  Magnetic order and the underlying magnetic model of the multiferroic pyroxene
NaFeGe2O6 are systematically investigated by neutron powder diffraction,
thermodynamic measurements, density-functional bandstructure calculations, and
Monte-Carlo simulations. Upon cooling, NaFeGe2O6 first reveals one-dimensional
spin-spin correlations in the paramagnetic state below about 50 K, revealed by
magnetic diffuse scattering. The sinusoidal spin-density wave with spins along
the a-direction sets in at 13 K, followed by the cycloidal configuration with
spins lying in the (ac) plane below 11.6 K. Microscopically, the strongest
magnetic coupling runs along the structural chains, J1 ' 12 K, which is likely
related to the one-dimensional spin-spin correlations. The interchain couplings
J2 ' 3:8K and J3 ' 2:1K are energetically well balanced and compete, thus
giving rise to the incommensurate order in sharp contrast to other
transition-metal pyroxenes, where one type of the interchain couplings
prevails. The magnetic model of NaFeGe2O6 is further completed by the weak
single-ion anisotropy along the a-direction. Our results resolve the earlier
controversies regarding the magnetic order in NaFeGe2O6 and establish relevant
symmetries of the magnetic structures. These results, combined with symmetry
analysis, enable us to identify the possible mechanisms of the magnetoelectric
coupling in this compound. We also elucidate microscopic conditions for the
formation of incommensurate magnetic order in pyroxenes.
