# Hectoborides: Crystal structure of NdB65 and ThB60

Gerda Rogl, Vilma Bursikova, Jiří Buršík, Gerald Giester, Henri Noel, Peter Rogl

► To cite this version:

Gerda Rogl, Vilma Bursikova, Jiří Buršík, Gerald Giester, Henri Noel, et al.. Hectoborides: Crystal structure of NdB65 and ThB60. Solid State Sciences, 2025, 162, pp.107826. 10.1016/j.solidstatesciences.2025.107826. hal-05039995

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# Hectoborides: Crystal structure of NdB<sub>65</sub> and ThB<sub>60</sub>

Gerda Rogl<sup>a</sup>, Vilma Bursikova<sup>b</sup>, Jiri Bursik<sup>c</sup>, Gerald Giester<sup>d</sup>, Henri Noel<sup>e</sup>, Peter Rogl<sup>a, *, f</sup>

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

**Keywords:**

Crystal structure

X-ray structure analysis

TEM

Phase diagram

Hardness

## ABSTRACT

The crystal structures of the "hectoborides" NdB<sub>65</sub> and ThB<sub>60</sub> have been elucidated from single crystal X-ray structure determination supported by TEM-SAED analyses. Both compounds crystallize in a face-centred cubic lattice with space group Fm$\bar{3}$c (No. 226) ($a$NdB<sub>65</sub> = 2.35385(4) nm, $R_F$ = 0.0491; $a$ThB<sub>60</sub> = 2.35170(5) nm, $R_F$ = 0.0489) isotypic with the aristo-type of hectoborides, namely YB<sub>66</sub>. With respect to the structure type of YB<sub>66</sub> in its original version by Richards and Kaspar [2], we observe for both compounds, NdB<sub>65</sub> and ThB<sub>60</sub>, a split position (total occupancies ~0.5) for the metal atoms in site 48f as well as a second metal atom position in site 8a at a reduced occupancy; furthermore, we only arrive at seven B-sites 192j and 5 B-sites 96i, but did not find a B-atom (B13) in site 64g (x = 0.23, x, x). In contrast to the earlier structure determination of ThB<sub>66</sub>(O) [11] our WDX analysis did not present any hints for oxygen contamination, therefore site 8a (¼,¼,¼) hosts a Th atom at low occupancy of 0.12(1) Th instead. Similarly, the 8a site is also partially occupied by 0.04(1) Nd in NdB<sub>65</sub>. Whereas the present investigation removes all doubts on the formation of NdB<sub>65</sub> (YB<sub>66</sub>-type), our analysis did not reveal the formation of isotypic "PrB<sub>-66</sub>". On the basis of our SEM data, revised phase relations for the three systems Nd-B, Pr-B and Th-B have been designed for the boron-rich part (>85 at.% B).

Whereas at room temperature hardness of ThB<sub>60</sub> (23.8 GPa) and for NdB<sub>65</sub> (24.3 GPa) fit well among hardness data for the corresponding rare earths hectoborides, the by-product material ThB<sub>99</sub> (βB-type) with 42 GPa is a superhard material.

## 1. Introduction

Smith and Gilles [1] coined the term “hectoborides” for the face-centred cubic boride compounds of the rare earth (RE) elements richest in boron content. From a detailed structure analysis by Richards and Kasper [2] the yttrium compound prompted the formula YB<sub>66</sub>, which from then on was transferred to all corresponding rare earth compounds: for a complete listing consult Pearson’s Crystal Data [3] and Fig. 3. PuB<sub>100</sub> remained the exception, as it was furthermore discussed with a primitive cubic symmetry [4,5]. The technical interest in YB<sub>66</sub> as a monochromator for soft X-rays in the 1–2 keV region [6] triggered deeper scientific interest and as a consequence the homogeneity region in context with structural details has been determined to range from YB<sub>48</sub> to YB<sub>62</sub> [7,8]. Whereas the changes of structural details within YB<sub>48-62</sub> are well documented, hitherto only the YbB<sub>-57</sub> compound

among isotypic RE-based borides has been studied from X-ray single-crystal data [9]. Among actinoids a corresponding thorium compound has been observed, but its X-ray single crystal structure analysis arrived at a site occupied by oxygen thus rendering the "hectoboride" to a thorium-oxyboride ThB<sub>66</sub>O<sub>0.33</sub> [10,11]. Among the elements with the largest metal atom radius to form isotypism with YB<sub>66</sub> we identify thorium and neodymium. The existence of the NdB<sub>-66</sub> compound was much disputed in the literature: for a discussion see the papers of [12–14]. Via a recent re-determination of the Th-B-C system [15] we came again across the Th-hectoboride and decided to have a closer look at the crystal structures of the thorium as well as the neodymium compound. The results of this investigation will be presented in the current paper together with a study of the phase relations in the systems Nb-B and Th-B for the boron rich region (>85 at.% B) including also the system Pr-B [16,17], for which a "PrB<sub>66</sub>" compound has been denied by

<footnote><sup>a</sup> Institute of Materials Chemistry, Universität Wien, Währingerstraße 42, A-1090 Wien, Austria</footnote>

<footnote><sup>b</sup> Institute of Physical Electronics, Masaryk University, Kotlářská 2, 61137 Brno, Czech Republic</footnote>

<footnote><sup>c</sup> Institute of Physics of Materials, Czech Academy of Sciences, Žižkova 22, 61600 Brno, Czech Republic</footnote>

<footnote><sup>d</sup> Institute of Mineralogy and Crystallography, Universität Wien, Josef-Holaubek-Platz 2, A-1090 Wien, Austria</footnote>

<footnote><sup>e</sup> Laboratoire de Chimie du Solide et Inorganique Moleculaire, UMR-CNRS 6511, Université de Rennes I, Avenue du Général Leclerc, Cedex, F-35042, Rennes, France</footnote>

<footnote>This article is part of a special issue entitled: ISBB2024 published in Solid State Sciences.</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: peter.franz.rogl@univie.ac.at (P. Rogl).</footnote>
Ref. [12], however, had been listed in an overview on "Higher Borides" [18].

## 2. Materials and methods

### 2.1. Sample preparation and characterisation techniques

Samples (each of a mass of about 1 g) were prepared by argon arc melting starting from ingots of neodymium, praseodymium (both Auer-Remy, D, 99.9 mass%), thorium (Cerac Inc., USA, 99.9 %), and crystalline B-chunks (H.C. Starck, D, 99.75 %). Elemental pieces were directly fused in an arc furnace in a Ti/Zr-gettered Ar atmosphere. All specimens (at nominal compositions ThB₁₂, ThB₂₃ and MB₄₃, MB₅₈, MB₇₃; M = Nd, Pr, Th) were melted at least three times to ensure homogeneity. Weight losses generally were less than 1 mass%. All samples were ground on SiC papers and polished with Al₂O₃ powders (down to 0.3 µm) via standard procedures and have been examined by light optical metallography (LOM) and by a Jeol JSM-6460 scanning electron microscope equipped with an energy dispersive X-ray (EDX) detector operated at 20 kV and by a Tescan LYRA 3XMH FEG/SEM scanning electron microscope equipped with an X-Max80 Oxford Instruments energy dispersive X-ray (EDX) detector operated at 20 kV. Quantitative analyses were also performed in a CAMEBAX SX50 microanalyser (XMA) with WDX-EPMA i.e. wavelength dispersive X-ray spectroscopy comparing the characteristic emissions of the elements in the alloys with those from standards of Th, ThB₄ and LaB₅.₈₅ after applying (i) a careful calibration of the light element emission next to the heavy element matrix and (ii) the PAP correction procedure [19]. The experimental parameters employed were: acceleration voltage of 12 kV, sample current of 100 nA and spectrometer crystals such as PET for Th-Mα₁, PC1 for C-Kα₁, PC3 for the B-Kα₁ radiation as well as LSM60 (O, C) and LSM80N (N) (PET = Pentaerythritol, LSM60 = W-Si superlattice, LSM80N = Ni-C superlattice).

X-ray powder diffraction (XPD) data were collected using a Huber Guinier camera with monochromatic Cu-Kα₁ radiation (λ = 0.154056 nm) and an image plate recording system (model G670) employing an internal standard of 99.9999 mass% pure Ge (aGe = 0.5657906 nm). For quantitative Rietveld refinements we applied the program FULLPROF [20] with the use of its internal tables for atomic form factors.

