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Van Vleck paramagnetism

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1296: 2080: 3407: 2522: 1096: 714: 1845: 3129: 2994: 2282: 1291:{\displaystyle \Delta E^{(1)}=\mu _{0}{\frac {\mu _{\rm {B}}}{\hbar }}\langle \mathrm {g} |(\mathbf {L} +g\mathbf {S} )\cdot \mathbf {H} |\mathrm {g} \rangle =g_{J}\mu _{0}{\frac {\mu _{\rm {B}}}{\hbar }}\langle \mathrm {g} |\mathbf {J} \cdot \mathbf {H} |\mathrm {g} \rangle } 542: 2075:{\displaystyle \Delta E^{\rm {(2)}}=\left({\frac {\mu _{0}\mu _{\rm {B}}}{\hbar }}\right)^{2}\sum _{i}{\frac {|\langle \mathrm {g} |(\mathbf {L} +g\mathbf {S} )\cdot \mathbf {H} |\mathrm {e} _{i}\rangle |^{2}}{E_{\mathrm {g} }^{(0)}-E_{\mathrm {e} ,i}^{(0)}}}} 79:, Van Vleck discovered an additional paramagnetic contribution of the same order as Langevin's diamagnetism. Van Vleck contribution is usually important for systems with one electron short of being half filled and this contribution vanishes for elements with 3120: 3402:{\displaystyle W_{i}^{\rm {(2)}}={\frac {1}{\hbar ^{2}}}\sum _{k(k\neq i)}{\frac {|\langle \mathrm {e} _{i}|L_{z}+gS_{z}|\mathrm {e} _{k}\rangle |^{2}}{\delta E_{i,k}}}\;;\;\delta E_{i,k}=E_{\mathrm {e} ,i}^{(0)}-E_{\mathrm {e} ,k}^{(0)}} 2724: 2517:{\displaystyle \chi _{\rm {VV}}=2n\mu _{0}\left({\frac {\mu _{\rm {B}}}{\hbar }}\right)^{2}\sum _{i(i\neq 0)}{\frac {g_{j}^{2}|\langle \mathrm {g} |L_{z}+gS_{z}|\mathrm {e} _{i}\rangle |^{2}}{E_{\mathrm {e} ,i}-E_{\rm {g}}}},} 2681: 1659: 2271: 2192: 1478: 306: 1398: 709:{\displaystyle {\mathcal {H}}={\mathcal {H}}_{0}+\mu _{0}{\frac {\mu _{\rm {B}}}{\hbar }}(\mathbf {L} +g\mathbf {S} )\cdot \mathbf {H} +\mu _{0}^{2}{\frac {e^{2}}{8m_{\rm {e}}}}r_{\perp }^{2}H^{2}} 3940: 3853: 3673: 3521: 151: 3606: 2120: 1742: 1329: 1542: 1052: 972: 3792: 3757: 3722: 374: 777: 500: 467: 3005: 994: 910: 884: 534: 115: 3550: 3449: 858: 407: 937: 238: 208: 2708: 434: 744: 4108: 4569: 2603: 2576: 1817: 1689: 1576: 1356: 1021: 174: 2989:{\displaystyle \chi _{\rm {para}}=\mu _{0}\mu _{\rm {B}}^{2}{\frac {n}{\sum _{i}p_{i}}}\sum _{i}p_{i}\left\;;\;p_{i}=\exp \left(-{\frac {E_{i}^{(0)}}{kT}}\right),} 4134: 4073: 2218: 1752:
While Curie and Larmor susceptibilities were well understood from experimental measurements, J.H. Van Vleck noticed that the calculation above was incomplete. If
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For a more general system (molecules, complex systems), the paramagnetic susceptibility for an ensemble of independent magnetic moments can be written as
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is very close to the ground state at 330 K and contributes through second order corrections as showed by Van Vleck. A similar effect is observed in
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paramagnetism), that leads to a positive magnetic susceptibility. For sufficiently large temperatures, this contribution is described by
1588: 2223: 2125: 4218:"On Dielectric Constants and Magnetic Susceptibilities in the New Quantum Mechanics Part III—Application to Dia- and Paramagnetism" 1422: 249: 1409: 511: 4347: 1365: 4297: 4414: 3881: 243:
Experimental measurements show that most non-magnetic materials have a susceptibility that behaves in the following way:
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is taken as the perturbation parameter, the calculation must include all orders of perturbation up to the same power of
3797: 3614: 3462: 1578:), leads to a negative response (magnetization that opposes the magnetic field). Usually known as Larmor or Langenvin 123: 3559: 2605:
are the projection of the spin and orbital angular momentum in the direction of the magnetic field, respectively.
1401: 887: 333: 4407: 4047: 2088: 48: 1548:. If the ground state has no total angular momentum there is no Curie contribution and other terms dominate. 4564: 1744:
is the mean squared radius of the atom. Note that Larmor susceptibility does not depend on the temperature.
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In this way, Van Vleck susceptibility comes from the second order energy correction, and can be written as
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to be in case (d) and obtained a more precise prediction of its susceptibility using the formula above.
