Complex magnetic ordering in RE 5 Pd 2 In 4 (RE = Tb-Tm) compounds investigated by neutron diffraction and magnetometric measurements

The compounds crystallize with the orthorhombic Lu 5 2 In 4 -type crystal structure ( Pbam space group). In this work we report results of structural and magnetic studies by means of X-ray and neutron diffraction as well as dc and ac magnetometric data. Magnetic susceptibility and neutron diffraction data revealed rare-earth moments order at low temperatures with complex magnetic structures showing a cascade of temperature-induced transitions. The magnetic ordering temperatures are found to be 97, 88, 28.5, 16.5 and 4.3 K for RE = Tb, Dy, Ho, Er and Tm, respectively. Magnetic structures related to the propagation vector = k [0, 0, 0] 1 are found just below the magnetic ordering temperatures in majority of the investigated com- pounds (RE = Tb-Er). Below the Curie temperature T C they have purely ferromagnetic character in Tb 5 Pd 2 In 4 and Dy 5 Pd 2 In 4 . A ferrimagnetic order finally sets at lower temperatures in Dy 5 Pd 2 In 4 , while in Ho 5 Pd 2 In 4 two magnetic phases related to k 1 are observed: the antiferromagnetic one (phase I) and the ferrimagnetic one (phase II, coexisting with phase I at lower temperatures). Er 5 Pd 2 In 4 is a canted antiferromagnet with additional ferromagnetic component developing at lower temperatures. A purely antiferromagnetic component of mag- netic structure with enlarged magnetic unit cell appears with decreasing temperature in Tb 5 Pd 2 In 4 ( = k [0, , 0] 2 1 2 and = k [0, , ] 3 1 2 1 2 ) while in Ho 5 Pd 2 In 4 such component ( = k [ , 0, 0] 4 1 4 ) is present within whole temperature range below the magnetic ordering temperature. Magnetic structure of Tm 5 Pd 2 In 4 , exceptionally, has no k 1 component, but is an antiferromagnetic incommensurate one related to two propagation vectors: = k [0.073(3), 0.451(1), ] 5 1 2 and = k [0, 0.335(2), ] 6 1 2 . In majority of the compounds (RE = Tb-Er) the first rare-earth 4 g site (noted as 4 g 1) orders at lower temperature than two remaining sites (2 a and 4 g 2). The direction of the magnetic moments depends on rare-earth element involved and indicates an influence of single-ion anisotropy in the crystalline electric field (CEF). © 2021 The Author(s). Published by Elsevier B.V.


a b s t r a c t
The RE 5 Pd 2 In 4 (RE = Tb-Tm) compounds crystallize with the orthorhombic Lu 5 Ni 2 In 4 -type crystal structure (Pbam space group). In this work we report results of structural and magnetic studies by means of X-ray and neutron diffraction as well as dc and ac magnetometric data. Magnetic susceptibility and neutron diffraction data revealed rare-earth moments order at low temperatures with complex magnetic structures showing a cascade of temperature-induced transitions. The magnetic ordering temperatures are found to be 97, 88, 28.5, 16.5 and 4.3 K for RE = Tb, Dy, Ho, Er and Tm, respectively. Magnetic structures related to the propagation vector

