Magnetic phase diagram of the solid solution LaMn2(Ge1−xSix)2 (0 ≤ x ≤ 1) unraveled by powder neutron diffraction

The structural and magnetic properties of the ThCr2Si2-type solid solution LaMn2(Ge1−xSix)2 (x = 0.0 to 1.0) have been investigated employing a combination of X-ray diffraction, magnetization and neutron diffraction measurements, which allowed establishing a magnetic composition-temperature phase diagram. Substitution of Ge by Si leads to a compression of the unit cell, which affects the magnetic exchange interactions. In particular, the magnetic structure of LaMn2(Ge1−xSix)2 is strongly affected by the unit cell parameter c, which is related to the distance between adjacent Mn layers. Commensurate antiferromagnetic layers and a canted ferromagnetic structure dominate the Si-rich part of the solid solution, whilst an incommensurate antiferromagnetic flat spiral and a conical magnetic structure are observed in the Si-poor part.

. Lattice parameters a and c, the cell ratio c/a and the unit cell volume V of LaMn 2 (Ge 1-x Si x ) 2 (x = 0, 0.05, 0.18, 0.33, 0.47, 0.58, 0.78, 1) from refined PXRD data. The Curie temperature T C and approximate values for the saturation magnetization M sat b at 2 K and 6 T were determined from the magnetization measurements. The numbers between parentheses show the error bars and represent 1σ. a Sample contains minor impurity, see "Methods" and Supplementary Information. b No saturation observed for x = 0.58, 0.78, 1 under the maximum applied fields. c The standard deviations are calculated from the fits. The experimental errors for M sat are estimated to be of the order of 10 −3 µ B /Mn.  (T) do not follow a linear behavior (not shown) up to 400 K and suggest that LaMn 2 Ge 2 and LaMn 2 Si 2 do not fully enter the paramagnetic regime up to the highest measured temperature. This is also supported by the powder neutron diffraction data presented below. Similar behavior of the DC magnetic susceptibility was also observed for the quaternary samples ( Supplementary Fig. S3). T C decreases from LaMn 2 Ge 2 to LaMn 2 Si 2 as a function of the Ge/Si mixing ( Table 1). The Curie temperatures of all samples were determined by fitting the first derivative of the susceptibility dχ DC /dT with a Gaussian peak function ( Supplementary Fig. S4a,b). The transition temperatures extracted from our data are in fair agreement with the literature 36 . Thus, the T C values reported in previous studies go from 323.3(2) K in LaMn 2 Ge 2 to 308.5(2) K in LaMn 2 Si 2 .
The isothermal magnetization curves of the ternary LaMn 2 Ge 2 and LaMn 2 Si 2 measured at 2 K and 250 K between − 6 and 6 T exhibit typical ferromagnetic behavior (Fig. 3 insets). For LaMn 2 Ge 2 , the M(H) curve reaches M sat = 1.3005(1) µ B /Mn at 2 K and M sat = 1.0976(1) µ B /Mn at 250 K. In contrast, the M(H) curve of LaMn 2 Si 2 shows no saturation under the same applied magnetic fields at 2 K, but reaches saturation with M sat = 0.99823(4) µ B /Mn at 250 K. A similar behavior to LaMn 2 Si 2 is also observed for the Si-rich quaternary samples with the composition x = 0.58 and 0.78 ( Supplementary Fig. S3): They reach saturation at 250 K, but not at 2 K. This lack of saturation possibly indicates the presence of an antiferromagnetic component 38 . Furthermore, the low saturation magnetization observed for all compositions points toward a magnetic structure different from that of a simple ferromagnet, which will be discussed below. Hysteresis loops were observed for all compositions. Supplementary  Fig. S5a,b highlight the isothermal magnetization of all samples between − 1 and 1 T. On raising the temperature from 2 to 250 K, the coercive field H c of the samples with compositions x = 0.33, 0.47, 0.58, 0.78 and 1 decreases clearly whereas H c remains nearly constant for x = 0, 0.05 and 0.18 (Supplementary Table S1). The change of the coercive field on warming hints at a magnetic phase transition occurring between 2 and 250 K which is also confirmed by the PND data presented below. The different coercive fields between samples are attributed to variations in particle size 39 . The magnetization data observed here are in line with the literature 15,38,40,41 . As the samples only contain small amounts of impurities, we believe the impurities do not affect the magnetization data in a significant way.
