3.1 Elastic constants and related mechanical properties
As well-known in literature [38-41], cubic materials have three characteristic independent elastic constants i.e. C11, C12 and C44. From these constants, C11 defines the longitudinal elastic behavior of a given cubic material, whereas C12 represents the off-diagonal and C44 points out the shear elastic characteristic of related material. In our work, first principles density functional theory (DFT) implemented in the CASTEP code has been used to calculate the above three independent typical cubic elastic constants of Rh2MnX alloys. Table 2. lists C11, C12 and C44 values for Rh2MnX alloys. From Table 2 it is easy to verify the presence of traditional Born mechanical (dynamical) stability conditions with C11 > 0, C44> 0, C11 − C12> 0, C11 + C12> 0 and cubic stability i.e. C12< 𝐵 < C12 where (B) is the bulk modulus of regarding materials [38-41]. Noteworthy to mention here, bulk modulus defines the resistance amount of a given material under external deformations where the shear modulus (G) describes materials resistance to the shape change under a shearing force. The Young’s modulus (E) describes the resistance of the material to uniaxial tensions and represents the stiffness degree, i.e., the higher the value of E, the stiffer is the material [42-47]. Table 2. also shows the calculated values of bulk modulus, shear modulus, Young’s modulus of Rh2MnX alloys. In addition to these three important moduli (B, G and E), the B/G ratio of any material identifies the brittle or ductile nature of the material. If the B/G ratio of a material is more or less than the limiting value of 1.75, then the material displays a ductile or brittle character [38-47]. It is also clear in Table 2. that the B/G ratio of all alloys is bigger than the limiting value which means that all of them are ductile. Moreover, Poisson’s ratio (ν) is the ratio between the transverse strain (et ) and longitudinal strain (el ) in the elastic loading direction. It provides detailed knowledge about the bonding character of solids. Usually, Poisson’s ratio values are about 0.1 for covalent materials, 0.25 for ionic materials and vary between 0.28 and 0.42 for metals [47]. Poisson’s ratio values of Rh2MnX alloys rise between 0.29 and 0.34 as in Table 2. So, it is apparent that all Rh2MnX alloys demonstrate metallic bonding. When compared with previous work of Ref. [26], although the metallic behavior of Rh2MnTi alloys correlates well with their findings, our elastic constants, elastic moduli and Vickers hardness (HV = 0.92(G/B)1.37.G0.708) data somewhat underestimate their findings for this alloy (Table 2). Also, when we compare our results with former results of Refs. [27,28] our elastic and mechanical data slightly overestimate the results of Refs. [27,28] (Table 2). Therefore, we can ensure that the differences between the results of this work and Ref. [26] as well as Refs. [27,28] may mostly arise from the three distinct calculation methods.

3.2 Electronic and Magnetic Properties

The electronic band structure of any periodic crystal corresponds to energy eigenvalues and helps us to figure out the Fermi levels of the material which determine the conducting, semiconducting, or insulating aspect of any material. Fig.2 exemplifies the electronic band structure of Rh2MnTi alloy as a precursor for spin down (↓) and spin up (↑) states. The band structure of Rh2MnTi alloy in Fig. 2, display a clear metallic nature in which the valence and conduction bands intersect each other along with the chosen high symmetry points. The same result is also valid for the rest of other concerned alloys like Rh2MnHf, Rh2MnSc, Rh2MnSc, Rh2MnzZr and Rh2MnZn. Further, our analysis on the orbital-selective partial density of states of Rh2MnTi revealed that the conductivity of these alloys mainly originates from the large contributions of the d-electrons of Rh and Mn atoms as seen in Fig. 3. The same findings are also true again for other alloys having Hf, Sc, Zn and Zr. Since s, p orbitals do not much contribute to the conductivity of these alloys we did not provide any results for them. Also, to ensure this neglected effect of s and p orbitals we also give the element-specific partial density of states of Rh2MnTi alloy with the total contributions of alloying elements in Fig.4. Moreover, in Fig.3 and Fig.4, the density of states plots either for spin down or spin up components display a non-symmetrical picture due to a clear polarization between the density of states of these components. In particular, our results for electronic band structure and partial density of states of Rh2MnTi alloy are in good agreement with findings of Ref. [26]
Since Rh2MnX alloys are declared as potential novel magnetic alloys in its original theoretical work [25], we computed the magnetic moment (µB) values of these alloys after spin-polarized calculations. Table 3. compares the calculated total magnetic moment (µB) values of Rh2MnX alloys with several former results. Our calculated values for the magnitudes of µB are in the range of Ti > Zr > Hf > Sc > Zn and this result compare well with the findings of Refs. [25-27] with ferromagnetic ordering.