Thomas Bonald

Nathan De Lara
email: nathan.delara@

Quentin Lutz
email: quentin.lutz@

Bertrand Charpentier

Scikit-network: Graph Analysis in Python

Keywords: Graph analysis, sparse matrices, Python, Cython, SciPy

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Scikit-learn (Pedregosa et al., 2011) is a machine learning package based on the popular Python language. It is well-established in today's machine learning community thanks to its versatility, performance and ease of use, making it suitable for both researchers, data scientists and data engineers. Its main assets are the variety of algorithms, the performance of their implementation and their common API.

Scikit-network is a Python package inspired by scikit-learn for graph analysis. The sparse nature of real graphs, with up to millions of nodes, prevents their representation as dense matrices and rules out most algorithms of scikit-learn. Scikit-network takes as input a sparse matrix in the CSR format of SciPy and provides state-of-the-art algorithms for ranking, clustering, classifying, embedding and visualizing the nodes of a graph.

The design objectives of scikit-network are the same as those having made scikit-learn a success: versatility, performance and ease of use. The result is a Python-native package, like Net-workX (Hagberg et al., 2008), that achieves the state-of-the-art performance of iGraph (Csardi and Nepusz, 2006) and graph-tool (Peixoto, 2014) (see the benchmark in section 5). Scikit-network uses the same API as Scikit-learn, with algorithms available as classes with the same methods (e.g., fit). * Also affiliated with Nokia Bell Labs, France 1 https://scikit-network.readthedocs.io/en/latest/ 1 BONALD, CHARPENTIER, DE LARA AND LUTZ It is distributed with the BSD license, with dependencies limited to NumPy (Walt et al., 2011) and SciPy (Virtanen et al., 2020).

Software features

The package is organized in modules with consistent API, covering various tasks:

• Data. Module for loading graphs from distant repositories, including Konect (Kunegis, 2013), parsing tsv files into graphs, and generating graphs from standard models, like the stochastic block model (Airoldi et al., 2008).

• Clustering. Module for clustering graphs, including a soft version that returns a nodecluster membership matrix.

• Hierarchy. Module for the hierarchical clustering of graphs, returning dendrograms in the standard format of SciPy. The module also provides various post-processing algorithms for cutting and compressing dendrograms.

• Embedding. Module for embedding graphs in a space of low dimension. This includes spectral embedding and standard dimension reduction techniques like SVD and GSVD, with key features like regularization.

• Ranking. Module for ranking the nodes of the graph by order of importance. This includes PageRank (Page et al., 1999) and various centrality scores.

• Classification. Module for classifying the nodes of the graph based on the labels of a few nodes (semi-supervised learning).

• Connectivity. Module relying on SciPy for the connectivity of the graph: shortest paths, graph traversals, connected components, etc.

• Visualization. Module for visualizing graphs and dendrograms in SVG (Scalable Vector Graphics) format. Examples are displayed in Figure 1.

These modules are only partially covered by existing graph softwares (see Table 1). Another interesting feature of scikit-network is its ability to work directly on bipartite graphs, represented by their biadjacency matrix.

Project Assets

Code quality and availability. Code quality is assessed by standard code coverage metrics. Today's coverage is at 98% for the whole package. Requirements are also kept up to date thanks to the PyUp tool. Scikit-network relies on TravisCI for continuous integration and cibuildwheel and manylinux for deploying on common platforms. OSX, Windows 32 or 64-bit and most Linux distributions (McGibbon and Smith, 2016) are supported for Python versions 3.6 and newer.


Sergey Nesterenko

Anna Tursina

Mathieu Pasturel

Sindisiwe Xhakaza

André Strydom

Two polymorphs of a new intermetallic Ce 2 Rh 2 Ga -crystal structure and physical properties

Keywords: cerium intermetallics, gallides, crystal structure, magnetic properties

Highlights

• Ce 2 Rh 2 Ga crystallizes in Pr 2 Co 2 Al-type (LT) and La 2 Ni 3 -type (HT) structures.

• The transition occurs at 864(5)°C

• Both structures can be described by different stacking of Ce-centered CeGa 4 building blocks.

• HT-Ce 2 Rh 2 Ga exhibits a phase transition, putatively of magnetic nature, at the unusually high temperature of 128.5

LT-Ce 2 Rh 2 Ga has a ground state of ferromagnetic nature which sets in at a paramagnetic-toferromagnetic phase transition at 5.2 K.

Introduction

The R 2 T 2 X series of compounds (with R = rare earth, T = transition metal and X = pmetal), the so-called 2:2:1 family, are known to exhibit a wide variety of physical properties at low temperature. Their crystallographic characteristics and magnetic data have been summarized in review articles (see e.g. [START_REF] Giovannini | Structural chemistry, magnetism and thermodynamic properties of R 2 Pd 2 In[END_REF][START_REF] Lukachuk | Intermetallic compounds with ordered U 3 Si 2 or Zr 3 Al 2 type structure-crystal chemistry, chemical bonding and physical properties[END_REF][START_REF] Pöttgen | Rare Earth-Transition Metal-Plumbides[END_REF]). In particular, in the ternary systems with X = In, the known R 2 T 2 X phases show ground states ranging from magnetically ordered Kondo compounds for T = Pd, Cu, and Ag via a heavy fermion behavior in Ce 2 Pt 2 In to intermediate valence for T = Ni and Rh [START_REF] Kaczorowski | Magnetic behavior in a series of cerium ternary intermetallics: Ce 2 T 2 In (T=Ni, Cu, Rh, Pd, Pt, and Au)[END_REF]. The Ce 2 Ni 2 Sn stannide behaves like a Kondo system and orders antiferromagnetic below T N = 4.7 K [START_REF] Chevalier | Magnetoresistivity of the antiferromagnetic Kondo stannide Ce 2 Ni 2 Sn[END_REF]. The cerium-based intermetallics Ce 2 Ni 1.88 Cd [START_REF] Niepmann | Ce 2 Ni 2 Cd -a new intermediate-valent cerium compound[END_REF][START_REF] Sampathkumaran | Heavy fermion behaviour in Ce 2 Ni 1.88 Cd[END_REF] as well as Ce 2 Ni 2 Ga [START_REF] Kaczorowski | Magnetic behaviour in CeNiGa 2 and Ce 2 Ni 2 Ga[END_REF] exhibit a fluctuating valence. Yb 2 Ni 2 Al and Yb 2 Pd 2 Sn were reported as heavy fermion systems with magnetically non-ordered ground states [START_REF] Geibel | Yb 2 Ni 2 Al: A prototypical Yb-based heavy-fermion system[END_REF][START_REF] Muramatsu | Reentrant quantum criticality in Yb 2 Pd 2 Sn[END_REF].

