Relationships between direct and reciprocal Bravais lattices

Shmueli, U.

doi:10.1107/97809553602060000552

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 1.4, pp. 105-106 | 1 | 2 |

Section 1.4.4.5. Relationships between direct and reciprocal Bravais lattices

U. Shmueli^a

1.4.4.5. Relationships between direct and reciprocal Bravais lattices

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Centred Bravais lattices in crystal space give rise to systematic absences of certain classes of reflections (IT I, 1952; IT A, 1983) and the corresponding points in the reciprocal lattice have accordingly zero weights. These absences are periodic in reciprocal space and their `removal' from the reciprocal lattice results in a lattice which – like the direct one – must belong to one of the fourteen Bravais lattice types. This must be so since the point group of a crystal leaves its lattice – and also the associated reciprocal lattice – unchanged. The magnitudes of the structure factors (the weight functions) are also invariant under the operation of this point group.

The correspondence between the types of centring in direct and reciprocal lattices is given in Table 1.4.4.1.

Table 1.4.4.1| top | pdf |
Correspondence between types of centring in direct and reciprocal lattices

Direct lattice		Reciprocal lattice
Lattice type(s)	Centring translations	Lattice type(s)	Restriction on hkl	Multiple unit cell
P, R		P, R		$[{\bf a}^{}]$ , $[{\bf b}^{}]$ , $[{\bf c}^{*}]$
A	0, $[{1 \over 2}]$ , $[{1 \over 2}]$	A		$[{\bf a}^{}]$ , $[2{\bf b}^{}]$ , $[2{\bf c}^{*}]$
B	$[{1 \over 2}]$ , 0, $[{1 \over 2}]$	B		$[2{\bf a}^{}]$ , $[{\bf b}^{}]$ , $[2{\bf c}^{*}]$
C	$[{1 \over 2}]$ , $[{1 \over 2}]$ , 0	C		$[2{\bf a}^{}]$ , $[2{\bf b}^{}]$ , $[{\bf c}^{*}]$
I	$[{1 \over 2}]$ , $[{1 \over 2}]$ , $[{1 \over 2}]$	F		$[2{\bf a}^{}]$ , $[2{\bf b}^{}]$ , $[2{\bf c}^{*}]$
F	0, $[{1 \over 2}]$ , $[{1 \over 2}]$	I		$[2{\bf a}^{}]$ , $[2{\bf b}^{}]$ , $[2{\bf c}^{*}]$
	$[{1 \over 2}]$ , 0, $[{1 \over 2}]$
	$[{1 \over 2}]$ , $[{1 \over 2}]$ , 0
$[R_{\rm hex}]$	$[{2 \over 3}]$ , $[{1 \over 3}]$ , $[{1 \over 3}]$	$[R_{\rm hex}]$		$[3{\bf a}^{}]$ , $[3{\bf b}^{}]$ , $[3{\bf c}^{*}]$
	$[{1 \over 3}]$ , $[{2 \over 3}]$ , $[{2 \over 3}]$

Notes:

(i) The vectors $[{\bf a}^{*}]$ , $[{\bf b}^{*}]$ and $[{\bf c}^{*}]$ , appearing in the definition of the multiple unit cell in the reciprocal lattice, define this lattice prior to the removal of lattice points with zero weights (absences). All the restrictions on hkl pertain to indexing on $[{\bf a}^{*}]$ , $[{\bf b}^{*}]$ and $[{\bf c}^{*}]$ .
(ii) The centring type of the reciprocal lattice refers to the multiple unit cell given in the table.
(iii) The centring type denoted by $[R_{\rm hex}]$ is a representation of the rhombohedral lattice R by a triple hexagonal unit cell, in the obverse setting (IT I, 1952), i.e. according to the transformation $[\eqalign{{\bf a} &= {\bf a}_{R} - {\bf b}_{R}\cr {\bf b} &= {\bf b}_{R} - {\bf c}_{R}\cr {\bf c} &= {\bf a}_{R} + {\bf b}_{R} + {\bf c}_{R},\cr} \eqno(1.4.4.14)]$ where $[{\bf a}_{R}]$ , $[{\bf b}_{R}]$ and $[{\bf c}_{R}]$ pertain to a primitive unit cell in the rhombohedral lattice R.

The corresponding multiple reciprocal cell , with centring denoted by $[R_{\rm hex}]$ , contains nine lattice points with coordinates 000, 021, 012, 101, 202, 110, 220, 211 and 122 – indexed on the usual reciprocal to the triple hexagonal unit cell defined by (1.4.4.14). Detailed derivations of these correspondences are given by Buerger (1942), and an elementary proof of the reciprocity of I and F lattices can be found, e.g., in pamphlet No. 4 of the Commission on Crystallographic Teaching (Authier, 1981). Intuitive proofs follow directly from the restrictions on hkl, given in Table 1.4.4.1.

References

International Tables for Crystallography (1983). Vol. A. Space-group symmetry, edited by Th. Hahn. Dordrecht: Reidel. (Present distributor Kluwer Academic Publishers, Dordrecht.)Google Scholar

International Tables for X-ray Crystallography (1952). Vol. I. Symmetry groups, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press.Google Scholar

Authier, A. (1981). The reciprocal lattice. Edited by the IUCr Commission on Crystallographic Teaching. Cardiff: University College Cardiff Press.Google Scholar

Buerger, M. J. (1942). X-ray crystallography. New York: John Wiley.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.4, pp. 105-106