TABLE I: Coordinates of the Fe 3+ ions in Bi 2 Fe 4 O 9 and spherical components of their magnetic moments in the first magnetic domain. The magnetic moments are defined in spherical coordinates by ( µ, θ, φ ). In cartesian coordinates (X,Y,Z) where X is along a , Y along b and Z along c , the mag- netic moment components are : M$_{X}$ = µ cosθ sinφ, M$_{Y}$ = µ sinθ sinφ, M$_{Z}$ = µ cosφ . FIG. 3: (color online) Linear magnetic susceptibility of Bi 2 Fe 4 O 9 as a function of temperature measured in magnetic fields of 0.1 T and 0.7 T below and above 300 K respectively. Inset: Temperature variation of the (1 / 2 , 3 / 2 , − 1 / 2) magnetic Bragg peak in zero field from neutron diffraction. (Bottom) Graphical representation of the magnetic structure refinement at T = 15 K in Bi 2 Fe 4 O 9 : observed versus calcu- lated intensities. 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. 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, 3 x y z µ θ φ Fe 1 1 0.3540 0.3361 0.5 3.52(1) 100.1(1) 90 Fe 1 2 0.6460 0.6639 0.5 3.52(1) θ 1 1 + 180 90 Fe 1 3 0.1460 0.8361 0.5 3.52(1) θ 1 1 - 90 90 Fe 1 4 0.8540 0.1639 0.5 3.52(1) θ 1 1 + 90 90 Fe 2 1 0 0.5 0.2582 3.73(1) -105.4(1) 90 Fe 2 2 0 0.5 0.7418 3.73(1) θ 1 2 90 Fe 2 3 0.5 0 0.7418 3.73(1) θ 1 2 +90 90 Fe 2 4 0.5 0 0.2582 3.73(1) θ 1 2 + 90 90