443:
31:
1148:
which is the case for X-ray scattering. Hence, in strong contrast to the case for nuclear scattering, the scattering object for magnetic scattering is far from a point source; it is still more diffuse than the effective size of the source for X-ray scattering, and the resulting
Fourier transform (the
1147:
Since these orbitals are typically of a comparable size to the wavelength of the free neutrons, the resulting form factor resembles that of the X-ray form factor. However, this neutron-magnetic scattering is only from the outer electrons, rather than being heavily weighted by the core electrons,
240:
549:
makes use of the variation of the form factor close to an absorption edge to vary the scattering power of specific atoms in the sample by changing the energy of the incident x-rays hence enabling the extraction of more detailed structural information.
849:
1154:) decays more rapidly than the X-ray form factor. Also, in contrast to nuclear scattering, the magnetic form factor is not isotope dependent, but is dependent on the oxidation state of the atom.
154:
1111:. In neutron scattering from condensed matter, magnetic scattering refers to the interaction of this moment with the magnetic moments arising from unpaired electrons in the outer
275:
115:
598:
635:
146:
1071:
of the spherical wave, which has dimensions of length. Hence, the amplitude of scattering that characterizes the interaction of a neutron with a given isotope is termed the
307:
726:
718:
417:
375:
329:
1142:
1010:
897:
678:
658:
525:
353:
1062:
1042:
903:
of the atom, and the electron form factor is the
Fourier transform of this. The electron form factors are normally calculated from X-ray form factors using the
484:
395:
955:) and cold neutrons (up to tens of Angstroms) typically used for such investigations is 4-5 orders of magnitude larger than the dimension of the nucleus (
534:
about the nucleus, and the form factor the
Fourier transform of this quantity. The assumption of a spherical distribution is usually good enough for
34:
X-ray atomic form factors of oxygen (blue), chlorine (green), Cl (magenta), and K (red); smaller charge distributions have a wider form factor.
1087:
of the same element. They may only be determined experimentally, since the theory of nuclear forces is not adequate to calculate or predict
17:
1067:
Since the interaction is nuclear, each isotope has a different scattering amplitude. This
Fourier transform is scaled by the
1200:
235:{\displaystyle f(\mathbf {Q} )=\int \rho (\mathbf {r} )e^{i\mathbf {Q} \cdot \mathbf {r} }\mathrm {d} ^{3}\mathbf {r} }
1338:
1313:
1284:
1247:
1184:
1176:
1021:
1235:
1024:
is unity; therefore, it is commonly said that neutrons "do not have a form factor;" i.e., the scattered amplitude,
660:
is the scattering angle between the incident x-ray beam and the detector measuring the scattered intensity, while
980:
907:. This formula takes into account both elastic electron-cloud scattering and elastic nuclear scattering.
249:
89:
1272:
1115:
of certain atoms. It is the spatial distribution of these unpaired electrons about the nucleus that is
1017:
559:
546:
844:{\displaystyle f(Q)=\sum _{i=1}^{4}a_{i}\exp \left(-b_{i}\left({\frac {Q}{4\pi }}\right)^{2}\right)+c}
603:
335:. As a result of the nature of the Fourier transform, the broader the distribution of the scatterer
120:
284:
541:
In general the X-ray form factor is complex but the imaginary components only become large near an
691:
400:
358:
312:
1360:
486:, of the atoms in a sample. As a result, X-rays are not very sensitive to light atoms, such as
1239:
1229:
904:
680:
is the wavelength of the X-rays. One interpretation of the scattering vector is that it is the
1276:
1118:
986:
873:
663:
640:
501:
535:
1264:
967:; for those that undergo nuclear scattering from a nucleus, the nucleus acts as a secondary
1150:
1068:
1013:
979:. (Although a quantum phenomenon, this can be visualized in simple classical terms by the
944:
459:
338:
63:
51:
8:
442:
67:
553:
Atomic form factor patterns are often represented as a function of the magnitude of the
1305:
1047:
1027:
900:
469:
380:
1334:
1309:
1280:
1265:
1243:
1180:
1072:
1012:
is the spatial density distribution of the nucleus, which is an infinitesimal point (
332:
71:
59:
1079:. Neutron scattering lengths vary erratically between neighbouring elements in the
924:
494:, and there is very little contrast between elements adjacent to each other in the
423:
83:
1108:
720:Ă…, the atomic form factor is well approximated by a sum of Gaussians of the form
542:
1016:), with respect to the neutron wavelength. The delta function forms part of the
30:
1225:
1112:
1080:
976:
920:
531:
495:
278:
79:
688:
with which the sample is observed. In the range of scattering vectors between
1354:
960:
952:
463:
54:
of a wave by an isolated atom. The atomic form factor depends on the type of
1100:
968:
58:, which in turn depends on the nature of the incident radiation, typically
956:
943:
Nuclear scattering of the free neutron by the nucleus is mediated by the
964:
948:
427:
55:
458:
X-rays are scattered by the electron cloud of the atom and hence the
447:
1084:
972:
916:
528:
487:
451:
1104:
431:
39:
27:
Measure of the scattering amplitude of a wave by an isolated atom
74:
of a spatial density distribution of the scattering object from
70:. The common feature of all form factors is that they involve a
923:. Both are used in the investigation structure and dynamics of
491:
1267:
Introduction to
Conventional Transmission Electron Microscopy
75:
422:
For crystals, atomic form factors are used to calculate the
1020:, by which the free neutron and the nuclei interact. The
1302:
Introduction to the Theory of
Thermal Neutron Scattering
1328:
1121:
1050:
1030:
989:
876:
729:
694:
666:
643:
606:
562:
504:
472:
403:
383:
361:
341:
315:
287:
252:
157:
123:
92:
86:). For an object with spatial density distribution,
1136:
1056:
1036:
1004:
915:There are two distinct scattering interactions of
891:
843:
712:
672:
652:
629:
592:
519:
478:
411:
389:
369:
347:
323:
301:
277:is the spatial density of the scatterer about its
269:
234:
140:
109:
419:; i.e., the faster the decay of the form factor.
