2905:
addition to rotations axes and mirror planes, space group may contain more complex symmetry elements – screw axes (combination of rotation and translation) and glide planes (combination of mirror reflection and translation). As a result, many different space groups can correspond to the same point group. For example, choosing different lattice types and glide planes one can generate 28 different space groups from point group mmm, e.g. Pmmm, Pnnn, Pccm, Pban, Cmcm, Ibam, Fmmm, Fddd, and so on. In some cases, a space group is generated when translations are simply added to a point group. In other cases there is no point around which the point group applies. The notation is somewhat ambiguous, without a table giving more information. For example, space groups I23 and I2
4279:
2994:
2987:
2980:
1979:
31:
3001:
1986:
4672:
2191:, which means that the group consists of 3-fold axis, three perpendicular 2-fold axes, and 3 vertical diagonal planes passing between these 2-fold axes, so it seems that the group can be denoted as 32m or 3m2. However, one should remember that, unlike Schoenflies notation, the direction of a plane in a Hermann–Mauguin symbol is defined as the direction perpendicular to the plane, and in the
2973:
4684:
2904:
with symbols specifying the symmetry elements. The symmetry elements are ordered the same way as in the symbol of corresponding point group (the group that is obtained if one removes all translational components from the space group). The symbols for symmetry elements are more diverse, because in
2073:
tertiary directions. For example, in the picture of a regular hexagon one can distinguish two sets of mirror planes – three planes go through two opposite vertexes, and three other planes go through the centers of opposite sides. In this case any of two sets can be chosen as
2106:
m2, ... . For symbols of point groups this order usually doesn't matter; however, it will be important for
Hermann–Mauguin symbols of corresponding space groups, where secondary directions are directions of symmetry elements along unit cell translations
226:
If two or more axes have the same direction, the axis with higher symmetry is shown. Higher symmetry means that the axis generates a pattern with more points. For example, rotation axes 3, 4, 5, 6, 7, 8 generate 3-, 4-, 5-, 6-, 7-, 8-point patterns, respectively.
2785:). The Hermann–Mauguin symbols were not designed for non-crystallographic groups, so their symbols are rather nominal and based on similarity to symbols of the crystallographic groups of a cubic crystal system. Group
4549:
4544:
194:
Hermann–Mauguin symbols show non-equivalent axes and planes in a symmetrical fashion. The direction of a symmetry element corresponds to its position in the
Hermann–Mauguin symbol. If a rotation axis
255:
generate 6-, 4-, 10-, 6-, 14-, 8-point patterns, respectively. If a rotation and a rotoinversion axis generate the same number of points, the rotation axis should be chosen. For example, the
2909:
3 (nos. 197 and 199) both contain two-fold rotational axes as well as two-fold screw axes. In the first, the two-fold axes intersect the three-fold axes, whereas in the second they do not.
4600:
830:). Analogously, symbols of non-crystallographic groups (with axes of order 5, 7, 8, 9, ...) can be constructed. These groups can be arranged in the following table
3082:. The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 2
399:, and 6 all generate 6-point patterns, as we can see on the figure in the right, but the latter should be used because it is a rotation axis – the symbol will be
2441:
positions. This can be done if the rotation axis can be unambiguously obtained from the combination of symmetry elements presented in the symbol. For example, the
187:
is equivalent to a mirror plane and usually notated as m. The direction of the mirror plane is defined as the direction perpendicular to it (the direction of the
4418:
4605:
4330:
2321:. All of them contain four diagonal 3-fold axes. These axes are arranged as 3-fold axes in a cube, directed along its four space diagonals (the cube has
3674:
Vainshtein, Boris K., Modern
Crystallography 1: Fundamentals of Crystals. Symmetry, and Methods of Structural Crystallography, Springer. 1994, page 93.
4496:
3788:
2006:
directions, perpendicular to higher-order axis, are symmetrically equivalent. For example, in the picture of a triangle all three mirror planes (
4595:
4587:
4648:
4626:
4641:
4491:
4157:
4022:
3871:
4631:
4529:
4225:
3878:
4653:
4511:
4481:
4410:
3724:
3661:
3580:
4363:
3683:
3760:
4636:
4559:
4433:
4032:
4393:
4715:
4471:
4486:
4476:
3781:
3622:
2749:
141:
notations, that shows rotation-reflection axes. The rotoinversion axes are represented by the corresponding number with a
3460:
is given by the position of the symbol in the
Hermann–Mauguin designation, just as with mirror planes. They are noted by
4610:
3883:
3861:
433:
138:
4288:
4162:
3916:
3811:
511:
34:
Three point groups with their respective
Hermann–Mauguin notation, stereographic projections, and symmetry elements.
4278:
4688:
4519:
3816:
2993:
2986:
2979:
4534:
4463:
3921:
3911:
3216:
2027:) are equivalent – all of them pass through one vertex and the center of the opposite side. For even-order axes
751:
461:
4047:
4710:
4676:
4400:
4296:
4169:
4132:
3926:
3906:
3774:
3711:
3226:
827:
441:
4084:
2597:
m. In groups containing one higher-order axis, this higher-order axis cannot be omitted. For example, symbols
2139:
2m denote two different space groups. This also applies to symbols of space groups with odd-order axes 3 and
4524:
4368:
4313:
4062:
4027:
3696:
3221:
3170:
2201:
group all mirror planes are perpendicular to 2-fold axes, so they should be written in the same position as
675:
4220:
2155:
or between them. Space groups P321 and P312 are examples of the former and the latter cases, respectively.
