within the sample, let there be a density of scatterers f(r); these scatterers should produce a scattered spherical wave of amplitude proportional to the local amplitude of the incoming wave times the number of scatterers in a small volume dV about r where S is the proportionality constant. Consider the fraction of scattered waves that leave with an outgoing wave-vector of kout and strike the screen at rscreen. Since no energy is lost (elastic, not inelastic scattering), the wavelengths are the same as are the magnitudes of the wave-vectors |kin|=|kout|. From the time that the photon is scattered at r until it is absorbed at rscreen, the photon undergoes a change in phase The net radiation arriving at rscreen is the sum of all the scattered waves throughout the crystal which may be written as a Fourier transform where q = kout – kin. The measured intensity of the reflection will be square of this amplitude Friedel and Bijvoet mates For every reflection corresponding to a point q in the reciprocal space, there is another reflection of the same intensity at the opposite point -q. This opposite reflection is known as the Friedel mate of the original reflection. This symmetry results from the mathematical fact that the density of electrons f(r) at a position r is always a real number. As noted above, f(r) is the inverse transform of its Fourier transform F(q); however, such an inverse transform is a complex number in general. To ensure that f(r) is real, the Fourier transform F(q) must be such that the Friedel mates F(−q) and F(q) are complex conjugates of one another. Thus, F(−q) has the same magnitude as F(q) but they have the opposite phase, i.e., φ(q) = −φ(-q) The equality of their magnitudes ensures that the Friedel mates have the same intensity |F|2. This symmetry allows one to measure the full Fourier transform from only half the reciprocal space, e.g., by rotating the crystal slightly more than 180° instead of a full 360° revolution. In crystals with significant symmetry, even more reflections may have the same intensity (Bijvoet mates); in such cases, even less of the reciprocal space may need to be measured. In favorable cases of high symmetry, sometimes only 90° or even only 45° of data are required to completely explore the reciprocal space. The Friedel-mate constraint can be derived from the definition of the inverse Fourier transform