S Biesenkamp

N Qureshi

Y Sidis

P Becker

L Bohatý

M Braden

Structural dimerization in the commensurate magnetic phases of NaFe(WO 4 ) 2 and MnWO 4

The structural distortion and magnetoelastic coupling induced through commensurate magnetism has been investigated by neutron diffraction in structurally related MnWO4 and NaFe(WO4)2. Both systems exhibit a competition of incommensurate spiral and commensurate spin up-up-downdown ordering along the magnetic chains. In the latter commensurate phases, the alternatingly parallel and antiparallel arrangement of Fe 3+ respectively Mn 2+ moments leads to sizeable bondangle modulation and thus to magnetic dimerization. For NaFe(WO4)2 this structural distortion has been determined to be strongest for the low-field up-up-down-down arrangement, and the structural refinement yields a bond-angle modulation of ±1.15 (16) degrees. In the commensurate phase of MnWO4, superstructure reflections signal a comparable structural dimerization and thus strong magneto-elastic coupling different to that driving the multiferroic order. Pronounced anharmonic second-and third-order reflections in the incommensurate and multiferroic phase of MnWO4 result from tiny commensurate fractions that can depin multiferroic domains.

I. INTRODUCTION

Since the discoveries of giant magnetoelectric coupling in TbMnO 3 1 and of functional responses in thin film BiFeO 3 2 , the research interest on multiferroic materials has been tremendously increased [START_REF] Fiebig | Revival of the magnetoelectric effect[END_REF][START_REF] Khomskii | Classifying multiferroics: Mechanisms and effects[END_REF][START_REF] Fiebig | The evolution of multiferroics[END_REF][START_REF] Spaldin | Advances in magnetoelectric multiferroics[END_REF] . In type-II multiferroics a ferroic arrangement directly induces a second ferroic order, which implies a strong coupling and thus the potential of manipulating one ferroic parameter by the conjugate field of the coexisting order [START_REF] Fiebig | Revival of the magnetoelectric effect[END_REF][START_REF] Khomskii | Classifying multiferroics: Mechanisms and effects[END_REF][START_REF] Fiebig | The evolution of multiferroics[END_REF] .

In many type-II multiferroics the microscopic coupling mechanism is described by the inverse Dzyaloshinskii-Moriya interaction (DMI) [START_REF] Dzyaloshinskii | A thermodynamic theory of weak ferromagnetism of antiferromagnetics[END_REF][START_REF] Moriya | Anisotropic Superexchange Interaction and Weak Ferromagnetism[END_REF] relating the ferroelectric polarization to the cross product of two spins and the direction of their connecting vector e ij : P ∝ e ij × (S i × S j ) [START_REF] Katsura | Spin current and magnetoelectric effect in noncollinear magnets[END_REF][START_REF] Mostovoy | Ferroelectricity in Spiral Magnets[END_REF][START_REF] Sergienko | Role of the Dzyaloshinskii-Moriya interaction in multiferroic perovskites[END_REF][START_REF] Kimura | Spiral Magnets as Magnetoelectrics[END_REF] . However, due to its origin from spin-orbit coupling, this effect and especially the induced ferroelectric polarization remain small compared to conventional ferroelectrics [START_REF] Khomskii | Classifying multiferroics: Mechanisms and effects[END_REF] . In contrast, larger ferroelectric polarization and more generally larger magnetoelastic coupling can be generated by symmetric-exchange striction in a collinear magnetic structure [START_REF] Sergienko | Ferroelectricity in the Magnetic E -Phase of Orthorhombic Perovskites[END_REF] . Indeed, exchange striction was observed to induce large ferroelectric polarization in rare-earth manganates RMnO 3 , where a small R ionic radius causes a change of the magnetic structure from an incommensurate spiral to commensurate E-type spin up-up-down-down (uudd) arrangement [START_REF] Sergienko | Ferroelectricity in the Magnetic E -Phase of Orthorhombic Perovskites[END_REF][START_REF] Ishiwata | Perovskite manganites hosting versatile multiferroic phases with symmetric and antisymmetric exchange strictions[END_REF][START_REF] Kimura | Magnetoelectric phase diagrams of orthorhombic RMnO3 (R = Gd, Tb, and Dy)[END_REF] .

The same kind of magnetic competition between incommensurate and commensurate magnetic order can also be found in the phase diagrams of the two tungstates MnWO 4 and NaFe(WO 4 ) 2 [START_REF] Ehrenberg | Magnetic phase diagrams of MnWO4[END_REF][START_REF] Heyer | A new multiferroic material: MnWO4[END_REF][START_REF] Lautenschläger | Magnetic phase transitions of MnWO4 studied by the use of neutron diffraction[END_REF][START_REF] Holbein | Strong magnetoelastic coupling at the transition from harmonic to anharmonic order in NaFe(WO 4 ) 2 with 3d 5 configuration[END_REF] . But only in MnWO 4 the incommensurate phase is multiferroic arising from the inverse DMI. In contrast, both materials exhibit strong magnetoelastic coupling in their commensurate phases with uudd spin arrangement. Characterizing the mag-netoelastic coupling associated with the commensurate uudd order in these tungstates is the aim of this work.

