#
Crystal Structure and Mechanical Properties of ThBC_{2}

^{*}

## Abstract

**:**

_{2}by the unbiased structure prediction method based on first-principles calculations. The dynamical and elastic stabilities of our proposed ThBC

_{2}are verified by the calculations of phonon spectrum and elastic constants. To study the mechanical properties fundamentally, we estimated the elastic anisotropy of ThBC

_{2}. The results show that the Young’s and shear moduli possess high degree of anisotropy. The ideal strength calculations reveal that ThBC

_{2}readily collapses upon applied stress due to small ideal strengths. The cleavage fracture probably occurs along the [111] direction while slip may easily appear along the [$\overline{1}10$] direction on the (111) plane for ThBC

_{2}. In addition, we provide an atomic explanation for the different characteristics of the strain–stress curves under different strains.

## 1. Introduction

_{2}C, ThBC

_{2,}and Th

_{2}BC

_{2}. The crystal structure of ThBC [2] shows a tetragonal symmetry with the P4

_{1}22 space group, which contains tetrahedra and trigonal prisms of Th atoms, and two boron and two carbon atoms form isolated zigzag chains. ThB

_{2}C [3] crystallizes in the rhombohedral space group ($R\overline{3}m$) with slightly puckered Th-metal layers, and boron atoms form B

_{6}hexagons connected by carbon atoms. Based on X-ray powder diffraction experiment, Rogl et al. reported the crystal structures of Th

_{3}B

_{2}C

_{3}[4] and Th

_{2}B

_{2}C

_{3}[5]. The monoclinic Th

_{3}B

_{2}C

_{3}belongs to the space group of P2/m, in which Th atoms form octahedra, tetrahedra, and trigonal prisms with B and C atoms occupying their centers, and non-metal atoms form isolated zigzag B–C chain fragments [4], which are similar to the B–C chains in ThBC [2]. The crystal structure of Th

_{2}B

_{2}C

_{3}consists of the infinite branched B–C chains and triangular prisms of Th atoms. Therefore, the flexible B–C framework leads to the complexity of the crystal structures for thorium borocarbide compounds. Consequently, the crystal structure of ThBC

_{2}, observed by Toth et al. long ago [1], remains an open question in the past years.

_{2}. The stability of ThBC

_{2}is verified by the phonon spectrum and elastic constants. To show the mechanical properties, we have systematically studied the elastic anisotropies of ThBC

_{2.}In addition, the failure modes under tensile and shear strains are investigated by the ideal strength calculations.

## 2. Computational Methods

_{2}was predicted using the PSO technique as implemented in the CALYPSO package [11]. To improve the prediction efficiency and avoid possible breakdown of structure optimization, the initial minimum inter-atomic distance between different types of atoms are 0.6 times the length of the sum of atomic covalent radii for the random structures. The proportion of the structures generated by PSO technique is set to 0.6. Other parameters are set as the default values of the CALYPSO package. All the self-consistent field calculations and structural relaxations were carried out within the framework of density functional theory using the Vienna ab initio simulation package (VASP) [18]. The generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof functional was employed to describe exchange–correlation interaction [19]. The interactions between electron and core were described by the frozen-core projector augmented wave (PAW) method [20], where the 6s

^{2}6p

^{6}6d

^{2}7s

^{2}, 2s

^{2}2p

^{1}, and 2s

^{2}2p

^{2}are valence electrons for Th, B, and C atoms, respectively. The PAW pseudo potential has been successfully used to study the compounds including Th atoms, such as thorium dicarbide [21], thorium dioxide [22], thorium dihydride [23], and thorium mononitrides [24]. The cutoff energy of 600 eV for plane wave and the 9 × 9 × 9 k-point meshes with Monkhorst–Pack method [25] were adopted, which can ensure that the total energy was converged to 1 meV per atom. All the structures were relaxed until the forces on each atom are below 0.001 eV/Å. To verify the dynamical stability, we performed the finite displacement method to calculate the phonon spectrum of ThBC

_{2}with 2 × 2 × 2 supercell of primitive cell and 4 × 4 × 4 k-point meshes as implemented in PHONOPY package [26]. The elastic constants were estimated by the strain–stress method described in our previous work [27], and a set of homogeneous strains with maximum strain value of 0.1% is applied. The elastic moduli and Pugh’s ratio were calculated using the Voigt–Reuss–Hill approximation [28]. To estimate the ideal tensile and shear strengths, the conventional cell of ThBC

