1726:
539:, see below) of a system of electrons describes how much energy is required to put an extra electron into the system, neglecting electrical potential energy. As the number of electrons in the system increases (with fixed temperature and volume), the internal chemical potential increases. This consequence is largely because electrons satisfy the
3080:
830:
2421:
1328:
2904:
electron fluid even for the non-interacting electron gas. It does not of course attempt to include electron-electron interaction effects. A simple form for an effective temperature which correctly recovers all the density-functional properties of even the
3205:
993:
is flat".) This balance requires that the variations in internal chemical potential are matched by equal and opposite variations in the electric potential energy. This gives rise to the "basic equation of nonlinear Thomas–Fermi theory":
2883:
In the effective temperature given above, the temperature is used to construct an effective classical model. However, this form of the effective temperature does not correctly recover the specific heat and most other properties of the
2936:
1522:
696:
2235:
368:
118:
1449:
3366:
2195:
1972:
1245:
2125:
246:
1678:
509:
2638:
1106:
2783:
1164:
905:
1609:
For the nonlinear Thomas–Fermi formula, solving these simultaneously can be difficult, and usually there is no analytical solution. However, the linearized formula has a simple solution (in
2556:
2468:
686:
2852:
1362:
3112:
1225:
3401:
1592:
966:
Nevertheless, the Thomas–Fermi model is likely to be a reasonably accurate approximation as long as the potential does not vary much over lengths comparable or smaller than 1 /
2509:
3496:
2230:
2036:
3430:
3255:
1791:
593:
3498:. Other mappings for the 3D case, and similar formulae for the effective temperature have been given for the classical map of the 2-dimensional electron gas as well.
628:
423:
3107:
2878:
2001:
1762:
1531:), the dielectric constant approaches infinity, reflecting the fact that charges get closer and closer to perfectly screened as you observe them from further away.
1454:
1188:
543:: only one electron may occupy an energy level and lower-energy electron states are already full, so the new electrons must occupy higher and higher energy states.
400:
1811:
3456:
3279:
2927:
2902:
2803:
564:
31:
when the wavevector (the reciprocal of the length-scale of interest) is much smaller than the Fermi wavevector, i.e. the long-distance limit. It is named after
289:
49:
1385:
3295:
1820:
171:
1616:
2561:
2930:
3075:{\displaystyle {\frac {T_{\rm {eff}}}{E_{\rm {F}}}}=\left({\frac {T^{2}}{E_{\rm {F}}^{2}}}+{\frac {T_{q}^{2}}{E_{\rm {F}}^{2}}}\right)^{1/2}}
997:
2643:
963:. Therefore, it cannot be possible to define a chemical potential at a single point, independent of the electron density at nearby points.
2130:
825:{\displaystyle n=2{\frac {1}{(2\pi )^{3}}}{\frac {4}{3}}\pi k_{\rm {F}}^{3}\quad ,\quad \mu ={\frac {\hbar ^{2}k_{\rm {F}}^{2}}{2m}}.}
1814:
3638:
2416:{\displaystyle {T_{\rm {eff}} \over T}={4 \over 3\Gamma (1/2)}{(E_{\rm {F}}/k_{\rm {B}}T)^{3/2} \over F_{-1/2}(\mu /k_{\rm {B}}T)}.}
2041:
1227:
at the points where the material is charge-neutral (the number of electrons is exactly equal to the number of ions), and similarly
428:
938:
1729:
Effective temperature for Thomas–Fermi screening. The approximate form is explained in the article, and uses the power p=1.8.
1131:
848:
634:, the internal chemical potential. The exact functional form depends on the system. For example, for a three-dimensional
1323:{\displaystyle \rho ^{\text{induced}}(\mathbf {r} )\approx -e^{2}{\frac {\partial n}{\partial \mu }}\phi (\mathbf {r} )}
916:
532:
1701:
the screening discussed here; for example due to the polarization of immobile core electrons. In that case, replace
3507:
2514:
2426:
986:
540:
3403:
or less, electron-electron interactions become negligible compared to the Fermi energy, then, using a value of
982:
641:
20:
3258:
2808:
1333:
514:
For more details and discussion, including the one-dimensional and two-dimensional cases, see the article on
275:). Such an approximation is valid for metals at room temperature, and the Thomas–Fermi screening wavevector
3281:
is the number of electrons in a unit volume using atomic units where the unit of length is the Bohr, viz.,
1193:
922:
3371:
1566:
3562:
2473:
842:
3461:
2200:
2006:
1375:
1116:) is the function discussed above (electron density as a function of internal chemical potential),
1234:
is defined as the internal chemical potential at the points where the material is charge-neutral.
3406:
3231:
1767:
834:(in this context—i.e., absolute zero—the internal chemical potential is more commonly called the
569:
373:
125:
981:
Finally, the Thomas–Fermi model assumes that the electrons are in equilibrium, meaning that the
3200:{\displaystyle {\frac {T_{q}}{E_{\rm {F}}}}={\frac {1}{a+b{\sqrt {r_{\rm {s}}}}+cr_{\rm {s}}}}}
3588:
3580:
1121:
141:
604:
3570:
1691:
1598:
566:, the highest occupied momentum state (at zero temperature) is known as the Fermi momentum,
408:
32:
3085:
2857:
1979:
1740:
1173:
378:
1796:
515:
403:
28:
24:
3435:
3566:
1242:
If the chemical potential does not vary too much, the above equation can be linearized:
3264:
2912:
2887:
2788:
1610:
1602:
1379:
599:
549:
43:
3632:
960:
147:
For the example of semiconductors that are not too heavily doped, the charge density
989:
of electrons is the same at all points". In semiconductor physics terminology, "the
835:
283:
261:
36:
1552:
in a solid, what field will it produce, taking electron screening into account?
