12
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All of them or only a subset?

This is a famous and fundamental result in solid state physics.

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  • $\begingroup$ If you don't mind I'll use this question as one of examples of good questions on a proposal for crystallography.SE $\endgroup$ – marcin Oct 16 '15 at 10:58
  • $\begingroup$ sure. go ahead. $\endgroup$ – Jiang-min Zhang Oct 17 '15 at 12:02
6
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According to Fizicheskaya Entsiklopediya (Physical Encyclopedia, in Russian, http://www.femto.com.ua/articles/part_2/3634.html ), no real crystals had been found for 4 space groups (Pcc2 and three others) as of the encyclopedia's publication in 1988-1999.

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  • $\begingroup$ Taking that at face value, we still have the problem that just because we haven't found it doesn't mean that such a crystal doesn't exist somewhere. It is hard to prove a negative, that it is impossible to place objects (molecules) of the required symmetry on the appropriate Bravais lattice to get those missing space group. $\endgroup$ – Jon Custer Jul 23 '14 at 13:30
  • $\begingroup$ @JonCuster Yes, but experimental results, and lack of them, are really the only way one can answer this question, aside from the citation of some physical principle which would otherwise rule out groups. $\endgroup$ – WetSavannaAnimal Sep 22 '17 at 5:48
2
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If there were only a few that had not been seen, I suspect that Ashcroft and Mermin would have gleefully pointed that out. However, that is not proof.

On the other hand, I see no way to prove that a given space group definitively could not exist in nature, given the essentially infinite number of molecules that could form crystals. We still spend lots of synchrotron beam time determining crystal structures of proteins - who is to say the next one would not have the magic unseen space group symmetry?

At best, one might say that, given the number of elements identified so far is less than 230, the STP equilibrium elemental crystals do not cover all of the space groups.

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  • $\begingroup$ The elements only cover 19 spacegroups (according to ElementData from Wolfram). You have to look at minerals to see a wide range of spacegroups represented. $\endgroup$ – alemi Jul 22 '14 at 22:37
  • $\begingroup$ @alemi - yup, lots of bcc and fcc metals in the elements. The huge number of minerals, not to mention organic molecules make up the rest. I would suspect that if there were a "unseen" space group, a good synthetic chemist could design and make something to have the right symmetry. $\endgroup$ – Jon Custer Jul 22 '14 at 22:45
2
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The first idea would be to do it ourselves but this is not so easy. Sure enough, one may come to the idea of exploring the Crystallographic Open Database (COD) as they provide a free access to their SQL database (or alternatively querying the Cambridge Structure Database (CSD) for those who have paid a subscription). So choose your preferred MySQL client and connect to host www.crystallography.net with user cod_reader, and issue the following SQL query:

select sg, count(*) from data group by sg

They do not provide the space group number unfortunately, but a bit of scripting (see the listing at the end of this answer) with the Computational Crystallographic ToolBox (cctbx) easily remedies that, and at the time of writing the answer, the only missing spacegroup is number 93. However the frequencies show that some are much rarer than others, as shown in the list at the end of this answer. One may seriously doubt whether the spacegroups appearing less than 0.5% of the time are actually mistakes or not. Here lies the problem. It is therefore necessary to painstakingly comb through the data and assess whether we have a genuine rare group or a mistake. No need to say that I don't quite have the time on my hands to do that!

But others have done precisely that. Urusov and Nadezhina [UN09} have analysed 100,444 mineral and inorganic structures from the CSD and they found 72 empty spacegroups. For organic compounds, the classic study is that of Mighell and Himes [MH83], which is pretty old by now but I haven't searched for a more up-to-date study. They studied about 30,000 organic structures in the NBS Crystal Data Identification File and they concluded that 35 spacegroups had no entry and 29 had only one entry.

[UN09] V. S. Urusov and T. N. Nadezhina. Frequency distribution and selection of space groups in inorganic crystal chemistry. Journal of Structural Chemistry, 50(1):22–37, Dec 2009.

[MH83] A. D. Mighell, V. L. Himes, and J. R. Rodgers. Space-group frequencies for organic compounds. Acta Crystallographica Section A, 39(5):737–740, Sep 1983.

