1450:
70:
1045:, to the right or to the left of the target, due to spin-orbit coupling. It can assume values ranging from -1 (spin-up electron is scattered with 100% probability to the left of the target) to +1 (spin-up electron is scattered with 100% probability to the right of the target). The value of the Sherman function depends on the energy of the incoming electron, evaluated via the parameter
989:
17:
1496:
is usually present, in order to accelerate the beam up to keV or to MeV energies. Since standard electron detectors count electrons being insensitive to their spin, after the scattering with the target any information about the original polarization of the beam is lost. Nevertheless, by measuring the
73:
Correction to
Coulomb potential due to spin-orbit coupling. The Coulomb potential, result of the interaction of the electron with the charged nucleus is shown in green. The new potential for spin up (blue) and spin down (red) electrons are shown after spin-orbit correctios have been taken into
477:
367:
1491:
To measure the polarization of an electron beam, a Mott detector is required. In order to maximize the spin-orbit coupling, it is necessary that the electrons arrive near to the nuclei of the target. To achieve this condition, a system of
801:
1497:
difference in the counts of the two detectors, the asymmetry can be evaluated and, if the
Sherman function is known from previous calibration, the polarization can be calculated by inverting the last formula.
738:
270:
1547:. If an electron beam with a 3:1 ratio of spin-up over spin-down electrons collide with the target, it will be splitted with a ratio 5:3, according to previous equation, with an asymmetry of 25%.
982:
1367:
1239:
378:
1797:
Berti, G.; Calloni, A.; Brambilla, A.; Bussetti, G.; Duò, L.; Ciccacci, F. (July 2014). "Direct observation of spin-resolved full and empty electron states in ferromagnetic surfaces".
1076:
992:
Sherman function as a function of the angle for mercury (Z = 80) with β = 0.2. Typically detectors are placed in a position where the effect is maximized, 120° for gold and mercury
20:
Schematic view of Mott scattering. An incident electron beam collides with the target and the electrons scattered to the left and to the right at a specific angle θ are detected
1668:
Nishide, Akinori; Takeichi, Yasuo; Okuda, Taichi; Taskin, Alexey A; Hirahara, Toru; Nakatsuji, Kan; Komori, Fumio; Kakizaki, Akito; Ando, Yoichi; Matsuda, Iwao (17 June 2010).
529:
1545:
1486:
1435:
1111:
615:
560:
502:
302:
1023:
146:
1390:
1043:
584:
106:
882:
855:
2029:
Petrov, V. N.; Landolt, M.; Galaktionov, M. S.; Yushenkov, B. V. (December 1997). "A new compact 60 kV Mott polarimeter for spin polarized electron spectroscopy".
696:
1372:
Plugging these formulas inside the definition of asymmetry, it is possible to obtain a simple expression for the evaluation of the asymmetry at a specific angle
902:
663:
639:
166:
52:
experiments. A correct evaluation of the
Sherman function associated to a particular experimental setup is of vital importance in experiments of spin polarized
1670:"Spin-polarized surface bands of a three-dimensional topological insulator studied by high-resolution spin- and angle-resolved photoemission spectroscopy"
372:
with the magnetic field that the electron sees, due to its orbital motion around the nucleus, whose expression in the non-relativistic limit is:
753:
2135:
1895:
1949:
Czyżewski, Zbigniew; MacCallum, Danny O’Neill; Romig, Alton; Joy, David C. (October 1990). "Calculations of Mott scattering cross section".
1449:
1870:
830:
at the same angle from the target, one on the right and one on the left, they will generally measure a different number of electrons
1778:
811:
with different probabilities at different angles. Since the spin-orbit coupling is enhanced when the involved nuclei possess a high
705:
174:
292:. Spin orbit interaction can be evaluated, in the rest reference frame of the electron, as the result of the interaction of the
281:
1753:
1441:
Theoretical calculations are available for different atomic targets and for a specific target, as a function of the angle.
910:
472:{\displaystyle \mathbf {B} ={\frac {1}{m_{\text{e}}ec^{2}}}{\frac {1}{r}}{\frac {\partial U(r)}{\partial r}}\mathbf {L} .}
1247:
1122:
1914:
Sherman, Noah; Nelson, Donald F. (15 June 1959). "Determination of
Electron Polarization by Means of Mott Scattering".
