3150:
28:
763:
20:
2238:
3543:
2185:
1374:
3555:
296:(the vector indicating a reciprocal lattice point from the reciprocal lattice origin) is the wavevector of a plane wave in the Fourier series of a spatial function (e.g., electronic density function) which periodicity follows the original Bravais lattice, so wavefronts of the plane wave are coincident with parallel lattice planes of the original lattice. Since a measured scattering vector in
1818:
1601:
2169:
970:
163:
366:
2425:. Thus, integer Miller indices implicitly include indices with all rational ratios. The reason why planes where the components (in the reciprocal-lattice basis) have rational ratios are of special interest is that these are the
2307:: the condensation of a material means that the atoms, ions or molecules are more stable if they are surrounded by other similar species; the surface tension of an interface thus varies according to the density on the surface
510:. In this case the integers are not necessarily in lowest terms, and can be thought of as corresponding to planes spaced such that the reflections from adjacent planes would have a phase difference of exactly one wavelength (2
2026:
instead. Thus as you can see, four-index zone indices in square or angle brackets sometimes mix a single direct-lattice index on the right with reciprocal-lattice indices (normally in round or curly brackets) on the left.
2557:"Ueber eine verbesserte Methode für die Bezeichnung der verschiedenen Flächen eines Krystallisationssystems, nebst Bemerkungen über den Zustand der Polarisierung der Seiten in den Linien der krystallinischen Structur"
1187:
1572:
272:
1320:
2456:. This construction corresponds precisely to the standard "cut-and-project" method of defining a quasicrystal, using a plane with irrational-ratio Miller indices. (Although many quasicrystals, such as the
1813:{\displaystyle h\mathbf {b} _{1}+k\mathbf {b} _{2}+\ell \mathbf {b} _{3}={\frac {2}{3a^{2}}}(2h+k)\mathbf {a} _{1}+{\frac {2}{3a^{2}}}(h+2k)\mathbf {a} _{2}+{\frac {1}{c^{2}}}(\ell )\mathbf {a} _{3}.}
2036:
983:
of the primitive reciprocal lattice vectors. Because the coordinates are integers, this normal is itself always a reciprocal lattice vector. The requirement of lowest terms means that it is the
1106:
403:
437:
1031:
of the intercepts of the plane, in the basis of the lattice vectors. If one of the indices is zero, it means that the planes do not intersect that axis (the intercept is "at infinity").
3159:
668:
3420:
875:
488:
3415:
1361:
lattices, the primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic
192:
459:
294:
583:
2955:
1499:(rather than reciprocal lattice vectors or planes) with four indices. However they don't operate by similarly adding a redundant index to the regular three-index set.
883:
551:
733:
79:
3471:
303:
2370:
Ordinarily, Miller indices are always integers by definition, and this constraint is physically significant. To understand this, suppose that we allow a plane (
629:
2024:
1922:
700:
2699:
1349:
such as {100} denote a family of plane normals which are equivalent due to symmetry operations, much the way angle brackets denote a family of directions.
774:, or as the inverse intercepts along the lattice vectors. Both definitions are given below. In either case, one needs to choose the three lattice vectors
1447:
This four-index scheme for labeling planes in a hexagonal lattice makes permutation symmetries apparent. For example, the similarity between (110) ≡ (11
754:
The Miller indices are defined with respect to any choice of unit cell and not only with respect to primitive basis vectors, as is sometimes stated.
2301:: adsorption and chemical reactions can occur at atoms or molecules on crystal surfaces, these phenomena are thus sensitive to the density of nodes;
1326:
Because of the symmetry of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes:
1121:
2269:
linking nodes. Some directions and planes have a higher density of nodes; these dense planes have an influence on the behavior of the crystal:
1509:
209:
206:
for the given
Bravais lattice. (Note that the plane is not always orthogonal to the linear combination of direct or original lattice vectors
1244:
3289:
3476:
3201:
3367:
2659:
2352:
the dislocation line tends to follow a dense direction, the dislocation line is often a straight line, a dislocation loop is often a
751:
since 1817. The method was also historically known as the
Millerian system, and the indices as Millerian, although this is now rare.
2615:
1046:
between adjacent lattice planes is related to the (shortest) reciprocal lattice vector orthogonal to the planes by the formula:
274:
because the direct lattice vectors need not be mutually orthogonal.) This is based on the fact that a reciprocal lattice vector
3466:
3458:
3519:
3497:
2460:, are formed by "cuts" of periodic lattices in more than three dimensions, involving the intersection of more than one such
2164:{\displaystyle d_{hk\ell }={\frac {a}{\sqrt {{\tfrac {4}{3}}\left(h^{2}+k^{2}+hk\right)+{\tfrac {a^{2}}{c^{2}}}\ell ^{2}}}}}
3512:
3362:
3028:
2893:
2742:
2634: – A collection of routines for rotation / orientation manipulation, including special tools for crystal orientations.
3502:
3400:
3096:
2749:
2483:
1338:
of directions which are equivalent due to symmetry operations, such as , , or the negative of any of those directions.
1208:
For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length (usually denoted
1049:
3524:
3382:
3352:
3281:
2594:
2224:
3559:
3234:
1492:
371:
408:
3507:
3430:
3304:
2903:
3342:
3264:
2531:
2334:
the dislocation core tends to spread on dense planes (the elastic perturbation is "diluted"); this reduces the
2206:
3357:
3347:
2652:
2202:
795:
that define the unit cell (note that the conventional unit cell may be larger than the primitive cell of the
641:
3481:
3129:
2754:
2732:
1462:
In the figure at right, the (001) plane has a 3-fold symmetry: it remains unchanged by a rotation of 1/3 (2
3033:
2787:
2682:
1386:
830:
468:
3390:
2687:
439:
as the incoming (toward the crystal lattice) X-ray wavevector, is equal to a reciprocal lattice vector
2556:
803:
illustrate). Given these, the three primitive reciprocal lattice vectors are also determined (denoted
168:
3405:
3334:
2792:
2782:
2339:
3586:
3547:
3271:
3167:
3040:
3003:
2918:
2797:
2777:
2645:
2422:
2362:
For all these reasons, it is important to determine the planes and thus to have a notation system.
