Local Structure Analysis of Heavy Fermion Ce₂Pt₆Ga₁₅ with a Honeycomb Structure Using Extended X-Ray Absorption Fine Structure

Abstract

Ce₂Pt₆Ga₁₅ is a heavy fermion compound near the quantum critical point (QCP). Its crystal structure may exhibit magnetic frustration due to a honeycomb arrangement; however, stacking faults in the crystal hinder structural analysis. As a local structure probe, extended X-ray absorption fine structure (EXAFS) is less sensitive to stacking faults and is a powerful tool for crystal structure determination. We synthesized single-crystal Ce₂Pt₆Ga₁₅, performed single-crystal and powder X-ray diffraction experiments, and conducted X-ray absorption spectroscopy (XAS) measurements. The composition of Ce₂Pt₆Ga₁₅ deviates from stoichiometry, suggesting Ce and Ga enrichment or Pt site deficiencies. A comparison of X-ray absorption near-edge structure (XANES) at the Ce L₃-edge with reference materials suggests that Ce valence is likely trivalent. To determine the exact structure, we simultaneously analyzed EXAFS spectra at the Ce L₃-, Pt L₃-, and Ga K-edges. The EXAFS spectra of Ce₂Pt₆Ga₁₅ are inconsistent with the hexagonal Sc_0.6Fe₂Si_4.9-type structure but are better explained by an orthorhombic structure with a honeycomb arrangement.

1. Introduction

The rare-earth and actinide atoms have localized magnetic moments because of the localized nature of the f electrons. The f electrons in rare-earth/actinide intermetallic compounds hybridize with the conduction electrons. This hybridization effect gives rise to the Kondo effect and the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction. The Kondo effect suppresses the magnetic order. On the other hand, the RKKY interaction stabilizes the magnetic order. The Kondo effect and the RKKY interaction are important for the exotic physical properties.

Unusual physical properties, such as non-Fermi liquid (NFL) behavior, heavy fermion characteristics, and non-Bardeen–Cooper–Schrieffer (BCS)-type superconductivity have been discovered in CeIn₃ [1], CeCoIn₅ [2,3,4,5], UTe₂ [6,7,8], etc. These properties occur around the quantum critical point (QCP). The quantum critical behavior is explained by the competition between the Kondo effect and the RKKY interaction, as depicted in the so-called Doniach phase diagram [9]. The QCP appears when the Kondo effect and the RKKY interaction are comparable in magnitude. The spin fluctuations are enhanced around the QCP. The behavior of the fluctuations around QCP can be explained by the Herts–Millis–Moriya (HMM) theory [10,11,12]. The unconventional spin fluctuations, which cannot be explained by the HMM theory, were observed in YbRh₂Si₂ [13,14], $\beta$-YbAlB₄ [15,16], etc. Local quantum criticality (LQC), where magnetic frustrations play a key role, was proposed to explain the unconventional fluctuations [17].