4037:
90:
67:
4142:
4577:
4373:
4335:
4094:
3828:
2866:. The next number is the rotational symmetry, as given above. The presence of mirror planes are denoted
3000:
432:
These groups may contain only two-fold axes, mirror planes, and/or an inversion center. These are the
4301:
4174:
4010:
3901:
3231:
2891:
2782:
2767:
2257:
2178:
836:
134:
82:
4318:
4306:
4181:
4147:
4127:
421:
2233:
complexes generate an inversion center, which combining with the 3-fold rotation axis generates a
4567:
4378:
4323:
3866:
142:
3545:
two glides with the same glide plane and translation along two (different) half-lattice vectors.
3572:
1978:
4501:
4340:
4268:
4248:
3968:
3838:
3720:
3576:
228:
130:
4539:
4345:
4263:
4253:
4052:
3985:
3956:
3949:
3564:
2972:
2871:
2851:
47:
2119:, while the tertiary directions correspond to the direction between unit cell translations
560:
is mmm.) If the symbol contains three positions, then they denote symmetry elements in the
4428:
4423:
4388:
4208:
4107:
4042:
4005:
4000:
3851:
3797:
2918:
2901:
86:
30:
3653:
3480:
glide, which is along a quarter of either a face or space diagonal of the unit cell. The
3539:
glide planes with translation along a quarter of a face diagonal or of a space diagonal.
4238:
4203:
4191:
4186:
4152:
4122:
4112:
4071:
4015:
3939:
3893:
3211:
2854:
can be depicted using the
Hermann–Mauguin system. The first letter is either lowercase
2755:
Besides five cubic groups, there are two more non-crystallographic icosahedral groups (
4704:
4383:
4196:
3995:
3630:
3565:
2748:
m. The full and short symbols for all 32 crystallographic point groups are given in
4089:
4079:
3973:
3856:
3512:
is used. (In these cases, centering entails that both glides occur.) To summarize:
3205:). This enantiomorphism results in 11 pairs of enantiomorphic space groups, namely
1985:
63:
2383:
Third position – diagonal directions between any two of the three coordinate axes
198:
and a mirror plane m have the same direction, then they are denoted as a fraction
3744:
3740:
4572:
4243:
4117:
3944:
3457:
2897:
2245:
59:
55:
51:
321:
already contains the mirror plane m). Analogously, in the case when both 3 and
4137:
3823:
3057:
2879:
27:
Notation to represent symmetry in point groups, plane groups and space groups
3846:
2863:
2361:
First position – symmetrically equivalent directions of the coordinate axes
3472:
depending on which axis (direction) the glide is along. There is also the
2143:. The perpendicular symmetry elements can go along unit cell translations
17:
4443:
4213:
3961:
39:
3476:
glide, which is a glide along the half of a diagonal of a face, and the
4453:
3485:
3597:
3598:"Chapter 10.1. Crystallographic and noncrystallographic point groups"
3484:
glide is often called the diamond glide plane as it features in the
2789:
can be denoted as 235, 25, 532, 53. The possible short symbols for
4448:
3766:
2373:. They are equivalent due to the presence of diagonal 3-fold axes.
1984:
1977:
29:
3763:– An introduction into the Hermann-Maguin notation for beginners.
93:
elements, and it specifies the directions of the symmetry axes.
3770:
420:
Finally, the
Hermann–Mauguin symbol depends on the type of the
428:
Groups without higher-order axes (axes of order three or more)
3527:
glide translation along half the lattice vector of this face.
70:(who modified it in 1931). This notation is sometimes called
3488:
structure. In cases where there are two possibilities among
3106:
is a 120° (threefold) rotation followed by a translation of
2900:
is defined by combining the uppercase letter describing the
2357:
symmetry). These symbols are constructed the following way:
2002:
the third position in symbol is always absent, because all
3086:
is a 180° (twofold) rotation followed by a translation of
133:, Hermann–Mauguin symbols show rotoinversion axes, unlike
4550:
3563:
Sands, Donald E. (1993). "Crystal
Systems and Geometry".
4545:
66:(who introduced it in 1928) and the French mineralogist
2415:
All Hermann–Mauguin symbols presented above are called
4438:
3571:. Mineola, New York: Dover Publications, Inc. p.
105:– 1, 2, 3, 4, 5, 6, 7, 8, ... (angle of rotation
3710:
Donald Sands (1975). "Crystal Systems and Geometry".
2419:. For many groups they can be simplified by omitting
1992:
It can be noticed that in groups with odd-order axes
2940:– Face centered (from the German "Flächenzentriert")
4619:
4586:
4558:
4510:
4462:
4409:
4356:
4287:
4070:
4061:
3984:
3892:
3837:
3804:
2934:– Body centered (from the German "Innenzentriert")
651:These are the crystallographic groups 3, 32, 3m,
81:The Hermann–Mauguin notation, compared with the
62:. It is named after the German crystallographer
3782:
3533:glide translation along half a face diagonal.
589:direction, assigned to the higher-order axis.
8:
2244: = ∞ are called limit groups or
279:generates 6 points, and 3 generates only 3,
74:, because it was adopted as standard by the
3741:the symmetry operations for space group 197
3694:Shubnikov, A.V., Belov, N.V. & others,
2256:These are the crystallographic groups of a
596:directions, which are perpendicular to the
592:Second position – symmetrically equivalent
4616:
4067:
3889:
3834:
3789:
3775:
3767:
2725:mm), but not to mmm; the short symbol for
621:Third position – symmetrically equivalent
367:generate four points. In the case of the
89:because it can easily be used to include
3700:, Oxford: Pergamon Press. 1964, page 70.
3602:International Tables for Crystallography
3207:
2882:do not exist in two-dimensional spaces.
2812:m. The possible symbols for limit group
832:
76:International Tables For Crystallography
3555:
629:directions. These can be 2, m, or
101:Rotation axes are denoted by a number
2252:Groups with several higher-order axes
7:
4683:
4023:Phase transformation crystallography
3761:Decoding the Hermann-Maguin Notation
2177:may be confusing; the corresponding
329:should be written. However we write
4530:Journal of Chemical Crystallography
3040:Rhombohedral in hexagonal setting,
2862:to represent primitive or centered
78:since their first edition in 1935.