The crystal and magnetic structures are illustrated in Fig. 1. Both materials are characterized by zigzag chains of edge-sharing MnO 6 -respectively FeO 6 octahedra along c that are adjacent along b. Along the third direction, a, the magnetic chains are separated by layers of WO 6 octahedra in MnWO 4 . This separation is enhanced in NaFe(WO 4 ) 2 , whose structure results from that of MnWO 4 by substituting the Mn in the zigzag chain layers alternatingly with Fe and non magnetic Na. Therefore, every second layer is magnetic in MnWO 4 , while only every fourth layer is magnetic in NaFe(WO 4 ) 2 . Concerning the magnetic structures it is most important to analyze the intrachain arrangements depicted in Fig. 1. MnWO 4 undergoes first a transition into a collinear incommensurate phase AF3 and then into an incommensurate spiral phase, AF2. This AF2 phase can be compared to the low-field incommensurate phase (LF-IC) that NaFe(WO 4 ) 2 directly reaches upon cooling. While the AF2 phase in MnWO 4 is multiferroic with finite ferroelectric polarization, the LF-IC phase remains paraelectric in NaFe(WO 4 ) 2 . The difference between the two magnetic structures concerns the rotation of the magnetic moments in the cycloid plane [START_REF] Ehrenberg | Magnetic phase diagrams of MnWO4[END_REF][START_REF] Lautenschläger | Magnetic phase transitions of MnWO4 studied by the use of neutron diffraction[END_REF][START_REF] Holbein | Strong magnetoelastic coupling at the transition from harmonic to anharmonic order in NaFe(WO 4 ) 2 with 3d 5 configuration[END_REF] . The upper and lower magnetic moments in one zigzag chain rotate in the same sense in MnWO 4 and in opposite sense in NaFe(WO 4 ) 2 . Therefore, the contributions to the polarization according to the inverse DMI cancel out for the Fe compound. MnWO 4 reaches a commensurate phase, AF1, at low temperature characterized by uudd order within a chain [START_REF] Ehrenberg | Magnetic phase diagrams of MnWO4[END_REF][START_REF] Lautenschläger | Magnetic phase transitions of MnWO4 studied by the use of neutron diffraction[END_REF] , and also NaFe(WO 4 ) 2 exhibits commensurate phases, low-field commensurate (LF-C) and high-field commensurate (HF-C), that however differ for high and low magnetic fields. In addition there is a strong hysteresis so that the magnetic structure depends on how a point in the B,T phase diagram (see inset of figure 3) is reached [START_REF] Holbein | Strong magnetoelastic coupling at the transition from harmonic to anharmonic order in NaFe(WO 4 ) 2 with 3d 5 configuration[END_REF] . In contrast to the E-type manganates, the uudd structures occurring in MnWO 4 and in NaFe(WO 4 ) 2 are not multiferroic as there is no finite polarization. Nevertheless, we will show that also these uudd phases show a strong magnetoelastic coupling characterized by prominent structural dimerization. Note, that the ferroelectric polarization in RMnO 3 only arises from the interplay between bond-angle driven dimerization and octahedron rotation, which causes a shift of all bridging oxygen ions in the same direction [START_REF] Sergienko | Ferroelectricity in the Magnetic E -Phase of Orthorhombic Perovskites[END_REF][START_REF] Ishiwata | Perovskite manganites hosting versatile multiferroic phases with symmetric and antisymmetric exchange strictions[END_REF] .

The first-order transition to the commensurate AF1 phase in MnWO 4 is accompanied by sizable thermal expansion anomalies ( ∆b b ≈ -1 × 10 -5 ) [START_REF] Chaudhury | Thermal expansion and pressure effect in MnWO4[END_REF] . Already in the multiferroic AF2 phase a considerable magneto-elastic coupling was observed [START_REF] Finger | Polarizedneutron-scattering studies on the chiral magnetism in multiferroic MnWO4[END_REF][START_REF] Finger | Electric-field control of the chiral magnetism of multiferroic MnWO4 as seen via polarized neutron diffraction[END_REF] in form of magnetic and nuclear anharmonic modulations. Every incommensurate modulated magnetic structure enforces a nuclear modulation of half the period in real space and of twice the propagation vector. Indeed a strong second order reflection appears in the multiferroic phase [START_REF] Taniguchi | Magnetic-field dependence of the ferroelectric polarization and spinlattice coupling in multiferroic MnWO4[END_REF] , and neutron polarization analysis separates the magnetic and nuclear contribution revealing a strong nuclear fraction and a small magnetic part [START_REF] Finger | Polarizedneutron-scattering studies on the chiral magnetism in multiferroic MnWO4[END_REF] . The same competition of incommensurate and uudd ordering also appears in the double tungstate NaFe(WO 4 ) 2 , in which transitions to the commensurate phases are accompanied by thermal expansion anomalies that are at least one order of magnitude larger ( ∆b b ≈ -2.6 × 10 -4 ) compared to the observed anomalies in MnWO 4 [START_REF] Holbein | Strong magnetoelastic coupling at the transition from harmonic to anharmonic order in NaFe(WO 4 ) 2 with 3d 5 configuration[END_REF][START_REF] Chaudhury | Thermal expansion and pressure effect in MnWO4[END_REF] . Independently of the absence of multiferroicity in NaFe(WO 4 ) 2 , the similar competition of incommensurate spiral and uudd ordering as well as the very strong magneto-elastic coupling to the commensurate phases render NaFe(WO 4 ) 2 an ideal candidate to investigate the magneto-elastic coupling in form of a structural dimerization effect, when entering the commensurate uudd phases.

The research field of multiferroics not only concerns fundamental aspects but also the development of new sensor and memory devices utilizing multiferroicity [START_REF] Scott | Multiferroic memories[END_REF][START_REF] Bibes | Multiferroics: Towards a magnetoelectric memory[END_REF][START_REF] Zhai | Detection of pico-Tesla magnetic fields using magnetoelectric sensors at room temperature[END_REF] . For memory devices the writing speed is an important factor, which in the case of a multiferroic material relates to the relaxation and inversion processes of antiferromagnetic domains, that are controlled by electric fields. The electric-field dependent behavior of multiferroic domains was studied intensively for the prototype multiferroic materials TbMnO 3 and MnWO 4 [START_REF] Arkenbout | Ferroelectricity in the cycloidal spiral magnetic phase of MnWO4[END_REF][START_REF] Meier | Observation and coupling of domains in a spin-spiral multiferroic[END_REF][START_REF] Niermann | Domain dynamics in the multiferroic phase of MnWO4[END_REF][START_REF] Baum | Kinetics of the multiferroic switching in MnWO4[END_REF][START_REF] Stein | Control of multiferroic domains by external electric fields in TbMnO3[END_REF][START_REF] Stein | Control of Chiral Magnetism Through Electric Fields in Multiferroic Compounds above the Long-Range Multiferroic Transition[END_REF] . A time-resolved neutron scattering investigation as well as second-harmonic generation imaging of multiferroic domain inversion in MnWO 4 revealed a rather complex switching behavior, as the inversion process is speeding up towards the lower first-order transition to the commensurate uudd phase [START_REF] Baum | Kinetics of the multiferroic switching in MnWO4[END_REF][START_REF] Hoffmann | Time-resolved imaging of magnetoelectric switching in multiferroic MnWO4[END_REF] .

