7.10 EXERCISES
daa
b
cFigure 7.17 A face-centred cubic crystal.points with coordinatesX 1 =(3, 2 ,2),X 2 =(2, 3 ,1),X 3 =(2, 1 ,3),X 4 =(3, 0 ,3)lies within the tetrahedron defined by the origin and the other three points.7.19 The vectorsa,bandcare not coplanar. The vectorsa′,b′andc′are the
associated reciprocal vectors. Verifythat the expressions (7.49)–(7.51) define a set
of reciprocal vectorsa′,b′andc′with the following properties:
(a) a′·a=b′·b=c′·c=1;
(b)a′·b=a′·c=b′·a etc = 0;
(c) [a′,b′,c′]=1/[a,b,c];
(d)a=(b′×c′)/[a′,b′,c′].7.20 Three non-coplanar vectorsa,bandc, have as their respective reciprocal vectors
the seta′,b′andc′. Show that the normal to the plane containing the points
k−^1 a,l−^1 bandm−^1 cis in the direction of the vectorka′+lb′+mc′.
7.21 In a crystal with a face-centred cubic structure, the basic cell can be taken as a
cube of edgeawith its centre at the origin of coordinates and its edges parallel
to the Cartesian coordinate axes; atoms aresited at the eight corners and at the
centre of each face. However, other basic cells are possible. One is the rhomboid
shown in figure 7.17, which has the three vectorsb,canddas edges.
(a) Show that the volume of the rhomboid is one-quarter that of the cube.
(b) Show that the angles between pairs of edges of the rhomboid are 60◦and that
the corresponding angles between pairs of edges of the rhomboid defined by
the reciprocal vectors tob,c,dare each 109. 5 ◦. (This rhomboid can be used
as the basic cell of a body-centred cubic structure, more easily visualised as
a cube with an atom at each corner and one at its centre.)
(c) In order to use the Bragg formula, 2dsinθ=nλ, for the scattering of X-rays
by a crystal, it is necessary to know the perpendicular distancedbetween
successive planes of atoms; for a given crystal structure,dhas a particular
value for each set of planes considered. For the face-centred cubic structure
find the distance between successive planes with normals in thek,i+jand
i+j+kdirections.