In this chapter, general geometrical formulae are given that describe (i) the relations between the lattice parameters and unit cells in direct and in reciprocal space and (ii) the relations between lattice vectors, point rows and net planes, and allow (iii) the calculation of various angles in direct and in reciprocal space (including the Miller formulae).
Keywords: angles in direct and reciprocal space; basis; direct and reciprocal lattices; lattices; Miller formulae; point rows.
In an ideal crystal structure, the arrangement of atoms is threedimensionally periodic. This periodicity is usually described in terms of point lattices
The vectors a, b, c form a primitive crystallographic basis
A primitive basis defines a primitive unit cell
Each vector lattice L and each primitive crystallographic basis a, b, c is uniquely related to a reciprocal vector lattice
In addition, the following equation holds: As all relations between direct and reciprocal lattices
For certain lattice types, it is usual in crystallography to refer to a `conventional' crystallographic basis
Such a conventional basis defines a conventional or centred unit cell
If m designates the number of centring lattice vectors
Equation (1.1.1.5)
Table 1.1.1.1

As a direct lattice and its corresponding reciprocal lattice do not necessarily belong to the same type of Bravais lattices
If the differences with respect to the coefficients of direct and reciprocallattice vectors are disregarded, all other relations discussed in Part 1 are equally true for primitive bases and for conventional bases.
The length t of a vector is given by Accordingly, the length of a reciprocallattice vector
The integer coefficients h, k, l of a vector are also the coordinates of a point of the corresponding reciprocal lattice and designate the Bragg reflection
Each vector r* is perpendicular to a family of equidistant parallel nets within a corresponding direct point lattice. If the coefficients h, k, l of r* are coprime, the symbol (hkl) describes that family of nets. The distance d(hkl) between two neighbouring nets is given by Parallel to such a family of nets, there may be a face or a cleavage plane of a crystal
The net planes (hkl) obey the equation Different values of n distinguish between the individual nets of the family; x, y, z are the coordinates of points on the net planes (not necessarily of lattice points). They are expressed in units a, b, and c, respectively.
Similarly, each vector with coprime coefficients u, v, w is perpendicular to a family of equidistant parallel nets within a corresponding reciprocal point lattice. This family of nets may be symbolized . The distance between two neighbouring nets can be calculated from A layer line on a rotation pattern or a Weissenberg photograph with rotation axis [uvw] corresponds to one such net of the family of the reciprocal lattice.
The nets obey the equation Equations (1.1.2.6)
A family of nets (hkl) and a point row with direction [uvw] out of the same point lattice are parallel if and only if the following equation is satisfied:
This equation is called the `zone equation'
Two (nonparallel) nets and intersect in a point row with direction [uvw] if the indices satisfy the condition The same condition must be satisfied for a zone axis
Three nets , , and intersect in parallel rows, or three faces with these indices belong to one zone if Two (nonparallel) point rows and in the direct lattice are parallel to a family of nets (hkl) if The same condition holds for a face (hkl) belonging to two zones and .
Three point rows , , and are parallel to a net (hkl), or three zones of a crystal with these indices have a common face (hkl) if A net (hkl) is perpendicular to a point row [uvw] if
The angles between the normal of a crystal face and the basis vectors a, b, c are called the direction angle
Similarly, the angles between a directlattice vector t and the reciprocal basis vectors , and are given by The angle between two directlattice vectors and or between two corresponding point rows and may be derived from the scalar product as Analogously, the angle between two reciprocallattice vectors and or between two corresponding point rows and or between the normals of two corresponding crystal faces and may be calculated as with
Finally, the angle between a first direction [uvw] of the direct lattice and a second direction [hkl] of the reciprocal lattice may also be derived from the scalar product of the corresponding vectors t and r*.
Consider four faces of a crystal that belong to the same zone in consecutive order: , , , and . The angles between the ith and the jth face normals
From the definition of , , and , it follows that all fractions in (1.1.4.1)