286:
example, the defect may result in an ion on its own ion site or a vacancy on the cation site. To complete the reactions, the proper number of each ion must be present (mass balance), an equal number of sites must exist (site balance), and the sums of the charges of the reactants and products must also be equal (charge balance).
129:
indicates the lattice site that the species occupies. For instance, Ni might occupy a Cu site. In this case, M would be replaced by Ni and S would be replaced by Cu. The site may also be a lattice interstice, in this case, the symbol "i" is used. A cation site can be represented by the symbols C or M
138:
corresponds to the electronic charge of the species relative to the site that it occupies. The charge of the species is calculated by the charge on the current site minus the charge on the original site. To continue the previous example, Ni often has the same valency as Cu, so the relative charge is
276:
When using Kröger–Vink notation for both intrinsic and extrinsic defects, it is imperative to keep all masses, sites, and charges balanced in each reaction. If any piece is unbalanced, the reactants and the products do not equal the same entity and therefore all quantities are not conserved as they
778:
From the chart above, there are total of four possible chemical reactions using Kröger–Vink
Notation depending on the intrinsic deficiency of atoms within the material. Assume the chemical composition is AX, with A being the cation and X being the anion. (The following assumes that X is a diatomic
285:
vacancies (Schottky) or cation/anion vacancies and interstitials (Frenkel). Otherwise, a compound is broken down into its respective cation and anion parts for the process to begin on each lattice. From here, depending on the required steps for the desired outcome, several possibilities occur. For
753:
tree for a simple ionic compound, AX, where A is a cation and X is an anion, summarizes the various ways in which intrinsic defects can form. Depending on the cation-to-anion ratio, the species can either be reduced and therefore classified as
277:
should be. The first step in this process is determining the correct type of defect and reaction that comes along with it; Schottky and
Frenkel defects begin with a null reactant (∅) and produce either
168:
672:
favored site, usually depending on temperature. For the two equations shown below, the right side is usually at high temperature as this allows for more movement of
1429:
Therefore, given a temperature and the formation energy of
Schottky defect, the intrinsic Schottky defect concentration can be calculated from the above equation.
1054:
Note that the vacancy on the Mg sublattice site has a −2 effective charge, and the vacancy on the oxygen sublattice site has a +2 effective charge. Using the
766:
1490:
1522:
1463:
35:. It is primarily used for ionic crystals and is particularly useful for describing various defect reactions. It was proposed by
779:
gas such as oxygen and therefore cation A has a +2 charge. Note that materials with this defect structure are often used in
1512:
738:(metal site oxidized, where B is an arbitrary cation having one more positive charge than the original atom on the site)
769:
Tree diagram representation of the oxidation and reduction states resulting from intrinsic defects in ionic compounds.
762:. Below, the tree is shown for a further explanation of the pathways and results of each breakdown of the substance.
1305:, the formula can be simplified into the following form where the enthalpy of formation can be directly calculated:
991:
28:
36:
999:
104:
41:
47:
1132:
1059:
147:
indicates a net single positive charge, while two would represent two net positive charges. Finally,
676:. The left side is usually at low temperature as the electrons lose their mobility due to loss in
1179:
1055:
987:
1517:
1486:
1459:
995:
24:
153:
1451:
319:
173:
signifies a net single negative charge, so two would indicate a net double negative charge.
1011:
765:
493:
489:
425:
363:
359:
315:
20:
52:
1063:
677:
660:
617:
421:
217:
98:
1455:
577:
Frenkel defect – forming an interstitial and vacancy pair on an anion or cation site:
1506:
1115:
Also, the equilibrium constant can be related to the Gibbs free energy of formation Δ
780:
116:
1480:
847:
In the reduced n-type, there is a deficiency of anions on the lattice sites:
192:— an aluminum ion sitting on an aluminum lattice site, with a neutral charge.
787:
In the reduced n-type, there are excess cations on the interstitial sites:
884:
In the oxidized p-type, there is cation deficiency on the lattice sites:
673:
511:
Schottky defect – forming a vacancy pair on both anion and cation sites:
110:
107:– i (although this is usually used to describe lattice site, not species)
1092:
Based on the above reaction, the stoichiometric relation is as follows:
244:— a chlorine anion on an interstitial site, with single negative charge.
1422:
934:
In the oxidized p-type, there are excess anions on interstitial sites:
669:
256:— an oxygen anion on an interstitial site, with double negative charge.
32:
130:(for metal), and an anion site can be represented by either an A or X.
204:— a nickel ion sitting on a copper lattice site, with neutral charge.
1446:
Kröger, F. A.; Vink, H. J. (1956). Seitz, F.; Turnbull, D. (eds.).
1002:) can be calculated given the defect concentration or vice versa.
956:
851:
764:
750:
507:
Assume that the cation C has +1 charge and anion A has −1 charge.