Frequencies of spacegroups in the COD

  • 14: 114611 (P 1 21/c 1)
  • 2: 85079 (P -1)
  • 15: 30821 (C 1 2/c 1)
  • 19: 19734 (P 21 21 21)
  • 4: 14607 (P 1 21 1)
  • 61: 10839 (P b c a)
  • 62: 7879 (P n m a)
  • 33: 4886 (P n a 21)
  • 12: 4522 (C 1 2/m 1)
  • 225: 4398 (F m -3 m)
  • 227: 3830 (F d -3 m :2)
  • 9: 3660 (C 1 c 1)
  • 148: 3132 (R -3 :H)
  • 60: 3046 (P b c n)
  • 1: 2892 (P 1)
  • 5: 2815 (C 1 2 1)
  • 13: 2627 (P 1 2/c 1)
  • 29: 2574 (P c a 21)
  • 11: 2362 (P 1 21/m 1)
  • 194: 2212 (P 63/m m c)
  • 139: 1912 (I 4/m m m)
  • 221: 1773 (P m -3 m)
  • 166: 1754 (R -3 m :H)
  • 88: 1625 (I 41/a :2)
  • 167: 1578 (R -3 c :H)
  • 63: 1572 (C m c m)
  • 7: 1543 (P 1 c 1)
  • 18: 1285 (P 21 21 2)
  • 43: 1272 (F d d 2)
  • 176: 1246 (P 63/m)
  • 56: 1176 (P c c n)
  • 191: 1033 (P 6/m m m)
  • 123: 901 (P 4/m m m)
  • 36: 795 (C m c 21)
  • 229: 760 (I m -3 m)
  • 129: 742 (P 4/n m m :2)
  • 64: 738 (C m c a)
  • 164: 732 (P -3 m 1)
  • 205: 687 (P a -3)
  • 92: 680 (P 41 21 2)
  • 216: 671 (F -4 3 m)
  • 82: 650 (I -4)
  • 58: 638 (P n n m)
  • 160: 637 (R 3 m :H)
  • 96: 616 (P 43 21 2)
  • 141: 613 (I 41/a m d :2)
  • 87: 608 (I 4/m)
  • 70: 604 (F d d d :2)
  • 20: 604 (C 2 2 21)
  • 186: 595 (P 63 m c)
  • 147: 573 (P -3)
  • 146: 566 (R 3 :H)
  • 230: 563 (I a -3 d)
  • 161: 563 (R 3 c :H)
  • 136: 555 (P 42/m n m)
  • 140: 538 (I 4/m c m)
  • 173: 536 (P 63)
  • 55: 532 (P b a m)
  • 86: 515 (P 42/n :2)
  • 198: 494 (P 21 3)
  • 57: 486 (P b c m)
  • 52: 473 (P n n a)
  • 220: 462 (I -4 3 d)
  • 74: 432 (I m m a)
  • 71: 428 (I m m m)
  • 152: 424 (P 31 2 1)
  • 223: 422 (P m -3 n)
  • 122: 422 (I -4 2 d)
  • 114: 410 (P -4 21 c)
  • 85: 380 (P 4/n :2)
  • 142: 369 (I 41/a c d :2)
  • 31: 368 (P m n 21)
  • 41: 366 (A b a 2)
  • 47: 362 (P m m m)
  • 154: 349 (P 32 2 1)
  • 59: 348 (P m m n :2)
  • 189: 335 (P -6 2 m)
  • 130: 331 (P 4/n c c :2)
  • 165: 314 (P -3 c 1)
  • 127: 314 (P 4/m b m)
  • 204: 305 (I m -3)
  • 113: 301 (P -4 21 m)
  • 217: 295 (I -4 3 m)
  • 76: 294 (P 41)
  • 163: 285 (P -3 1 c)
  • 8: 283 (C 1 m 1)
  • 193: 281 (P 63/m c m)
  • 78: 273 (P 43)
  • 65: 266 (C m m m)
  • 155: 259 (R 3 2 :H)
  • 72: 256 (I b a m)
  • 144: 246 (P 31)
  • 206: 240 (I a -3)
  • 145: 230 (P 32)
  • 159: 224 (P 3 1 c)
  • 187: 220 (P -6 m 2)
  • 169: 213 (P 61)
  • 192: 210 (P 6/m c c)
  • 54: 210 (P c c a)
  • 170: 207 (P 65)
  • 45: 202 (I b a 2)
  • 218: 201 (P -4 3 n)
  • 150: 196 (P 3 2 1)
  • 121: 175 (I -4 2 m)
  • 68: 158 (C c c a :2)
  • 34: 156 (P n n 2)
  • 197: 149 (I 2 3)
  • 182: 149 (P 63 2 2)
  • 143: 146 (P 3)
  • 137: 144 (P 42/n m c :2)
  • 110: 143 (I 41 c d)
  • 69: 143 (F m m m)
  • 51: 134 (P m m a)
  • 128: 133 (P 4/m n c)
  • 66: 130 (C c c m)
  • 26: 130 (P m c 21)
  • 203: 126 (F d -3 :2)
  • 46: 126 (I m a 2)
  • 190: 123 (P -6 2 c)
  • 178: 119 (P 61 2 2)
  • 53: 119 (P m n a)
  • 174: 116 (P -6)
  • 99: 116 (P 4 m m)
  • 40: 116 (A m a 2)
  • 199: 114 (I 21 3)
  • 126: 114 (P 4/n n c :2)
  • 10: 114 (P 1 2/m 1)
  • 215: 113 (P -4 3 m)
  • 179: 111 (P 65 2 2)
  • 79: 111 (I 4)
  • 80: 107 (I 41)
  • 73: 107 (I b c a)
  • 44: 107 (I m m 2)
  • 226: 103 (F m -3 c)
  • 201: 95 (P n -3 :2)
  • 185: 94 (P 63 c m)
  • 138: 94 (P 42/n c m :2)
  • 81: 93 (P -4)
  • 38: 93 (A m m 2)
  • 180: 91 (P 62 2 2)
  • 131: 86 (P 42/m m c)
  • 23: 85 (I 2 2 2)
  • 118: 83 (P -4 n 2)
  • 135: 82 (P 42/m b c)
  • 162: 78 (P -3 1 m)
  • 107: 78 (I 4 m m)
  • 213: 75 (P 41 3 2)
  • 212: 70 (P 43 3 2)
  • 32: 70 (P b a 2)
  • 156: 68 (P 3 m 1)
  • 224: 67 (P n -3 m :2)
  • 100: 66 (P 4 b m)
  • 219: 64 (F -4 3 c)
  • 202: 64 (F m -3)
  • 67: 63 (C m m a)
  • 222: 60 (P n -3 n :2)
  • 157: 60 (P 3 1 m)
  • 120: 60 (I -4 c 2)
  • 98: 59 (I 41 2 2)
  • 50: 59 (P b a n :2)
  • 95: 57 (P 43 2 2)
  • 119: 56 (I -4 m 2)
  • 94: 56 (P 42 21 2)
  • 228: 54 (F d -3 c :2)
  • 124: 54 (P 4/m c c)
  • 84: 54 (P 42/m)
  • 42: 54 (F m m 2)
  • 200: 52 (P m -3)
  • 125: 52 (P 4/n b m :2)
  • 151: 51 (P 31 1 2)
  • 6: 50 (P 1 m 1)
  • 175: 47 (P 6/m)
  • 3: 47 (P 1 2 1)
  • 158: 46 (P 3 c 1)
  • 196: 45 (F 2 3)
  • 104: 44 (P 4 n c)
  • 91: 44 (P 41 2 2)
  • 37: 43 (C c c 2)
  • 211: 42 (I 4 3 2)
  • 90: 42 (P 4 21 2)
  • 30: 42 (P n c 2)
  • 181: 41 (P 64 2 2)
  • 188: 40 (P -6 c 2)
  • 83: 40 (P 4/m)
  • 21: 40 (C 2 2 2)
  • 106: 39 (P 42 b c)
  • 75: 37 (P 4)
  • 214: 36 (I 41 3 2)
  • 17: 36 (P 2 2 21)
  • 132: 34 (P 42/m c m)
  • 77: 34 (P 42)
  • 133: 32 (P 42/n b c :2)
  • 134: 31 (P 42/n n m :2)
  • 102: 31 (P 42 n m)
  • 25: 31 (P m m 2)
  • 210: 30 (F 41 3 2)
  • 117: 30 (P -4 b 2)
  • 112: 30 (P -4 2 c)
  • 108: 30 (I 4 c m)
  • 35: 30 (C m m 2)
  • 24: 30 (I 21 21 21)
  • 149: 28 (P 3 1 2)
  • 97: 27 (I 4 2 2)
  • 39: 27 (A b m 2)
  • 116: 26 (P -4 c 2)
  • 22: 25 (F 2 2 2)
  • 48: 22 (P n n n :2)
  • 28: 21 (P m a 2)
  • 115: 20 (P -4 m 2)
  • 109: 20 (I 41 m d)
  • 209: 19 (F 4 3 2)
  • 171: 17 (P 62)
  • 195: 15 (P 2 3)
  • 103: 15 (P 4 c c)
  • 172: 14 (P 64)
  • 111: 13 (P -4 2 m)
  • 49: 13 (P c c m)
  • 27: 13 (P c c 2)
  • 16: 12 (P 2 2 2)
  • 207: 11 (P 4 3 2)
  • 183: 11 (P 6 m m)
  • 153: 10 (P 32 1 2)
  • 208: 9 (P 42 3 2)
  • 184: 6 (P 6 c c)
  • 177: 6 (P 6 2 2)
  • 105: 6 (P 42 m c)
  • 89: 6 (P 4 2 2)
  • 168: 5 (P 6)
  • 101: 4 (P 42 c m)

