Electron density as a Fourier series

Because the Fourier transform operation is reversible [Equations (5.10) and (5.11)], the electron density is in turn the transform of the structure factors, as 

This transform is a triple sum rather than a triple integral because the Fhkis represent a set of discrete entities the reflections of the diffraction pattern. The transform of a discrete function, such as the reciprocal lattice of measured intensities, is a summation of discrete values of the function. The transform of a continuous function, such as p(x,y,z), is an integral, which you can think of as a sum also, but a sum of an infinite number of infinitesimals. 

Superficially, except for the sign change (in the exponential term) that accompanies the transform operation, this equation appears identical to Eq. (5.9), a general three-dimensional Fourier series. But here, each Fhkl is not just one of many simple numerical amplitudes for a standard set of component waves in a Fourier series. Instead, each Fhkl is a structure factor, itself a Fourier series, describing a specific reflection in the diffraction pattern. ( Curiouser and curiouser, said Alice.) 

Equation (5.18) tells us, at last, how to obtain p(pc,y,z). We need merely to construct a Fourier series from the structure factors. The structure factors describe diffracted rays that produce the measured reflections. A full description of a diffracted ray, like any description of a wave, must include three parameters amplitude, frequency, and phase. In discussing data collection, however, I mentioned only two measurements the indices of each reflection and its intensity. Looking again at Eq. (5.18), you see that the indices of a reflection play the role of the three frequencies in one Fourier term. The only measurable variable remaining in the equation is Fhkf Does the measured intensity of a reflection, the only measurement we can make in addition to the indices, completely define Fhkp Unfortunately, the answer is no.  

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