1382:
76:), and denote the family of (parallel) lattice planes (of the given Bravais lattice) orthogonal to
442:
277:
3395:
3239:
3184:
2933:
2898:
2195:
965:{\displaystyle \mathbf {g} _{hk\ell }=h\mathbf {b} _{1}+k\mathbf {b} _{2}+\ell \mathbf {b} _{3}.}
740:
503:
2349:) is along a dense direction: the shift of one node in a dense direction is a lesser distortion;
158:{\displaystyle \mathbf {g} _{hk\ell }=h\mathbf {b} _{1}+k\mathbf {b} _{2}+\ell \mathbf {b} _{3}}
3149:
3091:
2908:
2298:
980:
766:
Examples of determining indices for a plane using intercepts with axes; left (111), right (221)
748:
556:
195:
770:
There are two equivalent ways to define the meaning of the Miller indices: via a point in the
361:{\displaystyle \Delta \mathbf {k} =\mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }}
3448:
3244:
3206:
3013:
2965:
2561:
Abhandlungen der physikalischen Klasse der Königlich-Preussischen
Akademie der Wissenschaften
2365:
1217:
712:
524:
507:
297:
2620:
2288:
thus varies according to the directions, whether the atoms are close or far; this gives the
3172:
3045:
2881:
2772:
1192:
That is, it uses the direct lattice basis instead of the reciprocal lattice. Note that is
1212:), as are those of the reciprocal lattice. Thus, in this common case, the Miller indices (
8:
3189:
3177:
3052:
3018:
2998:
2328:
2281:
2262:
1362:
1358:
1354:
1574:. For hexagonal crystals this may be expressed in terms of direct-lattice basis-vectors
3438:
3249:
3194:
2737:
2488:
2317:
2246:
771:
599:
203:
2610:
1935:
1830:
673:
3581:
3372:
3211:
3139:
3119:
2839:
2709:
2590:
2537:
2527:
2520:
2473:
2445:
2285:
2421:
appropriately: divide by the largest of the three numbers, and then multiply by the
27:
3410:
3216:
3134:
3124:
2923:
2856:
2827:
2820:
2405:
ratios, then the same family of planes can be written in terms of integer indices (
1224:
495:
3299:
3294:
3259:
3079:
2978:
2913:
2876:
2871:
2722:
2668:
2429:: they are the only planes whose intersections with the crystal are 2d-periodic.
2402:
2304:
796:
762:
199:
43:
39:
1027:, or some multiple thereof. That is, the Miller indices are proportional to the
3109:
3074:
3062:
3057:
3023:
2993:
2983:
2942:
2886:
2810:
2764:
2457:
2346:
1389:
631:
with square instead of round brackets, denotes a direction in the basis of the
462:
1506:) as suggested above can be written in terms of reciprocal lattice vectors as
3575:
3254:
3067:
2866:
2478:
2448:
ratios, on the other hand, the intersection of the plane with the crystal is
2426:
2289:
50:
19:
2960:
2950:
2844:
2453:
506:
should be 1. Miller indices are also used to designate reflections in
2541:
2366:
Integer versus irrational Miller indices: Lattice planes and quasicrystals
2237:
1182:{\displaystyle h\mathbf {a} _{1}+k\mathbf {a} _{2}+\ell \mathbf {a} _{3}.}
502:
for −3. The integers are usually written in lowest terms, i.e. their
3443:
3114:
2988:
2815:
2631:
2324:
2311:
1567:{\displaystyle h\mathbf {b} _{1}+k\mathbf {b} _{2}+\ell \mathbf {b} _{3}}
267:{\displaystyle h\mathbf {a} _{1}+k\mathbf {a} _{2}+\ell \mathbf {a} _{3}}
405:
as the outgoing (scattered from a crystal lattice) X-ray wavevector and
3008:
2694:
2461:
2294:
2209: in this section. Unsourced material may be challenged and removed.
2030:
And, note that for hexagonal interplanar distances, they take the form
465:, the measured scattered X-ray peak at each measured scattering vector
1315:{\displaystyle d_{hk\ell }={\frac {a}{\sqrt {h^{2}+k^{2}+\ell ^{2}}}}}
2717:
2493:
514:), regardless of whether there are atoms on all these planes or not.
2184:
3314:
3084:
2832:
2335:
2258:
739:
Miller indices were introduced in 1839 by the
British mineralogist
2637:
2626:
3324:
2353:
1466:/3 rad, 120°). The , and the directions are really similar. If
1373:
54:
1368:
2273:
1487:
3319:
2314:
tend to have straight grain boundaries following dense planes
2277:
2573:
1823:
Hence zone indices of the direction perpendicular to plane (
877:
denotes planes orthogonal to the reciprocal lattice vector:
2250:
53:
of a given (direct) Bravais lattice is determined by three
3421:
2254:
2174:
670:
denotes the set of all directions that are equivalent to
3416:
1200:) planes, except in a cubic lattice as described below.
2621:
Online tutorial about lattice planes and Miller indices
1459:
10) is more obvious when the redundant index is shown.
1440:
are identical to the corresponding Miller indices, and
3309:
2342:), the sliding occurs more frequently on dense planes;
2123:
2066:
1038:) planes intersecting one or more lattice points (the
635:
lattice vectors instead of the reciprocal lattice; and
527:
23:
Planes with different Miller indices in cubic crystals
2039:
1938:
1833:
1604:
1512:
1365:
and hence are again simply the
Cartesian directions.
