Magneto-crystalline anisotropy is one of the fundamental quantities for permanent magnets, but not only. The other example are magnetostriction and magnetoelastic effects. Utilizing the in-house developed approach the origin of the anisotropic magnetocrystalline energy (MAE) guided by the spin-orbit coupling (SOC) in the ordered Fe₅Ta₂ and L1₀-FePt phase is analyzed and discussed by means of theoretical calculations showing agreement with the known experimental studies. A systematic analysis of the MAE, magnetostriction, and magnetoelasticity by means of first principles plane-wave calculations and post processing of calculated eigenvalues (orbital energies) and functions (orbital occupancies) are done to establish their correlations. Our study includes the convolution of the projected wave function (density of states) of each orbital of the Fe and Pt sublattices into orbital energies. The current technique shows the orbital contributions to MAE and magnetoelasticity in accordance with the plane-wave total energies including SOC, which have not been discussed earlier. We also explore the complete anisotropic magnetostriction of FePt, finding a significant magnetostrictive (λ) performance of the order about λ ∼10^-4 - 10³ in some particular crystallographic directions of the ordered compound. However, the poly-crystalline model of L1₀-FePt based on the uniform stress approximation, leads to a sharp decline in the overall magnetostrictive behaviour due to the linear combination of the single crystal magnetostrictive coefficients, supported by the experimental measurements of the thin films and recent bulk one. However, this corresponds to the zero strain situation. For a zero stress situtation (realistic conditions used in experiments) a very small correction is found for the first anisotropy constant ΔK₁/K₁ = 0.07%, while a much more significant contribution is obtained for the second one ΔK₂/K₂= 21.86%. General analysis of this effect for tetragonal crystals is provided, finding that ΔK1$ will be always positive for any stable phase with this symmetry. We extend some of the analysis also for the first time to the antiferromagnetic system of MnPt. 

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