In this paper we report studies of the structural dimerization appearing in the commensurate uudd phases in NaFe(WO 4 ) 2 and in MnWO 4 . The paper is organized as follows: After the experimental methods, we first discuss the results for NaFe(WO 4 ) 2 , where a stronger dimerization is suggested by the larger thermal expansion anomalies [START_REF] Holbein | Strong magnetoelastic coupling at the transition from harmonic to anharmonic order in NaFe(WO 4 ) 2 with 3d 5 configuration[END_REF][START_REF] Chaudhury | Thermal expansion and pressure effect in MnWO4[END_REF] . Besides a qualitative proof of structural dimerization, we present a structural refinement that quantifies the structural distortion in NaFe(WO 4 ) 2 . In the last part of this paper we discuss the structural dimerization in MnWO 4 and its relation to the higher harmonic reflections in the incommensurate phase as well as its role in depinning multiferroic domain walls.

II. EXPERIMENTAL METHODS

The experiments on the structural dimerization in NaFe(WO 4 ) 2 were performed on the D10 diffractometer at the Institut Laue-Langevin (ILL). For the expected bond-angle modulation within the magnetic zigzag chain the shift of light oxygen ions is crucial, and therefore neutron diffraction is the appropriate tool. The effect is expected to be small and hence, it was essential to obtain the best possible signal to background ratio. Thus the instrument had to be equipped with the vertically focussing pyrolytic graphite analyzer, which reduces the background significantly, and a single He 3 detector was used. The commensurate phases of NaFe(WO 4 ) 2 are only accessible by applying a magnetic field along b. Thus, the sample was mounted in the (1 0 0)/(0 0 1) scattering plane and a vertical cryomagnet was used (µ 0 H < 6 T). All temperature and magnetic field dependent measurements have been performed with λ = 2.36 Å (pyrolithic graphite monochromator) yielding high flux, and the data collection for a structural refinement was done with λ = 1.26 Å (copper monochromator) in order to reach more reflections. The single crystal with dimensions 1 mm × 5 mm × 10 mm was grown as described in reference 19, which also presents the characterization by various macroscopic methods.

The measurements on MnWO 4 were performed on the triple-axis spectrometer 4F2 at the Laboratoire Léon Brillouin (LLB) using a wavelength λ = 4.05 Å extracted with a highly-oriented pyrolithic graphite doublemonochromator. Higher order contaminations were suppressed by a cooled Be filter and the sample was cooled in a closed-cycle refrigerator. The crystal had dimensions of 2 mm × 4 mm × 9 mm and was mounted in an aluminium can containing He exchange gas. The scattering geometry was (1 0 0)/(0 0 1). Information concerning the crystal growth and characterization are given in references 21, 22, and 32.

III. RESULTS AND DISCUSSION

A. Structural dimerization in NaFe(WO4)2

NaFe(WO 4 ) 2 crystallizes in the P 2/c space group with a ≈ 9.88 Å, b ≈ 5.72 Å, c ≈ 4.94 Å and β ≈ 90.33 • (at 298 K) [START_REF] Klevtsov | Single-crystal synthesis and investigation of the double tungstates NaR 3+ (WO4)2, where R 3+ = Fe, Sc, Ga, and In[END_REF] . The crystallographic unit cell of NaFe(WO 4 ) 2 is doubled along a with respect to the unit cell of MnWO 4 (see Fig. 1(a)). Due to the enhanced distance between adjacent [FeO 6 ]-octahedra chains along a, the magnetic interaction is significantly reduced in this direction, which provokes a more two-dimensional nature of magnetic ordering in NaFe(WO 4 ) 2 . Nevertheless, the system exhibits a cascade of magnetically ordered phases (see Fig. 1(a)) [START_REF] Holbein | Strong magnetoelastic coupling at the transition from harmonic to anharmonic order in NaFe(WO 4 ) 2 with 3d 5 configuration[END_REF] . First, below T N ≈ 4 K an incommensurate three-dimensional magnetic order with k = (0.485, 1 2 , 0.48) develops. The moments are rotating in a plane, which is spanned by the easy plane e ac and a small b component. However, as neighboring spirals rotate in the opposite sense with respect to each other, the DMI effect cancels and no multiferroic state develops in the low-field incommensurate (LF-IC*) phase. Below T ≈ 3 K emerging third-order reflections signal a squaring up of the magnetic structure but the spiral magnetic structure remains incommensurate in zero field at low temperatures (LF-IC phase). When applying a magnetic field at low temperatures (µ 0 H ≈ 2 T at T ≈ 2.5 K), the system undergoes a spin-flip transition to a highfield commensurate uudd arrangement with k = ( 1 2 , 1 2 , 1 2 ) (HF-C phase). While decrasing the magnetic field again (below µ 0 H ≈ 2 T at T ≈ 2.5 K), a spin-flop transition forces a spin-canting but the structure remains commensurate and still exhibits an uudd arrangement. This LF-C phase has been determined to be the groundstate of the system and all transitions from incommensurate to commensurate order are reported to be of first order [START_REF] Holbein | Strong magnetoelastic coupling at the transition from harmonic to anharmonic order in NaFe(WO 4 ) 2 with 3d 5 configuration[END_REF] . Astonishingly, it is not the onset of magnetic ordering that is accompanied by strong magnetoelastic anomalies but rather the emergence of anharmonic modulations and especially the transitions from incommensurate to the commensurate phases, which emphasizes a strong magnetoelastic coupling to them [START_REF] Holbein | Strong magnetoelastic coupling at the transition from harmonic to anharmonic order in NaFe(WO 4 ) 2 with 3d 5 configuration[END_REF] .

