Theory of thermal properties of magnetic materials with unknown entropy

Theoretical approaches to study thermal properties of magnetic materials typically require accurate models of magnetic interactions in order to define the entropy. Here we introduce a complementary approach for examining thermal properties in magnetic systems where an accepted model for such interactions does not exist. In place of a specific model for magnetic interactions, the approach integrates measurements of temperature dependent magnetization of the studied material into a first principles computational scheme. The approach calculates system pressure from thermally disordered microstates that properly incorporate vibrational and spin subsystems at each temperature as well as the coupling between these subsystems. We apply the approach to calculate phonon modes and to investigate the anomalously low thermal expansion of the classical Invar alloy ${\mathrm{Fe}}_{0.65}{\mathrm{Ni}}_{0.35}$. The calculated phonon dispersions for Invar are in excellent agreement with measured data. The Invar thermal expansion is shown to remain small between 50 K and room temperature, consistent with the experimentally observed low thermal expansion value in this same temperature range. This anomalously small thermal expansion is directly connected to a small positive contribution from lattice thermal disorder that is nearly canceled by a large negative magnetic disorder contribution. By contrast, calculations for bcc Fe show a much larger thermal expansion, consistent with experiment, which is dominated by a large contribution from lattice thermal disorder that is reduced only slightly by a small negative contribution from that of magnetism. These findings give insight into the unusual nature of magnetism and spin-lattice coupling in Invar and Fe. In addition, they give promising preliminary support to the presented methodology as a complementary way to investigate thermal properties of magnetic materials. The success achieved on Invar and Fe motivates future testing of the approach on other magnetic materials.

Figure 1

Example of thermal spin-lattice snapshots (microstates) for the Invar calculation. Spheres represent Fe (red) and Ni (grey) atoms while arrows represent corresponding magnetic moments. Top and bottom panels show ferromagnetic and room-temperature arrangements, respectively.

Figure 2

Calculated phonon dispersions for the classical Invar alloy at room temperature compared to measurement. Blue error bars are measurements of ${\mathrm{Fe}}_{0.7}{\mathrm{Ni}}_{0.3}$ from Ref. [30] while black error bars are measured data for ${\mathrm{Fe}}_{0.65}{\mathrm{Ni}}_{0.35}$ from Ref. [31].

Figure 3

Left panel: Calculated $P_{\mathfrak{M}}$ for Invar at ambient volume for $T=50\mathrm{K}$ and normalized magnetization, $\mathfrak{M}$, ranging from ferromagnetic, $\mathfrak{M}=1$, to paramagnetic, $\mathfrak{M}=0$ (see color bar for normalized magnetization scale). Calculated $P_{\mathfrak{M}}$ values at $T=296\mathrm{K}$ use the ambient volume at 50 K. Right panel: Same for bcc Fe for $T=300\mathrm{K}$ and $T=600\mathrm{K}$.

Figure 4

Calculated $P_{\mathfrak{M}}$ for a subset of values in Fig. 3, for Invar (left panel) and for bcc Fe (right panel). See Fig. 3 caption for axes/color-map definitions. Labeled points for Invar are as follows: point a, $P_{\mathfrak{M}}$ ($T=50\mathrm{K}$) using $\mathfrak{M}$ consistent with measured magnetization at 50 K; point b, $P_{\mathfrak{M}}$ ($T=296\mathrm{K}$) for same $\mathfrak{M}$ as at 50 K; point c, $P_{\mathfrak{M}}$ ($T=296\mathrm{K}$) with $\mathfrak{M}$ consistent with measurement at 296 K. Labeled points a, b, c for bcc Fe have same meaning as corresponding points for Invar but at 300 and 600 K. Point d, $P_{\mathfrak{M}}$ for a paramagnetic configuration.

Figure 5

Dependence of $P_{\mathcal{M}}$ for Invar upon degree of disorder $\mathfrak{M}$ and energy of the magnetic state. Values of $P_{\mathcal{M}}$ are given by color (see color bar for values in kbar). $E^{\mathrm{mag}}$ is the energy of the given magnetic configuration with atoms in the ideal lattice positions; $E_{\min }^{\mathrm{mag}}$ is the lowest energy of the plotted configurations. All points correspond to a lattice temperature of 50 K.

Figure 6

Dependency of $P_{\mathcal{M}}$ for Invar on degree of magnetic disorder $\mathfrak{M}$ for lattice temperatures of 50 and 296 K. The $T=50\mathrm{K}$ data is the same as in Fig. 5. For the $\mathfrak{M}$ range of interest, $P_{\mathcal{M}}$ is a well-determined linear function of $\mathfrak{M}$.