The $R_2$${\mathrm{Pt}}_6{X}_{15}$ system (R = rare-earth and actinide, X = Al, Ga, Si) is a candidate for the frustrated honeycomb lattice system, and its physical properties and crystal structures have been reported [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37]. First, the structure was identified as the hexagonal Sc_0.6Fe₂Si_4.9-type structure (space group 194 ($P{6}_3/mmc$)) in Figure 1a. This structure has the $R_2{X}_3$-layer and the $T{X}_2$-layer. The R ions in the $R_2{X}_3$-layer form a triangular lattice, and the occupancies of the R and X1 sites are 2/3 and 1/3, respectively, indicating that these sites are partially occupied. The distance between the nearest neighbor R-R atom is approximately 4 Å and the distance between $R_2{X}_3$-layers is about twice the R-R distance, indicating that the magnetic interaction can be two-dimensional. The neutron-scattering measurements of U₂Pt₆Ga₁₅ [34] and U₂Pt₆Al₁₅ [35] showed a two-dimensional magnetic interaction. The superstructure reflections were observed and the reflections had streaks in the system, indicating that the crystal structure cannot be explained by the hexagonal Sc_0.6Fe₂Si_4.9-type. Lately, the orthorhombic structure (space group 63 ($Cmcm$)) [19] in Figure 1b was proposed as an alternative possible structure. The R atoms in the $R_2{X}_3$-layer form a honeycomb lattice; this layer has stacking faults along the c-axis. The streak reflections can be attributed to a stacking disorder of the honeycomb lattices. As a result, the average structure maintains hexagonal symmetry. The X-ray analyses described in this paper were performed on the hexagonal unit cell. The $R_2$${\mathrm{Pt}}_6{X}_{15}$ system is a candidate for the magnetically frustrated system with a honeycomb lattice. The X-ray diffraction analysis of the streaks caused by the stacking faults is difficult; therefore, a different approach is needed to investigate the structure. For example, transmission electron microscopy was used to analyze the structure of ${\mathrm{Pr}}_2$${\mathrm{Pt}}_6$${\mathrm{Ga}}_{15}$ [31]. Extended X-ray absorption fine structure (EXAFS) is also a powerful structural analysis tool. Because of its sensitivity to local structure, stacking faults do not interfere with the structural analysis and it is able to distinguish whether the structure is the hexagonal Sc_0.6Fe₂Si_4.9-type model or not. The difference between hexagonal and orthorhombic models is whether the interatomic distances of 1.5 Å and 1.7 Å exist or not. For Ce₂Pt₆Ga₁₅, the Fourier transform (FT) of the experimental EXAFS data at the Ga K-edge was reported [26], but the local structure was not analyzed. In a previous work [36], we analyzed the EXAFS around the Ce L₃-edge and the Pt L₃-edge of Ce₂Pt₆Al₁₅ showing that the orthorhombic model is valid. However, we did not measure the EXAFS around the Al K-edge of Ce₂Pt₆Al₁₅. EXAFS analysis of the X site is important for revealing the structure.

Ce₂Pt₆Ga₁₅ is a suitable compound for studying quantum critical phenomena. The temperature (T) dependence of the susceptibility and the specific heat (C) of Ce₂Pt₆Ga₁₅ showed NFL-like behavior and C/T reached about 2.0 J/K² mole-Ce at 0.4 K. The effective mass of the conduction electrons is greatly enhanced. There is concern that Ce₂Pt₆Ga₁₅ is close to the quantum critical point (QCP) [22,27]. Because there are few candidate materials, the search for materials is necessary to research the quantum critical phenomena. If the $R_2$${\mathrm{Pt}}_6{X}_{15}$ is a magnetically frustrated system with a honeycomb structure, anomalous quantum fluctuations due to magnetic frustration can be possible in Ce₂Pt₆Ga₁₅. We synthesized the single-crystal Ce₂Pt₆Ga₁₅, performed single-crystal and powder X-ray diffraction experiments, and measured XAS spectra at the Ce L₃-edge, Pt L₃-edge, and Ga K-edge.

2. Experimental

Single crystals of Ce₂Pt₆Ga₁₅ were prepared via the Ga-flux method. The elemental purity was 3N (99.9 %): Ce (Nippon Yttrium Co., Ltd., Fukuoka, Japan), 3N: Pt (TANAKA PRECIOUS METAL TECHNOLOGIES Co., Ltd., Tokyo, Japan), and 5N: Ga (Thermo Fisher Scientific Inc., Waltham, MA, USA). The initial composition of Ce, Pt, and Ga was 1:2:30. The weighed elements were loaded into an alumina crucible and sealed in a quartz tube under a high vacuum. The tube was heated to 1000 °C, kept for 8 h, then cooled to 300 °C in 168 h. The Ga-flux was removed using a centrifuge. The typical sample length was about 1∼3 mm. Figure 2 shows a photograph of the single-crystal Ce₂Pt₆Ga₁₅

Powder X-ray diffraction was performed using a Rint 2100 (Rigaku, Tokyo, Japan) with Cu K$\alpha$ radiation at room temperature. The Rietveld analysis was performed using RIETAN-FP [38].