990:
536:
129:
3575:
3550:
3261:
corresponding to a sphere in atomic units containing one electron. That is, if
1725:
638:, a noninteracting electron gas, at absolute zero temperature, the relation is
1517:{\displaystyle k_{0}={\sqrt {4\pi e^{2}{\frac {\partial n}{\partial \mu }}}}.}
942:
3584:
1734:
635:
3592:
1559:
The Thomas–Fermi screening formula gives the charge density at each point
959:
is the Fermi wavenumber, i.e. a typical wavenumber for the states at the
272:
363:{\displaystyle k_{\rm {TF}}^{2}=4\left({\frac {3n}{\pi }}\right)^{1/3}.}
113:{\displaystyle k_{0}^{2}=4\pi e^{2}{\frac {\partial n}{\partial \mu }},}
1605:) relates the second derivative of the potential to the charge density.
1444:{\displaystyle \varepsilon (\mathbf {q} )=1+{\frac {k_{0}^{2}}{q^{2}}}}
2805:. A value that gives decent agreement with the exact result for all
941:. No electron can exist at a single point; each is spread out into a
1764:
can be expressed as a function of both temperature and Fermi energy
3361:{\displaystyle r_{\rm {s}}=\left({\frac {3}{4\pi n}}\right)^{1/3}.}
2909:
electron gas, including the pair-distribution functions at finite
2190:{\displaystyle k_{\rm {B}}T_{\rm {eff}}=n\partial \mu /\partial n}
1724:
985:
is the same at all points. (In electrochemistry terminology, "the
23:
by electrons in a solid. It is a special case of the more general
2640:
A simple approximate form that recovers both limits correctly is
1793:. The first step is calculating the internal chemical potential
527:
Relation between electron density and internal chemical potential
3618:
François Perrot and M. W. C. Dharma-wardana, Phys. Rev. Lett.
1967:{\displaystyle {\frac {\mu }{k_{\rm {B}}T}}=F_{1/2}^{-1}\left.}
1717:
is the relative permittivity due to these other contributions.
3432:
close to unity, we see that the CHNC effective temperature at
2120:{\displaystyle k_{0}^{2}=4\pi e^{2}n/k_{\rm {B}}T_{\rm {eff}}}
925:
is that there is an internal chemical potential at each point
241:{\displaystyle k_{0}^{2}={\frac {4\pi e^{2}n}{k_{\rm {B}}T}},}
2929:, has been given using the classical map hyper-netted-chain (
1673:{\displaystyle \phi (\mathbf {r} )={\frac {Q}{r}}e^{-k_{0}r}}
973:. This length usually corresponds to a few atoms in metals.
598:
Then the required relationship is described by the electron
504:{\displaystyle k_{0}^{2}=k_{\rm {TF}}^{2}(m_{e}/\hbar ^{2})}
27:; in particular, Thomas–Fermi screening is the limit of the
1374:
This relation can be converted into a wavevector-dependent
2633:{\displaystyle k_{\rm {B}}T_{\rm {eff}}=(2/3)E_{\rm {F}}.}
1737:(noninteracting electron gas), the screening wavevector
264:. In the opposite extreme, in the low-temperature limit
1555:
One seeks a self-consistent solution to two equations:
1101:{\displaystyle \rho ^{\text{induced}}(\mathbf {r} )=-e}
937:. This behaviour cannot be exactly true because of the
19:
is a theoretical approach to calculate the effects of
3464:
3438:
3409:
3374:
3298:
3267:
3234:
3115:
3088:
2939:
2915:
2890:
2860:
2811:
2791:
2778:{\displaystyle k_{\rm {B}}T_{\rm {eff}}=\left^{1/p},}
2646:
2564:
2517:
2476:
2429:
2238:
2203:
2133:
2044:
2009:
1982:
1823:
1799:
1770:
1743:
1619:
1569:
1457:
1388:
1336:
1248:
1196:
1176:
1134:
1000:
851:
699:
644:
607:
572:
552:
431:
411:
381:
292:
174:
52:
3549:
Stanton, Liam G.; Murillo, Michael S. (2016-04-08).
1159:{\displaystyle \rho ^{\text{induced}}(\mathbf {r} )}
900:{\displaystyle n(\mu )\propto e^{\mu /k_{\rm {B}}T}}
2880:, which has a maximum relative error of < 2.3%.
3490:
3450:
3424:
3395:
3360:
3273:
3249:
3199:
3101:
3074:
2921:
2896:
2872:
2846:
2797:
2777:
2632:
2550:
2503:
2462:
2415:
2224:
2189:
2119:
2030:
1995:
1966:
1805:
1785:
1756:
1672:
1586:
1516:
1443:
1356:
1322:
1219:
1182:
1158:
1100:
899:
824:
680:
622:
587:
558:
503:
417:
394:
362:
240:
112:
933:on the electron concentration at the same point
425:, the screening wavevector in Gaussian units is
3551:"Ionic transport in high-energy-density matter"
1697:Note that there may be dielectric permittivity
3561:(4). American Physical Society (APS): 043203.