1247:
1124:
1052:
886:
833:
715:
676:
644:
602:
559:
553:
denotes the set of all planes that are equivalent to
471:
445:
411:
374:
306:
280:
212:
171:
82:
2627:
MTEX – Free MATLAB toolbox for
Texture Analysis
2452:periodic. It forms an aperiodic pattern known as a
1827:) are, in suitably normalized triplet form, simply
1470:is the intercept of the plane with the axis, then
994:) denotes a plane that intercepts the three points
2587:Practical electron microscopy in materials science
2519:
2163:
2018:
1916:
1812:
1566:
1314:
1181:
1100:
987:reciprocal lattice vector in the given direction.
964:
869:
735:lacks any bracketing when designating a reflection
727:
694:
662:
623:
577:
545:
482:
453:
431:
397:
360:
288:
266:
186:
157:
2589:(N. V. Philips' Gloeilampenfabrieken, Eindhoven)
2386:(defined as above) are not necessarily integers.
979:) simply indicates a normal to the planes in the
3573:
1216:) and both simply denote normals/directions in
1101:{\displaystyle d=2\pi /|\mathbf {g} _{hk\ell }|}
747:) had already been used by German mineralogist
593:(not planes), the corresponding notations are:
2517:
2653:
2345:the perturbation carried by the dislocation (
1369:Case of hexagonal and rhombohedral structures
800:
398:{\displaystyle \mathbf {k} _{\mathrm {out} }}
2518:Ashcroft, Neil W.; Mermin, N. David (1976).
2280:"jumps" from one atom to the other with the
2261:) of a crystal. Similarly, crystallographic
1502:For example, the reciprocal lattice vector (
657:
645:
540:
528:
432:{\displaystyle \mathbf {k} _{\mathrm {in} }}
3490:
1203:
2660:
2646:
517:There are also several related notations:
16:Notation system for crystal lattice planes
2611:IUCr Online Dictionary of Crystallography
2225:Learn how and when to remove this message
2526:. New York: Holt, Rinehart and Winston.
2236:
1372:
761:
26:
18:
1928:are used for the zone normal to plane (
743:, although an almost identical system (
663:{\displaystyle \langle hk\ell \rangle }
3574:
2763:
2632:http://sourceforge.net/projects/orilib
2616:Miller index description with diagrams
2175:Crystallographic planes and directions
1932:), however, the literature often uses
2641:
2554:
2513:
2511:
2509:
827:Then, given the three Miller indices
3554:
2894:Phase transformation crystallography
2207:adding citations to reliable sources
2178:
3401:Journal of Chemical Crystallography
2667:
1495:literature) for indexing hexagonal
870:{\displaystyle h,k,\ell ,(hk\ell )}
706:Note, for Laue–Bragg interferences
483:{\displaystyle \Delta \mathbf {k} }
13:
2506:
1111:The related notation denotes the
472:
423:
420:
389:
386:
383:
352:
349:
332:
329:
326:
307:
14:
3598:
2604:
1396:system, which uses four indices (
1238:) lattice planes is (from above)
3553:
3542:
3541:
3148:
2574:Oxford English Dictionary Online
2555:Weiss, Christian Samuel (1817).
2245:Crystallographic directions are
2183:
1797:
1756:
1701:
1646:
1628:
1610:
1554:
1536:
1518:
1493:transmission electron microscopy
1166:
1148:
1130:
1077:
949:
931:
913:
889:
476:
447:
414:
377:
343:
320:
311:
282:
254:
236:
218:
187:{\displaystyle \mathbf {b} _{i}}
174:
145:
127:
109:
85:
2194:needs additional citations for
585:by the symmetry of the lattice.
3343:Bilbao Crystallographic Server
2579:
2567:
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2013:
2004:
1989:
1986:
1972:
1939:
1911:
1902:
1887:
1884:
1870:
1834:
1792:
1786:
1751:
1736:
1696:
1681:
1094:
1071:
1042:), the perpendicular distance
864:
852:
689:
677:
615:
603:
572:
560:
498:are written with a bar, as in
1:
2499:
2374:) where the Miller "indices"
2241:Dense crystallographic planes
757:
200:primitive translation vectors
1392:, it is possible to use the
454:{\displaystyle \mathbf {g} }
289:{\displaystyle \mathbf {g} }
7:
3391:Crystal Growth & Design
2683:Timeline of crystallography
2467:
1409:) that obey the constraint
49:In particular, a family of
10:
3603:
3202:Nuclear magnetic resonance
589:In the context of crystal
44:crystal (Bravais) lattices
38:form a notation system in
3537:
3457:
3429:
3406:Journal of Crystal Growth
3381:
3333:
3280:
3227:
3158:
3146:
2941:
2932:
2855:
2708:
2675:
1196:generally normal to the (
578:{\displaystyle (hk\ell )}
3272:Single particle analysis
3130:Hermann–Mauguin notation
2432:For a plane (abc) where
2423:least common denominator
1223:For cubic crystals with
1204:Case of cubic structures
638:similarly, the notation
546:{\textstyle \{hk\ell \}}
3396:Crystallography Reviews
3240:Isomorphous replacement
3034:Lomer–Cottrell junction
2276:: in condensed matter,
1334:such as ⟨100⟩ denote a
741:William Hallowes Miller
728:{\displaystyle hk\ell }
504:greatest common divisor
2909:Spinodal decomposition
2585:J. W. Edington (1976)
2242:
2165:
2020:
1918:
1814:
1568:
1444:is a redundant index.