Single-crystal X-ray diffraction was performed using an R-AXIS RAPID (Rigaku) diffractometer with Mo K$\alpha$ radiation at room temperature. The final atomic coordinates were refined via SHELXL97 [39].

XAS measurements were performed in transmission mode at the Ce L-edge (L₃: 5723 eV), Pt L-edge (L₃: 11,563 eV), and Ga K-edge (10,367 eV) at room temperature and performed at BL5S1 and BL11S2 in the Aichi Synchrotron Radiation Center. Ce₂Pt₆Ga₁₅ was crushed into powder and mixed with BN in a suitable ratio to create pellets for transmission measurements. Optimal thickness was calculated using the xasam program (Version 11.2023) from the GnXAS package (Version 11.2023), and we prepared two pellets: one optimized for the Ce L₃-edge and another for the Ga K- and Pt L₃-edges, since they had close values for optimal thickness. We also measured the XAS of CeF₃ and CeO₂ as reference compounds for ${\mathrm{Ce}}^{3+}$ and ${\mathrm{Ce}}^{4+}$, respectively. The EXAFS spectra were analyzed using the GnXAS package (Version 11.2023) [40,41], which allowed simultaneous fitting of multiple edges. Ab initio calculations of multiple scattering (MS) terms were performed in the muffin-tin approximation using a complex Hedin–Lundqvist potential. The fitting was conducted in a direct space using experimental raw data, without any pre-analysis. The background was refined together with the signal during the fitting procedure, accounting for possible multi-electron excitations.

3. Results and Discussion

3.1. Powder X-Ray Diffraction

We measured the powder X-ray diffraction of Ce₂Pt₆Ga₁₅. We analyzed the diffraction patterns using the Rietveld method with the hexagonal Sc_0.6Fe₂Si_4.9-type structure. The weighted profile R-factor $R_{\mathrm{wp}}$ and goodness of fit S were 20.29 and 1.40, respectively. The Rietveld plot of Ce₂Pt₆Ga₁₅ is shown in Figure 3. There were no impurity peaks and the diffraction patterns can be explained by the hexagonal Sc_0.6Fe₂Si_4.9-type structure.

3.2. Single-Crystal X-Ray Diffraction

We measured the single-crystal X-ray diffraction of Ce₂Pt₆Ga₁₅. We analyzed the crystal structure using the hexagonal Sc_0.6Fe₂Si_4.9-type structure model. The crystallographic data and the structure refinement results are shown in

Table 1. Crystallographic data and structure refinements of Ce₂Pt₆Ga₁₅ analyzed by the hexagonal Sc_0.6Fe₂Si_4.9-type structure.

Space group	P63/$mmc$ (194)
Lattice constants (Å)	a = 4.3334(9)
	c = 16.5473(15)
Formula units per cell, Z	1
Formula mass	2496.58
Calculated density (g ${\mathrm{cm}}^{-3}$)	15.405
Absorption coefficient $\mu$ (${\mathrm{cm}}^{-1}$)	1223.124
Crystal dimensions (mm)	0.038 × 0.025 × 0.022
Diffractometer	R-AXIS RAPID
Radiation	MoK$\alpha$ ($\lambda$ = 0.71075 Å)
Detector distance (mm)	127.40
Exposure rate (s/deg.)	90.0
$\theta$ range (deg.)	2.46–34.73
Range in $hkl$	−6 < h < 6
	−6 < k < 6
	−26 < l < 22
Total no. reflections	5892
Unique reflections	265
Reliability factor $R_{\mathrm{int}}$	0.0384
Reflection/parameter ratio	12.62
Goodness of fit	1.412
Reflections with I > 2$\sigma$(I)	257
$R_1$ (I > 2$\sigma$(I))	0.0133
R (All reflection)	0.0143
$w{R}_2$ (All reflection)	0.0265
Residual electron density ($\Delta {\rho }_{\max }$/$\Delta {\rho }_{\min }$ e Å−3)	0.84/−1.23
Temperature (K)	291

. The Wyckoff symbol, the fractional coordinates, occupancies, and equivalent atomic displacement parameters $U_{\mathrm{eq}}$ are shown in

Table 2. The Wyckoff symbol, fractional coordinates, occupancies, and equivalent atomic displacement parameters $U_{\mathrm{eq}}$ of Ce₂Pt₆Ga₁₅ analyzed by the hexagonal Sc_0.6Fe₂Si_4.9-type structure.

