Critical Point in the Curve of First-Order Magnetic Phase Transition

Abstract

A magnetic state diagram for ferromagnets exhibiting a first-order phase transition in a magnetic field has been calculated. Calculations have been made using the exchange-striction model of a ferromagnet. It has been shown that the curve of first-order magnetic phase transition ends up at a critical point (H_cr, T_cr) that is similar to the critical point of the liquid–gas transition on the (T, P) plane. Analytical expressions for thermodynamic quantities, namely, magnetic susceptibility, specific heat, and compressibility, which anomalously grow near the critical point, have been derived. Calculation results have been compared with available experimental data for La(Fe_0.88Si_0.12)₁₃ ferromagnet, which experiences the first-order magnetic phase transition.

APPENDIX

After substituting ω (3) into the formula J = J₀ + γω, the expression for x from (2) takes the form

$$x = d + am + b{{m}^{3}},$$

(A.1)

where

$$d = \frac{{2\mu sH}}{{kT}},\quad a = \frac{{3sT_{C}^{0}(P)}}{{(s + 1)T}},$$

$$b = 2{{s}^{4}}NK_{0}^{{ - 1}}\frac{{{{\gamma }^{2}}}}{{kT}},$$

and

$$T_{C}^{0}(P) = 2s(s + 1)({{J}_{0}} - K_{0}^{{ - 1}}P\gamma ){\text{/}}3k$$

is the pressure-dependent Curie point for the first-order PT.

The expansion of the Brillouin function into a series up x⁵ has the form

$${{B}_{S}}(x) = Ex - F{{x}^{3}} + G{{x}^{5}},$$

(A.2)

where

$$\begin{gathered} E = \frac{{s + 1}}{{3s}},\quad F = \frac{{{{{(2s + 1)}}^{4}} - 1}}{{45{{{(2s)}}^{4}}}}, \\ G = 2\frac{{{{{(2s + 1)}}^{5}} - 1}}{{945{{{(2s)}}^{5}}}}. \\ \end{gathered} $$

Then, the equation of magnetic state takes the form of (4) where

$$\begin{gathered} A = \frac{{T - T_{C}^{0}(P)}}{T},\quad B = {{a}^{3}}F - Eb, \\ C = 3F{{a}^{2}}b - G{{a}^{5}}. \\ \end{gathered} $$

(A.3)

In calculations, we used the values of quantities appearing in (A.1)–(A.3) for La(Fe_0.88Si_0.12)₁₃ ferromagnet: s = 1, N = 6 × 10²³ cm^–3, K₀ = 1.1 × 10¹² dyne/cm², γ = 4.89 × 10^–13 erg, and $T_{C}^{0}$ = 189 K (at P = 0). These values were determined from experimental data reported in [11, 15–17; see also [14]. The final values of $T_{C}^{0}$ and γ were taken from the coincidence condition for the calculated and experimental (T_C1 ≈ 196 K [15]) temperatures of first-order PTs.

From equations of state (3) and (4), we arrive at a formula to calculate thermodynamic quantities near critical point (T_cr, H_cr). For example, having differentiated Eq. (3) with respect to pressure, we obtain a formula for compressibility,

$$ - \frac{{\partial \omega }}{{\partial P}} = K_{0}^{{ - 1}} - N{{s}^{2}}m\gamma \frac{{\partial m}}{{\partial P}},$$

(A.4)

and from Eq. (4) it follows that

$$m\frac{{\partial A}}{{\partial P}} + (A + 3B{{m}^{2}} + 5C{{m}^{4}})\frac{{\partial m}}{{\partial P}} = 0,$$

and

$$\frac{{\partial m}}{{\partial P}} = - 2s(s\, + \,1)m\frac{\gamma }{{3kT{{K}_{0}}}}{{(A\, + \,3B{{m}^{2}}\, + \,5C{{m}^{4}})}^{{ - 1}}}.$$

(A.5)

Then,

$${{K}^{{ - 1}}} \equiv - {{\left( {\frac{{\partial \omega }}{{\partial P}}} \right)}_{T}}\, = \,K_{0}^{{ - 1}}\left[ {1 + \frac{{{{s}^{3}}(s + 1){{\gamma }^{2}}}}{{2{{\mu }^{2}}{{K}_{0}}}}\chi } \right],$$

(A.6)

where

$$\chi = {{\left( {\frac{{\partial M}}{{\partial H}}} \right)}_{T}} = \frac{{8{{\mu }^{2}}N}}{{(3kT)(A + 3B{{m}^{2}} + 5C{{m}^{4}})}}$$

(A.7)

is the magnetic susceptibility at T = const (M = 2μsNm).

The thermal expansion coefficient is found in the same way:

$$\alpha = {{\left( {\frac{{\partial \omega }}{{\partial T}}} \right)}_{P}} \approx \frac{{ - 8\mu {{s}^{3}}\gamma mNH}}{{3k{{K}_{0}}{{T}^{2}}(A + 3B{{m}^{2}} + 5C{{m}^{4}})}}.$$

(A.8)