8:
977:Electrons in equilibrium, nonlinear equation
2551:{\displaystyle k_{\rm {B}}T\ll E_{\rm {F}}}
2463:{\displaystyle k_{\rm {B}}T\gg E_{\rm {F}}}
845:at low to moderate electron concentration,
3531:
3529:
3527:
3525:
3523:
1690:(no screening), this becomes the familiar
3574:
3480:
3473:
3472:
3463:
3437:
3415:
3414:
3408:
3380:
3379:
3373:
3345:
3341:
3319:
3304:
3303:
3297:
3266:
3240:
3239:
3233:
3187:
3186:
3167:
3166:
3160:
3145:
3133:
3132:
3122:
3116:
3114:
3093:
3087:
3062:
3058:
3045:
3039:
3038:
3028:
3023:
3017:
3006:
3000:
2999:
2989:
2983:
2965:
2964:
2947:
2946:
2940:
2938:
2914:
2889:
2859:
2837:
2836:
2827:
2817:
2816:
2810:
2790:
2762:
2758:
2747:
2735:
2728:
2727:
2708:
2694:
2693:
2664:
2663:
2652:
2651:
2645:
2620:
2619:
2604:
2582:
2581:
2570:
2569:
2563:
2541:
2540:
2523:
2522:
2516:
2482:
2481:
2475:
2453:
2452:
2435:
2434:
2428:
2394:
2393:
2384:
2368:
2361:
2345:
2341:
2327:
2326:
2317:
2310:
2309:
2299:
2285:
2267:
2246:
2245:
2239:
2237:
2209:
2208:
2202:
2176:
2151:
2150:
2139:
2138:
2132:
2104:
2103:
2092:
2091:
2082:
2073:
2054:
2049:
2043:
2015:
2014:
2008:
1987:
1981:
1946:
1942:
1926:
1925:
1919:
1900:
1887:
1882:
1868:
1859:
1855:
1835:
1834:
1824:
1822:
1798:
1776:
1775:
1769:
1748:
1742:
1659:
1651:
1637:
1626:
1618:
1576:
1568:
1489:
1483:
1471:
1462:
1456:
1433:
1423:
1418:
1412:
1395:
1387:
1343:
1335:
1312:
1286:
1280:
1262:
1253:
1247:
1203:
1195:
1175:
1148:
1139:
1133:
1086:
1062:
1044:
1014:
1005:
999:
885:
884:
875:
871:
850:
802:
796:
795:
785:
778:
761:
755:
754:
737:
728:
709:
698:
681:{\displaystyle n(\mu )\propto \mu ^{3/2}}
668:
664:
643:
606:
578:
577:
571:
551:
492:
483:
477:
464:
455:
454:
441:
436:
430:
410:
386:
380:
347:
343:
324:
307:
298:
297:
291:
271:, electrons behave as quantum particles (
222:
221:
206:
193:
184:
179:
173:
87:
81:
62:
57:
51:
3605:Yu Liu and Jianzhong Wu, J. Chem. Phys.
2847:{\displaystyle k_{\rm {B}}T/E_{\rm {F}}}
1357:{\displaystyle \partial n/\partial \mu }
3519:
2933:) model of the electron fluid. That is
782:
489:
412:
3368:For a dense electron gas, e.g., with
2003:in terms of an effective temperature
1220:{\displaystyle \phi (\mathbf {r} )=0}
260:is given by the familiar formula for
7:
3396:{\displaystyle r_{\rm {s}}\approx 1}
1587:{\displaystyle \phi (\mathbf {r} )}
3474:
3416:
3381:
3305:
3241:
3188:
3168:
3134:
3040:
3001:
2966:
2954:
2951:
2948:
2838:
2818:
2729:
2695:
2671:
2668:
2665:
2653:
2621:
2589:
2586:
2583:
2571:
2542:
2524:
2489:
2486:
2483:
2454:
2436:
2395:
2328:
2311:
2276:
2253:
2250:
2247:
2216:
2213:
2210:
2181:
2170:
2158:
2155:
2152:
2140:
2111:
2108:
2105:
2093:
2022:
2019:
2016:
1927:
1891:
1836:
1813:, which involves the inverse of a
1777:
1721:Fermi gas at arbitrary temperature
1500:
1492:
1348:
1337:
1297:
1289:
1238:Linearization, dielectric function
886:
797:
756:
691:Proof: Including spin degeneracy,
579:
459:
456:
302:
299:
223:
136:is the electron concentration and
98:
90:
14:
3539:(Thomson Learning, Toronto, 1976)
3535:N. W. Ashcroft and N. D. Mermin,
2511:, while in the degenerate limit
1627:
1577:
1396:
1313:
1263:
1204:
1149:
1063:
1015:
939:Heisenberg uncertainty principle
42:The Thomas–Fermi wavevector (in
2504:{\displaystyle T_{\rm {eff}}=T}
1563:as a function of the potential
771:
767:
3491:{\displaystyle 2E_{\rm {F}}/3}
3458:approximates towards the form
3082:where the quantum temperature
2744:
2717:
2705:
2686:
2612:
2598:
2404:
2378:
2338:
2302:
2293:
2279:
1908:
1894:
1631:
1623:
1581:
1573:
1400:
1392:
1317:
1309:
1267:
1259:
1208:
1200:
1190:is defined in such a way that
1153:
1145:
1095:
1092:
1079:
1070:
1067:
1059:
1037:
1031:
1019:
1011:
861:
855:
725:
715:
654:
648:
617:
611:
498:
470:
168:is temperature. In this case,
1:
2225:{\displaystyle T_{\rm {eff}}}
2031:{\displaystyle T_{\rm {eff}}}
546:Given a Fermi gas of density
3425:{\displaystyle r_{\rm {s}}}
3250:{\displaystyle r_{\rm {s}}}
1786:{\displaystyle E_{\rm {F}}}
1371:and treated as a constant.
