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Ewald summation evaluates long-range interactions in a periodically repeated system by separating rapidly convergent real-space and reciprocal-space contributions. For a Coulomb interaction, the identity $1/r=\mathrm{erfc}(\alpha r)/r+\mathrm{erf}(\alpha r)/r$ splits a short-range term from a smooth long-range term.

The complementary-error-function term decays rapidly and is summed over nearby images. The smooth term is efficiently summed through Fourier modes, with a self-interaction correction removing the artificial screening contribution at each particle. The splitting parameter $\alpha$ changes computational work rather than the converged physical answer. Net charge, the zero mode and the boundary convention must be handled explicitly; an arbitrary unqualified infinite Coulomb sum need not converge uniquely.^[ref]

Paul Peter Ewald developed a method for calculating long-range electrostatic lattice potentials. Ewald summation separates the interaction into rapidly converging real-space and reciprocal-space contributions through a Gaussian splitting. The artificial splitting parameter changes the work assigned to each sum, while the complete physical result is unchanged when electric charge and boundary conventions are fixed. This makes lattice electrostatics computationally tractable without simply discarding distant electric charges, and connects crystal calculations with Fourier methods.^[ref]

A physicist who worked on crystallography and edited a participant-rich history of X-ray diffraction. The volume places experimental photographs, early theoretical arguments and personal recollections together, making it possible to trace how diffraction patterns became evidence for atomic arrangements in crystals. Its individual contributions have separate authors, while Ewald supplied the editorial framework.^[ref]