The main goal of the neutron experiments was to determine the structural distortion within the commensurate phases of the system. A bond-angle variation along the Fe-chain due to alternating exchange interaction violates the glide plane and the twofold rotation symmetry, but it does not break translation symmetry. Therefore, superstructure reflections 35 will appear at Q = (h 0 l) with l = odd. Fig. 2 displays recorded rocking-scans across several superstructure reflections. All reflections have been measured inside the HF-C phase at T ≈ 2.4 K and within the paramagnetic phase at T ≈ 5.2 K, while applying a magnetic field of µ 0 H = 6 T. Even in the paramagnetic phase a reflection is detectable, whose presence should actually be forbidden due to the P 2/c symmetry. The field induced moment cannot contribute a magnetic signal at these Q values, because it does not FIG. 3. NaFe(WO4)2: For analyzing the occurrence and strength of superstructure reflections in a particular phase, temperature and magnetic field dependent measurements have been performed exemplary for the superstructure reflection Q = (3 0 -1). The temperature and magnetic field dependent sweep path is here defined by the axes of the phase diagram (the white marked phase boundaries are extracted from [START_REF] Holbein | Strong magnetoelastic coupling at the transition from harmonic to anharmonic order in NaFe(WO 4 ) 2 with 3d 5 configuration[END_REF] ), which is plotted in the middle of this figure. The sweep path was traversed clockwise (blue path) and also anticlockwise (red path), as the accessibility of the LF-IC and LF-C phase depends on pre-history of applied fields. The Tand µ0H-sweep speed was low, as the weak signal required a counting time of ≈ 3 minutes per point.

break the glide-mirror symmetry. This is also visible in the magnetic field dependent measurements in Fig. 3 and 4 for Q = (3 0 -1) and Q = (4 0 1) respectively. This Bragg-signal in the paramagnetic phase can arise from either λ/2-contamination or multiple scattering. The λ/2-contamination was determined through rockingscans across the allowed (5 0 2) reflection and the resulting λ/2-contamination at Q = (2.5 0 1). The fraction of λ/2-contamination with respect to the main reflection amounts to ≈ 6 × 10 -4 and is too small to explain the observed Q = (3 0 -1) and Q = (4 0 1) intensities in the paramagnetic state. By measuring the intensity at Q = (60-2), we can conclude that the λ/2-contamination at Q = (30-1) only approaches ≈ 1.32 counts per seconds (see Fig. 2(a)). Thus, multiple scattering almost entirely causes the finite intensities at Q = (30-1) and Q = (401) outside the commensurate phases, but the enhancement of these signals in the commensurate phases cannot stem from multiple diffraction 36 . In Fig. 2, it can be seen that the integrated intensity of all reflections increases, when forcing the system to the HF-C phase. Therefore, we can conclude that the glide plane symmetry is broken, when entering the HF-C phase.

In order to investigate the occurrence and the strength of superstructure reflections in all phases, we performed temperature and magnetic-field dependent sweeps through the phase diagram. Due to the weakness of the signal and the corresponding long counting time, we restricted the data collection on recording only the intensity at the maximum position of the peak. Nevertheless, we checked by some rocking scans, whether the Bragg-peak shifts for different fields and temperatures. As this was not the case, we can exclude a maximum peak intensity modulation due to a shift of the Braggreflection in Q-space. Exemplary for the Q = (3 0 -1) superstructure reflection, the temperature and magneticfield dependent intensity is shown in Fig. 3. Because the commensurate or incommensurate character of the lowfield phases strongly depends on the pre-history of applied magnetic fields in this system, we performed clockwise and anticlockwise sweeps through the phase diagram of NaFe(WO 4 ) 2 . The temperature and magnetic field dependent sweep paths are marked by the axes of the B,T phase diagram in the middle of Fig. 3.

Starting the clockwise cycle (blue symbols in Fig. 3) in the paramagnetic phase at µ 0 H = 0 T and T ≈ 5.2 K, a reduction of the temperature does not change the peak intensity of the superstructure reflection, when passing the two transitions to the incommensurate phase (LF-IC * ) and to the anharmonic distorted incommensurate phase (LF-IC) respectively. Increasing the magnetic field at low temperatures leads to a sudden intensity enhancement at µ 0 H ≈ 1.8 T. At that field value, the system undergoes a transition from an incommensurate magnetic ordering to a commensurate uudd arrangement. The intensity enhancement and hence, the occurrence of a superstructure reflection coincides with the onset of an alternating ferro-and antiferromagnetic exchange along the magnetic zigzag chains. When increasing the temperature at high fields, the intensity drops back again to its initial value right at the transition to the paramagnetic phase at T ≈ 4 K and lowering the field at 5.2 K does not further change the intensity.

The anticlockwise sweep (red symbols in Fig. 3) through the phase diagram recovers the measured intensity of the clockwise sweep, when starting again from the paramagnetic phase in zero field at T ≈ 5.2 K ramping the field up and then cooling at high field. This is not surprising, as the same high-field phase is reached, when cooling below T ≈ 4 K in high fields at µ 0 H = 6 T. However, when decreasing the field at low temperatures, the magnetic arrangement does not return to the incommensurate and metastable low-field phase but a spinflop transition at µ 0 H ≈ 1.8 T leads to the LF-C phase with canted moments that are still ordered in a commensurate uudd arrangement. The superstructure intensity even further increases, when removing the applied field completely and hence, the superstructure reflection is strongest for the groundstate of the system, which leads to the assumption that also the bond-angle variation is strongest for this LF-C phase. The intensity sharply decreases, when the temperature is raised above the transition to the incommensurate LF-IC* phase recovering its initial value, which limits the presence of the structural distortion precisely to the commensurate phases. Qualitatively identical behavior is observed for the superstructure reflection Q = (4 0 1) (see Fig. 4).

B. Quantitative determination of the dimerization in NaFe(WO4)2

For a structural refinement and hence the quantitative analysis of the structural distortion, we collected a set of integrated intensities in the dimerized phase on the D10 diffractometer. As the superstructure signal is strongest within the LF-C phase, see Fig. 3 and4, a structural refinement seems to be most promising to execute within this commensurate phase in zero field. Moreover the data collection in zero field avoids an additional signal contribution from field induced moments to allowed main reflections. However, the application of a magnetic field was still necessary in order to drive the system to the LF-C phase.

