an optical microcavity".
549:
If we note that the equation of state for an ultra-relativistic quantum gas (which inherently describes photons) is given by
1458:
51:
2230:
1979:
1853:
588:
then we can combine the above formulas to produce an equation of state that looks much like that of an ideal gas:
1922:
2297:
2144:
1891:
1621:
1157:{\displaystyle P={\frac {1}{3}}\,{\frac {U}{V}}=\left({\frac {\pi ^{2}k^{4}}{45c^{3}\hbar ^{3}}}\right)\,T^{4}}
39:– including pressure, temperature, and entropy. The most common example of a photon gas in equilibrium is the
1439:) of the piston. This process can be continued at a constant temperature until the photon gas is at a volume
2329:
1350:
1305:
1343:
536:
2036:
727:
2281:
2245:
2109:
1931:
40:
134:, which is the energy of the radiation field per unit volume per unit frequency interval, given by:
1789:
1287:{\displaystyle S={\frac {4U}{3T}}=\left({\frac {4\pi ^{2}k^{4}}{45c^{3}\hbar ^{3}}}\right)\,VT^{3}}
546:
varies with the volume in a fixed manner, adjusting itself to have a constant density of photons.
2207:
2181:
1897:
1008:
120:
105:
79:
1943:
509:
2199:
2152:
2125:
2069:
2013:
1983:
1955:
1391:
1015:
681:
246:{\displaystyle u(\nu ,T)={\frac {8\pi h\nu ^{3}}{c^{3}}}~{\frac {1}{e^{\frac {h\nu }{kT}}-1}}}
2324:
2289:
2253:
2191:
2117:
1398:
555:
2009:
1971:
1917:
892:
788:
540:
289:
262:
71:
62:
with only one type of particle is uniquely described by three state functions such as the
2028:
997:{\displaystyle N=\left({\frac {2k^{3}\zeta (3)}{\pi ^{2}c^{3}\hbar ^{3}}}\right)\,VT^{3}}
91:
2285:
2249:
2113:
1935:
2097:
2065:
2005:
1951:
55:
2318:
1885:
2211:
1865:
127:
100:
with massive particles, the energy of the particles is distributed according to a
881:{\displaystyle U=\left({\frac {\pi ^{2}k^{4}}{15c^{3}\hbar ^{3}}}\right)\,VT^{4}}
63:
1452:) traveled yields the total work done to create this photon gas at this volume
2032:
109:
2129:
495:{\displaystyle N=\left({\frac {16\pi k^{3}\zeta (3)}{(hc)^{3}}}\right)VT^{3}}
1876:
97:
2203:
1871:
1298:
1043:
385:{\displaystyle U=\left({\frac {8\pi ^{5}k^{4}}{15(hc)^{3}}}\right)VT^{4}}
116:
59:
32:
2195:
1168:
2293:
2257:
2121:
1772:{\displaystyle W=-{\frac {bT^{4}Ax_{0}}{3}}=-{\frac {bT^{4}V_{0}}{3}}}
2055:"On the Theory of the Energy Distribution Law of the Normal Spectrum"
75:
67:
28:
1783:
The amount of heat that must be added in order to create the gas is
667:{\displaystyle PV={\frac {\zeta (4)}{\zeta (3)}}NkT\approx 0.9\,NkT}
126:
The thermodynamics of a black-body photon gas may be derived using
31:, which has many of the same properties of a conventional gas like
2186:
2037:"Zur Theorie des Gesetzes der Energieverteilung im Normalspectrum"
47:
2073:
1959:
36:
24:
396:
The derivation also yields the (expected) number of photons
1868:– derivation of distribution functions for all ideal gases
722:
Thermodynamic state functions for a black-body photon gas
284:
Integrating over frequency and multiplying by the volume,
1604:{\displaystyle b={\frac {8\pi ^{5}k^{4}}{15c^{3}h^{3}}}}
2042:
Verhandlungen der
Deutschen Physikalischen Gesellschaft
123:
of the photons must be zero for black-body radiation.
1792:
1686:
1624:
1540:
1516:{\displaystyle W=-\int _{0}^{x_{0}}P(A\mathrm {d} x)}
1461:
1401:
1353:
1308:
1178:
1053:
1018:
902:
798:
730:
684:
597:
558:
512:
409:
301:
143:
720:
46:Photons are part of a family of particles known as
1824:
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1414:
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527:
494:
384:
245:
2098:"Teaching the photon gas in introductory physics"
1978:. Vol. 3. English translation by Beck, A.
