503:
156:
498:{\displaystyle f_{\text{e}}(q,Z)={\frac {me^{2}}{32\pi ^{3}\hbar ^{2}\epsilon _{0}}}{\Bigg (}{\frac {Z-f_{x}(q,Z)}{q^{2}}}{\Bigg )}={\frac {1}{8\pi ^{2}a_{0}}}{\Bigg (}{\frac {Z-f_{x}(q,Z)}{q^{2}}}{\Bigg )}\approx (0.2393~{\textrm {nm}}^{-1})\cdot {\Bigg (}{\frac {Z-f_{x}(q,Z)}{q^{2}}}{\Bigg )}}
890:
The Mott–Bethe formula was originally derived for free atoms, and is rigorously true provided the X-ray scattering form factor is known exactly. However, in solids, the accuracy of the Mott–Bethe formula is best for large values of
809:, which is usually presented in dimensionless units. To perform a one-to-one comparison between the electron and X-ray form factors in the same units, the X-ray form factor should be multiplied by the square root of the
854:
1017:
762:
78:
609:
881:
807:
717:
124:
935:
578:
671:
640:
975:
955:
909:
554:
526:
1087:"An accurate parameterization for scattering factors, electron densities and electrostatic potentials for neutral atoms that obey all physical constraints"
957:(i.e. long distances) can deviate from the atomic distribution of electrons due the chemical bonds between atoms in a solid. For smaller values of
1019:
can be determined from tabulated values, such as those in the
International Tables for Crystallography using (non)relativistic
1266:
1143:
1064:
146:
for the charge density of an atom (including both the nucleus and electron cloud) in the
Fourier domain. Following the first
815:
17:
529:
1168:
Proceedings of the Royal
Society of London. Series A, Containing Papers of a Mathematical and Physical Character
1293:
980:
725:
41:
884:
1207:
1023:
calculations, or other numerical parameterizations of the calculated charge distribution of atoms.
765:
587:
859:
1288:
1043:
Cowley, J. M. (2006). "Electron diffraction and electron microscopy in structure determination".
1020:
770:
680:
87:
914:
810:
143:
563:
1219:
1175:
649:
618:
8:
612:
139:
131:
32:
1223:
1179:
1255:
960:
940:
894:
539:
511:
147:
135:
35:
1262:
1235:
1139:
1116:
1111:
1060:
1227:
1183:
1106:
1098:
1052:
533:
81:
1056:
581:
1086:
1102:
1282:
1239:
1231:
1120:
643:
557:
719:
is the dimensionless X-ray scattering form factor for the electron density.
1188:
1163:
764:
has units of length, as is typical for the scattering factor, unlike the
674:
1208:"Zur Theorie des Durchgangs schneller Korpuskularstrahlen durch Materie"
127:
1252:
937:Å) because the distribution of the charge density at smaller
1162:
Mott, Nevill
Francis; Bragg, William Lawrence (1930-06-02).
983:
963:
943:
917:
897:
862:
818:
773:
728:
683:
652:
621:
590:
566:
542:
514:
159:
90:
44:
134:both in 1930, and simply follows from applying the
1254:
1011:
969:
949:
929:
903:
875:
849:{\displaystyle {\sqrt {\sigma _{T}}}=r_{\text{e}}}
848:
801:
756:
711:
665:
634:
603:
572:
548:
520:
497:
118:
72:
490:
436:
394:
340:
298:
244:
1280:
1253:L. M. Peng; S. L. Dudarev; M. J. Whalen (2004).
1257:High-Energy Electron Diffraction and Microscopy
1084:
528:is the magnitude of the scattering vector of
31:is an approximation used to calculate atomic
126:. The formula was derived independently by
887:, to convert it back to a unit of length.
1261:. New York, NY: Oxford University Press.
1187:
1161:
1136:Advanced Computing in Electron Microscopy
1110:
1133:
1045:International Tables for Crystallography
138:for the scattering of electrons via the
1085:Lobato, I.; Van Dyck, D. (2014-11-01).
14:
1281:
1164:"The scattering of electrons by atoms"
1042:
1205:
1201:
1199:
1157:
1155:
1080:
1078:
1076:
1038:
1036:
1134:Kirkland, Earl J. (17 April 2013).