Introduction
The research in the field of magnetism of rare earth intermetallics concentrates nowadays on systems with complex crystal structures where localized magnetic moments are distributed among different Wyckoff sites. Such a distribution of magnetic moments leads to complex magnetic properties like presence of a cascade of temperature-and/or field-induced magnetic transitions, formation of complex non-collinear magnetic structures, coexistence of ferro-and antiferromagnetic components of magnetic structure, etc.
A series of ternary intermetallics of general composition RE 5 Pd 2 In 4 (RE = Sc [1], Y, Tb-Tm, Lu [2]) is a good example of such a system. The compounds are formed at 870 K and they crystallize in a complex crystal structure of the Lu 5 Ni 2 In 4 -type (Pearson symbol oP22, space group Pbam, No. 55, Z = 2) [3], where the rare-earth atoms occupy three nonequivalent crystallographic positions. No magnetic data have been yet reported for the RE 5 Pd 2 In 4 compounds, except for Sc 5 Pd 2 In 4 which is found to be a Pauli paramagnet [1].
In case of the isostructural RE 5 Ni 2 In 4 system, the magnetic properties have only been determined for selected members. Tb 5 Ni 2 In 4 is a ferro-/ferrimagnet below T C = 125 K with an additional antiferromagnetic component appearing below T N = 20 K [4]. Dy 5 Ni 2 In 4 has been reported to order ferromagnetically below T C = 105 K followed by a transformation to an antiferromagnetic state below T N = 30 K [5] or 20 K [6]. Ho 5 Ni 2 In 4 orders magnetically below 31 K [4,6] or 30 K [7]. The neutron diffraction data indicate formation of an antiferromagnetic structure which transforms to a ferrimagnetic one at 25 K [4] or 19 K [6]. In Er 5 Ni 2 In 4 a long-range magnetic ordering develops below 18.5 K [8] or 21 K [6]. Tm 5 Ni 2 In 4 is a complex antiferromagnet with the Néel temperature of 4.2 K [9] or 4.1 K [10]. Recent investigation of magnetocaloric effect in RE 5 Ni 2 In 4 showed magnetic entropy changes around T C equal to 4.7, 10.1 and 10.2 J/(kg ⋅ K) for RE = Dy, Ho and Er, respectively [6].
This work is a part of our broader study concerning magnetic properties of ternary indides. In this paper we focus on both the macroscopic (magnetic susceptibility, magnetization) as well as microscopic (crystal and magnetic structures) properties of the RE 5 Pd 2 In 4 (RE = Tb-Tm) system. The research involves different experimental techniques including magnetic measurements as well as X-ray and neutron diffraction. The obtained data are compared with those for the isostructural RE 5 Ni 2 In 4 compounds, in order to determine the influence of different transition metal elements (Ni and Pd) on the magnetic order in the rare earth sublattice.

Experimental details
The samples of RE 5 Pd 2 In 4 (RE = Tb-Tm) were prepared by arcmelting of high-purity metals (99.9 wt% purity for RE, 99.99 wt% for Pd and In) having total mass of about 5 g under a pure argon atmosphere. The buttons were sealed in evacuated silica tubes and annealed at 870 K for one month. Afterwards they were quenched into cold water. The crystal structure of the obtained samples was examined by X-ray powder diffraction at room temperature using a PANalytical X′Pert PRO or Empyrean diffractometer (Cu Kα-radiation, Bragg-Brentano geometry, measured angle interval 2θ = 10 − 130 ∘ , step scan mode, step size in 2θ = 0.03 ∘ , 15 s/step). The FullProf [11,12] program package was used for the Rietveld analysis of the collected X-ray data set.
The dc magnetic measurements were carried out on a VSM option of the Physical Property Measurement System (PPMS) by Quantum Design. The zero-field cooled (ZFC) and field cooled (FC) magnetic susceptibility vs. temperature curves were collected in a low external magnetic field of 50 Oe in order to determine temperatures of magnetic phase transitions. The ZFC magnetic susceptibility vs. temperature curves were recorded at 1 kOe in wide temperature range (1.9-390 K) in order to find the values of effective magnetic moments and paramagnetic Curie temperatures. Finally, hysteresis loops in applied magnetic fields up to 90 kOe were taken at a number of temperatures in order to find character of magnetic ordering and determine the critical and coercivity fields.
In addition to the dc magnetic measurements, the ac ones were completed with the use of an ACMS option of the Physical Property Measurement System (PPMS) by Quantum Design. The real ( ) and imaginary (χ″) magnetic susceptibility components were collected in function of temperature with the amplitude of oscillating magnetic field equal to 2 Oe. The dc component of magnetic field was fixed to zero during measurement.
Powder neutron diffraction patterns were collected on the E2 diffractometer (λ = 2.381 Å) located at the Helmholtz-Zentrum Berlin. The data were collected at a number of temperatures between 1.4 and 120 K. The Rietveld-type program FullProf [11,12] was used for processing the diffraction data.