Most of R 2 T 2 X phases crystallize in the four structure types: Mo 2 B 2 Fe (tetragonal, P4/mbm) [START_REF] Rieger | Die Kristallstruktur von Mo 2 FeB 2[END_REF], Er 2 Au 2 Sn or U 2 Pt 2 Sn (tetragonal, P4 2 /mnm) -a superstructure of Mo 2 B 2 Fe [START_REF] Pöttgen | Er 2 Au 2 Sn and other Ternary Rare Earth Metal Gold Stannides with Ordered Zr 3 Al 2 -Type Structure[END_REF][START_REF] Gravereau | Crystal structure of U 2 Pt 2 Sn: a new derivative of the tetragonal U 3 Si 2 -type structure[END_REF], W 2 B 2 Co (orthorhombic, Immm) [START_REF] Rieger | Die Kristallstruktur von W 2 CoB 2 und isotypen Phasen[END_REF], and Mn 2 B 2 Al (orthorhombic, Cmmm) [START_REF] Becher | Über das ternäre Borid Mn 2 AlB 2[END_REF].

Five less common types are encountered in literature. Two of these are monoclinic ones: HT-Pr 2 Co 2 Al or Ca 2 Ir 2 Si (C2/c) -stacking variant of the W 2 B 2 Co [START_REF] Pani | Structure and transport properties of the R 2 Co 2 Al compounds (R= Pr[END_REF][START_REF] Schoolaertm | Synthese und Kristallstrukturen der Calcium-Iridiumsilicide Ca 3 Ir 4 Si 4 und Ca 2 Ir 2 Si[END_REF] -and LT-Nd 2 Cu 2 Cd (C2/c, own structure type) -superstructure of Mn 2 B 2 Al [START_REF] Tappe | Dimorphic Nd 2 Cu 2 Cd -structures and magnetic properties of RE 2 Cu 2 Cd (RE -Ce, Pr, Nd)[END_REF]; the other three structures are orthorhombic: La 2 Ni 3 (Cmce) [START_REF] Van Vucht | The crystal structure of La 2 Ni 3[END_REF], o-La 2 Ni 2 In (Pbam) [START_REF] Pustovoychenko | Orthorhombic La 2 Ni 2 In-A new intergrown CsCl-and AlB 2 -type slabs[END_REF], and Ca 2 Pd 2 Ge (Fdd2) [START_REF] Doverbratt | Linear Metal Chains in Ca 2 M 2 X (M = Pd, Pt; X = Al, Ge): Origin of the Pairwise Distortion and Its Role in the Structure Stability[END_REF]. Only a few representatives of these types are reported to date [START_REF] Stegemann | Experimental and theoretical investigations of the polar intermetallics SrPt 3 Al 2 and Sr 2 Pd 2 Al[END_REF][START_REF] Stotskyi | Crystal structure of a solid solution in the Mg-Pd-Al system[END_REF][START_REF] Dzevenkoa | Large hydrogen capacity in hydrides R2Ni2In-H (R = La, Ce, Pr, Nd) with new structure type[END_REF][START_REF] Marushina | Crystal structure and unstable valence in a novel intermetallic phase Ce 2 Ru 2 Al[END_REF][START_REF] Mahon | R 2 Co 3-x Si x (R = Pr, Nd, Sm, Gd) and R 2 Ni 3-x Si x (R = Gd-Er), new series of La 2 Ni 3 -type phases[END_REF].

The structural diversity of the R 2 T 2 X family is frequently accompanied by polymorphic phenomena between these structure types. A structural transformation from Mo 2 B 2 Fe to W 2 B 2 Co-type was observed in the R 2 Ni 2 Sn (R = Ce, Pr, Nd, Sm, Gd -Tm, Lu) system as a function of the rare-earth size or external pressure [START_REF] Heying | The Stannides RE 2 Ni 2 Sn (RE = Pr, Ho, Er, Tm) -Structural Transition from the W 2 B 2 Co to the Mo 2 B 2 Fe Type as a Function of the Rare Earth Size[END_REF][START_REF] Heymann | Highpressure phases of Tb 2 Ni 2 Sn and Dy 2 Ni 2 Sn[END_REF]. Also, depending on the size of the rare earth element, several groups of indides and stannides crystallize with the Mo 2 B 2 Fe-type or the Er 2 Au 2 Sn structure [START_REF] Hulliger | On tetragonal M 2 Au 2 In and related compounds[END_REF][START_REF] Pöttgen | Ternary Rare Earth Metal Gold Stannides and Indides with Ordered U 3 Si 2 and Zr 3 Al 2 -Type Structure[END_REF]. A transition from the Mo 2 B 2 Fe-type to the Mn 2 B 2 Al-type, provided by geometrical reasons or temperature, was found in the RE 2 Ni 2 X (X = In or Cd) series. The latter transition is of a reconstructive nature [START_REF] Ya | The crystal structure of the compounds R 2 Ni 2 In, R 2 Ni 2-x In and R 2 Cu 2 In (R -La[END_REF][START_REF] Fickenscher | The Mo 2 FeB 2 -and Mn 2 AlB 2 -Type Modifications of RE 2 Ni 2 Cd (RE = La, Pr, Nd, Sm, Tb, Dy)[END_REF].

