Abstract
The present study shows that the electronic structure of the $\beta$-$AV_6O_{15}$ family of compounds ($A = Sr, Ca, Na ...$) is based on weakly interacting two-leg ladders, in contrast with the zig-zag chain model one could expect from their crystal structure. Spin dimer analysis, based on extended Hückel tight-binding calculations, was performed to determine the structure of the dominant transfer and magnetic interactions in the room temperature $\beta$-$SrV_6O_{15}$ phase. Two different two-legs ladders, associated with different charge/spin orders are proposed to describe these one-dimensional $\beta$-type systems. The antiferromagnetic ladders are packed in an 'IPN' geometry and coupled to each other through weak antiferromagnetic interactions. This arrangement of the dominant interactions explains the otherwise surprising similarities of the optical conductivity and Raman spectra for the one-dimensional $\beta$-type phases and the two-dimensional $\alpha$-type ones such as the well-known $\alpha^\prime$-$NaV_2O_5$ system.