CCTBX script to analyse the data

# Obtained from the COD and simply formatted as a Python dictionary
d={
'''NULL''':204,
'''?''':10,
'''???''':5,
'''????''':1,
'''?P?''':5,
'''(000 ???''':1,
'''A -1''':33,
'''A 1 1 2''':7,
'''A 1 1 2/a''':5,
'''A 1 1 2/m''':4,
'''A 1 1 m''':1,
'''A 1 2 1''':21,
'''A 1 2/a 1''':139,
'''A 1 2/m 1''':45,
'''A 1 2/n 1''':18,
'''A 1 a 1''':12,
'''A 1 n 1''':3,
'''A 2 1 1''':1,
'''A 2 a a''':2,
'''A 2 m m''':6,
'''A 2/m''':2,
'''A 21 2 2''':1,
'''A 21 a m''':64,
'''A 21 m a''':1,
'''A b a 2''':325,
'''A b m 2''':24,
'''A b m a''':23,
'''A b m m''':7,
'''A c 2 a''':1,
'''A c a m''':13,
'''A c m m''':4,
'''A e 2 a''':1,
'''A e a 2''':18,
'''A e a m''':1,
'''A e m a''':2,
'''A m 2 m''':1,
'''A m a 2''':99,
'''A m a 2 (b,-a,c)''':1,
'''A m a a''':9,
'''A m a m''':20,
'''A m m 2''':69,
'''A m m 2 (b,c,a)''':1,
'''A m m a''':18,
'''A m m m''':28,
'''A2/m''':1,
'''B -1''':8,
'''B 1''':1,
'''B 1 1 2''':5,
'''B 1 1 2/b''':30,
'''B 1 1 2/m''':23,
'''B 1 1 2/n''':3,
'''B 1 1 b''':9,
'''B 1 1 m''':11,
'''B 1 21 1''':26,
'''B 1 21/d 1''':4,
'''B 1 21/m 1''':1,
'''B 2 c b''':2,
'''B 2 m b''':2,
'''B 2 m m''':2,
'''B 21/c''':1,
'''B 21/d''':1,
'''B b 21 m''':27,
'''B b a 2''':3,
'''B b c b :2''':1,
'''B b c m''':8,
'''B b e m''':2,
'''B b m 2''':3,
'''B b m m''':53,
'''B m 2 m''':2,
'''B m a b''':59,
'''B m a m''':1,
'''B m m 2''':1,
'''B m m b''':12,
'''B2/m''':1,
'''Bmmm(0\b0)s00''':1,
'''C -1''':244,
'''C -4 2 b''':3,
'''C 1''':30,
'''C 1 1 2/a''':1,
'''C 1 1 21''':4,
'''C 1 1 21/d''':3,
'''C 1 2 1''':2636,
'''C 1 2 1 (a+2*c,a,b)''':6,
'''C 1 2/c 1''':29452,
'''C 1 2/c 1 (-a,a+c,b)''':1,
'''C 1 2/c 1 (-a,c,b)''':5,
'''C 1 2/c 1 (-a+c,c,b)''':2,
'''C 1 2/c 1 (c,a+c,b)''':1,
'''C 1 2/c 1 (c,b,-a)''':3,
'''C 1 2/c 1 (c,b,-a+c)''':19,
'''C 1 2/c 1 S''':1,
'''C 1 2/m 1''':4183,
'''C 1 2/m 1 (-a,c,b)''':4,
'''C 1 2/m 1 (a,b,a+2*c)''':4,
'''C 1 2/m 1 (a+c,b,c)''':21,
'''C 1 2/m 1 (c,b,-a)''':7,
'''C 1 2/n 1''':14,
'''C 1 21 1''':1,
'''C 1 c 1''':3522,
'''C 1 m 1''':248,
'''C 2 2 2''':40,
'''C 2 2 21''':603,
'''C 2 c b''':13,
'''C 2 c m''':11,
'''C 2 e b''':1,
'''C 2 m m''':6,
'''C 2/c''':1,
'''C 2/m 1 1''':1,
'''C 2/n 1 1''':1,
'''C 4 2 21''':1,
'''C c 2 a''':3,
'''C c c 2''':40,
'''C c c 2 (b,c,a)''':1,
'''C c c a''':6,
'''C c c a :2''':139,
'''C c c a :2 (-a,c,b)''':2,
'''C c c a (C c c e)''':2,
'''C c c b :1''':7,
'''C c c b :2''':1,
'''C c c e :1''':2,
'''C c c m''':119,
'''C c c m (b,c,a)''':2,
'''C c m 21''':25,
'''C c m b''':13,
'''C c m e''':1,
'''C c m m''':35,
'''C m 2 a''':3,
'''C m 2 m''':13,
'''C m c 21''':677,
'''C m c 21 (b,c,a)''':1,
'''C m c a''':560,
'''C m c a (-a,c,b)''':1,
'''C m c a (b,c,a)''':3,
'''C m c a (c,a,b)''':1,
'''C m c e''':51,
'''C m c m''':1420,
'''C m c m (-a,c,b)''':1,
'''C m c m (b,-a,c)''':5,
'''C m c m (b,c,a)''':4,
'''C m c m (c,a,b)''':2,
'''C m c m (c,b,-a)''':2,
'''C m m 2''':21,
'''C m m 2 (2*c,a,b)''':1,
'''C m m a''':47,
'''C m m a (b,c,a)''':2,
'''C m m a (c,a,b)''':1,