1378:
1377:Miller–Bravais indices
1316:
1183:
1102:
966:
871:
767:
749:Christian Samuel Weiss
729:
696:
664:
625:
579:
547:
484:
455:
433:
399:
362:
290:
268:
188:
159:
42:for lattice planes in
32:
31:Examples of directions
24:
3449:Gregori Aminoff Prize
3245:Molecular replacement
2340:Peierls–Nabarro force
2240:
2166:
2021:
1919:
1815:
1569:
1491:schemes (e.g. in the
1376:
1317:
1218:Cartesian coordinates
1184:
1103:
967:
872:
765:
730:
697:
665:
626:
580:
548:
508:X-ray crystallography
485:
456:
434:
400:
363:
298:X-ray crystallography
291:
269:
189:
160:
30:
22:
2755:Structure prediction
2576:(Consulted May 2007)
2203:improve this article
2037:
1936:
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372:
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210:
169:
80:
72:. They are written (
3019:Cottrell atmosphere
2999:Partial dislocation
2743:Restriction theorem
2522:Solid state physics
2329:plastic deformation
2282:Rayleigh scattering
1359:body-centered cubic
1355:face-centered cubic
3439:Carl Hermann Medal
3250:Molecular dynamics
3097:Defects in diamond
3092:Stone–Wales defect
2738:Reciprocal lattice
2700:Biocrystallography
2489:Reciprocal lattice
2274:optical properties
2243:
2161:
2146:
2075:
2016:
1914:
1810:
1564:
1379:
1312:
1234:between adjacent (
1179:
1098:
1034:Considering only (
962:
867:
772:reciprocal lattice
768:
725:
692:
660:
621:
575:
543:
480:
451:
429:
395:
358:
286:
264:
204:reciprocal lattice
184:
155:
33:
25:
3569:
3568:
3533:
3532:
3140:Thermal ellipsoid
3105:
3104:
3014:Frank–Read source
2974:
2973:
2840:Aperiodic crystal
2806:
2805:
2688:Crystallographers
2474:Crystal structure
2286:velocity of light
2235:
2234:
2227:
2159:
2158:
2145:
2074:
1784:
1734:
1679:
1310:
1309:
624:{\displaystyle ,}
496:negative integers
494:. By convention,
461:as stated by the
3594:
3557:
3556:
3545:
3544:
3488:
3487:
3411:Kristallografija
3265:Gerchberg–Saxton
3160:Characterisation
3152:
3135:Structure factor
2939:
2938:
2924:Ostwald ripening
2761:
2760:
2706:
2705:
2662:
2655:
2648:
2639:
2638:
2598:
2583:
2577:
2571:
2565:
2564:
2552:
2546:
2545:
2525:
2515:
2230:
2223:
2219:
2216:
2210:
2187:
2179:
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2168:
2167:
2162:
2160:
2157:
2156:
2147:
2144:
2143:
2134:
2133:
2124:
2118:
2114:
2104:
2103:
2091:
2090:
2076:
2067:
2064:
2060:
2055:
2054:
2025:
2023:
2022:
2019:{\displaystyle }
2017:
2012:
2011:
1999:
1982:
1923:
1921:
1920:
1917:{\displaystyle }
1915:
1910:
1909:
1897:
1880:
1819:
1817:
1816:
1811:
1806:
1805:
1800:
1785:
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1539:
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1526:
1521:
1465:
1458:
1454:
1450:
1321:
1319:
1318:
1313:
1311:
1308:
1307:
1295:
1294:
1282:
1281:
1272:
1268:
1263:
1262:
1225:lattice constant
1188:
1186:
1185:
1180:
1175:
1174:
1169:
1157:
1156:
1151:
1139:
1138:
1133:
1107:
1105:
1104:
1099:
1097:
1092:
1091:
1080:
1074:
1069:
971:
969:
968:
963:
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957:
952:
940:
939:
934:
922:
921:
916:
904:
903:
892:
876:
874:
873:
868:
745:Weiss parameters
734:
732:
731:
726:
701:
699:
698:
695:{\displaystyle }
693:
669:
667:
666:
661:
630:
628:
627:
622:
584:
582:
581:
576:
552:
550:
549:
544:
513:
501:
489:
487:
486:
481:
479:
460:
458:
457:
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450:
438:
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435:
430:
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417:
404:
402:
401:
396:
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380:
367:
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359:
357:
356:
355:
346:
337:
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335:
323:
314:
295:
293:
292:
287:
285:
273:
271:
270:
265:
263:
262:
257:
245:
244:
239:
227:
226:
221:
193:
191:
190:
185:
183:
182:
177:
164:
162:
161:
156:
154:
153:
148:
136:
135:
130:
118:
117:
112:
100:
99:
88:
3602:
3601:
3597:
3596:
3595:
3593:
3592:
3591:
3587:Crystallography
3572:
3571:
3570:
3565:
3529:
3486:
3453:
3425:
3377:
3329:
3300:CrystalExplorer
3276:
3260:Phase retrieval
3223:
3154:
3153:
3144:
3101:
3080:Schottky defect
2979:Perfect crystal
2970:
2966:Abnormal growth
2928:
2914:Supersaturation
2877:Miscibility gap
2858:
2851:
2802:
2759:
2723:Bravais lattice
2704:
2671:
2669:Crystallography
2666:
2607:
2602:
2601:
2584:
2580:
2572:
2568:
2553:
2549:
2534:
2516:
2507:
2502:
2470:
2368:
2305:surface tension
2249:linking nodes (
2231:
2220:
2214:
2211:
2200:
2188:
2177:
2152:
2148:
2139:
2135:
2129:
2125:
2122:
2099:
2095:
2086:
2082:
2081:
2077:
2065:
2059:
2044:
2040:
2038:
2035:
2034:
2007:
2003:
1995:
1978:
1937:
1934:
1933:
1905:
1901:
1893:
1876:
1832:
1829:
1828:
1801:
1796:
1795:
1778:
1774:
1769:
1760:
1755:
1754:
1727:
1723:
1719:
1714:
1705:
1700:
1699:
1672:
1668:
1664:
1659:
1650:
1645:
1644:
1632:
1627:
1626:
1614:
1609:
1608:
1603:
1600:
1599:
1594:
1587:
1580:
1558:
1553:
1552:
1540:
1535:
1534:
1522:
1517:
1516:
1511:
1508:
1507:
1497:lattice vectors
1485:There are also
1463:
1456:
1452:
1448:
1390:lattice systems
1371:
1303:
1299:
1290:
1286:
1277:
1273:
1267:
1252:
1248:
1246:
1243:
1242:
1206:
1170:
1165:
1164:
1152:
1147:
1146:
1134:
1129:
1128:
1123:
1120:
1119:
1093:
1081:
1076:
1075:
1070:
1065:
1051:
1048:
1047:
1022:
1011:
1000:
990:Equivalently, (
953:
948:
947:
935:
930:
929:
917:
912:
911:
893:
888:
887:
885:
882:
881:
832:
829:
828:
823:
816:
809:
797:Bravais lattice
794:
787:
780:
760:
714:
711:
710:
675:
672:
671:
643:
640:
639:
601:
598:
597:
558:
555:
554:
526:
523:
522:
511:
499:
475:
470:
467:
466:
446:
444:
441:
440:
419:
418:
413:
412:
410:
407:
406:
382:
381:
376:
375:
373:
370:
369:
348:
347:
342:
341:
325:
324:
319:
318:
310:
305:
302:
301:
281:
279:
276:
275:
258:
253:
252:
240:
235:
234:
222:
217:
216:
211:
208:
207:
178:
173:
172:
170:
167:
166:
149:
144:
143:
131:
126:
125:
113:
108:
107:
89:
84:
83:
81:
78:
77:
40:crystallography
17:
12:
11:
5:
3600:
3590:
3589:
3584:
3567:
3566:
3564:
3563:
3551:
3538:
3535:
3534:
3531:
3530:
3528:
3527:
3522:
3517:
3516:
3515:
3510:
3505:
3494:
3492:
3485:
3484:
3479:
3474:
3469:
3463:
3461:
3455:
3454:
3452:
3451:
3446:
3441:
3435:
3433:
3427:
3426:
3424:
3423:
3418:
3413:
3408:
3403:
3398:
3393:
3387:
3385:
3379:
3378:
3376:
3375:
3370:
3365:
3360:
3355:
3350:
3345:
3339:
3337:
3331:
3330:
3328:
3327:
3322:
3317:
3312:
3307:
3302:
3297:
3292:
3286:
3284:
3278:
3277:
3275:
3274:
3269:
3268:
3267:
3257:
3252:
3247:
3242:
3237:
3235:Direct methods
3231:
3229:
3225:
3224:
3222:
3221:
3220:
3219:
3214:
3204:
3199:
3198:
3197:
3192:
3182:
3181:
3180:
3175:
3164:
3162:
3156:
3155:
3147:
3145:
3143:
3142:
3137:
3132:
3127:
3122:
3120:Ewald's sphere
3117:
3112:
3106:
3103:
3102:
3100:
3099:
3094:
3089:
3088:
3087:
3082:
3072:
3071:
3070:
3065:
3063:Frenkel defect
3060:
3058:Bjerrum defect
3050:
3049:
3048:
3038:
3037:
3036:
3031:
3026:
3024:Peierls stress
3021:
3016:
3011:
3006:
3001:
2996:
2994:Burgers vector
2986:
2984:Stacking fault
2981:
2975:
2972:
2971:
2969:
2968:
2963:
2958:
2953:
2947:
2945:
2943:Grain boundary
2936:
2930:
2929:
2927:
2926:
2921:
2916:
2911:
2906:
2901:
2896:
2891:
2890:
2889:
2887:Liquid crystal
2884:
2879:
2874:
2863:
2861:
2853:
2852:
2850:
2849:
2848:
2847:
2837:
2836:
2835:
2825:
2824:
2823:
2818:
2807:
2804:
2803:
2801:
2800:
2795:
2790:
2785:
2780:
2775:
2769:
2767:
2758:
2757:
2752:
2750:Periodic table
2747:
2746:
2745:
2740:
2735:
2730:
2725:
2714:
2712:
2703:
2702:
2697:
2692:
2691:
2690:
2679:
2677:
2673:
2672:
2665:
2664:
2657:
2650:
2642:
2636:
2635:
2629:
2624:
2618:
2613:
2606:
2605:External links
2603:
2600:
2599:
2578:
2566:
2547:
2532:
2504:
2503:
2501:
2498:
2497:
2496:
2491:
2486:
2481:
2476:
2469:
2466:
2458:Penrose tiling
2427:lattice planes
2367:
2364:
2360:
2359:
2358:
2357:
2350:
2347:Burgers vector
2343:
2322:
2321:
2320:
2315:
2302:
2292:
2233:
2232:
2191:
2189:
2182:
2176:
2173:
2172:
2171:
2155:
2151:
2142:
2138:
2132:
2128:
2121:
2117:
2113:
2110:
2107:
2102:
2098:
2094:
2089:
2085:
2080:
2073:
2070:
2063:
2058:
2053:
2050:
2047:
2043:
2015:
2010:
2006:
2002:
1998:
1994:
1991:
1988:
1985:
1981:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1913:
1908:
1904:
1900:
1896:
1892:
1889:
1886:
1883:
1879:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1821:
1820:
1809:
1804:
1799:
1794:
1791:
1788:
1781:
1777:
1773:
1768:
1763:
1758:
1753:
1750:
1747:
1744:
1741:
1738:
1730:
1726:
1722:
1718:
1713:
1708:
1703:
1698:
1695:
1692:
1689:
1686:
1683:
1675:
1671:
1667:
1663:
1658:
1653:
1648:
1643:
1640:
1635:
1630:
1625:
1622:
1617:
1612:
1607:
1592:
1585:
1578:
1561:
1556:
1551:
1548:
1543:
1538:
1533:
1530:
1525:
1520:
1515:
1483:
1482:
1426:
1425:
1394:Bravais–Miller
1370:
1367:
1351:
1350:
1343:curly brackets
1339:
1332:angle brackets
1324:
1323:
1306:
1302:
1298:
1293:
1289:
1285:
1280:
1276:
1271:
1266:
1261:
1258:
1255:
1251:
1230:, the spacing
1205:
1202:
1190:
1189:
1178:
1173:
1168:
1163:
1160:
1155:
1150:
1145:
1142:
1137:
1132:
1127:
1096:
1090:
1087:
1084:
1079:
1073:
1068:
1064:
1061:
1058:
1055:
1040:lattice planes
1020:
1009:
998:
973:
972:
961:
956:
951:
946:
943:
938:
933:
928:
925:
920:
915:
910:
907:
902:
899:
896:
891:
866:
863:
860:
857:
854:
851:
848:
845:
842:
839:
836:
821:
814:
807:
801:examples below
792:
785:
778:
759:
756:
737:
736:
724:
721:
718:
704:
703:
691:
688:
685:
682:
679:
659:
656:
653:
650:
647:
636:
620:
617:
614:
611:
608:
605:
587:
586:
574:
571:
568:
565:
562:
542:
539:
536:
533:
530:
492:Miller indices
478:
474:
463:Laue equations
449:
425:
422:
416:
391:
388:
385:
379:
354:
351:
345:
340:
334:
331:
328:
322:
317:
313:
309:
284:
261:
256:
251:
248:
243:
238:
233:
230:
225:
220:
215:
181:
176:
152:
147:
142:
139:
134:
129:
124:
121:
116:
111:
106:
103:
98:
95:
92:
87:
70:Miller indices
51:lattice planes
36:Miller indices
15:
9:
6:
4:
3:
2:
3599:
3588:
3585:
3583:
3580:
3579:
3577:
3562:
3561:
3552:
3550:
3549:
3540:
3539:
3536:
3526:
3523:
3521:
3518:
3514:
3511:
3509:
3506:
3504:
3501:
3500:
3499:
3496:
3495:
3493:
3489:
3483:
3480:
3478:
3475:
3473:
3470:
3468:
3465:
3464:
3462:
3460:
3456:
3450:
3447:
3445:
3442:
3440:
3437:
3436:
3434:
3432:
3428:
3422:
3419:
3417:
3414:
3412:
3409:
3407:
3404:
3402:
3399:
3397:
3394:
3392:
3389:
3388:
3386:
3384:
3380:
3374:
3371:
3369:
3366:
3364:
3361:
3359:
3356:
3354:
3351:
3349:
3346:
3344:
3341:
3340:
3338:
3336:
3332:
3326:
3323:
3321:
3318:
3316:
3313:
3311:
3308:
3306:
3303:
3301:
3298:
3296:
3293:
3291:
3288:
3287:
3285:
3283:
3279:
3273:
3270:
3266:
3263:
3262:
3261:
3258:
3256:
3255:Patterson map
3253:
3251:
3248:
3246:
3243:
3241:
3238:
3236:
3233:
3232:
3230:
3226:
3218:
3215:
3213:
3210:
3209:
3208:
3205:
3203:
3200:
3196:
3193:
3191:
3188:
3187:
3186:
3183:
3179:
3176:
3174:
3171:
3170:
3169:
3166:
3165:
3163:
3161:
3157:
3151:
3141:
3138:
3136:
3133:
3131:
3128:
3126:
3125:Friedel's law
3123:
3121:
3118:
3116:
3113:
3111:
3108:
3107:
3098:
3095:
3093:
3090:
3086:
3083:
3081:
3078:
3077:
3076:
3073:
3069:
3068:Wigner effect
3066:
3064:
3061:
3059:
3056:
3055:
3054:
3053:Interstitials
3051:
3047:
3044:
3043:
3042:
3039:
3035:
3032:
3030:
3027:
3025:
3022:
3020:
3017:
3015:
3012:
3010:
3007:
3005:
3002:
3000:
2997:
2995:
2992:
2991:
2990:
2987:
2985:
2982:
2980:
2977:
2976:
2967:
2964:
2962:
2959:
2957:
2954:
2952:
2949:
2948:
2946:
2944:
2940:
2937:
2935:
2931:
2925:
2922:
2920:
2917:
2915:
2912:
2910:
2907:
2905:
2902:
2900:
2899:Precipitation
2897:
2895:
2892:
2888:
2885:
2883:
2880:
2878:
2875:
2873:
2870:
2869:
2868:
2867:Phase diagram
2865:
2864:
2862:
2860:
2854:
2846:
2843:
2842:
2841:
2838:
2834:
2831:
2830:
2829:
2826:
2822:
2819:
2817:
2814:
2813:
2812:
2809:
2808:
2799:
2796:
2794:
2791:
2789:
2786:
2784:
2781:
2779:
2776:
2774:
2771:
2770:
2768:
2766:
2762:
2756:
2753:
2751:
2748:
2744:
2741:
2739:
2736:
2734:
2731:
2729:
2726:
2724:
2721:
2720:
2719:
2716:
2715:
2713:
2711:
2707:
2701:
2698:
2696:
2693:
2689:
2686:
2685:
2684:
2681:
2680:
2678:
2674:
2670:
2663:
2658:
2656:
2651:
2649:
2644:
2643:
2640:
2633:
2630:
2628:
2625:
2622:
2619:
2617:
2614:
2612:
2609:
2608:
2596:
2595:1-878907-35-2
2592:
2588:
2582:
2575:
2570:
2562:
2558:
2551:
2543:
2539:
2535:
2529:
2524:
2523:
2514:
2512:
2510:
2505:
2495:
2492:
2490:
2487:
2485:
2482:
2480:
2479:Crystal habit
2477:
2475:
2472:
2471:
2465:
2463:
2459:
2455:
2451:
2447:
2443:
2439:
2435:
2430:
2428:
2424:
2420:
2416:
2412:
2409:) by scaling
2408:
2404:
2400:
2396:
2392:
2387:
2385:
2381:
2377:
2373:
2363:
2355:
2351:
2348:
2344:
2341:
2337:
2333:
2332:
2330:
2326:
2323:
2319:
2316:
2313:
2309:
2308:
2306:
2303:
2300:
2296:
2293:
2291:
2290:birefringence
2287:
2283:
2279:
2275:
2272:
2271:
2270:
2268:
2264:
2260:
2256:
2252:
2248:
2239:
2229:
2226:
2218:
2208:
2204:
2198:
2197:
2192:This section
2190:
2186:
2181:
2180:
2153:
2149:
2140:
2136:
2130:
2126:
2119:
2115:
2111:
2108:
2105:
2100:
2096:
2092:
2087:
2083:
2078:
2071:
2068:
2061:
2056:
2051:
2048:
2045:
2041:
2033:
2032:
2031:
2028:
2008:
2000:
1996:
1992:
1983:
1979:
1975:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1945:
1942:
1931:
1927:
1906:
1898:
1894:
1890:
1881:
1877:
1873:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1840:
1837:
1826:
1807:
1802:
1789:
1779:
1775:
1771:
1766:
1761:
1748:
1745:
1742:
1739:
1728:
1724:
1720:
1716:
1711:
1706:
1693:
1690:
1687:
1684:
1673:
1669:
1665:
1661:
1656:
1651:
1641:
1638:
1633:
1623:
1620:
1615:
1605:
1598:
1597:
1596:
1591:
1584:
1577:
1559:
1549:
1546:
1541:
1531:
1528:
1523:
1513:
1505:
1500:
1498:
1494:
1490:
1489:
1480:
1476:
1473:
1472:
1471:
1469:
1460:
1445:
1443:
1439:
1435:
1431:
1423:
1419:
1415:
1412:
1411:
1410:
1408:
1405:
1402:
1399:
1395:
1391:
1388:
1384:
1375:
1366:
1364:
1360:
1356:
1348:
1344:
1340:
1337:
1333:
1329:
1328:
1327:
1304:
1300:
1296:
1291:
1287:
1283:
1278:
1274:
1269:
1264:
1259:
1256:
1253:
1249:
1241:
1240:
1239:
1237:
1233:
1229:
1226:
1221:
1219:
1215:
1211:
1201:
1199:
1195:
1176:
1171:
1161:
1158:
1153:
1143:
1140:
1135:
1125:
1118:
1117:
1116:
1114:
1109:
1088:
1085:
1082:
1066:
1062:
1059:
1056:
1053:
1045:
1041:
1037:
1032:
1030:
1026:
1019:
1015:
1008:
1004:
997:
993:
988:
986:
982:
978:
959:
954:
944:
941:
936:
926:
923:
918:
908:
905:
900:
897:
894:
880:
879:
878:
861:
858:
855:
849:
846:
843:
840:
837:
834:
825:
820:
813:
806:
802:
798:
791:
784:
777:
773:
764:
755:
752:
750:
746:
742:
722:
719:
716:
709:
708:
707:
686:
683:
680:
654:
651:
648:
637:
634:
618:
612:
609:
606:
596:
595:
594:
592:
569:
566:
563:
537:
534:
531:
521:the notation
520:
519:
518:
515:
509:
505:
497:
493:
490:is marked by
464:
338:
315:
299:
259:
249:
246:
241:
231:
228:
223:
213:
205:
201:
197:
179:
150:
140:
137:
132:
122:
119:
114:
104:
101:
96:
93:
90:
75:
71:
67:
63:
59:
56:
52:
47:
45:
41:
37:
29:
21:
3558:
3546:
3491:Associations
3459:Organisation
2951:Disclination
2882:Polymorphism
2845:Quasicrystal
2788:Orthorhombic
2728:Miller index
2727:
2676:Key concepts
2597:, Appendix 2
2586:
2581:
2569:
2560:
2550:
2521:
2484:Kikuchi line
2454:quasicrystal
2449:
2441:
2437:
2433:
2431:
2418:
2414:
2410:
2406:
2398:
2394:
2390:
2388:
2383:
2379:
2375:
2371:
2369:
2361:
2325:dislocations
2312:crystallites
2266:
2244:
2221:
2212:
2201:Please help
2196:verification
2193:
2029:
1929:
1926:four indices
1925:
1824:
1822:
1589:
1582:
1575:
1503:
1501:
1496:
1486:
1484:
1478:
1474:
1467:
1461:
1446:
1441:
1437:
1433:
1429:
1427:
1421:
1417:
1413:
1406:
1403:
1400:
1397:
1393:
1387:rhombohedral
1380:
1352:
1346:
1342:
1335:
1331:
1325:
1235:
1231:
1227:
1222:
1213:
1209:
1207:
1197:
1193:
1191:
1112:
1110:
1043:
1039:
1035:
1033:
1028:
1024:
1017:
1013:
1006:
1002:
995:
991:
989:
984:
976:
974:
826:
818:
811:
804:
789:
782:
775:
769:
753:
744:
738:
705:
702:by symmetry.
632:
590:
588:
516:
491:
73:
69:
65:
61:
57:
48:
35:
34:
3444:Ewald Prize
3212:Diffraction
3190:Diffraction
3173:Diffraction
3115:Bragg plane
3110:Bragg's law
2989:Dislocation
2904:Segregation
2816:Crystallite
2733:Point group
1341:Indices in
1330:Indices in
64:, and
3576:Categories
3228:Algorithms
3217:Scattering
3195:Scattering
3178:Scattering
3046:Slip bands
3009:Cross slip
2859:transition
2793:Tetragonal
2783:Monoclinic
2695:Metallurgy
2563:: 286–336.