758:, or if the converse is true, the ionic species is classified as
92:
901:
832:
282:
278:
1014:, the Kröger–Vink defect reaction can be written as follows:
157:
554:
Schottky defect (charged) – forming an electron–hole pair:
232:— a calcium interstitial ion, with double positive charge.
982:
Relating chemical reactions to the equilibrium constant
156:
162:
101:– V or v (since V is also the symbol for vanadium)
19:is a set of conventions that are used to describe
267:— an electron. No site is normally specified.
8:
1479:Carter, C. Barry; Norton, M. Grant (2007).
1482:Ceramic Materials: Science and Engineering
155:
89:corresponds to the species. These can be
1438:
1122:according to the following relations,
7:
1197:
1124:
1094:
1068:
1016:
998:of formation, and the energy terms (
14:
1450:. Vol. 3. pp. 307–435.
139:zero. To indicate a null charge,
65:The notation follows the scheme:
503:Basic types of defect reactions
220:, with single positive charge.
1:
1456:10.1016/S0081-1947(08)60135-6
1421:is a constant containing the
1301:
1250:
1244:
1066:indicating concentration):
1010:For a Schottky reaction in
1539:
1523:Crystallographic defects
745:Oxidation–reduction tree
668:Associates – forming an
163:{\displaystyle \prime }
1485:. New York: Springer.
994:can be related to its
770:
164:
95:– e.g., Si, Ni, O, Cl,
1000:enthalpy of formation
768:
165:
37:Ferdinand Anne Kröger
1060:equilibrium constant
709:(metal site reduced)
154:
17:Kröger–Vink notation
1513:Chemical properties
1448:Solid State Physics
1242:Relating equations
1062:can be written as (
751:oxidation–reduction
1180:Boltzmann constant
1056:law of mass action
988:law of mass action
774:Schematic examples
771:
160:
143:is used. A single
1492:978-0-387-46270-7
1240:
1239:
1196:
1195:
1113:
1112:
1090:
1089:
1052:
1051:
996:Gibbs free energy
1530:
1497:
1496:
1476:
1470:
1469:
1443:
1416:
1413:
1411:
1410:
1398:
1395:
1382:
1374:
1371:
1369:
1368:
1359:
1356:
1343:
1341:
1340:
1328:
1325:
1312:
1294:
1291:
1289:
1288:
1277:
1274:
1261:
1234:
1198:
1190:
1170:
1167:
1165:
1164:
1153:
1150:
1137:
1125:
1107:
1095:
1084:
1069:
1046:
1037:
1036:
1028:
1027:
1017:
975:
974:
967:
966:
954:
953:
945:
944:
929:
928:
921:
920:
912:
911:
895:
894:
890:
879:
878:
871:
870:
862:
861:
842:
841:
826:
825:
821:
816:
815:
807:
806:
798:
797:
737:
736:
729:
728:
720:
719:
708:
707:
699:
698:
691:
690:
658:
657:
649:
648:
640:
639:
631:
630:
615:
614:
606:
605:
597:
596:
588:
587:
572:
571:
564:
563:
549:
548:
540:
539:
531:
530:
522:
521:
486:
485:
477:
476:
468:
467:
459:
458:
450:
449:
441:
440:
418:
417:
409:
408:
400:
399:
391:
390:
382:
381:
356:
355:
347:
346:
338:
337:
312:
311:
303:
302:
266:
265:
255:
254:
243:
242:
231:
230:
215:
214:
203:
202:
191:
190:
171:
169:
167:
166:
161:
78:
77:
56:
48:Hendrik Jan Vink
45:
21:electric charges
1538:
1537:
1533:
1532:
1531:
1529:
1528:
1527:
1503:
1502:
1501:
1500:
1493:
1478:
1477:
1473:
1466:
1445:
1444:
1440:
1435:
1414:
1406:
1399:
1396:
1391:
1387:
1386:
1384:
1380:
1372:
1367:
1360:
1357:
1352:
1348:
1347:
1345:
1336:
1329:
1326:
1321:
1317:
1316:
1314:
1310:
1299:Using equation
1292:
1284:
1278:
1275:
1270:
1266:
1265:
1263:
1259:
1232:
1222:
1211:
1204:
1188:
1177:
1168:
1160:
1154:
1151:
1146:
1142:
1141:
1139:
1135:
1118:
1105:
1082:
1064:square brackets
1058:, the reaction
1044:
1035:
1032:
1031:
1030:
1026:
1023:
1022:
1021:
1008:
984:
973:
971:
970:
969:
965:
962:
961:
960:
952:
949:
948:
947:
943:
940:
939:
938:
927:
925:
924:
923:
919:
916:
915:
914:
910:
907:
906:
905:
899:
892:
888:
887:
877:
875:
874:
873:
869:
866:
865:
864:
860:
857:
856:
855:
840:
838:
837:
836:
830:
823:
819:
818:
814:
811:
810:
809:
805:
802:
801:
800:
796:
793:
792:
791:
776:
747:
735:
733:
732:
731:
727:
724:
723:
722:
718:
715:
714:
713:
706:
703:
702:
701:
697:
695:
694:
693:
689:
686:
685:
684:
656:
653:
652:
651:
647:
644:
643:
642:
638:
635:
634:
633:
629:
626:
625:
624:
613:
610:
609:
608:
604:
601:
600:
599:
595:
592:
591:
590:
586:
583:
582:
581:
570:
568:
567:
566:
562:
560:
559:
558:
547:
544:
543:
542:
538:
535:
534:
533:
529:
526:
525:
524:
520:
517:
516:
515:
505:
490:Schottky defect
484:
481:
480:
479:
475:
472:
471:
470:
466:
463:
462:
461:
457:
454:
453:
452:
448:
445:
444:
443:
439:
436:
435:
434:
416:
413:
412:
411:
407:
404:
403:
402:
398:
395:
394:
393:
389:
386:
385:
384:
380:
377:
376:
375:
367:
360:Schottky defect
354:
351:
350:
349:
345:
342:
341:
340:
336:
333:
332:
331:
323:
316:Schottky defect
310:
307:
306:
305:
301:
298:
297:
296:
292:
274:
264:
262:
261:
260:
253:
250:
249:
248:
241:
238:
237:
236:
229:
226:
225:
224:
213:
210:
209:
208:
201:
198:
197:
196:
189:
186:
185:
184:
180:
152:
151:
149:
79:
76:
73:
72:
71:
63:
50:
39:
12:
11:
5:
1536:
1534:
1526:
1525:
1520:
1515:
1505:
1504:
1499:
1498:
1491:
1471:
1464:
1437:
1436:
1434:
1431:
1427:
1426:
1404:
1389:
1365:
1350:
1334:
1319:
1297:
1296:
1282:
1268:
1238:
1237:
1228:
1226:
1220:
1209:
1202:
1194:
1193:
1184:
1182:
1175:
1158:
1144:
1116:
1111:
1110:
1101:
1099:
1088:
1087:
1078:
1076:
1050:
1049:
1040:
1038:
1033:
1024:
1007:
1004:
983:
980:
979:
978:
977:
976:
972:
963:
950:
941:
932:
931:
930:
926:
917:
908:
897:
882:
881:
880:
876:
867:
858:
845:
844:
843:
839:
828:
812:
803:
794:
781:oxygen sensors
775:
772:
749:The following
746:
743:
742:
741:
740:
739:
734:
725:
716:
710:
704:
696:
687:
678:kinetic energy
666:
665:
664:
661:Frenkel defect
654:
645:
636:
627:
621:
618:Frenkel defect
611:
602:
593:
584:
575:
574:
573:
569:
561:
552:
551:
550:
545:
536:
527:
518:
504:
501:
500:
499:
498:
497:
482:
473:
464:
455:
446:
437:
431:
430:
429:
422:Frenkel defect
414:
405:
396:
387:
378:
372:
371:
370:
365:
352:
343:
334:
328:
327:
326:
321:
308:
299:
291:
288:
273:
270:
269:
268:
263:
257:
251:
245:
239:
233:
227:
221:
211:
205:
199:
193:
187:
179:
176:
175:
174:
159:
131:
122:
121:
120:
117:electron holes
114:
108:
102:
96:
81:
80:
74:
69:
62:
59:
13:
10:
9:
6:
4:
3:
2:
1535:
1524:
1521:
1519:
1516:
1514:
1511:
1510:
1508:
1494:
1488:
1484:
1483:
1475:
1472:
1467:
1465:9780126077032
1461:
1457:
1453:
1449:
1442:
1439:
1432:
1430:
1424:
1420:
1409:
1403:
1394:
1378:
1364:
1355:
1339:
1333:
1324:
1308:
1307:
1306:
1304:
1303:
1287:
1281:
1273:
1257:
1256:
1255:
1253:
1252:
1247:
1246:
1236:
1229:
1227:
1225:
1218:
1214:
1207:
1200:
1199:
1192:
1185:
1183:
1181:
1174:
1163:
1157:
1149:
1134:
1130:
1127:
1126:
1123:
1121:
1109:
1102:
1100:
1097:
1096:
1093:
1086:
1079:
1077:
1074:
1071:
1070:
1067:
1065:
1061:
1057:
1048:
1041:
1039:
1019:
1018:
1015:
1013:
1005:
1003:
1001:
997:
993:
992:concentration
990:, a defect's
989:
981:
958:
936:
935:
933:
903:
886:
885:
883:
853:
849:
848:
846:
834:
789:
788:
786:
785:
784:
782:
773:
767:
763:
761:
757:
752:
744:
711:
682:
681:
679:
675:
671:
667:
662:
622:
619:
579:
578:
576:
556:
555:
553:
513:
512:
510:
509:
508:
502:
495:
492:formation in
491:
488:
487:
432:
427:
424:formation in
423:
420:
419:
373:
368:
362:formation in
361:
358:
357:
329:
324:
318:formation in
317:
314:
313:
294:
293:
290:Example usage
289:
287:
284:
280:
271:
258:
246:
234:
222:
219:
216:— a chlorine
206:
194:
182:
181:
177:
172:
146:
142:
137:
136:
132:
128:
127:
123:
118:
115:
112:
109:
106:
105:interstitials
103:
100:
97:
94:
91:
90:
88:
87:
83:
82:
68:
67:
66:
60:
58:
54:
49:
43:
38:
34:
30:
27:positions of
26:
22:
18:
1481:
1474:
1447:
1441:
1428:
1418:
1407:
1401:
1392:
1376:
1362:
1353:
1337:
1331:
1322:
1300:
1298:
1285:
1279:
1271:
1249:
1243:
1241:
1230:
1223:
1216:
1212:
1205:
1186:
1172:
1161:
1155:
1147:
1128:
1119:
1114:
1103:
1091:
1080:
1072:
1053:
1042:
1009:
985:
835:) + 2 e
777:
759:
755:
748:
670:entropically
506:
275:
148:
144:
140:
134:
133:
125:
124:
85:
84:
64:
29:point defect
16:
15:
51: [
40: [
31:species in
1507:Categories
1433:References
1254:, we get:
986:Using the
968:+ 2 h
922:+ 2 h
872:+ 2 e
616:(cationic
348:+ 3 v
304:+ 2 v
674:electrons
659:(anionic
272:Procedure
158:′
111:electrons
99:vacancies
1518:Notation
1423:entropic
1417:, where
1171:, where
1006:Examples
178:Examples
61:Notation
33:crystals
1412:
1385:
1370:
1346:
1342:
1315:
1290:
1264:
1178:is the
1166:
1140:
896: X
891:⁄
827: X
822:⁄
483:surface
474:surface
218:vacancy
170:
150:
25:lattice
1489:
1462:
760:p-type
756:n-type
279:cation
1425:term.
1309:= exp
1020:∅ ⇌ v
959:) + X
904:) ⇌ v
854:) ⇌ A
623:∅ ⇌ v
580:∅ ⇌ v
557:∅ ⇌ e
514:∅ ⇌ v
364:BaTiO
330:∅ ⇌ v
295:∅ ⇌ v
283:anion
93:atoms
55:]
44:]
1487:ISBN
1460:ISBN
1248:and
955:⇌ A(
469:+ Mg
410:+ Mg
281:and
46:and
23:and
1452:doi
1379:exp
1258:exp
1208:= Δ
1133:exp
1029:+ v
1012:MgO
946:+ X
913:+ X
863:+ v
808:⇌ A
799:+ X
783:.)
730:+ e
721:→ B
700:→ M
692:+ e
650:+ X
641:⇌ v
632:+ A
607:+ M
598:⇌ v
589:+ C
565:+ h
541:+ v
532:⇌ v
523:+ v
494:MgO
478:+ O
460:+ v
451:⇌ v
442:+ O
426:MgO
401:+ v
392:⇌ O
383:+ O
339:+ v
320:TiO
119:– h
113:– e
1509::
1458:.
1375:=
1344:+
1295:=
1215:−
1131:=
1098:=
1075:=
1025:Mg
850:A(
817:+
680:.
456:Mg
438:Mg
433:Mg
415:Mg
379:Mg
374:Mg
344:Ti
335:Ba
300:Ti
235:Cl
223:Ca
212:Cl
200:Cu
195:Ni
188:Al
183:Al
57:.
53:nl
42:fr
1495:.
1468:.
1454::
1419:A
1415:)
1408:T
1405:B
1402:k
1400:2
1397:/
1393:H
1390:f
1388:Δ
1383:−
1381:(
1377:A
1373:)
1366:B
1363:k
1361:2
1358:/
1354:S
1351:f
1349:Δ
1338:T
1335:B
1332:k
1330:2
1327:/
1323:H
1320:f
1318:Δ
1313:−
1311:(
1302:5
1293:)
1286:T
1283:B
1280:k
1276:/
1272:G
1269:f
1267:Δ
1262:−
1260:(
1251:4
1245:2
1235:)
1233:5
1231:(
1224:S
1221:f
1219:Δ
1217:T
1213:H
1210:f
1206:G
1203:f
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145:•
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86:M
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