Bragg s law in reciprocal space

Now I will look at diffraction from within reciprocal space. I will show that the reciprocal-lattice points give the crystallographer a convenient way to compute the direction of diffracted beams from all sets of parallel planes in the crystalline lattice (real space). This demonstration entails showing how each reciprocal-lattice point must be arranged with respect to the X-ray beam in order to satisfy Bragg s law and produce a reflection from the crystal. 

Because P is a reciprocal lattice point, the length of the line OP is ldhkl, where h, k, and 1 are the indices of the set of planes represented by P. (Recall from the construction of the reciprocal lattice that the length of a line from O to a reciprocal-lattice point hkl is Vdhkl). So HOP = dhkl and 

In Fig. 4.10 , the crystal, and hence the reciprocal lattice, has been rotated until P, with indices h k I, touches the circle. The same construction as in Fig. 4.10a now shows that 

We can conclude that whenever the crystal is rotated so that a reciprocal-lattice point comes in contact with this circle of radius 1/A, Bragg s law is satisfied and a reflection occurs. What direction does the reflected beam take  

The conclusion that reflection occurs in the direction CP when reciprocal lattice-point P comes in contact with this circle also holds for all points on all circles produce by rotating the circle of radius 1/A about the X-ray beam. The figure that results, called the sphere cf reflection, is shown in Fig. 4.11 intersecting the reciprocal-lattice planes hOl and hi 1. In the crystal orientation shown, reciprocal-lattice point 012 is in contact with the sphere, so a diffracted ray R is diverging from the source beam in the direction defined by C and point 012. This ray would be detected as the 012 reflection. 

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