The structural study of heteropolyanions began in early thirtieth, but still many unsolved problems remain. The present review begins with a short historical glance upon early synthetic works by Marignac in nineteen century and theoretical structural elucidation by Pauling in 1929. The Section 2 contains nomenclature of heteropolyanions and explanation, by the aid of polyhedra model, of representative polyanions including well known Keggin structure PW12O403-, P2W18O626-, partially hydrolyzed polyanions, SiW11O408-, P2W17O6110, 9-heteropolyanion, H3PMo9O31 (OH) 33-, 5-molybdo-2-phosphate anion P2Mo5O236-, 12-molybdo-4-arsenate anion As4Mo12O508-. Formation conditions of arsenate and molybdate containing polyanions are discussed (see Table 5) . Another type is “substituted”polyanions, 12-and 18-heteropolyanions in which one or two Mo or W atoms are replaced by divalent or trivalent transition metal cations. Above mentioned polyanions contain the heteroatoms in tetrahedral oxygen coordination, MO4. But there are many heteropolyanions in which the heteroatoms are octahedrally coordinated by oxygen atoms. Examples shown are Co2Mo10O3810- NiV13 O387- and MnMo9O326-. The next section is devoted to explain recently solved structural problems of the geometrical isomers of 12-and 18-heteropolyacids. The Keggin structure SiW12O404- can have 4 isomers, α (Keggin structure itself), β, γ, δ, and e shown in Fig. 16. The crystal structure determinations of β-K4SiW12O409H2O and α-Ba2SiW12O4016H2O have shown that these structures correspond to β and α in Fig. 16. These five isomer models are successively made by 60° rotation of W3O13 unit in the Keggin structure. In the last section, as two representative cases of crystal structure of heteropolyanions, Na6Mo5P2O23⋅13H2O and α-Ba2SiW12O40⋅16H2O are illustrated.
Fast Fourier Transform (FFT) algorithm is very time-saving for the computation of the discrete Fourier transforms. This article briefly describes the principle of the algorithm and its application to crystallographic problems. If the crystallographic symmetry properties are incorporated into the algorithm, a great improvement is obtained. This is very effective for structure determination of macromolecules, for direct methods of crystal-structure analysis, and for some new approaches to the crystallographic problems such as the evaluation of structure factors from the electron density distribution.