1501:" lattice planes, which in turn have small spatial frequencies (g-values) and hence large lattice periodicities (d-spacings). A possible intuition behind this is that in electron microscopy, for electron beams to be directed down wide (i.e. easily visible) tunnels between columns of atoms in a crystal, directing the beam down a low-index (and by association high-symmetry) zone axis may help.
25:
100:
High-symmetry zone axes through a crystal lattice, in particular, often lie in the direction of tunnels through the crystal between planes of atoms. This is because, as we see below, such zone axis directions generally lie within more than one plane of atoms in the crystal.
458:
The term "zone axis" more specifically refers to the direction of a direct-space lattice vector. For example, since the and lattice vectors are parallel, their orientations both correspond the ⟨120⟩ zone of the crystal. Just as a set of
406:, for instance, ⟨100⟩ represents , , , , and because each of these vectors is symmetrically equivalent under a 90 degree rotation along an axis. A bar over a coordinate is equivalent to a negative sign (e.g.,
1002:
825:
563:
762:
698:
634:
266:
388:
1196:
1417:
75:
1062:
1478:
910:
303:
453:
569:) are used to refer to a symmetrically equivalent class of reciprocal lattice vectors, similar to angle brackets ⟨⟩ for classes of direct lattice vectors.
1281:
1585:
Zbigniew Dauter and
Mariusz Jaskolski (2010) "How to read (and understand) Volume A of International Tables for Crystallography: an introduction for nonspecialists",
1315:
1095:
869:
845:
1127:
1230:
565:
instead, generally between round brackets () (similar to square brackets for direct lattice vectors). Curly brackets {} (not to be confused with a mathematical
335:
1332:
in 3D", e.g. reciprocal lattices, is that the condition for a direct lattice vector (or zone axis) to be perpendicular to a reciprocal lattice vector (
43:
915:
767:
390:. Angle brackets ⟨⟩ are used to refer to a symmetrically equivalent class of lattice vectors (i.e. the set of vectors
497:
132:
703:
639:
575:
566:
221:
1497:
One result of this, as illustrated in the figure above, is that "low-index" zones are generally perpendicular to "low-
340:
1730:
1710:
1673:
1626:
61:
1540:
1166:
1339:
1770:
1420:
93:) of a crystal in three dimensions. It is therefore indexed with direct lattice indices, instead of with
1470:
radius (1/λ), so that electron diffraction generally "lights up" diffraction spots with g-vectors (
85:, a term sometimes used to refer to "high-symmetry" orientations in a crystal, most generally refers to
1760:
1419:. This is true even if, as is often the case, the basis vector set used to describe the lattice is not
1007:
878:
271:
1775:
409:
391:
136:
1455:
traveling along a lattice direction specified by the zone-axis indices . Because of their small
1133:
Bracket conventions for direct lattice and reciprocal lattice Miller indices in crystallography
1253:
128:
116:
1287:
1067:
854:
830:
1780:
1535:
1100:
403:
8:
1484:
1463:
1440:
1202:
1520:
464:
90:
308:
1765:
1726:
1706:
1669:
1622:
1467:
1129:
planes, measured in spatial frequency units, e.g. of cycles per angstrom (cycles/Å).
479:
468:
215:
472:
152:
39:
1689:
1510:
120:
268:
and is generally placed between square brackets . Thus a direct lattice vector
1493:
viewed down the zone (left), with the corresponding zone-axis pattern (right).
399:
1721:
P. Hirsch, A. Howie, R. Nicholson, D. W. Pashley and M. J. Whelan (1965/1977)
1686:
Transmission electron microscopy: Physics of image formation and microanalysis
1593:
1754:
872:
460:
1525:
1498:
1490:
491:
487:
395:
203:). Direct lattice vectors have components measured in distance units, like
110:
94:
1004:
has a direction perpendicular to a crystallographic plane and a magnitude
471:
and momenta, a "zone" is defined as a set of reciprocal lattice planes in
1530:
1436:
848:
1638:
cf. Charles W. Misner, Kip S. Thorne and John
Archibald Wheeler (1973)
1515:
1456:
1329:
124:
1444:
483:
208:
1703:
Transmission electron microscopy: A textbook for materials science
997:{\displaystyle (hkl)=h{\bf {a}}^{*}+k{\bf {b}}^{*}+l{\bf {c}}^{*}}
1487:
1481:
1452:
89:
direction referenced to the direct lattice (as distinct from the
820:{\displaystyle V_{c}={\bf {a}}\cdot ({\bf {b}}\times {\bf {c}})}
74:
478:
The reciprocal space analog to a zone axis is a "lattice plane
1448:
558:{\displaystyle \{{\bf {a}}^{*},{\bf {b}}^{*},{\bf {c}}^{*}\}}
204:
1477:
1621:(Addison-Wesley, paperback edition by Dover Books 1990)
482:" or "g-vector direction". Reciprocal lattice vectors (
1745:
475:
that corresponds to a lattice vector in direct space.