921:The main assumption in the
917:Local-density approximation
841:As another example, for an
588:{\displaystyle k_{\rm {F}}}
533:internal chemical potential
3655:
3576:10.1103/physreve.93.043203
2197:. The general result for
914:
164:is Boltzmann constant and
1170:. The electric potential
1166:is the induced charge at
987:electrochemical potential
541:Pauli exclusion principle
3639:Condensed matter physics
1733:For a three-dimensional
983:total chemical potential
21:electric field screening
2423:In the classical limit
1535:Example: A point charge
623:{\displaystyle n(\mu )}
3492:
3452:
3426:
3397:
3362:
3275:
3251:
3201:
3103:
3076:
2923:
2898:
2874:
2848:
2799:
2779:
2634:
2552:
2505:
2464:
2417:
2226:
2191:
2121:
2032:
1997:
1968:
1807:
1787:
1758:
1730:
1674:
1588:
1518:
1445:
1358:
1324:
1221:
1184:
1160:
1102:
901:
826:
682:
624:
589:
560:
505:
419:
418:{\displaystyle \hbar }
396:
364:
242:
114:
17:Thomas–Fermi screening
3508:Thomas–Fermi equation
3493:
3453:
3427:
3398:
3363:
3276:
3252:
3202:
3104:
3102:{\displaystyle T_{q}}
3077:
2924:
2899:
2875:
2873:{\displaystyle p=1.8}
2849:
2800:
2780:
2635:
2553:
2506:
2465:
2418:
2227:
2192:
2122:
2033:
1998:
1996:{\displaystyle k_{0}}
1969:
1808:
1788:
1759:
1757:{\displaystyle k_{0}}
1728:
1675:
1589:
1519:
1446:
1359:
1325:
1222:
1185:
1183:{\displaystyle \phi }
1161:
1128:is the position, and
1103:
902:
827:
683:
625:
590:
561:
506:
420:
397:
395:{\displaystyle m_{e}}
365:
243:
115:
3462:
3436:
3407:
3372:
3296:
3265:
3232:
3113:
3086:
2937:
2913:
2888:
2858:
2809:
2789:
2644:
2562:
2515:
2474:
2427:
2236:
2201:
2131:
2042:
2007:
1980:
1821:
1815:Fermi–Dirac integral
1806:{\displaystyle \mu }
1797:
1768:
1741:
1617:
1567:
1455:
1386:
1334:
1246:
1194:
1174:
1132:
998:
849:
843:n-type semiconductor
697:
642:
605:
570:
550:
535:(closely related to
429:
409:
379:
290:
172:
50:
3567:2016PhRvE..93d3203S
3537:Solid State Physics
3451:{\displaystyle T=0}
3259:Wigner–Seitz radius
3050:
3033:
3011:
2059:
1876:
1524:At long distances (
1428:
1376:dielectric function
911:Local approximation
807:
766:
469:
446:
312:
189:
67:
3488:
3448:
3422:
3393:
3358:
3271:
3247:
3197:
3099:
3072:
3034:
3019:
2995:
2919:
2894:
2870:
2844:
2795:
2775:
2630:
2548:
2501:
2460:
2413:
2222:
2187:
2117:
2045:
2028:
1993:
1964:
1851:
1803:
1783:
1754:
1731:
1670:
1584:
1539:If a point charge
1514:
1441:
1414:
1354:
1320:
1217:
1180:
1156:
1098:
923:Thomas–Fermi model
897:
822:
791:
750:
678:
620:
585:
556:
501:
450:
432:
415:
392:
372:If we restore the
360:
293:
238:
175:
126:chemical potential
110:
53:
44:Gaussian-cgs units
3555:Physical Review E
3335:
3274:{\displaystyle n}
3195:
3174:
3140:
3051:
3012:
2972:
2922:{\displaystyle T}
2897:{\displaystyle T}
2798:{\displaystyle p}
2408:
2297:
2262:
1936:
1912:
1846:
1645:
1509:
1507:
1439:
1304:
1256:
1142:
1122:elementary charge
1008:
817:
745:
735:
630:as a function of
559:{\displaystyle n}
337:
233:
142:elementary charge
105:
3646:
3623:
3616:
3610:
3603:
3597:
3596:
3578:
3546:
3540:
3533:
3497:
3495:
3494:
3489:
3484:
3479:
3478:
3477:
3457:
3455:
3454:
3449:
3431:
3429:
3428:
3423:
3421:
3420:
3419:
3402:
3400:
3399:
3394:
3386:
3385:
3384:
3367:
3365:
3364:
3359:
3354:
3353:
3349:
3340:
3336:
3334:
3320:
3310:
3309:
3308:
3291:
3289:
3286:
3280:
3278:
3277:
3272:
3256:
3254:
3253:
3248:
3246:
3245:
3244:
3227:
3220:
3213:
3206:
3204:
3203:
3198:
3196:
3194:
3193:
3192:
3191:
3175:
3173:
3172:
3171:
3161:
3146:
3141:
3139:
3138:
3137:
3127:
3126:
3117:
3108:
3106:
3105:
3100:
3098:
3097:
3081:
3079:
3078:
3073:
3071:
3070:
3066:
3057:
3053:
3052:
3049:
3044:
3043:
3032:
3027:
3018:
3013:
3010:
3005:
3004:
2994:
2993:
2984:
2973:
2971:
2970:
2969:
2959:
2958:
2957:
2941:
2928:
2926:
2925:
2920:
2903:
2901:
2900:
2895:
2879:
2877:
2876:
2871:
2853:
2851:
2850:
2845:
2843:
2842:
2841:
2831:
2823:
2822:
2821:
2804:
2802:
2801:
2796:
2784:
2782:
2781:
2776:
2771:
2770:
2766:
2757:
2753:
2752:
2751:
2739:
2734:
2733:
2732:
2713:
2712:
2700:
2699:
2698:
2676:
2675:
2674:
2658:
2657:
2656:
2639:
2637:
2636:
2631:
2626:
2625:
2624:
2608:
2594:
2593:
2592:
2576:
2575:
2574:
2557:
2555:
2554:
2549:
2547:
2546:
2545:
2529:
2528:
2527:
2510:
2508:
2507:
2502:
2494:
2493:
2492:
2469:
2467:
2466:
2461:
2459:
2458:
2457:
2441:
2440:
2439:
2422:
2420:
2419:
2414:
2409:
2407:
2400:
2399:
2398:
2388:
2377:
2376:
2372:
2355:
2354:
2353:
2349:
2333:
2332:
2331:
2321:
2316:
2315:
2314:
2300:
2298:
2296:
2289:
2268:
2263:
2258:
2257:
2256:
2240:
2231:
2229:
2228:
2223:
2221:
2220:
2219:
2196:
2194:
2193:
2188:
2180:
2163:
2162:
2161:
2145:
2144:
2143:
2126:
2124:
2123:
2118:
2116:
2115:
2114:
2098:
2097:
2096:
2086:
2078:
2077:
2058:
2053:
2037:
2035:
2034:
2029:
2027:
2026:
2025:
2002:
2000:
1999:
1994:
1992:
1991:
1973:
1971:
1970:
1965:
1960:
1956:
1955:
1954:
1950:
1941:
1937:
1932:
1931:
1930:
1920:
1913:
1911:
1904:
1883:
1875:
1867:
1863:
1847:
1845:
1841:
1840:
1839:
1825:
1812:
1810:
1809:
1804:
1792:
1790:
1789:
1784:
1782:
1781:
1780:
1763:
1761:
1760:
1755:
1753:
1752:
1689:
1679:
1677:
1676:
1671:
1669:
1668:
1664:
1663:
1646:
1638:
1630:
1599:Poisson equation
1593:
1591:
1590:
1585:
1580:
1551:
1544:
1530:
1523:
1521:
1520:
1515:
1510:
1508:
1506:
1498:
1490:
1488:
1487:
1472:
1467:
1466:
1450:
1448:
1447:
1442:
1440:
1438:
1437:
1427:
1422:
1413:
1399:
1364:is evaluated at
1363:
1361:
1360:
1355:
1347:
1329:
1327:
1326:
1321:
1316:
1305:
1303:
1295:
1287:
1285:
1284:
1266:
1258:
1257:
1254:
1226:
1224:
1223:
1218:
1207:
1189:
1187:
1186:
1181:
1165:
1163:
1162:
1157:
1152:
1144:
1143:
1140:
1107:
1105:
1104:
1099:
1091:
1090:
1066:
1049:
1048:
1018:
1010:
1009:
1006:
906:
904:
903:
898:
896:
895:
891:
890:
889:
879:
831:
829:
828:
823:
818:
816:
808:
806:
801:
800:
790:
789:
779:
765:
760:
759:
746:
738:
736:
734:
733:
732:
710:
687:
685:
684:
679:
677:
676:
672:
629:
627:
626:
621:
594:
592:
591:
586:
584:
583:
582:
565:
563:
562:
557:
510:
508:
507:
502:
497:
496:
487:
482:
481:
468:
463:
462:
445:
440:
424:
422:
421:
416:
401:
399:
398:
393:
391:
390:
369:
367:
366:
361:
356:
355:
351:
342:
338:
333:
325:
311:
306:
305:
270:
259:
247:
245:
244:
239:
234:
232:
228:
227:
226:
215:
211:
210:
194:
188:
183:
156:
119:
117:
116:
111:
106:
104:
96:
88:
86:
85:
66:
61:
33:Llewellyn Thomas
29:Lindhard formula
3654:
3653:
3649:
3648:
3647:
3645:
3644:
3643:
3629:
3628:
3627:
3626:
3622:, 206404 (2001)
3617:
3613:
3604:
3600:
3548:
3547:
3543:
3534:
3521:
3516:
3504:
3468:
3460:
3459:
3434:
3433:
3410:
3405:
3404:
3375:
3370:
3369:
3324:
3315:
3314:
3299:
3294:
3293:
3287:
3284:
3282:
3263:
3262:
3235:
3230:
3229:
3222:
3215:
3208:
3182:
3162:
3150:
3128:
3118:
3111:
3110:
3109:is defined as:
3089:
3084:
3083:
2985:
2982:
2978:
2977:
2960:
2942:
2935:
2934:
2911:
2910:
2886:
2885:
2856:
2855:
2832:
2812:
2807:
2806:
2787:
2786:
2743:
2723:
2704:
2689:
2685:
2681:
2680:
2659:
2647:
2642:
2641:
2615:
2577:
2565:
2560:
2559:
2536:
2518:
2513:
2512:
2477:
2472:
2471:
2448:
2430:
2425:
2424:
2389:
2357:
2356:
2337:
2322:
2305:
2301:
2272:
2241:
2234:
2233:
2204:
2199:
2198:
2146:
2134:
2129:
2128:
2099:
2087:
2069:
2040:
2039:
2010:
2005:
2004:
1983:
1978:
1977:
1976:We can express
1921:
1915:
1914:
1881:
1877:
1830:
1829:
1819:
1818:
1795:
1794:
1771:
1766:
1765:
1744:
1739:
1738:
1723:
1687:
1681:
1655:
1647:
1615:
1614:
1565:
1564:
1546:
1540:
1537:
1525:
1499:
1491:
1479:
1458:
1453:
1452:
1429:
1384:
1383:
1370:
1332:
1331:
1296:
1288:
1276:
1249:
1244:
1243:
1240:
1233:
1192:
1191:
1172:
1171:
1135:
1130:
1129:
1082:
1040:
1001:
996:
995:
979:
972:
958:
951:
919:
913:
880:
867:
847:
846:
809:
781:
780:
724:
714:
695:
694:
660:
640:
639:
603:
602:
573:
568:
567:
548:
547:
529:
524:
516:Lindhard theory
488:
473:
427:
426:
407:
406:
404:Planck constant
382:
