ABSTRACT
The experimental determination of the crystal structure of materials is usually performed in terms of the orientation of the sets of parallel planes that characterize the structure. We must therefore extend the vector representation of directions, Eq.(2.1), associated with lattice points, to describe the lattice planes. The two-dimensional array of points in a lattice plane can be represented by the vector product of any two vectors in the plane. The magnitude of the vector product is equal to the area of the parallelogram defined by the two vectors. Consider an arbitrary plane which intersects the axes of the crystal lattice as shown in Figure 3.1 at the points pa,qb and sc where p , q and s are integers. The triangular portion of the plane has sides:
pa - qb , qb - sc , sc - pa (3.1)
such that the vector product of any two of these gives:
(pa - qb) X (qb - sc) = qs(b X c ) + ps(c x a) + pq(a x b ) (3.2) It is usual to set:
h = qs, k - ps, I - pq
which leads to the ratio:
(3.3)
Figure 3.1. A set of parallel planes with integral intercepts on axes a,b and c of the direct lattice
The vector representation Eq.(3.2), of an arbitrary plane is defined for convenience as:
Km = — j/i(£ x c) + k(c x a ) + l(a x b )J (3.4)
where V0 is the primitive unit cell volume, Eq.(2.33). Hence Eq.(3.4) reduces to:
Khkl = ha* + kb* + Ic* (3.5)
which gives the desired vector representation of the lattice planes in terms of a so-called reciprocal base set, defined as:
~ b x c r* _ c Xa . a x ba =27i , b = 2k ——-— — , c - 2 k —— -— — (3.6) a (b Xc) b (c x a)
It is a straightforward matter to show, using Eq.(3.6), that:
a* ■ a =2/r, a* b =a* c = 0
b* b - 2 k , b* a - b * c = 0
c* c - 2 k , c* a - c * c - 0
c (a x b )
(3.7)
which is a set of equations often treated as formal definitions of the reciprocal base set. Equations (3.7) give both the magnitude of the components of the reciprocal base set:
and their orientation. The three components are normal to the faces A, B and C of the primitive unit cell respectively.