'''C m m b''':1,
'''C m m m''':218,
'''C m m m (c,a,b)''':20,
'''C-1''':1,
'''C-1(\a\b\g)0''':1,
'''C1(\a\b\g)0''':1,
'''C2:b1''':1,
'''C2/c(0\b0)s0''':2,
'''C2/m(\a0\g)-1s''':2,
'''C2/m(\a0\g)00''':2,
'''C2/m(\a0\g)0s''':5,
'''C2/m(0\b0)s0''':5,
'''Cc(\a0\g)0''':1,
'''Ccmm(\a00)00s''':1,
'''Cmca(00\g)s00''':3,
'''Cmcm(00\g)000''':1,
'''Cmmm(00\g)0s0''':1,
'''Cubic''':1,
'''F''':1,
'''F -1''':4,
'''F -4 3 c''':64,
'''F -4 3 m''':671,
'''F -4 d 2''':2,
'''F 1''':3,
'''F 1 -1 1''':1,
'''F 1 1 2''':1,
'''F 1 2/d 1''':2,
'''F 1 2/m 1''':3,
'''F 1 d 1''':6,
'''F 2 2 2''':25,
'''F 2 3''':45,
'''F 2 d d''':32,
'''F 2 m m''':1,
'''F 4 3 2''':19,
'''F 4/m m m''':4,
'''F 41 3 2''':30,
'''F 41/a d c''':1,
'''F d -3''':5,
'''F d -3 :1''':5,
'''F d -3 :2''':115,
'''F d -3 c''':10,
'''F d -3 c :2''':44,
'''F d -3 m''':88,
'''F d -3 m :1''':1136,
'''F d -3 m :2''':2604,
'''F d -3 m {origin @ -3 m}''':1,
'''F d -3 m {origin @ centre (-3m)}''':1,
'''F d 2 d''':9,
'''F d 3 m''':1,
'''F d d''':2,
'''F d d 2''':1229,
'''F d d 2 S''':2,
'''F d d d''':59,
'''F d d d :1''':44,
'''F d d d :2''':501,
'''F d d d {origin @ -1 @ d d d}''':2,
'''F d d d {origin @ 2 2 2}''':1,
'''F m -3''':64,
'''F m -3 c''':103,
'''F m -3 m''':4385,
'''F m -3m''':2,
'''F m 2 m''':5,
'''F m m 2''':46,
'''F m m 2 (b,c,a)''':2,
'''F m m m''':143,
'''F_d_d_2''':1,
'''F2(\a0\g)0''':4,
'''F2/m(\a0\g)0s''':4,
'''Fd-3m:1''':1,
'''Fd3''':1,
'''Fdd2''':1,
'''Fddd(00\g)ss0''':2,
'''Fm-3m''':2,
'''Fm3m''':9,
'''Fmmm(\a00)0s0''':3,
'''I''':1,
'''I -1''':54,
'''I -4''':649,
'''I -4 2 d''':421,
'''I -4 2 d S''':1,
'''I -4 2 m''':175,
'''I -4 3 d''':462,
'''I -4 3 m''':295,
'''I -4 c 2''':60,
'''I -4 m 2''':56,
'''I -4(\a\-b0,\b\a0)00''':1,
'''I 1''':1,
'''I 1 1 2''':1,
'''I 1 1 2/a''':3,
'''I 1 1 2/b''':13,
'''I 1 1 2/m''':4,
'''I 1 1 b''':2,
'''I 1 2 1''':139,
'''I 1 2/a 1''':955,
'''I 1 2/a 1 S''':1,
'''I 1 2/c 1''':141,
'''I 1 2/m 1''':207,
'''I 1 a 1''':103,
'''I 1 c 1''':9,
'''I 1 m 1''':23,
'''I 2 1 1 2''':1,
'''I 2 2 2''':85,
'''I 2 3''':149,
'''I 2 c b''':4,
'''I 2 c m''':6,
'''I 2 m b''':23,
'''I 2 m m''':4,
'''I 2/b 1 1''':2,
'''I 2/c 1 1''':12,
'''I 2/m''':11,
'''I 21 21 21''':30,
'''I 21 3''':113,
'''I 4''':111,
'''I 4 1/a (origin at -1)''':1,
'''I 4 2 2''':27,
'''I 4 3 2''':42,
'''I 4 c m''':30,
'''I 4 m m''':78,
'''I 4/m''':608,
'''I 4/m c m''':538,
'''I 4/m c m S''':1,
'''I 4/m m m''':1910,
'''I 4/m m m (a+b,-a+b,c)''':1,
'''I 41''':107,
'''I 41 2 2''':59,
'''I 41 3 2''':36,
'''I 41 c d''':143,
'''I 41 m d''':20,
'''I 41/a''':157,
'''I 41/a :1''':79,
'''I 41/a :2''':1389,
'''I 41/a (origin @ -1 on glide pla''':1,
'''I 41/a c d''':30,
'''I 41/a c d :1''':25,
'''I 41/a c d :2''':314,
'''I 41/a m d''':31,
'''I 41/a m d :1''':175,
'''I 41/a m d :2''':407,
'''I 41/a m d 1''':1,
'''I 41/a Z1''':1,
'''I 43''':1,
'''I a -3''':240,
'''I a -3 d''':562,
'''I b a 2''':197,
'''I b a m''':247,
'''I b a m (b,c,a)''':1,
'''I b a m S''':1,
'''I b c a''':106,
'''I b m 2''':10,
'''I b m m''':17,
