Taking Derivatives of the Bragg Equation

Several important relationships in crystallography directly follow from a derivative of the Bragg equation [Equation (5)]. First we rewrite Bragg s law making the /-spacing the subject of the equation  

The uncertainty of the measured lattice spacing is given by the total derivative Ad, which can be written according to the chain rule as  

This equation allows us to discuss several physically important phenomena. 

When a crystal is strained, the i-spacings vary. A macroscopic strain changes the interplanar spacing by giving rise to a shift in the average position of the diffraction peak of A0, while microscopic strains give a distribution of /-spacings which broaden the peak by 66. This is discussed in detail in Chapters 12 and 13. 

There are many geometrical contributions to the angular resolution (c.g., angular width of the receiving slit in front of the detector). Another contribution comes from finite wavelength spread of the incident beam A2. From Equation (40) we get the angular dispersion to be  

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