409:
is the nominal frequency shift of that mode divided by its mode number (most software outputs normalized frequency shift by default). Because the film is treated as an extension of thickness, Sauerbrey’s equation only applies to systems in which the following three conditions are met: the deposited
54:
The equation is derived by treating the deposited mass as though it were an extension of the thickness of the underlying quartz. Because of this, the mass to frequency correlation (as determined by
Sauerbrey’s equation) is largely independent of electrode geometry. This has the benefit of allowing
46:
with the mass deposited on it. He simultaneously developed a method for measuring the characteristic frequency and its changes by using the crystal as the frequency determining component of an oscillator circuit. His method continues to be used as the primary tool in
645:
1205:
946:
147:
1070:
The
Sauerbrey equation was developed for oscillation in air and only applies to rigid masses attached to the crystal. It has been shown that quartz crystal microbalance measurements can be performed in liquid, in which case a
1430:
1401:
473:
439:
1262:
1235:
974:
839:
812:
320:
289:
1059:
1030:
1003:
761:
407:
380:
349:
234:
209:
736:
707:
678:
180:
481:
1282:
1081:
861:
783:
256:
866:
64:
1475:
1388:
1518:
1508:
1358:
48:
36:
1473:
Kanazawa, K. Keiji; Gordon II, Joseph G. (July 1985). "Frequency of a quartz microbalance in contact with liquid".
475:> 0.05, the Z-match method must be used to determine the change in mass. The formula for the Z-match method is:
1397:
1313:
55:
mass determination without calibration, making the set-up desirable from a cost and time investment standpoint.
1513:
1439:
1326:
259:
1301:
32:
1350:
183:
447:
413:
1240:
1213:
952:
817:
790:
298:
267:
51:(QCM) experiments for conversion of frequency to mass and is valid in nearly all applications.
1342:
1037:
1008:
981:
743:
389:
358:
327:
216:
191:
1484:
1447:
1334:
714:
685:
656:
640:{\displaystyle {\frac {\Delta m}{A}}\ ={\frac {N_{q}\rho _{q}}{\pi Zf_{L}}}\tan ^{-1}\left}
410:
mass must be rigid, the deposited mass must be distributed evenly and the frequency change
158:
1322:
1200:{\displaystyle \Delta f={-\ f_{0}^{3/2}(\eta _{l}\rho _{l}/\pi \rho _{q}\mu _{q}n)^{1/2}}}
1443:
1426:"Quartz-crystal microbalance study for characterizing atomic oxygen in plasma ash tools"
1330:
941:{\displaystyle ={\sqrt {\left({\frac {\rho _{q}\mu _{q}}{\rho _{f}\mu _{f}}}\ \right)}}}
1267:
846:
768:
241:
1502:
1425:
1354:
352:
40:
142:{\displaystyle \Delta f=-{\frac {2f_{0}^{2}}{A{\sqrt {\rho _{q}\mu _{q}}}}}\Delta m}
39:, Germany. It is a method for correlating changes in the oscillation frequency of a
1373:(NB. This was partially presented at Physikertagung in Heidelberg in October 1957.)
17:
1306:"Verwendung von Schwingquarzen zur Wägung dünner Schichten und zur Mikrowägung"
1305:
1346:
1072:
1424:
Srivastava, Aseem Kumar; Sakthivel, Palanikumaran (January–February 2001).