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are the energies of the excited states and the ground state, respectively, the sum excludes the state
4229: 3115:{\displaystyle W_{i}^{(1)}=g_{J}^{(i)}\langle \mathrm {e} _{i}|J_{z}|\mathrm {e} _{i}\rangle /\hbar } 339: 211: 753: 472: 439: 4504: 4011: 1059: 747: 977: 893: 867: 517: 98: 4489: 4461: 3553: 3526: 3416: 861: 834: 804: 379: 44: 4267: 3455:. Van Vleck summarizes the results of this formula in four cases, depending on the temperature: 915: 217: 187: 2686: 412: 4449: 4383: 4353: 4323: 4245: 1359: 940: 828: 722: 4078: 974:
is the unperturbed Hamiltonian without the magnetic field, the second one is proportional to
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can be positive, negative or null. Van Vleck paramagnetism often refers to systems where
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First order perturbation theory on the second term of the Hamiltonian (proportional to
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The first perturbation theory on the third term of the Hamiltonian (proportional to
4444: 1579: 76: 52: 4217: 2683:, as the signs of Larmor and Van Vleck susceptibilities are opposite, the sign of 4317: 1090:) for electrons bound to an atom, gives a positive correction to energy given by 943:
orthogonal to the magnetic field. The Hamiltonian has three terms, the first one
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between the 1920s and the 1930s to explain the magnetic response of gaseous
4241: 4541: 4152: 4137: 2676:{\displaystyle \chi _{0}\approx \chi _{\rm {VV}}+\chi _{\rm {Larmor}}} 1792:. As Larmor diamagnetism comes from first order perturbation of the 1484:
a susceptibility that is inversely proportional to the temperature
1023:. In order to obtain the ground state of the system, one can treat 184:
material, its magnetization is parallel to the magnetic field and
1654:{\displaystyle \chi _{\rm {Larmor}}=-C_{2}\langle r^{2}\rangle } 4403: 2266:{\displaystyle |\mathrm {e} _{0}\rangle =|\mathrm {g} \rangle } 2187:{\displaystyle E_{\mathrm {e} ,i}^{(0)},E_{\mathrm {g} }^{(0)}} 1473:{\displaystyle \chi _{\rm {Curie}}\approx {\frac {C_{1}}{T}}} 4399: 1054:
exactly, and treat the magnetic field dependent terms using
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of a material, derived from second order corrections to the
3998:, deviates slightly from it. In 1927, Van Vleck considered 3675:, the susceptibility is independent of the temperature; 514:
for an electron in a static homogeneous magnetic field
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where the sum goes over all excited degenerate states
1819:, one must calculate second order perturbation of the 1404:). This correction leads to what is known as Langevin 1393:{\displaystyle \mathbf {J} =\mathbf {L} +\mathbf {S} } 4116: 4081: 4055: 3948: 3884: 3861: 3800: 3765: 3730: 3684: 3617: 3562: 3529: 3465: 3419: 3132: 3008: 2727: 2689: 2614: 2584: 2557: 2533: 2285: 2226: 2200: 2128: 2091: 1848: 1825: 1798: 1778: 1758: 1717: 1697: 1670: 1591: 1557: 1510: 1490: 1425: 1368: 1337: 1307: 1099: 1076: 1029: 1002: 980: 949: 918: 896: 870: 837: 813: 789: 756: 725: 545: 520: 475: 442: 415: 382: 342: 318: 252: 220: 190: 162: 126: 101: 95:
of a material under an external small magnetic field
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Nolting, Wolfgang; Ramakanth, Anupuru (2009-10-03).
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The Theory of Electric and Magnetic Susceptibilities
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The standard example of Van Vleck paramagnetism are
3935:{\displaystyle |\delta E_{i,k}|\approx k_{\rm {B}}T} 2710:
depends on the specific properties of the material.
