Crystal and magnetic structures of R 2 Ni 1.78 In compounds ( R = Tb, Ho, Er and Tm)

and Tm. The obtained magnetic structures are discussed on the basis of symmetry analysis. The rare earth magnetic moments, determined from neutron diffraction data collected at 1.6 K, are 6.5 (1) (cid:2) B (Er) and 6.09 (4) (cid:2) B (Tm), while in the incommensurate modulated magnetic structure in Ho 2 Ni 1.78 In the amplitude of modulation of the Ho magnetic moment is 7.93 (8) (cid:2) B . All these values are smaller than those expected for the respective free R 3+ ions. A of the Tb Ni 1.78 below Ne´el moments localized solely nickel atoms remain reduction rare in ordered in 2 = and and in direction indicate ﬁeld on stability magnetic order in the investigated


Introduction
Intermetallic compounds of the R-Ni-In systems are the subject of intensive studies with respect to their crystal structures, chemical bonds and magnetic properties (Kalychak et al., 2004). Within these systems, the compounds with a stoichiometry close to 2:2:1 have been observed to crystallize in two different crystal structures: (i) A tetragonal one of the Mo 2 FeB 2 type (space group P4/mbm) found for both the R 2 Ni 2 In (R = La-Nd) stoichiometric composition and the R 2 Ni 2Àx In (x = 0.22, R = Y, Sm, Gd-Tm, Lu) nonstoichiometric composition (Kalychak et al., 1990).
Studies of the physical properties of R 2 Ni 2 In reveal that Ce 2 Ni 2 In is a non-magnetic intermediate-valence system (Hauser et al., 1997;Kaczorowski et al., 1996), while Nd 2 Ni 2 In orders antiferromagnetically below 8 K (Maskova et al., 2014). Magnetic and thermodynamic studies of R 2 Ni 2 In (R = Gd-Tm) report an antiferromagnetic order with Né el temperatures between 5 K (Tm) and 40 K (Tb) (Szytuła et al., 2015).
A recent paper on magnetism in the 2:2:1 compounds reports a ferromagnetic order for R = Pr and Nd, and an antiferromagnetic order for R = Dy and Ho. A considerable magnetocaloric effect is detected in the vicinity of the magnetic transition in all four compounds (Zhang et al., 2019).
This work is a continuation of our investigation of the magnetic properties of the compounds belonging to the R-Ni-In systems. We report here for the first time the lowtemperature magnetic structures of R 2 Ni 1.78 In (R = Ho, Er, Tm) as determined from powder neutron diffraction. The validity of the structures is verified by symmetry analysis. In addition, we present a symmetry analysis of the Tb 2 Ni 1.78 In magnetic structure which was missing from the original paper (Szytuła et al., 2013). The magnetic properties of Tm 2 Ni 1.78 In, as derived from DC magnetic measurements, are also reported.

Experimental details
Polycrystalline samples of R 2 Ni 1.78 In (R = Ho, Er and Tm) of a total weight of 5 g each were obtained by arc melting of the constituent elements (purity 99.9 wt% or better) under an argon atmosphere. The obtained ingots were encapsulated in evacuated silica tubes and annealed at 870 K for one month, then quenched in cold water. Finally, the samples' quality was checked by X-ray powder diffraction performed at room temperature using a PANalytical X'Pert PRO diffractometer (Cu K radiation).
DC magnetic measurements were carried out using a commercial vibrating sample magnetometer (VSM) option installed on the PPMS platform by Quantum Design. The data were collected in the temperature range 2-300 K in magnetic fields up to 90 kOe.
Powder neutron diffraction patterns were taken at low temperature (1.6 K) and in the paramagnetic state (above the respective Né el temperature) on the E6 diffractometer at Helmholtz-Zentrum Berlin fü r Materialien und Energie GmbH. The incident neutron wavelength was 2.432 Å .
For Rietveld analysis of X-ray and neutron diffractograms the computer program FULLPROF was utilized (Rodríguez-Carvajal, 1993, 2001, while for symmetry analysis the computer program BASIREPS, which is distributed together with FULLPROF, was used. No new data were collected for the Tb 2 Ni 1.78 In sample. The previously reported neutron diffraction patterns (Szytuła et al., 2013) were re-analyzed on the basis of symmetry analysis.