Atom	Wyckoff Symbol	x	y	z	${\mathit{U}}_{\mathbf{eq}}$ (Å2)	Occupancy
Ce	2c	1/3	2/3	1/4	0.0060(2)	0.693(3)
Pt	4f	1/3	2/3	0.60788(2)	0.00676(9)	1
Ga1	6h	0.5340(2)	0.0680(5)	1/4	0.0073(4)	0.332(3)
Ga2	4e	0	0	0.13562(4)	0.0092(2)	1.030(4)
Ga3	4f	1/3	2/3	0.04606(4)	0.0088(2)	1.027(4)

. The lattice constants of Ce₂Pt₆Ga₁₅ are a = 4.3334(9) Å and c = 16.5473(15) Å and these values are close to the Kwei et al. results [26].

The parameters were analyzed assuming the occupancy of the Pt site as 1 and the occupancies of other sites as fitting parameters. The occupancies of the Ce and Ga sites of Ce₂Pt₆Ga₁₅ deviated from the stoichiometric ratio. The composition obtained by the single X-ray diffraction was ${\mathrm{Ce}}_{2.079\left(9\right)}$${\mathrm{Pt}}_6$${\mathrm{Ga}}_{15.330\left(43\right)}$, indicating that the composition of Ce and Ga atoms in Ce₂Pt₆Ga₁₅ was rich or the Pt site was defective. Similar compositional deviations were observed in the single-crystalline Ce₂Pt₆Al₁₅ synthesized by the Al-flux method [36]. The composition of the $R_2$${\mathrm{Pt}}_6{X}_{15}$ system, synthesized using the flux method, may tend to be non-stoichiometric.

The single-crystal X-ray diffraction results were converted from the hexagonal model to the orthorhombic model using the procedure by Lutsyshyn et al. [18]. The atomic parameters of Ce₂Pt₆Ga₁₅ for the orthorhombic model are listed in

Table 3. Atomic parameters of the orthorhombic structure model for Ce₂Pt₆Ga₁₅, which were converted from the crystal parameters listed in Table 2 and analyzed under the Sc_0.6Fe₂Si_4.9-type structure. The lattice constants of orthorhombic Ce₂Pt₆Ga₁₅ are a = 13.0002 Å, b = 7.5057 Å, and c = 16.5473 Å.

Atom	x	y	z
Ce	1/6	1/6	1/4
Pt1	1/3	1/3	0.1079
Pt2	0	1/3	0.1079
Ga1	1/6	1/6	0.0461
Ga2	1/3	0	0.1356
Ga3	0.3997	0.2670	1/4
Ga4	0	0	0.1356
Ga5	0	1/3	0.5461
Ga6	0	0.4660	1/4

. The lattice constants of Ce₂Pt₆Ga₁₅ with the orthorhombic model are a = 13.0002 Å, b = 7.5057 Å, and c = 16.5473 Å. The number of Pt sites changes from 1 to 2, and the number of Ga sites changes from 3 to 6. To analyze EXAFS, the interatomic distances and coordination numbers (N) of neighboring atoms around the Ce, Pt1, Pt2, Ga1, Ga2, Ga3, Ga4, Ga5, and Ga6 sites in Ce₂Pt₆Ga₁₅ $R<$ 4 Å are shown in

Table 4. Interatomic distance R and coordination number (N) of neighboring atoms around the Ce, Pt1, Pt2, Ga1, Ga2, Ga3, Ga4, Ga5, and Ga6 sites of Ce₂Pt₆Ga₁₅ of the orthorhombic model $R<$ 4 Å. The circles denote the interatomic scattering paths used for the EXAFS analysis.