2533:0030839939
2500:References
2462:hyperplane
2446:irrational
2310:Pores and
2299:reactivity
2295:adsorption
975:That is, (
758:Definition
591:directions
3335:Databases
2798:Triclinic
2778:Hexagonal
2718:Unit cell
2710:Structure
2494:Zone axis
2259:molecules
2215:July 2019
2150:ℓ
2052:ℓ
1970:ℓ
1961:−
1955:−
1868:ℓ
1790:ℓ
1642:ℓ
1550:ℓ
1451:0) and (1
1383:hexagonal
1363:supercell
1301:ℓ
1260:ℓ
1162:ℓ
1113:direction
1089:ℓ
1063:π
945:ℓ
901:ℓ
862:ℓ
847:ℓ
799:, as the
723:ℓ
687:ℓ
658:⟩
655:ℓ
646:⟨
613:ℓ
570:ℓ
538:ℓ
473:Δ
339:−
308:Δ
250:ℓ
141:ℓ
97:ℓ
3582:Geometry
3548:Category
3383:Journals
3315:OctaDist
3310:JANA2020
3282:Software
3168:Electron
3085:F-center
2872:Eutectic
2833:Fiveling
2828:Twinning
2821:Equiaxed
2468:See also
2403:rational
2336:friction
2318:cleavage
1029:inverses
985:shortest
194:are the
165:, where
55:integers
3560:Commons
3508:Germany
3185:Neutron
3075:Vacancy
2934:Defects
2919:GP-zone
2765:Systems
2354:polygon
1924:. When
1455:0) ≡ (1
202:of the
3503:France
3498:Europe
3431:Awards
2961:Growth
2811:Growth
2593:
2542:934604
2540:
2530:
2284:; the
2267:planes
2263:planes
1488:ad hoc
1347:braces
1336:family
1016:, and
817:, and
788:, and
633:direct
68:, the
3525:Japan
3472:IOBCr
3325:SHELX
3320:Olex2
3207:X-ray
2857:Phase
2773:Cubic
2444:have
2401:have
2278:light
2251:atoms
2247:lines
1428:Here
1381:With
981:basis
368:with
196:basis
3467:IUCr
3368:ICDD
3363:ICSD
3348:CCDC
3295:Coot
3290:CCP4
3041:Slip
3004:Kink
2591:ISBN
2538:OCLC
2528:ISBN
2440:and
2417:and
2397:and
2382:and
2297:and
2265:are
2255:ions
1588:and
1477:= 1/
1436:and
1424:= 0.
1385:and
1357:and
1353:For
3482:DMG
3477:RAS
3373:PDB
3358:COD
3353:CIF
3305:DSR
3029:GND
2956:CSL
2464:.)
2450:not
2407:hkℓ
2389:If
2372:abc
2257:or
2205:by
1930:hkℓ
1825:hkℓ
1595:as
1504:hkℓ
1345:or
1236:hkℓ
1214:hkℓ
1198:hkℓ
1194:not
1036:hkℓ
992:hkℓ
977:hkℓ
824:).
198:or
74:hkℓ
3578::
3520:US
3513:UK
2559:.
2536:.
2508:^
2436:,
2413:,
2393:,
2378:,
2331:)
2253:,
1581:,
1432:,
1420:+
1416:+
1220:.
1115::
1108:.
1005:,
810:,
781:,
300:,
60:,
46:.
2661:e
2654:t
2647:v
2623:.
2544:.
2442:c
2438:b
2434:a
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2415:b
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2356:.
2338:(
2327:(
2228:)
2222:(
2217:)
2213:(
2199:.
2154:2
2141:2
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2131:2
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2120:+
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2112:k
2109:h
2106:+
2101:2
2097:k
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2084:h
2079:(
2072:3
2069:4
2062:a
2057:=
2049:k
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2042:d
2014:]
2009:2
2005:)
2001:c
1997:/
1993:a
1990:(
1987:)
1984:2
1980:/
1976:3
1973:(
1967:,
1964:k
1958:h
1952:,
1949:k
1946:,
1943:h
1940:[
1912:]
1907:2
1903:)
1899:c
1895:/
1891:a
1888:(
1885:)
1882:2
1878:/
1874:3
1871:(
1865:,
1862:k
1859:2
1856:+
1853:h
1850:,
1847:k
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1841:h
1838:2
1835:[
1808:.
1803:3
1798:a
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1787:(
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1740:h
1737:(
1729:2
1725:a
1721:3
1717:2
1712:+
1707:1
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1697:)
1694:k
1691:+
1688:h
1685:2
1682:(
1674:2
1670:a
1666:3
1662:2
1657:=
1652:3
1647:b
1639:+
1634:2
1629:b
1624:k
1621:+
1616:1
1611:b
1606:h
1593:3
1590:a
1586:2
1583:a
1579:1
1576:a
1560:3
1555:b
1547:+
1542:2
1537:b
1532:k
1529:+
1524:1
1519:b
1514:h
1481:.
1479:S
1475:i
1468:S
1464:π
1457:2
1453:2
1449:2
1442:i
1438:ℓ
1434:k
1430:h
1422:i
1418:k
1414:h
1407:ℓ
1404:i
1401:k
1398:h
1322:.
1305:2
1297:+
1292:2
1288:k
1284:+
1279:2
1275:h
1270:a
1265:=
1257:k
1254:h
1250:d
1232:d
1228:a
1210:a
1177:.
1172:3
1167:a
1159:+
1154:2
1149:a
1144:k
1141:+
1136:1
1131:a
1126:h
1095:|
1086:k
1083:h
1078:g
1072:|
1067:/
1060:2
1057:=
1054:d
1044:d
1025:ℓ
1023:/
1021:3
1018:a
1014:k
1012:/
1010:2
1007:a
1003:h
1001:/
999:1
996:a
960:.
955:3
950:b
942:+
937:2
932:b
927:k
924:+
919:1
914:b
909:h
906:=
898:k
895:h
890:g
865:)
859:k
856:h
853:(
850:,
844:,
841:k
838:,
835:h
822:3
819:b
815:2
812:b
808:1
805:b
793:3
790:a
786:2
783:a
779:1
776:a
720:k
717:h
690:]
684:k
681:h
678:[
652:k
649:h
619:,
616:]
610:k
607:h
604:[
573:)
567:k
564:h
561:(
541:}
535:k
532:h
529:{
512:π
500:3
477:k
448:g
424:n
421:i
415:k
390:t
387:u
384:o
378:k
353:n
350:i
344:k
333:t
330:u
327:o
321:k
316:=
312:k
283:g
260:3
255:a
247:+
242:2
237:a
232:k
229:+
224:1
219:a
214:h
180:i
175:b
151:3
146:b
138:+
133:2
128:b
123:k
120:+
115:1
110:b
105:h
102:=
94:k
91:h
86:g
66:ℓ
62:k
58:h
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