Before we collected respective superstructure reflections, we measured (with a microstrip detector and without additional analyzer) 108 unique allowed nuclear reflections of which 56 are independent within the LF-IC phase in order to refine the isotropic extinction coeffi-cient. The refinement was done by using the software JANA [START_REF] Petříček | Crystallographic computing system JANA2006: General features[END_REF] , which utilizes the Becker-Coppens formalism for the extinction correction. For the corresponding refinement we used the structural parameters that have been obtained by Holbein et al. [START_REF] Holbein | Strong magnetoelastic coupling at the transition from harmonic to anharmonic order in NaFe(WO 4 ) 2 with 3d 5 configuration[END_REF] . Hereafter we measured with a single He 3 -detector and with the analyzer option to improve the intensity over background ratio. In order to obtain the appropriate scaling factor for the refinement of the superstructure, we measured 13 unique allowed reflections, of which 7 are independent at T ≈ 2 K and µ 0 H = 0 T.

It was already discussed that the intensity of superstructure peaks superposes the multiple scattering signal and the tiny λ/2-contamination and thus, it was necessary to measure the respective superstructure reflections inside and outside the commensurate phases in order to extract the integrated intensity that is purely originating from the structural distortion. Hence, we measured these superstructure reflections in the LF-C phase at T ≈ 2 K, µ 0 H = 0 T as well as in the LF-IC phase (also at T ≈ 2 K, µ 0 H = 0 T). This procedure and the weak signal itself limit the number of collectible reflections and we were able to collect 26 superstructure reflections of which 24 are independent. The required vertical cryomagnet with its magnetic field along b and the analyzer option prevented the collection of reflections with a non-zero kcomponent and thus the structural refinement will only testify a structural distortion within the ac-plane.

The left part of Table I contains the structural parameters that have been refined by Holbein et al. in the paramagnetic phase [START_REF] Holbein | Strong magnetoelastic coupling at the transition from harmonic to anharmonic order in NaFe(WO 4 ) 2 with 3d 5 configuration[END_REF] . All Fe ions are surrounded by edge sharing oxygen octahedra, and when considering the P 2/c space group, all oxygen ions along the Fe-chain are generated from the general site O2. The distance between two O2's belonging to the same edge alternate in the dimerized phase resulting in alternating enhancement and reduction of the Fe-O2-Fe bond angle and thus in an alternation of the magnetic exchange energy. In the simple picture neglecting ligands, a ferromagnetic arrangement prefers a 90 • Fe-O-Fe bond-angle, whereas an antiferromagnetic bond requires the bond angle to approach 180 • . Both O2 ions on the same edge are connected through inversion symmetry, and as they elongate or shorten their distance uniformly, the inversion symmetry is preserved in a dimerized state. However, an alternating exchange interaction along the chain and the corresponding bondangle modulation violate the two-fold axis and the glide mirror symmetry giving rise to the observed superstructure reflections. Thus, the refinement of the superstructure within the LF-C phase was done by considering a transition from space group P 2/c to P 1. The middle part of Table I displays the structural parameters transformed from P 2/c to P 1 setting and the right part the refined distortion in the lower symmetry.

The transformation to space group P 1 splits the site of O2 in P2/c into the two independent sites O2a and O2b. The Fe-O-Fe bonding along the chains passes alternatingly through O2a-O2a and O2b-O2b edges. In TABLE I. The left part of this table contains the structural parameters that have been determined in the paramagnetic phase at T = 12 K by Holbein et al. [START_REF] Holbein | Strong magnetoelastic coupling at the transition from harmonic to anharmonic order in NaFe(WO 4 ) 2 with 3d 5 configuration[END_REF] . All positions have be converted to space group P 1 as given in the middle part of this table. The right part consists of the refined structural parameters within the LF-C phase and by assuming P 1 symmetry. For the refinement 26 unique superstructure reflection of which 24 are independent have been used. Moreover 13 unique allowed reflections of which 7 are independent have been added to the refinement. In order to reduce the number of variables, the following constraints have been utilized: the refinements we kept the average O2 position fixed and allowed only for alternating O2 shifts. In order to further reduce the number of variables, equivalent constraints have been also stated for the W1 and W2 positions. 6 variables have been refined. The right part of Table I contains the refined parameters of the P 1 structure within the LF-C phase. For the refinement, the isotropic B parameters have been adopted from [START_REF] Holbein | Strong magnetoelastic coupling at the transition from harmonic to anharmonic order in NaFe(WO 4 ) 2 with 3d 5 configuration[END_REF] . At first glance, the obtained R-values (see Table I) may appear large but one has to keep in mind that only very weak superstructure reflections (almost 4 orders of magnitude weaker than strong fundamental Bragg peaks) intervene in these refinements. In Fig. 5 we plot the observed structure factors against the calculated ones documenting the reliability of this analysis. The O2a and O2b sites are shifting significantly in opposite directions by 0.0195(32) Å. The Fe ions are only moving along z by 0.0093(31) Å and the position of W1 and W2 is also shifting because of their bonding to the O2a and O2b sites. Within the commensurate LF-C phase, the calculated bond angles amount α 1 = 97.20 (14) • for the Fe-O2a-Fe bond and α 2 = 99.49 (14) • for the Fe-O2b-Fe bond, whereas the bond angle outside the commensurate phase amounts α = α 1 = α 2 = 98.35(7) •19 . Hence, a bondangle variation of ≈ ±1.15 (16) • can be observed, when entering the commensurate uudd phase. This modulation includes the shift of oxygen and iron ions within the ac-plane. A respective shift along b cannot be studied with the chosen scattering geometry.

