Table 9.3.4.1

Gruber, B.

doi:10.1107/97809553602060000519

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

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International Tables for Crystallography (2006). Vol. A. ch. 9.3, p. 758

Table 9.3.4.1

B. Gruber^a ^‡

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

Table 9.3.4.1 | top | pdf |
Conventional cells

Bravais type	Centring mode of the cell (a, b, c)	Conditions
cP	P	$[\matrix{a = b = c,\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}]$
cI	I	$[\matrix{a = b = c,\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}]$
cF	F	$[\matrix{a = b = c,\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}]$
tP	P	$[\matrix{a = b \neq c,\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}]$
tI	I	$[\matrix{c/\sqrt{2} \neq a = b \neq c,\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}]$ ^†
oP	P	$[\matrix{a \lt b \lt c,\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}]$ ^‡
oI	I	$[\matrix{a \lt b \lt c,\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}]$
oF	F	$[\matrix{a \lt b \lt c,\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}]$
oC	C	$[\matrix{a \lt b \neq a\sqrt{3},\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}]$ ^§
hP	P	$[\matrix{a = b,\hfill\cr \alpha = \beta = 90^{\circ},\ \gamma = 120^{\circ}\hfill\cr}]$
hR	P	$[\matrix{a = b = c,\hfill\cr \alpha = \beta = \gamma,\hfill\cr \alpha \neq 60^{\circ},\ \alpha \neq 90^{\circ},\ \alpha \neq \omega\hfill\cr}]$ ^¶
mP	P	$[\matrix{-2c \cos\beta \lt a \lt c,\hfill\cr \alpha = \gamma = 90^{\circ} \lt \beta\hfill\cr}]$ ^††
mI	I	$[\matrix{-c \cos \beta \lt a \lt c,\hfill\cr\alpha = \gamma = 90^{\circ} \lt \beta,{\hbox to 5.pc{}}(9.3.4.2)\cr}\hfill]$ ^‡‡
		$[\matrix{{\hbox{but not}}&a^{2} + b^{2} = c^{2},\hfill\cr& a^{2} + ac \cos \beta = b^{2},&{\hbox to -.15pc{}}(9.3.4.3)\hfill\cr}]$ ^§§
		$[\matrix{\hbox{nor}&a^{2} + b^{2} = c^{2},\hfill\cr&b^{2} + ac \cos \beta = a^{2},&{\hbox to 1.15pc{}}(9.3.4.4)\hfill\cr}\hfill]$ ^¶¶
		$[\matrix{{\hbox{nor}}&c^{2} + 3b^{2} = 9a^{2},\hfill\cr&c = -3a \cos \beta,&{\hbox to 2.6pc{}}(9.3.4.5)\cr}\hfill]$ ^†††
		$[\matrix{{\hbox{nor}}&a^{2} + 3b^{2} = 9c^{2},\hfill\cr&a = -3c \cos \beta&{\hbox to 2.7pc{}}(9.3.4.6)\hfill\cr}\hfill]$

Note: All remaining cases are covered by Bravais type aP.

^†For $[a = c/\sqrt{2}]$ , the lattice is cF with conventional basis vectors $[{\bf c}, {\bf a}+{\bf b}, {\bf a}-{\bf b}]$ .
^‡The labelling of the basis vectors according to their length is the reason for unconventional Hermann–Mauguin symbols: for example, the Hermann–Mauguin symbol Pmna may be changed to Pncm, Pbmn, Pman, Pcnm or Pnmb. Analogous facts apply to the oI, oC, oF, mP and mI Bravais types.
^§For $[b = a\sqrt{3}]$ , the lattice is hP with conventional vectors $[{\bf a}, ({\bf b}-{\bf a})/2, {\bf c}]$ .
^¶ $[\omega = \arccos(-1/3) = 109^{\circ}28'16'']$ . For $[\alpha = 60^{\circ}]$ , the lattice is cF with conventional vectors $[-{\bf a}+{\bf b}+{\bf c}]$ , $[{\bf a}-{\bf b}+{\bf c}]$ , $[{\bf a}+{\bf b}-{\bf c}]$ ; for $[\alpha = \omega]$ , the lattice is cI with conventional vectors $[{\bf a}+{\bf b}]$ , $[{\bf a}+{\bf c}]$ , $[{\bf b}+{\bf c}]$ .
^††This means that a, c are shortest non-coplanar lattice vectors in their plane.
^‡‡This means that a, c are shortest non-coplanar lattice vectors in their plane on condition that the cell (a, b, c) is body-centred.
^§§If (9.3.4.2)

and (9.3.4.3)

hold, the lattice is hR with conventional vectors $[{\bf a}, ({\bf a}+{\bf b}-{\bf c})/2, ({\bf a}-{\bf b}-{\bf c})/2]$ , making the rhombohedral angle smaller than 60°.
^¶¶If (9.3.4.2)

and (9.3.4.4)

hold, the lattice is hR with conventional vectors $[{\bf a}, ({\bf a}+{\bf b}+{\bf c})/2, ({\bf a}-{\bf b}+{\bf c})/2]$ , making the rhombohedral angle between 60 and 90°.
^†††If (9.3.4.2)

and (9.3.4.5)

hold, the lattice is hR with conventional vectors $[-{\bf a}, ({\bf a}+{\bf b}+{\bf c})/2, ({\bf a}-{\bf b}+{\bf c})/2]$ , making the rhombohedral angle between 90° and ω.
^‡‡‡If (9.3.4.2)

and (9.3.4.6)

hold, the lattice is hR with conventional vectors $[-{\bf c}, ({\bf a}+{\bf b}+{\bf c})/2, ({\bf a}-{\bf b}+{\bf c})/2]$ , making the rhombohedral angle greater than ω.