1342:
1328:
A useful and quite general rule of crystallographic "
1290:
1256:
1205:
1169:
1103:
1070:
1010:
918:
881:
857:
833:
770:
757:{\displaystyle {\bf {c}}^{*}\equiv (a\times b)/V_{c}}
706:
693:{\displaystyle {\bf {b}}^{*}\equiv (c\times a)/V_{c}}
642:
629:{\displaystyle {\bf {a}}^{*}\equiv (b\times c)/V_{c}}
578:
500:
412:
343:
311:
274:
224:
34:
may be too technical for most readers to understand
1666:Practical electron microscopy in materials science
1411:
1309:
1275:
1224:
1190:
1121:
1089:
1056:
996:
904:
863:
839:
819:
756:
692:
628:
557:
494:using coordinates in the reciprocal lattice basis
447:
382:
329:
297:
260:
1668:(N. V. Philips' Gloeilampenfabrieken, Eindhoven)
261:{\displaystyle \{{\bf {a}},{\bf {b}},{\bf {c}}\}}
1752:
1695:
383:{\displaystyle u{\bf {a}}+v{\bf {b}}+w{\bf {c}}}
1701:David B. Williams and C. Barry Carter (1996)
155:, i.e. the magnitudes of the vectors, called
78:Nanocrystal (left) zone-axis patterns (right)
1269:
1257:
1183:
1170:
552:
501:
255:
225:
1658:
1598:
1725:(Butterworths/Krieger, London/Malabar FL)
1645:
1632:
1611:
1579:
1553:
1306:
1272:
1221:
1187:
62:Learn how and when to remove this message
46:, without removing the technical details.
1746:International Tables for Crystallography
1566:
1476:
73:
1678:
1336:) can be written with a dot product as
1191:{\displaystyle \langle {uvw\rangle }\,}
1064:equal to the reciprocal of the spacing
1753:
1715:
1412:{\displaystyle \cdot (hkl)=uh+vk+wl=0}
167:, and the angles between them, called
16:High symmetry orientation of a crystal
1426:
467:vector in the complementary space of
104:
44:make it understandable to non-experts
1723:Electron microscopy of thin crystals
875:). Thus a reciprocal lattice vector
18:
1576:(Cambridge U. Press, Cambridge UK).
1561:Mathematical methods for physicists
218:in the direct lattice basis system
214:A lattice vector is indexed by its
13:
1642:(W. H. Freeman, San Francisco CA).
1574:Principles of the theory of solids
14:
1792:
1739:
1057:{\displaystyle g_{hkl}=1/d_{hkl}}
463:in direct space corresponds to a
1541:Transmission electron microscopy
1462:, high energy electrons used in
983:
963:
943:
885:
809:
799:
786:
764:, where the unit cell volume is
710:
646:
582:
541:
524:
507:
440:
375:
362:
349:
278:
250:
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230:
23:
905:{\displaystyle {\bf {g}}_{hkl}}
298:{\displaystyle {\bf {s}}_{uvw}}
1474:) that are perpendicular to .
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1:
1684:Ludwig Reimer (1997 4th ed)
1546:
1240:
1153:
421:
7:
1655:(North-Holland, Amsterdam).
1563:(Academic Press, New York).
1504:
1131:
448:{\displaystyle =-{\bf {a}}}
10:
1797:
1572:J. M. Ziman (1972 2nd ed)
108:
1606:X-ray diffraction methods
1276:{\displaystyle \{hkl\}\,}
151:, or equivalently by the
123:is described by a set of
117:translational invariance
1319:reciprocal space, e.g.
1310:{\displaystyle (hkl)\,}
1237:contravariant or polar
1155:zone or lattice-vector
1090:{\displaystyle d_{hkl}}
864:{\displaystyle \times }
1664:J. W. Edington (1976)
1651:John M. Cowley (1975)
1604:E. W. Nuffield (1966)
1494:
1439:pattern taken with an
1413:
1311:
1277:
1226:
1192:
1123:
1091:
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998:
906:
865:
841:
840:{\displaystyle \cdot }
821:
758:
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559:
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384:
331:
299:
262:
79:
1587:J. Appl. Crystallogr.