Crystal structure
The X-ray diffraction data collected at room temperature confirm the previously reported tetragonal crystal structure of the Mo 2 FeB 2 type. Detailed analysis of the powder neutron diffraction patterns taken in the paramagnetic state provided information on both the samples' phase composition and their crystal structure parameters. The investigated samples contain about 80% of the R 2 Ni 1.78 In phase with a small content of the 2:2:1 and 5:2:4 phases. In addition, in the Er-and Tm-based samples small amounts of the R 2 O 3 oxides were detected (see Table 1).

Figure 1
The neutron diffraction pattern of A typical neutron powder diffraction pattern in the paramagnetic state (taken at T = 7.2 K for Tm 2 Ni 1.78 In) is shown in Fig. 1. Similar patterns are observed for the other investigated samples. The tetragonal crystal structure of the Mo 2 FeB 2 type (space group P4/mbm) is shown in Fig. 2. The determined unit-cell parameters and atomic positional parameters are listed in Table 2. They are in good agreement with the previously reported data (Kalychak et al., 1990;.

Magnetic properties
Magnetic measurements were performed for the R 2 Ni 1.78 In (R = Ho, Er and Tm) samples. The data for R = Ho and Er are in agreement with those reported previously  and so they are not presented in this work. The results of magnetic measurements for Tm 2 Ni 1.78 In are presented in Fig. 3. At high temperatures the reciprocal magnetic susceptibility obeys the Curie-Weiss law with a negative value of the paramagnetic Curie temperature (À9.8 K) and the effective magnetic moment per Tm atom (7.77 B ) being close to that predicted for the free Tm 3+ ion (7.56 B ). In the low-temperature range (see the upper inset in Fig. 3) a maximum typical of an antiferro-to paramagnetic transition is clearly visible at 4.5 K. The lower inset in Fig. 3 presents the hysteresis loop of magnetization versus external magnetic field up to 90 kOe. The data were taken at 2.0 K. It is worth noting that the value of the magnetic moment per Tm atom, as derived from the measurements at T = 2.0 K and H = 90 kOe, equals 3.4 B and is significantly smaller than that of the free Tm 3+ ion, which equals 7.0 B . The initial magnetization curve shows a metamagnetic transition at the critical field of 13.7 kOe.

Magnetic structures
The magnetic moments in R 2 Ni 1.78 In (R = Tb, Ho, Er, Tm) are expected to be localized on the rare earth atoms, as the effective magnetic moments per rare earth atom R are close to the values predicted for the free R 3+ ions [see Section 3.2, as well as the data in Table II Table 2 Crystal structure parameters for R 2 Ni 1.78 In (R = Tb, Ho, Er, Tm; space group P4/mbm) determined by Rietveld refinement from neutron diffraction patterns collected in the paramagnetic state.
For R = Tb, Er the nearest-neighbour distances (NN) are between atoms R1 and R4 and between atoms R2 and R3, and the next-nearest neighbour distance (NNN) is equal to the c lattice parameter. For R = Ho, Tm there is a reverse relationship: NN = c and NNN = R1-R4 and R2-R3 distances.

Figure 2
The crystal unit cell of Tm 2 Ni 1.78 In, typical of other R 2 Ni 1.78 In compounds (tetragonal crystal structure of the Mo 2 FeB 2 type, space group P4/mbm).