Polycrystalline Mo 2 FeB 2 -type U 2 Pt 2 In transforms via a displacive phase transition to the Er 2 Au 2 Sn-type in the process of single crystal growth [START_REF] Estrela | Structural, magnetic and transport properties of single-crystalline U 2 Pt 2 In[END_REF]. Pr 2 Co 2 Al is dimorphic with a W 2 B 2 Co-type low-temperature modification and a monoclinic (space-group C2/c) hightemperature modification [START_REF] Pani | Structure and transport properties of the R 2 Co 2 Al compounds (R= Pr[END_REF]. According to [START_REF] Pustovoychenko | Orthorhombic La 2 Ni 2 In-A new intergrown CsCl-and AlB 2 -type slabs[END_REF], La 2 Ni 2 In crystallizes with the orthorhombic (Pbam) or tetragonal Mo 2 B 2 Fe (P4/mbm) modification depending on annealing temperature.

Interestingly, the same transformation takes place in the course of the hydrogenation of La 2 Ni 2 In, during which the symmetry changes from tetragonal to orthorhombic [START_REF] Dzevenko | Large hydrogen capacity in hydrides R 2 Ni 2 In-H (R = La, Ce, Pr, Nd) with new structure type[END_REF]. Finally, in the Gd-Co-Ga system, the compound with the W 2 B 2 Co-type structure is formed at the stoichiometric composition, but the Ga-deficient compound with Gd 2 Co 2.9 Ga 0.1 composition crystallizes in the Mn 2 B 2 Al-type structure [START_REF] Canepa | Structure and magnetism of Gd 2 Co 2 Ga, Gd 2 Co 2 Al and Gd 14 Co 3 In 2.7[END_REF]36], and in the Dy-Ni-Si system the variation of alloy composition from Dy 2 Ni 2.3 5Si 0.65 to Dy 2 Ni 2.5 Si 0.5 leads to crystallization of W 2 B 2 Co-or La 2 Ni 3 -type compounds [36].

In the course of our investigations on the ternary Ce-Rh-Ga system, we have discovered a new compound, namely Ce 2 Rh 2 Ga. As it turned out, it is dimorphic, with the high temperature form adopting the orthorhombic La 2 Ni 3 -type structure and the low temperature form crystallizing in the monoclinic Pr 2 Co 2 Al-type, and shows a new type of phase transition in the R 2 T 2 X family. Synthesis, crystal structures determination and LT physical properties for Ce 2 Rh 2 Ga polymorphs and La 2 Rh 2 Ga are reported herein. Preliminary data were presented at the SCTE 2018 conference [START_REF] Strydom | Unusual order in the new compound Ce 2 Rh 2 Ga, Book of abstract SCTE˗18[END_REF].

Experimental

The alloys were prepared by arc-melting (Arc Melter AM/0.5, Edmund Bühler GmbH) appropriate amounts ((Ce or La)40Rh40Ga20 (at.%)) of the constituent elements under a highpurity argon atmosphere in a water-cooled copper hearth. The purities of the starting materials were 99.85 % for Ce and La, 99.95 % for Rh, and 99.999 % for Ga. The ingots were remelted three times to ensure homogeneity. Then both the resulting cerium and lanthanum ingots were divided into two parts. First halves of Ce based and La based samples were annealed at 700 and the second ones at 900°C for 30 days in evacuated quartz tubes and then quenched in cold water.

Both the composition and the homogeneity of the annealed samples were checked by microprobe analysis using a Carl Zeiss LEO EVO 50XVP scanning electron microscope equipped with an INCA Energy 450 (Oxford Instruments) EDX-spectrometer. X-ray powder diffraction was performed at room temperature with the use of a STOE STADI-P transmission diffractometer with CuKα 1 radiation (λ = 1.54056 Å) in the 10 -90° 2θ range with a step size of 0.01°.

Single crystals of HT-Ce 2 Rh 2 Ga and La 2 Rh 2 Ga were separated from the respective ascast samples, whereas single crystal of LT-Ce 2 Rh 2 Ga was found in the annealed at 700°C sample. Single crystal X-ray diffraction data were collected at room temperature either on a Bruker Apex II (Mo Kα radiation) or on a CAD4 Enraf Nonius (Ag K α radiation) diffractometers.

The crystal structures were refined by the full-matrix least-squares method in the anisotropic approximation for all atoms using either the SHELXL-97 software [START_REF] Sheldrick | A short history of SHELX[END_REF] or SHELXL-2015/1 one.

Further details of the crystal structure investigations are listed in Table 1 and may be obtained from the Fachinformationszentrum Karlsruhe, 76344 Eggenstein-Leopoldshafen, Germany (Fax: +49-7247-808-666; E-Mail: crysdata@fiz-karlsruhe.de, http://www.fiz-karlsruhe.de/request_for_ deposited_data.html) on quoting the depository numbers CSD 434348 (LT-Ce 2 Rh 2 Ga) and CSD 434386 (HT-Ce 2 Rh 2 Ga). The crystal structure of La 2 Rh 2 Ga was deposited to the joint CCDC/FIZ Karlsruhe database via www.ccdc.cam.ac.uk with the reference number 1941455.