Crystal structure
X-ray powder diffraction data of the RE 5 Pd 2 In 4 samples collected at room temperature indicate that they crystallize in the orthorhombic Lu 5 Ni 2 In 4 -type structure (Pearson symbol oP22, space group Pbam, No. 55, Z = 2) [3]. Fig. 1 shows, as an example, a diffraction pattern of the Er 5 Pd 2 In 4 sample with corresponding crystal unit cell.
The rare-earth atoms, which are situated at three different Wyckoff positions, occupy one layer of the structure (z = 0) while the second one ( = z 1 2 ) is occupied by the palladium and indium atoms. Refined lattice parameters and atomic coordinates are presented in Table 1 and they are in good agreement with the previous results reported in Ref. [2] (see Fig. 2). Further details are given in Section 5. Fig. 2 presents dependences of the lattice parameters a, b and c and the unit cell volume V vs. effective ionic radius of rare-earths [13]. The dependences are almost linear and follow the lanthanide contraction. Fig. 3  The distributions of Er atoms at different sites in Er 5 Pd 2 In 4 and atomic nets of erbium are shown in Fig. 4a. The shortest interatomic Er-Er (Fig. 4a) as well as Pd-In and In-In (Fig. 4b) distances in Å, according to refined crystallographic parameters (see Table 1), have been emphasized. It should be mentioned that in this two-layered structure the atoms of same sites are at the distance of the shortest unit cell parameter c = 3.5971 Å.

Magnetic properties
The results of dc magnetic measurements (RE = Tb-Tm) are presented in Fig. 5 while the ac data (RE = Tb-Er) in Fig. 6. The parameters characterizing the magnetic order, as determined from the above mentioned data, are listed in Table 2. A typical para-to ferro-/ferrimagnetic transition in RE 5 Pd 2 In 4 (RE = Tb-Er) manifests itself in presence of inflection point in the χ dc (T) curve collected at 50 Oe. The transition is confirmed by maxima found in the χ ac (T) data (H ac = 2 Oe, H dc = 0 Oe). The Curie temperatures range from about 16 K for RE = Er up to almost 100 K for RE = Tb. A number of additional anomalies in both the dc and ac magnetic susceptibility vs. temperature curves, identified either as maxima or extra inflection points, are found (see Table 2 for details). Tm 5 Pd 2 In 4 is an exception as its χ dc (T) curve has at 4.3 K a maximum typical of para-to antiferromagnetic transition. For all investigated compounds a significant discrepancy between the ZFC and FC curves is noticeable below the respective critical temperature (Curie or Néel) of magnetic ordering.
Reciprocal magnetic susceptibility, collected at H = 1 kOe, becomes linear and therefore follows the Curie-Weiss law at high enough temperatures. The paramagnetic Curie temperatures are found positive for RE = Tb-Er or equal to zero in Tm 5 Pd 2 In 4 . The effective magnetic moments per RE atoms are close to the values predicted for free RE 3+ ions. The moments in the ordered state, as derived from   530.17 (7) 524.06 (7) 519.65 (11) 514.40 (10) 508 (11) 0.1181 (7) 0.1196 (6) Pd at 4h (x, y, (7) In2 at 4h (x, y, 1 0.33(9) 0.14(9) 0.22 (11) 0.56 (10)  magnetization measurements taken at T = 2.0 K and H = 90 kOe, are significantly smaller than those of free RE 3+ ions. The magnetization hysteresis loops have been recorded between H = − 90 and 90 kOe at T = 2.0 K and selected higher temperatures. The primary curves taken at 2.0 K show metamagnetic transitions, characteristic of antiferromagnetic contribution to the magnetic structure, for RE = Tb, Dy and Tm. The hysteresis loops at the same temperature are characterized by coercivity fields which decrease with increasing number of the 4f electrons from 12.4 kOe for RE = Tb down to 0.10 kOe for RE = Tm.