Powder neutron diffraction measurements. Powder neutron diffraction (PND) patterns were collected for a series of samples with the composition LaMn 2 (Ge 1−x Si x ) 2 (x = 0, 0.05, 0.18, 0.33, 0.47, 0.58, 0.78, 1). Following previous works 11,28,29,[31][32][33]42 , we will describe the observed magnetic structures based on elementary magnetic components. The latter can be readily identified from characteristic magnetic reflections: 1. The antiferromagnetic flat spiral (AFfs) can be described as an antiferromagnetic alignment of magnetic moments within the square lattice for each Mn layer. The spin motive of each layers is the same but the moments are rotated by an angle φ in adjacent layers, by 2φ in case of the next-nearest layer and so on. Therefore, the magnetic moments in AFfs form a flat spiral along the c-axis. (Fig. 4a) The incommensurate propagation vector (0, 0, k z ) describes the length of the spiral or how many crystal unit cells are necessary until the magnetic moments have reached a full rotation. In the diffraction patterns, the AFfs can be identified by pairs of low intensity, magnetic modulation peaks appearing around reflections with the diffraction Figure 3. Zero field-cooled (ZFC, red) and field-cooled (FC, blue) magnetic susceptibilities and isothermal magnetization (insets) of LaMn 2 Ge 2 (a) and LaMn 2 Si 2 (b). All error bars are shown and represent 1σ. However, the error bars may be smaller than the symbol.  (101) and (103), respectively. 2. The structure of antiferromagnetic layers (AFl) consists of the same antiferromagnetic arrangement of magnetic moments within the square lattice as in the case of AFfs. However, the moments in adjacent layers are rotated by 180° along c. (Fig. 4b) The magnetic reflections of AFl can be indexed with a k-vector of (0, 0, 0) and add intensity to nuclear Bragg peaks with the reflection condition h + k = 2n + 1. The magnetic signal of the AFl contribution is especially visible for (101) and (103). 3. In the ferromagnetic (FM) component, all magnetic moments are aligned along c (Fig. 4c). The FM contribution is found on nuclear Bragg peaks fulfilling the reflection conditions h + k = 2n and l = 2n. Therefore, the FM Bragg peaks increase the intensity of nuclear Bragg peaks. This is most noticeable for the reflections (002) and (112).
Before presenting the results of the PND studies, we would like to provide some general comments about the magnetic structures: Reflection condition (1) points to incommensurate magnetic modulation (IC), while conditions (2) and (3) indicate commensurate magnetic reflections (C). The Bragg markers corresponding to the magnetic phases in the PND patterns presented below are separated into IC and C contributions. The sets of magnetic peaks corresponding to either of the magnetic components (1) or (2) can be observed in PND patterns in the absence of other magnetic reflections, suggesting that these two elementary components represent actual magnetic structures. In addition, more complex magnetic arrangements result from combinations of the elementary contributions listed above: 4. The ferromagnetic mixed incommensurate structure (Fmi) is a superposition of the in-plane component (1) and the out-of-plane component (3), and is characterized by a conical magnetic structure with the cone axis parallel to c (Fig. 4d). This type of structure is referred to as conical as the magnetic moments appear to rotate in a conical fashion. Due to the FM contribution to Fmi, all magnetic moments lie parallel to c which results in an overall non-zero net moment. Additionally, there is the non-zero contribution of the AFfs with an antiferromagnetic arrangement in the basal plane. Similar to AFfs, the magnetic moments of Fmi are rotated by an angle φ from layer to layer. 5. The ferromagnetic mixed commensurate state (Fmc) is a superposition of (2) and (3)-the resulting structure is similar to AFl with the same antiferromagnetic in-plane arrangement and antiferromagnetic coupling between neighboring layers, but the magnetic moments are canted out-of-plane. Thus, Fmc exhibits an additional ferromagnetic coupling along c (Fig. 4e).