Final atomic coordinates and equivalent isotropic displacement parameters are listed in Table 2, and the most important bond distances for Ce 2 Rh 2 Ga polymorphs are given in Table 3.

The annealed samples of Ce 2 Rh 2 Ga were analyzed by differential thermal analysis (DTA) using a NETZSCH Leading Thermal Analysis STA 449 F1 Jupiter Platinum RT apparatus. The polycrystalline samples (~30 mg) were put in a small Al 2 O 3 crucible, and the subsequent measurement was conducted under a continuous He flow of 30 mL min -1 . The cooling and heating rates were 20 °C min -1 . An empty Al 2 O 3 crucible was used as a reference.

The magnetic susceptibility and heat capacity were measured as function of temperature in high resolution using a Dynacool Physical Properties Measurement System from Quantum Design (San Diego, USA). In the case of magnetic susceptibility, data were recorded in both warming and cooling modes in order to investigate the reversibility of the phase transition.

Results

Phase analysis

Microprobe analysis on both annealed Ce 2 Rh 2 Ga samples confirmed the chemical composition and homogeneous distribution of the elements with negligible amount of CeRhGa phase [START_REF] Hulliger | On new rare-earth compounds LnIrGa and LnRhGa[END_REF]. As this equiatomic ternary phase shows intermediate valent behavior without magnetic ordering [START_REF] Chevalier | Hydrogenation of the Ce(Rh 1-x Ir x )Ga System: Occurrence of Antiferromagnetic Ordering in the Hydrides Ce(Rh 1-x Ir x )GaH 1.8[END_REF][START_REF] Goraus | Electronic and thermal properties of non-magnetic CeRhGa[END_REF] we expect its presence in our magnetic study on Ce 2 Rh 2 Ga to be of no consequence.

The X-ray powder patterns of these samples were different depending on the thermal treatment used, and at the same time the crystal structure of the sample annealed at 900°C does not differ from the as-cast ones. DTA was used to establish the reason for this difference (Fig. 1, and Fig. S1 in Supplementary). When heating of the sample annealed at 700°C, one relatively weak endothermic peak was detected at 864(5)°C, that may be attributed to the low temperature (LT) -high temperature (HT) polymorphous transformation, followed by melting at 970(5)°C. It is worth noticing that the reversible polymorphic reaction is not detected in the cooling branch of DTA scan; this could be due to the small amount of heat and slow kinetics associated to this reaction.

The XRD data refinement of polycrystalline samples reveals that the sample annealed at 700°C consisted solely of the LT phase, whereas the as-cast samples as well as the one annealed at 900°C contain exclusively the HT phase (Fig. 2). Other phases were not detected. The Rietveld refinements were performed with the MRIA program [START_REF] Zlokazov | MRIA -a program for a full profile analysis of powder multiphase neutron-diffraction time-of-flight (direct and Fourier) spectra[END_REF] using the crystallographic data obtained from the single crystal XRD experiments (Tables 1,2) for orthorhombic HT-Ce 2 Rh 2 Ga and for monoclinic LT-Ce 2 Rh 2 Ga.

Crystal structures

The HT-Ce 2 Rh 2 Ga crystallizes with the orthorhombic La 2 Ni 3 -type structure [START_REF] Van Vucht | The crystal structure of La 2 Ni 3[END_REF], while the low-temperature polymorph adopts the monoclinic Pr 2 Co 2 Al (also known as Ca 2 Ir 2 Si) type [START_REF] Pani | Structure and transport properties of the R 2 Co 2 Al compounds (R= Pr[END_REF][START_REF] Schoolaertm | Synthese und Kristallstrukturen der Calcium-Iridiumsilicide Ca 3 Ir 4 Si 4 und Ca 2 Ir 2 Si[END_REF]. The non-f-electron counterpart La 2 Rh 2 Ga was found to crystallize with the orthorhombic La 2 Ni 3 -type structure.

The coordination polyhedra of all the atoms are presented in Fig. 3 and Fig. S2 in the Supplementary Materials. The environment of Ce atoms in the two polymorphs is rather similar and comprises six Rh, four Ga and five Ce atoms. The resulting polyhedron can be regarded as a derivative of a pentagonal prism centered on one basal face and on all side faces. In both structures, the Rh atom is coordinated by six cerium atoms and two gallium atoms in the form of a distorted square antiprism with two additional rhodium atoms capping the basal faces of the prism. Finally, the arrangement of the eight cerium atoms and four rhodium atoms around Ga results in a severely distorted icosahedron.

The Ce-Ce distances are ranging from 3.43 to 3.70 Å for the HT polymorph -within the similar range observed in the isostructural Ce 2 Ru 2 Al with d Ce-Ce =3.32 and 3.64 Å [START_REF] Marushina | Crystal structure and unstable valence in a novel intermetallic phase Ce 2 Ru 2 Al[END_REF] -and from 3.61 to 3.71 Å for the LT one (Table 3). These distances correspond well to twice the metallic radius of this element (r Ce = 1.825 Å [START_REF] Teatum | Compilation of calculated data useful in predicting metallurgical behavior of the elements in binary alloy systems[END_REF]). Likewise, the interatomic distances Ce-Rh and Ce-Ga in HT modification are in the interval of corresponding values in the LT modification. Nevertheless, consideration of the Ga and Rh metallic radii (r Ga = 1.411 Å and r Rh = 1.345 Å [START_REF] Teatum | Compilation of calculated data useful in predicting metallurgical behavior of the elements in binary alloy systems[END_REF]) highlights short Ce-Rh distances and rather elongated Ce-Ga ones. The Rh-Ga contacts of 2.56 Å in HT-Ce 2 Rh 2 Ga and from 2.58 to 2.61 Å in the LT-Ce 2 Rh 2 Ga are significantly smaller than the sum of the metallic radii, also suggesting a strong bonding between these elements. Opposite, more or less elongated Rh-Rh distances are observed and Ga atoms are well separated from each other and produce non-bonding contacts in both phases.