Magnetic structures
The neutron powder diffraction data collected in the paramagnetic state confirm the crystal structure of the Lu 5 Ni 2 In 4 -type (see Fig. 7a: diffraction pattern for Tb 5 Pd 2 In 4 taken at T = 119.8 K, which was chosen as a representative). Determined values of the lattice parameters a, b and c as well as the unit cell volume V are in good agreement with those determined from X-ray data at room temperature. At low temperatures additional peaks of magnetic origin are detected (Figs. 7b-d, 8a, 9a-b, 10a, 11a). According to the magnetic data (the values of effective magnetic moments are very close to those predicted for the RE 3+ ions), the magnetism of RE 5 Pd 2 In 4 is related to the rare-earth atoms which occupy three Wyckoff sites: 2a sublattice: RE1_1 (0, 0, 0) and RE1_2 ( , , 0) , and RE2_4 , and RE3_4 The atomic positional parameters obtained from the refinements are listed in Table 1.
The possible magnetic structures can be determined using the formalism of the basis vectors (BV) of irreducible representations (IR), which depend on the crystal space group, magnetic propagation vector and the Wyckoff site of the magnetic atom. The representations allowed for each Wyckoff site occupied by RE atoms in RE 5 Pd 2 In 4 were calculated using BasIreps from the FullProf [11,12] package (the details of symmetry analysis can be found in the supplementary material). A Rietveld refinement of magnetic structures was performed for difference patterns, obtained by subtraction of pattern taken in the paramagnetic state from the patterns collected below the magnetic ordering temperature. The only exception is for Dy 5 Pd 2 In 4 , where the magnetic structure was determined based on the full diffraction patterns due to high level of noise caused by a large absorption coefficient of Dy.

Tb 5 Pd 2 In 4
The above presented complex magnetic properties of Tb 5 Pd 2 In 4 are in agreement with thermal evolution of magnetic contribution to the powder neutron diffraction pattern (see Fig. 7b-7d). The Bragg reflections of magnetic origin at 69.8 K (see Fig. 7b) overlap with those of nuclear origin (compare with Fig. 7a) indicating a propa- i.e. the magnetic unit cell has the same size as the nuclear one. For the space group Pbam (No. 55), symmetry analysis gives four one-dimensional irreducible representations for the 2a site and eight one-dimensional irreducible representations for the 4g1 and 4g2 sites. Atoms occupying each particular sublattice belong to one orbit, which means that their magnetic moments are constrained by symmetry. The best agreement with the experimental data was obtained for magnetic structure model with magnetic moments in the 2a and 4g2 sites corresponding to the basis vectors of the IR3 representation (representation numbers follow the output files of BasIreps), which resulted in the ferromagnetic order along the c-axis, while the magnetic moments in the 4g1 sublattice remained disordered. The results of the Rietveld refinement are presented in Table 3, and the corresponding visualization of the magnetic structure at 69.8 K is shown in Fig. 7e. The magnetic moment at the 2a site is equal to 6.7(2) μ B , and it is larger than 4.1(1) μ B found at the 4g2 site.
The magnetic reflections at 39.7 K (Fig. 7c) are observed at different positions than those at higher temperature ( Fig. 7b) indicating the existence of an additional magnetic phase transition between 40 and 70 K. The magnetic reflections at 39.7 K can be indexed with the , which corresponds to an antiferromagnetic order with magnetic unit cell doubled along the baxis when compared with the crystal unit cell. Symmetry analysis gives two representations for each sublattice, with magnetic moments ordered along the c-direction (IR1) or in the ab-plane (IR2). The agreement with the experimental pattern is obtained for magnetic moments in the 2a and 4g2 sites ordered antiferromagnetically along the c-axis (Fig. 7f ) plus additional group of reflections related to the  Table 1), have been emphasized: (a) Er-Er, (b) Pd-In and In-In.
propagation vector 2 (see Fig. 7d). The latter vector corresponds to the magnetic unit cell doubled along both the b-and c-directions. The magnetic ordering at 1.4 K described by , however, the Rietveld refinement proves that the k 3 vector describes ordering in the ab-plane ( Table 5). The main contribution to the antiferromagnetic structure related to

