90:
67:, no two electrons can occupy the same state, and it is not possible for all the electrons to have zero kinetic energy. Instead, the confinement makes the allowed energy levels quantized, and the electrons fill them from the bottom upwards. If many electrons are confined to a small volume, on average the electrons have a large kinetic energy, and a large pressure is exerted.
50:
In metals and white dwarf stars, electrons can be modeled as a gas of non-interacting electrons confined to a finite volume. In reality, there are strong electromagnetic forces between the negatively charged electrons. However, these are balanced by the positive nuclei, and neglected in the simplest
70:
In white dwarf stars, the positive nuclei are completely ionized ā disassociated from the electrons ā and closely packed ā a million times more dense than the Sun. At this density gravity exerts immense force pulling the nuclei together. This force is balanced by the electron degeneracy pressure
308:
74:
In metals, the positive nuclei are partly ionized and spaced by normal interatomic distances. Gravity has negligible effect; the positive ion cores are attracted to the negatively charge electron gas. This force is balanced by the electron degeneracy pressure.
463:
200:
343:
478:(the number of free electrons per unit volume). For the case of a metal, one can prove that this equation remains approximately true for temperatures lower than the Fermi temperature, about
171:
541:. In the appropriate systems, the free electron pressure can be calculated; it can be shown that this pressure is an important contributor to the compressibility or
101:) in three dimensions. Pauli repulsion in fermions gives them an additional pressure over an equivalent classical gas, most significantly at low temperature.
59:. The degeneracy pressure is most prominent at low temperatures: If electrons were classical particles, the movement of the electrons would cease at
565:
star from collapsing. A star exceeding this limit and without significant thermally generated pressure will continue to collapse to form either a
633:
An english translation of the original work of Enrico Fermi on the quantization of the monoatomic ideal gas, is given in this paper
787:
303:{\displaystyle P={\frac {2}{3}}{\frac {E_{\text{tot}}}{V}}={\frac {2}{3}}{\frac {p_{\text{F}}^{5}}{10\pi ^{2}m\hbar ^{3}}},}
128:
63:
and the pressure of the electron gas would vanish. However, since electrons are quantum mechanical particles that obey the
803:
Mazzali, P. A.; Rƶpke, F. K.; Benetti, S.; Hillebrandt, W. (2007). "A Common
Explosion Mechanism for Type Ia Supernovae".
870:
533:, several approximations are carefully justified to treat the electrons as independent particles. Usual models are the
762:
670:
125:, each particle can be treated independently with a single-fermion energy given by the purely kinetic term,
596:
458:{\displaystyle P_{\text{e}}={\frac {(3\pi ^{2})^{2/3}\hbar ^{2}}{5m_{\text{e}}}}{\rho _{\text{e}}}^{5/3},}
875:
538:
493:
118:
553:
Electron degeneracy pressure will halt the gravitational collapse of a star if its mass is below the
64:
489:
185:
its mass. Every possible momentum state of an electron within this volume up to the Fermi momentum
24:
695:
507:
levels, a modified formula is required. The relativistic degeneracy pressure is proportional to
573:, because the degeneracy pressure provided by the electrons is weaker than the inward pull of
880:
659:
822:
707:
586:
8:
554:
534:
826:
711:
321:
is the total energy of the ensemble. Specifically for the electron degeneracy pressure,
846:
812:
731:
622:
504:
32:
838:
783:
758:
735:
723:
719:
676:
666:
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44:
36:
850:
830:
715:
654:
601:
89:
475:
56:
864:
754:
727:
650:
60:
834:
621:
Zannoni, Alberto (1999). "On the
Quantization of the Monoatomic Ideal Gas".
842:
566:
558:
542:
337:
121:. In general, for an ensemble of non-interacting fermions, also known as a
20:
680:
817:
627:
591:
562:
496:
close to the zero-temperature limit (temperatures much smaller than the
43:
solids. It is a manifestation of the more general phenomenon of quantum
570:
122:
94:
93:
Pressure vs temperature curves of classical and quantum ideal gases (
84:
98:
52:
574:
530:
114:
106:
482:
197:
The degeneracy pressure at zero temperature can be computed as
110:
802:
40:
105:
Electrons are members of a family of particles known as
346:
203:
131:
658:
457:
340:, so the electron degeneracy pressure is given by
302:
165:
35:effect critical to understanding the stability of
778:Taylor, John Robert; Zafiratos, Chris D. (1991).
862:
693:
687:
645:
643:
641:
777:
649:
638:
488:The term 'degenerate' here is not related to
55:exerted by the electrons is related to their
336:and the Fermi momentum is obtained from the
78:
780:Modern physics for scientists and engineers
796:
816:
748:
626:
561:). This is the pressure that prevents a
782:. Englewood Cliffs, N.J: Prentice Hall.
665:. New York: Holt, Rinehart and Winston.
88:
694:Koester, D; Chanmugam, G (1990-07-01).
620:
863:
314:is the total volume of the system and
166:{\displaystyle E={\frac {p^{2}}{2m}},}
500:which is about 10000 K for metals).