Hardness with loads below 5 N was measured at room temperature with a Hysitron TI980 nanoindenter (Bruker) equipped with a Berkovich-type diamond tip (diameter of ~50 nm) applying rapid mechanical property mapping (XPM) at a constant loading rate of 100 mN/s. The error is <5 %. For each final value 25 to 100 imprints were evaluated. Hardness data with loads from 0.5 N to 10 N at room temperature were obtained from a Zwickl/Roell EMCO-TEST equipment: the values were the average of at least five measurements. With this equipment it is also possible to get information on the Vickers fracture resistance. The error is in the range of about 5 %.

### 2.2. Single crystal X-ray structure analyses

Single crystal (SC) specimens of NdB₆₅ and ThB₆₀ suitable for X-ray structure determination were extracted from arc melted alloys NdB₇₃ and ThB₄₃ from which rather "spherical" single crystal fragments with "diameters" in the range of 50–90 µm were selected for X-ray analyses. They were inspected on an AXS D8-GADDS texture goniometer, which assured high crystal quality, unit cell dimensions and Laue symmetry of the single crystal specimens prior to X-ray intensity data collection at room temperature on a four-circle BRUKER-APEX II diffractometer equipped with a CCD area detector and an Incoatec Microfocus Source IμS (30 W, multilayer mirror, Mo-Kα; λ = 0.071069 nm; detector distance of 4 cm; full sphere; 2θ < 2θ < 72°). Orientation matrix and unit cell parameters derived for a cubic system as well as data integration and scaling were performed using the program APEX4 [21]. Besides the general treatment of absorption effects using the multi-scan technique

(SADABS; redundancy of integrated reflections ~8), no individual absorption correction was necessary because of the rather regular crystal shape and small dimensions of the investigated specimens (~50 × 70 × 90 µm³). The structures were solved with the aid of the SHELXS-97 program and refined against *F*² (Program SHELXL-97-2) within the program OSCAIL [22]. Finally, the crystal structure was standardized with program TYPIX [23].

A thin lamella (lateral dimensions of about 5 × 3 µm²) was prepared from the polished sample NdB₇₃ using a focused ion beam (FIB) technique in a TESCAN LYRA 3 XMU FEG/SEM × FIB scanning electron microscope. A ThermoFisher Scientific Thalos F200i transmission electron microscope (TEM) with FEG operated at 200 kV was employed in diffraction mode to support information about the correct symmetry of the crystal structure.

## 3. Results and discussion

### 3.1. Crystal structures and structural chemistry of NdB₆₅ and ThB₆₀

Interest in X-ray single-crystal structure analyses of the hectoborides of Nd and Th was triggered by the fact that neodymium and thorium are the largest metal atoms [24] to enter the three-dimensional framework cages in the YB₆₅-type structure. X-ray single-crystal intensity data of crystals from both compounds were completely indexed on the basis of a face-centred cubic lattice with *a* ~2.35 nm and extinctions (*hhℓ*) for *h*,*ℓ* = 2n+1 suggesting space groups *Fm$\overline{3}$c (No. 226) and *F*4c (No. 219) out of which centrosymmetric *Fm$\overline{3}$c with the highest symmetry was chosen for further refinement. Structure solution with direct methods prompted thorium (neodymium) atoms in 48*f* and 9 boron atoms with full occupation of their sites (4 B in sites 192j and 5 B in sites 96i). Large atom displacement factors (ADPs) for the Th(Nd)-atom prompted a defect site with occ. ~ 0.5. The difference Fourier synthesis revealed three additional 192j sites partially occupied by boron atoms as well as a pronounced anisotropy of thermal parameters for the Th(Nd)-site suggesting a split position. A refinement with free site occupation factors for B05-B07 and the split Th(Nd)-positions resulted in a reliability factor as low as *R*<sub>F</sub> ~ 0.052 and site occupancies: occ.(Th11) = 0.31, occ. (Th12) = 0.22 and occ.(Nd11) = 0.41, occ.(Nd12) = 0.09, respectively. At this stage however, a significant intensity of 7 × 10³ electron/nm³ appeared at 0.240 nm from B07. Filling this position with 0.12 Th2 (0.04 Nd2) finally lead to a reliability factor *R*<sub>F</sub> = 0.049 with residual densities of 1.7 × 10³ electron/nm³ (at 0.076 nm from Th2) or 1.9 × 10³ electron/nm³ (at 0.083 nm from Nd2). Consequently, the refinements lead to structural chemical formulae of Th<sub>26.1</sub>B<sub>1557.1</sub> = ThB₅₉.₇ and Nd<sub>24.2</sub>B<sub>1567.1</sub> = NdB₆₄.₇, which in the phase diagrams locate at Th1.7B98.3 at.% and Nd1.5B98.5 at.%, respectively. It should be noted that about 25 systematic absence violations (out of a total of 152538 reflections) were reported for (*hhℓ*) odd planes in ThB₆₀ as well as for NdB₆₅; their maximal intensities <22 (4 σ) however, amount to only 2‰ of the strongest reflections recorded and suggest Renninger effects. Such reflections violate the *c*-glide in *Fm$\overline{3}$c and would prompt a lower symmetry *Fm$\overline{3}$c. A smaller crystal for both compounds (about 1/8 to 1/10 of the volume of the first crystal), however, did not reveal any symmetry violations (odd (*hhℓ*) reflections). Therefore, lattice parameters, symmetry and atom sites clearly indicated isotypism for both, NdB₆₅ and ThB₆₀, with the structure of the so-called "hectoborides" [25] with prototype YB₆₅ [5] in the highest symmetric space group *Fm$\overline{3}$c. Space group *Fd$\overline{3}$c (No. 228; *h*00 only observed for *h* = 4n) was ruled out as reflections (*h*00) with *h* = 2n proved to be strong reflections.

Crystallographic data are listed in Table 1a. Interatomic distances are given in Table 1b (due to the similarity with ThB₆₀ we declined from a presentation of the interatomic distances for NdB₆₅). With respect to the structure type of YB₆₅ in its original version by Richards [2], we observe a split position for the metal atoms in 48*f* as well as a second metal atom position in 8*a* at a reduced occupancy; furthermore, we only arrive at
seven B-sites 192j and 5 B-sites 96i, but did not find any hints for the B-atom B13 in site 64g (x = 0.23, x, x) as for instance observed in Y-hectoborides [2,7,8,26].

The split position in 48f was already reported by Tanaka [26] (with

0.08 B13 in 64g) and later confirmed by Hossain [8]. A split position was also recognized by Sologub et al. for YbB$_{56.7}$ [9]. Interestingly the site of B13 is absent in ThB$_{66}$O$_{0.33}$ [11], where vice-versa the atom in 8a (¼,¼,¼) was considered as “oxygen”. As our WDX analysis did not

<caption>**Table 1a**</caption>

Structure analyses¹ from X-ray single-crystal intensity data at 300 K for NbB$_{65}$ and ThB$_{60}$; YB$_{66}$-type, space group Fm$\bar{3}$c (No. 226); origin at centre; structure standardized with program *Structure Tidy* [23]. Anisotropic atomic displacements Uij in $[10^2 \text{ nm}^2]$.