'''I c 2 a''':1,
'''I c a b''':1,
'''I c m a''':1,
'''I c m a S''':1,
'''I c m m''':18,
'''I m -3''':305,
'''I m -3 m''':759,
'''I m 2 a''':4,
'''I m 2 m''':10,
'''I m a 2''':82,
'''I m a 2 (b,-a,c)''':1,
'''I m a m''':14,
'''I m c b''':7,
'''I m c m''':3,
'''I m m 2''':92,
'''I m m 2 (b,c,a)''':1,
'''I m m a''':374,
'''I m m a (-a,c,b)''':2,
'''I m m a (c,a,b)''':2,
'''I m m b''':2,
'''I m m m''':428,
'''I-4''':1,
'''I-42d''':1,
'''I2(1)3''':1,
'''I2/a''':1,
'''I2/b(\a\b0)00''':1,
'''I2/m''':5,
'''I4/m m m''':1,
'''Ia3d''':1,
'''Im-3m''':1,
'''Imma(00\g)s00''':2,
'''P -1''':85046,
'''P -1 (-a+b+c,a-b+c,a+b-c)''':2,
'''P -3''':573,
'''P -3 1 c''':285,
'''P -3 1 c S''':1,
'''P -3 1 m''':78,
'''P -3 c 1''':313,
'''P -3 m 1''':732,
'''P -4''':93,
'''P -4 2 c''':30,
'''P -4 2 m''':13,
'''P -4 21 c''':410,
'''P -4 21 m''':301,
'''P -4 21/c''':1,
'''P -4 3 m''':113,
'''P -4 3 n''':201,
'''P -4 b 2''':30,
'''P -4 c 2''':26,
'''P -4 m 2''':20,
'''P -4 n 2''':83,
'''P -6''':116,
'''P -6 2 c''':123,
'''P -6 2 m''':335,
'''P -6 c 2''':40,
'''P -6 m 2''':220,
'''P 1''':2888,
'''P 1 (-a+c,-b,a+c)''':1,
'''P 1 (b+c,a+c,a+b)''':2,
'''P 1 1 2''':5,
'''P 1 1 2/a''':1,
'''P 1 1 2/b''':5,
'''P 1 1 2/m''':7,
'''P 1 1 2/n''':3,
'''P 1 1 21''':56,
'''P 1 1 21/a''':33,
'''P 1 1 21/b''':75,
'''P 1 1 21/m''':30,
'''P 1 1 21/n''':44,
'''P 1 1 a''':1,
'''P 1 1 b''':2,
'''P 1 1 m''':6,
'''P 1 2 1''':41,
'''P 1 2/a 1''':98,
'''P 1 2/c 1''':1599,
'''P 1 2/c 1 (a,2*b,c)''':1,
'''P 1 2/m 1''':106,
'''P 1 2/n 1''':917,
'''P 1 21 1''':14543,
'''P 1 21 1 (a-1/4,b,c)''':1,
'''P 1 21 1 (c,2*a+c,b)''':1,
'''P 1 21/a 1''':2929,
'''P 1 21/c 1''':65464,
'''P 1 21/c 1 (2*a+c,b,c)''':3,
'''P 1 21/c 1 (2*c,2*a+c,b)''':14,
'''P 1 21/m 1''':2330,
'''P 1 21/m 1 S''':1,
'''P 1 21/n 1''':45963,
'''P 1 a 1''':25,
'''P 1 c 1''':889,
'''P 1 m 1''':41,
'''P 1 n 1''':624,
'''P 2 1 1''':1,
'''P 2 2 2''':12,
'''P 2 2 21''':36,
'''P 2 21 21''':62,
'''P 2 3''':15,
'''P 2 a n''':5,
'''P 2 c b''':3,
'''P 2 c m''':2,
'''P 2 m m''':8,
'''P 2 n n''':6,
'''P 2/c 1 1''':1,
'''P 2/m 1 1''':1,
'''P 2/n''':1,
'''P 21''':1,
'''P 21 1 1''':5,
'''P 21 2 21''':14,
'''P 21 21 2''':1209,
'''P 21 21 21''':19726,
'''P 21 21 21 (origin shift x,y,z+1''':1,
'''P 21 3''':493,
'''P 21 a b''':13,
'''P 21 a m''':8,
'''P 21 c a''':30,
'''P 21 c n''':79,
'''P 21 m a''':16,
'''P 21 m n''':8,
'''P 21 n b''':58,
'''P 21 n m''':29,
'''P 21/b''':8,
'''P 21/b 1 1''':60,
'''P 21/c''':3,
'''P 21/c 1 1''':3,
'''P 21/m 1 1''':2,
'''P 21/N''':9,
'''P 21/n 1 1''':8,
'''P 3''':146,
'''P 3 1 2''':28,
'''P 3 1 c''':224,
'''P 3 1 m''':60,
'''P 3 2 1''':196,
'''P 3 c 1''':45,
'''P 3 m 1''':68,
'''P 31''':246,
'''P 31 1 2''':51,
'''P 31 2 1''':424,
'''P 31 2 1 S''':1,
'''P 32''':230,
'''P 32 1 2''':10,
'''P 32 2 1''':349,
'''P 32 2 1 S''':3,
'''P 3c1''':1,
'''P 4''':37,
'''P 4 2 2''':6,
'''P 4 21 2''':42,
'''P 4 3 2''':11,
'''P 4 b m''':64,
'''P 4 b m (a,b,2*c)''':2,
'''P 4 c c''':15,
'''P 4 m m''':115,
'''P 4 n c''':44,