It should be noted that the FM component (3) is only observed in combination with AFfs (1) and AFl (2) in the Fmi and Fmc structures, and thus, is not an independent magnetic structure of LaMn 2 (Ge 1−x Si x ) 2 . The superposition of magnetic components can be understood as an addition of vectors. Adding an out-of-plane and an in-plane magnetic component will result in a canted magnetic structure. The canting angle of such a noncollinear structure is defined by the ratio of the vector lengths. The individual components are therefore projections onto either the ab-plane (AFfs and AFl) or the c-axis (FM). Figure 4 illustrates the spin arrangements for all three diffraction conditions and the two observed superpositions of the elementary magnetic contributions. www.nature.com/scientificreports/ LaMn 2 Ge 2 . Neutron diffraction data of the ternary LaMn 2 Ge 2 were collected between 28 and 500 K. Refinements confirm that LaMn 2 Ge 2 is paramagnetic at 430 K. Below 420 K, magnetic satellite peaks consistent with diffraction condition (1) occur around the (101) and (103) reflections (Fig. 5). They can be indexed with the propagation vector (0, 0, k z ), and their intensities as well as k z increase with decreasing temperatures. The magnetic structure is a pure antiferromagnetic flat spiral (AFfs) 28 (Fig. 4a). The ordering temperature observed here is in good agreement with previous studies 28,34 . At 330 K, slightly above a ferromagnetic-like transition observed in the magnetic susceptibility, the nuclear peaks following reflection condition (3) gain intensity. This increase is most clearly visible on the (112) reflection, as its nuclear contribution is negligible. Diffraction condition (3) describes the ferromagnetic contribution (FM), in which the moments align parallel to c (Fig. 4c). The magnetic signal attributed to FM co-exists with the satellite peaks (101) − /(101) + and (103) − /(103) + of AFfs down to low temperatures. As discussed above, the superposition of an in-plane AFfs and an out-of-plane FM contribution forms the ferromagnetic mixed incommensurate structure (Fmi, Fig. 4d), reported previously 15 . Figure 6a shows the temperature dependence of the total magnetic moment µ tot of LaMn 2 Ge 2 and its partial components µ AFfs and µ FM derived from the data refinements. The magnetic transition temperatures from PND were defined where an abrupt drop in the magnetic moment is observed, as can be seen in Fig. 6 and is indicated by the vertical dash-dotted line. The same methodology was applied to all samples. As it is an approximate value, error propagation is not considered. At 28 K, LaMn 2 Ge 2 reaches magnetic moments of µ tot ≈ 3.13(3) µ B , µ AFfs ≈  Magnetic and structural parameters of LaMn 2 Ge 2 derived from PND data refinements: (a) Temperature dependence of the total magnetic moment µ tot , its partial components µ AFfs , µ FM , and the propagation vector k z (inset). (b) Change of the lattice parameters a and c, the unit cell volume V and the cell ratio c/a as a function of temperature. All error bars are shown and represent 1σ. However, the error bars may be smaller than the symbol. The dashed lines connecting neighboring points were added to guide the eye. The vertical dash-dotted lines indicate the magnetic transition temperatures.  (2). At this temperature, the magnetic moment is canted from the c-axis by an angle of α ≈ 59.1(4)°. The value of µ FM refined from the PND data is slightly larger than the M sat value of 1.3005(1) observed in the isothermal magnetization but is in line with the approximately 1.5 µ B /Mn reported in the literature 15,40,[43][44][45] . The lower value of M sat determined here might be explained by a non-magnetic amorphous impurity or hindered domain wall motion preventing complete saturation of the magnetization. µ tot decreases for increasing temperatures, makes a stronger downturn close to T C before it vanishes abruptly at 430 K. The intermediate dip at T C also occurs in µ AFfs and k z (Fig. 6a inset).  Temperature dependence of the total magnetic moment µ tot , its partial components µ AFfs , µ FM , µ AFl , and the propagation vector k z (inset) derived from the PND refinements. All error bars are shown and represent 1σ. However, the error bars may be smaller than the symbol. The dashed lines connecting neighboring points were added to guide the eye. The vertical dash-dotted lines indicate the magnetic transition temperatures. . In contrast to related solid solutions with a co-existence of the Fmc structure and the antiferromagnetic mixed commensurate phase (AFmc) 11,29,31,33,46 , a co-refinement of the Fmi and Fmc structures could not be performed for LaMn 2 (Ge 0.95 Si 0.05 ) 2 . Since both these magnetic phases share the FM contribution, FM could not be unambiguously partitioned between Fmi and Fmc. Instead, the elementary AFfs, AFl, and FM components were refined individually.