Dy 5 Pd 2 In 4
Magnetic peaks in the patterns of Dy 5 Pd 2 In 4 are indexed by the propagation vector. The magnetic structure at all studied temperatures below the Curie point is described by the basis vectors of the IR3 representation, indicating presence of the ferromagnetic order along the c-axis (Table 3). A representative diffraction pattern is shown in Fig. 8a. At 80.0, 65.0 and 40.0 K, the magnetic moments in the 4g1 site are found disordered while the moments in the 2a and 4g2 sites are parallel to each other. Therefore at these temperatures Dy 5 Pd 2 In 4 is a ferromagnet. The magnetic moments in 2a and 4g2 sites increase with decreasing temperature, reaching at 1.5 K the values 5.4(7) μ B and 3.9(5) μ B , respectively. The parallel orientation of magnetic moments in the 2a and 4g2 sublattices is preserved down to 1.5 K. Additionally, non-zero localized magnetic moments of 1.7(4) μ B in the 4g1 sublattice, oriented in the opposite direction to the moments in the 2a and 4g2 sites, are derived from Rietveld refinement at 1.5 K. The resultant structure at 1.5 K is thus a ferrimagnetic one, as presented in Fig. 8b.

Ho 5 Pd 2 In 4
The magnetic peaks in the diffraction patterns of Ho 5 Pd 2 In 4 can be indexed using at least two different propagation vectors. Most of , implying an additional modulation of magnetic moments superimposed on the main structure ( Fig. 9a-b). At 19.9 and 14.9 K, the experimental diffraction pattern (Fig. 9a) Fig. 9c). The magnetic moments in the 4g1 sublattice remain disordered. At and χ″ refer to the real and imaginary components, respectively.

Table 2
Parameters characterizing magnetic order in RE 5 Pd 2 In 4 (RE = Tb-Tm) as derived from dc and/or ac magnetometric measurements. T C and T N denote the Curie and Néel temperatures, respectively. T t refers to additional anomalies detected in the magnetic susceptibility vs. temperature curves. θ p and μ eff are the paramagnetic Curie temperature and effective magnetic moments obtained from reciprocal magnetic susceptibility. μ is the magnetic moment in the ordered state, measured at T = 2.0 K and H = 90 kOe. "exp." and "theor." denote the experimental and theoretical values, respectively. The critical fields (H cr ) indicating metamagnetic transitions have been determined from primary magnetization curves while the coercivity fields (H coer ) from hysteresis loops taken at selected temperatures.
i -inflection point; m -maximum; f -determined from the FC curve lower temperatures, 7.9 and 1.4 K, the diffraction peaks arising from the antiferromagnetic phase of Ho 5 Pd 2 In 4 mentioned above and denoted as phase I, are still present, but the refined magnetic moments are lower than those at 14.9 K. Therefore the phase I gradually disappears with decreasing temperature, which is confirmed by simultaneous appearance of new magnetic peaks (Fig. 9b). The new peaks can be also indexed by = k [0, 0, 0] 1 , but they correspond to another magnetic phase, denoted as phase II, co-existing with phase I at lower temperatures. The magnetic moments in phase II are ordered according to combination of the basis vectors of BV1 of IR3 (ordering along the c-direction) and those of BV2 of IR5 (ordering along the b-direction). The resultant structure of phase II is ferrimagnetic: moments in the 2a and 4g2 sites are ordered ferromagnetically within the bc-plane, while the moments in the 4g1 site are oriented entirely along the b-axis, in almost opposite direction to the moments in the 2a and 4g2 sublattices (Fig. 9d).   Table 6). The magnetic moments in the 2a site (Ho1_1 and Ho1_2) are constrained by symmetry while the moments in the 4g2 site split into two orbits: the first one containing Ho3_1, Ho3_4 and the second one containing Ho3_2 and Ho3_3. As symmetry allows different magnitudes of magnetic moments in different orbits, the C i coefficients for the first and second orbit were refined independently (see Table 6). Magnetic moments in the 2a site are found more than twice as large as those at each particular 4g2 orbit. The magnetic unit cell, with the size of four crystal unit cells quadrupled along the a-axis, is presented in Fig. 9e. At lower temperatures, where phase I is gradually disappearing, the satellites significantly broaden indicating a decrease in magnetic domain size. Due to small signal-to-background ratio for the satellites found in the patterns taken at 7.9 K and 1.4 K, the magnetic moments are determined with large uncertainties and therefore they are not listed in Table 6.