'''P 4/m''':40,
'''P 4/m b m''':312,
'''P 4/m c c''':54,
'''P 4/m m m''':901,
'''P 4/m n c''':133,
'''P 4/mbm''':2,
'''P 4/n''':45,
'''P 4/n :1''':18,
'''P 4/n :2''':317,
'''P 4/n b m''':5,
'''P 4/n b m :1''':11,
'''P 4/n b m :2''':36,
'''P 4/n c c''':31,
'''P 4/n c c :1''':24,
'''P 4/n c c :2''':276,
'''P 4/n m m''':21,
'''P 4/n m m :1''':285,
'''P 4/n m m :2''':436,
'''P 4/n n c''':20,
'''P 4/n n c :1''':5,
'''P 4/n n c :2''':89,
'''P 4/n n c {origin @ -1 @ n n (n,''':2,
'''P 4/n n c Z1''':1,
'''P 41''':294,
'''P 41 2 2''':44,
'''P 41 21 2''':680,
'''P 41 3 2''':75,
'''P 42''':34,
'''P 42 21 2''':56,
'''P 42 3 2''':9,
'''P 42 b c''':39,
'''P 42 c m''':4,
'''P 42 m c''':6,
'''P 42 n m''':31,
'''P 42/m''':54,
'''P 42/m b c''':82,
'''P 42/m c m''':34,
'''P 42/m m c''':86,
'''P 42/m m c S''':1,
'''P 42/m n m''':554,
'''P 42/mnm''':1,
'''P 42/n''':47,
'''P 42/n :1''':28,
'''P 42/n :2''':440,
'''P 42/n b c''':6,
'''P 42/n b c :1''':1,
'''P 42/n b c :2''':25,
'''P 42/n c m''':9,
'''P 42/n c m :1''':10,
'''P 42/n c m :2''':75,
'''P 42/n m c''':13,
'''P 42/n m c :1''':44,
'''P 42/n m c :2''':85,
'''P 42/n m c S''':2,
'''P 42/n n m''':5,
'''P 42/n n m :1''':9,
'''P 42/n n m :2''':17,
'''P 43''':273,
'''P 43 2 2''':57,
'''P 43 21 2''':616,
'''P 43 3 2''':70,
'''P 4mm''':1,
'''P 6''':5,
'''P 6 2 2''':6,
'''P 6 c c''':6,
'''P 6 m m''':11,
'''P 6/m''':47,
'''P 6/m c c''':210,
'''P 6/m c c S''':1,
'''P 6/m m m''':1033,
'''P 61''':213,
'''P 61 2 2''':119,
'''P 62''':17,
'''P 62 2 2''':91,
'''P 63''':533,
'''P 63 2 2''':149,
'''P 63 c m''':94,
'''P 63 m c''':595,
'''P 63 m c S''':1,
'''P 63/m''':1242,
'''P 63/m c m''':281,
'''P 63/m m c''':2198,
'''P 64''':14,
'''P 64 2 2''':41,
'''P 65''':207,
'''P 65 2 2''':111,
'''P a -3''':686,
'''P b -3''':1,
'''P b 2 n''':1,
'''P b 21 a''':12,
'''P b 21 m''':8,
'''P b a 2''':67,
'''P b a a''':3,
'''P b a m''':521,
'''P b a n''':10,
'''P b a n :1''':4,
'''P b a n :2''':42,
'''P b c 21''':91,
'''P b c a''':10582,
'''P b c b''':4,
'''P b c m''':440,
'''P b c n''':2826,
'''P b m 2''':3,
'''P b m a''':19,
'''P b m m''':12,
'''P b m n''':19,
'''P b n 21''':55,
'''P b n a''':58,
'''P b n b''':11,
'''P b n m''':1193,
'''P b n m S''':1,
'''P b n n''':11,
'''P c 21 b''':37,
'''P c 21 n''':89,
'''P c a 2 1''':1,
'''P c a 21''':2388,
'''P c a a''':1,
'''P c a b''':256,
'''P c a m''':11,
'''P c a n''':40,
'''P c c 2''':13,
'''P c c a''':201,
'''P c c b''':1,
'''P c c m''':13,
'''P c c n''':1133,
'''P c m 21''':4,
'''P c m a''':3,
'''P c m b''':9,
'''P c m n''':80,
'''P c n 2''':1,
'''P c n b''':27,
'''P c n m''':1,
'''P c n n''':12,
'''P m -3''':52,
'''P m -3 m''':1770,
'''P m -3 n''':422,
'''P m 1 1''':3,
'''P m 2 a''':3,
'''P m 2 m''':4,
'''P m 21 b''':3,
'''P m 21 n''':14,
'''P m 3 m''':1,
'''P m a 2''':13,
'''P m a b''':5,
'''P m a m''':9,
'''P m a n''':10,
'''P m c 21''':91,
'''P m c a''':2,
'''P m c b''':7,
'''P m c m''':4,
'''P m c n''':258,
'''P m c n S1''':1,
'''P m c n S2''':1,
'''P m m 2''':19,
'''P m m a''':99,