Ce				Pt1				Pt2
Use	Site	$\mathit{R}$ (Å)	$\mathit{N}$	Use	Site	$\mathit{R}$ (Å)	$\mathit{N}$	Use	Site	$\mathit{R}$ (Å)	$\mathit{N}$
○	Ga3	3.121	2	○	Ga2	2.544	2	○	Ga4	2.544	1
○	Ga6	3.121	1	○	Ga4	2.544	1	○	Ga2	2.544	2
○	Ga2	3.137	4	○	Ga1	2.547	1	○	Ga5	2.547	1
○	Ga4	3.137	2	○	Ga3	2.554	1	○	Ga6	2.554	1
	Ga1	3.375	2	○	Ga1	2.703	2	○	Ga1	2.703	2
○	Pt1	3.434	4	○	Ga5	2.703	1	○	Ga5	2.703	1
○	Pt2	3.434	2	○	Ce	3.434	2	○	Ce	3.434	2
Ga1				Ga2				Ga3
Use	Site	$\mathit{R}$(Å)	$\mathit{N}$	Use	Site	$\mathit{R}$(Å)	$\mathit{N}$	Use	Site	$\mathit{R}$(Å)	$\mathit{N}$
○	Pt1	2.547	1	○	Pt1	2.544	2	○	Pt1	2.554	2
○	Pt1	2.703	2	○	Pt2	2.544	1		Ga3	2.609	1
○	Pt2	2.703	1	○	Ga6	2.888	1		Ga6	2.609	1
○	Ga4	2.908	1	○	Ga3	2.888	1	○	Ga4	2.888	2
○	Ga2	2.908	2	○	Ga1	2.908	2	○	Ga2	2.888	2
○	Ga5	2.93	1	○	Ga5	2.908	1	○	Ce	3.121	2
○	Ga1	2.93	2	○	Ce	3.137	2	○	Ga5	3.696	2
	Ce	3.375	1	○	Ga2	3.785	1	○	Ga2	3.977	2
○	Ga4	3.911	1	○	Ga1	3.911	2
○	Ga2	3.911	2	○	Ga5	3.911	1
				○	Ga3	3.977	1
Ga4				Ga5				Ga6
Use	Site	$\mathit{R}$(Å)	$\mathit{N}$	Use	Site	$\mathit{R}$(Å)	$\mathit{N}$	Use	Site	$\mathit{R}$(Å)	$\mathit{N}$
○	Pt2	2.544	1	○	Pt2	2.547	1	○	Pt2	2.554	2
○	Pt1	2.544	2	○	Pt1	2.703	2		Ga3	2.609	2
○	Ga3	2.888	2	○	Pt2	2.703	1	○	Ga2	2.888	4
○	Ga1	2.908	2	○	Ga2	2.908	2	○	Ce	3.121	2
○	Ga5	2.908	1	○	Ga4	2.908	1	○	Ga5	3.696	2
○	Ce	3.137	2	○	Ga1	2.93	2	○	Ga4	3.977	2
○	Ga4	3.785	1	○	Ga5	2.93	1
○	Ga1	3.911	2	○	Ga3	3.696	2
○	Ga5	3.911	1	○	Ga6	3.696	1
○	Ga6	3.977	1	○	Ga2	3.911	2
				○	Ga4	3.911	1

. The Ce, Pt1, Pt2, Ga1, Ga2, Ga3, Ga4, Ga5, and Ga6 sites with the surrounding neighboring atoms are shown in Figure 4.