1352:
1170:
1262:
441:
29:
1299:
1099:Although neutral, neutrons also have a
865:
14:
1353:
1224:
1094:
1091:from other properties of the nucleus.
910:
1329:Dobrzynski, L.; K. Blinowski (1994).
1022:Fourier transform of a delta function
938:
931:(sometimes also termed chemical) and
437:
450:of the atomic scattering factor of
377:, the narrower the distribution of
270:{\displaystyle \rho (\mathbf {r} )}
110:{\displaystyle \rho (\mathbf {r} )}
24:
217:
25:
1372:
1177:Blackwell Scientific Publications
593:{\displaystyle Q=2k\sin(\theta )}
1331:Neutrons and Solid State Physics
630:{\displaystyle k=2\pi /\lambda }
405:
363:
317:
289:
260:
228:
209:
201:
185:
165:
131:
100:
141:{\displaystyle f(\mathbf {Q} )}
1322:
1293:
1256:
1218:
1193:
1164:
1131:
1125:
999:
993:
886:
880:
739:
733:
587:
581:
514:
508:
302:{\displaystyle \mathbf {r} =0}
264:
256:
189:
181:
169:
161:
135:
127:
104:
96:
13:
1:
1173:Essentials of Crystallography
1157:
1107:and hence have an associated
527:in the above equation is the
462:of X-rays increases with the
446:The energy dependence of the
1171:McKie, D.; C. McKie (1992).
862:, and c are tabulated here.
713:{\displaystyle 0<Q<25}
412:{\displaystyle \mathbf {Q} }
370:{\displaystyle \mathbf {r} }
324:{\displaystyle \mathbf {Q} }
7:
870:The relevant distribution,
10:
1377:
1273:Cambridge University Press
959:). The free neutrons in a
547:Anomalous X-ray scattering
1333:. Ellis Horwood Limited.
1144:for magnetic scattering.
981:Huygens–Fresnel principle
1300:Squires, Gordon (1996).
1137:{\displaystyle \rho (r)}
1005:{\displaystyle \rho (r)}
975:scattered neutrons as a
892:{\displaystyle \rho (r)}
673:{\displaystyle \lambda }
653:{\displaystyle 2\theta }
520:{\displaystyle \rho (r)}
498:. For X-ray scattering,
48:atomic scattering factor
18:Atomic scattering factor
1263:De Graef, Marc (2003).
1103:. They are a composite
637:is the wavenumber and
1138:
1058:
1038:
1006:
901:potential distribution
893:
845:
765:
714:
674:
654:
631:
594:
521:
480:
455:
413:
391:
371:
349:
325:
303:
271:
236:
142:
111:
50:, is a measure of the
35:
1238:Publishing. pp.