Below 300 K, the AFl contribution vanishes, and only the Fmi magnetic structure remains. The Mn magnetic moment obtained from the refinements at 14 K drops with respect to LaMn 2 Ge 2 down to µ tot ≈ 2.99(4) µ B and µ AFfs ≈ 2.52(3) µ B . Similarly, the refined value of k z ≈ 0.2907(3) is marginally smaller than the value observed in LaMn 2 Ge 2 . The ferromagnetic moment, on the other hand, stays relatively constant at µ FM ≈ 1.61(6) µ B , which leads to a slightly smaller angle α ≈ 57.5(6)°. The temperature dependence of µ tot , µ AFfs and k z shows a strong resemblance to the ternary LaMn 2 Ge 2 (Fig. 7b). In the temperature region of the co-existing AFl and AFfs phases, µ tot was numerically calculated from µ AFl and µ AFfs and the refinement errors were estimated using the error propagation formula. The in-plane moment of the co-existing µ AFl and µ AFfs were calculated by averaging over the vector sum using the integral: The integral averages over all possible angles ω between µ AFl and µ AFfs . For the co-existing Fmi and Fmc phases, the in-plane component calculated using the integral above was combined with the ferromagnetic out-of-plane µ FM component using the Pythagorean equation. 0.18, 0.33, 0.47, 0.58). A further increase in the Si fraction in LaMn 2 (Ge 1−x Si x ) 2 leads to a continuous increase of T N and the disappearance of the AFfs structure. In the samples with compositions LaMn 2 (Ge 1−x Si x ) 2 (x = 0.18, 0.33, 0.47, 0.58), only AFl could be observed above T C (Fig. 8). LaMn 2 (Ge 0.67 Si 0.33 ) 2 retains this magnetic structure up to at least 475 K. In LaMn 2 (Ge 0.53 Si 0.47 ) 2 , the paramagnetic regime was not reached even at 500 K. For the other two samples (x = 0.18, 0.58) no data were collected at such high tempera- AFl + µ 2 AFfs + 2µ AFl µ AFfs cosωdω In LaMn 2 (Ge 0.82 Si 0.18 ) 2 , the simultaneous appearance of the satellite peaks (101) − /(101) + and the intensity increase of the (112) reflection below 320 K indicate a transition to the Fmi structure below T C (Fig. 8a). However, as the magnetic scattering contribution on the (101) reflection does not disappear at the same time, the Fmi co-exists with the Fmc phase down to at least 290 K. Finally, LaMn 2 (Ge 0.82 Si 0.18 ) 2 transforms into the Fmi structure at even lower temperatures. Since only a few data points were collected for this sample, the exact transition temperature is not known but it occurs somewhere between 200 K < T c/i N2 < 290 K. Taking the transition temperatures of the neighboring samples into account, T c/i N2 is expected to be at around 250 K. Below T C , the samples with the compositions x = 0.33, 0.47, 0.58 undergo a transition to the Fmc phase, identified by magnetic peaks consistent with diffraction conditions (2) and (3). Below 275 K, 250 K and 210 K, respectively, modulation peaks following condition (1) appear, which suggests co-existence of the Fmc and Fmi structures. Interestingly, the temperature range in which this co-existence is observed increases with the amount of Si in LaMn 2 (Ge 1−x Si x ) 2 : at 275 K < T < 250 K for x = 0.33, at 250 K < T < 200 K for x = 0.47, and at 210 K < T < 70 K for x = 0.58. Figure 8b shows the PND patterns of LaMn 2 (Ge 0.53 Si 0.47 ) 2 at 320 K, 300 K, 225 K and 29 K.