Figure 3
The reciprocal magnetic susceptibility for Tm 2 Ni 1.78 In (circles), together with a fitted line representing the Curie-Weiss law. The upper inset shows the low-temperature behaviour, while the lower one shows the hysteresis loop of the magnetization versus external magnetic field up to 90 kOe. The vertical line indicates the metamagnetic transition.
rare earths in R 2 Ni 1.78 In occupy the 4h Wyckoff site in the crystal unit cell (space group P4/mbm, see Fig. 2) (Szytuła et al., 2013). The coordinates of the four R atoms in the unit cell are as follows: . Propagation vectors describing the magnetic ordering were obtained based on the positions of magnetic reflections in the neutron diffraction patterns of R 2 Ni 1.78 In. Subsequently, the allowed magnetic structures were calculated using the BASIREPS program from the FULLPROF package. The program generates possible irreducible representations (IRs) for a given propagation vector, space group and Wyckoff position of magnetic atoms. The proper IRs, describing the magnetic structure for a selected R 2 Ni 1.78 In compound, were chosen as those giving the best agreement between the experimental and calculated patterns (the full symmetry analysis is available in the supporting information). The final magnetic structures in R 2 Ni 1.78 In (R = Tb, Ho, Tm) were obtained by Rietveld refinement applied to the difference patterns (i.e. those obtained by subtraction of the neutron diffraction pattern in the paramagnetic state from the low-temperature pattern) as containing purely the magnetic contribution. The only exception was made for R = Er, where the full low-temperature neutron diffraction pattern was used for Rietveld refinement of the magnetic structure because of the high noise in the difference pattern.
3.3.1. Tb 2 Ni 1.78 In. The low-temperature magnetic structure has been reported by Szytuła et al. (2013). What is missing from the original paper is a symmetry analysis of the magnetic structure. Thus, a detailed symmetry analysis with a new refinement of the neutron powder diffraction data is presented below. A comparison with the previously reported data is presented at the end of this section.
The magnetic reflections in the neutron diffraction pattern registered at 1.6 K can be indexed by the propagation vector k 1 = [ 1 4 , 1 4 , 1 2 ], as shown for the difference pattern 1.6 K À 29.9 K in Fig. 4(a). The large magnetic unit cell consists of 32 crystal unit cells. It is enlarged four times along the a and b directions and twice along the c direction. For the k 1 = [ 1 4 , 1 4 , 1 2 ] propagation vector and the 4h site of the P4/mbm space group, the Tb atoms are divided into three independent orbits: atom Tb1 in orbit I, a (Tb2, Tb3) pair in orbit II and atom Tb4 in orbit III. Therefore the magnetic ordering of the (Tb2, Tb3) atoms is constrained by symmetry, while the magnetic moments of Tb1 and Tb4 are independent.   Theory predicts four irreducible representations, namely IR1 and IR3, related to the ordering of magnetic moments within the ab plane, and IR2 and IR4, related to the ordering along the c axis. The best agreement with the experimental pattern [ Fig. 4(a)] is obtained for an antiferromagnetic ordering along the c direction, described by the IR2 representation (Table 3). Although the symmetry analysis allows for different amplitudes of modulation for the magnetic moments of atoms belonging to different orbits, it was reasonable to assume an identical absolute value of the C 1 parameter (see Table 3) for all Tb atoms.
As only the relative phase shifts between atoms influence the intensities of the magnetic peaks, the absolute value of the magnetic phase factor of the Tb1 reference atom can be chosen arbitrarily, taking into account the physical validity of the proposed magnetic structure. For the zero phase factor of atom Tb1 and the magnetic phase factors of Tb2, Tb3, Tb4 fixed to 1 2 , 1 4 , 1 4 (in 2 units), respectively, the magnetic moment on a particular Tb atom, as determined for T = 1.6 K, is equal either to zero or to the modulation amplitude |C 1 | = 10.67 (7) B .
The magnetic structure presented above for Tb 2 Ni 1.78 In coincides with the structure reported originally by Szytuła et al. (2013), although the new refinement leads to a better value of the reliability factor (R magn = 0.054) than the one originally reported (0.081).  Table 4 The magnetic structure of Ho 2 Ni 1.78 In described by the IR2 representation for the 4h Wyckoff site of the P4/mbm space group and propagation vector k 2 = [k x , k x , 1 2 ], where k x = 0.3074 (4) at 1.6 K.
C 1 and È are, respectively, an amplitude and a phase factor of modulation of the magnetic moment along the BV1 vector, max is the maximum magnetic moment of the rare earth atom and R magn is the magnetic reliability factor of the Rietveld refinement. The parameter is defined as = 2k x .
C 1 and È are, respectively, an amplitude and a phase factor of modulation of the magnetic moment along the BV1 vector, is the magnetic moment of the rare earth atom, and R magn is the magnetic reliability factor of the Rietveld refinement.