Er 5 Pd 2 In 4
Magnetic structure of Er 5 Pd 2 In 4 is again described by the pro- (Fig. 10b), but it corresponds to different set of basis vectors of irreducible representations than for the above reported magnetic structures in RE 5 Pd 2 In 4 (RE = Tb, Dy, Ho) (see Table 3). The magnetic moments in the 2a and 4g2 sublattices are ordered antiferromagnetically and are related to the basis vectors of IR7. For the 2a site, the contribution to the total magnetic moment along the b-axis is about twice as large as the contribution along the a-axis, while for the 4g2 site the whole contribution to the magnetic moment is solely along the b-axis. The magnetic moments in the 4g1 sublattice are disordered at 29.9 and 11.9 K, while at 7.9 K and 1.5 K they are ordered ferromagnetically along the b-axis, and they correspond to BV2 of IR5. Magnetic ordering at 7.9 K is presented in Fig. 10b.

Tm 5 Pd 2 In 4
The magnetic reflections observed for Tm 5 Pd 2 In 4 at 1.4 K (Fig. 11a)  has shown that all magnetic atoms are independent with respect to their magnitudes and orientations (symmetry does not impose any extra constrains) and two representations are possible: IR1 corresponding to magnetic ordering along the c-axis and IR2 corresponding to ordering within the ab-plane. Rietveld refinement for the Tm 5 Pd 2 In 4 data has shown that reasonable results are only possible for IR2. In order to reduce the number of fitting variables, the same amplitude of modulation of magnetic moment was assumed for all moments occupying the same Wyckoff site (see Table 7). Symmetry analysis performed for = k [0, 0.34, ] 6 1 2 results in four different representations. Rietveld refinement favors the magnetic structure related to the basis vectors of IR4 which involve magnetic moments within the ab-plane. The Tm1_1 and Tm1_2 atoms are constrained by symmetry, while each of the 4g1 and 4g2 sites split into two orbits: Tm2_1, Tm2_3 and Tm2_2, Tm2_4 (and the same for the Tm3 site). Again, the amplitude of modulation of magnetic moment has been constrained to be equal within each particular sublattice involving two orbits ( Table 8). The refined magnetic structures are presented in Fig. 11b-