1236:North-Holland Physics
1201:"Atomic form factors"
1139:
1059:
1039:
1018:Fermi pseudopotential
1007:
894:
854:where the values of a
846:
745:
715:
675:
655:
632:
595:
536:X-ray crystallography
522:
481:
445:
414:
392:
372:
350:
348:{\displaystyle \rho }
326:
304:
272:
237:
143:
112:
33:
1151:magnetic form factor
1119:
1048:
1044:, is independent of
1028:
987:
951:of thermal (several
945:strong nuclear force
874:
866:Electron form factor
727:
692:
664:
641:
604:
560:
502:
470:
460:scattering amplitude
401:
381:
359:
339:
313:
285:
250:
155:
121:
90:
52:scattering amplitude
1231:Diffraction Physics
1095:Magnetic scattering
911:Neutron form factor
117:, the form factor,
1306:Dover Publications
1134:
1054:
1034:
1002:
939:Nuclear scattering
927:: they are termed
905:Mott–Bethe formula
889:
841:
710:
670:
650:
627:
590:
517:
476:
456:
438:X-ray form factors
409:
387:
367:
345:
321:
299:
267:
232:
138:
107:
44:atomic form factor
36:
1073:scattering length
1057:{\displaystyle Q}
1037:{\displaystyle b}
818:
555:scattering vector
479:{\displaystyle Z}
390:{\displaystyle f}
333:momentum transfer
72:Fourier transform
16:(Redirected from
1368:
1345:
1344:
1326:
1320:
1319:
1297:
1291:
1290:
1270:
1260:
1254:
1253:
1222:
1216:
1215:
1213:
1211:
1197:
1191:
1190:
1168:
1143:
1141:
1140:
1135:
1063:
1061:
1060:
1055:
1043:
1041:
1040:
1035:
1011:
1009:
1008:
1003:
983:.) In this case
925:condensed matter
898:
896:
895:
890:
850:
848:
847:
842:
834:
830:
829:
828:
823:
819:
817:
806:
799:
798:
775:
774:
764:
759:
719:
717:
716:
711:
679:
677:
676:
671:
659:
657:
656:
651:
636:
634:
633:
628:
623:
599:
597:
596:
591:
526:
524:
523:
518:
485:
483:
482:
477:
424:structure factor
418:
416:
415:
410:
408:
396:
394:
393:
388:
376:
374:
373:
368:
366:
354:
352:
351:
346:
330:
328:
327:
322:
320:
308:
306:
305:
300:
292:
276:
274:
273:
268:
263:
241:
239:
238:
233:
231:
226:
225:
220:
214:
213:
212:
204:
188:
168:
148:, is defined as
147:
145:
144:
139:
134:
116:
114:
113:
108:
103:
84:reciprocal space
21:
1376:
1375:
1371:
1370:
1369:
1367:
1366:
1365:
1351:
1350:
1349:
1348:
1341:
1327:
1323:
1316:
1308:. p. 260.
1298:
1294:
1287:
1261:
1257:
1250:
1226:Cowley, John M.
1223:
1219:
1209:
1207:
1199:
1198:
1194:
1187:
1169:
1165:
1160:
1120:
1117:
1116:
1109:magnetic moment
1097:
1049:
1046:
1045:
1029:
1026:
1025:
988:
985:
984:
941:
913:
875:
872:
871:
868:
861:
857:
824:
810:
805:
801:
800:
794:
790:
786:
782:
770:
766:
760:
749:
728:
725:
724:
693:
690:
689:
665:
662:
661:
642:
639:
638:
619:
605:
602:
601:
561:
558:
557:
543:absorption edge
503:
500:
499:
471:
468:
467:
440:
404:
402:
399:
398:
382:
379:
378:
362:
360:
357:
356:
340:
337:
336:
316:
314:
311:
310:
288:
286:
283:
282:
259:
251:
248:
247:
227:
221:
216:
215:
208:
200:
196:
192:
184:
164:
156:
153:
152:
130:
122:
119:
118:
99:
91:
88:
87:
82:(also known as
28:
23:
22:
15:
12:
11:
5:
1374:
1364:
1363:
1361:Atomic physics
1347:
1346:
1339:
1321:
1314:
1292:
1285:
1255:
1248:
1217:
1192:
1185:
1162:
1161:
1159:
1156:
1133:
1130:
1127:
1124:
1096:
1093:
1081:periodic table
1053:
1033:
1014:delta function
1001:
998:
995:
992:
977:spherical wave
940:
937:
912:
909:
888:
885:
882:
879:
867:
864:
859:
855:
852:
851:
840:
837:
833:
827:
822:
816:
813:
809:
804:
797:
793:
789:
785:
781:
778:
773:
769:
763:
758:
755:
752:
748:
744:
741:
738:
735:
732:
709:
706:
703:
700:
697:
669:
649:
646:
626:
622:
618:
615:
612:
609:
589:
586:
583:
580:
577:
574:
571:
568:
565:
532:charge density
516:
513:
510:
507:
496:periodic table
475:
439:
436:
407:
386:
365:
355:in real space
344:
319:
298:
295:
291:
279:center of mass
266:
262:
258:
255:
244:
243:
230:
224:
219:
211:
207:
203:
199:
195:
191:
187:
183:
180:
177:
174:
171:
167:
163:
160:
137:
133:
129:
126:
106:
102:
98:
95:
80:momentum space
26:
9:
6:
4:
3:
2:
1373:
1362:
1359:
1358:
1356:
1342:
1340:0-13-617192-3
1336:
1332:
1325:
1317:
1315:0-486-69447-X
1311:
1307:
1303:
1296:
1288:
1286:0-521-62995-0
1282:
1278:
1274:
1269:
1268:
1259:
1251:
1249:0-444-86121-1
1245:
1241:
1237:
1233:
1232:
1227:
1221:
1206:
1202:
1196:
1188:
1186:0-632-01574-8
1182:
1178:
1174:
1167:
1163:
1155:
1153:
1152:
1145:
1128:
1122:
1114:
1110:
1106:
1102:
1092:
1090:
1086:
1082:
1078:
1074:
1070:
1065:
1051:
1031:
1023:
1019:
1015:
996:
990:
982:
978:
974:
970:
966:
962:
958:
954:
950:
946:
936:
934:
930:
926:
922:
918:
908:
906:
902:
883:
877:
863:
838:
835:
831:
825:
820:
814:
811:
807:
802:
795:
791:
787:
783:
779:
776:
771:
767:
761:
756:
753:
750:
746:
742:
736:
730:
723:
722:
721:
707:
704:
701:
698:
695:
687:
683:
667:
647:
644:
624:
620:
616:
613:
610:
607:
584:
578:
575:
572:
569:
566:
563:
556:
551:
548:
544:
539:
537:
533:
530:
511:
505:
497:
493:
489:
473:
465:
464:atomic number
461:
453:
449:
444:
435:
433:
429:
425:
420:
384:
342:
334:
296:
293:
280:
253:
222:
205:
197:
193:
178:
175:
172:
158:
151:
150:
149:
124:
93:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
32:
19:
1330:
1324:
1301:
1295:
1266:
1258:
1230:
1220:
1208:. Retrieved
1204:
1195:
1172:
1166:
1149:
1146:
1101:nuclear spin
1098:
1088:
1083:and between
1076:
1066:
969:point source
963:travel in a
942:
935:scattering.
932:
928:
914:
869:
853:
685:
681:
554:
552:
540:
457:
426:for a given
421:
245:
47:
43:
37:
1275:. pp.
957:femtometres
1158:References
965:plane wave
949:wavelength
682:resolution
428:Bragg peak
76:real space
56:scattering
1123:ρ
1069:amplitude
991:ρ
953:ångströms
878:ρ
815:π
788:−
780:
747:∑
686:yardstick
668:λ
648:θ
625:λ
617:π
600:. Herein
585:θ
579:
506:ρ
448:real part
343:ρ
254:ρ
206:⋅
179:ρ
176:∫
94:ρ
1355:Category
1228:(1981).
1113:orbitals
1085:isotopes
973:radiates
933:magnetic
917:neutrons
529:electron
488:hydrogen
452:chlorine
64:electron
1205:TU Graz
1105:fermion
929:nuclear
899:is the
432:crystal
331:is the
309:), and
68:neutron
40:physics
1337:
1312:
1283:
1246:
1183:
971:, and
947:. The
921:nuclei
492:helium
246:where
42:, the
1210:3 Jul
430:of a
60:X-ray
46:, or
1335:ISBN
1310:ISBN
1281:ISBN
1244:ISBN
1212:2018
1181:ISBN
961:beam
705:<
699:<
490:and
1277:113
919:by
858:, b
777:exp
684:or
576:sin
397:in
78:to
66:or
38:In
1357::
1304:.
1279:.
1271:.
1242:.
1240:78
1234:.
1203:.
1179:.
1175:.
1075:,
1064:.
708:25
545:.
538:.
466:,
434:.
62:,
1343:.
1318:.
1289:.
1252:.
1214:.
1189:.
1132:)
1129:r
1126:(
1089:b
1077:b
1052:Q
1032:b
1000:)
997:r
994:(
887:)
884:r
881:(
860:i
856:i
839:c
836:+
832:)
826:2
821:)
812:4
808:Q
803:(
796:i
792:b
784:(
772:i
768:a
762:4
757:1
754:=
751:i
743:=
740:)
737:Q
734:(
731:f
702:Q
696:0
645:2
621:/
614:2
611:=
608:k
588:)
582:(
573:k
570:2
567:=
564:Q
515:)
512:r
509:(
474:Z
454:.
406:Q
385:f
364:r
318:Q
297:0
294:=
290:r
281:(
265:)
261:r
257:(
242:,
229:r
223:3
218:d
210:r
202:Q
198:i
194:e
190:)
186:r
182:(
173:=
170:)
166:Q
162:(
159:f
136:)
132:Q
128:(
125:f
105:)
101:r
97:(
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
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