The total magnetic moment of LaMn 2 (Ge 0.53 Si 0.47 ) 2 reaches µ tot ≈ 2.66(4) µ B per Mn atom, with partial components of µ AFfs ≈ 2.24(3) µ B and µ FM ≈ 1.44(5) µ B at 29 K and a resulting angle of α ≈ 59.2(6)°. Thus, both magnetic contributions drop compared to the Ge-richer samples. In a similar fashion, a lower k z value of 0.1983(3) is observed. The temperature dependence of µ tot , the partial magnetic moments µ AFfs , µ AFl , µ FM , and k z are plotted in Fig. 9a. It is noteworthy that µ AFl rises while µ AFfs drops in the temperature region where Fmi and Fmc co-exist. Meanwhile, µ FM appears to be unperturbed. Thus, the co-existence occurs in the temperature region where the phase transition from Fmc to Fmi takes place. In comparison to LaMn 2 Ge 2 and LaMn 2 (Ge 0.95 Si 0.05 ) 2 , k z decreases less abruptly in LaMn 2 (Ge 0.53 Si 0.47 ) 2 (Fig. 9a inset). Figure 9b shows the change of the lattice parameters as a function of temperature. The non-linear behavior of c/a with a slope increase around T C noted earlier for LaMn 2 Ge 2 can also be seen in LaMn 2 (Ge 0.53 Si 0.47 ) 2 but becomes less pronounced with increasing Si-concentration. Interestingly, LaMn 2 Si 2 orders in the same magnetic structures as LaMn 2 (Ge 0.22 Si 0.78 ) 2 : LaMn 2 Si 2 remains in the Fmc phase down to 70 K, and the co-existence of Fmi and Fmc sets in at 50 K (Fig. 10a). In previous PND measurements of LaMn 2 Si 2 , the satellite reflections (101) − /(101) + were detected as a broadening at the foot of the (101) peak 15,35 . We can clearly distinguish the satellite peaks thanks to the higher resolution of our data. The incommensurate peaks were refined with k z ≈ 0.0710(5) at 3 K which is even smaller than the value of 0.09 reported earlier 15,35 . The temperature dependence of the total magnetic moment µ tot and its partial components µ AFfs , µ AFl and µ FM is shown in Fig. 10b. The magnetic moments vary substantially from previous studies. The value for µ AFfs derived from our refinements is 1.35(3) µ B at 3 K, which is significantly larger than the 0.8 µ B and 0.5 µ B published by Venturini et al. 15 and Hofmann et al., respectively 35 . µ FM ≈ 1.72(4) µ B is in line with their www.nature.com/scientificreports/ results but µ AFl ≈ 1.43(3) µ B is slightly lower. Nevertheless, µ tot reaches similar values in all cases. We attribute the discrepancies between the partial moments to our improved data resolution. As we were able to resolve the modulation peaks, it is much easier to refine the accurate values of k z and the partial magnetic moments. The compositional and thermal variation of the magnetic phases is plotted in Fig. 11. Below T C , the Ge-rich part of the solid solution is dominated by the Fmi phase, the Si-rich-by Fmc. In-between, the co-existence of Fmi and Fmc originally observed in LaMn 2 Si 2 15,35 spreads from low Si concentrations at high temperatures to the Si-rich compositions at low temperatures. Fmi and Fmc co-exist in every quaternary sample as well as in LaMn 2 Si 2 . As the FM component vanishes at T C , the magnetic moments align within the plane of the Mn square net. The AFl phase prevails for a wide range of compositions at T C < T < T N , while AFfs is favored by LaMn 2 Ge 2 . In the narrow Si-poor window around x = 0.05, AFfs and AFl co-exist above T C . As the AFfs contribution vanishes faster than the AFl, the pure AFl structure is detected at higher temperatures. The thick black lines in Fig. 11

Discussion
