X-ray diffraction by small crystals

A general method of calculating the intensity of X-ray diffraction from small crystalline particles whose boundary is defined by a shape function is discussed. The intensity formula which is generally given by a double sum over the reciprocal-lattice points is simplified into the form of a single sum, using 'the random-shift treatment' which assumes that the position of the boundary relative to the crystal lattice varies at random from crystal to crystal. By the use of Fourier theorems, the intensity formulas are also converted into a single sum over the direct lattice. Although the electron distribution in the particle has been defined in various ways by the shape function, a more reasonable expression of the electron density appropriate to small crystals is introduced. The intensity formulas derived on the basis of the new form of the electron density are compared with other intensity formulas which have so far been proposed.