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The Magnetic Polarizability Dyadic and Point Symmetry
Published in Carl E. Baum, Detection and Identification of Visually Obscured Targets, 2019
Octahedral symmetry has four 3-fold axes, three 4-fold axes, and four 2-fold axes. The cube and regular octahedron are examples of such symmetry. Icosahedral symmetry has six 5-fold axes, 10 3-fold axes, and 15 2-fold axes. The dodecahedron and icosahedron are examples of such symmetry. For both O and Y the rotation axes are also sufficient to enforce the isotropic form for the magnetic-polarizability dyadic as in (7.48).
First-principles computation of new series of quaternary Heusler alloys CoScCrZ (Z = Al, Ga, Ge, In): a study of structural, magnetic, elastic and thermal response for spintronic devices
Published in Molecular Physics, 2020
M. Shakil, Hafsa Arshad, M. Zafar, M. Rizwan, S. S. A. Gillani, Shabbir Ahmed
Figures 8–11 indicated that Fermi level is positioned inside BG of spin-down channel and crosses orbitals in spin up channel for CoScCrZ (Z = Al, Ga, Ge, In). This location of Fermi level in both spin states leads to character of half-metallicity and gives 100% SP. The d-orbitals of transition metals (Co, Sc and Cr) are present near Fermi level. Firstly, d-d hybridisation of Co and Sc creates five bonding (obeys tetrahedral-symmetry) and five non-bonding orbitals (obeys octahedral-symmetry). The five-bonding d-orbitals further split up into double de-generated eg and triple de-generated t2g bonding orbitals. The octahedral symmetry of non-bonding orbitals breaks up into double de-generated eu and triple degenerated t1u states. Due to octahedral symmetry of five non-bonding states, the hybridisation among Co-Sc and Cr cannot take place. But the tetrahedral symmetry leads to d-d hybridisation of Co-Sc and Cr which creates five bonding (double de generated eg and triple de-generated t2g) and five anti-bonding (double de-generated eg* and triple de-generated t2g* ) orbitals. The energy gap lies among unoccupied eu and occupied t1u non-bonding states. The BG energy in spin-down channel can be determined by subtracting energies of CBM and VBM. The electronic contribution from sp element mainly influences size of BG and position of Fermi level. This BG is basically originated due to hybridisation of d-d orbital states of transition metals. This is also known as d-d BG and it gives the origin of HM BG. The covalent hybridisation among higher-valent atom, and lower-valent atom leads to the formation of BG. The presence of sp element also causes p-d hybridisation among transition metals and sp elements. The values of p orbitals near Fermi level are almost negligible, hence these states do not directly contribute in the creation of BG but defines the degree of occupation of p-d states. It can be concluded as width and formation of BG are affected by p-d hybridisation.