600:-axis. These can be 2, m, or
25:
3623:"(International Tables) Abstract"
3064:, where the angle of rotation is
2376:Second position – diagonal 3 or
2078:directions, the rest set will be
576:Groups with one higher-order axis
4682:
4671:
4670:
4277:
3684:Point groups in three dimensions
3664:from the original on 2012-04-15.
2999:
2992:
2985:
2978:
2971:
2892:List of space groups § list
3713:Introduction to Crystallography
3567:Introduction to Crystallography
2958:– Base centered on C faces only
2952:– Base centered on B faces only
2946:– Base centered on A faces only
2693:can be simplified to 4/mmm (or
2094:2m, ... can be written as
4472:Bilbao Crystallographic Server
3125:The possible screw axes are: 2
1:
2750:crystallographic point groups
434:crystallographic point groups
283:should be written instead of
271:combination is equivalent to
3342:
3236:
3006:
1658:
1604:
1532:
1370:
1337:
1293:
1168:
1117:
1078:
1008:
976:
931:
880:
834:
625:directions, passing between
383:combination, where 2, 3, 6,
4520:Crystal Growth & Design
3812:Timeline of crystallography
2921:types in three dimensions:
4732:
4331:Nuclear magnetic resonance
3654:"Семейства точечных групп"
3332:
3323:
3314:
3306:
3298:
3289:
3280:
3272:
3259:
3250:
3242:
2889:
2158:The symbol of point group
924:
4666:
4535:Journal of Crystal Growth
4275:
3745:those for space group 199
3225:
3220:
3215:
2998:
2991:
2984:
2977:
2970:
2082:directions. Hence groups
2055:secondary directions and
1556:
1513:
1333:
1294:
1060:
1018:
972:
932:
572:direction, respectively.
46:is used to represent the
4401:Single particle analysis
4259:Hermann–Mauguin notation
3102:of the lattice vector. 3
2770:) and two limit groups (
2395:. These can be 2, m, or
2131:. For example, symbols P
442:triclinic crystal system
44:Hermann–Mauguin notation
4525:Crystallography Reviews
4369:Isomorphous replacement
4163:Lomer–Cottrell junction
3596:Hahn, Th.; Klapper, H.
3122:of the lattice vector.
2423:-fold rotation axes in
676:trigonal crystal system
391:axes are present, axes
4038:Spinodal decomposition
3060:is noted by a number,
2816:are ∞∞ or 2∞, and for
1989:
1982:
514:). (The short form of
91:translational symmetry
72:international notation
68:Charles-Victor Mauguin
35:
4716:Chemical nomenclature
4578:Gregori Aminoff Prize
4374:Molecular replacement
3456:The orientation of a
2890:Further information:
1988:
1981:
363:, because both 4 and
33:
3884:Structure prediction
3171:enantiomorphic pairs
2783:Schoenflies notation
2768:Schoenflies notation
2258:cubic crystal system
2237:rotoinversion axis.
754:), and 6, 622, 6mm,
83:Schoenflies notation
4148:Cottrell atmosphere
4128:Partial dislocation
3872:Restriction theorem
183:, ... .
4568:Carl Hermann Medal
4379:Molecular dynamics
4226:Defects in diamond
4221:Stone–Wales defect
3867:Reciprocal lattice
3829:Biocrystallography
2709:mm) and 6/mmm (or
2179:Schoenflies symbol
1990:
1983:
325:axes are present,
131:improper rotations
85:, is preferred in
36:
4698:
4697:
4662:
4661:
4269:Thermal ellipsoid
4234:
4233:
4143:Frank–Read source
4103:
4102:
3969:Aperiodic crystal
3935:
3934:
3817:Crystallographers
3449:
3448:
3049:
3048:
2874:are only denoted
2872:glide reflections
1976:
1975:
581:First position –
229:Improper rotation
48:symmetry elements
16:(Redirected from
4723:
4686:
4685:
4674:
4673:
4617:
4540:Kristallografija
4394:Gerchberg–Saxton
4289:Characterisation
4281:
4264:Structure factor
4068:
4053:Ostwald ripening
3890:
3835:
3791:
3784:
3777:
3768:
3748:
3737:
3731:
3730:
3718:
3707:
3701:
3697:Colored Symmetry
3692:
3686:
3681:
3675:
3672:
3666:
3665:
3649:
3643:
3642:
3640:
3638:
3629:. Archived from
3619:
3613:
3612:
3610:
3608:
3593:
3587:
3586:
3570:
3560:
3208:
3121:
3119:
3118:
3115:
3112:
3101:
3099:
3098:
3095:
3092:
3081:
3079:
3078:
3073:
3070:
3032:Body centered,
3003:
2996:
2989:
2982:
2975:
2968:
2967:
2896:The symbol of a
2842:
2838:
2836:
2835:
2832:
2829:
2811:
2807:
2803:
2799:
2747:
2743:
2741:
2740:
2737:
2734:
2728:
2724:
2722:
2721:
2718:
2715:
2708:
2706:
2705:
2702:
2699:
2692:
2691:
2689:
2688:
2685:
2682:
2676:
2674:
2673:
2670:
2667:
2661:
2659:
2658:
2655:
2652:
2644:
2643:
2641:
2640:
2637:
2634:
2628:
2626:
2625:
2622:
2619:
2613:
2611:
2610:
2607:
2604:
2596:
2592:
2591:
2589:
2588:
2585:
2582:
2576:
2573:
2571:
2570:
2567:
2564:
2556:
2554:
2553:
2550:
2547:
2540:
2539:
2537:
2536:
2533:
2530:
2524:
2522:
2521:
2518:
2515:
2509:
2507:
2506:
2503:
2500:
2492:
2491:
2489:
2488:
2485:
2482:
2476:
2474:
2473:
2470:
2467:
2461:
2459:
2458:
2455:
2452:
2440:
2438:
2437:
2434:
2431:
2410:
2408:
2407:
2404:
2401:
2379:
2356:
2355:
2353:
2352:
2349:
2346:
2340:
2337:
2335:
2334:
2331:
2328:
2320:
2319:
2317:
2316:
2313:
2310:
2304:
2301:
2299:
2298:
2295:
2292:
2284:
2280:
2279:
2276:
2274:
2273:
2270:
2267:
2236:
2232:
2230:
2229:
2226:
2223:
2217:. Second, these
2216:
2214:
2213:
2210:
2207:
2176:
2174:
2173:
2170:
2167:
2161:
2142:
2138:
2134:
2105:
2101:
2097:
2093:
2089:
2085:
2072:
2070:
2069:
2066:
2063:
2054:
2052:
2051:
2048:
2045:
2035:
2000:
1972:
1971:
1969:
1968:
1965:
1962:
1956:
1954:
1953:
1950:
1947:
1941:
1939:
1938:
1935:
1932:
1921:
1920:
1918:
1917:
1914:
1911:
1905:
1903:
1902:
1899:
1896:
1890:
1888:
1887:
1884:
1881:
1870:
1869:
1867:
1866:
1863:
1860:
1854:
1852:
1851:
1848:
1845:
1839:
1837:
1836:
1833:
1830:
1819:
1818:
1816:
1815:
1812:
1809:
1803:
1801:
1800:
1797:
1794:
1788:
1786:
1785:
1782:
1779:
1768:
1767:
1765:
1764:
1761:
1758:
1752:
1750:
1749:
1746:
1743:
1737:
1735:
1734:
1731:
1728:
1717:
1716:
1714:
1713:
1710:
1707:
1701:
1699:
1698:
1695:
1692:
1686:
1684:
1683:
1680:
1677:
1650:
1637:
1624:
1622:
1621:
1618:
1615:
1600:
1587:
1575:
1567:
1561:
1552:
1550:
1549:
1546:
1543:
1528:
1526:
1525:
1522:
1519:
1509:
1507:
1506:
1503:
1500:
1493:
1485:
1483:
1482:
1479:
1476:
1470:
1463:
1461:
1460:
1457:
1454:
1448:
1441:
1439:
1438:
1435:
1432:
1426:
1419:
1417:
1416:
1413:
1410:
1404:
1399:
1397:
1396:
1393:
1390:
1384:
1290:
1288:
1287:
1284:
1281:
1271:
1269:
1268:
1265:
1262:
1252:
1250:
1249:
1246:
1243:
1233:
1231:
1230:
1227:
1224:
1214:
1212:
1211:
1208:
1205:
1195:
1193:
1192:
1189:
1186:
1161:
1150:
1137:
1135:
1134:
1131:
1128:
1114:
1103:
1092:
1075:
1073:
1072:
1069:
1066:
1056:
1049:
1042:
1035:
1028:
1023:
833:
825:
824:
822:
821:
818:
815:
809:
807:
806:
803:
800:
794:
792:
791:
788:
785:
777:
775:
774:
771:
768:
761:
757:
749:
748:
746:
745:
742:
739:
733:
731:
730:
727:
724:
718:
716:
715:
712:
709:
701:
699:
698:
695:
692:
685:
681:
678:), 4, 422, 4mm,
673:
671:
670:
667:
664:
658:
654:
647:
645:
643:
642:
639:
636:
618:
616:
614:
613:
610:
607:
559:
557:
556:
553:
550:
544:
542:
541:
538:
535:
529:
527:
526:
523:
520:
509:
507:
506:
503:
500:
494:
492:
491:
488:
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158:
154:
149:
128:
127:
125:
124:
119:
116:
21:
4731:
4730:
4726:
4725:
4724:
4722:
4721:
4720:
4711:Crystallography
4701:
4700:
4699:
4694:
4658:
4615:
4582:
4554:
4506:
4458:
4429:CrystalExplorer
4405:
4389:Phase retrieval
4352:
4283:
4282:
4273:
4230:
4209:Schottky defect
4108:Perfect crystal
4099:
4095:Abnormal growth
4057:
4043:Supersaturation
4006:Miscibility gap
3987:
3980:
3931:
3888:
3852:Bravais lattice
3833:
3800:
3798:Crystallography
3795:
3757:
3752:
3751:
3738:
3734:
3727:
3716:
3709:
3708:
3704:
3693:
3689:
3682:
3678:
3673:
3669:
3658:www.chem.msu.su
3651:
3650:
3646:
3636:
3634:
3621:
3620:
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3604:
3595:
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3583:
3562:
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3557:
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3454:
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3426:
3424:
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3409:
3408:
3401:
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3392:
3390:
3383:
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3375:
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3369:
3362:
3360:
3353:
3352:
3345:
3338:
3336:
3329:
3327:
3320:
3318:
3311:
3310:
3303:
3302:
3295:
3293:
3286:
3284:
3277:
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3267:
3263:
3256:
3254:
3247:
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3239:
3204:
3200:
3196:
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3188:
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3180:
3176:
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3152:
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3144:
3140:
3136:
3132:
3128:
3116:
3113:
3110:
3109:
3107:
3105:
3096:
3093:
3090:
3089:
3087:
3085:
3074:
3071:
3068:
3067:
3065:
3054:
3024:Face centered,
3016:Base centered,
2919:Bravais lattice