539:. Note that for a particular temperature, the
115:A very important difference between a generic
8:
1920:(1917). "Zur Quantentheorie der Strahlung".
1847:Photon gases with tunable chemical potential
1446:. Integrating the force over the distance (
1666:{\displaystyle P(x)={\frac {bT^{4}}{3}}\,}
2185:
2151:. Springer Science & Business Media.
1821:
1815:
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192:
181:
165:
142:
86:Thermodynamics of a black body photon gas
2265:Herrmann, F.; WĂĽrfel, P. (August 2005).
1894:– the total flux emitted by a black body
1888:as a function of frequency or wavelength
128:quantum statistical mechanical arguments
2267:"Light with nonzero chemical potential"
1976:The Collected Papers of Albert Einstein
1909:
1254:
1127:
964:
848:
734:
277: is the Boltzmann constant, and
7:
2091:
2089:
2087:
2085:
2083:
1944:"On the Quantum Theory of Radiation"
1882:Planck's law of black-body radiation
1380:{\displaystyle A=-{\frac {1}{3}}\,U}
1677:Integrating, the work done is just
1332:{\displaystyle H={\frac {4}{3}}\,U}
1503:
714:, which is independent of volume (
112:distribution at that temperature.
14:
74:. However, for a black body, the
2231:"The elusive chemical potential"
760:{\displaystyle (\hbar =h/2\pi )}
2229:Baierlein, Ralph (April 2001).
1531: has been used. Defining
2096:Leff, Harvey S. (2002-07-12).
2018:(2nd ed.). W. H. Freeman.
1634:
1628:
1510:
1496:
938:
932:
754:
731:
633:
627:
619:
613:
522:
516:
463:
453:
448:
442:
353:
343:
269: is the speed of light,
159:
147:
102:Maxwell–Boltzmann distribution
92:Planck's law § Photon Gas
16:Gas-like collection of photons
1:
2143:Schwabl, Franz (2006-06-13).
1825:{\displaystyle Q=U-W=H_{0}\,}
292:of a black-body photon gas:
2274:American Journal of Physics
2238:American Journal of Physics
2102:American Journal of Physics
2351:
1980:Princeton University Press
1854:Bose-Einstein condensation
1428:Isothermal transformations
89:
1923:Physikalische Zeitschrift
528:{\displaystyle \zeta (n)}
273: is the frequency,
1856:of photons is observed.
1032:{\displaystyle \mu =0\,}
707:{\displaystyle P=bT^{4}}
52:Bose–Einstein statistics
50:, particles that follow
1527:where the relationship
281: is temperature.
2062:The Old Quantum Theory
1948:The Old Quantum Theory
1884:– the distribution of
1826:
1773:
1667:
1605:
1517:
1416:
1381:
1333:
1288:
1158:
1033:
998:
882:
761:
708:
668:
578:
529:
496:
386:
247:
2335:Statistical mechanics
2149:Statistical Mechanics
2053:ter Haar, D. (1967).
1942:ter Haar, D. (1967).
1827:
1774:
1668:
1606:
1518:
1417:
1415:{\displaystyle G=0\,}
1382:
1344:Helmholtz free energy
1334:
1289:
1159:
1034:
999:
883:
762:
709:
669:
579:
577:{\displaystyle U=3PV}
537:Riemann zeta function
530:
497:
387:
248:
1954:. pp. 167–183.