_{2}is deformed incrementally and continuously along the direction of the applied strain. Then, the atomic basis vectors orthogonal to the applied strain and the positions of all atoms are relaxed simultaneously. As a result, the strain–stress curves for deducing the ideal strength are obtained [29,30]. Considering that the spin-orbit coupling (SOC) plays an important role on the compounds including heavy atoms [31], we tested the influence of SOC on the crystal structure and elastic constants of ThBC

_{2}. The calculated results [see the Supplementary Material (SM)] imply that SOC has little impact on the structure and mechanical properties of ThBC

_{2}. Therefore, we can ignore the SOC effect in the following. For orthorhombic structure, the Young’s modulus E along the [hkl] direction is described by the expression [32]

_{ij}are the elastic compliance constants of orthorhombic symmetry, α, β, and γ denote the direction cosines of the [hkl] direction. The shear modulus G along the arbitrary [uvw] directions on the (hkl) shear plane can be written as [32].

_{1}, β

_{1}, and γ

_{1}are the direction cosines of the [uvw] direction, while α

_{2}, β

_{2}, and γ

_{2}are the direction cosines of the normal direction of the (hkl) shear plane. The anisotropy of Young’s modulus and shear modulus are discussed based on above two expressions.

## 3. Results and Discussion

_{2}, we performed the structural search by the unbiased PSO technique [11] with the cell up to 8 formula units (f.u.). As a result, we obtained the ground state structure of ThBC

_{2}according to comparing their total energies. As shown in Figure 1a,b, the predicted crystal structure of ThBC

_{2}belong to the space group of Immm (no. 71) within orthorhombic symmetry (16 atoms/cell), in which Th atoms occupy the 4e (0.26226, 0, 0) Wyckoff position, B atoms occupy 4j (1/2, 0, 0.36170) Wyckoff position, and C atoms occupy 8l (0, 0.17557, 0.26910) Wyckoff position, respectively. The optimized lattice constants of the conventional cell are a = 7.2403 Å, b = 3.9978 Å, and c = 6.7686 Å. Clearly, we can see that the B and C atoms form the armchair-like B–C chains along (010) direction (see Figure 1c). These armchair-like B–C chains are separated by the adjacent Th atomic layers. Note that the armchair-like B–C chains had been uncovered in other thorium borocarbide compounds, such as ThBC [2,7]. One Th atom and four C atoms can constitute a square pyramid (see Figure 1d), which serves as a basic building-block of ThBC

_{2}. The average B–C bond length of ThBC

_{2}is 1.57 Å, which is shorter than those of t-BC

_{4}(1.66 Å) [27,33] and d-BC

_{3}(1.61 Å) [34], but is slightly longer than those of ThBC (1.54 Å) [2] and ThB

_{2}C (1.49 Å) [3]. The interatomic distance between adjacent B atoms is 1.87 Å, which is also larger than those of ThBC (1.77 Å) [2] and ThB

_{2}C (1.85 Å) [3]. In addition, to facilitate the possible experimental synthesis, we discuss the metastable ThBC

_{2}in the SM, and the crystal structures of the most stable form and other four predicted forms are available in the SM.

_{2}. As shown in Figure 2a, it is found that there is no imaginary frequency in the Brillouin zone along the high-symmetry direction, which indicates the dynamical stability of ThBC

_{2}. Moreover, due to the large differences of atomic masses among Th, B, and C atoms, the optical branches and acoustic branches of ThBC

_{2}are separated distinctly, leading to large acoustic-optical branch gap. Meanwhile, the high-frequency optical phonon bands are also discrete. The formation energy can reflect the relative stability of the compound and whether the compound can be readily synthesized experimentally. Here, we choose the face-centered cubic thorium, rhombohedral α-boron and graphite as the reference phases. Therefore, the formation energy $\Delta {E}_{f}$ of ThBC

_{2}can be defined by

_{2}, and $E\left(\mathrm{Th}\right)$, $E\left(\mathrm{B}\right)$, and $E\left(\mathrm{C}\right)$ denote the total energies of bulk thorium, α-boron, and graphite, respectively. The calculated formation energy of ThBC

_{2}is −1.454 eV/f.u., suggesting the thermodynamic stability and possible synthesis of ThBC

_{2}in experiment. In addition, the electronic band structure calculation reveals that ThBC

_{2}is metallic, as shown in Figure 2b,c. The density of states (DOS) for ThBC

_{2}implies that the Th and C atoms mainly contribute to the total density of states around the Fermi level.