2915:
2908:
2894:
2888:
2849:
2840:
2833:
2830:
2827:
2826:
2824:
2822:
2809:
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2277:
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2234:
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2224:
2221:
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2218:
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2208:
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2202:
2200:
2190:
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2168:
2165:
2164:
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2159:
2140:
2136:
2132:
2103:
2099:
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2087:
2083:
2067:
2064:
2059:
2058:
2056:
2049:
2046:
2041:
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2033:
2026:
2019:
2012:
1998:
1966:
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1190:
1187:
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1176:
1159:
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1129:
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1123:
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1101:
1090:
1085:
1070:
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1063:
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1054:
1047:
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1019:
1016:
939:
887:
819:
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813:
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804:
801:
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795:
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578:
554:
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524:
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480:
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454:
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396:
392:
388:
384:
377:
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371:
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368:
364:
357:
354:
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349:
348:
346:
339:
336:
333:
332:
330:
326:
322:
318:
311:
308:
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303:
302:
300:
293:
290:
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286:
284:
280:
276:
272:
265:
262:
259:
258:
256:
252:
248:
244:
240:
236:
232:
220:
210:
207:
202:
201:
199:
188:
184:
180:
176:
172:
168:
164:
160:
156:
152:
147:
120:
117:
114:
113:
111:
106:
99:
87:crystallography
28:
23:
22:
15:
12:
11:
5:
4729:
4727:
4719:
4718:
4713:
4703:
4702:
4696:
4695:
4693:
4692:
4680:
4667:
4664:
4663:
4660:
4659:
4657:
4656:
4651:
4646:
4645:
4644:
4639:
4634:
4623:
4621:
4614:
4613:
4608:
4603:
4598:
4592:
4590:
4584:
4583:
4581:
4580:
4575:
4570:
4564:
4562:
4556:
4555:
4553:
4552:
4547:
4542:
4537:
4532:
4527:
4522:
4516:
4514:
4508:
4507:
4505:
4504:
4499:
4494:
4489:
4484:
4479:
4474:
4468:
4466:
4460:
4459:
4457:
4456:
4451:
4446:
4441:
4436:
4431:
4426:
4421:
4415:
4413:
4407:
4406:
4404:
4403:
4398:
4397:
4396:
4386:
4381:
4376:
4371:
4366:
4364:Direct methods
4360:
4358:
4354:
4353:
4351:
4350:
4349:
4348:
4343:
4333:
4328:
4327:
4326:
4321:
4311:
4310:
4309:
4304:
4293:
4291:
4285:
4284:
4276:
4274:
4272:
4271:
4266:
4261:
4256:
4251:
4249:Ewald's sphere
4246:
4241:
4235:
4232:
4231:
4229:
4228:
4223:
4218:
4217:
4216:
4211:
4201:
4200:
4199:
4194:
4192:Frenkel defect
4189:
4187:Bjerrum defect
4179:
4178:
4177:
4167:
4166:
4165:
4160:
4155:
4153:Peierls stress
4150:
4145:
4140:
4135:
4130:
4125:
4123:Burgers vector
4115:
4113:Stacking fault
4110:
4104:
4101:
4100:
4098:
4097:
4092:
4087:
4082:
4076:
4074:
4072:Grain boundary
4065:
4059:
4058:
4056:
4055:
4050:
4045:
4040:
4035:
4030:
4025:
4020:
4019:
4018:
4016:Liquid crystal
4013:
4008:
4003:
3992:
3990:
3982:
3981:
3979:
3978:
3977:
3976:
3966:
3965:
3964:
3954:
3953:
3952:
3947:
3936:
3933:
3932:
3930:
3929:
3924:
3919:
3914:
3909:
3904:
3898:
3896:
3887:
3886:
3881:
3879:Periodic table
3876:
3875:
3874:
3869:
3864:
3859:
3854:
3843:
3841:
3832:
3831:
3826:
3821:
3820:
3819:
3808:
3806:
3802:
3801:
3796:
3794:
3793:
3786:
3779:
3771:
3765:
3764:
3756:
3755:External links
3753:
3750:
3749:
3732:
3725:
3719:. p. 72.
3702:
3687:
3676:
3667:
3644:
3633:on 4 July 2013
3614:
3588:
3581:
3554:
3553:
3551:
3548:
3547:
3546:
3540:
3534:
3528:
3508:), the letter
3453:
3450:
3447:
3446:
3440:
3437:
3431:
3428:
3422:
3419:
3414:
3411:
3406:
3403:
3397:
3394:
3388:
3385:
3380:
3377:
3371:
3367:
3364:
3358:
3355:
3350:
3347:
3341:
3340:
3334:
3331:
3325:
3322:
3316:
3313:
3308:
3305:
3300:
3297:
3291:
3288:
3282:
3279:
3274:
3271:
3265:
3261:
3258:
3252:
3249:
3244:
3241:
3235:
3234:
3229:
3224:
3219:
3214:
3212:Crystal system
3202:
3198:
3194:
3190:
3186:
3182:
3178:
3174:
3169:. There are 4
3166:
3162:
3158:
3154:
3150:
3146:
3142:
3138:
3134:
3130:
3126:
3103:
3083:
3053:
3050:
3047:
3046:
3038:
3030:
3022:
3014:
3005:
3004:
2997:
2990:
2983:
2976:
2966:
2965:
2964:– Rhombohedral
2959:
2953:
2947:
2941:
2935:
2929:
2917:These are the
2914:
2911:
2906:
2887:
2884:
2848:
2845:
2820:
2793:
2778:
2763:
2413:
2412:
2381:
2374:
2253:
2250:
2195:
2185:
2024:
2017:
2010:
1974:
1973:
1924:
1922:
1873:
1871:
1822:
1820:
1771:
1769:
1720:
1718:
1667:
1661:
1657:
1656:
1654:
1652:
1645:
1643:
1641:
1639:
1633:
1631:
1629:
1627:
1607:
1603:
1602:
1595:
1593:
1591:
1589:
1583:
1581:
1579:
1577:
1571:
1569:
1555:
1535:
1531:
1530:
1512:
1510:
1488:
1486:
1466:
1464:
1444:
1442:
1422:
1420:
1400:
1378:
1373:
1369:
1368:
1365:
1363:
1360:
1358:
1355:
1353:
1350:
1348:
1345:
1343:
1336:
1335:
1332:
1330:
1327:
1325:
1322:
1320:
1317:
1315:
1312:
1310:
1307:
1301:
1296:
1292:
1291:
1274:
1272:
1255:
1253:
1236:
1234:
1217:
1215:
1198:
1196:
1177:
1171:
1167:
1166:
1164:
1162:
1157:
1155:
1153:
1151:
1146:
1144:
1142:
1140:
1120:
1116:
1115:
1110:
1108:
1106:
1104:
1099:
1097:
1095:
1093:
1088:
1086:
1081:
1077:
1076:
1059:
1057:
1052:
1050:
1045:
1043:
1038:
1036:
1031:
1029:
1024:
1017:
1011:
1007:
1006:
1003:
1001:
998:
996:
993:
991:
988:
986:
983:
981:
975:
974:
971:
969:
966:
964:
961:
959:
956:
954:
951:
949:
946:
940:
934:
930:
929:
926:
923:
920:
917:
914:
911:
908:
905:
902:
899:
896:
893:
888:
883:
879:
878:
875:
872:
869:
866:
863:
860:
857:
854:
851:
848:
845:
842:
839:
649:
648:
619:
590:
577:
574:
429:
426:
98:
95:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4728:
4717:
4714:
4712:
4709:
4708:
4706:
4691:
4690:
4681:
4679:
4678:
4669:
4668:
4665:
4655:
4652:
4650:
4647:
4643:
4640:
4638:
4635:
4633:
4630:
4629:
4628:
4625:
4624:
4622:
4618:
4612:
4609:
4607:
4604:
4602:
4599:
4597:
4594:
4593:
4591:
4589:
4585:
4579:
4576:
4574:
4571:
4569:
4566:
4565:
4563:
4561:
4557:
4551:
4548:
4546:
4543:
4541:
4538:
4536:
4533:
4531:
4528:
4526:
4523:
4521:
4518:
4517:
4515:
4513:
4509:
4503:
4500:
4498:
4495:
4493:
4490:
4488:
4485:
4483:
4480:
4478:
4475:
4473:
4470:
4469:
4467:
4465:
4461:
4455:
4452:
4450:
4447:
4445:
4442:
4440:
4437:
4435:
4432:
4430:
4427:
4425:
4422:
4420:
4417:
4416:
4414:
4412:
4408:
4402:
4399:
4395:
4392:
4391:
4390:
4387:
4385:
4384:Patterson map
4382:
4380:
4377:
4375:
4372:
4370:
4367:
4365:
4362:
4361:
4359:
4355:
4347:
4344:
4342:
4339:
4338:
4337:
4334:
4332:
4329:
4325:
4322:
4320:
4317:
4316:
4315:
4312:
4308:
4305:
4303:
4300:
4299:
4298:
4295:
4294:
4292:
4290:
4286:
4280:
4270:
4267:
4265:
4262:
4260:
4257:
4255:
4254:Friedel's law
4252:
4250:
4247:
4245:
4242:
4240:
4237:
4236:
4227:
4224:
4222:
4219:
4215:
4212:
4210:
4207:
4206:
4205:
4202:
4198:
4197:Wigner effect
4195:
4193:
4190:
4188:
4185:
4184:
4183:
4182:Interstitials
4180:
4176:
4173:
4172:
4171:
4168:
4164:
4161:
4159:
4156:
4154:
4151:
4149:
4146:
4144:
4141:
4139:
4136:
4134:
4131:
4129:
4126:
4124:
4121:
4120:
4119:
4116:
4114:
4111:
4109:
4106:
4105:
4096:
4093:
4091:
4088:
4086:
4083:
4081:
4078:
4077:
4075:
4073:
4069:
4066:
4064:
4060:
4054:
4051:
4049:
4046:
4044:
4041:
4039:
4036:
4034:
4031:
4029:
4028:Precipitation
4026:
4024:
4021:
4017:
4014:
4012:
4009:
4007:
4004:
4002:
3999:
3998:
3997:
3996:Phase diagram
3994:
3993:
3991:
3989:
3983:
3975:
3972:
3971:
3970:
3967:
3963:
3960:
3959:
3958:
3955:
3951:
3948:
3946:
3943:
3942:
3941:
3938:
3937:
3928:
3925:
3923:
3920:
3918:
3915:
3913:
3910:
3908:
3905:
3903:
3900:
3899:
3897:
3895:
3891:
3885:
3882:
3880:
3877:
3873:
3870:
3868:
3865:
3863:
3860:
3858:
3855:
3853:
3850:
3849:
3848:
3845:
3844:
3842:
3840:
3836:
3830:
3827:
3825:
3822:
3818:
3815:
3814:
3813:
3810:
3809:
3807:
3803:
3799:
3792:
3787:
3785:
3780:
3778:
3773:
3772:
3769:
3762:
3759:
3758:
3754:
3746:
3742:
3736:
3733:
3728:
3726:0-486-67839-3
3722:
3715:
3714:
3706:
3703:
3699:
3698:
3691:
3688:
3685:
3680:
3677:
3671:
3668:
3663:
3659:
3655:
3652:Zorky, Petr.
