Tables for
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A. ch. 9.3, pp. 757-758

Section 9.3.5. Conventional characters

B. Grubera

aDepartment of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, CZ-11800 Prague 1, Czech Republic

9.3.5. Conventional characters

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Lattice characters were defined in Section 9.3.2[link] by dividing the Niggli image of a certain Bravais type [{\scr T}] into components. Doing the same – instead of with the Niggli points – with the parameters of conventional cells5 of lattices of the Bravais type [{\scr T}] we obtain a division of the range6 of these parameters into components. This leads to a further division of lattices of the Bravais type [{\scr T}] into equivalence classes. We call these classes – in analogy to the Niggli characters – conventional characters. There are 22 of them.

Two lattices of the same Bravais type belong to the same conventional character if and only if one lattice can be deformed into the other in such a way that the conventional parameters of the deformed lattice change continuously from the initial to the final position without change of the Bravais type. The word `continuously' cannot be replaced by the stronger term `linearly' because the range of conventional parameters of the monoclinic centred lattices is not convex.

Conventional characters form a superdivision of the lattice characters. Therefore, no special notation of conventional characters need be invented: we write them simply as sets of lattice characters which constitute the conventional character. Denoting the lattice characters by integral numbers from 1 to 44 (according to the convention in Section 9.2.5[link] ), we obtain for the conventional characters symbols like [\{8,19,42\}] or [\{7\}].

Conventional characters are described in Table[link].

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Conventional characters

Bravais typeConditionsConventional character
cP   {3}
cI   {5}
cF   {1}
tP [a \lt c] {11}
  [c \lt a] {21}
tI [a \lt c/\sqrt{2}] {15}
  [c/\sqrt{2} \lt a \lt c] {7}
  [c \lt a] {6, 18}
oP   {32}
oI   {8, 19, 42}
oF   {16, 26}
oC [b \lt a\sqrt{3}] {13, 23}
  [a\sqrt{3} \lt b] {36, 38, 40}
hP   {12, 22}
hR [\alpha \lt 60^{\circ}] {9}
  [60^{\circ} \lt \alpha \lt 90^{\circ}] {2}
  [90^{\circ} \lt \alpha \lt \omega] {4}
  [\omega \lt \alpha] {24}
mP   {33, 34, 35}
mC   {10, 14, 17, 20, 25, 27, 28, 29, 30, 37, 39, 41, 43}
aP [\alpha \lt 90^{\circ}] {31}
  [90^{\circ} \leq \alpha] {44}
The angle α refers to the rhombohedral description of the hR lattices.
[\omega = \arccos (-1/3) = 109^{\circ}28'16''].

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