24:
25:
1305:
1196:
1152:
1073:
1033:
1012:{\displaystyle f_{\text{e}}(q,Z)}
757:{\displaystyle f_{\text{e}}(q,Z)}
73:{\displaystyle f_{\text{e}}(q,Z)}
1091:Acta Crystallographica Section A
536:(in units of inverse distance),
722:The electron scattering factor
530:momentum-transfer cross section
1246:
1127:
1006:
994:
796:
784:
751:
739:
706:
694:
472:
460:
428:
402:
376:
364:
280:
268:
182:
170:
113:
101:
67:
55:
13:
1:
1026:
604:{\displaystyle \epsilon _{0}}
1057:10.1107/97809553602060000558
876:{\displaystyle r_{\text{e}}}
7:
10:
1310:
802:{\displaystyle f_{x}(q,Z)}
712:{\displaystyle f_{x}(q,Z)}
119:{\displaystyle f_{x}(q,Z)}
1112:10067/1221030151162165141
1103:10.1107/S205327331401643X
885:classical electron radius
1232:10.1002/andp.19303970303
930:{\displaystyle q>0.5}
136:first Born approximation
1189:10.1098/rspa.1930.0082
1013:
971:
951:
931:
905:
877:
850:
803:
758:
713:
667:
636:
605:
574:
573:{\displaystyle \hbar }
550:
522:
499:
120:
74:
1014:
972:
952:
932:
906:
878:
851:
811:Thomson cross section
804:
759:
714:
668:
666:{\displaystyle a_{0}}
642:is the electron rest
637:
635:{\displaystyle m_{0}}
606:
575:
551:
523:
500:
121:
75:
981:
961:
941:
915:
895:
860:
816:
771:
726:
681:
650:
619:
588:
564:
540:
512:
157:
88:
42:
1224:1930AnP...397..325B
1180:1930RSPSA.127..658M
613:vacuum permittivity
140:Coulomb interaction
33:electron scattering
1212:Annalen der Physik
1206:Bethe, H. (1930).
1009:
967:
947:
927:
901:
873:
846:
799:
754:
709:
663:
632:
601:
570:
546:
518:
495:
148:Born approximation
142:together with the
116:
70:
29:Mott–Bethe formula
18:Mott-Bethe formula
1294:Scattering theory
1268:978-0-19-850074-2
1145:978-1-4757-4406-4
1066:978-0-7923-6592-1
991:
970:{\displaystyle q}
950:{\displaystyle q}
904:{\displaystyle q}
870:
843:
831:
766:X-ray form factor
736:
549:{\displaystyle Z}
521:{\displaystyle q}
486:
416:
410:
390:
336:
294:
240:
167:
52:
16:(Redirected from
1301:
1273:
1272:
1260:
1250:
1244:
1243:
1203:
1194:
1193:
1191:
1174:(806): 658–665.
1159:
1150:
1149:
1131:
1125:
1124:
1114:
1082:
1071:
1070:
1040:
1018:
1016:
1015:
1010:
993:
992:
989:
976:
974:
973:
968:
956:
954:
953:
948:
936:
934:
933:
928:
910:
908:
907:
902:
882:
880:
879:
874:
872:
871:
868:
855:
853:
852:
847:
845:
844:
841:
832:
830:
829:
820:
808:
806:
805:
800:
783:
782:
763:
761:
760:
755:
738:
737:
734:
718:
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715:
710:
693:
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670:
669:
664:
662:
661:
641:
639:
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633:
631:
630:
610:
608:
607:
602:
600:
599:
579:
577:
576:
571:
555:
553:
552:
547:
534:reciprocal space
527:
525:
524:
519:
504:
502:
501:
496:
494:
493:
487:
485:
484:
475:
459:
458:
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440:
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363:
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346:
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307:
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267:
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250:
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241:
239:
238:
237:
228:
227:
218:
217:
204:
203:
202:
189:
169:
168:
165:
144:Poisson equation
125:
123:
122:
117:
100:
99:
82:X-ray scattering
79:
77:
76:
71:
54:
53:
50:
21:
1309:
1308:
1304:
1303:
1302:
1300:
1299:
1298:
1279:
1278:
1277:
1276:
1269:
1251:
1247:
1204:
1197:
1160:
1153:
1146:
1132:
1128:
1083:
1074:
1067:
1041:
1034:
1029:
988:
984:
982:
979:
978:
962:
959:
958:
942:
939:
938:
916:
913:
912:
896:
893:
892:
867:
863:
861:
858:
857:
840:
836:
825:
821:
819:
817:
814:
813:
778:
774:
772:
769:
768:
733:
729:
727:
724:
723:
688:
684:
682:
679:
678:
657:
653:
651:
648:
647:
626:
622:
620:
617:
616:
595:
591:
589:
586:
585:
582:Planck constant
565:
562:
561:
541:
538:
537:
513:
510:
509:
489:
488:
480:
476:
454:
450:
443:
441:
435:
434:
419:
413:
412:
411:
393:
392:
384:
380:
358:
354:
347:
345:
339:
338:
329:
325:
319:
315:
311:
306:
297:
296:
288:
284:
262:
258:
251:
249:
243:
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233:
229:
223:
219:
213:
209:
205:
198:
194:
190:
188:
164:
160:
158:
155:
154:
95:
91:
89:
86:
85:
49:
45:
43:
40:
39:
23:
22:
15:
12:
11:
5:
1307:
1297:
1296:
1291:
1289:Atomic physics
1275:
1274:
1267:
1245:
1218:(3): 325–400.
1195:
1151:
1144:
1126:
1097:(6): 636–649.