1556:
1566:
53:
56:, which is an experimental technique which allows to obtain information about the magnetic behaviour of a sample.
1984:
Nelson, D. F.; Pidd, R. W. (1 May 1959). "Measurement of the Mott
Asymmetry in Double Scattering of Electrons".
1641:
Proceedings of the Royal
Society of London. Series A, Containing Papers of a Mathematical and Physical Character
1113:, spin-up electrons will be scattered with the same probability to the right and to the left of the target.
1048:
2130:
285:
1512:
In the panel it is shown an example of the working principle of a Mott detector, supposing a value for
2098:
2038:
1993:
1958:
1923:
1806:
1718:
1681:
1606:
507:
293:
1515:
1456:
362:{\displaystyle {\boldsymbol {\mu }}_{S}=-g_{\text{s}}\mu _{\text{B}}{\frac {\mathbf {S} }{\hbar }},}
2140:
1597:
Sherman, Noah (15 September 1956). "Coulomb
Scattering of Relativistic Electrons by Point Nuclei".
1402:
808:
289:
276:
It is known that, when an electron collides against a nucleus, the scattering event is governed by
29:
1081:
593:
538:
485:
41:
1840:
618:
563:
1771:
Polarized
Sources, Targets and Polarimetry : Proceedings of the 13th International Workshop
999:
69:
1500:
In order to characterize completely the in-plane polarization, setups are available, with four
288:
coupling can be taken into account and the effect on the
Hamiltonian can be evaluated with the
1891:
1866:
1832:
1774:
1749:
642:
115:
45:
1375:
1028:
2106:
2046:
2009:
2001:
1966:
1931:
1822:
1814:
1726:
1689:
1648:
1614:
741:
699:
569:
277:
81:
1025:
is a measure of the probability of a spin-up electron to be scattered, at a specific angle
860:
833:
747:
Due to spin orbit coupling, a new term will appear in the
Hamiltonian, whose expression is
1561:
1493:
672:
587:
49:
2102:
2042:
1997:
1962:
1927:
1810:
1722:
1694:
1685:
1669:
1610:
887:
666:
648:
624:
151:
33:
2124:
2110:
2065:
812:
532:
1844:
1504:, two devoted to the left-right measure and two devoted to the up-right measure.
818:, the target is usually made of heavy metals, such as mercury, gold and thorium.
1501:
827:
988:
1730:
16:
1935:
1618:
2005:
1836:
1653:
1636:
796:{\displaystyle V_{\text{SO}}=-{\boldsymbol {\mu _{s}}}\cdot \mathbf {B} }
109:
37:
2014:
1827:
2050:
1818:
1970:
2089:
Ladislas Wiza, Joseph (June 1979). "Microchannel plate detectors".
1448:
987:
148:, exists. The unbalance can be evaluated through the polarization
78:
When an electron beam is polarized, an unbalance between spin-up,
68:
15:
1769:
Ciullo, Giuseppe; Contalbrigo, Marco; Lenisa, Paolo (2009).
733:{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }
265:{\displaystyle P={\frac {n_{up}-n_{down}}{n_{up}+n_{down}}}}
40:. It was first evaluated theoretically by the physicist
1865:(Second ed.). World scientific. pp. 756–757.
1773:. World Scientific Publishing Co Pte Ltd. p. 337.
1709:
Mayne, K. I. (July 1969). "Polarized electron beams".
884:. Consequently it is possible to define the asymmetry
1518:
1459:
1405:
1378:
1250:
1125:
1084:
1051:
1031:
1002:
913:
890:
863:
836:
756:
708:
675:
651:
627:
596:
572:
541:
510:
488:
381:
305:
177:
154:
118:
84:
977:{\displaystyle A={\frac {n_{R}-n_{L}}{n_{R}+n_{L}}}}
1362:{\displaystyle n_{L}={\frac {n_{up}+n_{down}}{2}}.}
1539:
1480:
1429:
1384:
1361:
1234:{\displaystyle n_{R}={\frac {n_{up}+n_{down}}{2}}}
1233:
1105:
1070:
1037:
1017:
976:
896:
876:
849:
795:
732:
690:
657:
633:
609:
578:
554:
523:
496:
471:
361:
264:
160:
140:
100:
1863:Handbook of accelerator physics and engineering
8:
1909:
1907:
1861:Chao, Alexander W.; Mess, Karl H. (2013).