In the new compounds, the four Ga neighbors around each Ce atom form a slightly distorted CeGa 4 tetrahedron with Ce-Ga contacts being in the range of 3.34 -3.39 Å and 3.30 -3.35 Å for HT-and LT-Ce 2 Rh 2 Ga, respectively. Cerium centered gallium tetrahedra can be regarded as the main common building blocks. In the HT-Ce 2 Rh 2 Ga, CeGa 4 tetrahedra are merged by edge-sharing with two neighboring ones forming rows alternating in a ''checkerboard'' pattern running along the a-axis (Fig. 4a). Each CeGa 4 tetrahedron has three more shared edges with adjacent rows.

The CeGa 4 tetrahedra are delimiting tetrahedral voids centered by Rh atoms (Fig. 4b).

Thus the Ga environment of Rh atoms is a RhGa 4 tetrahedron which is severely distorted in contrast to that of Ce-centered. Within the Rh-centered tetrahedron, two Rh-Ga contacts are bonding -each of 2.555 Å -and two others are approximately 2 Å longer -4.540 Å and 4.862 Å. Each RhGa 4 tetrahedron contacts with six neighbors by edge-sharing generating a 3D framework. The superposition of the Ce-centered tetrahedra and the Rh-centered tetrahedra nets results in a three-dimensional network by sharing triangular faces composed of Ga atoms (see Fig. 4c).

In the LT-Ce 2 Rh 2 Ga, the CeGa 4 -tetrahedra framework is built by edge-sharing of each of the six edges of a CeGa 4 tetrahedron (Fig. 5a) which is delimiting octahedral voids centered by a Rh-Rh dumbbell (d Rh-Rh = 2.75 Å). In this structure, each Rh atom resides in a distorted squarebased pyramid RhGa 5 with Rh-Ga interatomic distances of 2.578 Å, 2.607 Å, 4.579 Å, 4.633 Å, and 4.668 Å. Two neighboring RhGa 5 pyramids share their basal faces to form a Rh 2 Ga 6 octahedron (Fig. 5b). The resulting structure is produced by the CeGa 4 -and Rh 2 Ga 6 -networks linked by the mutual triangular faces of the Ce-centered tetrahedra and of the Rh-doublecentered octahedra (Fig. 5c).

Physical properties

LT-Ce 2 Rh 2 Ga. Figure 6 shows the variation of the Ce-molar magnetic susceptibility of LT-Ce 2 Rh 2 Ga with temperature on a logarithmic temperature scale. Below 400 K χ(T) assumes small values but follows a regular Curie-Weiss law, χ(T)=C/(T-θ P ) down to 100 K as is shown by the line fitted to the experimental data (inset to Fig. 6). Here C =Nµ eff 2 /Mk B in terms of Avogadro's number N, the substance molar mass M and Boltzmann's constant k B . The fit parameters are the Weiss temperature θ P and the effective magnetic moment µ eff 2 =g J 2 J(J+1) in terms of the Landé factor g J =1+{[J(J+1)+S(S+1)-L(L+1)]/[2J(J+1)]} as function of the spin (S), orbital (L) and total (J) angular momentum quantum numbers. A good fit is achieved using an effective moment value of µ eff =2.54 µ B Ce -1 . Interestingly, the obtained Weiss temperature is negative; θ P = -30 K, which suggests dominant antiferromagnetic exchange in this compound.

However, a steep and ferromagnetic-like increase in χ(T) is found upon cooling below 10 K. The point of steepest incline at this anomaly is marked with an arrow labelled as the Curie point T C =5.2 K in Fig. 6.

The specific heat C p per formula unit of LT-Ce 2 Rh 2 Ga is shown on a logarithmic temperature axis in the main panel of Fig. 7. Near room temperature C p ≈125 J mol fu -1 K -1 for this compound which closely resembles the phenomenological Dulong-Petit value for a compound with five independent atoms in its formula unit [START_REF] Petit | Recherches sur quelques points importants de la Théorie de la Chaleur[END_REF]. C p decreases steadily below room temperature but between 6 K and 5 K C p rises sharply into a peak of nearly 20 J mol fu -1 K -1 before decreasing further towards lower temperatures. The calculated configurational entropy S per mole Ce is shown in inset (a) of Fig. 7, where ܵ(ܶሻ ൌ  ܥ( ܶ′ሻdܶ′ ⁄

் . An entropy amount of 4.2 J mol Ce -1 K is released at the phase transition which is somewhat lower than the value Rln(2)=5.76 J mol -1 K -1 attributable to long-range magnetic ordering out of a doublet ground state.

