FIG. 2: (color online) The Bi 2 Fe 4 O 9 compound (space group Pbam ; a = 7.965(4), b = 8.448(5), c = 6.007(3) ˚ A at 300 K). (left): View of the unit cell with the environment of the two Fe ions. Bi atoms are purple, Fe yellow and O red. Projection of the structure on the ( a , c ) plane (middle) and on the ( a , b ) plane (right). The atoms labels are in blue. Black lines materialize the magnetic exchange interactions. volves only Fe$_{2}$ atoms in the octahedron columns and is achieved through two interactions, J$_{1}$ between the atoms within the unit cell and J$_{2}$ between the atoms of adjacent cells. Within a layer, each Fe$_{1}$ interacts with its nearest neighbour Fe$_{1}$ via J$_{4}$ and with two nearest neighbour Fe$_{2}$ pair (located above and below the mean plane) via J$_{3}$ and J$_{5}$ . The projection of this layer along c forms a pentagonal lattice with three slightly different bonds per pentagon (Fig. 2). This geometry is equivalent to a distorted Cairo lattice, the perfect one involving convex equilateral pentagons with equal-length sides, but with different associated angles (Fig. 1). What should be expected from such a geometry and what is actually observed ? Up to now, no experimental studies and very few theoretical calculations are available in the literature [6–8]. They only concern a very particu- lar pentagonal tiling, derived from the hexagonal lattice (see lattice c of Fig.1). Calculations within the antiferro- magnetic Ising model on this lattice yields a disordered ground state with a finite entropy per spin [6, 8], whereas in the Heisenberg model the classical ground state of the perfect pentagonal lattice has not yet been solved [7]. Single crystals of Bi$_{2}$Fe$_{4}$O$_{9}$ were grown by the high temperature solution growth method using a flux of Bi$_{2}$O$_{3}$ [9]. The temperature and magnetic field depen- dences of the magnetization along the principal crystallo- graphic axes were measured using a SQUID magnetome- ter (Quantum Design) in the temperature range 2-380 K and in fields up to 5.5 T. The reported data, measured on the very same crystal used for the neutron diffraction study (2.5 x 2 x 1.5 mm $^{3}$), were corrected for the demag- netizing field effect. The DC susceptibility between 300 K and 800 K was measured on a home made Faraday-type magnetic balance in a magnetic field of 0.7 T, using an as- sembly of several randomly oriented single crystals. The reported susceptibility was corrected from diamagnetism using the Pascal’s constants χ$_{dia}$ = -198 10 $_{−}$6 emu/mol. Above room temperature, the magnetic susceptibility of When decreasing the temperature, new reflections, characteristic of an antiferromagnetic order, appear around T$_{N}$ = 244(2) K. They can be indexed with a propagation vector k = (1 / 2 , 1 / 2 , 1 / 2), that is a doubling of the magnetic cell compared to the nuclear one in the three directions. The temperature variation of a mag- netic Bragg peak is presented in Fig. 3. The magnetic structure was solved from the 654 magnetic Bragg peaks collected at T= 15 K taking into account the presence of two magnetic domains [9]. There are four Fe$_{1}$ atoms and four Fe$_{2}$ in the unit cell, whose coordinates are re- Further insights concerning the magnetic arrangement below T$_{N}$ were obtained using neutron diffraction. The neutron diffraction experiments were performed on the two CEA-CRG diffractometers D23 and D15 at the Insti- tut Laue-Langevin (Grenoble, France) using wavelengths of respectively λ = 1 . 280 ˚ A and 1 . 173 ˚ A . Complete data collections have been made at two different temperatures T = 15 K and T = 300 K. The experimental data were corrected from absorption using the CCSL library [12]. The structural and magnetic arrangements were refined using the least-square programs MXD [13] and FULL- PROF [14], including extinction corrections [15]. Bi$_{2}$Fe$_{4}$O$_{9}$ is found to obey a Curie-Weiss law with an ex- trapolated paramagnetic temperature θ$_{p}$ ≈ - 1670 K and an effective magnetic moment µ$_{eff}$ = 6.3(3) µ$_{B}$ per iron atom (see Fig. 3). This later value is compatible with the 5.9 µ$_{B}$ expected for Fe 3+ ions ( J = S =5/2). When decreasing the temperature, an evident drop in the ( a , b ) plane susceptibility and a well pronounced increase of the c axis susceptibility are observed at T$_{N}$ = 238(2) K, in- dicating a transition toward a magnetic long range order. This N´eel temperature is in agreement with the results on single crystal reported in ref. 10 but some 20 K lower that the one reported in refs. 4, 5, 11 for sintered sam- ples. The observed ratio between the N´eel temperature and the paramagnetic one, θ$_{p}$/ T$_{N}$ ≈ 7, suggests, as ex- pected from the geometry, a large degree of frustration. 2