1431:
Journal of Vacuum
Science & Technology A: Vacuum, Surfaces, and Films
1488:
1338:
292:
43:
1452:
785:– Piezoelectrically active crystal area (Area between electrodes, cm)
1075:
related decrease in the resonant frequency will be observed:
738:– Frequency constant for AT-cut quartz crystal (1.668x10Hz·Å)
709:– Frequency of unloaded crystal, i.e. Resonant frequency (Hz)
1390:
QCM100 – Quartz
Crystal Microbalance Theory and Calibration
444:
If the change in frequency is greater than 5%, that is,
1270:
1243:
1216:
1084:
1040:
1011:
984:
955:
869:
849:
820:
793:
771:
746:
717:
688:
659:
484:
450:
416:
392:
361:
330:
301:
270:
244:
219:
194:
161:
67:
1276:
1256:
1229:
1199:
1061:– Shear modulus of film (Varies: units are g·cm·s)
1053:
1024:
997:
968:
940:
855:
833:
806:
777:
755:
730:
701:
672:
639:
467:
433:
401:
374:
343:
314:
283:
250:
228:
203:
174:
141:
262:active crystal area (Area between electrodes, cm)
35:in 1959, while working on his doctoral thesis at
976:– Density of the film (Varies: units are g/cm)
8:
1451:
1269:
1248:
1242:
1221:
1215:
1186:
1182:
1169:
1159:
1147:
1141:
1131:
1114:
1110:
1105:
1094:
1083:
1045:
1039:
1016:
1010:
989:
983:
960:
954:
919:
909:
897:
887:
880:
873:
868:
848:
825:
819:
798:
792:
770:
745:
722:
716:
693:
687:
664:
658:
619:
608:
595:
588:
554:
541:
523:
513:
506:
485:
483:
457:
449:
423:
415:
391:
366:
360:
335:
329:
306:
300:
275:
269:
243:
218:
193:
166:
160:
122:
112:
106:
95:
90:
80:
66:
1293:
58:The Sauerbrey equation is defined as:
1383:
1381:
1379:
7:
1264:is the viscosity of the liquid, and
1085:
747:
680:– Frequency of loaded crystal (Hz)
488:
451:
417:
393:
220:
211:– normalized frequency change (Hz)
195:
133:
68:
25:
1407:from the original on 2019-02-27
1400:/ Lambda Photometrics Limited,
1364:from the original on 2019-02-26
1237:is the density of the liquid,
1179:
1124:
355:of quartz for AT-cut crystal (
37:TTechnische Universität Berlin
1:
863:– Z-Factor of film material
650:Equation 2 – Z-match method
186:of the fundamental mode (Hz)
31:was developed by the German
1005:– Shear modulus of quartz (
49:quartz crystal microbalance
1535:
468:{\displaystyle \Delta f/f}
434:{\displaystyle \Delta f/f}
1398:Stanford Research Systems
1257:{\displaystyle \eta _{l}}
1230:{\displaystyle \rho _{l}}
969:{\displaystyle \rho _{f}}
834:{\displaystyle \rho _{q}}
807:{\displaystyle \rho _{q}}
386:The normalized frequency
315:{\displaystyle \rho _{q}}
284:{\displaystyle \rho _{q}}
1054:{\displaystyle \mu _{f}}
1025:{\displaystyle \mu _{q}}
998:{\displaystyle \mu _{q}}
756:{\displaystyle \Delta m}
402:{\displaystyle \Delta f}
375:{\displaystyle \mu _{q}}
344:{\displaystyle \mu _{q}}
229:{\displaystyle \Delta m}
204:{\displaystyle \Delta f}
1314:Zeitschrift für Physik
1302:Sauerbrey, Günter Hans
1278:
1258:
1231:
1201:
1055:
1026:
999:
970:
942:
857:
835:
808:
779:
757:
732:
703:
674:
641:
469:
435:
403:
376:
345:
316:
285:
252:
230:
205:
176:
143:
1279:
1259:
1232:
1202:
1056:
1027:
1000:
971:
943:
858:
836:
814:– Density of quartz (
809:
780:
758:
733:
731:{\displaystyle N_{q}}
704:
702:{\displaystyle f_{U}}
675:
673:{\displaystyle f_{L}}
642:
470:
436:
404:
377:
346:
317:
286:
253:
231:
206:
177:
175:{\displaystyle f_{0}}
144:
1519:Weighing instruments
1509:Electrical phenomena
1476:Analytical Chemistry
1284:is the mode number.
1268:
1241:
1214:
1082:
1038:
1009:
982:
953:
867:
847:
818:
791:
769:
744:
715:
686:
657:
482:
448:
414:
390:
359:
328:
299:
268:
242:
217:
192:
159:
65:
1489:10.1021/ac00285a062
1444:2001JVSTA..19...97S
1331:1959ZPhy..155..206S
1123:
100:
1339:10.1007/BF01337937
1274:
1254:
1227:
1197:
1101:
1051:
1032:= 2.947x10 g·cm·s)
1022:
995:
966:
938:
853:
831:
804:
775:
753:
728:
699:
670:
637:
465:
431:
399:
382:= 2.947x10 g·cm·s)
372:
341:
312:
281:
248:
226:
201:
184:Resonant frequency
172:
139:
86:
29:Sauerbrey equation
18:Sauerbrey constant
1453:10.1116/1.1335681
1277:{\displaystyle n}
1100:
936:
930:
926:
856:{\displaystyle Z}
778:{\displaystyle A}
763:– Mass change (g)
625:
548:
502:
498:
260:Piezoelectrically
251:{\displaystyle A}
236:– Mass change (g)
131:
128:
16:(Redirected from
1526:
1493:
1492:
1483:(8): 1770–1771.