_{2}. The orthorhombic ThBC

_{2}has nine independent elastic constants, which are C

_{11}= 315 GPa, C

_{12}= 146 GPa, C

_{13}= 40 GPa, C

_{23}= 67 GPa, C

_{22}= 286 GPa, C

_{33}= 576 GPa, C

_{44}= 40 GPa, C

_{55}= 49 GPa, and C

_{66}= 117 GPa. Clearly, these elastic constants satisfy the elastic stability criteria for a stable orthorhombic structure, that is, C

_{11}> 0, C

_{44}> 0, C

_{55}> 0, C

_{66}> 0, C

_{11}C

_{22}> C

_{12}

^{2}, C

_{11}C

_{22}C

_{33}+ 2C

_{12}C

_{13}C

_{23}− C

_{11}C

_{23}

^{2}− C

_{22}C

_{13}

^{2}− C

_{33}C

_{12}

^{2}> 0 [35]. The estimated bulk modulus (B) and shear modulus (G) of ThBC

_{2}are 184 and 86 GPa, which are larger than the bulk modulus (123 GPa) and shear modulus (61 GPa) of tetragonal ThBC [7]. That is to say, our predicted orthorhombic ThBC

_{2}has better ability to resist the tensile and shear deformations than that of ThBC. Accordingly, we can obtain the Pugh’s modulus ratio B/G, which is an indicator for determining whether material is ductile or brittle; if Pugh’s ratio is larger than 1.75, a material will tend to be ductile, otherwise brittle [36,37]. The calculated Pugh’s ratio of 2.13 implies that our predicted ThBC

_{2}tends to be ductile.

_{2}along different orientations present a high degree of anisotropy. In particular, we can clearly see that the maximum Young’s modulus is along the [001] direction, which is significantly larger than the Young’s modulus along other directions. This can be ascribed to the fact that there exist a number of strong covalent B–C bonds along the [001] direction, which facilitates ThBC

_{2}to resist tensile deformation and results in largest Young’s modulus along this direction.

_{[001]}= 560 GPa), which shows excellent consistency with the three-dimensional (3D) plots of Young’s modulus. It is found that Young’s modulus along principal crystal directions possess the sequence of E

_{[101]}< E

_{[011]}< E

_{[010]}< E

_{[100]}< E

_{[110]}< E

_{[001]}. To measure the ability of resisting shear deformation for ThBC

_{2}, the dependences of shear moduli on the directions are illustrated in Figure 3d. We can see that the largest shear modulus (G = 117 GPa) appears along the [001] direction on the (100) shear plane. Meanwhile, the obtained smallest shear modulus is along the [100] direction on the (001) shear plane, i.e., G

_{(001)[100]}= 40 GPa, which is equal to the elastic constant of C

_{44}.

_{2}to further explore its mechanical property. The tensile and shear strains are continuously applied upon ThBC

_{2}to obtain the strain–stress relationships, and consequently the ideal strength can be deduced. The calculated strain–stress relationships are shown in Figure 4. It is found that the largest ideal tensile strength is σ = 59 GPa along the [010] direction. The main reason is that the armchair-like B–C chains are along [010] direction, which can form strong covalent B–C against the tensile deformation. The lowest ideal tensile strength appears in the [111] direction, i.e., σ

_{[001]}= 7 GPa, suggesting that the cleavage fracture can easily occur along this direction. An inspection of the Th-C bond length suggests that the distance between the Th4 and C6 increases from 2.66 Å to 3.01 Å when the [111] tensile strain increases from 0 to 0.07. This fact implies these Th-C bonds are the main load bearing component along the [111] direction. Therefore, the weak Th-C bonds along the [111] direction is responsible for this low tensile strength. Besides the [111] direction, the [011] and [110] tensile deformation also easily stretch the weak Th-C bonds and result in small ideal tensile strength along these directions. It is worth noting that the ideal shear strengths of ThBC

_{2}on different shear planes are all very small, which is the inevitable consequence of the layered structure of ThBC

_{2}. We can see that the highest ideal shear strength is along the [100] direction on the (001) plane, i.e., τ

_{(001)[100]}= 18 GPa, and the obtained lowest ideal shear strength occurs within the [$\overline{1}10$] direction on the (111) plane, indicating that ThBC

_{2}can readily slip under shear along this direction.

