Ewald Sphere and Reciprocal Space

The scattering vector s or q has dimension of length-1 and spans a three-dimensional reciprocal space. The origin of the reciprocal space is located at the same origin O of the real space spanned by r. The information on the structure of the sample is contained in the function p(r) in real space, and its Fourier transform gives the amplitude A(s), a function in reciprocal space. If A(s) could be determined over all s, the real space function p(r) is recovered by the inverse Fourier transform 

The information contents of p(r) and A(s) are therefore entirely equivalent to each other. Since, in Equation (1.78), the intensity I(s) in reciprocal space is given by the Fourier transform of (r) in real space, one can also derive the latter from the former 

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Fig. 1.8 Experiment geometry, Ewald sphere, and reciprocal space. The stimple is placed in the center of the sphere. Its structure is defined in the real-space coordinates, r = (ri, r2, r3>. The origin of reciprocal space, s = (s 1, 2,. 3). is attached to the surface of the Ewtdd sphere. The orientation of its axes is strictly fixed to the orientation of the sample space r. Each point on a plane detector (for clarity shifted to a position in front of the sphere) ctin be addressed by the two coordinates scattering angle, 20, and azimuthed angle, (p. The mapping of these two coordinates into reciprocal space is mediated by means of the surface of the Ewald sphere [60]...

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