Non-primitive crystallographic bases

Koch, E.

doi:10.1107/97809553602060000572

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. C. ch. 1.1, p. 3

Section 1.1.1.2. Non-primitive crystallographic bases

E. Koch^a

^a Institut für Mineralogie, Petrologie und Kristallographie, Universität Marburg, Hans-Meerwein-Strasse, D-35032 Marburg, Germany

1.1.1.2. Non-primitive crystallographic bases

| top | pdf |

For certain lattice types, it is usual in crystallography to refer to a `conventional' crystallographic basis $[{\bf a}_c,{\bf b}_c,{\bf c}_c]$ instead of a primitive basis a, b, c. In that case, $[{\bf a}_c]$ , $[{\bf b}_c]$ , and $[{\bf c}_c]$ with all their integral linear combinations are lattice vectors again, but there exist other lattice vectors $[{\bf t}\in{\bf L}]$ , $[{\bf t}=t_1{\bf a}_c+t_2{\bf b}_c+t_3{\bf c}_c,]$ with at least two of the coefficients [t_1] , [t_2] , [t_3] being fractional.

Such a conventional basis defines a conventional or centred unit cell for a corresponding point lattice, the volume [V_c] of which may be calculated by analogy with V by substituting $[{\bf a}_c,{\bf b}_c,{\bf c}_c]$ for a, b, and c in (1.1.1.1).

If m designates the number of centring lattice vectors t with $[0\le t_1,t_2,t_3\lt1]$ , [V_c] may be expressed as a multiple of the primitive unit-cell volume V: $[V_c=mV.\eqno (1.1.1.10)]$ With the aid of equations (1.1.1.2) and (1.1.1.3), the reciprocal basis $[{\bf a}^*_c,{\bf b}^*_c,{\bf c}^*_c]$ may be derived from $[{\bf a}_c,{\bf b}_c,{\bf c}_c]$ . Again, each reciprocal-lattice vector $[{\bf r}^*=h{\bf a}^*_c+k{\bf b}^*_c+l{\bf c}^*_c\in{\bf L}^*]$ is an integral linear combination of the reciprocal basis vectors, but in contrast to the use of a primitive basis only certain triplets h, k, l refer to reciprocal-lattice vectors.

Equation (1.1.1.5) also relates [V_c] to [V^*_c] , the reciprocal cell volume referred to $[{\bf a}^*_c,{\bf b}^*_c,{\bf c}^*_c]$ . From this it follows that $[V^*_c={1\over m}V^*.\eqno (1.1.1.11)]$

Table 1.1.1.1 contains detailed information on `centred lattices' described with respect to conventional basis systems.

Table 1.1.1.1| top | pdf |
Direct and reciprocal lattices described with respect to conventional basis systems

Direct lattice			Reciprocal lattice
$[{\bf a}_c, {\bf b}_c, {\bf c}_c]$			$[{\bf a}^_c, {\bf b}^_c, {\bf c}^*_c]$
Bravais letter	Centring vectors	Unit-cell volume	Conditions for reciprocal-lattice vectors $[h{\bf a}^_c+k{\bf b}^_c+l{\bf c}^*_c]$	Unit-cell volume	Bravais letter
A	$[{1\over2}{\bf b}_c+{1\over2}{\bf c}_c]$	2V		$[{1\over2}V^*]$	A
B	$[{1\over2}{\bf a}_c+{1\over2}{\bf c}_c]$	2V		$[{1\over2}V^*]$	B
C	$[{1\over2}{\bf a}_c+{1\over2}{\bf b}_c]$	2V		$[{1\over2}V^*]$	C
I	$[{1\over2}{\bf a}_c+{1\over2}{\bf b}_c+{1\over2}{\bf c}_c]$	2V		$[{1\over2}V^*]$	F
F	$[{1\over2}{\bf a}_c+{1\over2}{\bf b}_c,]$ $[{1\over2}{\bf a}_c+{1\over2}{\bf c}_c,]$ $[{1\over2}{\bf b}_c+{1\over2}{\bf c}_c]$	4V		$[{1\over4}V^*]$	I
R	$[{1\over3}{\bf a}_c+{2\over3}{\bf b}_c+{2\over3}{\bf c}_c]$ , $[{2\over3}{\bf a}_c+{1\over3}{\bf b}_c+{1\over3}{\bf c}_c]$	3V		$[{1\over3}V^*]$	R

As a direct lattice and its corresponding reciprocal lattice do not necessarily belong to the same type of Bravais lattices [IT A (2005, Section 8.2.5 )], the Bravais letter of $[{\bf L}^*]$ is given in the last column of Table 1.1.1.1. Except for P lattices, a conventionally chosen basis for $[{\bf L}^*]$ coincides neither with a*, b*, c* nor with $[{\bf a}^*_c,{\bf b}^*_c,{\bf c}^*_c]$ . This third basis, however, is not used in crystallography. The designation of scattering vectors and the indexing of Bragg reflections usually refers to $[{\bf a}^*_c,{\bf b}^*_c,{\bf c}^*_c]$ .

If the differences with respect to the coefficients of direct- and reciprocal-lattice vectors are disregarded, all other relations discussed in Part 1 are equally true for primitive bases and for conventional bases.

References

International Tables for Crystallography (2005). Vol. A, Space-group symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 1.1, p. 3