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ported in Table I. The best refinement led to a residual factor Rw = 6 . 4%, with two nearly equally populated do- mains (42/58%). The magnetic moment parameters for the first domain are reported in Table I. The correspond- ing pictures of the magnetic arrangements are presented in Fig. 4. The moments on all the atoms are found re- stricted in the ( a , b ) plane. On each site, the four atoms are gathered into two pairs, antiferromagnetically cou- pled in the case of Fe$_{1}$ and ferromagnetically coupled for Fe$_{2}$. The magnetic moments of each pair are oriented at 90 ◦ , that is perpendicular to the moments of the other pair on the same site. The phase between the two sublattices Fe$_{1}$ and Fe$_{2}$ is 155 ◦ . The moment amplitudes on the two sites Fe$_{1}$ and Fe$_{2}$ are slightly different, and both much smaller than the expected value for a Fe 3+ ion (5 µ$_{B}$ ). A spin transfer process from the iron to the neighbouring oxygen ions [16] could explain this reduc- tion. 

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As it can be seen on Fig. 4, this non-collinear mag- netic structure forms imbricated rectangles of spins. This is quite unusual in such an orthorhombic system, where all the directions are non-equivalent and where there is no fourfold axis. This peculiarity certainly has an ori- gin in the frustration of the magnetic interactions. One should also notice that for this structure the magnetic space group is centrosymmetric, that is non polar. This result is a priori not compatible with ferroelectricity, and this shed some doubts on the claimed ferroelectricity ob- served in polycrystalline Bi$_{2}$Fe$_{4}$O$_{9}$ [4]. 

To check the relevance of the Cairo pentagon lattice model and the role of geometric frustration on the ob- served magnetic structure of Bi$_{2}$Fe$_{4}$O$_{9}$, we have under- taken a two step theoretical approach based on a lo- calized spin model. Calculations have been made both for an ideal Cairo lattice (including the perfect case J$_{3}$ = J$_{4}$ = J$_{5}$ ), and in the real geometry. The angles between magnetic moments in a system is imposed by the set of magnetic interactions. In a first step, an exact Fourier space analysis at zero temperature is performed, assuming a single-k like ordering, which allows to find the periodicity of the magnetic structure [17]. Once the pe- riodicity is known, an energy minimization in real space gives the respective orientation of the magnetic moments within the magnetic unit cell. For these calculations, the five exchange interaction parameters J$_{1}$ to J$_{5}$ already de- scribed have been considered, using a Heisenberg Hamil- tonian H = − 1 2 ∑$_{i,j}$ J$_{k}$ ⃗ S$_{i}$ · ⃗ S$_{j}$ . All the moments were assumed to be of equal amplitudes. 

In the Cairo pentagonal lattice, a large set of J$_{i}$ values yield the observed (1/2, 1/2, 1/2) propagation vector. The doubling of the magnetic cell along c alone requires however J$_{2}$ to be negative and J$_{1}$ positive or slightly neg- ative. Whatever the k = (1 / 2 , 1 / 2 , 1 / 2) solution found, 