Figure 5
Magnetic structures in R 2 Ni 1.78 In as described in Tables 3, 4 and 5: (a) R = Tb, (b) R = Ho and (c) R = Er, Tm (presented here for R = Tm). The magnetic unit cell is shown for commensurate structures in panels (a) and (c), while for the incommensurate structure in (b) a representative fragment of the structure is shown. Atoms R 1 , R 2 , R 3 and R 4 are depicted in red, black, yellow and blue, respectively.
3.3.2. Ho 2 Ni 1.78 In. The magnetic contribution to the Ho 2 Ni 1.78 In diffraction pattern at T = 1.6 K is shown in Fig. 4(b). The propagation vector indexing the magnetic reflections observed for Ho 2 Ni 1.78 In is an incommensurate one, k 2 = [k x , k x , 1 2 ], where k x = 0.3074 (4). The symmetry analysis shows that the Ho atoms are divided into three orbits, namely, Ho1 in orbit I, the (Ho2, Ho3) pair in orbit II and Ho4 in orbit III, and that the ordering of magnetic moments is allowed both within the ab plane (IR1, IR3) and along the c direction (IR2, IR4). The agreement with the experimental difference pattern 1.6 K À 12.2 K [ Fig. 4(b)] is obtained for the IR2 irreducible representation, i.e. the ordering of magnetic moments along the c axis (see Table 4).
The amplitude of modulation of the magnetic moments, constrained to be identical for all Ho atoms, is equal to C 1 = 7.93 (8) B . As for an incommensurate magnetic structure, the absolute values of the magnetic phase factors È i have no physical meaning. Thus, the phase factor of Ho1 was fixed to zero in order to define a point of reference, while the phase factors of all remaining atoms were refined. The magnetic phase factors for Ho3 and Ho4 have opposite signs, but absolute values close to 1 4 (while assuming È 1 = 0), and for Ho2 the phase factor is È 3 + k x ' 1 2 . A fragment of the antiferromagnetic structure of Ho 2 Ni 1.78 In (a magnetic unit cell cannot be defined due to incommensurability) is presented in Fig. 5(b).  Table 5 The magnetic structure of R 2 Ni 1.78 In (R = Er, Tm) described by the pair of propagation vectors k 1 = [ 1 4 , 1 4 , 1 2 ] (atoms R1 and R4) and k 000 1 = [À 1 4 , 1 4 , À 1 2 ] (atoms R2 and R3).