<table><thead><tr><th>Parameter/compound</th><th>NdB<sub>65</sub></th><th>ThB<sub>60</sub></th></tr></thead><tbody><tr><td>Phase composition (EDX, at.%)</td><td>Nd~1.5B~98.5</td><td>Th1.7B98.3 (WDX) = ThB<sub>57.8</sub></td></tr><tr><td>Refinement composition (at.%)</td><td>Nd<sub>24.2</sub>B<sub>156.7</sub> = NdB<sub>64.7</sub> = (Nd1.5B98.5)</td><td>Th<sub>26.1</sub>B<sub>1557.1</sub> = Th<sub>1.09</sub>B<sub>64.87</sub> = ThB<sub>59.5</sub> (Th1.65B98.35)</td></tr><tr><td>a [nm]</td><td>2.35385(4)</td><td>2.35170(5)</td></tr><tr><td>Density ρ [Mg/m<sup>3</sup>]</td><td>2.60</td><td>2.89</td></tr><tr><td>θ range [deg]</td><td>2.0 ≤ 2θ ≤ 70.1</td><td>2.1 ≤ 2θ ≤ 72.7</td></tr><tr><td>Crystal size</td><td>75 × 85 × 90 µm<sup>3</sup></td><td>55 × 65 × 75 µm<sup>3</sup></td></tr><tr><td>Refl. In refinement [total]</td><td>1039 ≥ 4σ(F<sub>o</sub>) of 1287 [167148]</td><td>1231 ≥ 4σ(F<sub>o</sub>) of 1410 [167164]</td></tr><tr><td>Number of variables</td><td>108</td><td>108</td></tr><tr><td>Mosaicity</td><td>&lt;0.51</td><td>&lt;0.49</td></tr><tr><td>Linear absorption [mm<sup>-1</sup>]</td><td>2.52</td><td>7.22</td></tr><tr><td>R<sup>2</sup><sub>F</sub> = Σ|F<sub>o</sub><sup>2</sup> - F<sub>c</sub><sup>2</sup>| / ΣF<sub>o</sub><sup>2</sup></td><td>0.0491</td><td>0.0489</td></tr><tr><td>wR2</td><td>0.1405</td><td>0.1133</td></tr><tr><td>R<sub>Int</sub></td><td>0.089</td><td>0.068</td></tr><tr><td>GOF</td><td>1.052</td><td>1.140</td></tr><tr><td>Extinction (Zachariasen)</td><td>0.0002(1)</td><td>0.0004(1)</td></tr><tr><td>M1</td><td>0.41(3) Nd11 in 48f (0.0538(3), ¼, ¼)</td><td>0.28(3) Th11 in 48f (0.0565(4), ¼, ¼)</td></tr><tr><td>Uij</td><td>U<sub>11</sub>=0.012(1); U<sub>22</sub>=U<sub>33</sub>=0.0107(7); U<sub>23</sub>=U<sub>13</sub>=U<sub>12</sub>=0</td><td>U<sub>11</sub>=0.013(1); U<sub>22</sub>=U<sub>33</sub>=0.0100(8); U<sub>23</sub>=U<sub>13</sub>=U<sub>12</sub>=0</td></tr><tr><td>U<sub>23</sub>=U<sub>13</sub>=U<sub>12</sub>=0</td><td>0.09(3) Nd12 in 48f (0.0641(18), ¼, ¼)</td><td>0.24(3) Th12 in 48f (0.0663(8), ¼, ¼)</td></tr><tr><td>M2</td><td>U<sub>11</sub>=0.016(7); U<sub>22</sub>=U<sub>33</sub>=0.032(6); U<sub>23</sub>=U<sub>13</sub>=U<sub>12</sub>=0</td><td>U<sub>11</sub>=0.023(3); U<sub>22</sub>=U<sub>33</sub>=0.0251(11); U<sub>23</sub>=U<sub>13</sub>=U<sub>12</sub>=0</td></tr><tr><td>Uij</td><td>0.04(1) Nd2 in 8a (¼, ¼, ¼)</td><td>0.12(1) Th2 in 8a (¼, ¼, ¼)</td></tr><tr><td>U<sub>11</sub>=U<sub>22</sub>=U<sub>33</sub>=0.011(3); U<sub>23</sub>=U<sub>13</sub>=U<sub>12</sub>=0</td><td>0.0004(1)</td><td>0.0004(1)</td></tr><tr><td>B01</td><td>1.00(1) B01 in 192j (0.0386(1), 0.1398(1), 0.1216(1))</td><td>1.00(1) B01 in 192j (0.0389(1), 0.1392(1), 0.1215(1))</td></tr><tr><td>Uij</td><td>U<sub>11</sub>=0.016(1); U<sub>22</sub>=0.0107(9); U<sub>33</sub>=0.0109(9); U<sub>23</sub>=0.015(7); U<sub>13</sub>=0.0032(7); U<sub>12</sub>=0.0012(7)</td><td>U<sub>11</sub>=0.015(1); U<sub>22</sub>=0.0081(9); U<sub>33</sub>=0.0089(9); U<sub>23</sub>=0.029(7); U<sub>13</sub>=0.0025(7); U<sub>12</sub>=-0.0002(8)</td></tr><tr><td>B02</td><td>1.00(1) B02 in 192j (0.0393(1), 0.0814(1), 0.2290(1))</td><td>1.00(1) B02 in 192j (0.0395(1), 0.0810(1), 0.2288(1))</td></tr><tr><td>Uij</td><td>U<sub>11</sub>=0.018(1); U<sub>22</sub>=0.0091(9); U<sub>33</sub>=0.0097(9); U<sub>23</sub>=0.0004(8); U<sub>13</sub>=-0.0029(8); U<sub>12</sub>=0.0002(7)</td><td>U<sub>11</sub>=0.014(1); U<sub>22</sub>=0.0075(9); U<sub>33</sub>=0.0079(9); U<sub>23</sub>=0.0001(8); U<sub>13</sub>=-0.0024(8); U<sub>12</sub>=0.0009(8)</td></tr><tr><td>B03</td><td>1.00(1) B03 in 192j (0.0629(1), 0.0773(1), 0.1582(1))</td><td>1.00(1) B03 in 192j (0.0628(1), 0.0770(1), 0.1583(1))</td></tr><tr><td>Uij</td><td>U<sub>11</sub>=0.0113(9); U<sub>22</sub>=0.0118(9); U<sub>33</sub>=0.0125(9); U<sub>23</sub>=-0.0001(7); U<sub>13</sub>=0.0035(7); U<sub>12</sub>=0.0011(7)</td><td>U<sub>11</sub>=0.0081(9); U<sub>22</sub>=0.0083(9); U<sub>33</sub>=0.0103(9); U<sub>23</sub>=-0.0004(7); U<sub>13</sub>=0.0023(7); U<sub>12</sub>=0.0017(7)</td></tr><tr><td>B04</td><td>1.00(1) B04 in 192j (0.0633(1), 0.1451(1), 0.1944(1))</td><td>1.00(1) B04 in 192j (0.0633(1), 0.1449(1), 0.1940(1))</td></tr><tr><td>Uij</td><td>U<sub>11</sub>=0.0120(9); U<sub>22</sub>=0.0095(9); U<sub>33</sub>=0.0198(11); U<sub>23</sub>=0.0006(8); U<sub>13</sub>=-0.0006(8); U<sub>12</sub>=-0.0007(7)</td><td>U<sub>11</sub>=0.011(1); U<sub>22</sub>=0.0086(9); U<sub>33</sub>=0.0172(11); U<sub>23</sub>=0.0001(8); U<sub>13</sub>=-0.0005(9); U<sub>12</sub>=-0.0011(8)</td></tr><tr><td>B05</td><td>0.78(2) B05 in 192j (0.1317(3), 0.1740(3), 0.1970(3))</td><td>0.30(2) B05 in 192j (0.1269(6), 0.1726(5), 0.2409(6))</td></tr><tr><td>Uij</td><td>U<sub>11</sub>=0.038(3); U<sub>22</sub>=0.050(4); U<sub>33</sub>=0.049(4); U<sub>23</sub>=-0.025(3); U<sub>13</sub>=-0.009(2); U<sub>12</sub>=-0.009(2)</td><td>U<sub>11</sub>=0.021(5); U<sub>22</sub>=0.021(5); U<sub>33</sub>=0.034(8); U<sub>23</sub>=0.008(4); U<sub>13</sub>=-0.0004(4); U<sub>12</sub>=-0.005(4)</td></tr><tr><td>B06</td><td>0.30(2) B06 in 192j (0.1272(5), 0.1722(5), 0.2410(6))</td><td>0.79(2) B06 in 192j (0.1320(4), 0.1744(4), 0.1971(4))</td></tr><tr><td>Uij</td><td>U<sub>11</sub>=0.026(5); U<sub>22</sub>=0.023(5); U<sub>33</sub>=0.037(7); U<sub>23</sub>=0.016(4); U<sub>13</sub>=-0.003(4); U<sub>12</sub>=-0.001(4)</td><td>U<sub>11</sub>=0.046(4); U<sub>22</sub>=0.059(5); U<sub>33</sub>=0.054(5); U<sub>23</sub>=-0.032(4); U<sub>13</sub>=-0.010(3); U<sub>12</sub>=-0.013(3)</td></tr><tr><td>B07</td><td>0.58(2) B07 in 192j (0.1609(4), 0.2320(4), 0.1985(4))</td><td>0.53(2) B07 in 192j (0.1638(5), 0.2306(5), 0.1977(5))</td></tr><tr><td>Uij</td><td>U<sub>11</sub>=0.050(5); U<sub>22</sub>=0.047(5); U<sub>33</sub>=0.050(5); U<sub>23</sub>=-0.002(3); U<sub>13</sub>=-0.001(4); U<sub>12</sub>=0.004(4)</td><td>U<sub>11</sub>=0.042(5); U<sub>22</sub>=0.043(5); U<sub>33</sub>=0.053(6); U<sub>23</sub>=-0.016(4); U<sub>13</sub>=-0.012(4); U<sub>12</sub>=0.015(4)</td></tr><tr><td>B08</td><td>1.00(1) B08 in 96i (0, 0.0371(1), 0.0592(1))</td><td>1.00(1) B08 in 96i (0, 0.0374(2), 0.0592(2))</td></tr><tr><td>Uij</td><td>U<sub>11</sub>=0.010(1); U<sub>22</sub>=0.010(1); U<sub>33</sub>=0.009(1); U<sub>23</sub>=-0.0002(9); U<sub>13</sub>=U<sub>12</sub>=0</td><td>U<sub>11</sub>=0.008(1); U<sub>22</sub>=0.009(1); U<sub>33</sub>=0.007(1); U<sub>23</sub>=-0.0004(9); U<sub>13</sub>=U<sub>12</sub>=0</td></tr><tr><td>B09</td><td>1.00(1) B09 in 96i (0, 0.0388(1), 0.1806(2))</td><td>1.00(1) B09 in 96i (0, 0.0388(2), 0.1807(2))</td></tr><tr><td>Uij</td><td>U<sub>11</sub>=0.014(1); U<sub>22</sub>=0.008(1); U<sub>33</sub>=0.010(1); U<sub>23</sub>=0.001(1); U<sub>13</sub>=U<sub>12</sub>=0</td><td>U<sub>11</sub>=0.010(1); U<sub>22</sub>=0.005(1); U<sub>33</sub>=0.009(1); U<sub>23</sub>=-0.001(1); U<sub>13</sub>=U<sub>12</sub>=0</td></tr><tr><td>B10</td><td>1.00(1) B10 in 96i (0, 0.0757(1), 0.1167(1))</td><td>1.00(1) B10 in 96i (0, 0.0757(2), 0.1167(2))</td></tr><tr><td>Uij</td><td>U<sub>11</sub>=0.013(1); U<sub>22</sub>=0.010(1); U<sub>33</sub>=0.009(1); U<sub>23</sub>=-0.001(1); U<sub>13</sub>=U<sub>12</sub>=0</td><td>U<sub>11</sub>=0.012(1); U<sub>22</sub>=0.008(1); U<sub>33</sub>=0.006(1); U<sub>23</sub>=0.001(1); U<sub>13</sub>=U<sub>12</sub>=0</td></tr><tr><td>B11</td><td>1.00(1) B11 in 96i (0, 0.1477(1), 0.2420(1))</td><td>1.00(1) B11 in 96i (0, 0.1470(2), 0.2417(2))</td></tr><tr><td>Uij</td><td>U<sub>11</sub>=0.013(1); U<sub>22</sub>=0.009(1); U<sub>33</sub>=0.010(1); U<sub>23</sub>=0.001(1); U<sub>13</sub>=U<sub>12</sub>=0</td><td>U<sub>11</sub>=0.011(1); U<sub>22</sub>=0.008(1); U<sub>33</sub>=0.005(1); U<sub>23</sub>=0.004(1); U<sub>13</sub>=U<sub>12</sub>=0</td></tr><tr><td>B12</td><td>1.00(1) B12 in 96i (0, 0.1852(1), 0.1708(1))</td><td>1.00(1) B12 in 96i (0, 0.1849(1), 0.1705(2))</td></tr><tr><td>Uij</td><td>U<sub>11</sub>=0.012(1); U<sub>22</sub>=0.008(1); U<sub>33</sub>=0.011(1); U<sub>23</sub>=0.001(1); U<sub>13</sub>=U<sub>12</sub>=0</td><td>U<sub>11</sub>=0.011(1); U<sub>22</sub>=0.005(1); U<sub>33</sub>=0.007(1); U<sub>23</sub>=0.002(1); U<sub>13</sub>=U<sub>12</sub>=0</td></tr><tr><td>Residual electron density; max; min in (electron/nm<sup>3</sup>) × 10<sup>3</sup></td><td>1.9 (0.08 nm from Nd2); -0.54</td><td>1.7 (0.00 nm from Th2); -1.71</td></tr></tbody></table>

