Disorder phenomena in the incommensurate compound ${\mathrm{Bi}}_{2+\mathrm{x}}$${\mathrm{Sr}}_{3\mathrm{-}\mathrm{x}}$${\mathrm{Fe}}_2$${\mathrm{O}}_{9+\mathrm{\delta }}$

The refinement of the modulated structure of ${\mathrm{Bi}}_{2+\mathrm{x}}$ ${\mathrm{Sr}}_{3\mathrm{-}\mathrm{x}}$ ${\mathrm{Fe}}_2$ ${\mathrm{O}}_{9+\mathrm{\delta }}$, using the four-dimensional formalism and single-crystal x-ray-diffraction data, has been performed. Different models, already used for the copper compound, have been tested to describe the [BiO] layers. A discussion pointing out the qualities and the failings of each one led us to consider a model assuming a static disorder both on the bismuth and oxygen atom positions. Then, an extra oxygen atom has been located within the [BiO] layers and its occupancy probability has been refined. Because of this model, two disordered regions, governing the behavior of all the atoms in the crystal, have been evidenced. A comparison with the supercell description previously reported shows the interest of the four-dimensional formalism in this case. Finally, a Mössbauer study of ${\mathrm{Bi}}_{2+\mathrm{x}}$ ${\mathrm{Sr}}_{3\mathrm{-}\mathrm{x}}$ ${\mathrm{Fe}}_2$ ${\mathrm{O}}_{9+\mathrm{\delta }}$ specifies the trivalent state of iron and confirms the various surroundings in oxygen atoms of the iron atoms. The correlation between the incommensurate character of the structure and the Mössbauer spectroscopy data is discussed.