Spin dynamics in natural multiferroic pyroxene NaFeSi$_2$O$_6$

Spin dynamics in the natural mineral aegirine, NaFeSi$_2$O$_6$, a member of the pyroxene family, was studied by elastic and inelastic neutron scattering. Magnetization and specific heat measurements as well as single-crystal neutron diffraction maps, taken in the temperature range 2 - 20 K, confirm two successive magnetic transitions at 8.8 and 5.8 K, consistent with previous studies. The observed spin-wave excitations emerge from the incommensurate magnetic Bragg peaks corresponding to the propagation vector $k_{\rm ICM} = (0, 0.77, 0)$, and extend up to energies of about 1.5 meV. In the low-temperature helical phase, the spin dynamics of the Fe$^{3+}$ ions is well described by a simple linear spin-wave model. The observed excitations can be modeled using a spin Hamiltonian that includes three primary exchange interactions - intrachain coupling $J=0.142(2)$ meV, interchain couplings $J_1=0.083(1)$ meV and $J_2=0.186(1)$ meV - and an easy-plane anisotropy $D=0.020(6)$ meV. Our results show that no single exchange interaction dominates the spin dynamics. The similar strengths of the intrachain and interchain couplings point to the fact that the magnetic interactions in aegirine are three-dimensional rather than confined along one direction. As a result, the system cannot be considered quasi-one-dimensional, as previously suggested, and calls for further investigations.