'''P m m a (2*b,c,a)''':1,
'''P m m a (2*b+1/4,c,a-1/3)''':4,
'''P m m b''':5,
'''P m m m''':361,
'''P m m m (2*a,2*b,c)''':1,
'''P m m n''':21,
'''P m m n :1''':48,
'''P m m n :2''':228,
'''P m n 21''':291,
'''P m n 21 S''':1,
'''P m n a''':74,
'''P m n b''':75,
'''P m n b S''':3,
'''P m n m :1''':8,
'''P m n m :2''':20,
'''P m n n''':27,
'''P n -3''':4,
'''P n -3 :1''':29,
'''P n -3 :2''':62,
'''P n -3 m :1''':22,
'''P n -3 m :2''':45,
'''P n -3 n''':6,
'''P n -3 n :1''':1,
'''P n -3 n :2''':53,
'''P n 1 1''':2,
'''P n 2 n''':3,
'''P n 21 a''':119,
'''P n 21 m''':15,
'''P n a 21''':4484,
'''P n a a''':32,
'''P n a b''':28,
'''P n a m''':344,
'''P n a n''':16,
'''P n c 2''':35,
'''P n c a''':67,
'''P n c b''':1,
'''P n c b :1''':2,
'''P n c m''':14,
'''P n c n''':20,
'''P n m 21''':11,
'''P n m a''':5925,
'''P n m a (c,a-1/4,b)''':3,
'''P n m b''':1,
'''P n m m''':1,
'''P n m m :1''':1,
'''P n m m :2''':21,
'''P n m n''':25,
'''P n n 2''':147,
'''P n n a''':412,
'''P n n b''':2,
'''P n n m''':582,
'''P n n n''':6,
'''P n n n :1''':1,
'''P n n n :2''':15,
'''P_-1''':1,
'''P_1_21/c_1''':1,
'''p_21_c_a''':1,
'''p_21/a''':1,
'''P_63''':3,
'''P_63/m''':3,
'''P-1''':30,
'''P-1(\a\b\g)0''':1,
'''P-3c1''':1,
'''P-4-h(10)g(2)dcba''':1,
'''P-42_1m(a,a,0)00s(-a,a,0)000''':1,
'''P1''':1,
'''P112/n''':1,
'''P2_1/n''':1,
'''P2(1)2(1)2(1)''':2,
'''P2/m(\a1/2\g)00''':1,
'''P21 21 21''':1,
'''P21 or P21/m''':1,
'''P21.a''':1,
'''P21(\a0\g)0''':2,
'''P21/c(\a0\g)00''':1,
'''P21/c(0\b0)s0''':1,
'''P21/m(\a0\g)00''':4,
'''P21/m(\a0\g)0s''':1,
'''P212121''':5,
'''P213''':1,
'''P3c&''':1,
'''P42 mnm''':1,
'''P43-n''':1,
'''P63/m''':1,
'''P63/mmc''':14,
'''Pa3''':1,
'''Pbam''':1,
'''Pbc2''':2,
'''Pbca''':1,
'''Pca21''':1,
'''Pm-3m''':1,
'''Pm21n(\a00)000''':1,
'''Pm3m''':1,
'''Pmcn(00\g)ss0''':2,
'''Pmmm(\a,1/2,0)000(1/2,\b,0)000''':1,
'''Pmn2''':3,
'''Pna2(1)''':1,
'''Pna21''':1,
'''Pnma''':1,
'''Pnnm''':4,
'''R -3''':193,
'''R -3 :H''':2761,
'''R -3 :R''':176,
'''R -3 c''':94,
'''R -3 c :H''':1416,
'''R -3 c :R''':66,
'''R -3 c RS''':7,
'''R -3 cr {rhombohedral axes}''':1,
'''R -3 HR''':1,
'''R -3 m''':52,
'''R -3 m :H''':1530,
'''R -3 m :R''':172,
'''R -3 m HR''':1,
'''R -3:H''':1,
'''R -3:r''':1,
'''R 1 2/c 1''':1,
'''R 3''':52,
'''R 3 :H''':481,
'''R 3 :R''':32,
'''R 3 2''':24,
'''R 3 2 :H''':221,
'''R 3 2 :R''':13,
'''R 3 c''':19,
'''R 3 c :H''':527,
'''R 3 c :R''':17,
'''R 3 m''':15,
'''R 3 m :H''':580,
'''R 3 m :R''':42,
'''R 3:r''':1,
'''R 32''':1,
'''R-3(00\g)0''':13,
'''r-3ch''':2,
'''R-3m(00\g)''':1,
'''R-3m(00\g)00''':3,
'''R-3m(00\g)0s''':1,
'''R???''':1,
'''R3(00\g)t''':2,
'''R31(00\g)ts''':8,
'''tetragonal''':1,
'''unknown''':1,
'''X c''':1,
'''X-3(00\g)0''':1,
'''X-3c1(00\g)000''':1,
'''X2/m''':2,
'''X2/m(\a0\g)0s''':13,
'''X21(000)0''':1,
'''X21(1/21/2\g)0''':4,
'''X3(00\g)0''':1,
'''X4bm''':8,
'''Xmc21''':1,
'''Xmcm(0\b0)s0s''':1,
'''Xmcm(00\g)000''':3,
'''Xmmm(\a00)000''':1,
'''Xmmm(00\g)0s0''':2,
'''Xmnm(\a00)00s''':1,
}