3.3. X-Ray Absorption Spectroscopy

Figure 5 shows the Ce L₃-edge X-ray absorption near-edge structure (XANES) spectra of Ce₂Pt₆Ga₁₅, CeF₃, and CeO₂. CeF₃ is a trivalent reference compound of Ce, and CeO₂ is a tetravalent reference compound of Ce. The XAS edge energy of Ce₂Pt₆Ga₁₅ is around 5720 eV, close to that of CeF₃ with ${\mathrm{Ce}}^{3+}$, while the edge energy of CeO₂ with ${\mathrm{Ce}}^{4+}$ shifts to higher energy. The Ce valence in Ce₂Pt₆Ga₁₅ is, therefore, estimated to be trivalent and Ce possesses the local magnetic moment at room temperature. This result is consistent with previous measurements of magnetic susceptibility [27]. The temperature dependence of the susceptibility follows the Curie–Weiss law at high temperatures. The effective Bohr magneton ${\mu }_{\mathrm{eff}}$ of Ce₂Pt₆Ga₁₅ is 2.53 ${\mu }_{\mathrm{B}}$, which is close to the estimated value of 2.54 ${\mu }_{\mathrm{B}}$ for the free ${\mathrm{Ce}}^{3+}$ ion.

The EXAFS analysis of Ce₂Pt₆Ga₁₅ was performed using GnXAS. First, the initial interatomic distances, R, and coordination numbers, N, required for EXAFS analysis were taken from the orthorhombic model. Due to the complexity of the structure, there are many scattering paths. The EXAFS analysis was performed using two-body signals for which the interatomic distance $R<$ 4 Å; we combined paths with close R values and we did not consider the paths with small scattering amplitudes. The scattering paths used in the EXAFS analysis are shown in Table 4. The scattering paths for two-body scattering—used as initial values for the interatomic distance (R) and coordination number (N)—are shown in

Table 5. Averaged interatomic distance (R) and its coordination number of neighboring atoms (N) for EXAFS fitting of Ce₂Pt₆Ga₁₅ with the orthorhombic model. The averaged scattering paths are obtained from Table 4.

Ce				Pt				Ga
Path	Atom	$\mathit{R}$ (Å)	$\mathit{N}$	Path	Atom	$\mathit{R}$ (Å)	$\mathit{N}$	Path	Atom	$\mathit{R}$ (Å)	$\mathit{N}$
${\mathrm{Ce}}_{\mathrm{i}}$	Ga	3.132	9	${\mathrm{Pt}}_{\mathrm{i}}$	Ga	2.547	5	${\mathrm{Ga}}_{\mathrm{i}}$	Pt	2.547	2
${\mathrm{Ce}}_{\mathrm{ii}}$	Pt	3.434	6	${\mathrm{Pt}}_{\mathrm{ii}}$	Ga	2.703	3	${\mathrm{Ga}}_{\mathrm{ii}}$	Pt	2.703	1.2
				${\mathrm{Pt}}_{\mathrm{iii}}$	Ce	3.434	2	${\mathrm{Ga}}_{\mathrm{iii}}$	Ga	2.907	5.2
								${\mathrm{Ga}}_{\mathrm{iv}}$	Ce	3.132	1.2
								${\mathrm{Ga}}_{\mathrm{v}}$	Ga	3.726	1.2
								${\mathrm{Ga}}_{\mathrm{vi}}$	Ga	3.928	3.2

