Elastic properties of paramagnetic austenitic steel at finite temperature: Longitudinal spin fluctuations in multicomponent alloys

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.

Figure 1

Schematic of the multicomponent alloy model. The dashed lines illustrate the lattice planes and $i=1,...,N$ is the site index. Different filling patterns stand for different alloy components A, B, C and any other alloy elements (illustrated by the empty site). The corresponding areas sketch the on-site concentration of different alloy species. The arrows sketch the spin-up and spin-down orientations of each alloy component (DLM state), whereas their lengths represent the local magnetic moment magnitudes, which may vary with alloy species and site. It should be noted that the spin state is only sketched for two sites but they actually may exist on all sites. A single spin impurity of species A on site $i$ is sketched.

Figure 2

On-site LSF energy $E_{\mathrm{LSF}}$ of PM fcc Fe as a function of the spin impurity magnetic moment $\mu$ evaluated with alternative approaches: a single fluctuating spin embedded in a supercell with two distinct magnetic moments on neighboring sites (i.e., ${\mu }^{\mathrm{NB}}=0\text{or}1.84{\mu }_{\mathrm{B}}$), the IF formalism combined with the OSA (static local magnetic moment $1.84{\mu }_{\mathrm{B}}$), and the FMA. All calculations were performed at the Wigner-Seitz radius $w_{1400\mathrm{K}}=2.6951\mathrm{bohrs}$.

Figure 3

Mean magnetic moment $m_{\mathrm{sf}}^I$ of alloy component $I$ in PM Fe15Cr15Ni as a function of the local magnetic moments of the other two alloy species in the effective medium, where $I$ is Fe, Cr, and Ni for panels (a), (b), and (c), respectively. While the magnetic moment of one species in the medium is constrained to the value in parentheses, the magnetic moment of the other one is varied. The shaded horizontal and vertical bars indicate the self-consistent solution, the black arrows indicate the static magnetic moment and the OSA mean magnetic moment for Fe, whereas the green arrows are explained in the text. All calculations were performed for 1600 K at the Wigner-Seitz radius $w_{1600\mathrm{K}}=2.6987\mathrm{bohrs}$.

Figure 4

On-site LSF energies $E_{\mathrm{LSF}}^I$ as a function of the local magnetic moment $\mu$ for LSFs on Fe, Cr, or Ni sites (lower panel) and the corresponding LSF density distributions at $1600\mathrm{K}$ (upper panel) in PM Fe15Cr15Ni. $E_{\mathrm{LSF}}^I$ was calculated for each alloy component $I$ adopting the FMA. All calculations were performed at the same Wigner-Seitz radius $w_{1600\mathrm{K}}=2.6987\mathrm{bohrs}$. The solid lines in the lower panel are the fitted polynomial Landau expansions to the data (points) and were used to establish the continuous LSF distributions as sketched by the transparent areas.

Figure 5

Temperature dependence of the mean local magnetic moment $m_{\mathrm{sf}}^I$ and the static local magnetic moment ${\mu }_0^I$ for Fe, Cr, or Ni in PM Fe15Cr15Ni taking into account thermal expansion.

Figure 6

Temperature dependence of the single-crystal elastic constants $c^{\prime }$ and $c_{44}$ of PM Fe15Cr15Ni calculated with different schemes. In addition to the LSF results, elastic constants adopting the FS and $c$-FS schemes along with available experimental data from (a) Ref. [56] and (b) Ref. [57] are presented for comparison. It should be noted that the error bars indicated for two experimental points in each case apply to all data from (b) Ref. [57].

Figure 7

$c^{\prime }, c_{44}$, and the SED for PM Fe15Cr15Ni as a function of the local magnetic moment ${\mu }^{\mathrm{Cr}}$ or ${\mu }^{\mathrm{Ni}}$ with the moment of the other component constrained to zero. The local magnetic moment of Fe is fixed to $m_{\mathrm{sf}}^{\mathrm{Fe}}=1.717{\mu }_{\mathrm{B}}$ in all cases. The calculations were performed at $700\mathrm{K}$ for the Wigner-Seitz radius $w_{700\mathrm{K}}=2.6499$ bohrs.