## 4. Conclusions

_{2}and calculated the mechanical properties of ThBC

_{2}. Our predicted ThBC

_{2}consists of the Th atomic layers and the armchair-like B–C chains in parallel with Th atomic layers. Based on the phonon spectra and elastic constants, we verified the stability of ThBC

_{2}. We also calculated the anisotropy of Young’s modulus and shear modulus for ThBC

_{2}. The obtained results show that ThBC

_{2}exhibits significant anisotropy. The calculated strain–stress curves suggest ThBC

_{2}has low ideal strengths, implying that the cleavage fracture and slip can easily occur in ThBC

_{2}. We expect that our findings can favor the identification of structure for ThBC

_{2}, and provide insight understanding into the mechanical properties of ThBC

_{2}.

## Supplementary Materials

_{2}, Figure S2: Crystal structure and phonon spectrum of C2/m ThBC

_{2}, Figure S3: Crystal structure and phonon spectrum of $\mathrm{I}\overline{4}m2$ ThBC

_{2}, Figure S4: Crystal structure and phonon spectrum of Amm2 ThBC

_{2}, Table S1: Lattice parameters, atomic positions and the total energies per atom of four metastable ThBC

_{2}, Table S2: Calculated elastic constants C

_{ij}of the most stable ThBC

_{2}without and with SOC, most stable structure.cif (The most stable structure of ThBC

_{2}), no.47.cif (Crystal structure of Pmmm ThBC

_{2}), no.12.cif (Crystal structure of C2/m ThBC

_{2}), no.119.cif (Crystal structure of $\mathrm{I}\overline{4}m2$ ThBC

_{2}), no.38.cif (Crystal structure of Amm2 ThBC

_{2}).

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Side (

**a**) and top (

**b**) views of the crystal structure of ThBC

_{2}, in which the purple (large), red (middle), and blue (small) spheres denote Th, C, and B atoms, respectively. (

**c**) The infinite armchair-like B–C chain along y-axis direction. (

**d**) The square pyramid building block consisting of one Th and four C atoms.

**Figure 2.**(

**a**) Phonon spectrum of ThBC

_{2}along the high-symmetry lines. (

**b**) Band structure of ThBC

_{2}along the high-symmetry lines. (

**c**) DOS of ThBC

_{2}. Here the vertical blue dashed lines represent the positions of high-symmetry point in the Brillouin zone, and the horizontal blue dashed line denotes the Fermi level.

**Figure 3.**3D plots of Young’s modulus (

**a**) and its plane projections (

**b**) as the function of the directions, in which the distances between the origin and the surface are the values of Young’s moduli along corresponding directions. Orientation dependence of Young’s modulus (

**c**) and shear modulus (

**d**), in which the principal crystal orientations [110], [101], and [011] are marked by arrows with different color.

**Figure 4.**The stress as the function of tensile (

**a**) and shear (

**b**) strains along different tensile and shear strain directions.

**Figure 5.**Bond lengths as a function of strain under the [010] (

**a**) and [111] (

**d**) tensile modes. The ELF for ThBC

_{2}before (

**b**) and after (

**c**) the critical [010] tensile strain.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Zhou, X.; Zheng, B. Crystal Structure and Mechanical Properties of ThBC_{2}. *Crystals* **2019**, *9*, 389.
https://doi.org/10.3390/cryst9080389

**AMA Style**

Zhou X, Zheng B. Crystal Structure and Mechanical Properties of ThBC_{2}. *Crystals*. 2019; 9(8):389.
https://doi.org/10.3390/cryst9080389

**Chicago/Turabian Style**

Zhou, Xinchun, and Baobing Zheng. 2019. "Crystal Structure and Mechanical Properties of ThBC_{2}" *Crystals* 9, no. 8: 389.
https://doi.org/10.3390/cryst9080389