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Alongside other magnetic effects like 8: 3272: 3209: 3101: 3054: 2458: 2402: 2260: 2244: 2109: 1997: 1937: 1731: 1718: 1648: 1635: 1400:is the total angular momentum operator (see 1318: 1285: 1249: 1204: 1151: 214:, the magnetization opposes the field, and 4110:) vanishes, the first excited state with 4422: 4408: 4400: 3315: 3311: 2917: 2913: 4382:. Cambridge: Cambridge University Press. 4352:. Springer Science & Business Media. 4115: 4092: 4086: 4080: 4054: 3947: 3922: 3921: 3909: 3897: 3885: 3883: 3860: 3828: 3806: 3805: 3799: 3774: 3773: 3764: 3739: 3738: 3729: 3709: 3697: 3685: 3683: 3655: 3654: 3642: 3630: 3618: 3616: 3590: 3568: 3567: 3561: 3556:, the susceptibility follows Curie law: 3535: 3534: 3528: 3503: 3502: 3490: 3478: 3466: 3464: 3429: 3424: 3418: 3387: 3375: 3374: 3355: 3343: 3342: 3323: 3296: 3281: 3276: 3266: 3261: 3255: 3249: 3233: 3224: 3218: 3213: 3204: 3201: 3180: 3168: 3159: 3143: 3142: 3137: 3131: 3104: 3095: 3090: 3084: 3078: 3069: 3063: 3058: 3042: 3037: 3018: 3013: 3007: 2956: 2951: 2945: 2922: 2896: 2891: 2865: 2849: 2844: 2833: 2822: 2812: 2799: 2789: 2779: 2773: 2767: 2766: 2756: 2733: 2732: 2726: 2694: 2688: 2651: 2650: 2633: 2632: 2619: 2613: 2589: 2583: 2562: 2556: 2532: 2501: 2500: 2480: 2479: 2467: 2462: 2452: 2447: 2441: 2435: 2419: 2410: 2405: 2397: 2391: 2386: 2379: 2358: 2348: 2332: 2331: 2325: 2314: 2291: 2290: 2284: 2255: 2250: 2238: 2233: 2227: 2225: 2199: 2172: 2166: 2165: 2146: 2134: 2133: 2127: 2115:{\displaystyle |\mathrm {e} _{i}\rangle } 2103: 2098: 2092: 2090: 2057: 2045: 2044: 2025: 2019: 2018: 2006: 2001: 1991: 1986: 1980: 1975: 1964: 1953: 1945: 1940: 1932: 1929: 1923: 1913: 1896: 1895: 1885: 1878: 1857: 1856: 1847: 1824: 1803: 1797: 1777: 1757: 1725: 1716: 1711:the number of atoms per unit volume, and 1696: 1675: 1669: 1642: 1629: 1597: 1596: 1590: 1562: 1556: 1528: 1515: 1509: 1489: 1459: 1453: 1431: 1430: 1424: 1385: 1377: 1369: 1367: 1342: 1336: 1313: 1308: 1306: 1280: 1275: 1270: 1262: 1257: 1252: 1237: 1236: 1230: 1224: 1214: 1199: 1194: 1189: 1178: 1167: 1159: 1154: 1139: 1138: 1132: 1126: 1107: 1098: 1075: 1038: 1032: 1031: 1028: 1007: 1001: 981: 979: 958: 952: 951: 948: 923: 917: 897: 895: 871: 869: 843: 842: 836: 812: 788: 762: 761: 755: 730: 724: 700: 690: 685: 671: 670: 656: 650: 644: 639: 627: 616: 605: 590: 589: 583: 577: 564: 558: 557: 547: 546: 544: 521: 519: 480: 474: 447: 441: 420: 414: 387: 381: 360: 347: 341: 317: 292: 274: 268: 251: 219: 189: 161: 138: 127: 125: 102: 100: 3994:follows Curie law as in case (a), while 1408:(the quantum theory is sometimes called 180:. When a magnetic field is applied to a 4190: 3165: 3109: 2339: 1904: 1244: 1146: 996:, and the third one is proportional to 597: 4570:Electric and magnetic fields in matter 2714:General formula and Van Vleck criteria 1748:Second order: Van Vleck susceptibility 1058:Note that for strong magnetic fields, 1737:{\displaystyle \langle r^{2}\rangle } 1324:{\displaystyle |\mathrm {g} \rangle } 7: 4371: 4369: 4341: 4339: 4291: 4289: 4287: 4261: 4259: 4211: 4209: 1691:is another constant proportional to 3942:, there is no simple dependence on 4198:Van Vleck, John Hasbrouck (1932). 3923: 3816: 3813: 3810: 3807: 3775: 3740: 3656: 3578: 3575: 3572: 3569: 3536: 3504: 3376: 3344: 3262: 3214: 3091: 3059: 2768: 2743: 2740: 2737: 2734: 2667: 2664: 2661: 2658: 2655: 2652: 2637: 2634: 2502: 2481: 2448: 2406: 2333: 2295: 2292: 2256: 2234: 2167: 2135: 2099: 2046: 2020: 1987: 1941: 1897: 1849: 1613: 1610: 1607: 1604: 1601: 1598: 1537:{\displaystyle C_{0}\approx C_{1}} 1444: 1441: 1438: 1435: 1432: 1314: 1281: 1253: 1238: 1200: 1155: 1140: 1100: 1047:{\displaystyle {\mathcal {H}}_{0}} 967:{\displaystyle {\mathcal {H}}_{0}} 844: 763: 672: 591: 14: 4316:Marder, Michael P. (2010-11-17). 4268:"John H. van Vleck Nobel Lecture" 35:-independent contribution to the 4379:Magnetism and Magnetic Materials 3982:are similar paramagnetic gases, 3787:{\displaystyle \ll k_{\rm {B}}T} 3752:{\displaystyle \gg k_{\rm {B}}T} 3717:{\displaystyle |\delta E_{i,k}|} 1976: 1965: 1954: 1386: 1378: 1370: 1271: 1263: 1190: 1179: 1168: 982: 898: 872: 628: 617: 606: 522: 139: 128: 103: 4216:Van Vleck, J. H. (1928-04-01). 3451:is the LandĂ© g-factor of state 1066:First order perturbation theory 369:{\displaystyle C_{0},\chi _{0}} 117:is approximately described by 4575:Eponymous equations of physics 3910: 3886: 3710: 3686: 3643: 3619: 3491: 3467: 3436: 3430: 3394: 3388: 3362: 3356: 3277: 3256: 3225: 3205: 3196: 3184: 3150: 3144: 3085: 3070: 3049: 3043: 3025: 3019: 2963: 2957: 2903: 2897: 2856: 2850: 2463: 2442: 2411: 2398: 2374: 2362: 2251: 2228: 2179: 2173: 2153: 2147: 2093: 2064: 2058: 2032: 2026: 2002: 1981: 1969: 1950: 1946: 1933: 1864: 1858: 1309: 1276: 1258: 1195: 1183: 1164: 1160: 1114: 1108: 772:{\displaystyle \mu _{\rm {B}}} 621: 602: 495:{\displaystyle \chi _{0}>0} 462:{\displaystyle C_{0}\approx 0} 262: 256: 1: 4510:ferromagnetic superconductor 4296:Anderson, Philip W. (1987). 