Maximum Entropy and the Phase Problem in Protein Crystallography

Abstract

X-ray diffraction from a crystal enables the intensity of the Fourier transform of the scattering electron density to be measured. From these data we wish to deduce the atomic coordinates of the crystallised molecule. A crystal is an object which is translationally periodic, which allows a unit cell to be defined by three non-coplanar vectors a₁, a₂, a₃, such that the density is the same after a translation by any of these vectors. Using the translation invariance, the Fourier transform F(k) = ∫ ρ(r) exp(2πir.k)d ³ r of the electron density ρ is then non-zero only at points k such thath _i º k. a_i- are integers for each i. The vectors {aj} reciprocal to {a_i}, so that aj. a_j = d_ij, form a basis for the space of k, usually termed reciprocal space, with k = Σ_ih_ia. The points given by integral hi are called the reciprocal lattice. It is generally convenient to use fractional cell coordinates x, so that r = Σ_ic_ia_i, and hence r. k = x. h.