Longitudinal spin fluctuation contribution to thermal lattice expansion of paramagnetic Fe

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.

Figure 1

Total energy as a function of the local magnetic moment (lower panels) and the corresponding spin density distributions (upper panels). The left figure (a) is for PM bcc Fe at 1100 K and the right figure (b) for PM fcc Fe at 1400 K. Results are shown for the smallest ($w_{\mathrm{bcc}}=2.65$ bohrs and $w_{\mathrm{fcc}}=2.58$ bohrs) and the largest ($w_{\mathrm{bcc}}=2.75$ bohrs and $w_{\mathrm{fcc}}=2.88$ bohrs) Wigner-Seitz radii employed here. The solid lines in the lower panels indicate the Landau expansion used to establish the continuous spin density distributions as sketched by the transparent areas.

Figure 2

Contour plots of the quadratic mean magnetic moment ($m_{\text{sf}}$) for PM bcc (left panels) and fcc (right panels) phases. The evolution of the magnetic moment in (b) and (d) involves the effects from the thermal spin fluctuations and the lattice expansion. For comparison, the equilibrium local magnetic moment ($m_0$) in the ground state (GS) in the FS calculations are shown in the upper panels (a) and (b) for the bcc and fcc structures. The dotted and dashed lines in (a) and (b) indicate the deviations ($\mathrm{\Delta }m$) between $m_{\text{sf}}$ and $m_0$ at various Wigner-Seitz radii for temperatures shown in the legend.

Figure 3

Predicted thermal lattice expansions for PM bcc and fcc Fe taken into account LSFs. For comparison, results from the FS calculations and the available experimental data (${}^{\text{a}}$Ref. [52] and ${}^{\text{b}}$Ref. [53]) are also included.

Figure 4

The Helmholtz free energies ($\mathrm{\Delta }F$, expressed relative to the minimum value) calculated through the LSF and FS schemes as a function of the Wigner-Seitz radius for PM fcc Fe at 1200 K (lower panel). The energy contributions from the vibrational free energy ($\mathrm{\Delta }{F}^{\mathrm{vib}}$), magnetic entropy term ($-T\mathrm{\Delta }{S}^{\mathrm{mag}}$), and the internal energy ($\mathrm{\Delta }{E}^{\mathrm{int}}$) to $\mathrm{\Delta }F$ are specified in the upper panel for both the LSF and FS calculations.

Figure 5

The bulk modulus $B$ and the Debye temperature $\mathrm{\Theta }$ of PM bcc and fcc Fe as a function temperature. Both the theoretical results from the LSF and FS calculations are presented. For comparison, the polycrystalline bulk moduli evaluated from the experimental single crystal elastic constants (${}^{\text{a}}$Ref. [54], ${}^{\text{b}}$Ref. [55], ${}^{\text{c}}$Ref. [56], and ${}^{\text{d}}$Ref. [57]) and the DMFT prediction (${}^{\text{e}}$Ref. [11]) are also shown.