(continued on next page)
<table><caption>**Table 1a (continued)**</caption><thead><tr><th>Parameter/compound</th><th colspan="3">NdB<sub>65</sub></th><th colspan="3">ThB<sub>60</sub></th></tr></thead><tbody><tr><td rowspan="15">Principal mean square atomic displacements U'<sub>ii</sub></td><td>Nd11</td><td>0.0124</td><td>0.0106</td><td>0.0106</td><td>Th11</td><td>0.0134</td><td>0.0100</td><td>0.0100</td></tr><tr><td>Nd12</td><td>0.0318</td><td>0.0318</td><td>0.0162</td><td>Th12</td><td>0.0251</td><td>0.0251</td><td>0.0230</td></tr><tr><td>Nd2</td><td>0.0108</td><td>0.0108</td><td>0.0108</td><td>Th2</td><td>0.2664</td><td>0.2664</td><td>0.2664</td></tr><tr><td>B01</td><td>0.0180</td><td>0.0121</td><td>0.0081</td><td>B01</td><td>0.0159</td><td>0.0108</td><td>0.0052</td></tr><tr><td>B02</td><td>0.0186</td><td>0.0094</td><td>0.0085</td><td>B02</td><td>0.0145</td><td>0.0076</td><td>0.0068</td></tr><tr><td>B03</td><td>0.0155</td><td>0.0119</td><td>0.0081</td><td>B03</td><td>0.0118</td><td>0.0092</td><td>0.0056</td></tr><tr><td>B04</td><td>0.0199</td><td>0.0122</td><td>0.0093</td><td>B04</td><td>0.0173</td><td>0.0117</td><td>0.0082</td></tr><tr><td>B05</td><td>0.0754</td><td>0.0452</td><td>0.0164</td><td>B05</td><td>0.0890</td><td>0.0536</td><td>0.0163</td></tr><tr><td>B06</td><td>0.0473</td><td>0.0259</td><td>0.0126</td><td>B06</td><td>0.0381</td><td>0.0237</td><td>0.0142</td></tr><tr><td>B07</td><td>0.0539</td><td>0.0490</td><td>0.0441</td><td>B07</td><td>0.0749</td><td>0.0358</td><td>0.0265</td></tr><tr><td>B08</td><td>0.0100</td><td>0.0098</td><td>0.0087</td><td>B08</td><td>0.0092</td><td>0.0079</td><td>0.0073</td></tr><tr><td>B09</td><td>0.0143</td><td>0.0103</td><td>0.0083</td><td>B09</td><td>0.0105</td><td>0.0095</td><td>0.0051</td></tr><tr><td>B10</td><td>0.0129</td><td>0.0105</td><td>0.0082</td><td>B10</td><td>0.0121</td><td>0.0084</td><td>0.0061</td></tr><tr><td>B11</td><td>0.0128</td><td>0.0097</td><td>0.0092</td><td>B11</td><td>0.0114</td><td>0.0085</td><td>0.0049</td></tr><tr><td>B12</td><td>0.0124</td><td>0.0108</td><td>0.0080</td><td>B12</td><td>0.0111</td><td>0.0079</td><td>0.0041</td></tr></tbody></table>

<caption>**Table 1b**</caption>

Interatomic distances (in nm) in ThB<sub>60</sub>; standard deviation in brackets. Distances from or to a site with partial atom occupation are indicated by an asterisk (*).