1470:
1464:
1463:
1461:
1460:
1455:
1421:
1415:
1414:
1413:
1412:
1406:
1395:
1385:
1374:
1372:
1370:
1369:
1363:
1310:
1298:
1283:
1281:
1280:
1275:
1263:
1261:
1260:
1255:
1253:
1252:
1236:
1234:
1233:
1228:
1226:
1225:
1206:
1204:
1203:
1198:
1196:
1195:
1194:
1190:
1174:
1173:
1164:
1163:
1151:
1146:
1145:
1136:
1135:
1122:
1118:
1109:
1098:
1060:
1058:
1057:
1052:
1050:
1049:
1031:
1029:
1028:
1023:
1021:
1020:
1004:
1002:
1001:
996:
994:
993:
975:
973:
972:
967:
965:
964:
947:
945:
944:
939:
937:
935:
931:
928:
927:
925:
924:
923:
914:
913:
903:
902:
901:
892:
891:
881:
874:
862:
860:
859:
854:
840:
838:
837:
832:
830:
829:
813:
811:
810:
805:
803:
802:
784:
782:
781:
776:
762:
760:
759:
754:
737:
735:
734:
729:
727:
726:
708:
706:
705:
700:
698:
697:
679:
677:
676:
671:
669:
668:
646:
644:
643:
638:
636:
632:
631:
627:
626:
624:
623:
614:
613:
612:
600:
599:
589:
562:
561:
549:
547:
546:
545:
529:
528:
527:
518:
517:
507:
500:
499:
494:
486:
474:
472:
471:
466:
461:
440:
438:
437:
432:
427:
408:
406:
405:
400:
381:
379:
378:
373:
371:
370:
350:
348:
347:
342:
340:
339:
321:
319:
318:
313:
311:
310:
290:
288:
287:
282:
280:
279:
257:
255:
254:
249:
235:
233:
232:
227:
210:
208:
207:
202:
181:
179:
178:
173:
171:
170:
148:
146:
145:
140:
132:
130:
129:
127:
126:
117:
116:
107:
101:
99:
94:
81:
33:Günter Sauerbrey
21:
1534:
1533:
1529:
1528:
1527:
1525:
1524:
1523:
1499:
1498:
1497:
1496:
1472:
1471:
1467:
1458:
1456:
1423:
1422:
1418:
1410:
1408:
1404:
1393:
1387:
1386:
1377:
1367:
1365:
1361:
1323:Springer-Verlag
1308:
1304:(April 1959) .
1300:
1299:
1295:
1290:
1266:
1265:
1244:
1239:
1238:
1217:
1212:
1211:
1178:
1165:
1155:
1137:
1127:
1080:
1079:
1068:
1041:
1036:
1035:
1012:
1007:
1006:
985:
980:
979:
956:
951:
950:
915:
905:
904:
893:
883:
882:
879:
875:
865:
864:
845:
844:
821:
816:
815:
794:
789:
788:
767:
766:
742:
741:
718:
713:
712:
689:
684:
683:
660:
655:
654:
615:
604:
591:
590:
584:
580:
570:
566:
550:
537:
530:
519:
509:
508:
487:
480:
479:
446:
445:
412:
411:
388:
387:
362:
357:
356:
331:
326:
325:
302:
297:
296:
271:
266:
265:
240:
239:
215:
214:
190:
189:
162:
157:
156:
118:
108:
102:
82:
63:
62:
23:
22:
15:
12:
11:
5:
1532:
1530:
1522:
1521:
1516:
1511:
1501:
1500:
1495:
1494:
1465:
1416:
1375:
1292:
1291:
1289:
1286:
1273:
1251:
1247:
1224:
1220:
1208:
1207:
1193:
1189:
1185:
1181:
1177:
1172:
1168:
1162:
1158:
1154:
1150:
1144:
1140:
1134:
1130:
1126:
1121:
1117:
1113:
1108:
1104:
1097:
1093:
1090:
1087:
1067:
1064:
1063:
1062:
1048:
1044:
1033:
1019:
1015:
992:
988:
977:
963:
959:
948:
934:
922:
918:
912:
908:
900:
896:
890:
886:
878:
872:
852:
842:
828:
824:
801:
797:
786:
774:
764:
752:
749:
739:
725:
721:
710:
696:
692:
681:
667:
663:
648:
647:
635:
630:
622:
618:
611:
607:
603:
598:
594:
587:
583:
579:
576:
573:
569:
565:
560:
557:
553:
544:
540:
536:
533:
526:
522:
516:
512:
505:
497:
493:
490:
464:
460:
456:
453:
430:
426:
422:
419:
398:
395:
384:
383:
369:
365:
338:
334:
323:
309:
305:
278:
274:
263:
247:
237:
225:
222:
212:
200:
197:
187:
169:
165:
150:
149:
138:
135:
125:
121:
115:
111:
105:
98:
93:
89:
85:
79:
76:
73:
70:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1531:
1520:
1517:
1515:
1512:
1510:
1507:
1506:
1504:
1490:
1486:
1482:
1478:
1477:
1469:
1466:
1454:
1449:
1445:
1441:
1438:(1): 97–100.