In inset (b) to Fig. 7 Weiss temperature θ p = -120 K. However, upon cooling below the predictable Curie-Weiss range, at a temperature labelled T N = 128.5 K χ drops precipitously and continues to decrease losing about 25% of its magnitude at T N when ~80 K is reached. Below this temperature χ rises again towards low temperatures. The shape of the anomaly in χ(T) at T N is not unlike what may be expected at a paramagnetic-to-antiferromagnetic phase transition, which is corroborated by the dominant antiferromagnetic nature of the magnetic exchange obtained from the calculated negative value of the Weiss temperature for this compound. In the inset of Fig. 8 the region near T N is shown on expanded scales. Between data collected while slowly cooling the sample (black symbols) and warming the sample (red symbols) a thermal hysteresis amounting to ∆T N = 1 K is found. Outside of the region of ∆T N the warming and cooling data curves coincide. Figure 9 shows the results of magnetization measurements at a number of fixed temperatures. The three isotherms in panel (a) were collected at three temperatures close to the phase transition temperature. The field-dependent magnetization in this temperature region proceeds linearly in field up to 7 T. The magnetic moment extracted from magnetization measurements even at low temperature and deep into the ordered region is small -see Fig. 9(b). An applied field of 9 T yields only 0.34 µ B Ce -1 for the sample at 1.7 K. It is noteworthy, however, that correlated electron Ce-based compounds often yield only small magnetization values [START_REF] Janka | Cerium intermetallics with TiNiSi-type structure[END_REF][START_REF] Pöttgen | Cerium intermetallics with ZrNiAl-type structure -a review[END_REF].

Finally, we turn to the specific heat of HT-Ce 2 Rh 2 Ga as illustrated in the main panel of Fig. 10 on a per-chemical formula unit basis. Below room temperature we notice that C p is nearly constant at a value close to the Dulong-Petit value for this compound. In the Debye model of the specific heat of solids this value is achieved when the substance is at a temperature suitably higher than its Debye temperature θ D . At 130 K however, the monotonous behavior in C p (T) ceases where C p rises very sharply and achieves 424 J mol fu -1 K -1 at 127.5 K. In the inset of Fig. 10 we cast the specific heat data below 20 K in the form C p (T)/(T) against the square of temperature. The line on the experimental points is drawn with γ=172.5 mJ mol Ce -1 K -1 and β=1.217 x 10 -3 mJ mol Ce -1 K -4 . By contrast, in the specific heat of the comparable nonmagnetic compound La 2 Rh 2 Ga (see Fig. 11a) there are no anomalies observed below room temperature, as would be expected for a simple metal. For La 2 Rh 2 Ga the Sommerfeld coefficient is found to be γ =13.45 mJ mol La -1 K -2 according to the least-squares fit shown in panel (b) of Fig. 11.

The unusual behavior of HT-Ce 2 Rh 2 Ga at 128.5 K prompted us to search for a possible crystal structure phase transition. The first attempts to determine structural changes in the transition through T N =128 K using single crystal X-ray diffraction showed that the structure changes are too small to be reliably described quantitatively. Changes in cell dimensions and cell volume as well as the appearance of a small monoclinic distortion when cooling a single crystal of HT-Ce 2 Rh 2 Ga from room temperature down to 80 K are presented in Figure S3 (Supplementary materials). Further crystal structure determination was not possible due to the twinning of the crystals induced by the structural transition.

Discussion

The two new Ce-based compounds LT-Ce 2 Rh 2 Ga and HT-Ce 2 Rh 2 Ga show contrasting behaviours despite being alike in their chemical compound formulas. In both structures Ce atoms occupy a unique crystallographic site however in LT-Ce 2 Rh 2 Ga it is a general position 8f (site symmetry 1) and in HT-Ce 2 Rh 2 Ga it is a special position 8f (site symmetry m). Magnetically, both compounds can be described in terms of the full trivalent magnetic state of the Ce ion over a wide range below room temperature, which makes both compounds amenable to magnetic order. In LT-Ce 2 Rh 2 Ga the Ce ions order in a spin arrangement at a temperature of 5.2 K that imparts a ferromagnetic character to its temperature-dependent magnetic susceptibility as well as its field-dependent magnetization. This magnetic phase transition temperature is typically in the range of many cerium-based compounds. The single 4f-electron of cerium in its trivalent state being predisposed to an antiferromagnetic on-site Kondo exchange in metals, however, means that by comparison ferromagnetic order in cerium intermetallic compounds is much less ubiquitous than antiferromagnetic order.

Taken together, the results of magnetic susceptibility and specific heat for the phase transition found in this study on the compound HT-Ce 2 Rh 2 Ga bear the signatures of antiferromagnetic order, but at the extraordinary high phase transition temperature of 128.5 K.

With the present state of investigation and results available on this compound we cannot rule out the possibility of a crystal structure modification accompanying the magnetic phase transition at T N . This phase transition is temperature hysteretic furthermore which is unusual since fluctuations in the order parameter is a common feature of magnetic order driven by an antiparallel spin arrangement.

In conclusion, the novel Ce 2 Rh 2 Ga intermetallics crystallize in two polymorphs adopting rather unusual structure-types: an orthorhombic ordered La The solid line is a Curie-Weiss fit to the data as explained in the text.

  specific heat in the form C p (T)/T is plotted against the square of temperature. The linear behaviour above T C enables to estimate the electronic contribution on account of the different temperature dependencies of the electronic γ and the lattice β specific heats; C p (T)/T = γ + βT 2 . The solid line on the data in inset (b) is the result of a least-square fit with γ =75 mJ mol Ce -1 K -2 and β =0.547 mJ mol Ce -1 K -4 . HT-Ce 2 Rh 2 Ga. Figure 8 (main panel) shows the per-mole Ce magnetic susceptibility of HT-Ce 2 Rh 2 Ga measured in a static field of 0.1 T. The line on the experimental points is the result of fitting the data to the Curie-Weiss law. Between 400 K and 200 K the experimental points may thus be described using an effective magnetic moment value of 2.54 µ B Ce -1 and a

Fig. 1

 1 Fig. 1 DTA curve of the annealed at 700°C sample (the arrows show the course of temperature change).

Fig. 2 Fig. 3 .