<table><thead><tr><th>Central Atom</th><th>Ligand</th><th>Distance</th><th>Central Atom</th><th>Ligand</th><th>Distance</th><th>Central Atom</th><th>Ligand</th><th>Distance</th></tr></thead><tbody><tr><td>*Th11-</td><td>*Th12</td><td>0.023</td><td>*Th12-</td><td>*Th11</td><td>0.0232</td><td>Th2-</td><td>*24B07</td><td>0.240(1)</td></tr><tr><td>-</td><td>*4B06</td><td>0.247(1)</td><td>-</td><td>*4B06</td><td>0.232(2)</td><td>-</td><td>-</td><td>-</td></tr><tr><td>-</td><td>*Th11</td><td>0.266(2)</td><td>-</td><td>*4B07</td><td>0.265(2)</td><td>-</td><td>-</td><td>-</td></tr><tr><td>-</td><td>4B12</td><td>0.2759(5)</td><td>-</td><td>*4B05</td><td>0.266(1)</td><td>-</td><td>-</td><td>-</td></tr><tr><td>-</td><td>4B11</td><td>0.2771(5)</td><td>-</td><td>4B04</td><td>0.280(1)</td><td>-</td><td>-</td><td>-</td></tr><tr><td>-</td><td>*4B05</td><td>0.2801(5)</td><td>-</td><td>4B12</td><td>0.288(1)</td><td>-</td><td>-</td><td>-</td></tr><tr><td>-</td><td>4B04</td><td>0.2804(5)</td><td>-</td><td>4B11</td><td>0.289(1)</td><td>-</td><td>-</td><td>-</td></tr><tr><td>-</td><td>*4B07</td><td>0.2853(6)</td><td>-</td><td>*Th11</td><td>0.2900(6)</td><td>-</td><td>-</td><td>-</td></tr><tr><td>-</td><td>*Th12</td><td>0.290(2)</td><td>-</td><td>*Th12</td><td>0.3132(6)</td><td>-</td><td>-</td><td>-</td></tr><tr><td>B01-</td><td>B03</td><td>0.1706(3)</td><td>B02-</td><td>B02</td><td>0.1704(5)</td><td>B03-</td><td>B01</td><td>0.1706(3)</td></tr><tr><td>-</td><td>B10</td><td>0.1754(4)</td><td>-</td><td>B03</td><td>0.1749(4)</td><td>-</td><td>B02</td><td>0.1749(4)</td></tr><tr><td>-</td><td>B03</td><td>0.1791(4)</td><td>-</td><td>B09</td><td>0.1768(4)</td><td>-</td><td>B10</td><td>0.1772(3)</td></tr><tr><td>-</td><td>B04</td><td>0.1805(4)</td><td>-</td><td>B04</td><td>0.1799(4)</td><td>-</td><td>B01</td><td>0.1791(4)</td></tr><tr><td>-</td><td>B12</td><td>0.1824(4)</td><td>-</td><td>B11</td><td>0.1834(4)</td><td>-</td><td>B04</td><td>0.1805(4)</td></tr><tr><td>-</td><td>B01</td><td>0.1829(6)</td><td>-</td><td>B08</td><td>0.1858(6)</td><td>-</td><td>B09</td><td>0.1807(3)</td></tr><tr><td>B04-</td><td>*B05</td><td>0.1755(8)</td><td>*B05-</td><td>*B06</td><td>0.103(2)</td><td>*B06-</td><td>*B05</td><td>0.103(2)</td></tr><tr><td>-</td><td>B02</td><td>0.1799(4)</td><td>-</td><td>*B07</td><td>0.154(2)</td><td>-</td><td>*B07</td><td>0.125(2)</td></tr><tr><td>-</td><td>B03</td><td>0.1805(4)</td><td>-</td><td>*B07</td><td>0.175(1)</td><td>-</td><td>*B06</td><td>0.156(3)</td></tr><tr><td>-</td><td>B01</td><td>0.1806(4)</td><td>-</td><td>B04</td><td>0.1755(8)</td><td>-</td><td>*B07</td><td>0.169(2)</td></tr><tr><td>-</td><td>B12</td><td>0.1845(3)</td><td>-</td><td>2B05</td><td>0.191(2)</td><td>-</td><td>*B07</td><td>0.193(2)</td></tr><tr><td>-</td><td>B11</td><td>0.1863(4)</td><td>-</td><td>*B07</td><td>0.193(1)</td><td>-</td><td>B04</td><td>0.197(1)</td></tr><tr><td>-</td><td>*B06</td><td>0.197(1)</td><td>-</td><td>*B06</td><td>0.205(2)</td><td>-</td><td>*B05</td><td>0.205(2)</td></tr><tr><td>-</td><td>*Th12</td><td>0.2801(3)</td><td>-</td><td>*Th12</td><td>0.2661</td><td>-</td><td>*Th12</td><td>0.232(2)</td></tr><tr><td>-</td><td>*Th11</td><td>0.2804(3)</td><td>-</td><td>*Th11</td><td>0.2801</td><td>-</td><td>*Th11</td><td>0.247(1)</td></tr><tr><td>*B07-</td><td>*B06</td><td>0.125(2)</td><td>B08-</td><td>B10</td><td>0.1623(5)</td><td>B11-</td><td>B12</td><td>0.1813(5)</td></tr><tr><td>-</td><td>*B07</td><td>0.142(3)</td><td>-</td><td>4B08</td><td>0.1727(4)</td><td>-</td><td>2B02</td><td>0.1834(4)</td></tr><tr><td>-</td><td>*B05</td><td>0.154(1)</td><td>-</td><td>B08</td><td>0.1762(7)</td><td>-</td><td>2B04</td><td>0.1863(4)</td></tr><tr><td>-</td><td>*B06</td><td>0.169(2)</td><td>B09-</td><td>B10</td><td>0.1739(5)</td><td>-</td><td>B12</td><td>0.1894(5)</td></tr><tr><td>-</td><td>*B05</td><td>0.175(1)</td><td>-</td><td>2B02</td><td>0.1768(4)</td><td>-</td><td>*2Th11</td><td>0.2771(5)</td></tr><tr><td>-</td><td>*2B07</td><td>0.186(2)</td><td>-</td><td>2B03</td><td>0.1807(3)</td><td>-</td><td>*2Th12</td><td>0.289(1)</td></tr><tr><td>-</td><td>*2B07</td><td>0.192(2)</td><td>-</td><td>B09</td><td>0.1826(7)</td><td>B12-</td><td>B11</td><td>0.1813(5)</td></tr><tr><td>-</td><td>*B05</td><td>0.193(1)</td><td>B10-</td><td>B08</td><td>0.1623(5)</td><td>-</td><td>2B01</td><td>0.1824(4)</td></tr><tr><td>-</td><td>*B06</td><td>0.193(2)</td><td>-</td><td>B09</td><td>0.1739(5)</td><td>-</td><td>2B04</td><td>0.1845(3)</td></tr><tr><td>-</td><td>*Th2</td><td>0.240(1)</td><td>-</td><td>2B01</td><td>0.1754(4)</td><td>-</td><td>B11</td><td>0.1894(5)</td></tr><tr><td>-</td><td>*Th12</td><td>0.265(1)</td><td>-</td><td>2B03</td><td>0.1772(3)</td><td>-</td><td>*2Th11</td><td>0.2759(5)</td></tr><tr><td>-</td><td>*Th11</td><td>0.285(1)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>*2Th12</td><td>0.288(1)</td></tr></tbody></table>

present any hints for oxygen contamination, the 8a site in our refinement hosts a Th atom at low occupancy of 0.12(1) Th replacing the B13 atom. Similarly, the 8a site is also partially occupied by 0.04(1) Nd in NdB<sub>65</sub> also here replacing B13. With respect to the symmetry reducing effects via doping of YB<sub>62</sub> by a series of smaller atoms (Ti to Ni and Zr to Mo), as reported by Tanaka [27], we have tested all our alloys via long term SEM-EDX exposures (100 s) but did not track any traces of Fe, Cu, W, which eventually could have been incorporated from either the boron starting material (Fe), or from the Cu-die-cast and the W-electrode in the arc melting process (Fig. 1).

The occasional appearance of faint odd (hhf') reflections prompted us to investigate NdB<sub>65</sub> with electron diffraction in order to decide on the proper crystal symmetry: either Fm&#773;3c or Fm&#773;3 (without the c-glide, centrosymmetric space group Fm&#773;3 is the maximal non-isomorphic subgroup of Fm&#773;3c). Selected area electron diffraction (SAED) patterns were collected from thin lamellae of the NdB<sub>73</sub> sample (see Fig. 1). All diffraction patterns were fully indexed on the basis of an fcc lattice using the parameters obtained from the single crystal refinement revealing NdB<sub>65</sub> (Table 1). No further superstructure spots were detected - thus no ordering exists among the randomly distributed Nd-atoms and vacancies.

<footnote>a B13-atom (as observed in YB<sub>66</sub>) was not accepted in the refinement.</footnote>
<figure><img src="image_4.png" /><figcaption>Fig. 1. SAED patterns of NdB₆₅ at various sample tilts together with results of simulation.</figcaption></figure>

The upper panel in Fig. 1 shows examples of low index zone-axis diffraction patterns recorded at various sample tilts together with simulated SAED patterns in the lower panel obtained using the software JEMS [28,29]. In order to rule out double diffraction, which may cause weak (*hh*′) reflections (T31 and T11) in the SAED pattern along [211], we carefully analysed the SAED pattern along [110], where no traces of odd (*hh*′) reflections can be detected. Therefore, electron diffraction confirms the space group type *Fm$\bar{3}$c chosen from single crystal X-ray diffraction. The crystal structure of ThB₆₀ is essentially composed of so-called boron super-icosahedra i.e. 12 icosahedra around a central icosahedron creating a B₁₅₆ cluster. Two types of these super-icosahedra (rotated by 90° to each other) form two interpenetrating *fcc* sublattices providing space for Th-atoms (split Th11 + Th12) as well as cages filled with irregular clusters of B₇₂-goups around a central Th2-atom with rather low occupation. As the B₇₂ group (i.e. 192 B atoms from three

sites 192j at 8 locations in the *fcc* lattice: (192 × 3)/8 = 72) is made of boron defect sites with random distribution, it is in fact composed of a total of 39 B atoms only.