377:
376:
326:
320:
319:
288:
287:
281:
265:
258:
251:
217:
216:
202:
195:
170:
169:
163:
148:
97:
89:
77:
48:
47:
25:Lindhard theory
12:
11:
5:
3652:
3650:
3642:
3641:
3631:
3630:
3625:
3624:
3611:
3598:
3541:
3518:
3517:
3515:
3512:
3511:
3510:
3503:
3500:
3487:
3483:
3476:
3471:
3467:
3447:
3444:
3441:
3418:
3413:
3392:
3389:
3383:
3378:
3357:
3352:
3348:
3344:
3339:
3333:
3330:
3327:
3323:
3318:
3313:
3307:
3302:
3270:
3243:
3238:
3190:
3185:
3181:
3178:
3170:
3165:
3159:
3156:
3153:
3149:
3144:
3136:
3131:
3125:
3121:
3096:
3092:
3069:
3065:
3061:
3056:
3048:
3042:
3037:
3031:
3026:
3022:
3016:
3009:
3003:
2998:
2992:
2988:
2981:
2976:
2968:
2963:
2956:
2953:
2950:
2945:
2918:
2893:
2869:
2866:
2863:
2840:
2835:
2830:
2826:
2820:
2815:
2794:
2785:for any power
2774:
2769:
2765:
2761:
2756:
2750:
2746:
2742:
2738:
2731:
2726:
2722:
2719:
2716:
2711:
2707:
2703:
2697:
2692:
2688:
2684:
2679:
2673:
2670:
2667:
2662:
2655:
2650:
2629:
2623:
2618:
2614:
2611:
2607:
2603:
2600:
2597:
2591:
2588:
2585:
2580:
2573:
2568:
2544:
2539:
2535:
2532:
2526:
2521:
2500:
2497:
2491:
2488:
2485:
2480:
2456:
2451:
2447:
2444:
2438:
2433:
2412:
2406:
2403:
2397:
2392:
2387:
2383:
2380:
2375:
2371:
2367:
2364:
2360:
2352:
2348:
2344:
2340:
2336:
2330:
2325:
2320:
2313:
2308:
2304:
2295:
2292:
2288:
2284:
2281:
2278:
2275:
2271:
2266:
2261:
2255:
2252:
2249:
2244:
2218:
2215:
2212:
2207:
2186:
2183:
2179:
2175:
2172:
2169:
2166:
2160:
2157:
2154:
2149:
2142:
2137:
2113:
2110:
2107:
2102:
2095:
2090:
2085:
2081:
2076:
2072:
2068:
2065:
2062:
2057:
2052:
2048:
2024:
2021:
2018:
2013:
1990:
1986:
1963:
1959:
1953:
1949:
1945:
1940:
1935:
1929:
1924:
1918:
1910:
1907:
1903:
1899:
1896:
1893:
1890:
1886:
1880:
1874:
1871:
1866:
1862:
1858:
1854:
1850:
1844:
1838:
1833:
1828:
1802:
1779:
1774:
1751:
1747:
1722:
1719:
1699:in addition to
1685:
1667:
1662:
1658:
1654:
1650:
1644:
1641:
1636:
1633:
1629:
1625:
1622:
1607:
1606:
1601:(derived from
1595:
1594:at that point.
1583:
1579:
1575:
1572:
1536:
1533:
1513:
1505:
1502:
1497:
1494:
1486:
1482:
1478:
1475:
1470:
1465:
1461:
1436:
1432:
1426:
1421:
1417:
1411:
1408:
1405:
1402:
1398:
1394:
1391:
1368:
1353:
1350:
1346:
1342:
1339:
1319:
1315:
1311:
1308:
1302:
1299:
1294:
1291:
1283:
1279:
1275:
1272:
1269:
1265:
1261:
1252:
1239:
1236:
1231:
1216:
1213:
1210:
1206:
1202:
1199:
1179:
1155:
1151:
1147:
1138:
1097:
1094:
1089:
1085:
1081:
1078:
1075:
1072:
1069:
1065:
1061:
1058:
1055:
1052:
1047:
1043:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1017:
1013:
1004:
978:
975:
970:
956:
949:
945:of size ≈ 1 /
912:
909:
894:
888:
883:
878:
874:
870:
866:
863:
860:
857:
854:
821:
815:
812:
805:
799:
794:
788:
784:
777:
774:
770:
764:
758:
753:
749:
744:
741:
731:
727:
723:
720:
717:
713:
708:
705:
702:
675:
671:
667:
663:
659:
656:
653:
650:
647:
619:
616:
613:
610:
600:number density
581:
576:
555:
528:
525:
523:
520:
500:
495:
491:
486:
480:
476:
472:
467:
461:
458:
453:
449:
444:
439:
435:
414:
389:
385:
359:
354:
350:
346:
341:
336:
332:
329:
323:
318:
315:
310:
304:
301:
296:
279:
256:
237:
231:
225:
220:
214:
209:
205:
201:
198:
192:
187:
182:
178:
161:
109:
103:
100:
95:
92:
84:
80:
76:
73:
70:
65:
60:
56:
13:
10:
9:
6:
4:
3:
2:
3651:
3640:
3637:
3636:
3634:
3621:
3615:
3612:
3609:064115 (2014)
3608:
3602:
3599:
3594:
3590:
3586:
3582:
3577:
3572:
3568:
3564:
3560:
3556:
3552:
3545:
3542:
3538:
3532:
3530:
3528:
3526:
3524:
3520:
3513:
3509:
3506:
3505:
3501:
3499:
3485:
3481:
3469:
3465:
3445:
3442:
3439:
3411:
3390:
3387:
3376:
3355:
3350:
3346:
3342:
3337:
3331:
3328:
3325:
3321:
3316:
3311:
3300:
3268:
3260:
3236:
3225:
3218:
3211:
3183:
3179:
3176:
3163:
3157:
3154:
3151:
3147:
3142:
3129:
3123:
3119:
3094:
3090:
3067:
3063:
3059:
3054:
3046:
3035:
3029:
3024:
3020:
3014:
3007:
2996:
2990:
2986:
2979:
2974:
2961:
2943:
2932:
2916:
2908:
2891:
2881:
2867:
2864:
2861:
2833:
