Chirality and helicity in the frustrated langasite Ba$_{3}$NbFe$_{3}$Si$_{2}$O$_{14}$*

Ba$_{3}$NbFe$_{3}$Si$_{2}$O$_{14}$ cristallizes in the trigonal non-centrosymmetric space group P321. The Fe$^{3+}$ carry a spin 5/2 and form a triangular lattice of equilateral triangles. This compound orders magnetically at T$_{N}$=27K, well below the Curie-Weiss temperature of $\theta$=-171K, which is a fingerprint of magnetic frustration. Neutron diffraction experiments allowed solving the magnetic structure. This consists in a ferro-chiral configuration of equal moments lying in the plane perpendicular to the 3- fold c axis, oriented at 120$^{\circ}$ from each other on each triangle of Fe$^{3+}$ (due to the intra-plane geometrical frustration), with the same chirality for all the triangles, the whole helically modulated (due to the inter-plane frustration of interactions) along the 3-fold c axis with a $\approx$ 2$\pi$/7 period. Additional neutron polarimetry experiments revealed that a single helicity is selected and that, in turn, this selects a single chirality, in agreement with group representation analysis. A mean field analysis shows that this unique feature is due to the twist in the spin-spin exchange path from plane to plane.