989:{\displaystyle \mathbf {H} } 905:{\displaystyle \mathbf {S} } 879:{\displaystyle \mathbf {L} } 529:{\displaystyle \mathbf {H} } 110:{\displaystyle \mathbf {H} } 4349:Quantum Theory of Magnetism 4266:van Vleck, John H. (1977). 3545:{\displaystyle k_{\rm {B}}} 3444:{\displaystyle g_{J}^{(i)}} 853:{\displaystyle m_{\rm {e}}} 402:{\displaystyle C_{0}\geq 0} 4591: 4075:and Curie's contribution ( 1544:is the material dependent 932:{\displaystyle r_{\perp }} 233:{\displaystyle \chi <0} 203:{\displaystyle \chi >0} 4322:. John Wiley & Sons. 4177:which have a localized 5f 2703:{\displaystyle \chi _{0}} 888:angular momentum operator 429:{\displaystyle \chi _{0}} 31:refers to a positive and 4319:Condensed Matter Physics 4299:John Hasbrouck Van Vleck 4048:azimuthal quantum number 1362:of the ground state and 939:is the component of the 739:{\displaystyle \mu _{0}} 49:John Hasbrouck Van Vleck 47:theory was developed by 4467:Van Vleck paramagnetism 4376:Coey, J. M. D. (2010). 4103:{\displaystyle C_{0}/T} 3966:While molecular oxygen 178:magnetic susceptibility 37:magnetic susceptibility 29:Van Vleck paramagnetism 4242:10.1103/PhysRev.31.587 4130: 4104: 4069: 3956: 3936: 3869: 3849: 3788: 3753: 3718: 3669: 3602: 3546: 3517: 3457: 3445: 3403: 3116: 2990: 2704: 2677: 2599: 2572: 2541: 2518: 2267: 2214: 2188: 2116: 2076: 1833: 1813: 1786: 1766: 1738: 1705: 1685: 1655: 1572: 1538: 1498: 1474: 1394: 1352: 1325: 1292: 1084: 1048: 1017: 990: 968: 933: 906: 880: 854: 821: 797: 773: 740: 710: 530: 496: 463: 430: 403: 370: 326: 302: 234: 204: 170: 147: 111: 4131: 4105: 4070: 3957: 3937: 3870: 3850: 3789: 3754: 3719: 3670: 3603: 3547: 3518: 3446: 3404: 3117: 2991: 2705: 2678: 2600: 2598:{\displaystyle L_{z}} 2573: 2571:{\displaystyle S_{z}} 2542: 2519: 2268: 2215: 2189: 2117: 2077: 1834: 1814: 1812:{\displaystyle H^{2}} 1787: 1767: 1739: 1706: 1686: 1684:{\displaystyle C_{2}} 1656: 1573: 1571:{\displaystyle H^{2}} 1539: 1499: 1475: 1402:Wigner–Eckart theorem 1395: 1353: 1351:{\displaystyle g_{J}} 1331:is the ground state, 1326: 1293: 1085: 1049: 1018: 1016:{\displaystyle H^{2}} 991: 969: 934: 907: 881: 855: 822: 798: 774: 741: 711: 531: 497: 464: 431: 404: 371: 327: 303: 235: 205: 171: 169:{\displaystyle \chi } 148: 112: 4114: 4079: 4053: 3946: 3882: 3859: 3798: 3763: 3728: 3682: 3615: 3560: 3527: 3463: 3417: 3130: 3006: 2725: 2687: 2612: 2582: 2555: 2531: 2283: 2224: 2198: 2126: 2089: 1846: 1823: 1796: 1776: 1756: 1715: 1695: 1668: 1589: 1555: 1508: 1488: 1423: 1366: 1335: 1305: 1097: 1074: 1056:perturbation theory. 1027: 1000: 978: 947: 916: 894: 868: 835: 811: 787: 754: 723: 543: 518: 473: 440: 413: 380: 340: 316: 250: 218: 212:diamagnetic material 188: 160: 124: 99: 4537:amorphous magnetism 4505:superferromagnetism 4234:1928PhRv...31..587V 4129:{\displaystyle j=1} 4068:{\displaystyle j=0} 4012:europium(III) oxide 4006:Systems of interest 3440: 3398: 3366: 3155: 3053: 3029: 2967: 2907: 2860: 2778: 2396: 2213:{\displaystyle i=0} 2183: 2157: 2068: 2036: 1060:Paschen-Back effect 748:vacuum permeability 695: 649: 4490:antiferromagnetism 4462:superparamagnetism 4202:. Clarendon Press. 4126: 4100: 4065: 3952: 3932: 3865: 3845: 3784: 3749: 3714: 3665: 3598: 3554:Boltzmann constant 3542: 3513: 3441: 3420: 3399: 3370: 3338: 3200: 3133: 3112: 3033: 3009: 2986: 2947: 2887: 2840: 2817: 2794: 2762: 2700: 2673: 2595: 2568: 2537: 2514: 2382: 2378: 2263: 2210: 2184: 2161: 2129: 2112: 2072: 2040: 2014: 1928: 1829: 1809: 1782: 1762: 1734: 1701: 1681: 1651: 1568: 1534: 1494: 1470: 1390: 1348: 1321: 1288: 1080: 1044: 1013: 986: 964: 929: 902: 876: 850: 817: 793: 769: 736: 706: 681: 635: 526: 492: 459: 426: 399: 376:are constant, and 366: 322: 298: 230: 200: 166: 143: 107: 45:quantum mechanical 41:Zeeman interaction 4552: 4551: 4450:superdiamagnetism 4438:Magnetic response 4389:978-0-521-81614-4 4359:978-3-540-85416-6 4329:978-0-470-94994-8 4046:that has a total 3978:and nitric oxide 