1856:
1854:
28:describes the dependence of electron-atom
2013:
1826:
1693:
1652:
1592:
1590:
1588:
1586:
1584:
1582:
1517:
1458:
1453:Mott scattering of an electron beam with
1404:
1377:
1311:
1271:
1264:
1255:
1249:
1186:
1146:
1139:
1130:
1124:
1083:
1058:
1050:
1030:
1001:
965:
952:
940:
927:
920:
912:
889:
868:
862:
841:
835:
788:
778:
773:
761:
755:
725:
717:
709:
707:
674:
650:
626:
601:
595:
571:
546:
540:
515:
509:
489:
487:
461:
432:
422:
413:
400:
390:
382:
380:
346:
344:
338:
328:
312:
307:
304:
244:
228:
207:
191:
184:
176:
153:
123:
117:
89:
83:
2066:"MeV Mott Polarimetry at Jefferson Lab"
1792:
1790:
1748:(2nd ed.). Pearson Prentice Hall.
1578:
779:
775:
573:
351:
308:
1637:"The scattering of electrons by atoms"
1630:
1628:
807:Due to this effect, electrons will be
1635:Mott, Nevill Francis (January 1997).
1071:{\displaystyle \beta ={\frac {v}{c}}}
7:
65:Polarization and spin-orbit coupling
452:
435:
280:. This is the leading term in the
14:
1746:Introduction to quantum mechanics
44:and it allows the measurement of
2031:Review of Scientific Instruments
1799:Review of Scientific Instruments
789:
726:
718:
710:
490:
462:
383:
347:
2091:Nuclear Instruments and Methods
1890:. Springer-Verlag. p. 49.
524:{\displaystyle \mu _{\text{B}}}
1540:{\displaystyle S(\theta )=0.5}
1528:
1522:
1481:{\displaystyle S(\theta )=0.5}
1469:
1463:
1424:
1418:
1347:
1344:
1338:
1326:
1301:
1298:
1292:
1280:
1222:
1219:
1213:
1201:
1176:
1173:
1167:
1155:
1094:
1088:
1012:
1006:
504:is the spin angular-momentum,
447:
441:
1:
1695:10.1088/1367-2630/12/6/065011
1430:{\displaystyle A=PS(\theta )}
1116:Then it is possible to write
2136:Foundational quantum physics
2111:10.1016/0029-554X(79)90734-1
1744:Griffiths, Davis J. (2005).
1106:{\displaystyle S(\theta )=0}
610:{\displaystyle m_{\text{e}}}
555:{\displaystyle g_{\text{s}}}
497:{\displaystyle \mathbf {S} }
2157:
1951:Journal of Applied Physics
1567:Photoemission spectroscopy
1018:{\displaystyle S(\theta )}
284:, but a correction due to
54:photoemission spectroscopy
1886:Joachim, Kessler (1976).
1731:10.1080/00107516908204794
1936:10.1103/PhysRev.114.1541
1619:10.1103/physrev.103.1601
141:{\displaystyle n_{down}}
2006:10.1103/PhysRev.114.728
1385:{\displaystyle \theta }
1038:{\displaystyle \theta }
48:of an electron beam by
1674:New Journal of Physics
1654:10.1098/rspa.1930.0082
1557:Spin–orbit interaction
1541:
1488:
1482:
1431:
1386:
1363:
1235:
1107:
1072:
1039:
1019:
993:
978:
898:
878:
851:
797:
734:
692:
659:
635:
611:
580:
579:{\displaystyle \hbar }
556:
525:
498:
473:
363:
266:
162:
142:
102:
101:{\displaystyle n_{up}}
75:
21:
1542:
1483:
1452:
1432:
1387:
1364:
1236:
1108:
1073:
1040:
1020:
996:The Sherman function
991:
979:
899:
879:
877:{\displaystyle n_{L}}
852:
850:{\displaystyle n_{R}}
798:
735:
693:
660:
636:
612:
581:
557:
526:
499:
482:In these expressions
474:
364:
267:
163:
143:
103:
72:
19:
1711:Contemporary Physics
1516:
1457:
1403:
1376:
1248:
1123:
1082:
1049:
1029:
1000:
911:
888:
861:
834:
754:
706:
702:of the electron and
691:{\displaystyle U=eV}
673:
649:
625:
594:
570:
539:
508:
486:
379:
303:
294:spin magnetic moment
175:
152:
116:
82:
2103:1979NucIM.162..587L
2043:1997RScI...68.4385P
1998:1959PhRv..114..728N
1963:1990JAP....68.3066C
1928:1959PhRv..114.1541S
1888:Polarized electrons
1811:2014RScI...85g3901B
1723:1969ConPh..10..387M
1686:2010NJPh...12f5011N
1611:1956PhRv..103.1601S
290:perturbation theory
278:Coulomb interaction
1537:
1489:
1478:
1427:
1382:
1359:
1231:
1103:
1068:
1035:
1015:
994:
974:
894:
874:
847:
793:
730:
688:
655:
631:
607:
576:
552:
521:
494:
469:
359:
262:
158:
138:
98:
76:
22:
2051:10.1063/1.1148400
2037:(12): 4385–4389.
