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

International Tables for Crystallography (2006). Vol. C, ch. 1.1, pp. 2-3

Section 1.1.1. General relations between direct and reciprocal lattices

E. Kocha

aInstitut für Mineralogie, Petrologie und Kristallographie, Universität Marburg, Hans-Meerwein-Strasse, D-35032 Marburg, Germany

1.1.1. General relations between direct and reciprocal lattices

| top | pdf |

1.1.1.1. Primitive crystallographic bases

| top | pdf |

The vectors a, b, c form a primitive crystallographic basis of the vector lattice L, if each translation vector may be expressed as with u, v, w being integers.

A primitive basis defines a primitive unit cell for a corresponding point lattice. Its volume V may be calculated as the mixed product (triple scalar product) of the three basis vectors: Here a, b and c designate the lengths of the three basis vectors and , and the angles between them.

Each vector lattice L and each primitive crystallographic basis a, b, c is uniquely related to a reciprocal vector lattice and a primitive reciprocal basis a*, b*, c*:The lengths , and of the reciprocal basis vectors and the angles , and are given by: a*, b*, c* define a primitive unit cell in a corresponding reciprocal point lattice. Its volume V* may be expressed by analogy with V [equation (1.1.1.1)]:

In addition, the following equation holds: As all relations between direct and reciprocal lattices are symmetrical, one can calculate a, b, c from a*, b*, c*: The unit-cell volumes V and V* may also be obtained from:

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 instead of a primitive basis a, b, c. In that case, , , and with all their integral linear combinations are lattice vectors again, but there exist other lattice vectors , with at least two of the coefficients , , being fractional.

Such a conventional basis defines a conventional or centred unit cell for a corresponding point lattice, the volume of which may be calculated by analogy with V by substituting for a, b, and c in (1.1.1.1).

If m designates the number of centring lattice vectors t with , may be expressed as a multiple of the primitive unit-cell volume V: With the aid of equations (1.1.1.2) and (1.1.1.3), the reciprocal basis may be derived from . Again, each reciprocal-lattice vector 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 to , the reciprocal cell volume referred to . From this it follows that

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 latticeReciprocal lattice

Bravais letterCentring vectorsUnit-cell volume Conditions for reciprocal-lattice vectors Unit-cell volume Bravais letter
A 2V A
B 2V B
C 2V C
I 2V F
F 4V I
R , 3V 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 is given in the last column of Table 1.1.1.1. Except for P lattices, a conventionally chosen basis for coincides neither with a*, b*, c* nor with . This third basis, however, is not used in crystallography. The designation of scattering vectors and the indexing of Bragg reflections usually refers to .

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.