3955:{\displaystyle T} 3868:{\displaystyle c} 3309: 3176: 3174: 2976: 2879: 2808: 2806: 2785: 2540:{\displaystyle n} 2509: 2354: 2342: 2070: 1919: 1907: 1832:{\displaystyle B} 1785:{\displaystyle H} 1765:{\displaystyle H} 1704:{\displaystyle n} 1497:{\displaystyle T} 1468: 1247: 1149: 1083:{\displaystyle H} 941:position operator 829:elementary charge 820:{\displaystyle e} 796:{\displaystyle g} 679: 600: 325:{\displaystyle T} 283: 16:Magnetic property 4582: 4424: 4417: 4410: 4401: 4394: 4393: 4373: 4364: 4363: 4343: 4334: 4333: 4313: 4307: 4306: 4304: 4293: 4282: 4281: 4279: 4278: 4263: 4254: 4253: 4213: 4204: 4203: 4195: 4176: 4175: 4174: 4165: 4164: 4163: 4150: 4149: 4148: 4135: 4133: 4132: 4127: 4109: 4107: 4106: 4101: 4096: 4091: 4090: 4074: 4072: 4071: 4066: 4045: 4044: 4043: 4034: 4033: 4032: 4024: 4023: 4001: 3997: 3993: 3992: 3991: 3981: 3977: 3976: 3975: 3961: 3959: 3958: 3953: 3941: 3939: 3938: 3933: 3928: 3927: 3926: 3913: 3908: 3907: 3889: 3874: 3872: 3871: 3866: 3854: 3852: 3851: 3846: 3832: 3821: 3820: 3819: 3793: 3791: 3790: 3785: 3780: 3779: 3778: 3758: 3756: 3755: 3750: 3745: 3744: 3743: 3723: 3721: 3720: 3715: 3713: 3708: 3707: 3689: 3674: 3672: 3671: 3666: 3661: 3660: 3659: 3646: 3641: 3640: 3622: 3607: 3605: 3604: 3599: 3594: 3583: 3582: 3581: 3551: 3549: 3548: 3543: 3541: 3540: 3539: 3522: 3520: 3519: 3514: 3509: 3508: 3507: 3494: 3489: 3488: 3470: 3450: 3448: 3447: 3442: 3439: 3428: 3408: 3406: 3405: 3400: 3397: 3386: 3379: 3365: 3354: 3347: 3334: 3333: 3310: 3308: 3307: 3306: 3287: 3286: 3285: 3280: 3271: 3270: 3265: 3259: 3254: 3253: 3238: 3237: 3228: 3223: 3222: 3217: 3208: 3202: 3199: 3175: 3173: 3172: 3160: 3154: 3153: 3141: 3121: 3119: 3118: 3113: 3108: 3100: 3099: 3094: 3088: 3083: 3082: 3073: 3068: 3067: 3062: 3052: 3041: 3028: 3017: 2995: 2993: 2992: 2987: 2982: 2978: 2977: 2975: 2966: 2955: 2946: 2927: 2926: 2912: 2908: 2906: 2895: 2880: 2878: 2870: 2869: 2864: 2859: 2848: 2834: 2827: 2826: 2816: 2807: 2805: 2804: 2803: 2793: 2780: 2777: 2772: 2771: 2761: 2760: 2748: 2747: 2746: 2709: 2707: 2706: 2701: 2699: 2698: 2682: 2680: 2679: 2674: 2672: 2671: 2670: 2642: 2641: 2640: 2624: 2623: 2604: 2602: 2601: 2596: 2594: 2593: 2577: 2575: 2574: 2569: 2567: 2566: 2546: 2544: 2543: 2538: 2523: 2521: 2520: 2515: 2510: 2508: 2507: 2506: 2505: 2492: 2491: 2484: 2473: 2472: 2471: 2466: 2457: 2456: 2451: 2445: 2440: 2439: 2424: 2423: 2414: 2409: 2401: 2395: 2390: 2380: 2377: 2353: 2352: 2347: 2343: 2338: 2337: 2336: 2326: 2319: 2318: 2300: 2299: 2298: 2272: 2270: 2269: 2264: 2259: 2254: 2243: 2242: 2237: 2231: 2219: 2217: 2216: 2211: 2193: 2191: 2190: 2185: 2182: 2171: 2170: 2156: 2145: 2138: 2121: 2119: 2118: 2113: 2108: 2107: 2102: 2096: 2081: 2079: 2078: 2073: 2071: 2069: 2067: 2056: 2049: 2035: 2024: 2023: 2012: 2011: 2010: 2005: 1996: 1995: 1990: 1984: 1979: 1968: 1957: 1949: 1944: 1936: 1930: 1927: 1918: 1917: 1912: 1908: 1903: 1902: 1901: 1900: 1890: 1889: 1879: 1869: 1868: 1867: 1838: 1836: 1835: 1830: 1818: 1816: 1815: 1810: 1808: 1807: 1791: 1789: 1788: 1783: 1771: 1769: 1768: 1763: 1743: 1741: 1740: 1735: 1730: 1729: 1710: 1708: 1707: 1702: 1690: 1688: 1687: 1682: 1680: 1679: 1660: 1658: 1657: 1652: 1647: 1646: 1634: 1633: 1618: 1617: 1616: 1577: 1575: 1574: 1569: 1567: 1566: 1543: 1541: 1540: 1535: 1533: 1532: 1520: 1519: 1503: 1501: 1500: 1495: 1479: 1477: 1476: 1471: 1469: 1464: 1463: 1454: 1449: 1448: 1447: 1399: 1397: 1396: 1391: 1389: 1381: 1373: 1357: 1355: 1354: 1349: 1347: 1346: 1330: 1328: 1327: 1322: 1317: 1312: 1297: 1295: 1294: 1289: 1284: 1279: 1274: 1266: 1261: 1256: 1248: 1243: 1242: 1241: 1231: 1229: 1228: 1219: 1218: 1203: 1198: 1193: 1182: 1171: 1163: 1158: 1150: 1145: 1144: 1143: 1133: 1131: 1130: 1118: 1117: 1089: 1087: 1086: 1081: 1053: 1051: 1050: 1045: 1043: 1042: 1037: 1036: 1022: 1020: 1019: 1014: 1012: 1011: 995: 993: 992: 987: 985: 973: 971: 970: 965: 963: 962: 957: 956: 938: 936: 935: 930: 928: 927: 911: 909: 908: 903: 901: 885: 883: 882: 877: 875: 859: 857: 856: 851: 849: 848: 847: 826: 824: 823: 818: 802: 800: 799: 794: 778: 776: 775: 770: 768: 767: 766: 745: 743: 742: 737: 735: 734: 715: 713: 712: 707: 705: 704: 694: 689: 680: 678: 677: 676: 675: 661: 660: 651: 648: 643: 631: 620: 609: 601: 