1559:George Arfken (1970)
1480:
1414:
1312:
1278:
1227:
1193:
1124:
1122:{\displaystyle (hkl)}
1092:
1059:
999:
907:
866:
842:
822:
759:
695:
631:
560:
450:
402:). In the case of a
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332:
300:
263:
77:
1617:B. E. Warren (1969)
1536:Electron diffraction
1464:electron microscopes
1340:
1288:
1254:
1203:
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1068:
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916:
879:
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576:
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309:
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222:
1771:Electron microscopy
1705:(Plenum Press, NY)
1688:(Springer, Berlin)
1653:Diffraction Physics
1485:face centered cubic
1322:covariant or axial
1234:direct space, e.g.
1134:
469:spatial frequencies
1521:Reciprocal lattice
1495:
1466:have a very large
1427:Zone-axis patterns
1409:
1330:dual vector spaces
1307:
1273:
1242:plane or g-vector
1225:{\displaystyle \,}
1222:
1188:
1132:
1119:
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1054:
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861:
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555:
465:reciprocal lattice
445:
380:
327:
295:
258:
153:lattice parameters
105:Zone-axis indexing
91:reciprocal lattice
80:
1761:Materials science
1619:X-ray diffraction
1608:(John Wiley, NY).
1433:zone-axis pattern
1431:By extension, a
1326:
1325:
1141:Equivalence Class
424:
398:of the lattice's
127:, direct lattice
72:
71:
64:
1788:
1733:
1719:
1713:
1699:
1693:
1682:
1676:
1662:
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1649:
1643:
1636:
1630:
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1609:
1602:
1596:
1583:
1577:
1570:
1564:
1557:
1418:
1416:
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1410:
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1314:
1313:
1308:
1282:
1280:
1279:
1274:
1231:
1229:
1228:
1223:
1197:
1195:
1194:
1189:
1186:
1135:
1128:
1126:
1125:
1120:
1096:
1094:
1093:
1088:
1086:
1085:
1063:
1061:
1060:
1055:
1053:
1052:
1037:
1026:
1025:
1003:
1001:
1000:
995:
993:
992:
987:
986:
973:
972:
967:
966:
953:
952:
947:
946:
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909:
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901:
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870:
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389:
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366:
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337:, is defined as
336:
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330:{\displaystyle }
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1776:Crystallography
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1511:Crystallography
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1200:
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1150:Transformation
1144:Specific Vector
1102:
1099:
1098:
1075:
1071:
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1042:
1038:
1033:
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637:
620:
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611:
587:
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580:
579:
577:
574:
573:
546:
540:
539:
538:
529:
523:
522:
521:
512:
506:
505:
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473:frequency space
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438:
416:
411:
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407:
374:
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361:
360:
348:
347:
342:
339:
338:
310:
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277:
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273:
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219:
121:crystal lattice
113:
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68:
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51:
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40:help improve it
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24:
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12:
11:
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1740:External links
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95:Miller indices
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873:cross product
858:
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834:
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781:
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749:
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404:cubic lattice
401:
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290:
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133:contravariant
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129:basis vectors
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32:This article
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1592:, 1150–1171
1589:
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1526:Miller index
1499:Miller index
1496:
1491:atom cluster
1471:
1468:Ewald sphere
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111:Miller index
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52:October 2014
49:
33:
1781:Diffraction
1640:Gravitation
1531:Diffraction
1437:diffraction
1435:(ZAP) is a
849:dot product
216:coordinates
1755:Categories
1516:Dual basis
1457:wavelength
847:denotes a
1547:Footnotes
1445:electrons
1421:Cartesian
1359:⋅
1184:⟩
1171:⟨
990:∗
970:∗
950:∗
859:×
835:⋅
805:×
792:⋅
731:×
722:≡
717:∗
667:×
658:≡
653:∗
603:×
594:≡
589:∗
548:∗
531:∗
514:∗
436:−
422:¯
392:generated
209:angstroms
195:(between
183:(between
171:(between
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125:unit cell
83:Zone axis
1766:Crystals
1505:See also
1453:neutrons
1441:incident
484:one-form
1690:preview
1488:silicon
1482:Diamond
207:(m) or
191:), and
38:Please
1729:
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1460:λ
1449:X-rays
1138:Object
700:, and
572:Here,
490:) are
480:normal
396:action
394:by an
205:meters
193:γ
181:β
169:α
147:, and
1147:Units
488:axial
305:, or
211:(Å).
137:polar
119:of a
1727:ISBN
1707:ISBN
1670:ISBN
1623:ISBN
851:and
199:and
187:and
175:and
163:and
115:The
1594:pdf
1472:hkl
1451:or
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567:set
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