x[O2a]=x[O2b]-0.78224, x[W1]=x[W2]-0.52592, z[O2a]=z[O2b]-0.
The implication of the bond angle, φ, on the magnetic exchange interaction has been intensively studied for RM eO 3 with M e=Fe and Mn finding a sin 4 (φ/2) relation [START_REF] Zhou | Intrinsic structural distortion in orthorhombic perovskite oxides[END_REF][START_REF] Dong | Origin of multiferroic spiral spin order in the rmno3 perovskites[END_REF] . This strong bond-angle dependence of the magnetic interaction drives the magnetoelectric coupling in E-type RMnO 3 [START_REF] Sergienko | Ferroelectricity in the Magnetic E -Phase of Orthorhombic Perovskites[END_REF][START_REF] Ishiwata | Perovskite manganites hosting versatile multiferroic phases with symmetric and antisymmetric exchange strictions[END_REF] and causes the dynamic magneto- electric coupling and the strongest electromagnon modes in chiral RMnO 3 [START_REF] Valdés Aguilar | Origin of Electromagnon Excitations in Multiferroic RMnO3[END_REF] . However, this sin 4 (φ/2) relation applies to a bond angle, which is not too far below 180 • , while the O-Fe-O angles in NaFe(WO 4 ) 2 amount to 97.2 and 99.5 degrees. Therefore, the arrangement more closely resembles the dimerization in the spin-Peierls compound CuGeO 3 , in which a bond-angle alternation of only ±0.5 • around the average value of ∼99 • results in a sizeable variation of the magnetic exchange by ±20 percent [START_REF] Braden | Structural analysis of CuGeO3: Relation between nuclear structure and magnetic interaction[END_REF] . Unfortunately, the magnetic exchange parameters in NaFe(WO 4 ) 2 are not known. For MnWO 4 the nearest-neighbor intra-chain is not dominating and several parameters are of the order of 0.3 to 0.4 meV [START_REF] Ye | Long-range magnetic interactions in the multiferroic antiferromagnet mnwo4[END_REF][START_REF] Tian | Magnetic structure and ferroelectric polarization of MnWO4 investigated by density functional calculations and classical spin analysis[END_REF] explaining the considerable frustration. The antiferromagnetic Curie-Weiss temperature in NaFe(WO 4 ) 2 amounts to -8.5 K (compared to -75 K for MnWO 4 [START_REF] Tian | Magnetic structure and ferroelectric polarization of MnWO4 investigated by density functional calculations and classical spin analysis[END_REF] ), which transforms to a nearest-neighbor intrachain interaction of only 0.06 meV when neglecting all other interactions. NaFe(WO 4 ) 2 seems to be less frustrated with a ratio of Néel and Weiss temperatures of only ∼2, but the nextnearest neighbor interaction along the chains will stabilize the uudd magnetic order. Since the magnetic interaction in NaFe(WO 4 ) 2 is quite weak the relative change induced by the sizeable structural dimerization through the bond-angle alternation is expected to be rather strong.

The strong reduction of the b lattice parameter upon entering the commensurate magnetic order in NaFe(WO 4 ) 2 [START_REF] Holbein | Strong magnetoelastic coupling at the transition from harmonic to anharmonic order in NaFe(WO 4 ) 2 with 3d 5 configuration[END_REF] is not contributing to the dimerization, which can be considered as the primary order parameter of the structural distortion. Note, that the strain does not break the glide mirror symmetry. Since both distortions correspond to zone-center phonon modes, one would expect a bilinear coupling, but the coupling with the magnetism renders the analysis more complex. The commensurate magnetic order clearly is the the primary order parameter and both structural distortions, dimerization and strain along b, are coupled to it. The dimerization modulates the nearest-neighbor interaction, thereby rendering the uudd structure more favorable. The strain along b acts on the ratio between nearest and next-nearest neighbor interaction. One expects an enhanced second-nearest neighbor interaction for smaller b, which again will favor the commensurate uudd order. Since the two types of distortion act differently, one can understand their different relative strengths in NaFe(WO 4 ) 2 and MnWO 4 .

C. Structural dimerization in MnWO4

MnWO 4 crystallizes in the monoclinic space group P 2/c with a ≈ 4.823 Å, b ≈ 5.753 Å, c ≈ 4.992 Å and β ≈ 91.08 • (at 1.5 K) [START_REF] Lautenschläger | Magnetic phase transitions of MnWO4 studied by the use of neutron diffraction[END_REF] . Within the bc-plane, magnetic Mn 2+ ions are forming edge-sharing [MnO 6 ] octahedra chains along c and those layers, containing the chains are separated along the a direction by nonmagnetic [WO 6 ]octahedra chains [START_REF] Weitzel | Kristallstrukturverfeinerung von Wolframiten und Columbiten[END_REF] . The crystallographic unit cell of the parent wolframite structure is bisected along a with respect to the structure of NaFe(WO 4 ) 2 due to the missing [NaO 6 ]-layer. Similar to NaFe(WO 4 ) 2 , the system exhibits also a sequence of magnetic phases with competing commensurate and incommensurate ordering [START_REF] Ehrenberg | Magnetic phase diagrams of MnWO4[END_REF][START_REF] Lautenschläger | Magnetic phase transitions of MnWO4 studied by the use of neutron diffraction[END_REF] . First, below T N 3 ≈ 13.5 K an incommensurate modulated spindensity wave with k = (-0.214, 1 2 , 0.457) emerges (AF3phase). The magnetic moments point along the easy direction, which lies within the ac-plane and forms an angle of about ≈ 35.5 • with the a-direction. In this configuration, the magnetic moments cannot saturate and thus due to entropy reasons a spiral with an additional component along the b-direction develops at T N 2 ≈ 12.3 K but the modulation remains incommensurate (AF2-phase) [START_REF] Ehrenberg | Magnetic phase diagrams of MnWO4[END_REF][START_REF] Lautenschläger | Magnetic phase transitions of MnWO4 studied by the use of neutron diffraction[END_REF] . The spiral arrangements allows for the occurrence of a ferroelectric polarization following DMI and indeed, it was possible to measure an electric polarization along b, which is fully switchable by external electric fields [START_REF] Niermann | Domain dynamics in the multiferroic phase of MnWO4[END_REF][START_REF] Baum | Kinetics of the multiferroic switching in MnWO4[END_REF] .