Discussion
The work reports the results of X-ray and neutron diffraction as well as dc and ac magnetic measurements for the RE 5 Pd 2 In 4 (RE = Tb-Tm) series of compounds. X-ray and neutron diffraction data confirm the orthorhombic crystal structure of the Lu 5 Ni 2 In 4type. The crystal structure is a typical layered structure, where the layers are formed by the rare-earth atoms (z = 0) and they are separated by layers containing Pd and In atoms ( = z 1 2 ). The rare-earth atoms are located at three nonequivalent Wyckoff sites: the 2a site and two 4g sites with different atomic positional parameters.
Magnetic and neutron diffraction data indicate that the compounds have complex magnetic properties and the magnetic moments localized solely on the rare-earth atoms. The determined effective magnetic moment values in the paramagnetic state are close to those of the theoretical ones of the free RE 3+ ions while the magnetic moments in the ordered state, as found from magnetization as well as from the neutron diffraction data, are smaller than those of free RE 3+ ions. The latter discrepancy indicates a significant influence of the Crystalline Electric Field (CEF) on the physical properties of RE 5 Pd 2 In 4 . The positive values of the paramagnetic Curie temperatures, only with the exception for Tm where θ p equals zero, indicate that ferromagnetic interactions are dominant. Fig. 12 shows the critical temperatures of magnetic ordering (T C,N ) together with paramagnetic Curie temperatures (θ p ) plotted against the de Gennes factor for both the Ni-and Pd-based RE 5 T 2 In 4 (RE = Tb-Tm, T = Ni, Pd) systems. The observed deviations from linear dependence are another evidence of significant influence of the crystalline electric field (CEF) on stability of the magnetic order in both series of compounds [18]. Similar deviations, attributed to the influence of CEF, have been observed in RERh 4 B 4 (RErare earth element) [19] and RE 11 T 4 In 9 (RE = Gd-Er; T = Ni, Pd) [20,21].
The low temperature experimental data presented in this work show that the RE 5 Pd 2 In 4 (RE = Tb-Tm) intermetallics undergo a number of temperature-and field-induced magnetic transitions. For the compounds with RE=Tb-Er some common features can be found, namely: 1. Below the Curie temperature a main contribution to the magnetic structure arises from a component of magnetic structure related to = k [0, 0, 0] 1 and formed by the localized rare-earth magnetic moments occupying the RE1 and RE3 sites. Further decrease of temperature leads to appearance of the magnetic ordering at the RE2 site. Additional antiferromagnetic components of magnetic structures with enlarged magnetic unit cell are observed in ). 2. The direction of magnetic moments is strictly correlated with the electronic properties of the rare-earth elements, indicating especially a strong influence of the crystalline electric field (CEF). Namely, just below the Curie point the magnetic moments of Such a behavior is commonly found in rareearth intermetallics and it is related to the sign of the Stevens operator α J which is negative for Tb, Dy and Ho but positive for Er [22]. Interestingly, the magnetic moments in RE 5 Pd 2 In 4 (RE = Tb-Tm) undergo a reorientation with decreasing temperature and at low temperatures they are aligned within the ab-plane for Tb and Er or within the bc-plane for Ho, while they are parallel to the caxis for Dy. 3. Similarities between magnetic structures in the isostructural RE 5 Ni 2 In 4 compounds are noticeable. The magnetic structures have already been reported for the Tb-, Ho- [4] and Er-based compounds [8]. A ferro-/ferrimagnetic order, related to In the investigated compounds the rare-earth magnetic moments occupying different sublattices order at different temperatures. This behavior is strongly correlated with the distribution of the rare-earth atoms in the crystal unit cell. The framework of the crystal structure in RE 5 Pd 2 In 4 is determined by the shortest Pd-In and In-In bonds. These elements form the chains of distorted pentagonal prisms [REPd 4 In 6 ] running along the b-axis (see Fig. 4b). The RE2 moments are situated in the center of the polyhedron, while the RE1 and RE3 ones are positioned between the chains. In the investigated compounds a clear hierarchy of ordering temperatures related to different rare-earth sublattices is observed. In most cases the moments in the RE1 and RE3 sites order at higher temperatures than those in the RE2 site. It is worth noting that the shortest rare-earth interatomic distances are those between the RE1-RE3 atoms (~3.4 Å) and  . The magnetic unit cell is not defined due to incommensurability of the propagation vectors, therefore representative fragments of each structure are shown instead.

Table 3
Component of the magnetic structures in RE 5 Pd 2 In 4 (RE = Tb-Er) described by the propagation vector k 1 = [0, 0, 0]. IR denote irreducible representations while BV are their basis vectors obtained by symmetry analysis for the 2a and 4g (identical for 4g1 and 4g2) Wyckoff sites of Pbam space group. The total number of possible IR for the 4g site equals eight but for better clarity those not used in the refinement are omitted. C i coefficients denote the contribution along each BV to the total magnetic moment of an atom.