596: 595: 594: 584: 582: 581: 569: 568: 563: 562: 552: 551: 535: 533: 532: 527: 525: 501: 499: 498: 493: 485: 484: 468: 466: 465: 460: 452: 451: 435: 433: 432: 427: 425: 424: 408: 406: 405: 400: 392: 391: 375: 373: 372: 367: 365: 364: 352: 351: 332:is the absolute 331: 329: 328: 323: 307: 305: 304: 299: 297: 296: 284: 279: 278: 269: 239: 237: 236: 231: 209: 207: 206: 201: 175: 173: 172: 167: 152: 150: 149: 144: 142: 131: 116: 114: 113: 108: 106: 67:'s formulas for 58: 21:condensed matter 4590: 4589: 4585: 4584: 4583: 4581: 4580: 4579: 4555: 4554: 4553: 4548: 4478:Magnetic states 4473: 4433: 4428: 4398: 4397: 4390: 4375: 4374: 4367: 4360: 4345: 4344: 4337: 4330: 4315: 4314: 4310: 4302: 4295: 4294: 4285: 4276: 4274: 4265: 4264: 4257: 4222:Physical Review 4215: 4214: 4207: 4197: 4196: 4192: 4187: 4181:configuration. 4180: 4173: 4171: 4170: 4169: 4167: 4162: 4160: 4159: 4158: 4156: 4147: 4145: 4144: 4143: 4141: 4112: 4111: 4082: 4077: 4076: 4051: 4050: 4042: 4040: 4039: 4038: 4036: 4031: 4028: 4027: 4026: 4022: 4019: 4018: 4017: 4015: 4008: 3999: 3995: 3990: 3987: 3986: 3985: 3983: 3979: 3974: 3971: 3970: 3969: 3967: 3944: 3943: 3917: 3893: 3880: 3879: 3857: 3856: 3801: 3796: 3795: 3769: 3761: 3760: 3734: 3726: 3725: 3693: 3680: 3679: 3650: 3626: 3613: 3612: 3563: 3558: 3557: 3530: 3525: 3524: 3498: 3474: 3461: 3460: 3415: 3414: 3319: 3292: 3288: 3275: 3260: 3245: 3229: 3212: 3203: 3164: 3128: 3127: 3089: 3074: 3057: 3004: 3003: 2968: 2941: 2937: 2918: 2871: 2836: 2835: 2832: 2828: 2818: 2795: 2784: 2752: 2728: 2723: 2722: 2716: 2690: 2685: 2684: 2646: 2628: 2615: 2610: 2609: 2585: 2580: 2579: 2558: 2553: 2552: 2529: 2528: 2496: 2475: 2474: 2461: 2446: 2431: 2415: 2381: 2327: 2321: 2320: 2310: 2286: 2281: 2280: 2232: 2222: 2221: 2196: 2195: 2124: 2123: 2097: 2087: 2086: 2013: 2000: 1985: 1931: 1891: 1881: 1880: 1874: 1873: 1852: 1844: 1843: 1821: 1820: 1799: 1794: 1793: 1774: 1773: 1754: 1753: 1750: 1721: 1713: 1712: 1693: 1692: 1671: 1666: 1665: 1638: 1625: 1592: 1587: 1586: 1558: 1553: 1552: 1524: 1511: 1506: 1505: 1486: 1485: 1455: 1426: 1421: 1420: 1364: 1363: 1338: 1333: 1332: 1303: 1302: 1232: 1220: 1210: 1134: 1122: 1103: 1095: 1094: 1072: 1071: 1068: 1030: 1025: 1024: 1003: 998: 997: 976: 975: 950: 945: 944: 919: 914: 913: 912:the spin and 892: 891: 886:is the orbital 866: 865: 838: 833: 832: 809: 808: 785: 784: 757: 752: 751: 726: 721: 720: 696: 666: 662: 652: 585: 573: 556: 541: 540: 516: 515: 508: 476: 471: 470: 443: 438: 437: 416: 411: 410: 383: 378: 377: 356: 343: 338: 337: 314: 313: 288: 270: 248: 247: 216: 215: 186: 185: 158: 157: 122: 121: 97: 96: 89: 56: 17: 12: 11: 5: 4588: 4586: 4578: 4577: 4572: 4567: 4565:Atomic physics 4557: 4556: 4550: 4549: 4547: 4546: 4545: 4544: 4539: 4529: 4527:mictomagnetism 4524: 4519: 4514: 4513: 4512: 4507: 4500:ferromagnetism 4497: 4495:ferrimagnetism 4492: 4487: 4485:altermagnetism 4481: 4479: 4475: 4474: 4472: 4471: 4470: 4469: 4464: 4454: 4453: 4452: 4441: 4439: 4435: 4434: 4429: 4427: 4426: 4419: 4412: 4404: 4396: 4395: 4388: 4365: 4358: 4335: 4328: 4308: 4283: 4255: 4228:(4): 587–613. 4205: 4189: 4188: 4186: 4183: 4178: 4172: 4161: 4151:ions). In the 4146: 4125: 4122: 4119: 4099: 4095: 4089: 4085: 4064: 4061: 4058: 4041: 4029: 4020: 4007: 4004: 3988: 3972: 3964: 3963: 3951: 3931: 3925: 3920: 3916: 3912: 3906: 3903: 3900: 3896: 3892: 3888: 3876: 3875:is a constant; 3864: 3844: 3841: 3838: 3835: 3831: 3827: 3824: 3818: 3815: 3812: 3809: 3804: 3783: 3777: 3772: 3768: 3748: 3742: 3737: 3733: 3712: 3706: 3703: 3700: 3696: 3692: 3688: 3676: 3664: 3658: 3653: 3649: 3645: 3639: 3636: 3633: 3629: 3625: 3621: 3609: 3597: 3593: 3589: 3586: 3580: 3577: 3574: 3571: 3566: 3538: 3533: 3512: 3506: 3501: 3497: 3493: 3487: 3484: 3481: 3477: 3473: 3469: 3438: 3435: 3432: 3427: 3423: 3411: 3410: 3396: 3393: 3390: 3385: 3382: 3378: 3373: 3369: 3364: 3361: 3358: 3353: 3350: 3346: 3341: 3337: 3332: 3329: 3326: 3322: 3318: 3314: 3305: 3302: 3299: 3295: 3291: 3284: 3279: 3274: 3269: 3264: 3258: 3252: 3248: 3244: 3241: 3236: 3232: 3227: 3221: 3216: 3211: 3207: 3198: 3195: 3192: 3189: 3186: 3183: 3179: 3171: 3167: 3163: 3158: 3152: 3149: 3146: 3140: 3136: 3124: 3123: 3111: 3107: 3103: 3098: 3093: 3087: 3081: 3077: 3072: 3066: 