1072:
1065:
1031:
1030:
1028:
1025:
1008:
1005:
1002:
999:
996:
987:
966:
946:
926:
923:
920:
900:
866:
839:
835:
828:
824:
798:
795:
792:
789:
786:
781:
777:
753:
750:
747:
744:
741:
732:
708:
705:
702:
699:
696:
691:
687:
660:
656:
629:
625:
598:
594:
569:
545:
517:
506:
505:
492:
483:
479:
474:
471:
468:
465:
462:
457:
453:
449:
446:
438:
433:
430:
425:
422:
407:
404:
401:
396:
387:
383:
378:
375:
372:
369:
366:
361:
357:
353:
350:
342:
332:
328:
322:
318:
314:
310:
305:
300:
291:
287:
282:
279:
276:
273:
270:
265:
261:
257:
254:
246:
236:
232:
226:
222:
216:
212:
208:
201:
197:
193:
187:
184:
181:
178:
175:
172:
163:
115:
112:
109:
106:
103:
98:
94:
84:form factors,
80:, from atomic
69:
66:
63:
60:
57:
48:
9:
6:
4:
3:
2:
1306:
1295:
1292:
1290:
1287:
1286:
1284:
1270:
1264:
1259:
1258:
1249:
1241:
1237:
1233:
1229:
1225:
1221:
1217:
1213:
1209:
1202:
1200:
1190:
1185:
1181:
1177:
1173:
1169:
1165:
1158:
1156:
1147:
1141:
1137:
1130:
1122:
1118:
1113:
1108:
1104:
1100:
1096:
1092:
1088:
1081:
1079:
1077:
1068:
1062:
1058:
1054:
1050:
1046:
1039:
1037:
1032:
1024:
1022:
1003:
1000:
997:
985:
964:
944:
924:
921:
918:
898:
888:
886:
864:
837:
833:
826:
822:
812:
793:
790:
787:
779:
775:
767:
748:
745:
742:
730:
720:
703:
700:
697:
689:
685:
676:
658:
654:
645:
627:
623:
614:
596:
592:
583:
567:
560:of the atom,
559:
558:atomic number
543:
535:
531:
515:
481:
477:
469:
466:
463:
455:
451:
447:
444:
431:
423:
420:
405:
399:
385:
381:
373:
370:
367:
359:
355:
351:
348:
330:
326:
320:
316:
312:
308:
303:
289:
285:
277:
274:
271:
263:
259:
255:
252:
234:
230:
224:
220:
214:
210:
206:
199:
195:
191:
185:
179:
176:
173:
161:
153:
152:
151:
149:
145:
141:
137:
133:
129:
110:
107:
104:
96:
92:
83:
64:
61:
58:
46:
37:
34:
30:
19:
1256:
1248:
1215:
1211:
1171:
1167:
1138:. Springer.
1135:
1129:
1094:
1090:
1048:
1044:
1021:Hartree–Fock
889:
721:
507:
132:Neville Mott
36:form factors
28:
26:
1051:: 276–345.
675:Bohr Radius
1283:Categories
1027:References
128:Hans Bethe
1240:1521-3889
1121:2053-2733
823:σ
593:ϵ
568:ℏ
448:−
432:⋅
421:−
400:≈
352:−
317:π
256:−
231:ϵ
221:ℏ
211:π
856:, where
1220:Bibcode
1176:Bibcode
883:is the
673:is the
611:is the
580:is the
1265:
1238:
1142:
1119:
1063:
677:, and
615:, and
508:Here,
409:
406:0.2393
1263:ISBN
1236:ISSN
1140:ISBN
1117:ISSN
1061:ISBN
922:>
644:mass
556:the
130:and
27:The
1228:doi
1216:397
1184:doi
1172:127
1107:hdl
1099:doi
1053:doi
925:0.5
532:in
1285::
1234:.
1226:.
1214:.
1210:.
1198:^
1182:.
1170:.
1166:.
1154:^
1115:.
1105:.
1095:70
1093:.
1089:.
1075:^
1059:.
1047:.
1035:^
977:,
646:,
584:,
415:nm
207:32
150:,
38:,
1271:.
1242:.
1230::
1222::
1192:.
1186::
1178::
1148:.
1123:.
1109::
1101::
1069:.
1055::
1049:B
1007:)
1004:Z
1001:,
998:q
995:(
990:e
986:f
965:q
945:q
919:q
911:(
899:q
869:e
865:r
842:e
838:r
834:=
827:T
797:)
794:Z
791:,
788:q
785:(
780:x
776:f
752:)
749:Z
746:,
743:q
740:(
735:e
731:f
707:)
704:Z
701:,
698:q
695:(
690:x
686:f
659:0
655:a
628:0
624:m
597:0
544:Z
516:q
491:)
482:2
478:q
473:)
470:Z
467:,
464:q
461:(
456:x
452:f
445:Z
437:(
429:)
424:1
403:(
395:)
386:2
382:q
377:)
374:Z
371:,
368:q
365:(
360:x
356:f
349:Z
341:(
331:0
327:a
321:2
313:8
309:1
304:=
299:)
290:2
286:q
281:)
278:Z
275:,
272:q
269:(
264:x
260:f
253:Z
245:(
235:0
225:2
215:3
200:2
196:e
192:m
186:=
183:)
180:Z
177:,
174:q
171:(
166:e
162:f
114:)
111:Z
108:,
105:q
102:(
97:x
93:f
68:)
65:Z
62:,
59:q
56:(
51:e
47:f
20:)
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