Now we are in a position to combine Bragg's law which gives the distance between the adjacent planes in the crystal to the Miller's characterization of the planes through the indices ( h, k, l ). The unit cell edge-lengths are a, b and c.
Consider a square Lattice as shown in Fig 18.5
Figure 18.5 Calculation of the separation between planes.
In Fig 18.5, two adjacent hkl planes are shown. If a is the edgelength of the lattice, then the hkl plane intersects the axes at a / h, b / k, and c / l. For example, if ( hkl ) was (1,1,1), then this Miller plane would intersect the three axes in a cubic lattice at (a,a,a). If ( hkl ) is (1,0,0), the intersections would be at (a,
, ).
Label the two planes in Fig. (18.5) as 1 and 2. Plane 2 intersects the x axis at a / h and the y axis at b / h. The distance perpendicular to the planes 1 and 2 is d h k. For the planar lattice, l = 0.
sin
= dh k/ (a / .k) and cos
= dh k/ ( a / h )
(18.2)
squaring and adding
or
(18.3)
(18.4)
Extending the analogy to three dimensions and for a plane where l 0, the distance d hkl will now be given by
(18.5)
or
(18.6)
For a general orthorhombic lattice wherein a b c, the expression corresponding to (18.5) is
, 
). 
= d
= d
or 
0, the distance d hkl will now be given by 