2828:
2824:
2813:
2792:
2772:
2767:
2763:
2759:
2754:
2748:
2740:
2736:
2724:
2720:
2714:
2709:
2701:
2690:
2682:
2677:
2660:
2648:
2627:
2616:
2609:
2605:
2601:
2595:
2578:
2566:
2537:
2533:
2530:
2519:
2498:
2495:
2478:
2449:
2445:
2442:
2431:
2410:
2401:
2390:
2385:
2381:
2373:
2369:
2365:
2362:
2358:
2350:
2346:
2342:
2334:
2323:
2318:
2306:
2290:
2286:
2282:
2273:
2269:
2264:
2259:
2242:
2205:
2184:
2177:
2173:
2167:
2164:
2147:
2135:
2100:
2088:
2083:
2079:
2074:
2070:
2066:
2063:
2060:
2055:
2050:
2046:
2011:
1988:
1984:
1974:
1961:
1957:
1951:
1947:
1943:
1938:
1933:
1922:
1916:
1905:
1901:
1897:
1888:
1884:
1878:
1872:
1869:
1864:
1860:
1856:
1852:
1848:
1842:
1831:
1826:
1816:
1800:
1772:
1749:
1745:
1736:
1727:
1720:
1718:
1716:
1712:
1708:
1704:
1700:
1695:
1693:
1692:Coulomb's law
1684:
1665:
1660:
1656:
1652:
1648:
1642:
1639:
1634:
1620:
1612:
1604:
1600:
1596:
1570:
1562:
1558:
1557:
1556:
1553:
1549:
1545:is placed at
1543:
1534:
1532:
1528:
1511:
1503:
1495:
1484:
1480:
1476:
1473:
1468:
1463:
1459:
1434:
1430:
1424:
1419:
1415:
1409:
1406:
1403:
1389:
1381:
1377:
1372:
1367:
1351:
1344:
1340:
1306:
1300:
1292:
1281:
1277:
1273:
1270:
1250:
1237:
1235:
1230:
1214:
1211:
1197:
1177:
1169:
1136:
1127:
1123:
1119:
1115:
1111:
1087:
1083:
1076:
1073:
1056:
1053:
1050:
1045:
1041:
1034:
1028:
1025:
1022:
1002:
992:
988:
984:
976:
974:
969:
964:
962:
961:Fermi surface
955:
948:
944:
940:
936:
932:
929:that depends
928:
924:
918:
910:
908:
892:
881:
876:
872:
868:
864:
858:
852:
844:
839:
837:
832:
819:
813:
810:
803:
792:
786:
775:
772:
768:
762:
751:
747:
742:
739:
729:
721:
718:
711:
706:
703:
700:
692:
689:
673:
669:
665:
661:
657:
651:
645:
637:
633:
614:
608:
601:
596:
574:
553:
544:
542:
538:
534:
526:
521:
519:
517:
512:
493:
484:
478:
474:
465:
451:
447:
442:
437:
433:
405:
387:
383:
375:
374:electron mass
370:
357:
352:
348:
344:
339:
334:
330:
327:
321:
316:
313:
308:
294:
285:
278:
274:
268:
263:
255:
248:
235:
229:
218:
212:
207:
203:
199:
196:
190:
185:
180:
176:
167:
160:
155:
151:
145:
143:
139:
135:
131:
127:
123:
107:
101:
93:
82:
78:
74:
71:
68:
63:
58:
54:
45:
40:
38:
34:
30:
26:
22:
18:
3619:
3614:
3606:
3601:
3558:
3554:
3544:
3536:
3223:
3216:
3209:
2906:
2882:
1975:
1732:
1714:
1710:
1706:
1702:
1698:
1696:
1682:
1611:cgs-Gaussian
1608:
1560:
1554:
1547:
1541:
1538:
1526:
1380:cgs-Gaussian
1373:
1365:
1241:
1228:
1167:
1125:
1117:
1113:
1109:
980:
967:
965:
953:
946:
934:
930:
926:
920:
840:
836:Fermi energy
833:
693:
690:
631:
597:
545:
530:
513:
371:
284:atomic units
276:
266:
262:Debye length
253:
249:
165:
158:
153:
149:
146:
137:
133:
121:
41:
37:Enrico Fermi
16:
15:
2907:interacting
1603:Gauss's law
991:Fermi level
537:Fermi level
130:Fermi level
3514:References
3290:10 cm
2470:, we find
943:wavepacket
915:See also:
522:Derivation
3585:2470-0045
3388:≈
3329:π
3219:= −0.3160
2534:≪
2446:≫
2382:μ
2363:−
2277:Γ
2182:∂
2174:μ
2171:∂
2067:π
1892:Γ
1870:−
1827:μ
1801:μ
1735:Fermi gas
1653:−
1621:ϕ
1571:ϕ
1504:μ
1501:∂
1493:∂
1477:π
1390:ε
1352:μ
1349:∂
1338:∂
1307:ϕ
1301:μ
1298:∂
1290:∂
1274:−
1271:≈
1251:ρ
1198:ϕ
1178:ϕ
1137:ρ
1084:μ
1074:−
1057:ϕ
1042:μ
1026:−
1003:ρ
873:μ
865:∝
859:μ
783:ℏ
773:μ
748:π
722:π
662:μ
658:∝
652:μ
636:Fermi gas
615:μ
490:ℏ
413:ℏ
335:π
282:given in
200:π
102:μ
99:∂
91:∂
75:π
3633:Category
3593:27176414
3502:See also
3226:= 0.0240
2558:we find
1713:, where
1613:units):
952:, where
402:and the
273:fermions
157:, where
3563:Bibcode
3292:, then
3257:is the
3228:. Here
3212:= 1.594
2884:finite-
1382:units)
1255:induced
1141:induced
1120:is the
1007:induced
140:is the
124:is the
3591:
3583:
3207:where
1451:where
1378:: (in
1330:where
1108:where
120:where
3283:5.291
2127:, or
1680:With
250:i.e.
46:) is
3589:PMID
3581:ISSN
2931:CHNC
1597:The
931:only
531:The
286:is
35:and
3607:141
3571:doi
2868:1.8
2854:is
2232:is
1705:by
1688:= 0
1550:= 0
1529:→ 0
838:).