Figure 6
Ho 2 Ni 1.78 In: temperature dependence of (a) the amplitude of modulation of the magnetic moment (C 1 parameter) and (b) the k x component of the In the case of Ho 2 Ni 1.78 In, the thermal evolution of its neutron diffraction pattern was also recorded. Based on these data, the temperature dependences of the amplitude of modulation (C 1 parameter) and the k x component of the propagation vector were determined and are presented in Fig. 6. The temperature dependence of C 1 yields T N = 7.5 K. It is also worth noting that the k x component of the propagation vector increases noticeably while approaching T N .
3.3.3. Er 2 Ni 1.78 In and Tm 2 Ni 1.78 In. The magnetic contributions to the neutron patterns of Er 2 Ni 1.78 In and Tm 2 Ni 1.78 In are similar to one another, indicating the same magnetic structure in both compounds, with the only difference being in the absolute values of the magnetic moments. The magnetic reflections observed at T = 1.6 K for Er 2 Ni 1.78 In [Fig. 4(c)] and Tm 2 Ni 1.78 In [Fig. 4(d)] can be indexed by the propagation vector k 1 = [ 1 4 , 1 4 , 1 2 ]. In contrast to what was found for R = Tb, agreement with the experimental patterns for R = Er, Tm was obtained for the magnetic moments constrained to the ab plane according to the IR1 representation. However, this initial model of the magnetic structure in R 2 Ni 1.78 In (R = Er, Tm) leads to unreliable values of the magnetic moments, i.e. following the results of Rietveld refinement, the moments on atoms R1 and R4 equal 9.2 (2) and 8.62 (5) B for R = Er and Tm, respectively, while atoms R2 and R3 do not contribute at all to the intensity of the magnetic peaks as they have zero magnetic moments. This non-physical result points to the necessity of taking into consideration additional propagation vectors, similar to what was previously reported for TmAgGe (Baran et al., 2009).
The k 1 = [ 1 4 , 1 4 , 1 2 ] vector forms a star containing three other propagation vectors, namely, k 0 1 = [À 1 4 , À 1 4 , 1 2 ], k 00 1 = [ 1 4 , À 1 4 , 1 2 ] and k 000 1 = [À 1 4 , 1 4 , À 1 2 ]; all these propagation vectors provide the same positions of Bragg reflections of the magnetic origin. k 0 1 is equivalent to Àk 1 while k 000 1 is equivalent to Àk 00 1 . It is worth noting that taking a combination of the propagation vectors from the first (k 1 , k 0 1 ) and second (k 00 1 , k 000 1 ) pairs leads to new magnetic structures that cannot be obtained with the use of the propagation vectors originating from a single pair. The Rietveld refinement shows that both good agreement with the experimental patterns and reasonable values of the magnetic moments can be obtained for a combination of the k 1 and k 000 1 vectors. The final structure is presented in Fig. 5(c).
The magnetic ordering of atoms R1 and R4 is described by k 1 = [ 1 4 , 1 4 , 1 2 ], while that involving atoms R2 and R3 is related to k 000 1 = [À 1 4 , 1 4 , À 1 2 ] (see Table 5). According to the symmetry analysis reported by the BASIREPS program, in the discussed structure the magnetic moments of R1, . . . , R4 are independent with respect to the amplitude of modulation and the magnetic phase factor. However, in order to get a physically valid magnetic structure the amplitudes of modulation of all R i were constrained to be equal. The Rietveld refinement yields a difference of 1 4 between the magnetic phase factors of atoms R2 and R4 and those of atoms R1 and R3. In order to obtain identical absolute values of the magnetic moments for all rare earth atoms, equal to |C 1 |, the magnetic phases were fixed to 1 8 for R1 and R3 and to 3 8 for R2 and R4. The magnetic moments derived from refinement of the neutron diffraction patterns collected at T = 1.6 K equal 6.5 (1) B for R = Er and 6.09 (4) B for R = Tm.

Discussion
The X-ray and neutron diffraction data confirm that the R 2 Ni 1.78 In (R = Tb, Ho, Er and Tm) compounds crystallize in the tetragonal Mo 2 FeB 2 -type structure (space group P4/mbm) in a broad temperature range down to 1.6 K. In this structure, the rare earth atoms occupy the 4h Wyckoff site in the ab plane (z = 1 2 ) separated by planes containing the Ni and In atoms (z = 0). The structure is highly anisotropic with a short c lattice parameter (the c/a ratio is close to 0.5). The atomic arrangement is formed from slabs of CsCl-type (RIn) and AlB 2 -type (RNi 2 ). The rare earth atoms form triangles and squares within the ab plane (see Fig. 7). The resultant natural anisotropic multilayer structure suggests a high anisotropy of physical properties, especially those of magnetic character.
Magnetic susceptibility measurements, taken in the paramagnetic state, show that the effective magnetic moments per rare earth atom are close to the free R 3+ ion values, and therefore suggest that only the rare earth atoms carry magnetic moments. This result is in agreement with the results of Rietveld refinement of the low-temperature neutron diffraction data. Therefore, the magnetism in the investigated compounds is related to the rare earth magnetic moments,