 23 Fig. 2 Rietveld refined X-ray diffraction patterns for (up) HT-Ce 2 Rh 2 Ga (χ 2 = 1.78, R p = 0.024, R exp = 0.022) and (down) LT-Ce 2 Rh 2 Ga (χ 2 = 3.69, R p = 0.024, R exp = 0.015). The experimental diffraction profile is indicated by black dots. The calculated diffraction profile is shown as the upper blue line, the difference profile is shown as the bottom red line and the vertical bars correspond to the calculated Bragg positions.Fig.3. Coordination polyhedra of the three atoms in the crystal structure of HT-Ce 2 Rh 2 Ga (upper row) and LT-Ce 2 Rh 2 Ga (bottom row). Ce-atoms are drawn as large green balls, Rh-atoms as small blue ball, and Ga-atoms as middle size pink balls.

Fig. 4 .

 4 Fig. 4. (a) CeGa 4 and (b) RhGa 4 tetrahedra arrangement in the unit cell of HT-Ce 2 Rh 2 Ga. (c) The packing of the CeGa 4 and RhGa 4 tetrahedra. CeGa 4 tetrahedra are highlighted in green, RhGa 4 tetrahedra are highlighted in blue.

Fig. 5 .

 5 Fig. 5. (a) CeGa 4 tetrahedra and (b) double centered Rh octahedra arrangement in the unit cell of LT-Ce 2 Rh 2 Ga. (c) The packing of the CeGa 4 tetrahedra and Rh 2 Ga 6 octahedra. CeGa 4 tetrahedra are highlighted in green. Rh 2 Ga 6 -octahedra are emphasized in blue.

Fig. 6 .

 6 Fig. 6. (main panel) Semi-log plot of dc-magnetic susceptibility per mole Ce of LT-Ce 2 Rh 2 Ga measured in a static applied field of 0.1 T, with the arrow marking the paramagnetic-toferromagnetic phase transition temperature T C . (inset) Inverse susceptibility against temperature.

Fig. 7 .

 7 Fig. 7. (main panel) Semi-log plot of the specific heat C p per mole of Ce of LT-Ce 2 Rh 2 Ga. The arrow indicates the ferromagnetic phase transition temperature. Dashed line is a guide to the eye connecting the experimental points. Inset (a): Calculated entropy per mole of Ce. Inset (b): Specific heat in the form C p (T)/T against the square of temperature. The dashed line is a guide to the eye. Solid line illustrates a fit to extract the Sommerfeld coefficient γ= 75 mJ mol Ce -1 K -2 .

Fig. 8 .

 8 Fig. 8. (main panel) Magnetic dc-susceptibility of HT-Ce 2 Rh 2 Ga measured in a static applied field of 0.1 T. The solid line on the data is obtained through a Curie-Weiss fit. The dashed line illustrates the continuation of the Curie-Weiss fit below the fitted temperature range which is intercepted at the phase transition marked T N at 128.5 K. (inset) Magnetic susceptibility on expanded scales near T N shows the temperature hysteretic behaviour between data obtained during cooling (black symbols) and warming (red symbols) of the sample.

Fig. 9 .

 9 Fig. 9. dc-Magnetization per Ce atom of HT-Ce 2 Rh 2 Ga at temperatures (a) close to T N and (b) at low temperature. The magnetization is linear in field at all investigated temperatures except for the T=1.70 K isotherm where the high-field region permits to calculate a saturation magnetization of 0.19 µ B /Ce.

Fig. 10 .

 10 Fig. 10. (main panel) Temperature dependence of specific heat per formula unit of HT-Ce 2 Rh 2 Ga. The dashed line is a guide to the eye to connect the data points through the sharp phase transition marked by an arrow at T N =128.5 K. (inset) Specific heat per mole of Ce of HT-Ce 2 Rh 2 Ga in the form C p (T)/T plotted against the square of temperature. The solid line on the data is a fit to extract the electronic specific heat coefficient γ=172.5 mJ mol Ce -1 K -2 . An upturn of unknown origin is found to occur in C p (T)/T below about 7 K.

Fig. 11 .

 11 Fig. 11. (a) Specific heat of the Pauli-paramagnetic compound La 2 Rh 2 Ga over a wide range of temperature. (b) Specific heat in the form C p (T)/T against the square of temperature. A value γ=13.45 mJ mol La -1 K -2 is obtained for the Sommerfeld coefficient as illustrated by the solid line on the data points.

Table 1 .

 1 3 Ni 2 -type for samples annealed above 867°C and a monoclinic Pr 2 Co 2 Al-type for samples annealed below this temperature. Both structure types can be built from CeGa 4 cerium centered gallium tetrahedral bricks. Both allotropes show phase transitions of a magnetic nature and further studies are needed to determine the order parameter. In HT-Ce 2 Rh 2 Ga in particular it would be interesting to clarify what property of the material is driving the temperature hysteresis that accompanies the magnetic ordering. Selected single-crystal data collection and structure refinement parameters for La 2 Rh 2 Ga , HT-Ce 2 Rh 2 Ga, and LT-Ce 2 Rh 2 Ga.

