Theoretical study of Spinel based Transition Metal Oxides is reported. The Density Functional Theory approach is adopted for computational investigation using LAPW code. The structural, elastic properties of (XAl2O4 for X=Mg, Mn, Fe, Co, Ni, Cu, Zn), (MGa2O4 for M=Mg, Mn, Fe, Co, Ni, Cu, Zn, Cd), (NIn2O4 for N=Mg, Zn, In) & (ZnP2O4 for P=V, Cr, Fe, Rh, Sn) along with electronic & magnetic properties of cubic spinel oxides (XAl2O4) and (MGa2O4) have calculated. The calculated structural parameters conform to other experimental and theoretical evidences. Fulfillment of elastic stability criteria proves the reliability of the reported data. The calculated elastic constants show that all materials abide its stability opposing elastic deformations and the Cauchy’s pressure confirms the dynamical stability of these materials. The high values of bulk moduli show the hardness of these materials while the Young moduli data reveal that MgAl2O4 is the stiffer material among all. Further, Poisson’s ratios testify the compressibility, connote stableness to external deformation of these materials and reveal dominance of central forces being the interatomic forces. Out of 24 materials, only 05 are brittle and the rest are ductile with elastic anisotropy. MgAl2O4 is the most brittle and CdGa2O4 is the most ductile material. The electronic structure (Total DOS) show all XAl2O4, MGa2O4 (M=Fe, Co, Zn, Cd) have antiferromagnetic, NiGa2O4 & CuGa2O4 have FM state and the valence near Fermi level due to the transition metal 3d state. The calculation of direct exchange energy Δx(d) shows that the exchange mechanism of electrons is dominant in introducing magnetism and enormous exchange splitting is observed in these materials. The ferromagnetism in the group MGa2O4 of spinel oxide compounds is due to hybridization between the electrons in the 3d-states cation and 2p states anion. Appearance of magnetic moments in these materials is due to unpaired electrons in 3d states of Transition metal ions. The negative value of indirect exchange energy confirms the magnetism by spin of electrons.
The biggest challenge of 21 st century for the humanity is to make this world a peaceful abode. The human beings are threatened by the dangerous weapons of mass destruction, invented by their own hands. On the other hand, the Islamic principles of peace are the excellent ones for the promotion of peace. The prophets (ﷺ) of Allāh Almighty always focus on the establishment of a pious and peace loving society. In this paper the author explores the measures taken by Islām for the promotion of peace, for example, Islām forbids abusing the false gods, just because it creates hatred; it does not allow to use force to coerce someone to change his or her faith; Islām teaches to do trade, share social norms, and cultural rites with other nations; it is imperative, to respect all religions; a true Muslim gives due regards to others honor, life and property, which is the key to a peaceful living; Islām stresses its followers to help each other for the noble deeds and the welfare of society. The advancement in the science and technology has transformed the world into a global village. The mutual cooperation is far more necessary for the prosperity and welfare of the human beings, now. This dream is only possible through a worldwide peace program. This program is Islām. This paper explores such possibilities in Islām for the promotion of peace and harmony in the human society.
In this dissertation, we have prospected the idea of (de)homogenization technique in Grabner Basis for modules concerning variable T'. Cases of (de)homogenization of Grabner Basis for modules are explained in a precise way to extricate how things work when (de)homogenization is applied to Grobner basis in modules