We investigate the impact of longitudinal thermal spin fluctuations on the temperature dependence of the elastic constants of paramagnetic body-centered-cubic (bcc) and face-centered-cubic (fcc) Fe. Based on a series of constrained local magnetic moment calculations, the spin fluctuation distribution is established using Boltzmann statistics and involving the Jacobian weight, and a temperature-dependent quadratic mean moment is introduced that accurately represents the spin fluctuation state as a function of temperature. We show that with increasing temperature, $c^{\prime }$ and $c_{44}$ for the fcc phase and $c_{44}$ for the bcc phase decrease at different rates due to different magnetoelastic coupling strengths. In contrast, $c^{\prime }$ in the bcc phase exhibits relatively high thermal stability. Longitudinal thermal spin fluctuations diminish the softening of both elastic constants in either phase and have comparatively large contributions in the fcc phase. In both bcc and fcc Fe, $c_{44}$ has a larger temperature factor than $c^{\prime }$. On the other hand, $c^{\prime }$ is more sensitive to the longitudinal thermal spin fluctuations, which balance the volume-induced softening by 21.6% in fcc Fe.

High entropy alloys based on 3d transition metals display rich and promising magnetic characteristics for various high-technology applications. Understanding their behavior at finite temperature is, however, limited by the incomplete experimental data for single-phase alloys. Here we use first-principles alloy theory to investigate the magnetic structure of polymorphic CoCrFeMnNi in the paramagnetic state by accounting for the longitudinal spin fluctuations (LSFs) as a function of temperature. In both face-centered cubic (fcc) and hexagonal close-packed (hcp) structures, the LSFs induce sizable magnetic moments for Co, Cr and Ni. The impact of LSFs is demonstrated on the phase stability, stacking fault energy and the fcc-hcp interfacial energy. The hcp phase is energetically preferable to the fcc one at cryogenic temperatures, which results in negative stacking fault energy at these conditions. With increasing temperature, the stacking fault energy increases, suppressing the formation of stacking faults and nano-twins. Our predictions are consistent with recent experimental findings.

Using an efficient first-principles computational scheme for paramagnetic body-centered cubic (bcc) and face-centered cubic (fcc) Fe, we investigate the impact of thermal longitudinal spin fluctuations (LSFs) on the thermal lattice expansion. The equilibrium physical parameters are derived from the self-consistent Helmholtz free energy, in which the LSFs are considered within the adiabatic approximation and the anharmonic lattice vibration effect is included using the Debye-Grüneisen model taking into account the interplay between thermal, magnetic, and elastic degrees of freedom. Thermal LSFs are energetically more favorable in the fcc phase than in the bcc one giving a sizable contribution to the linear thermal expansion of $\gamma$-Fe. The present scheme leads to accurate temperature-dependent equilibrium Wigner-Seitz radius, bulk modulus, and Debye temperature within the stability fields of the two phases and demonstrates the importance of thermal spin fluctuations in paramagnetic Fe.

We propose a first-principles framework for longitudinal spin fluctuations (LSFs) in disordered paramagnetic (PM) multicomponent alloy systems and apply it to investigate the influence of LSFs on the temperature dependence of two elastic constants of PM austenitic stainless steel Fe15Cr15Ni. The magnetic model considers individual fluctuating moments in a static PM medium with first-principles-derived LSF energetics in conjunction with describing chemical disorder and randomness of the transverse magnetic component in the single-site alloy formalism and disordered local moment (DLM) picture. A temperature-sensitive mean magnetic moment is adopted to accurately represent the LSF state in the elastic-constant calculations. We make evident that magnetic interactions between an LSF impurity and the PM medium are weak in the present steel alloy. This allows gaining accurate LSF energetics and mean magnetic moments already through a perturbation from the static DLM moments instead of a tedious self-consistent procedure. We find that LSFs systematically lower the cubic shear elastic constants $c^{\prime }$ and $c_{44}$ by $\sim 6\mathrm{GPa}$ in the temperature interval 300–1600 K, whereas the predominant mechanism for the softening of both elastic constants with temperature is the magneto-volume coupling due to thermal lattice expansion. We find that non-negligible local magnetic moments of Cr and Ni are thermally induced by LSFs, but they exert only a small influence on the elastic properties. The proposed framework exhibits high flexibility in accurately accounting for finite-temperature magnetism and its impact on the mechanical properties of PM multicomponent alloys.

Elucidating the evolution law of the elastic properties of the matrix phase is of great significance for the control of steel properties and quality during continuous casting and subsequent heat treatment. In this paper, thermal expansion experiments and ab initio calculations are used to study the elastic properties of the interstitial free (IF) steel matrix phase in different magnetic states and crystal structures. The results show that the bulk modulus B and the tetragonal shear elastic constant C’ for the entire temperature range decrease with increasing temperature, but C₄₄ is the opposite. While from paramagnetic (PM) to ferromagnetic (FM) state, C’(C₄₄) have changed ~188% (~27%), B increases by ~55% during the crystal structure change (fcc→bcc). With the FM to PM state, the Zener anisotropy parameter increases sharply, and Young’s modulus decreases significantly in the [001] direction; the maximum difference is ~76 GPa. The evolution rate of average Young’s modulus in single bcc-phase FM (fcc-phase PM) range reaches ~5.5(~5.6) × 10⁻² GPa K⁻¹. The research provides an effective method for ab initio calculation of the elastic properties of interstitial free and ultra-low carbon steels at high temperature, also furnishing a basis for the application of ab initio calculations to the high temperature performance of steel materials.

The hexagonal close-packed (hcp) phase of iron is unstable under ambient conditions. The limited amount of existing experimental data for this system has been obtained by extrapolating the parameters of hcp Fe–Mn alloys to pure Fe. On the theory side, most density functional theory (DFT) studies on hcp Fe have considered non-magnetic or ferromagnetic states, both having limited relevance in view of the current understanding of the system. Here, we investigate the equilibrium properties of paramagnetic hcp Fe using DFT modelling in combination with alloy theory. We show that the theoretical equilibrium $c/a$ and the equation of state of hcp Fe become consistent with the experimental values when the magnetic disorder is properly accounted for. Longitudinal spin fluctuation effects further improve the theoretical description. The present study provides useful data on hcp Fe at ambient and hydrostatic pressure conditions, contributing largely to the development of accurate thermodynamic modelling of Fe-based alloys.