1892:Stefan–Boltzmann law
1790:
1684:
1622:
1538:
1459:
1399:
1351:
1306:
1176:
1051:
1016:
900:
796:
728:
682:
595:
556:
510:
407:
299:
141:
41:black-body radiation
27:-like collection of
2286:2005AmJPh..73..717H
2250:2001AmJPh..69..423B
2196:10.1038/nature09567
2114:2002AmJPh..70..792L
1936:1917PhyZ...18..121E
1492:
769:
72:number of particles
1898:Radiation pressure
1822:
1769:
1663:
1601:
1513:
1471:
1412:
1377:
1329:
1284:
1154:
1029:
1009:Chemical potential
994:
878:
757:
721:
704:
664:
574:
525:
492:
382:
243:
121:chemical potential
106:two-photon physics
80:chemical potential
2294:10.1119/1.1904623
2258:10.1119/1.1336839
2122:10.1119/1.1479743
1989:978-0-691-10250-4
1767:
1729:
1660:
1599:
1425:
1424:
1392:Gibbs free energy
1371:
1323:
1264:
1203:
1137:
1079:
1068:
974:
858:
637:
473:
363:
241:
231:
202:
198:
54:and with integer
2342:
2311:
2309:
2308:
2302:
2296:. Archived from
2271:
2261:
2235:
2216:
2215:
2189:
2169:
2163:
2162:
2145:"4.5 Photon gas"
2140:
2134:
2133:
2093:
2078:
2077:
2059:
2050:
2026:
2020:
2019:
2002:
1996:
1993:
1963:
1939:
1914:
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1820:
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1778:
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1711:
1710:
1697:
1672:
1670:
1669:
1664:
1661:
1656:
1655:
1654:
1641:
1615:The pressure is
1610:
1608:
1607:
1602:
1600:
1598:
1597:
1596:
1587:
1586:
1573:
1572:
1571:
1562:
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1548:
1522:
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1514:
1506:
1491:
1490:
1489:
1479:
1451:
1438:
1421:
1419:
1418:
1413:
1386:
1384:
1383:
1378:
1372:
1364:
1338:
1336:
1335:
1330:
1324:
1316:
1293:
1291:
1290:
1285:
1283:
1282:
1269:
1265:
1263:
1262:
1261:
1252:
1251:
1238:
1237:
1236:
1227:
1226:
1213:
1204:
1202:
1194:
1186:
1163:
1161:
1160:
1155:
1153:
1152:
1142:
1138:
1136:
1135:
1134:
1125:
1124:
1111:
1110:
1109:
1100:
1099:
1089:
1080:
1072:
1069:
1061:
1038:
1036:
1035:
1030:
1003:
1001:
1000:
995:
993:
992:
979:
975:
973:
972:
971:
962:
961:
952:
951:
941:
928:
927:
914:
887:
885:
884:
879:
877:
876:
863:
859:
857:
856:
855:
846:
845:
832:
831:
830:
821:
820:
810:
775:State function (
770:
768:
766:
764:
763:
758:
747:
718:is a constant).
713:
711:
710:
705:
703:
702:
673:
671:
670:
665:
638:
636:
622:
608:
583:
581:
580:
575:
534:
532:
531:
526:
501:
499:
498:
493:
491:
490:
478:
474:
472:
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451:
438:
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421:
391:
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381:
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368:
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361:
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338:
337:
336:
327:
326:
313:
252:
250:
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244:
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233:
232:
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214:
204:
200:
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196:
187:
186:
185:
166:
2350:
2349:
2345:
2344:
2343:
2341:
2340:
2339:
2315:
2314:
2306:
2304:
2300:
2269:
2264:
2233:
2228:
2225:
2223:Further reading
2220:
2219:
2171:
2170:
2166:
2159:
2142:
2141:
2137:
2095:
2094:
2081:
2057:
2052:
2031:
2027:
2023:
2015:Thermal Physics
2010:Herbert Kroemer
2006:Kittel, Charles
2004:
2003:
1999:
1990:
1970:
1941:
1916:
1915:
1911:
1906:
1886:photon energies
1862:
1849:
1841:
1811:
1788:
1787:
1753:
1743:
1739:
1715:
1702:
1698:
1682:
1681:
1646:
1642:
1620:
1619:
1588:
1578:
1574:
1563:
1553:
1549:
1536:
1535:
1481:
1457:
1456:
1447:
1444:
1434:
1430:
1397:
1396:
1349:
1348:
1304:
1303:
1274:
1253:
1243:
1239:
1228:
1218:
1214:
1208:
1195:
1187:
1174:
1173:
1144:
1126:
1116:
1112:
1101:
1091:
1090:
1084:
1049:
1048:
1014:
1013:
984:
963:
953:
943:
942:
919:
915:
909:
898:
897:
893:Particle number
868:
847:
837:
833:
822:
812:
811:
805:
794:
793:
789:Internal energy
726:
725:
723:
694:
680:
679:
623:
609:
593:
592:
554:
553:
541:particle number
508:
507:
482:
462:
452:
429:
422:
416:
405:
404:
372:
352:
339:
328:
318:
314:
308:
297:
296:
290:internal energy
263:Planck constant
223:
215:
209:
208:
188:
177:
167:
139:
138:
96:In a classical
94:
88:
17:
12:
11:
5:
2348:
2346:
2338:
2337:
2332:
2330:Thermodynamics
2327:
2317:
2316:
2313:
2312:
2280:(8): 717–723.
2262:
2244:(4): 423–434.
