  The crystal and magnetic structure of multiferroic LiFe(WO$_4$)$_2$ were
investigated by temperature and magnetic-field dependent specific heat,
susceptibility and neutron diffraction experiments on single crystals.
Considering only the two nearest-neighbour magnetic interactions, the system
forms a $J_1$, $J_2$ magnetic chain but more extended interactions are
sizeable. Two different magnetic phases exhibiting long-range incommensurate
order evolve at $T_{\text{N}1}\approx 22.2 \text{ K}$ and $T_{\text{N}2}\approx
19 \text{ K}$. First, a spin-density wave develops with moments lying in the
$ac$ plane. In its multiferroic phase below $T_{\text{N}2}$, LiFe(WO$_4$)$_2$
exhibits a spiral arrangement with an additional spin-component along $b$.
Therefore, the inverse Dzyaloshinskii-Moriya mechanism fully explains the
multiferroic behavior in this material. A partially unbalanced multiferroic
domain distribution was observed even in the absence of an applied electric
field. For both phases only a slight temperature dependence of the
incommensurability was observed and there is no commensurate phase emerging at
low temperature or at finite magnetic fields up to $6\text{ T}$.
LiFe(WO$_4$)$_2$ thus exhibits a simple phase diagram with the typical sequence
of transitions for a type-II multiferroic material.