Co-existence of different magnetic phases is frequently observed in solid solutions. In the ThCr 2 Si 2 -type structure alone it has been found for, e.g., La 1−x Y x Mn 2 Si 2 29,46,48 , La 1−x Pr x Mn 2 Si 2 33 , CeMn 2 (Ge 1−x Si x ) 2 11 and PrMn 2 (Ge 1−x Si x ) 2 31 . In all these examples, such co-existence is reported in a limited composition region. LaMn 2 (Ge 1−x Si x ) 2 sets itself apart from all these cases as the co-existence occurs in all quaternary samples and the ternary LaMn 2 Si 2 . In the literature, the origin of magnetic phase co-existence is usually explained by a chemical phase separation of the quaternary samples into regions with nearly identical compositions and, thus, nearly identical lattice parameters. In CeMn 2 (Ge 1−x Si x ) 2 , for example, compositional inhomogeneity was suggested based on high-resolution synchrotron PXRD studies 11 . In La 1−x Y x Mn 2 Si 2 , a peak splitting could even be observed in the PND data 48 . The coexistence of two or more magnetic phases found in the PND measurements is supported by the non-saturation of the isothermal magnetization found for LaMn 2 Si 2 and some of the quaternary samples at 2 K under 6 T external field. This behavior of the isothermal magnetization indicates the existence of more than one magnetic component, as explained above.
In the quaternary samples in our study, a broadening of certain peaks in the PXRD data at room temperature can be detected, which likely indicates a somewhat inhomogeneous distribution of Si and Ge. Since this behavior is especially visible for some (hkl) reflections with non-zero l, these small inhomogeneities must have a stronger impact on c. Figure 12 shows the PXRD patterns in the 2θ region around the (105) reflection. The peak broadening and the asymmetry is pronounced in some of the quaternary samples. In LaMn 2 (Ge 0.82 Si 0.18 ) 2 and LaMn 2 (Ge 0.42 Si 0.58 ) 2 , the (105) reflection even appears to be split. LaMn 2 Si 2 , however, exhibits the smallest reflection width, rendering any significant chemical inhomogeneity (such as related to intrinsic defects) improbable. Future high-resolution PXRD measurements at a synchrotron source may shed light on the crystal structural origin of the co-existing magnetic phases in LaMn 2 Si 2 .
The partial substitution of Ge by Si leads to a minor decrease of T C from 326.10(4) K in LaMn 2 Ge 2 to 308.37(6) K in LaMn 2 Si 2 and was reported previously 36 . The values for T C we observe from magnetization and PND measurements are in agreement with each other and match those from the literature 28,36,49 . Considering the strong composition and temperature dependence of the AFfs and AFl components ( Fig. 11 and Supplementary  Fig. S6) it is noteworthy that FM, and therefore T C , remains nearly constant throughout the solid solution. A similar effect was also noted for T N , which increases monotonically with increasing Si content from approximately 420 K in LaMn 2 Ge 2 34 to 470 K in LaMn 2 Si 2 according to the literature 34,35,49,50 . The Neél temperature of LaMn 2 Ge 2 is in line with the values reported earlier 28,34 . Although we did not investigate the high temperature behavior for all samples, the three quaternary samples (x = 0.05, 0.33, 0.47) for which we collected PND data up to 500 K confirm the trend observed earlier: T N increases with increasing Si content 34 . Our data suggests, however, that the actual ordering temperatures may be higher than reported previously 34,35,50 . This is especially visible for the sample with the composition x = 0.47 which did not even reach the paramagnetic regime up to 500 K. Additional measurements at elevated temperatures may be required to confirm if T N is indeed higher than the values reported in the literature.