3648:
3645:
3632:
3628:
3624:
3618:
3615:
3603:
3599:
3592:
3589:
3584:
3582:0-486-67839-3
3578:
3574:
3569:
3568:
3559:
3556:
3549:
3544:
3541:
3538:
3535:
3532:
3529:
3526:
3522:
3518:
3515:
3514:
3513:
3511:
3507:
3503:
3499:
3495:
3491:
3487:
3483:
3479:
3475:
3471:
3467:
3463:
3459:
3451:
3438:
3429:
3420:
3412:
3404:
3395:
3386:
3378:
3365:
3356:
3348:
3346:Group Number
3343:
3240:Group Number
3237:
3233:
3230:
3228:
3223:
3218:
3213:
3210:
3209:
3206:
3172:
3123:
3077:
3063:
3059:
3051:
3045:
3044:
3039:
3037:
3036:
3031:
3029:
3028:
3023:
3021:
3020:
3015:
3013:
3012:
3007:
3002:
2995:
2988:
2981:
2974:
2969:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2930:
2927:
2924:
2923:
2922:
2920:
2913:Lattice types
2912:
2910:
2903:
2899:
2893:
2885:
2883:
2881:
2877:
2873:
2869:
2865:
2861:
2857:
2853:
2846:
2844:
2819:
2815:
2792:
2788:
2784:
2777:
2773:
2769:
2762:
2758:
2753:
2751:
2444:
2430:
2422:
2418:
2394:
2390:
2386:
2382:
2375:
2372:
2368:
2364:
2360:
2359:
2358:
2259:
2251:
2249:
2247:
2243:
2238:
2199:
2194:
2189:
2184:
2180:
2156:
2154:
2153:
2148:
2147:
2130:
2129:
2124:
2123:
2118:
2117:
2112:
2111:
2081:
2077:
2062:
2044:
2036:
2030:
2023:
2016:
2009:
2005:
2001:
1995:
1987:
1980:
1925:
1923:
1874:
1872:
1823:
1821:
1772:
1770:
1721:
1719:
1676:
1668:
1664:
1659:
1655:
1653:
1646:
1644:
1642:
1640:
1634:
1632:
1630:
1628:
1614:
1605:
1596:
1594:
1592:
1590:
1584:
1582:
1580:
1578:
1572:
1570:
1566:
1560:
1542:
1533:
1511:
1489:
1487:
1467:
1465:
1445:
1443:
1423:
1421:
1401:
1383:
1379:
1376:
1371:
1366:
1364:
1361:
1359:
1356:
1354:
1351:
1349:
1346:
1344:
1341:
1338:
1331:
1328:
1326:
1323:
1321:
1318:
1316:
1313:
1311:
1308:
1305:
1302:
1299:
1275:
1273:
1256:
1254:
1237:
1235:
1218:
1216:
1199:
1197:
1185:
1178:
1174:
1169:
1165:
1163:
1158:
1156:
1154:
1152:
1147:
1145:
1143:
1141:
1127:
1118:
1111:
1109:
1107:
1105:
1100:
1098:
1096:
1094:
1089:
1087:
1084:
1079:
1058:
1053:
1051:
1046:
1044:
1039:
1037:
1032:
1030:
1025:
1022:
1015:
1009:
1004:
1002:
999:
997:
994:
992:
989:
987:
984:
982:
980:
977:
970:
967:
965:
962:
960:
957:
955:
952:
950:
947:
944:
941:
937:
927:
921:
918:
915:
912:
909:
906:
903:
900:
897:
894:
892:
889:
886:
881:
876:
873:
870:
867:
864:
861:
858:
855:
852:
849:
846:
843:
840:
838:
835:
831:
829:
753:
677:
628:
624:
620:
599:
595:
591:
588:
584:
580:
579:
575:
573:
571:
567:
563:
513:
463:
444:), 2, m, and
443:
435:
427:
425:
423:
418:
230:
224:
219:
205:
197:
192:
150:
144:
140:
136:
132:
123:
109:
104:
96:
94:
92:
88:
84:
79:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
32:
19:
4687:
4675:
4620:Associations
4588:Organisation
4258:
4080:Disclination
4011:Polymorphism
3974:Quasicrystal
3917:Orthorhombic
3857:Miller index
3805:Key concepts
3735:
3712:
3705:
3695:
3690:
3679:
3670:
3657:
3647:
3635:. Retrieved
3631:the original
3626:
3617:
3605:. Retrieved
3601:
3591:
3566:
3558:
3542:
3536:
3530:
3524:
3520:
3516:
3509:
3505:
3501:
3497:
3493:
3489:
3481:
3477:
3473:
3469:
3465:
3461:
3455:
3452:Glide planes
3344:Second group
3124:
3075:
3061:
3055:
3042:
3041:
3034:
3033:
3026:
3025:
3018:
3017:
3010:
3009:
2961:
2955:
2949:
2943:
2937:
2931:
2925:
2916:
2902:lattice type
2895:
2886:Space groups
2875:
2867:
2859:
2855:
2852:Plane groups
2850:
2847:Plane groups
2817:
2813:
2790:
2786:
2775:
2771:
2760:
2756:
2754:
2557:mm, and for
2493:is mmm, for
2443:short symbol
2442:
2428:
2420:
2417:full symbols
2416:
2414:
2392:
2388:
2384:
2370:
2366:
2362:
2255:
2246:Curie groups
2241:
2240:Groups with
2239:
2197:
2192:
2187:
2182:
2157:
2151:
2150:
2145:
2144:
2127:
2126:
2121:
2120:
2115:
2114:
2109:
2108:
2079:
2075:
2060:
2042:
2032:
2028:
2021:
2014:
2007:
2003:
1997:
1993:
1991:
1674:
1662:
1612:
1564:
1558:
1540:
1381:
1374:
1339:
1303:
1297:
1183:
1172:
1125:
1082:
1020:
1013:
978:
942:
935:
890:
884:
650:
626:
622:
597:
593:
586:
585:direction –
582:
569:
565:
561:
512:orthorhombic
464:), and 222,
431:
419:
225:
217:
203:
195:
193:
146:
121:
107:
102:
100:
97:Point groups
80:
75:
71:
64:Carl Hermann
60:space groups
56:plane groups
52:point groups
43:
37:
4573:Ewald Prize
4341:Diffraction
4319:Diffraction
4302:Diffraction
4244:Bragg plane
4239:Bragg's law
4118:Dislocation
4033:Segregation
3945:Crystallite
3862:Point group
3627:it.iucr.org
3607:December 5,
3458:glide plane
3238:First group
3173:of axes: (3
3008:Primitive,
2928:– Primitive
2898:space group
2260:: 23, 432,
837:Schoenflies
510:, and mm2 (
135:Schoenflies
4705:Categories
4357:Algorithms
4346:Scattering
4324:Scattering
4307:Scattering
4175:Slip bands
4138:Cross slip
3988:transition
3922:Tetragonal
3912:Monoclinic
3824:Metallurgy
3637:2 February
3550:References
3217:Tetragonal
3058:screw axis
3052:Screw axes
2880:Screw axes
2864:unit cells
2037:there are
841:H–M symbol
752:tetragonal
462:monoclinic
317:, because
18:H–M Symbol
4464:Databases
3927:Triclinic
3907:Hexagonal
3847:Unit cell
3839:Structure
3500:(such as
3227:Hexagonal
3197:), and (6
2076:secondary
828:hexagonal
627:secondary
594:secondary
139:Shubnikov
4677:Category
4512:Journals
4444:OctaDist
4439:JANA2020
4411:Software
4297:Electron
4214:F-center
4001:Eutectic
3962:Fiveling
3957:Twinning
3950:Equiaxed
3739:Compare
3662:Archived
3222:Trigonal
2870:, while
2843:or ∞∞m.
