Restriction of the number of terms in the Sayre–Hughes equation connected with a criterion to establish the absence of atomic overlap in projection

In the Sayre-Hughes equation the variable reciprocal-lattice vector ranges over the whole reciprocal net. It is shown that under the condition that a projection along a zone axis has no overlapping atoms the variable reciprocal-lattice vector can be restricted to range over a plane of the reciprocal net. A fortiori if a projection on an axis is free from overlap the variable vector can be restricted to range over a row of the reciprocal net. Criteria to determine the projections for which these conditions are best fulfilled are given. These criteria involve the absolute values of the structure factors only. The theory is illustrated by a test employing normalized structure factors calculated from atomic coordinates.