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The crystal structure of the double tungstate NaFe(WO$_4$)$_2$ arises fromthat of the spin-driven multiferroic MnWO$_4$ by inserting non-magnetic Nalayers. NaFe(WO$_4$)$_2$ exhibits a three-dimensional incommensuratespin-spiral structure at low temperature and zero magnetic field, which,however, competes with commensurate order induced by magnetic field. Theincommensurate zero-field phase corresponds to the condensation of a singleirreducible representation but it does not imply ferroelectric polarizationbecause spirals with opposite chirality coexist. Sizable anharmonic modulationsemerge in this incommensurate structure, which are accompanied by largemagneto-elastic anomalies, while the onset of the harmonic order is invisiblein the thermal expansion coefficient. In magnetic fields applied along themonoclinic axis, we observe a first-order transition to a commensuratestructure that again is accompanied by large magneto-elastic effects. The largemagnetoelastic coupling, a reduction of the $b$ lattice parameter, is thusassociated only with the commensurate order. Upon releasing the field at lowtemperature, the magnetic order transforms to another commensurate structurethat considerably differs from the incommensurate low-temperature phaseemerging upon zero-field cooling. The latter phase, which exhibits a reducedordered moment, seems to be metastable.

Strong magnetoelastic coupling at the transition from harmonic to anharmonic order in NaFe(WO$_4$)$_2$ with 3d$^5$ configuration