The natural mineral linarite PbCuSO_4(OH)_2 has been characterized as a frustrated quantum spin chain system, with a nearest-neighbor ferromagnetic coupling J_1~100K along a b-axis oriented chain, and a next-nearest-neighbor chain coupling J_2~30K. While the magnetic phase diagram has been characterized extensively for fields B || b axis by micro- and macroscopic techniques, for the a and c axis a microscopic characterization is lacking. Here, we propose to carry out field dependent studies by means of neutron scattering of the magnetically ordered phase(s) of linarite with the field applied along a and c axis.