	Empirical formula	La 2 Rh 2 Ga	HT-Ce 2 Rh 2 Ga	LT-Ce 2 Rh 2 Ga
	Molar mass, g⋅mol -1	553.36		555.78
	Space group	Cmce (64)	Cmce (64)	C 2/c (15)
	Structure type	La 2 Ni 3	La 2 Ni 3	Pr 2 Co 2 Al
	a (Å)	5.9251(17)	5.851(2)	10.0903(6)
	b(Å)	9.8402(17)	9.618 (2)	5.6041(3)
				β=104.995(3)°
	c (Å)	7.5019(14)	7.487 (2)	7.8153(4)
	Cell volume (Å 3 )	437.39(17)	421.3(2)	426.88(4)
	Z	4	4	4
	D calc (g*cm -3 )	8.403	8.762	8.648
	Abs. coeff µ (mm -1 )	17.046	18.437	34.464
	Radiation (Å)	AgKα,	AgKα,	MoKα,
		0.56087	0.56087	0.71073
	Index range Θ range	-8≤ h ≤8, 0≤ k ≤14, 0≤ l ≤11 3.27 -24.5	0≤ h ≤11, -19≤ k ≤0, -15≤ l ≤15 3.34° -34.96°	-16≤ h ≤16, 0≤ k ≤9, 0≤ l ≤13 4.18° -36.53°
	Number of measured reflections	803	983	1060
	Number of reflections with	361	597	937
	I ≥ 2σ(I)
	Number of refined parameters	17	17	25
	GooF on F 2	1.116	0.972	1.088
	R[F 2 >2σ(F 2 )]	0.0310	0.0389	0.0239
	wR(F 2 )	0.0843	0.0641	0.0537

Table 2 .

 2 Atomic coordinates and equivalent isotropic displacement parameters U eq (Å 2 ) for La 2 Rh 2 Ga , HT-Ce 2 Rh 2 Ga, and LT-Ce 2 Rh 2 Ga.

	Atom	Wyckoff site	x/a	y/b	z/c	U eqv
				La 2 Rh 2 Ga
	La	8f	0	0.33857(4)	0.09757(6)	0.0132(2)
	Rh	8e	1/4	0.09407(5)	1/4	0.0146(2)
	Ga	4a	0	0	0	0.0128(3)
				HT-Ce 2 Rh 2 Ga
	Ce	8f	0	0.33904(4)	0.09881(6)	0.0123(1)
	Rh	8e	1/4	0.09769(6)	1/4	0.0143 (2)
	Ga	4a	0	0	0	0.0118(2)
				LT-Ce 2 Rh 2 Ga
	Ce	8f	0.35324(2)	0.14759(4)	0.35191(3)	0.01041(7)
	Rh	8f	0.13144(3)	0.13464(5)	0.00765(4)	0.01100(8)
	Ga	4e	0	0.1398(2)	1/4	0.0098(2)

Table 3 .

 3 Interatomic distances (Å) in the HT-and LT-Ce 2 Rh 2 Ga

		HT-Ce 2 Rh 2 Ga			LT-Ce 2 Rh 2 Ga
	Atom To atom	d	Atom To atom	d	Atom To atom	d
	Ce	2 Rh	2.9680(8) Ce	Rh	2.9310(4) Rh	Ga	2.5781(3)
		2 Rh	3.0545(7)		Rh	3.0227(4)		Ga	2.6070(5)
		2 Rh	3.1000(8)		Rh	3.0227(4)		Rh	2.7494(6)
		Ga	3.3438(7)		Rh	3.0666(4)		Ce	2.9310(4)
		Ga	3.3793(8)		Rh	3.1040(4)		Ce	3.0137(4)
		2 Ga	3.391(1)		Rh	3.2252(4)		Ce	3.0227(4)
		Ce	3.432(1)		Ga	3.3028(3)		Rh	3.0270(6)
		2 Ce	3.699(1)		Ga	3.3235(6)		Ce	3.0667(4)
		2 Ce	3.699(2)		Ga	3.3965(6)		Ce	3.1040(4)
	Rh	2 Ga	2.5546(5)		Ga	3.4455(3)		Ce	3.2252(4)
		2 Rh	2.925 (2)		Ce	3.6072(3) Ga	2 Rh	2.5782(4)
		2 Ce	2.9680(8)		Ce	3.6073(3)		2 Rh	2.6071(5)
		2 Ce	3.0545(7)		Ce	3.6495(5)		2 Ce	3.3028(3)
		2 Ce	3.1000(8)		Ce	3.6759(5)		2 Ce	3.3235(6)
	Ga	4 Rh	2.5546(5)		Ce	3.7081(5)		2 Ce	3.3965(6)
		2 Ce	3.3438(8)					2 Ce	3.4455(3)
		2 Ce	3.3793(8)
		4 Ce	3.391 (1)

Acknowledgements

AMS thanks the URC/FRC of UJ and the SA NRF (93549) for generous financial assistance, and the Max Planck Institute CPfS in Dresden for their hospitality where parts of this research were conducted. This study was supported by the Russian Foundation for Basic Research (Grant No 18-03-00656a and 19-03-00135).

HIGHLIGHTS

Manuscript: Two polymorphs of a new intermetallic Ce2Rh2Ga -crystal structure and physical properties The existence of the new compound Ce2Rh2Ga is reported. Ce2Rh2Ga is dimorphic with the high temperature form (HT) adopting an ordered version of the orthorhombic La2Ni3-type structure and the low temperature form crystallizing in the monoclinic Pr2Co2Al-type In this manuscript focus is placed on the HT version of Ce2Rh2Ga. A phase transition at the exceptionally high temperature of 130 K is found in the orthorhombic version of Ce2Rh2Ga. The transition is temperature hysteretic and involves at the same temperature magnetic phase transition features in the temperature dependence of magnetic susceptibility, a huge and sharp peak in the heat capacity, and a modification in the crystal structure.

DECLARATION of INTEREST STATEMENT

The authors declare that they have no competing interest with the contents or subject of the manuscript.