The polarization is small but sizeable (P ≈ 60 µC m -2 ) and its existence is limited to the spiral phase AF2 [START_REF] Arkenbout | Ferroelectricity in the cycloidal spiral magnetic phase of MnWO4[END_REF][START_REF] Taniguchi | Ferroelectric Polarization Flop in a Frustrated Magnet MnWO4 Induced by a Magnetic Field[END_REF] . However, due to single ion anisotropy the system preferably aligns along its easy direction, which provokes the transition to a commensurate uudd arrangement with k = (-1 4 , 1 2 , 1 2 ) at T N 1 ≈ 7.5 K [START_REF] Ehrenberg | Magnetic phase diagrams of MnWO4[END_REF][START_REF] Lautenschläger | Magnetic phase transitions of MnWO4 studied by the use of neutron diffraction[END_REF] . As in NaFe(WO 4 ) 2 , the transition to the commensurate phase is of first order and accompanied by a thermal expansion anomaly. It is expected that a structural distortion is here also caused by a bond-angle modulation due to the alternating ferro-and antiferromagnetic exchange interaction along the magnetic Mn 2+ chain. The observed thermal expansion anomaly in MnWO 4 is one order of magnitude smaller compared to NaFe(WO 4 ) 2 but its commensurate phase is accessible without the application of a magnetic field. The symmetry consideration from NaFe(WO 4 ) 2 can be adopted to MnWO 4 and it becomes obvious that corresponding superstructure reflections occur at Q = (h+ 1 2 0 l) with l=odd for MnWO 4 . X-ray measurements have already qualitatively confirmed the presence of a respective signal in the context of second harmonic order reflections within the AF1 phase [START_REF] Taniguchi | Magnetic-field dependence of the ferroelectric polarization and spinlattice coupling in multiferroic MnWO4[END_REF] .

Neutron scattering experiments were performed on the cold triple-axis spectrometer 4F2, where we first executed longitudinal scans over the Q-space position of the superstructure reflection Q = (1.5 0 1) for different temperatures. At these noninteger positions no multiple scattering can contribute to the signal. From the measured data, which is displayed in Fig. 6(a) it can be clearly stated that at the transition to the low-lying commensurate phase, a superstructure reflection evolves. We measured the peak intensity, while lowering the temperature from above T N 1 to far below the transition of the commensurate phase (see Fig. 6(b)). This temperature dependent sweep demonstrates that the superstructure reflection is exactly evolving, when passing the latter mentioned transition. Hence, also in MnWO 4 the occurrence of the commensurate uudd arrangement coincides with an evolving superstructure. In view of the equivalent behavior in NaFe(WO 4 ) 2 , we conclude that also in MnWO 4 a bond-angle modulation is coupled to the onset of commensurate uudd ordering. Comparing the intensities of superstructure reflections Q = (2h 0 l) for NaFe(WO 4 ) 2 and Q = (h 0 l) for MnWO 4 (h=1.5) with respect to the particular strongest allowed reflections (Q = (0 2 3) 46 in NaFe(WO 4 ) 2 and Q = (1 0 0) in MnWO 4 ) results in almost equal ratios of a few 10 -4 . This indicates a bondangle modulation of the same order of magnitude in the two systems.

Locally, the commensurate uudd structure can also evolve within the multiferroic phase and it was proposed for NaFe(WO 4 ) 2 that an anharmonic modulation of the incommensurate structure within its incommensurate LF-IC phase corresponds to tiny fragments of commensurate order [START_REF] Holbein | Strong magnetoelastic coupling at the transition from harmonic to anharmonic order in NaFe(WO 4 ) 2 with 3d 5 configuration[END_REF] that are regularly arranged. This assumption is supported by the occurrence of third order reflections, which signal a squaring up of the cycloid [START_REF] Holbein | Strong magnetoelastic coupling at the transition from harmonic to anharmonic order in NaFe(WO 4 ) 2 with 3d 5 configuration[END_REF] . Fig. 7(a)-(c) presents rocking scans of the higher harmonic reflections as a function temperature for MnWO 4 . In Fig. 7(d) and (e), the peak intensity is plotted as a function of temperature. The temperature dependence of the first and of the second order reflections resembles reported measurements 21 but the occurrence of a third order reflection has not been reported so far. Fig. 7(e) displays the temperature dependence of the measured peak intensity of the third order reflection Q = (-0.642 1 2 1.371). At low temperatures this reflection becomes even stronger than the second-order reflection and its existence is not limited to the multiferroic phase but a finite signal persists in the AF3 phase. While the second-order intensity clearly follows the intensity of the main magnetic peak, the third-order reflection exhibits a different temperature dependence. The first-and secondorder peaks tend to saturate towards the lower temperature limit of the multiferroic AF2 phase, but the growth of the third-order peak continuously increases showing a negative second temperature derivative. This strongly resembles the behavior in NaFe(WO 4 ) 2 [START_REF] Holbein | Strong magnetoelastic coupling at the transition from harmonic to anharmonic order in NaFe(WO 4 ) 2 with 3d 5 configuration[END_REF] and this diverging third-order peak must be considered as a precursor of the phase transition into the commensurate uudd phase. Tiny commensurate fragments with the corresponding discommensurations strongly increase upon cooling towards the low-temperature limit of the AF2 phase. These perturbations of the harmonic multiferroic magnetic structure can explain the depinning of multiferroic ordering reported in the various studies on the the dynamics of multiferroic domains [START_REF] Niermann | Domain dynamics in the multiferroic phase of MnWO4[END_REF][START_REF] Baum | Kinetics of the multiferroic switching in MnWO4[END_REF]47 .

IV. SUMMARY AND CONCLUSION

We present a comprehensive study of a dimerization effect in NaFe(WO 4 ) 2 and MnWO 4 . It was possible to qualitatively observe the existence of superstructure reflections documenting the loss of symmetry, when entering the respective commensurate magnetic phases of both materials. This loss of symmetry is connected to a bond-angle modulation along the magnetic zigzag chains due to the alternating ferromagnetic and antiferromagnetic arrangement in the respective commensurate uudd phases. For NaFe(WO 4 ) 2 a bond-angle modulation is present in the HF-C as well as in the LF-C phase and it turned out that the dimerization effect is strongest for the commensurate phase in zero field. In this phase, a structural refinement of neutron data yields a bond-angle modulation of ≈ ±1.15 (16) • but it has to be noted that this specified bond-angle modulation doesn't include a possible modulation along the b-direction, as the chosen scattering plane and the utilized cryomagnet and analyzer prevented the collection of reflections with a finite k-component.

