A geometrical-approach to solving crystal structures

The Karle & Hauptman and Goedkoop matrices are regarded as metric tensors in Hilbert space. This enables an E value to be expressed in terms of a finite number of other E's. It is shown that the knowledge of a finite number of E's is theoretically sufficient to determine all the atomic coordinates. This number is smaller than 8(N + 3) divided by the order of the point group, counting a complex E twice (N is the number of atoms in the unit cell). The phase problem is analyzed from a geometrical point of view and it is shown how probability becomes certainty with a finite number of data. The Σ₂ relationship is obtained as a particular approximation of an exact relationship between E's. This theory enables a criterion to be established which is equivalent to Tsoucaris's maximum determinant rule but more restrictive. The theory is valid for structures with both equal and unequal atoms.