2285:3m, and
2135:m2 and P
2080:tertiary
623:tertiary
275:. Since
216:or
40:geometry
4689:Commons
4637:Germany
4314:Neutron
4204:Vacancy
4063:Defects
4048:GP-zone
3894:Systems
3486:diamond
3165:, and 6
3120:
3108:
3100:
3088:
3080:
3066:
2837:
2825:
2742:
2730:
2723:
2711:
2707:
2695:
2690:
2678:
2675:
2663:
2660:
2648:
2642:
2630:
2627:
2615:
2612:
2600:
2590:
2578:
2572:
2560:
2555:
2543:
2538:
2526:
2523:
2511:
2508:
2496:
2490:
2478:
2475:
2463:
2460:
2448:
2439:
2425:
2409:
2397:
2354:
2342:
2336:
2324:
2318:
2306:
2300:
2288:
2275:
2263:
2231:
2219:
2215:
2203:
2175:
2163:
2071:
2057:
2053:
2039:
1970:
1958:
1955:
1943:
1940:
1928:
1919:
1907:
1904:
1892:
1889:
1877:
1868:
1856:
1853:
1841:
1838:
1826:
1817:
1805:
1802:
1790:
1787:
1775:
1766:
1754:
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1527:
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1496:
1484:
1472:
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1440:
1428:
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764:
747:
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705:
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672:
660:
644:
632:
615:
603:
583:primary
558:
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543:
531:
528:
516:
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481:
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214:
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191:axis).
129:). For
126:
112:
4632:France
4627:Europe
4560:Awards
4090:Growth
3940:Growth
3723:
3579:
3496:, and
2839:∞ or m
2752:page.
2391:, and
2369:, and
1362:(10)22
778:, and
702:, and
655:, and
436:1 and
387:, and
345:, not
143:macron
4654:Japan
4601:IOBCr
4454:SHELX
4449:Olex2
4336:X-ray
3986:Phase
3902:Cubic
3743:with
3717:(PDF)
3523:, or
3468:, or
3232:Cubic
3189:), (6
3181:), (4
2796:are m
2380:axes.
1562:2m =
1329:(11)2
1005:12mm
422:group
299:(not
231:axes
4596:IUCr
4497:ICDD
4492:ICSD
4477:CCDC
4424:Coot
4419:CCP4
4170:Slip
4133:Kink
3721:ISBN
3639:2022
3609:2022
3577:ISBN
3445:212
3436:181
3427:179
3418:172
3410:170
3402:153
3393:154
3384:145
3339:213
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3321:178
3312:171
3304:169
3296:151
3287:152
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3069:360°
3056:The
2823:are
2774:and
2759:and
2645:and
2593:is m
2445:for
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2125:and
2113:and
2102:m2,
2098:m2,
2090:2m,
2086:2m,
2031:and
1996:and
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1000:10mm
762:m2,
686:2m,
223:/m.
137:and
115:360°
58:and
4611:DMG
4606:RAS
4502:PDB
4487:COD
4482:CIF
4434:DSR
4158:GND
4085:CSL
3504:or
3434:22
3425:22
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3391:12
3376:96
3363:95
3361:22
3354:78
3328:22
3319:22
3294:21
3285:12
3270:92
3257:91
3255:22
3248:76
3201:– 6
3193:– 6
3185:– 4
3177:– 3
3161:, 6
3157:, 6
3153:, 6
3149:, 6
3145:, 4
3141:, 4
3137:, 4
3133:, 3
3129:, 3
2858:or
2808:m,
2804:, m
2800:, m
2781:in
2766:in
2744:is
2541:is
2181:is
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1357:822
1352:622
1347:422
1334:∞2
995:8mm
990:6mm
985:4mm
979:nmm
973:∞m
968:11m
925:...
874:...
50:in
38:In
4707::
4649:US
4642:UK
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3396:P3
3387:P3
3379:P3
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3349:P4
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1319:72
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948:3m
928:∞
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877:∞
871:12
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395:,
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3262:1
3253:1
3245:1
3203:4
3199:2
3195:5
3191:1
3187:3
3183:1
3179:2
3175:1
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3163:4
3159:3
3155:2
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3147:3
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3111:1
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