The x-T phase diagram of LaMn 2 (Ge 1−x Si x ) 2 exhibits certain similarities to those of La 1−x Y x Mn 2 Ge 2 28 , CeMn 2 (Ge 1−x Si x ) 2 11 and PrMn 2 (Ge 1−x Si x ) 2 31 . In all these solid solutions, the Fmc structure is observed in a similar www.nature.com/scientificreports/ composition range as Fmi. Analysis of the unit cell and magnetic phase evolution indicates that the Fmi structure dominates the samples with longer lattice parameters and at lower temperatures, while Fmc is found for the samples with shorter lattice parameters and at higher temperatures 11,28,31 . The same tendency is observed in LaMn 2 (Ge 1−x Si x ) 2 and suggests a correlation to the lattice dimensions. In previous studies, the intra-planar Mn-Mn distance was proposed as one of the important crystal structure parameters that help rationalize the magnetic phase diagram of the REMn 2 X 2 systems, as was discussed in the Introduction. We note, however, that the distance between adjacent Mn square nets also appears to be a significant factor for stabilization of certain magnetic phases. As the composition dependence of c does not follow Vegard's law but appears to make a kink in Ge-rich side of the solid solution (Fig. 2), the composition region where magnetic incommensurability is most pronounced even above room temperature (Fig. 11), the transition from incommensurate to commensurate structure must be governed by the Mn-Mn interlayer spacing d inter . Figure 13 shows the d inter -T phase diagram of the different magnetic structures in LaMn 2 (Ge 1−x Si x ) 2 which can be assigned to clearly defined regions. The data points of the co-existing phases were excluded for this consideration. Published results for other solid solutions series were added to the phase diagram in order to put the d inter -T trend found from our data into perspective. Interestingly, each data point from these other solid solutions fits perfectly into the d inter -T phase diagram of LaMn 2 (Ge 1−x Si x ) 2 . Thus, the occurrence of the commensurate Fmc and incommensurate Fmi structures can be directly correlated to the inter-planar Mn-Mn distances and the temperature. Therefore, Fig. 13 represents a "universal" phase diagram for the REMn 2 X 2 systems. Although it may not enable prediction of all possible magnetic phases in these materials, the uncovered relationship between the magnetism and the crystal structure can be used to target magnetic incommensurability, which can be of significance for design of functional magnetic materials.

Conclusions
The influence of the substitution of Ge by Si in LaMn 2 (Ge 1−x Si x ) 2 on the structural and magnetic properties has been investigated by PXRD, magnetization and PND measurements between 3 and 500 K, which allowed establishing a magnetic phase diagram. Replacing Ge with Si leads to a compression of the unit cell. The nonlinear lattice contraction in the Ge-richer samples at room temperature suggests strong magnetovolume effects. The magnetic structures of LaMn 2 (Ge 1−x Si x ) 2 are strongly affected by the change of the unit cell parameter c, which is reflective of the interlayer separation. In the x-T phase diagram, the commensurate Fmc and AFl structures dominate the Si-richer part of the solid solution mostly at higher temperatures, while the incommensurate Fmi and AFfs prevail in the Si-poorer part at lower temperatures. Thus, the transition from commensurate to incommensurate phases is linked to a combination of both inter-planar Mn-Mn distances and temperature. Coexistence of magnetic phases is observed in all quaternary samples and LaMn 2 Si 2 . Peak broadening of certain reflections in the PXRD pattern of the quaternary samples suggests the existence of compositional inhomogeneities as a result of the Ge/Si mixing. This effect could be the origin of the magnetic phase co-existence in the quaternary compositions. However, the same cannot be true for LaMn 2 Si 2 . High-resolution PXRD measurements might shed light on the origin of the co-existence of magnetic phases in LaMn 2 Si 2 . Comparison of the data on the LaMn 2 (Ge 1−x Si x ) 2 series and related solid solutions reported in the literature allows construction of a universal phase diagram relating the emergence of magnetic incommensurability to the inter-planar Mn-Mn distance.   (IPEN). Starting elements were molten in an argon atmosphere purified with a hot titanium getter. La pieces with 99.9% of purity and Mn, Ge and Si pieces with 99.999% of purity were added in the stoichiometric ratio. A little excess of Mn (around 5% by mass fraction) was used to compensate the weight loss by evaporation during reaction. After melting, the resulting ingot of each sample was sealed in an evacuated fused silica tube under reduced pressure of 10 -2 Pa and annealed at 1073 K for 24 h.
