Landau-type parametrization of the equation of state of a ferromagnet

An approximate equation of state is proposed for ferromagnets, with a particular reference to magnetic refrigerants such as gadolinium. The equation applies for $T\lesssim T_C$ and an arbitrary magnetic field and is based on Landau’s theory of second-order phase transitions, with Bloch’s 3/2 power law built in one of Landau’s coefficients. The displacement of the maximum in the specific heat under an applied magnetic field is demonstrated to be nonmonotonic: the maximum shifts toward lower temperatures in a weak field but toward higher temperatures in a strong field.

Figure 1

(Color online) Reduced temperature vs square of reduced magnetization in the spin-$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ Heisenberg model on the fcc lattice. The open circles are numerical data from Table III of Ref. 16 (Padé approximants); the line represents Eq. (12). The line thickness reflects the degree of confidence expressed in Ref. 16—the thin line in the top-left corner is drawn through poorly converged data points.

Figure 2

Fitting experimental data on Gd (Ref. 22). All parameters are as in Table unless indicated otherwise. (a) Reduced spontaneous magnetization vs reduced temperature: filled circles—experiment; solid line—Eq. (18). (b) Reciprocal susceptibility vs reduced temperature: circles—experiment [($○$) $a$ axis, ($●$) $c$ axis]; solid line—Eq. (20) with $a_0=0.9\text{MOe}$; and dotted line—Eq. (20) with $a_0=1.0\text{MOe}$. (c) Reduced magnetization vs reduced temperature at three indicated values of internal magnetic field: circles—experiment; solid lines—Eqs. (16, 17).

Figure 3

Fitting Ni data. All parameters are as in Table . (a) Reduced spontaneous magnetization vs reduced temperature: open circles—experiment (Ref. 23); filled circles—experiment (Ref. 24); and solid line—Eq. (18). (b) Reciprocal susceptibility vs reduced temperature: open circles—experiment (Ref. 25); filled circles—experiment (Ref. 26); and solid line—Eq. (20). (c) Reduced magnetization vs internal magnetic field at various fixed temperatures: circles—experiment (Ref. 24); solid lines—Eq. (15).

Figure 4

Fitting Fe data. All parameters are as in Table . (a) Reduced spontaneous magnetization vs reduced temperature: open circles—experiment (Ref. 23); solid line—Eq. (18). (b) Reciprocal susceptibility vs reduced temperature: filled circles—experiment (Ref. 26); solid line—Eq. (20).

Figure 5

Fitting Co data. All parameters are as in Table . (a) Reduced spontaneous magnetization vs reduced temperature: data points—experiment [Refs. 27 $(○)$, 28 $(△)$, and 29 $(●)$]; solid line—Eq. (18). (b) Reciprocal susceptibility vs reduced temperature: filled circles—experiment (Ref. 26); solid line—Eq. (20).

Figure 6

(a) Negative magnetic entropy change induced in Gd by a change in magnetic field from 0 to 20 or to 50 kOe, computed by means of Eq. (29). (b) Square of reduced magnetization of Gd vs reduced temperature, which is obtained from Eq. (18) for $H=0$ and from Eqs. (16, 17) for $H=20\text{kOe}$.

Figure 7

Calculated molar specific heat of gadolinium. (a) $H=0$, lattice specific heat (dashed-dotted line), total specific heat (solid line), and total specific heat smeared out by inhomogeneities (dotted line). (b) Magnetic specific heat at various values of applied magnetic field.

Figure 8

Position of the maximum in the specific heat of Gd as a function of applied magnetic field. Solid line—calculations, this work. Data points—experiment, Refs. 37 $(\nabla )$, 38 $(\times )$, and 20 $\left(○\cdots \cdot ○\right)$.

Figure 9

Position of the maximum in the magnetic specific heat of a ferromagnet against rescaled magnetic field, which is computed by using Eqs. (52, 53).

Figure 10

Lowering of the heat-capacity peak in a magnetic field. The redistribution of the entropy content leads to a shift of the maximum to the left in a weak field (a), but to the right in a strong field (b).