Also for MnWO 4 , a structural dimerization of comparable size was detected by the appearance of superstructure reflections in its commensurate phase. Moreover, strong harmonic third order reflections have been observed within the multiferroic phase that perfectly correspond to the anharmonic modulation reported for NaFe(WO 4 ) 2 [START_REF] Holbein | Strong magnetoelastic coupling at the transition from harmonic to anharmonic order in NaFe(WO 4 ) 2 with 3d 5 configuration[END_REF] . The emerging third-order reflection is connected to a squaring up of the magnetic ordering and indicates the presence of small commensurate fragments already in the incommensurate and multiferroic phase. As the magnetic arrangement of MnWO 4 already starts to resemble the uudd arrangement within the incommensurate spiral phase, it can lead to depinning of the mul-tiferroic domains and of the corresponding domain walls. The pinning of the multiferroic domains bases on the harmonic modulation, which causes the ferroelectric polarization through the inverse DMI. Therefore, the anomalous speeding up of the multiferroic domain inversion can be attributed to the emergence of the anharmonic modulations. Such effects can even be beneficial for the application of multiferroics.

Single-crystal neutron diffraction data from the D10 diffractometer are available 48 .

FIG. 1

 1 FIG. 1. (a) and (b) display the nuclear and magnetic structure of NaFe(WO4)2 and MnWO4, respectively. The unit cells are marked by black boxes and arrows indicate the direction of magnetic moments within the respective magnetic phases. Note the similarities between the incommensurate spiral phases AF2 and LF-IC, as well as between the commensurate uudd phases AF1 and HF-C / LF-C.

FIG. 2 .

 2 FIG.2. NaFe(WO4)2: (a) -(d) display rocking-scans over the Q-space position of superstructure reflections. For all measured reflections, the respective rocking scans have been executed in the high-field paramagnetic phase (HF-PM) at T ≈ 10 K and µ0H = 6 T as well as within the high-field commensurate phase (HF-C) at T ≈ 2.4 K and µ0H = 6 T.

FIG. 4 .

 4 FIG. 4. NaFe(WO4)2: Similar to Fig. 3 this figure presents the temperature and magnetic field dependence of the the peak intensity for the superstructure reflection Q = (4 0 1) in NaFe(WO4)2 . The blue and red arrows indicate the clockwise or anticlockwise paths in the B,T space respectively.

FIG. 5 .

 5 FIG.5. The observed structure factors are plotted against the calculated ones for the final refinement of the structural dimerization within the LF-C phase.

FIG. 6 .

 6 FIG. 6. MnWO4 : Panel (a) shows the longitudinal scans over Q = (H 0 L) for four different temperatures. The measured peak intensity for Q = (1.5 0 1) is displayed in panel (b) as a function of temperature.

FIG. 7 .

 7 FIG. 7. MnWO4 : Panels (a), (b) and (c) present rocking scans over the first (Q = (-0.214 1 2 0.457)), second (Q = (-0.428 1 0.914)) and third harmonic reflection (Q = (-0.642 1 2 1.371)) for different temperatures. The respective temperature dependence of the peak intensities is shown in (d) and (e).

  3154 and z[W1]=z[W2]+0.0144. The structural refinement yielded reliability factors of R obs = 7.07, wR obs = 16.44, R all = 9.13 and wR all = 18.54 for structure factors.

		Paramagnetic phase (spacegroup P 2/c)	Paramagnetic phase (spacegroup P 1) LF-C phase (spacegroup P 1)
		x	y	z	Biso	x	y	z	x	y	z
	Fe	0.0	0.67074(19)	0.25	0.04(2) Fe	0.0	0.67074(19)	0.25	-0.0002(2) 0.67074 0.2525(7)
	Na	0.5	0.6971(6)	0.25	0.35(5) Na	0.5	0.6971(6)	0.25	0.5	0.6971	0.25
	W 0.23704(14) 0.1831(2) 0.2572(3) 0.12(3) W1 0.23704(14) 0.1831(2)	0.2572(3)	0.2383(3) 0.1831 0.2562(8)
	O1 0.35385(12) 0.3813(3) 0.3816(3) 0.25(3) W2 0.76296(14) 0.1831(2)	0.2428(3)	0.7642(3) 0.1831 0.2418(8)
	O2 0.10888(13) 0.6226(3) 0.5923(3) 0.22(2) O1a 0.35385(12) 0.3813(3)	0.3816(3)	0.35385 0.3813	0.3816
	O3 0.33177(13) 0.0897(2) 0.9533(3) 0.22(3) O1b 0.64615(12) 0.3813(3)	0.1184(3)	0.64615 0.3813	0.1184
	O4 0.12606(13) 0.1215(3) 0.5757(3) 0.17(2) O2a 0.10888(13) 0.6226(3)	0.5923(3)	0.1111(2) 0.6226 0.5935(8)
					O2b 0.89112(13) 0.6226(3)	0.9077(3)	0.8933(2) 0.6226 0.9089(8)
					O3a 0.33177(13) 0.0897(2)	0.9533(3)	0.33177 0.0897	0.9533
					O3b 0.66823(13) 0.0897(2)	0.5467(3)	0.66823 0.0897	0.5467
					O4a 0.12606(13) 0.1215(3)	0.5757(3)	0.12606 0.1215	0.5757
					O4b 0.87394(13) 0.1215(3)	0.9243(3)	0.87394 0.1215	0.9243

* e-mail: braden@ph2.uni-koeln.de 1 T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, and Y. Tokura, "Magnetic control of ferroelectric polariza-

V. ACKNOWLEDGEMENTS

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Project number 277146847 -CRC 1238, projects A02 and B04.