269:= 0
132:),
3635::
3620:87
3587:.
3579:.
3569:.
3559:93
3557:.
3553:.
3522:^
3285:77
3221:,
3214:,
2038::
1817:,
1694:.
1124:,
907:.
688:.
595:.
518:.
511:.
280:TF
252:1/
152:∝
144:.
39:.
3595:.
3573::
3565::
3486:3
3482:/
3475:F
3470:E
3466:2
3446:0
3443:=
3440:T
3417:s
3412:r
3391:1
3382:s
3377:r
3356:.
3351:3
3347:/
3343:1
3338:)
3332:n
3326:4
3322:3
3317:(
3312:=
3306:s
3301:r
3288:×
3269:n
3242:s
3237:r
3224:c
3217:b
3210:a
3189:s
3184:r
3180:c
3177:+
3169:s
3164:r
3158:b
3155:+
3152:a
3148:1
3143:=
3135:F
3130:E
3124:q
3120:T
3095:q
3091:T
3068:2
3064:/
3060:1
3055:)
3047:2
3041:F
3036:E
3030:2
3025:q
3021:T
3015:+
3008:2
3002:F
2997:E
2991:2
2987:T
2980:(
2975:=
2967:F
2962:E
2955:f
2952:f
2949:e
2944:T
2917:T
2892:T
2865:=
2862:p
2839:F
2834:E
2829:/
2825:T
2819:B
2814:k
2793:p
2773:,
2768:p
2764:/
2760:1
2755:]
2749:p
2745:)
2741:3
2737:/
2730:F
2725:E
2721:2
2718:(
2715:+
2710:p
2706:)
2702:T
2696:B
2691:k
2687:(
2683:[
2678:=
2672:f
2669:f
2666:e
2661:T
2654:B
2649:k
2628:.
2622:F
2617:E
2613:)
2610:3
2606:/
2602:2
2599:(
2596:=
2590:f
2587:f
2584:e
2579:T
2572:B
2567:k
2543:F
2538:E
2531:T
2525:B
2520:k
2499:T
2496:=
2490:f
2487:f
2484:e
2479:T
2455:F
2450:E
2443:T
2437:B
2432:k
2411:.
2405:)
2402:T
2396:B
2391:k
2386:/
2379:(
2374:2
2370:/
2366:1
2359:F
2351:2
2347:/
2343:3
2339:)
2335:T
2329:B
2324:k
2319:/
2312:F
2307:E
2303:(
2294:)
2291:2
2287:/
2283:1
2280:(
2274:3
2270:4
2265:=
2260:T
2254:f
2251:f
2248:e
2243:T
2217:f
2214:f
2211:e
2206:T
2185:n
2178:/
2168:n
2165:=
2159:f
2156:f
2153:e
2148:T
2141:B
2136:k
2112:f
2109:f
2106:e
2101:T
2094:B
2089:k
2084:/
2080:n
2075:2
2071:e
2064:4
2061:=
2056:2
2051:0
2047:k
2023:f
2020:f
2017:e
2012:T
1989:0
1985:k
1962:.
1958:]
1952:2
1948:/
1944:3
1939:)
1934:T
1928:F
1923:E
1917:(
1909:)
1906:2
1902:/
1898:3
1895:(
1889:3
1885:2
1879:[
1873:1
1865:2
1861:/
1857:1
1853:F
1849:=
1843:T
1837:B
1832:k
1778:F
1773:E
1750:0
1746:k
1715:ε
1711:ε
1709:/
1707:Q
1703:Q
1686:0
1683:k
1666:r
1661:0
1657:k
1649:e
1643:r
1640:Q
1635:=
1632:)
1628:r
1624:(
1582:)
1578:r
1574:(
1561:r
1548:r
1542:Q
1527:q
1512:.
1496:n
1485:2
1481:e
1474:4
1469:=
1464:0
1460:k
1435:2
1431:q
1425:2
1420:0
1416:k
1410:+
1407:1
1404:=
1401:)
1397:q
1393:(
1369:0
1366:μ
1345:/
1341:n
1318:)
1314:r
1310:(
1293:n
1282:2
1278:e
1268:)
1264:r
1260:(
1232:0
1229:μ
1215:0
1212:=
1209:)
1205:r
1201:(
1168:r
1154:)
1150:r
1146:(
1126:r
1118:e
1114:μ
1112:(
1110:n
1096:]
1093:)
1088:0
1080:(
1077:n
1071:)
1068:)
1064:r
1060:(
1054:e
1051:+
1046:0
1038:(
1035:n
1032:[
1029:e
1023:=
1020:)
1016:r
1012:(
971:F
968:k
957:F
954:k
950:F
947:k
935:r
927:r
893:T
887:B
882:k
877:/
869:e
862:)
856:(
853:n
820:.
814:m
811:2
804:2
798:F
793:k
787:2
776:=
769:,
763:3
757:F
752:k
743:3
740:4
730:3
726:)
719:2
716:(
712:1
707:2
704:=
701:n
674:2
670:/
666:3
655:)
649:(
646:n
632:μ
618:)
612:(
609:n
580:F
575:k
554:n
499:)
494:2
485:/
479:e
475:m
471:(
466:2
460:F
457:T
452:k
448:=
443:2
438:0
434:k
388:e
384:m
358:.
353:3
349:/
345:1
340:)
331:n
328:3
322:(
317:4
314:=
309:2
303:F
300:T
295:k
277:k
267:T
257:0
254:k
236:,
230:T
224:B
219:k
213:n
208:2
204:e
197:4
191:=
186:2
181:0
177:k
166:T
162:B
159:k
154:e
150:n
138:e
134:n
128:(
122:μ
108:,
94:n
83:2
79:e
72:4
69:=
64:2
59:0
55:k
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