Table 4
Component of the magnetic structure in Tb 5 Pd 2 In 4 described by the IR1 representation obtained by symmetry analysis for the 2a and 4g2 Wyckoff sites of the Pbam space group for the propagation vector k 2 = [0, 1 2 , 0]. C i coefficients denote the contribution along each basis vector (BV) to the total magnetic moment, Δϕ is the magnetic phase factor calculated with respect to Tb1_1.
S. Baran the RE3-RE3 ones (~3.5 Å) within the ab-plane (see Fig. 4). Both these distances are smaller than the sum of atomic radii equal to about 3.55 Å. Such a result suggests that the exchange interactions within the ab-plane involving the RE1 and RE3 atoms are mostly driven by the Campbell model-type mechanism [23]. The model assumes local interaction between the f-and d-orbitals (within individual rare-earth atom) and subsequently the direct exchange interactions between different atoms via the d-d orbitals. The interactions have local character as they are limited by the sum of radii of the atoms involved. The RE2 sublattice is more separated as the shortest RE1-RE2, RE2-RE2 and RE2-RE3 distances equal about 4.4, 3.6 and 3.7 Å, respectively. Such large interatomic distances exclude any direct interactions and suggest indirect exchange interactions via conduction electrons of the RKKY-type [24]. These interactions have more long-range character than those predicted by the Campbell model and therefore the RKKY-type interactions support appearance with decreasing temperature of the antiferromagnetic phases related to (RE = Ho). Table 5 Component of the magnetic structure in Tb 5 Pd 2 In 4 at 1.4 K described by the IR2 representation obtained by symmetry analysis for the 2a and 4g (identical for 4g1 and 4g2) Wyckoff sites of the Pbam space group for the propagation vector k3 = [0, 1 2 , 1 2 ]. C i coefficients (given in μ B ) denote the contribution along each basis vector (BV) to the total magnetic moment, Δϕ is the magnetic phase factor calculated with respect to Tb1_1.

Table 6
Component of the magnetic structure in Ho 5 Pd 2 In 4 described by the IR1 representation obtained by symmetry analysis for the 2a and 4g2 Wyckoff sites of the Pbam space group for the propagation vector k 4 = [ 1 4 , 0, 0]. C i coefficients denote the contribution along each basis vector (BV) to the total magnetic moment, α = |cos(δπ)|, β = |sin(δπ)| where δ = ¼.

Table 7
Component of the magnetic structure in Tm 5 Pd 2 In 4 at 1.4 K described by the IR2 representation obtained by symmetry analysis for the 2a and 4g (identical for 4g1 and 4g2) Wyckoff sites of the Pbam space group for the propagation vector k5 = [0.073(3), 0.451(1), 1 2 ]. C i coefficients denote the contribution along each basis vector (BV) to the total magnetic moment, while Δϕ is the magnetic phase factor calculated with respect to Tm1_1.

Conclusions
The RE 5 Pd 2 In 4 compounds (RE = Tb-Tm) with complex crystal structure in which the rare-earth atoms occupy three nonequivalent sites, present interesting magnetic properties including complex magnetic structures formed by the magnetic moments localized on the rare-earth atoms. Magnetic moments at different sublattices (Wyckoff sites) have different magnitudes and order at different temperatures. Just below the critical temperature a magnetic ordering described by = k [0, 0, 0] 1 is observed. For RE = Tb and Ho an antiferromagnetic component of the magnetic structure with enlarged magnetic unit cell develops. Tm 5 Pd 2 In 4 is an exception as its magnetic structure is purely antiferromagnetic and incommensurate within the whole temperature range below the critical temperature. Observed magnetic structures result from complex magnetic interactions involving the direct local exchange interactions related to the Campbell model and the indirect long-range ones of the RKKY-type. An influence of the crystalline electric field (CEF) on the direction of magnetic moments is noticeable.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Table 8 Component of the magnetic structure in Tm 5 Pd 2 In 4 at 1.4 K described by the IR4 representation obtained by symmetry analysis for the 2a and 4g (identical for 4g1 and 4g2) Wyckoff sites of the Pbam space group for the propagation vector k6 = [0, 0.335(2), 1 2 ]. C i coefficients denote the contribution along each basis vector (BV) to the total magnetic moment, Δϕ is the magnetic phase factor calculated with respect to Tm1_1, α = |cos(δπ)|, β = |sin(δπ)| where δ = 0.335 (2). Fig. 12. Critical temperatures of magnetic ordering (T C,N ) together with paramagnetic Curie temperatures (θ p ) vs. de Gennes factor for RE 5 T 2 In 4 (RE = Tb-Tm, T = Ni, Pd). The data for T = Ni have been taken from Refs. [5][6][7][8][9][10] while those for T = Pd are reported in this work.