Fig. 2 depicts the B₇₂ unit centred by Th2. The difference to the B₈₀ group described in the literature [2, 18 and others] is due to the B13 atoms (site 64g) which in ThB₆₀ are replaced by Th2 in 8a.

The bond lengths observed (Table 1b) are in general agreement with the YB₆₆ structure type: the 12 bonds between the central icosahedron and its 12 engulfing icosahedra are short (d<sub>B·B</sub> = 0.1623 nm), whereas the B₁₅₆ units are interlinked at distances of 0.1704 nm ≤ d<sub>B·B</sub> ≤ 0.1813 nm. Although all Th-sites are defect, the Th-B distances are close to the sum of radii [24]: R<sub>Th</sub>(0.180 nm) + R<sub>B</sub>(0.088 nm) = 0.268 nm. We note, that despite the high degree of disorder in the 192j sites of B05, B06, B07 (random distribution of B-atoms and vacancies), the positional parameters of these atoms are rather invariant for all the single crystal analyses available for hectoborides [2, 7–9, 11, 26]. Also, the level of individual occupancy remains rather uniform in these three defect sites. Secondly - except for the hectoborides of Y and Yb as well as the metal doped variants Y<sub>1-x</sub>M<sub>x</sub>B<sub>~66</sub> [27] - ThB<sub>60</sub> and NdB<sub>65</sub> reveal the absence of the B13-atoms: in all the latter crystals the appearance of (RE,Th) in the 8a-site suppresses the formation of B13-B13 dumbbells.

<figure><img src="image_5.png" /><figcaption>Fig. 2. Atoms surrounding the Th2-atom (yellow) in ThB₆₀. Defect boron sites B05 (red), B06 (blue) and B07 (purple) form the B₇₂ units with a total concentration of 39 B. All atoms are presented with ADPs from single crystal refinement at 96 % probability level. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)</figcaption></figure>

## 3.2. On the formation of hectoborides

Fig. 3 displays the lattice parameters of “hectoborides” for Th, Pu and RE elements. Values were collected from this work, Chemical Abstract Service and Pearson’s Crystal Data [3]. Except for a few metals we observe a rather large spread among data for each compound, which probably cannot be taken as a reliable signature of a significant homogeneity region. Whereas for instance Higashi et al. [7] and Tanaka et al. [26] have determined composition and lattice parameters of the homogeneity region of YBₓ, grown by self-flux, to range within x = 62 (a = 2.34364 nm) and x = 56 (a = 2.34600 nm), Hossain et al. [8] extended the region to x = 48 (a = 2.34307 nm). The much lower value by Roger [30] i.e. a = 2.3390 nm (no quantitative analyses of composition) would indicate a significantly different (lower?) Y-content in the phase. Earlier values of Richards [2] and Schwetz [25], however, are only slightly lower than Higashi’s data. Strong deviations from the general trend given by the lanthanide contraction are likely due to typos such as the value of Yu [31] for ErB~66~. Single crystal data are surprisingly rare: YB~66~ [2], YBₓ (56 < x < 62; [7], YB₄₈ [8], Y₁ₓMₓB~66~ [27], YbB₅₆.₇ [9], ThB~66~O~0.33~ [11], ThB₆₀ and NdB₆₅, (both this work). The largest volumes among “hectoborides” are encountered among Th, Nd, and Sm
<figure><img src="image_6.png" /><figcaption>Fig. 3. Lattice parameters of {RE,Th,Pu}B$_{60}$ with data from this work, from Chemical Abstract Service and Pearson's Crystal Data [3]; [33–38].</figcaption></figure>

atoms, which also are among the largest metal atoms [24] to form this
class of compounds. It should be mentioned that synthesis at high
temperature (2000 °C) and high pressure (6.5 GPa) produced a Sm-rich
compound SmB$_{66}$ (composition not evaluated) concomitant with a
larger lattice parameter [32]. PuB$_{100}$ (a = 2.343 nm [4]) is a special
case, as its symmetry was judged [5] to be Pn$\overline{3}$c instead of Fm$\overline{3}$c.

3.3. *The boron-rich part of the phase diagrams Nd-B, Pr-B and Th-B*

Microstructures (SEM backscatter images) in Fig. 4 document the
phase relations for the boron-rich part of the systems Th-B, Nd-B and
Pr-B. In all three systems large crystals of congruently melting
hexaboride dominate the microstructures. Area scans in Fig. 4a define a

<figure><img src="image_7.png" /><figcaption>Fig. 4. Microstructures and compositions (at.%) of various as cast alloys ThBₓ (top panel), NdBₓ (middle panel) and PrBₓ (bottom panel). Top panel (compositions from WDX): (a) ThB₁₂: bright Th11.5B88.5 (Th₁ₓB₆), dark matrix Th1.6B98.4 (ThB₆₀), eutectic Th~4B~96; (b) ThB₂₃: eutectic Th~1.3B~98.7 with bright ThB₆₀, dark phase βB (ThB₉₉); (c) ThB₄₃: bright ThB₆₀, dark matrix βB(ThB₉₉). Middle panel: (d) NdB₄₃: bright Nd₁ₓB₆, dark grey NdB₆₅, black βB(Nd₀.₀₅B₁₀₀); (e) NdB₅₈: bright Nd₁ₓB₆, dark grey NdB₆₅, black βB(Nd₀.₀₅B₁₀₀); (f) NdB₇₃: dark grey NdB₆₅, eutectic with black βB(Nd₀.₀₅B₁₀₀); Bottom panel: (g) PrB₄₃: bright Pr₁ₓB₆, eutectic with black βB(Pr₀.₀₅B₁₀₀); (h) PrB₅₈: bright Pr₁ₓB₆, eutectic with black βB(Pr₀.₀₅B₁₀₀); (i) PrB₇₃: bright Pr₁ₓB₆, eutectic with black βB(Pr₀.₀₅B₁₀₀).</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Partial T-x phase diagrams in the boron rich region (>85 at.%B) for the systems Th-B (left panel), Nd-B (middle panel) and Pr-B (right panel). Note the absence of "PrB<sub>66</sub>". Solubilities of metal in βB are from WDX (Th) and EDX (Nd, Pr). The boron-rich boundary of Nd<sub>1-x</sub>B<sub>6</sub> is from Ref. [44], which was also assumed for Pr<sub>1-x</sub>B<sub>6</sub>. Melting temperatures are according to Ref. [39].</figcaption></figure>

eutectic L ↔ ThB<sub>6</sub> + ThB<sub>60</sub> at ~4 at.% Th with ThB<sub>60</sub> as the dark phase. Furthermore a eutectic at ~1.3 at.% Th (L ↔ ThB<sub>60</sub> + (βB)) is evident from Fig. 4b with βB dissolving ~1 at.% Th as the dark phase. Whereas the eutectic at ~4 at.% Th corresponds well to the value shown in the Pauling File [39] (5 at.% Th), the eutectic (βB) + ThB<sub>60</sub>, located by WDX at 1.3 at.% Th, is at a significantly higher Th concentration than anticipated in Ref. [39] (0.3 at.% Th). The reason for this discrepancy might lie in the fact that no solid solubility of Th in βB was accepted in Ref. [39], despite WDX, in agreement with X-ray single crystal data, unambiguously define a solid solubility of about 1 at.% Th in βB [40]. The B-rich composition of the hexaboride Th<sub>1-x</sub>☐<sub>x</sub>B<sub>6</sub> appeared at Th11.5B88.5 at.% in agreement with its homogeneity region 0 ≤ x < 0.22 [40,41] (☐ denotes a vacancy). The phase relations as derived from the present work for the B-rich part of Th-B are summarized in the left panel of Fig. 5.

Fig. 4d clearly depicts three phases in the as cast alloy of NdB<sub>43</sub>: bright NdB<sub>6</sub>, dark grey NdB<sub>65</sub> and black βB. No typical peritectic formation for NdB<sub>65</sub> can be seen: we thus follow the phase relations as summarized for the thermodynamic assessment of the entire system Nd-B by Refs. [42,43] relying on a rather depleted peritectic formation of NdB<sub>65</sub> and denying any solubility of Nd in βB. Therefore, the essential innovation is the limited solid solution of about 0.05 at.% Nd in βB (EDX value), which demands a eutectic reaction L ↔ (NdB<sub>99.95</sub>) + NdB<sub>65</sub> at about 0.4-0.6 at.% Nd. It is interesting to note that the EDX composition of Nd<sub>1-x</sub>☐<sub>x</sub>B<sub>6</sub> in our as cast alloy is measured at 8.5 at.% Nd in good agreement to the boron-rich phase boundary derived by E. Storms [43] (NdB<sub>11</sub> = Nd<sub>0.55</sub>B<sub>6</sub> = Nd8.4B91.6 at.% at 2200 K). The middle panel of Fig. 5 reveals the revised version of the boron-rich phase diagram Nd-B.