1437:
1433:
1432:
1427:
1420:
1417:
1403:
1399:
1392:
1391:
1384:
1382:
1380:
1376:
1360:
1356:
1352:
1348:
1344:
1340:
1336:
1332:
1328:
1324:
1320:
1317:(in German).
1316:
1315:
1307:
1303:
1297:
1294:
1287:
1285:
1271:
1249:
1245:
1222:
1218:
1191:
1187:
1183:
1175:
1170:
1166:
1160:
1156:
1152:
1148:
1142:
1138:
1132:
1128:
1119:
1115:
1111:
1106:
1102:
1095:
1091:
1088:
1078:
1077:
1076:
1074:
1065:
1046:
1042:
1034:
1017:
1013:
990:
986:
978:
961:
957:
949:
932:
920:
916:
910:
906:
898:
894:
888:
884:
876:
870:
850:
843:
841:= 2.648 g/cm)
826:
822:
799:
795:
787:
772:
765:
750:
740:
723:
719:
711:
694:
690:
682:
665:
661:
653:
652:
651:
633:
628:
620:
616:
609:
605:
601:
596:
592:
585:
581:
577:
574:
571:
567:
563:
558:
555:
551:
542:
538:
534:
531:
524:
520:
514:
510:
503:
495:
491:
478:
477:
476:
462:
458:
454:
442:
428:
424:
420:
396:
367:
363:
354:
353:Shear modulus
336:
332:
324:
322:= 2.648 g/cm)
307:
303:
294:
276:
272:
264:
261:
245:
238:
223:
213:
198:
188:
185:
167:
163:
155:
154:
153:
136:
123:
119:
113:
109:
103:
96:
91:
87:
83:
77:
74:
71:
61:
60:
59:
56:
52:
50:
45:
42:
41:piezoelectric
38:
34:
30:
19:
1480:
1474:
1468:
1457:. Retrieved
1435:
1429:
1419:
1409:, retrieved
1389:
1366:. Retrieved
1318:
1312:
1296:
1209:
1069:
649:
443:
385:
151:
57:
53:
28:
26:
1514:Transducers
1325:: 206–222.
1066:Limitations
441:< 0.05.
295:of quartz (
1503:Categories
1459:2019-02-27
1411:2019-02-27
1368:2019-02-26
1288:References
1355:122855173
1347:0044-3328
1246:η
1219:ρ
1167:μ
1157:ρ
1153:π
1139:ρ
1129:η
1096:−
1086:Δ
1073:viscosity
1043:μ
1014:μ
987:μ
958:ρ
917:μ
907:ρ
895:μ
885:ρ
823:ρ
796:ρ
748:Δ
602:−
586:π
578:
564:
556:−
532:π
521:ρ
489:Δ
452:Δ
418:Δ
394:Δ
364:μ
333:μ
304:ρ
273:ρ
221:Δ
196:Δ
134:Δ
120:μ
110:ρ
78:−
69:Δ
1402:archived
1359:Archived
1440:Bibcode
1327:Bibcode
293:Density
152:where:
44:crystal
1353:
1345:
1210:where
1099:
929:
501:
1405:(PDF)
1394:(PDF)
1362:(PDF)
1351:S2CID
1321:(2).
1309:(PDF)
1343:ISSN
27:The
1485:doi
1448:doi
1335:doi
1319:155
575:tan
552:tan
1505::
1481:57
1479:.
1446:.
1436:19
1434:.
1428:.
1396:,
1378:^
1357:.
1349:.
1341:.
1333:.
1311:.
351:–
291:–
258:–
182:–
1491:.
1487::
1462:.
1450::
1442::
1371:.
1337::
1329::
1272:n
1250:l
1223:l
1192:2
1188:/
1184:1
1180:)
1176:n
1171:q
1161:q
1149:/
1143:l
1133:l
1125:(
1120:2
1116:/
1112:3
1107:0
1103:f
1092:=
1089:f
1047:f
1018:q
991:q
962:f
933:)
921:f
911:f
899:q
889:q
877:(
871:=
851:Z
827:q
800:q
773:A
751:m
724:q
720:N
695:U
691:f
666:L
662:f
634:]
629:)
621:U
617:f
610:L
606:f
597:U
593:f
582:(
572:Z
568:[
559:1
543:L
539:f
535:Z
525:q
515:q
511:N
504:=
496:A
492:m
463:f
459:/
455:f
429:f
425:/
421:f
397:f
368:q
337:q
308:q
277:q
246:A
224:m
199:f
168:0
164:f
137:m
124:q
114:q
104:A
97:2
92:0
88:f
84:2
75:=
72:f
20:)
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