Conventional cells

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. 757

Section 9.3.4. Conventional cells

B. Gruber^a ^‡

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

9.3.4. Conventional cells

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Conventional cells are dealt with in Chapter 9.1 . They are illustrated in Fig. 9.1.7.1 and described in Table 9.1.7.2 . This description, however, is not exhaustive enough for determining the Bravais type. In mathematical terms, the conditions in Table 9.1.7.2 are necessary but not sufficient. For example, the C-centred cell with $[{a = 6,\quad b = 8,\quad c = 5,\quad \cos \beta = -7/15,\quad \alpha = \gamma = 90^{\circ}} \eqno(9.3.4.1)]$ has the typical shape of a conventional cell of an mC lattice. But the lattice generated by the C-centred cell (9.3.4.1) is actually hR with the conventional rhombohedral basis vectors $[{\bf c},\quad ({\bf a} + {\bf b})/2,\quad ({\bf a} - {\bf b})/2.]$

It is a natural goal to establish a system of conditions for the conventional cells which would be not only necessary but also sufficient. This is done in Table 9.3.4.1. In order to make the conditions as simple as possible, the usual mC description of the monoclinic centred lattices is replaced by the mI description. The relation between the two descriptions is simple: $[{\bf a}_{I} = -{\bf c}_{C},\quad {\bf b}_{I} = {\bf b}_{C},\quad {\bf c}_{I} = {\bf a}_{C} + {\bf c}_{C}.]$ The exact meaning of Table 9.3.4.1 is as follows: Suppose that a Bravais type different from aP is given and that its symbol appears in column 1 in the ith entry of Table 9.3.4.1. Then a lattice L is of this Bravais type if and only if there exists a cell (a, b, c) in L such that

(i) the centring of (a, b, c) agrees with the centring mode given in column 2 in the ith entry, and

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.

(ii) the parameters of the cell (a, b, c) fulfil the conditions listed in column 3 in the ith entry of Table 9.3.4.1.

References

International Tables for Crystallography (2006). Vol. A. ch. 9.3, p. 757