327:is substituted by the electron mass
179:is the momentum of one particle and
16:Repulsive force in quantum mechanics
13:
14:
892:
751:Introduction to Quantum Mechanics
753:(Second ed.). London, UK:
548:
117:, follow Pauli's principle and
771:
742:
700:Reports on Progress in Physics
696:"Physics of white dwarf stars"
614:
380:
363:
1:
607:
529:For the case of electrons in
503:When particle energies reach
29:electron degeneracy pressure
7:
580:
519:
10:
897:
720:10.1088/0034-4885/53/7/001
539:nearly free electron model
82:
871:Pauli exclusion principle
524:
79:From the Fermi gas theory
71:keeping the star stable.
65:Pauli exclusion principle
597:BoseāEinstein condensate
490:degenerate energy levels
25:condensed matter physics
835:10.1126/science.1136259
494:FermiāDirac statistics
459:
304:
167:
119:FermiāDirac statistics
102:
474:is the free electron
460:
305:
168:
109:. Fermions, like the
92:
587:Exchange interaction
344:
201:
129:
827:2007Sci...315..825M
712:1990RPPh...53..837K
661:Solid state physics
555:Chandrasekhar limit
535:free electron model
266:
45:degeneracy pressure
749:Griffiths (2005).
455:
300:
252:
163:
103:
33:quantum mechanical
876:Stellar evolution
811:(5813): 825ā828.
789:978-0-13-589789-8
757:. Equation 5.46.
655:Mermin, N. David.
651:Neil W., Ashcroft
531:crystalline solid
498:Fermi temperature
434:
423:
419:
354:
295:
259:
248:
235:
229:
218:
158:
37:white dwarf stars
888:
855:
854:
820:
818:astro-ph/0702351
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775:
769:
768:
746:
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691:
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628:cond-mat/9912229
618:
515:
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279:
265:
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257:
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241:
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231:
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194:being occupied.
193:
184:
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157:
149:
148:
139:
896:
895:
891:
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861:
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858:
801:
797:
790:
776:
772:
765:
747:
743:
692:
688:
673:
648:
639:
619:
615:
610:
602:Nuclear density
583:
551:
527:
522:
514:
508:
479:
472:
466:
442:
438:
431:
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351:
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328:
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288:
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275:
271:
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226:
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186:
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174:
150:
144:
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130:
127:
126:
87:
81:
17:
12:
11:
5:
894:
884:
883:
878:
873:
857:
856:
795:
788:
770:
763:
741:
706:(7): 837ā915.
686:
671:
637:
612:
611:
609:
606:
605:
604:
599:
594:
589:
582:
579:
550:
547:
526:
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521:
518:
512:
470:
454:
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445:
441:
430:
415:
411:
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368:
365:
359:
350:
332:
318:
299:
291:
287:
283:
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264:
255:
247:
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239:
234:
225:
217:
214:
209:
206:
190:
162:
156:
153:
147:
143:
137:
134:
83:Main article:
80:
77:
57:kinetic energy
15:
9:
6:
4:
3:
2:
893:
882:
879:
877:
874:
872:
869:
868:
866:
852:
848:
844:
840:
836:
832:
828:
824:
819:
814:
810:
806:
799:
791:
785:
781:
774:
766:
760:
756:
755:Prentice Hall
752:
745:
737:
733:
729:
725:
721:
717:
713:
709:
705:
701:
697:
690:
682:
678:
674:
668:
663:
662:
656:
652:
646:
644:
642:
634:
629:
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603:
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598:
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584:
578:
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572:
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560:
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546:
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511:
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484:
477:
469:
452:
447:
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339:
331:
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317:
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276:
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245:
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195:
189:
183:
177:
160:
154:
151:
145:
141:
135:
132:
124:
120:
116:
112:
108:
100:
96:
91:
86:
76:
72:
68:
66:
62:
61:absolute zero
58:
54:
48:
46:
42:
38:
34:
30:
26:
22:
881:White dwarfs
808:
804:
798:
779:
773:
750:
744:
703:
699:
689:
660:
632:
616:
567:neutron star
559:solar masses
552:
549:White dwarfs
543:bulk modulus
528:
509:
505:relativistic
502:
487:
467:
338:Fermi energy
329:
323:
315:
311:
196:
187:
181:
175:
104:
73:
69:
51:models. The
49:
28:
21:astrophysics
18:
592:Fermi level
563:white dwarf
545:of metals.
865:Categories
764:0131244051
672:0030839939
608:References
571:black hole
736:250915046
728:0034-4885
492:, but to
429:ρ
399:ℏ
371:π
286:ℏ
273:π
123:Fermi gas
95:Fermi gas
85:Fermi gas
851:16408991
843:17289993
657:(1976).
581:See also
537:and the
520:Examples
107:fermions
99:Bose gas
53:pressure
823:Bibcode
805:Science
708:Bibcode
575:gravity
476:density
115:neutron
113:or the
849:
841:
786:
761:
734:
726:
681:934604
679:
669:
557:(1.44
525:Metals
483:kelvin
465:where
310:where
173:where
111:proton
847:S2CID
813:arXiv
732:S2CID
623:arXiv
41:metal
31:is a
839:PMID
784:ISBN
759:ISBN
724:ISSN
677:OCLC
667:ISBN
39:and
23:and
831:doi
809:315
716:doi
569:or
319:tot
228:tot
47:.
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867::
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353:e
349:P
333:e
330:m
324:m
316:E
312:V
298:,
290:3
282:m
277:2
263:5
258:F
254:p
246:3
243:2
238:=
233:V
224:E
216:3
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208:=
205:P
191:F
188:p
182:m
176:p
161:,
155:m
152:2
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142:p
136:=
133:E
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