The sample composition was confirmed by powder X-ray diffraction (PXRD) and revealed, aside from the targeted pseudo ternary, small amounts of impurities for the samples with the composition x = 0, 0.05, 0.  4 54 in x = 0.58 [1.49(19) % by mass fraction]. The compositions for the respective LaMn 2 (Ge 1−x Si x ) 2 samples were refined from the PXRD data and are used throughout the text to identify the samples.
Powder X-ray diffraction (PXRD). Powder X-ray diffraction patterns were collected at room temperature using a Panalytical X'Pert PRO diffractometer (Panalytical, Netherlands) operated in Bragg-Brentano geometry. The instrument is equipped with a Johansson Ge monochromator to generate pure Cu K α1 radiation (λ = 1.54059 Å). The samples were measured on zero-background Si sample holders. Rietveld refinements of the PXRD patterns were performed by Fullprof 55 . Phase analysis yielded only small amounts of impurities. Three representative PXRD patterns of the samples with the composition x = 0, 0.47 and 1 are plotted in Supplementary  Fig. S2a-c.

Magnetic measurements. Magnetization was measured utilizing a Quantum Design Physical Property
Measurement System (PPMS, Quantum Design, USA). A Vibrating Sample Magnetometer (VSM) option was employed to collect zero-field cooled (ZFC) and field-cooled (FC) magnetization data between 2 and 400 K in static magnetic fields (DC). Isothermal magnetization was measured at 2 K and 250 K up to 6 T. Polycrystalline samples were loaded into polypropylene (PP) sample containers which were subsequently mounted in brass sample holders.
Powder neutron diffraction (PND). Powder neutron diffraction patterns were acquired during two beamtimes at the neutron sources of the Canadian Neutron Beam Centre (CNBC, Chalk River, Ontario, Canada) and the Center for High Resolution Neutron Scattering (CHRNS) at the National Institute of Standards and Technology (NIST Center for Neutron Research (NCNR), Gaithersburg, MD, USA), respectively. At the CNBC, diffraction patterns for the LaMn 2 (Ge 1−x Si x ) 2 samples with x = 0.18, 0.58, 0.78 and 1 were collected on the High Resolution Powder Diffractometer C2 in the angular range 2θ between 18.9° to 99° with a neutron wavelength of λ = 2.3722(17) Å in a He-cryostat (3 K to 290 K) and a dedicated furnace (320 K to 380 K). At NCNR, the measurements with the compositions x = 0, 0.05, 0.33 and 0.47 took place at the High Resolution Neutron Powder Diffractometer BT-1 56 equipped with 32 3 He detectors covering an angular range of 3° ≤ 2θ ≤ 166° with a step size of 0.050°. The data were collected using a Ge (311) monochromator wavelength of λ = 2.0787(2) Å and in pile collimation of 60 min per arc. Closed Cycle Refrigerators (CCRs) were used to cover the temperature range of 14 K to 500 K. Rietveld refinements of the PND patterns were carried out for magnetic structure determination employing Fullprof for all samples 55 .

Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.