. The N and R in Table 5 were obtained by averaging the scattering paths. ${\mathrm{Ce}}_{\mathrm{i}}$ and ${\mathrm{Ce}}_{\mathrm{ii}}$ are the averaged paths of Ce around 3.1 Å (Ce-Ga2, Ce-Ga3, Ce-Ga4, and Ce-Ga6), and 3.4 Å (Ce-Pt1 and Ce-Pt2), respectively. ${\mathrm{Pt}}_{\mathrm{i}}$, ${\mathrm{Pt}}_{\mathrm{ii}}$, and ${\mathrm{Pt}}_{\mathrm{iii}}$ represent the weighted average paths of Pt around 2.5 Å (Pt1-Ga1, Pt1-Ga2, Pt1-Ga3, Pt1-Ga4, Pt2-Ga1, Pt2-Ga2, Pt2-Ga3, and Pt2-Ga4), 2.7 Å (Pt1-Ga1, Pt1-Ga5, Pt2-Ga1, and Pt2-Ga5), and 3.4 Å (Pt1-Ce and Pt2-Ce), respectively. ${\mathrm{Ga}}_{\mathrm{i}}$, ${\mathrm{Ga}}_{\mathrm{ii}}$, ${\mathrm{Ga}}_{\mathrm{iii}}$, ${\mathrm{Ga}}_{\mathrm{iv}}$, ${\mathrm{Ga}}_{\mathrm{v}}$, and ${\mathrm{Ga}}_{\mathrm{vi}}$ represent the averaged paths of Ga around 2.5 Å (Ga1-Pt1, Ga2-Pt1, Ga3-Pt1, Ga4-Pt1, Ga4-Pt2, Ga5-Pt2, and Ga6-Pt2), 2.7 Å (Ga1-Pt1, Ga1-Pt2, Ga5-Pt1, and Ga5-Pt2), 2.9 Å (Ga1-Ga1, Ga1-Ga2, Ga1-Ga4, Ga1-Ga5, Ga2-Ga3, Ga2-Ga5, Ga2-Ga6, Ga3-Ga4, Ga4-Ga5, and Ga5-Ga5), 3.1 Å (Ga2-Ce, Ga3-Ce, Ga4-Ce, and Ga6-Ce), 3.7 Å (Ga2-Ga2, Ga3-Ga5, Ga4-Ga4, and Ga5-Ga6), and 3.9 Å (Ga1-Ga2, Ga1-Ga4, Ga2-Ga3, Ga2-Ga5, Ga4-Ga5, and Ga4-Ga6), respectively.

Figure 6 shows the $k^2$-weighted $\chi$(k) EXAFS oscillations of Ce₂Pt₆Ga₁₅, and Figure 7 shows the EXAFS signals and Fourier transform (FT) for the Ce L₃-edge, Pt L₃-edge, and Ga K-edge of Ce₂Pt₆Ga₁₅, together with the results of the fitting procedure. In GnXAS, the fitting is performed in energy (k) space, while the FT is just for visualization. The refined structural parameters (N), the Debye–Waller factor (${\sigma }^2$), and R are shown in

Table 6. The refined crystal parameters for Ce₂Pt₆Ga₁₅ obtained by the EXAFS analysis. The path is the averaged scattering path, N denotes the number of neighboring atoms, ${\sigma }^2$ denotes the Debye–Waller factor, and R denotes interatomic distances between Ce, Pt, or Ga sites and the neighboring scattering atoms.

Ce
Path	Atom	$\mathit{N}$	$\mathit{R}$ (Å)	${\mathbf{\sigma }}^{\mathbf{2}}$ (Å2)
${\mathrm{Ce}}_{\mathrm{i}}$	Ga	9	3.11(3)	0.010(4)
${\mathrm{Ce}}_{\mathrm{ii}}$	Pt	6	3.43(1)	0.010(2)
Pt
Path	Atom	$\mathit{N}$	$\mathit{R}$(Å)	${\mathbf{\sigma }}^{\mathbf{2}}$(Å2)
${\mathrm{Pt}}_{\mathrm{i}}$	Ga	5	2.557(2)	0.0043(3)
${\mathrm{Pt}}_{\mathrm{ii}}$	Ga	3	2.721(6)	0.008(1)
${\mathrm{Pt}}_{\mathrm{iii}}$	Ce	2	3.43(1)	0.010(2)
Ga
Path	Atom	$\mathit{N}$	$\mathit{R}$(Å)	${\mathbf{\sigma }}^{\mathbf{2}}$(Å2)
${\mathrm{Ga}}_{\mathrm{i}}$	Pt	2	2.557(2)	0.0043(3)
${\mathrm{Ga}}_{\mathrm{ii}}$	Pt	1.2	2.721(6)	0.008(1)
${\mathrm{Ga}}_{\mathrm{iii}}$	Ga	5.2	2.868(5)	0.0120(4)
${\mathrm{Ga}}_{\mathrm{iv}}$	Ce	1.2	3.11(3)	0.010(2)
${\mathrm{Ga}}_{\mathrm{v}}$	Ga	1.2	3.70(2)	0.005(2)
${\mathrm{Ga}}_{\mathrm{vi}}$	Ga	3.2	3.91(1)	0.009(1)