1897:978-3-662-12721-6
1819:10.1063/1.4885447
1354:
1229:
1066:
972:
897:{\displaystyle A}
764:
658:{\displaystyle c}
643:elementary charge
634:{\displaystyle e}
604:
549:
518:
459:
430:
420:
403:
354:
341:
331:
260:
161:{\displaystyle P}
36:of the scattered
2148:
2115:
2114:
2097:(1–3): 587–601.
2086:
2080:
2079:
2077:
2075:
2070:
2064:Steigerwald, M.
2061:
2055:
2054:
2026:
2020:
2019:
2017:
1981:
1975:
1974:
1971:10.1063/1.346400
1957:(7): 3066–3072.
1946:
1940:
1939:
1922:(6): 1541–1542.
1911:
1902:
1901:
1883:
1877:
1876:
1858:
1849:
1848:
1830:
1794:
1785:
1784:
1766:
1760:
1759:
1741:
1735:
1734:
1706:
1700:
1699:
1697:
1665:
1659:
1658:
1656:
1647:(806): 658–665.
1632:
1623:
1622:
1605:(6): 1601–1607.
1594:
1546:
1544:
1543:
1538:
1487:
1485:
1484:
1479:
1436:
1434:
1433:
1428:
1391:
1389:
1388:
1383:
1368:
1366:
1365:
1360:
1355:
1350:
1325:
1324:
1279:
1278:
1265:
1260:
1259:
1240:
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1237:
1232:
1230:
1225:
1200:
1199:
1154:
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1140:
1135:
1134:
1112:
1110:
1109:
1104:
1077:
1075:
1074:
1069:
1067:
1059:
1044:
1042:
1041:
1036:
1024:
1022:
1021:
1016:
983:
981:
980:
975:
973:
971:
970:
969:
957:
956:
946:
945:
944:
932:
931:
921:
903:
901:
900:
895:
883:
881:
880:
875:
873:
872:
856:
854:
853:
848:
846:
845:
826:If we place two
802:
800:
799:
794:
792:
784:
783:
782:
766:
765:
762:
742:angular momentum
739:
737:
736:
731:
729:
721:
713:
700:potential energy
697:
695:
694:
689:
664:
662:
661:
656:
640:
638:
637:
632:
616:
614:
613:
608:
606:
605:
602:
585:
583:
582:
577:
561:
559:
558:
553:
551:
550:
547:
530:
528:
527:
522:
520:
519:
516:
503:
501:
500:
495:
493:
478:
476:
475:
470:
465:
460:
458:
450:
433:
431:
423:
421:
419:
418:
417:
405:
404:
401:
391:
386:
368:
366:
365:
360:
355:
350:
345:
343:
342:
339:
333:
332:
329:
317:
316:
311:
296:of the electron
271:
269:
268:
263:
261:
259:
258:
257:
236:
235:
222:
221:
220:
199:
198:
185:
167:
165:
164:
159:
147:
145:
144:
139:
137:
136:
108:, and spin-down
107:
105:
104:
99:
97:
96:
26:Sherman function
2156:
2155:
2151:
2150:
2149:
2147:
2146:
2145:
2121:
2120:
2119:
2118:
2088:
2087:
2083:
2073:
2071:
2068:
2063:
2062:
2058:
2028:
2027:
2023:
1986:Physical Review
1983:
1982:
1978:
1948:
1947:
1943:
1916:Physical Review
1913:
1912:
1905:
1898:
1885:
1884:
1880:
1873:
1860:
1859:
1852:
1796:
1795:
1788:
1781:
1768:
1767:
1763:
1756:
1743:
1742:
1738:
1708:
1707:
1703:
1667:
1666:
1662:
1634:
1633:
1626:
1599:Physical Review
1596:
1595:
1580:
1575:
1562:Mott scattering
1553:
1514:
1513:
1510:
1494:electron optics
1455:
1454:
1447:
1401:
1400:
1374:
1373:
1307:
1267:
1266:
1251:
1246:
1245:
1182:
1142:
1141:
1126:
1121:
1120:
1080:
1079:
1047:
1046:
1027:
1026:
998:
997:
961:
948:
947:
936:
923:
922:
909:
908:
886:
885:
864:
859:
858:
837:
832:
831:
824:
774:
757:
752:
751:
704:
703:
671:
670:
647:
646:
623:
622:
597:
592:
591:
588:Planck constant
586:is the reduced
568:
567:
542:
537:
536:
511:
506:
505:
484:
483:
451:
434:
409:
396:
395:
377:
376:
334:
324:
306:
301:
300:
240:
224:
223:
203:
187:
186:
173:
172:
150:
149:
119:
114:
113:
85:
80:
79:
67:
62:
50:Mott scattering
12:
11:
5:
2154:
2152:
2144:
2143:
2138:
2133:
2123:
2122:
2117:
2116:
2081:
2056:
2021:
1992:(3): 728–735.