Although "PrB<sub>66</sub>" has been listed in a compilation of "Higher Borides" by T. Mori [18], micrographs for boron-rich as cast alloys in Fig. 4g,h,i do not reveal any hints for its formation (in agreement with a statement on the absence of "PrB<sub>66</sub>" by Spear [12]). Consequently, we measured (EDX) a maximal solid solubility of 0.02 at.% Pr in βB and the corresponding eutectic composition L ↔ (Pr<sub>0.02</sub>B<sub>100</sub>) + Pr<sub>1-x</sub>B<sub>6</sub> at 0.3 to 0.4 at.% Pr. As the eutectic is located very close to βB, the eutectic temperature at 2025 °C [12,17] seems unfeasibly low and needs to be raised at least to 2050 °C. With similar vaporisation characteristics the boron-rich boundaries of the hexaborides Nd<sub>1-x</sub>B<sub>6</sub> [44] and Pr<sub>1-x</sub>B<sub>6</sub> were assumed to be practically identical, also based on EDX data of our as cast alloy at 8.3 at.% Pr. The

<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Upper panel: Hardness of thorium and rare earth hectoborides. Lower panel: Hardness of ThB<sub>99</sub> (this work) in comparison with various 3d-metals forming solid solutions with beta boron (from Carlsson [45]).</figcaption></figure>
revised phase diagram Pr-B is shown in the right panel of Fig. 5.

### 3.4. Hardness

Hardness data were collected from measurements with two equipments (see the experimental part) in order to have a full dependence of hardness, HV, on the applied load in the range from 0.15 N to 10 N. For $ThB_{60}$ as well as for $NdB_{65}$ the results are depicted in the inset of the upper panel of Fig. 6. The values of HV = 23.8 ± 1 GPa for $ThB_{60}$ and HV = 24.3 ± 1 GPa for $NdB_{65}$ fit very well in the average hardness range of HV = 23.8 GPa for rare earths as published by Schwetz et al. [25]. It should be mentioned that proper hardness values can only be obtained at loads >2 N. Crack lengths from the indentations provide the average Vickers fracture resistance, which for both compounds is in the range of 3.9 MPam$^{1/2}$.

With 42 ± 2 GPa $ThB_{99}$ (solution of Th in beta boron; see bottom panel of Fig. 6) qualifies as a superhard material slightly harder than the βB solution phases containing 3d-elements [45].

### 4. Conclusion

Single crystal X-ray structure determination supported by TEM-SAED analyses documented that both "hectoborides" $NdB_{65}$ and $ThB_{60}$ crystallize in a face-centred cubic lattice with space group $Fm\bar{3}c$ (No. 226) ($a_{NdB_{65}} = 2.35385(4)$ nm, $R_f = 0.0491$; $a_{ThB_{60}} = 2.35170(5)$ nm, $R_s = 0.0489$) isotypic with the $YB_{66}$ type structure. Both compounds $NdB_{65}$ and $ThB_{60}$ are characterized by a split position for the metal atoms in site 48f at a combined occupancy of ~0.5 as well as by a second metal atom position in site 8a at a reduced occupancy. It should be emphasized that our WDX analysis of $ThB_{60}$ did not reveal any hints for oxygen contamination: in contrast to the earlier structure determination of thorium-oxyboride $ThB_{66}O_{0.33}$ [11], site 8a (¼,¼,¼) hosts a Th atom at low occupancy of 0.12(1) Th instead of an overpopulation of 8.1 oxygen atoms. Similarly, the 8a site is partially occupied by 0.04(1) Nd in $NdB_{65}$. Furthermore, a final difference Fourier electron density map with a maximum of 1.9 (electron/nm$^3$) × 10$^3$ (near Th2 or Nd2, respectively) neither for $ThB_{60}$ nor for $NdB_{65}$ detected a B-atom in site 64g (x = 0.23, x, x), which for $YB_{66}$ was labelled as B13 with occ. 0.279 (for details see Ref. [23], vol. 4, page 1581). In both cases metal atoms in the 8a site replaced the B13 atoms. Whereas from our investigations there is no doubt about the existence of $NdB_{65}$ with $YB_{66}$ type structure, our analysis did not reveal an isotypic "PrB̄_{66}”. Additionally, phase relations for the three systems Nd-B, Pr-B and Th-B have been presented for the boron-rich part (>85 at.% B) essentially based on our SEM data.

Room temperature hardness values for $ThB_{60}$ (23.8 GPa) and for $NdB_{65}$ (24.3 GPa) fit well among the corresponding rare earths hectoborides, whereas $ThB_{99}$ (βB-type) with 41 GPa is a superhard material.

### CRediT authorship contribution statement

**Gerda Rogl:** Validation, Investigation. **Vilma Bursikova:** Validation, Investigation. **Jiri Bursik:** Validation, Investigation. **Gerald Giester:** Validation, Investigation. **Henri Noel:** Validation, Investigation. **Peter Rogl:** Writing – review & editing, Writing – original draft, Validation, Supervision, Investigation, Conceptualization.

### Declaration of competing interest

No conflicts of interest exist.

### Acknowledgements

The research reported herein was in part supported by the Austrian OEAD within the WTZ-project Austria-Czech Republic, project no. CZ-01/2019 and MEYS, Czech Republic for the Mobility project

8J19AT011. EPMA-WDX measurements for Th-B were carried out at the Centre de la Microsonde Electronique de l'Ouest, IFREMER, Plouzane, France - we thank M. Bohn for his expertise in conducting calibration and measurements.

### Data availability

Data will be made available on request.

### References

[1] P.K. Smith, P.W. Gilles, Rare earth hectoborides, J. Inorg. Nucl. Chem. 26 (8) (1964) 1465-1467.

[2] S.M. Richards, J.S. Kaspar, The crystal structure of $YB_{66}$, Acta Crystallogr. B25 (1969) 237-251, https://doi.org/10.1107/S056774086900207X.

[3] P. Villars, K. Censual, Pearson's Crystal Data: Crystal Structure Database for Inorganic Compounds, ASM International, Materials Park: Ohio, USA, Release 2022/23.

[4] H.A. Eick, Plutonium borides, Inorg. Chem. 4 (1965) 1237-1239.

[5] J.S. Kasper, Status of $YB_{66}$-type structures, J. Less Common Metals 47 (1976) 17-21.

[6] J. Wong, T. Tanaka, M. Rowen, F. Schafers, B.R. Muller, Z.U. Rek, $YB_{66}$ - a new soft X-ray monochromator for synchrotron radiation. II. Characterization, J. Synchrotron Radiation 6 (6) (1999) 1086-1095.

[7] I. Higashi, K. Kobayashi, T. Tanaka, Y. Ishizawa, Structure refinement of $YB_{62}$ and $YB_{56}$ of the $YB_{66}$-type structure, J. Solid State Chem. 133 (1997) 16-20, https://doi.org/10.1006/jssc.1997.7308.

[8] M.A. Hossain, I. Tanaka, T. Tanaka, A.U. Khan, T. Mori, $YB_{48}$, the metal rich boundary of $YB_{66}$; Crystal growth and thermoelectric properties, J. Phys. Chem. Solids 87 (2015) 221-227, https://doi.org/10.1016/j.jpcs.2015.08.015.

[9] O. Sologub, L.P. Salamakha, B. Stöger, P.F. Rogl, T. Mori, G. Eguchi, H. Michor, E. Bauer, On the boron rich phases in the Yb-B system, J. Solid State Chem. 255 (2017) 172-177, https://doi.org/10.1016/J.JSSC.2017.08.007.

[10] J. Etourneau, R. Naslain, Chimie Minérale. Sur quelques nouvelles phases riches en boré dans le système bore-thorium (in French), C.R. Seances Acad, Sci. (Ser. C) (1968) 266-264.

[11] R. Naslain, J. Etourneau, Structure cristalline de la phase du système bore-thorium; Un nouveau borure a structure type $YB_{66}$, J. Solid State Chemistry 3 (1971) 101-111 (in French).

[12] K.E. Spear, G.I. Solovyev, High boron content rare-earth borides, NBS spec. publication, Proc. 5th Mater. Res. Symp. October 18-21, 1971, Gaithersburg, Maryland, USA, 364 (1972) 597-604; see also: K.E. Spear, II - phase behavior and related properties of rare-earth borides, Mater. Sci. Technol. 4 (1976) 91-159, https://doi.org/10.1016/B978-0-12-053204-9.950010-2.

