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Besides, elastic constants are also related to thermodynamic properties such as specific heat, thermal expansion, Debye temperature, Gruneisen parameter and melting temperature. Therefore, determination of elastic constants is essential to characterise a solid material. Three independent elastic constants viz. Cn, C12, and C44 are required to analyse the elastic properties of a cubic crystal. The elastic constant C11 measure the compression along the principle crystallographic axis whereas C44 measure the resistance to shear deformation across plane in the  direction.
It can be noticed from Table 2 that C11 is larger than C44 for all lutetium monopnictides which infers that these materials show weaker resistance to pure shear deformation than the unidirectional compression resistance. Furthermore, the criterion for mechanical stability of a cubic crystal is that the stress energy must be positive.
This means that the elastic constants must satisfy the following conditions for mechanical stability of a crystal. Table 3. From Table 2, it is clear that the elastic constants of LuX compounds satisfy all these conditions demonstrating the mechanical stability of these compounds. The Zener anisotropy factor A defined by Eqn. Thus, anisotropy of a material is measured by simply subtracting the value of A from unity.
As shown in Table 2, the calculated anisotropy factor A differs from unity which indicates the elastically anisotropic behaviour of LuX. The mechanical properties such as Young's modulus E , shear modulus G and Poisson's ratio v of LuX compounds are obtained from elastic constants employing Voigt-Reuss-Hill approximation32 given in Eqns The subscript v and R stands for Voigt and Reuss notation respectively. Results obtained for E, G and v have been assembled in Table 2 together with the anisotropy factor.checkout.midtrans.com/conocer-gente-soltera-en-san-mateo-de-gllego.php
Pnictides and Chalcogenides III (Actinide monopnictides) : Robert Troc :
The constants G and E are essential to describe the stiffness of an isotropic material. It is clear from the calculated values of G and E that these materials are stiff and their stiffness decreases as the radius of pnictide ion increases. The high stiffness of these materials may be due to their covalent bonding as predicted below. The Cauchy pressure35 CC44 describing the angular character of a compound has also been computed. Negative value of Cauchy pressure indicates that in conjunction with angular character the material has directional covalent bonding too while the positive value shows the metallic bonding.
Static and Dynamical Properties of heavy actinide Monopnictides of Lutetium
In the present study, Cauchy pressure predicted for all lutetium monopnictides are negative which reflects their covalent character. A material with negative value of Cauchy pressure is expected to be brittle whereas a positive value displays the ductile behaviour. From the results presented in Table 2, we found that the lutetium monopnictides show brittle character.
Frantsevich et al. The results obtained for v further assures the brittle character of LuX. The Poisson's ratio may also be used to obtain the information about the nature of bonding in a material. The value of Poisson's ratio varies from 0. Covalently bonded materials have small value for v 33, for ionic crystals the critical value is 0.
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In present study, the value of v was found to be less than the critical value 0. Furthermore, the melting temperature of lutetium monopnictides has also been calculated employing the empirical relation Eqn-9 proposed by Fine et al? The results obtained are listed in Table 3.
It is observed that the predicted temperature is maximum for LuN and the melting temperature decreases as the mass of pnictide ion increases. Figure 1. The longitudinal and transverse wave velocities along several crystallographic directions for these materials were also computed employing equations Here, p is the computed mass density.
Static and Dynamical Properties of heavy actinide Monopnictides of Lutetium
In the case of , the displacement of particles are along  direction and perpendicular to K vector for shear velocity vs whereas for longitudinal wave velocity vh the displacement is along  direction and parallel to the K vector. The wave velocities calculated along several crystallographic directions are listed in Table 3. Phonon Properties. The dispersion of phonons is an interesting property to understand the stability of a crystal besides knowing their thermal behaviour, superconductivity, Raman and thermal spectroscopy.
Since, we found that there was not a significant change. Phonon frequencies calculated were found positive throughout the Brillouin zone which indicates the dynamical stability of these compounds.
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For LuN, it is clear from Fig. It can be also seen from Fig. Also, transverse acoustic TA phonon modes are found to be nondegenerate along the r — K and K — L symmetry direction only. Besides, we observe that the longitudinal acoustic LA phonon modes meet the optical phonon modes along r — L symmetry directions for LuSb and LuBi. The calculated optical zone centre frequencies were 5.
We found that the highest observed optical frequency was largest for LuN and the frequency decreases as the mass of the pnictide ion increases. The SOC interactions have been included to correctly predict the structural and elastic properties. Our results indicate that all lutetium monopnictides are mechanically stable and elastically anisotropic. The mechanical properties such as Young's modulus E, Poisson's ratio v, shear modulus G, and Pugh's ratio have also been computed. From these results, we predict that all LuX compounds are brittle in nature and possess directional bonding.
The melting temperature calculated is maximum for LuN and was found to decrease as the mass of pnictide ion increases. It is also found that the longitudinal wave velocities are larger than the shear wave velocities along the given directions for all lutetium monopnictides. The phonon frequencies of all lutetium monopnictides have been found positive throughout the Brillouin zone which indicates that these compounds are dynamically stable. We used spin-polarized formalism in the Perdew, Burke, and Ernzerhof PBE parameterization for the exchange and correlation functional The interaction between electrons and ion cores was described by projector-augmented wave method PAW The value assumed for Ueff, now onwards called as U only has been taken from ref.
The wave functions were expanded in a plane-wave basis set up to a kinetic energy cutoff of eV. We have checked the energy cutoff convergence. Going from a cutoff of eV to eV, does not change the results. The integration within the Brillouin zone was performed with 9 x 9 x 9 grid using Monkhrost-Pack scheme We have checked the convergence in k-points.
Going from 9 x 9 x 9 to 11 x 11 x 11 even to 15 x 15 x 15 does not change the results. The atomic geometry of the system obtained was fully relaxed until the Hellmann-Feynman forces exerting on all atoms were less than 0. The elastic tensor was calculated by performing six finite distortions in the order of 0. Elastic constants were obtained then from strain-stress48 relationship with and without spin orbit coupling effect. Afterwards, the lattice dynamical properties were calculated by using finite displacement method as implemented in Phonopy49,50 with the support of VASP.
The VASP interface was used to calculate the force constant matrix. Ludbrook, B. Growth and Properties of Epitaxial GdN. Griebel, M. Jha, P. Lattice Vibrations in Yb-pnictide Compounds.
Larson, P. B - Condens. Matter Mater. Pagare, G. Legar, J.
Shirotani, I. Solid State Commun. Errandonea, D. Charifi, Z. De, M. Sheng, Q. Soyalp, F. Gupta, S. Gupta, D. Hasegawa, A. Energy Band Structures of Gd-Pnictides. Gokoglu, G. Korozlu, N. Mankad, V. Results Phys.