1976:
1941:
1903:
1896:
1878:
1872:978-9814415859
1871:
1850:
1786:
1779:
1761:
1754:
1736:
1717:(4): 387–412.
1701:
1660:
1624:
1577:
1576:
1574:
1571:
1570:
1569:
1564:
1559:
1552:
1549:
1536:
1533:
1530:
1527:
1524:
1521:
1509:
1506:
1477:
1474:
1471:
1468:
1465:
1462:
1446:
1443:
1439:
1438:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1381:
1370:
1369:
1358:
1353:
1349:
1346:
1343:
1340:
1337:
1334:
1331:
1328:
1323:
1320:
1317:
1314:
1310:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1277:
1274:
1270:
1263:
1258:
1254:
1242:
1241:
1228:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1198:
1195:
1192:
1189:
1185:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1152:
1149:
1145:
1138:
1133:
1129:
1102:
1099:
1096:
1093:
1090:
1087:
1065:
1062:
1057:
1054:
1034:
1014:
1011:
1008:
1005:
986:
985:
968:
964:
960:
955:
951:
943:
939:
935:
930:
926:
919:
916:
893:
871:
867:
844:
840:
823:
820:
805:
804:
791:
787:
781:
777:
772:
769:
760:
728:
724:
720:
716:
712:
687:
684:
681:
678:
667:speed of light
654:
630:
600:
575:
545:
514:
492:
480:
479:
468:
464:
457:
454:
449:
446:
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440:
437:
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426:
416:
412:
408:
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394:
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349:
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327:
323:
320:
315:
310:
274:
273:
256:
253:
250:
247:
243:
239:
234:
231:
227:
219:
216:
213:
210:
206:
202:
197:
194:
190:
183:
180:
157:
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2132:
2131:Electron beam
2129:
2128:
2126:
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2108:
2104:
2100:
2096:
2092:
2085:
2082:
2067:
2060:
2057:
2052:
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2011:
2007:
2003:
1999:
1995:
1991:
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1980:
1977:
1972:
1968:
1964:
1960:
1956:
1952:
1945:
1942:
1937:
1933:
1929:
1925:
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1917:
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714:
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619:electron mass
598:
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543:
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533:Bohr magneton
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282:Hamiltonian
168:defined as
2141:Scattering
2125:Categories
1755:0131118927
1573:References
286:spin-orbit
60:Background
30:scattering
1526:θ
1467:θ
1422:θ
1380:θ
1342:θ
1296:θ
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1217:θ
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1171:θ
1092:θ
1053:β
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1010:θ
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828:detectors
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574:ℏ
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336:μ
322:−
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201:−
110:electrons
38:electrons
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1837:25085146
1551:See also
564:g-factor
2099:Bibcode
2074:25 June
2039:Bibcode
1994:Bibcode
1959:Bibcode
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1508:Example
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