First-principles modeling of the Invar effect in ${\mathrm{Fe}}_{65}{\mathrm{Ni}}_{35}$ by the spin-wave method

Thermal lattice expansion of the Invar ${\mathrm{Fe}}_{0.65}{\mathrm{Ni}}_{0.35}$ alloy is investigated in first-principles calculations using the spin-wave method, which is generalized here for the ferromagnetic state with short-range order. It is shown that magnetic short-range order effects make a substantial contribution to the equilibrium lattice constant and cannot be neglected in the accurate ab initio modeling of the thermal expansion in Fe-Ni alloys. We also demonstrate that at high temperatures, close to and above the magnetic transition, magnetic entropy associated with transverse and longitudinal spin fluctuations yields a noticeable contribution to the equilibrium lattice constant. The obtained theoretical results for the temperature dependent lattice constant are in semiquantitative agreement with the experimental data apart from the region close the magnetic transition.

Figure 1

LSF energy of Ni and Fe in random ${\mathrm{Fe}}_{65}{\mathrm{Ni}}_{35}$ at 1000 K.

Figure 2

Magnetic exchange interactions in random ${\mathrm{Fe}}_{65}{\mathrm{Ni}}_{35}$ alloy in the FM state with different magnetizations and in the paramagnetic state with LSF on Ni at 500 K, which is the Curie temperature of ${\mathrm{Fe}}_{65}{\mathrm{Ni}}_{35}$.

Figure 3

Total energy of random ${\mathrm{Fe}}_{65}{\mathrm{Ni}}_{35}$ in the FM and PM states. The latter is obtained in the CPA-DLM (diamonds) and SWM (circles) calculations. The arrows show the positions of equilibrium WS radii.

Figure 4

Local magnetic moment of Fe in ${\mathrm{Fe}}_{65}{\mathrm{Ni}}_{35}$ as a function of the WS radius in the FM and paramagnetic states.

Figure 5

Reduced magnetization as a function of reduced temperature in ${\mathrm{Fe}}_{65}{\mathrm{Ni}}_{35}$: experiment [57] vs present theoretical modeling.

Figure 6

Average spin-spin correlation function in ${\mathrm{Fe}}_{65}{\mathrm{Ni}}_{35}$ at different temperatures.

Figure 7

Zero-temperature lattice constant obtained in the SWM and PDLM calculations as a function of magnetic state translated to the corresponding temperature shown on the $x$ axis.

Figure 8

Temperature dependence of the lattice constant of the IPM of ${\mathrm{Fe}}_{65}{\mathrm{Ni}}_{35}$ obtained in the Debye-Grüneisen model using (1) the total energy of from the DLM-CPA calculations (E, DLM) and (2) the total energy from the SWM calculations (E, SWM ). The other results are obtained using the Helmholtz free energy at the corresponding temperatures with contributions from the one-electron excitations, LSF using (21), and TSF using (20) in different combinations. Liot and Hooley EMTO-CPA results are from Ref. [26].

Figure 9

Lattice constant of ${\mathrm{Fe}}_{65}{\mathrm{Ni}}_{35}$ obtained from the SWM total energy calculations (SWM) without and with the PDLM entropy (SWM + $S^{\mathrm{PDLM}}$), from DLM calculations with LSF on Ni and TSF on Fe and LSF on both elements. The experimental data are taken from Ref. [64]. Liot and Hooley EMTO-CPA results are from Ref. [26]. The arrow ($a_0$) indicates the equilibrium lattice constant obtained in the FM state without the zero point lattice vibration contribution.