[13] M.E. Schlesinger, The lesser-known B-In (Boron-Lanthanide) systems: B-Dy (Boron-Dysprosium), B-Ho (Boron-Holmium), B-Lu (Boron-Lutetium), B-Pr (Boron-Praseodymium), B-Tm (Boron-Thulium), and B-Yb (Boron Ytterbium), J. Phase Equilibria 19 (1) (1998) 49-55, https://doi.org/10.1361/105497198770342742.

[14] M.A. Ende, I.H. Jung, Y.H. Kim, T.S. Kim, Thermodynamic optimization of the Dy-Nd-Fe-B system and application in the recovery and recycling of rare earth metals from NdFeB magnet, Green Chem. 17 (4) (2015) 2246-2262, https://doi.org/10.1039/C4GC02232G.

[15] P.F. Rogl, R. Podloucky, H. Noel, G. Giester, Crystal structure of $Th_2B_2C_3$ with unique mixed B-C structural units, Acta Mater. 144 (2018) 484-495.

[16] T.B. Massalski, in: Binary Alloy Phase Diagrams, 2nd Edition, Materials Information Soc., Materials Park, Ohio, ASM International Materials Park, OH, USA, 1990, p. 44073. Pauling File Binaries Editions, Version 1.0, Release 2002/1.

[17] C.P. Wang, Y. Shi, D. Wang, Y. Lu, D.L. Zhao, X.J. Liu, Thermodynamic assessment of the B-Ce and B-Pr systems, Calphad 41 (2013) 150-155.

[18] T. Mori, Higher Borides, Chapter 238 in Handbook on the Physics and Chemistry of the Rare Earths, vol. 38, 2008, pp. 105-173, https://doi.org/10.1016/S0168-1273 (07)38003-3.

[19] L. Pouchon, F. Pichoir, Electron probe microanalysis applied to thin surface films and deposits, J. Microsc. Spectrosc. Electron. 10 (1985) 279-305.

[20] J. Rodriguez-Carvajal, FULLPROF, a program for Rietveld refinement and pattern matching analysis, in: Abstract of the Satellite Meeting on Powder Diffraction of the XV Congress, 1990, p. 127. Int. Union of Crystallography, Talence, France. See also: Recent developments of the program FULLPROF, Comm. Powder Diffr. IUCr. Newsl. 26 (2001) 12-19.

[21] Bruker Advanced X-ray solutions. APEX4 User Manual. Version 1.22. 2004, Bruker AXS Inc. For APEX4, SAINT and SADABS See 2021, Bruker Analytical X-Ray Instruments, Inc., Madison, Wisconsin, USA.

[22] G.M. Sheldrick, Shelx - program for crystal structure refinement, Acta Crystallogr., A64, in: P. McArdle, K. Gilligan, D. Cunningham, R. Dark, M. Mahon (Eds.), Windows Version OSCAIL by, Cryst. Eng. Comm. 6, 2008, pp. 112-122, 2004) 303-309.

[23] E. Parthé, L. Gelato, B. Chabot, M. Penzo, K. Censual, R. Gladyshevskii, Typix - standardized data and crystal chemical characterization of inorganic structure types, in: Gmelin Handbook of Inorganic and Organometallic Chemistry (8th Ed., Heidelberg: Springer-Verlag, Berlin, Heidelberg, 1994 see also J. Appl. Crystallogr. 20 (1987)139.

[24] W.B. Pearson, The Crystal Chemistry and Physics of Metals and Alloys, New York, USA: Wiley-Interscience (1972). The 12-coordinated metallic radii in Table 4-4 of
this book have been taken from: E. Teatum, K. Gschneidner, J. Waber, Report LA-2345, US Department of Commerce, Washington, D.C., USA, 1960.

[25] K. Schwetz, P. Ettmayer, R. Kieffer, A. Lipp, Über die Hektoboridphasen der Lanthaniden und Aktiniden, J. Less Common Met. 26 (1972) 99–104, https://doi.org/10.1016/0022-5088(72)90012-4 (in German).

[26] T. Tanaka, K. Kamiya, T. Numazawa, A. Sato, S. Takenouchi, The effect of transition metal doping on thermal conductivity of YB<sub>66</sub>, Z. f. Kristallogr. 221 (2006) 472–476.

[27] T. Tanaka, Y. Shi, T. Mori, A. Leithe-Jasper, Effect of transition metal doping in YB<sub>66</sub>, J. Solid State Chemistry 154 (2000) 54–60.

[28] P.A. Stadelmann. JEMS: Java Electron Microscopy Software, 2024. http://www.jems-swiss.ch. Switzerland.

[29] P.A. Stadelmann, Ems - a software package for electron diffraction analysis and HREM image simulation in materials science, Ultramicroscopy 21 (1987) 131–145.

[30] J. Roger, V.S. Babishetzkii, T. Guizouarn, K. Hiebl, R. Guerin, J.F. Halet, The ternary RE-Si-B systems (RE = Dy, Ho, Er and Y) at 1270 K: Solid state phase equilibria and magnetic properties of the solid solution REB<sub>2-x</sub>Si<sub>x</sub> (RE= Dy and Ho), J. Alloys Compd. 417 (2006) 72–84.

[31] Y. Yu, T. Lundström, A study of the boron-rich corner of the Er-Al-B system, J. Alloys Compd. 210 (1994) 191–196.

[32] J.F. Cannon, T. Hall, High pressure synthesis of lanthanide-boron and actinide-boron compounds, rare earth mod. Sci. Technol. (1978) 219–224.

[33] G.I. Solovyev, K.E. Spear, Phase behavior in the Sm-B system, J. Am. Ceram. Soc. 55 (1972) 475–479.

[34] M.C. Nichols, R.W. Mar, Erbium hectoboride, Inorg. Chem. 12 (1973) 1710–1711.

[35] J.H. Blanks, Phase behavior in the Gadolinium – Boron system and calculation of metal boride equilibria, Diss. Abstr. Int. B40 (1980) 402.

[36] Y.B. Paderno, T. Lundström, On the homogeneity ranges of LaB<sub>6</sub>, EuB<sub>6</sub> and SmB<sub>6</sub>, Acta Chem. Scand. 37 (1983) 609–612.

[37] V.V. Novikov, D.V. Avdashchenko, N.V. Mitroshenkov, A.V. Matovnikov, S. L. Bud'ko, Thermal expansion and lattice dynamics of RB<sub>66</sub> compounds at low temperatures, Phys. Solid State 56 (10) (2014) 2069–2076.

[38] P. Sauerschnig, K. Tsuchiyab, T. Tanaka, Y. Michiue, O. Sologub, S. Yin, A. Yoshikawa, T. Shishido, T. Mori, On the thermoelectric and magnetic properties, hardness, and crystal structure of the higher boride YbB<sub>66</sub>, J. Alloys Compounds 813 (2020) 1–8, 152182.

[39] P. Villars, K. Cenzual, J.L.C. Daams, F. Hulliger, T. Massalski, H. Okamoto, K. Osaki, A. Prince, S. Iwata (Eds.), Pauling File, Inorganic Materials Database, Binaries Edition Version 1.0, ASM International, Materials Park, Ohio, USA, 2002.

[40] S. Steiner, G. Rogl, H. Flandorfer, H. Noël, A. Pereira Gonçalves, G. Giester, P. F. Rogl, The system Thorium-Palladium-Boron: a DFT study on the stability and properties of Th<sub>2</sub>Pd<sub>15</sub>B<sub>5</sub>, J. Alloys Compounds 811 (2019) 151578, 1–12.

[41] J. Etourneau, R. Naslain, S. LaPlaca, L'hexaborure de thorium non-stoechiometrique Th<sub>1-x</sub>B<sub>6</sub>, J. Less Common Met. 24 (2) (1971) 183–193 (in French).

[42] P.K. Liao, K.E. Spear, M.E. Schlesinger, The B-Nd (Boron-Neodymium) system, J. Phase Equilibria 17 (1996) 335, https://doi.org/10.1007/BF02665560.

[43] T.L. Chen, J. Wang, C.P. Guo, C.R. Li, Z.M. Dub, G.H. Rao, H.Y. Zhou, Thermodynamic description of the Nd-Fe-B ternary system, Calphad 66 (2019) 1–14, 101627.

[44] E.K. Storms, Phase relationship, vaporization, and thermodynamic properties of Neodymium hexaboride, J. Phys. Chem. 85 (1981) 1536–1540.

[45] J.O. Carlsson, T. Lundström, Solution hardening of β-rhombohedral boron, J. Less-Common Metals 22 (1970) 317–320.