3061: 3056: 3051: 3048: 3045: 3040: 3036: 3032: 3027: 3024: 3021: 3016: 3012: 2997: 2996: 2985: 2981: 2974: 2971: 2965: 2962: 2959: 2954: 2950: 2944: 2940: 2936: 2933: 2930: 2925: 2921: 2916: 2911: 2905: 2902: 2899: 2894: 2890: 2886: 2883: 2877: 2874: 2868: 2863: 2858: 2855: 2852: 2847: 2843: 2839: 2831: 2825: 2821: 2815: 2811: 2802: 2798: 2792: 2788: 2783: 2776: 2770: 2765: 2759: 2755: 2751: 2745: 2742: 2739: 2736: 2731: 2715: 2712: 2697: 2693: 2669: 2666: 2663: 2660: 2657: 2654: 2649: 2645: 2639: 2636: 2631: 2627: 2622: 2618: 2592: 2588: 2565: 2561: 2549:number density 2536: 2525: 2524: 2513: 2504: 2499: 2495: 2490: 2487: 2483: 2478: 2470: 2465: 2460: 2455: 2450: 2444: 2438: 2434: 2430: 2427: 2422: 2418: 2413: 2408: 2404: 2400: 2394: 2389: 2385: 2376: 2373: 2370: 2367: 2364: 2361: 2357: 2351: 2346: 2341: 2335: 2330: 2324: 2317: 2313: 2309: 2306: 2303: 2297: 2294: 2289: 2262: 2258: 2253: 2249: 2246: 2241: 2236: 2230: 2209: 2206: 2203: 2181: 2178: 2175: 2169: 2164: 2160: 2155: 2152: 2149: 2144: 2141: 2137: 2132: 2111: 2106: 2101: 2095: 2083: 2082: 2066: 2063: 2060: 2055: 2052: 2048: 2043: 2039: 2034: 2031: 2028: 2022: 2017: 2009: 2004: 1999: 1994: 1989: 1983: 1978: 1974: 1971: 1967: 1963: 1960: 1956: 1952: 1948: 1943: 1939: 1935: 1926: 1922: 1916: 1911: 1906: 1899: 1894: 1888: 1884: 1877: 1872: 1866: 1863: 1860: 1855: 1851: 1828: 1806: 1802: 1781: 1761: 1749: 1746: 1733: 1728: 1724: 1720: 1700: 1678: 1674: 1662: 1661: 1650: 1645: 1641: 1637: 1632: 1628: 1624: 1621: 1615: 1612: 1609: 1606: 1603: 1600: 1595: 1565: 1561: 1546:Curie constant 1531: 1527: 1523: 1518: 1514: 1493: 1482: 1481: 1467: 1462: 1458: 1452: 1446: 1443: 1440: 1437: 1434: 1429: 1388: 1384: 1380: 1376: 1372: 1360:LandĂ© g-factor 1345: 1341: 1320: 1316: 1311: 1299: 1298: 1287: 1283: 1278: 1273: 1269: 1265: 1260: 1255: 1251: 1246: 1240: 1235: 1227: 1223: 1217: 1213: 1209: 1206: 1202: 1197: 1192: 1188: 1185: 1181: 1177: 1174: 1170: 1166: 1162: 1157: 1153: 1148: 1142: 1137: 1129: 1125: 1121: 1116: 1113: 1110: 1106: 1102: 1079: 1067: 1064: 1041: 1035: 1010: 1006: 984: 961: 955: 926: 922: 900: 874: 846: 841: 816: 792: 765: 760: 733: 729: 717: 716: 703: 699: 693: 688: 684: 674: 669: 665: 659: 655: 647: 642: 638: 634: 630: 626: 623: 619: 615: 612: 608: 604: 599: 593: 588: 580: 576: 572: 567: 561: 555: 550: 524: 507: 504: 491: 488: 483: 479: 458: 455: 450: 446: 423: 419: 398: 395: 390: 386: 363: 359: 355: 350: 346: 321: 310: 309: 295: 291: 287: 282: 277: 273: 267: 264: 261: 258: 255: 229: 226: 223: 199: 196: 193: 165: 154: 153: 141: 137: 134: 130: 105: 88: 85: 25:atomic physics 15: 13: 10: 9: 6: 4: 3: 2: 4587: 4576: 4573: 4571: 4568: 4566: 4563: 4562: 4560: 4543: 4540: 4538: 4535: 4534: 4533: 4530: 4528: 4525: 4523: 4522:metamagnetism 4520: 4518: 4517:helimagnetism 4515: 4511: 4508: 4506: 4503: 4502: 4501: 4498: 4496: 4493: 4491: 4488: 4486: 4483: 4482: 4480: 4476: 4468: 4465: 4463: 4460: 4459: 4458: 4457:paramagnetism 4455: 4451: 4448: 4447: 4446: 4443: 4442: 4440: 4436: 4432: 4425: 4420: 4418: 4413: 4411: 4406: 4405: 4402: 4391: 4385: 4381: 4380: 4372: 4370: 4366: 4361: 4355: 4351: 4350: 4342: 4340: 4336: 4331: 4325: 4321: 4320: 4312: 4309: 4301: 4300: 4292: 4290: 4288: 4284: 4273: 4269: 4262: 4260: 4256: 4251: 4247: 4243: 4239: 4235: 4231: 4227: 4223: 4219: 4212: 4210: 4206: 4201: 4194: 4191: 4184: 4182: 4154: 4139: 4123: 4120: 4117: 4097: 4093: 4087: 4083: 4062: 4059: 4056: 4049: 4013: 4005: 4003: 3949: 3929: 3918: 3914: 3904: 3901: 3898: 3894: 3890: 3877: 3862: 3842: 3839: 3836: 3833: 3829: 3825: 3822: 3802: 3781: 3770: 3766: 3746: 3735: 3731: 3704: 3701: 3698: 3694: 3690: 3677: 3662: 3651: 3647: 3637: 3634: 3631: 3627: 3623: 3610: 3595: 3591: 3587: 3584: 3564: 3555: 3531: 3510: 3499: 3495: 3485: 3482: 3479: 3475: 3471: 3458: 3456: 3454: 3433: 3425: 3421: 3391: 3383: 3380: 3371: 3367: 3359: 3351: 3348: 3339: 3335: 3330: 3327: 3324: 3320: 3316: 3312: 3303: 3300: 3297: 3293: 3289: 3282: 3267: 3250: 3246: 3242: 3239: 3234: 3230: 3219: 3193: 3190: 3187: 3181: 3177: 3169: 3161: 3156: 3147: 3138: 3134: 3126: 3125: 3105: 3096: 3079: 3075: 3064: 3046: 3038: 3034: 3030: 3022: 3014: 3010: 3002: 3001: 3000: 2983: 2979: 2972: 2969: 2960: 2952: 2948: 2942: 2938: 2934: 2931: 2928: 2923: 2919: 2914: 