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We report a study of the spin dynamics of the frustrated ferromagnetic$J_1$-$J_2$ chain compound linarite, PbCuSO$_4$(OH)$_2$, in applied magneticfields up to field polarization. By means of an extreme-environment inelasticneutron scattering experiment, we have measured the low-energy spin excitationsof linarite in fields up to 8.8 T for $\mathbf{H} \parallel b$ axis. We haverecorded the spin excitation spectra along $h$, $k$ and $l$ for thefield-induced magnetic phases IV, V and the field polarized state close tosaturation. By employing first-principles calculations, we estimate the leadingmagnetic exchanges out of the $bc$ plane and model the dispersion relationsusing linear spin-wave theory. In this way, we find evidence for a (very weak)residual magnetic exchange coupling out of the $bc$ plane. Together with thepreviously established dominant intrachain couplings $J_1$ and $J_2$ and theinterchain coupling $J_3$, we derive an effective set of exchange couplings fora microscopic description of linarite. Further, we find that the peculiarcharacter of phase V manifests itself in the measured spin dynamics.

physical review. b./physical review. b

Low-energy spin excitations of the frustrated ferromagnetic $J_1$-$J_2$  chain material linarite, PbCuSO$_4$(OH)$_2$, in applied magnetic fields  $\mathbf{H} \parallel b$ axis