. The coordination numbers, N, were kept fixed during the fitting procedure. The EXAFS spectra around the Ce L₃-edge were fitted using the ${\mathrm{Ce}}_{\mathrm{i}}$ and ${\mathrm{Ce}}_{\mathrm{ii}}$ paths. Even if optimized for transmission measurements, the X-ray absorption jump for the Ce L₃-edge was quite small due to the low number of atoms and the presence of other heavy ions, like Pt. Therefore, the EXAFS spectra were noisier with respect to Pt L₃- and Ga K-edges, as can be seen from the residuals. Therefore, the EXAFS signal of the Ce L₃-edge was noisier than the EXAFS signals of Pt L₃- and Ga K-edges, as can be seen from the residuals (Figure 7b). The absence of a peak in the FT of the Ce L₃-edge data at 1.5 and 1.7 Å indicates that the hexagonal ScSc_0.6Fe₂Si_4.9-type structure is not valid. The EXAFS spectra of Pt L₃-edge were fitted using the ${\mathrm{Pt}}_{\mathrm{i}}$, ${\mathrm{Pt}}_{\mathrm{ii}}$, and ${\mathrm{Pt}}_{\mathrm{iii}}$ paths. Some oscillations remained in the residual of Pt L₃-edge, caused by the presence of additional contributions around 4.5 Å in the FT. Since we only considered paths with $R<4$ Å, these were not fitted. The EXAFS spectra of the Ga K-edge were fitted using the ${\mathrm{Ga}}_{\mathrm{i}}$, ${\mathrm{Ga}}_{\mathrm{ii}}$, ${\mathrm{Ga}}_{\mathrm{iii}}$, ${\mathrm{Ga}}_{\mathrm{iv}}$, ${\mathrm{Ga}}_{\mathrm{v}}$, and ${\mathrm{Ga}}_{\mathrm{vi}}$ paths. The FT data of the Ga K-edge are similar to previously reported data [26]. Debye–Waller factors ${\sigma }^2$ obtained from the fitting are similar to what we found in our previous study on Ce₂Pt₆Al₁₅ [36]. These EXAFS spectra were well explained through the calculation of the orthorhombic model.

4. Conclusions

We synthesized single-crystal Ce₂Pt₆Ga₁₅ using the Ga-flux method. From the powder X-ray diffraction, the samples of Ce₂Pt₆Ga₁₅ were confirmed to be single-phase. Single-crystal X-ray diffraction was carried out, and the crystal parameters of Ce₂Pt₆Ga₁₅ were obtained for the EXAFS analysis. We performed XAS measurements on the Ce₂Pt₆Ga₁₅ The edge energy of the Ce L₃-edge for Ce₂Pt₆Ga₁₅ was near that of CeF₃ which had ${\mathrm{Ce}}^{3+}$, suggesting that the valence of the sample was trivalent. We analyzed the EXAFS spectra of the Ce L₃-edge, Pt L₃-edge, and Ga K-edge for Ce₂Pt₆Ga₁₅. We found that the EXAFS spectra of Ce₂Pt₆Ga₁₅ cannot be explained by the hexagonal Sc_0.6Fe₂Si_4.9-type structure, but the orthorhombic structure with a honeycomb structure is appropriate. The unconventional QCP, spin liquid, etc., is possible in the $R_2$${\mathrm{Pt}}_6{X}_{15}$ system. Further physical property measurements at very low temperatures will be required for the $R_2$${\mathrm{Pt}}_6{X}_{15}$ system, especially for Ce₂Pt₆Ga₁₅.