2224:
2221:
2218:
2217:
2164:
2157:
2135:
2108:(8): 792–797.
2079:
2068:. p. 82.
2066:Pergamon Press
2051:Translated in
2021:
2012:(1980-01-15).
1997:
1995:
1994:
1988:
1952:Pergamon Press
1940:Translated in
1908:
1907:
1905:
1902:
1901:
1900:
1895:
1889:
1879:
1874:
1869:
1861:
1858:
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1839:
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1814:
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1807:
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1801:
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1714:
1709:
1705:
1701:
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1675:
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1659:
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1645:
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1627:
1613:
1612:
1595:
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1570:
1566:
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1546:
1543:
1525:
1524:
1512:
1509:
1505:
1501:
1498:
1495:
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1484:
1478:
1474:
1470:
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1442:
1429:
1426:
1423:
1422:
1410:
1407:
1404:
1394:
1388:
1387:
1376:
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1367:
1362:
1359:
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1346:
1340:
1339:
1328:
1322:
1319:
1314:
1311:
1301:
1295:
1294:
1281:
1277:
1273:
1268:
1260:
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1250:
1246:
1242:
1235:
1231:
1225:
1221:
1217:
1211:
1207:
1201:
1198:
1193:
1190:
1184:
1181:
1171:
1165:
1164:
1151:
1147:
1141:
1133:
1129:
1123:
1119:
1115:
1108:
1104:
1098:
1094:
1087:
1083:
1078:
1075:
1067:
1064:
1059:
1056:
1046:
1040:
1039:
1027:
1024:
1021:
1011:
1005:
1004:
991:
987:
983:
978:
970:
966:
960:
956:
950:
946:
940:
937:
934:
931:
926:
922:
918:
912:
908:
905:
895:
889:
888:
875:
871:
867:
862:
854:
850:
844:
840:
836:
829:
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819:
815:
808:
804:
801:
791:
785:
784:
773:
756:
753:
750:
746:
742:
739:
736:
733:
701:
697:
693:
690:
687:
676:
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663:
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653:
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629:
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621:
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612:
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603:
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586:
585:
573:
570:
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19:In physics, a
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60:gas of bosons
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2305:. Retrieved
2298:the original
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2237:
2177:
2173:
2167:
2148:
2138:
2105:
2101:
2061:
2046:
2040:
2029:Planck's law
2024:
2014:
2000:
1975:
1972:Einstein, A.
1947:
1927:
1921:
1918:Einstein, A.
1912:
1866:Gas in a box
1850:
1837:
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1782:
1676:
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1526:
1448:
1440:
1435:
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780:
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587:
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288:, gives the
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283:
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266:
258:
256:
131:
125:
114:
95:
45:
20:
18:
2180:: 545–548.
1930:: 121–128.
64:temperature
2319:Categories
2307:2012-06-29
2049:: 237–245.
2033:Planck, M.
1904:References
110:black-body
90:See also:
70:, and the
21:photon gas
2187:1007.4088
2130:0002-9505
1964:See also
1877:Fermi gas
1803:−
1735:−
1694:−
1555:π
1473:∫
1469:−
1361:−
1255:ℏ
1220:π
1128:ℏ
1093:π
1020:μ
965:ℏ
945:π
930:ζ
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814:π
752:π
735:ℏ
649:≈
625:ζ
611:ζ
514:ζ
440:ζ
427:π
320:π
235:−
220:ν
179:ν
172:π
151:ν
98:ideal gas
2204:21107426
2074:66029628
2035:(1900).
1974:(1993).
1960:66029628
1872:Bose gas
1860:See also
1299:Enthalpy
1044:Pressure
117:Bose gas
33:hydrogen
2325:Photons
2282:Bibcode
2246:Bibcode
2212:4349640
2110:Bibcode
1932:Bibcode
1169:Entropy
767:
724:
535:is the
261:is the
29:photons
2210:
2202:
2174:Nature
2155:
2128:
2072:
1986:
1958:
1836:where
1529:V = Ax
506:where
257:where
201:
76:energy
68:volume
48:bosons
2301:(PDF)
2270:(PDF)
2234:(PDF)
2208:S2CID
2182:arXiv
2058:(PDF)
23:is a
2200:PMID
2153:ISBN
2126:ISSN
2070:LCCN
1984:ISBN
1956:LCCN
58:. A
56:spin
37:neon
2290:doi
2254:doi
2192:doi
2178:468
2118:doi
652:0.9
35:or
25:gas
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