2909: 2900: 2892: 2888: 2884: 2881: 2875: 2872: 2866: 2861: 2853: 2845: 2841: 2837: 2829: 2823: 2819: 2813: 2809: 2800: 2796: 2790: 2786: 2781: 2774: 2763: 2757: 2753: 2749: 2729: 2721: 2720: 2719: 2713: 2711: 2695: 2691: 2647: 2643: 2629: 2625: 2620: 2616: 2608:In this way, 2606: 2590: 2586: 2563: 2559: 2550: 2534: 2511: 2497: 2493: 2488: 2485: 2476: 2468: 2453: 2436: 2432: 2428: 2425: 2420: 2416: 2392: 2387: 2383: 2371: 2368: 2365: 2359: 2355: 2349: 2344: 2328: 2322: 2315: 2311: 2307: 2304: 2301: 2287: 2279: 2278: 2277: 2274: 2247: 2239: 2207: 2204: 2201: 2176: 2162: 2158: 2150: 2142: 2139: 2130: 2104: 2061: 2053: 2050: 2041: 2037: 2029: 2015: 2007: 1992: 1972: 1961: 1958: 1924: 1920: 1914: 1909: 1892: 1886: 1882: 1875: 1870: 1861: 1853: 1842: 1841: 1840: 1826: 1804: 1800: 1779: 1759: 1747: 1745: 1726: 1722: 1698: 1676: 1672: 1643: 1639: 1630: 1626: 1622: 1619: 1593: 1585: 1584: 1583: 1581: 1563: 1559: 1549: 1547: 1529: 1525: 1521: 1516: 1512: 1491: 1465: 1460: 1456: 1450: 1427: 1419: 1418: 1417: 1415: 1411: 1407: 1406:paramagnetism 1403: 1382: 1374: 1361: 1343: 1339: 1267: 1233: 1225: 1221: 1215: 1211: 1207: 1186: 1175: 1172: 1135: 1127: 1123: 1119: 1111: 1104: 1093: 1092: 1091: 1077: 1065: 1063: 1061: 1057: 1039: 1008: 1004: 959: 942: 924: 920: 889: 863: 862:electron mass 839: 830: 814: 806: 790: 782: 781:Bohr magneton 758: 749: 731: 727: 701: 697: 691: 686: 682: 667: 663: 657: 653: 645: 640: 636: 632: 624: 613: 610: 586: 578: 574: 570: 565: 553: 539: 538: 537: 513: 505: 503: 489: 486: 481: 477: 456: 453: 448: 444: 421: 417: 396: 393: 388: 384: 361: 357: 353: 348: 344: 335: 319: 293: 289: 285: 280: 275: 271: 265: 259: 253: 246: 245: 244: 241: 227: 224: 221: 213: 197: 194: 191: 183: 179: 163: 135: 132: 120: 119: 118: 94: 93:magnetization 86: 84: 82: 81:closed shells 78: 74: 70: 69:paramagnetism 66: 65:Paul Langevin 62: 54: 50: 46: 42: 38: 34: 30: 26: 22: 4466: 4445:diamagnetism 4378: 4348: 4318: 4311: 4298: 4275:. Retrieved 4271: 4225: 4221: 4199: 4193: 4009: 3965: 3452: 3412: 2998: 2717: 2607: 2526: 2275: 2084: 1751: 1663: 1580:diamagnetism 1550: 1483: 1300: 1069: 718: 509: 311: 242: 182:paramagnetic 155: 90: 77:diamagnetism 53:nitric oxide 28: 18: 4272:Nobel Prize 1414:Curie's law 1062:dominates. 512:Hamiltonian 334:temperature 87:Description 73:Curie's law 33:temperature 4559:Categories 4532:spin glass 4277:2020-10-18 4185:References 3724:is either 506:Derivation 61:rare-earth 4431:Magnetism 4250:0031-899X 4153:actinides 3915:≈ 3891:δ 3823:∝ 3803:χ 3767:≪ 3732:≫ 3691:δ 3648:≫ 3624:δ 3585:∝ 3565:χ 3496:≪ 3472:δ 3368:− 3317:δ 3290:δ 3273:⟩ 3210:⟨ 3191:≠ 3178:∑ 3166:ℏ 3110:ℏ 3102:⟩ 3055:⟨ 2943:− 2935:⁡ 2882:− 2810:∑ 2787:∑ 2764:μ 2754:μ 2730:χ 2692:χ 2648:χ 2630:χ 2626:≈ 2617:χ 2494:− 2459:⟩ 2403:⟨ 2369:≠ 2356:∑ 2340:ℏ 2329:μ 2312:μ 2288:χ 2261:⟩ 2245:⟩ 2110:⟩ 2038:− 1998:⟩ 1973:⋅ 1938:⟨ 1921:∑ 1905:ℏ 1893:μ 1883:μ 1850:Δ 1732:⟩ 1719:⟨ 1649:⟩ 1636:⟨ 1623:− 1594:χ 1522:≈ 1451:≈ 1428:χ 1410:Brillouin 1319:⟩ 1286:⟩ 1268:⋅ 1250:⟨ 1245:ℏ 1234:μ 1222:μ 1205:⟩ 1187:⋅ 1152:⟨ 1147:ℏ 1136:μ 1124:μ 1101:Δ 925:⊥ 759:μ 728:μ 687:⊥ 637:μ 625:⋅ 598:ℏ 587:μ 575:μ 478:χ 454:≈ 418:χ 394:≥ 358:χ 290:χ 266:≈ 254:χ 222:χ 210:. For a 192:χ 164:χ 136:χ 59:) and of 4542:spin ice 4138:samarium 3523:, where 2220:, where 1504:, where 805:g-factor 409:, while 4230:Bibcode 4140:salts ( 3878:if all 3678:if all 3611:if all 3459:if all 2999:where 2547:is the 1358:is the 860:is the 827:is the 803:is the 779:is the 746:is the 176:is the 4386:  4356:  4326:  4248:  3855:where 2551:, and 2527:where 2122:, and 1839:term: 1664:where 1301:where 719:where 312:where 156:where 75:) and 43:. 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Index

condensed matter
atomic physics
temperature
magnetic susceptibility
Zeeman interaction
quantum mechanical
John Hasbrouck Van Vleck
nitric oxide
rare-earth
Paul Langevin
paramagnetism
Curie's law
diamagnetism
closed shells
magnetization
magnetic susceptibility
paramagnetic
diamagnetic material
temperature
Hamiltonian
vacuum permeability
Bohr magneton
g-factor
elementary charge
electron mass
angular momentum operator
position operator
perturbation theory.
Paschen-Back effect
Landé g-factor

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