from cctbx import sgtbx
n = {}
for hm, count in d.iteritems():
  try:
    sgi = sgtbx.space_group_info(hm)
    nb = sgi.type().number()
  except Exception, err:
    nb = 0
  n.setdefault(nb, 0)
  n[nb] += count
print "missing: {}".format(set(xrange(1,231))-set(n.keys()))
print "buggy: {}".format(n[0])
for count, nb in sorted(((count, nb) for nb, count in n.iteritems()
                         if nb != 0), 
                        reverse=True):
  sgi = sgtbx.space_group_info(number=nb)
  print "- {:3d}: {:>7} ({})".format(
    nb, count, sgi.type().universal_hermann_mauguin_symbol())
$\endgroup$
1
$\begingroup$

Actually for all 230 of them, according to this Q/A on Earth Science.SE , which refers to a nice table given at this blog post.

$\endgroup$
  • $\begingroup$ Do you know anything about that blog, or can you give a reproduction of the same information in the peer reviewed literature? That's a really significant answer if true. $\endgroup$ – WetSavannaAnimal Sep 22 '17 at 5:52
  • $\begingroup$ I don't know the blogger personally. But the info looks trustworthy. Firstly, that blog post has a link to a peer-reviewed journal for each space group. I haven't had a look into every linked journal article, so I can't say the blogger hasn't fabricated fake entries, but this is almost as close to having a peer-reviewed literature concerning this question. Secondly, for the entry #89 and #93, the blogger has a detailed comment, actually correcting the data in the peer-reviewed literature. So I am almost certain that the blogger is herself/himself a crystallographer. $\endgroup$ – Yuji Sep 22 '17 at 6:03
  • $\begingroup$ And the "about" page of the blog says it's based on a university course at Universität Hamburg, with an accompanying Youtube channel: youtube.com/channel/UCts9FTFNInqTMvcFpdyap7w $\endgroup$ – Yuji Sep 22 '17 at 6:07
  • $\begingroup$ I agree, it does look authentic. This then would be a definitive answer to the OP's question. Most fascinating information, thank you!\ $\endgroup$ – WetSavannaAnimal Sep 22 '17 at 8:23
0
$\begingroup$

I would be surprised if there wasn't an example of each.

For instance, you can search the American Mineralogist Crystal Structure Database by space group. Click on Cell Parameters and Symmetry and select whichever you like. I'll leave it to you to go through all 230 of them.

$\endgroup$
  • $\begingroup$ I too chose not to brute force it but thought I'd approach from a slightly different direction. $\endgroup$ – Jon Custer Jul 22 '14 at 22:26
  • 1
    $\begingroup$ PCC2 (mentioned by akhmeteli as not being observed as of 1999) is not in the dropdown menu. Also "The list box does not contain all possible space groups, but only those represented in the database." Interpret that how you will. $\endgroup$ – user10851 Jul 23 '14 at 1:00
0
$\begingroup$

Probably all of them and more besides. The 230 does not include quasicrystals, while nature does.

$\endgroup$

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