Uniaxial ferromagnet

Borovik-Romanov, A. S.; Grimmer, H.

doi:10.1107/97809553602060000632

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.5, p. 123

Section 1.5.3.3.1. Uniaxial ferromagnet

A. S. Borovik-Romanov^a ^‡ and H. Grimmer^b ^*

^a P. L. Kapitza Institute for Physical Problems, Russian Academy of Sciences, Kosygin Street 2, 119334 Moscow, Russia, and ^bLabor für Neutronenstreuung, ETH Zurich, and Paul Scherrer Institute, CH-5234 Villigen PSI, Switzerland
Correspondence e-mail: hans.grimmer@psi.ch

1.5.3.3.1. Uniaxial ferromagnet

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The temperature of transition from the paramagnetic to the ferromagnetic state is called the Curie temperature. The thermodynamic treatment of the behaviour of uniaxial ferromagnets in the neighbourhood of the Curie temperature [T_c] is given below.

In the case of a ferromagnet $[({\bf L}=0)]$ , the thermodynamic potential (1.5.3.27) near [T_c] including the magnetic energy $[-{\bf MH}]$ is given by (see 1.5.3.25) $[\tilde{\Phi} = \Phi_0 + (B/2){\bf M}^2 + (b/2)(M_x^2 + M_y^2) + (C/4){\bf M}^4 - {\bf MH}, \eqno(1.5.3.29)]$ where $[\tilde{\Phi}]$ is used to designate the thermodynamic potential in variables $[p, T, {\bf H}]$ [instead of $[\Phi(p, T, {\bf M})]$ ]; at the given field, $[\tilde{\Phi}]$ should be a minimum. The equilibrium value of the magnetization $[{\bf M}]$ is found by minimizing the thermodynamic potential.

First consider the ferromagnet in the absence of the external field $[({\bf H} = 0)]$ . The system of equations $[{\partial}\tilde{\Phi}/{\partial}{\bf M}=0]$ has three solutions: $[\displaylines{{\rm (I)} \hfill M_x = M_y = M_z = 0 \hfill(1.5.3.30)\cr {\rm (II)} \hfill M_z = 0;\quad M_x^2 + M_y^2 = M_{\perp}^2 = -{{B + b}\over{C}}\hfill(1.5.3.31)\cr {\rm (III)} \hfill M_x = M_y = 0;\quad M_z^2 = -{{B}\over{C}}. \hfill(1.5.3.32)}%fd1.5.3.32]$

In the whole range of temperatures $[T\,\gt\, T_c]$ when $[B\,\gt\, 0]$ , the minimum of the potential is determined by solution (I) (i.e. absence of a spontaneous magnetization). The realization of the second or third state depends on the sign of the coefficient b. If $[b\,\gt\,0]$ , then the third state is realized, the magnetization $[{\bf M}]$ being directed along the axis. In this case, the transition from the paramagnetic into the ferromagnetic state will take place at [T_c = T_0] (when [B=0] ). If $[b \,\lt\, 0]$ , the magnetization is directed perpendicular to the axis. In this case, the Curie temperature is $[T_c = T_0 - b/\lambda]$ (when [B+b = 0] ). In the absence of a magnetic field, the difference between the two values of [T_c] has no physical meaning, since it only means another value of the coefficient B [see (1.5.3.25)]. In a magnetic field, both temperatures may be determined experimentally, i.e. when B becomes zero and when [B + b] becomes zero.

If a magnetic field $[{\bf H}]$ is applied parallel to the z axis and $[b\,\gt\,0]$ , the minimization of the thermodynamic potential $[\Phi]$ leads to $[H/M = CM^2 + B. \eqno(1.5.3.33)]$ This relation has been verified in many experiments and the corresponding graphical representations are known in the literature as Arrott–Belov–Kouvel plots (see Kouvel & Fisher, 1964). Putting $[B=\lambda(T-T_c)]$ according to (1.5.3.14), equations (1.5.3.32) and (1.5.3.33) may be used to derive expressions for the initial magnetic susceptibilities (for $[H \rightarrow 0]$ ): $[\eqalignno{ \chi_0 &= {{1}\over{2\lambda (T_c - T)^{\gamma}}}, \quad T \,\lt\, T_c, &(1.5.3.34)\cr \chi_0 &= {{1}\over{ \lambda (T - T_c)^{\gamma}}}, \quad \,\,\, T\,\gt\, T_c, &(1.5.3.35)}%fd1.5.3.35]$ where $[\gamma = 1]$ .

The Landau theory of phase transitions does not take account of fluctuations of the order parameter . It gives qualitative predictions of all the possible magnetic structures that are allowed for a given crystal if it undergoes a second-order transition . The theory also explains which of the coefficients in the expression for the thermodynamic potential is responsible for the corresponding magnetic structure. It describes also quantitative relations for the magnetic properties of the material if $[1 \gg (T-T_c)/T_c \gg T_cB^2/b{\alpha ^3}, \eqno(1.5.3.36)]$ where $[\alpha]$ is the coefficient in the term which describes the gradient energy. In this chapter, we shall not discuss the behaviour of the material in the fluctuation region. It should be pointed out that, in this region, $[\gamma]$ in relations (1.5.3.34) and (1.5.3.35) depends on the dimensionality of the structure n and equals 1.24 for [n=1] , 1.31 for [n=2] and 1.39 for [n=3] . Similar considerations are relevant to the relations (1.5.3.31) and (1.5.3.32), which describe the temperature dependence of spontaneous magnetization.

The relations (1.5.3.31) and (1.5.3.32) describe the behaviour of the ferromagnet in the `saturated' state when the applied magnetic field is strong enough to destroy the domain structure. The problem of the domains will be discussed later (see Section 1.5.4).

The transition from the paramagnetic to the ferromagnetic state is a second-order transition , provided that there is no magnetic field. In the presence of a magnetic field that is parallel to the easy axis of magnetization, the magnetic symmetry of the crystal is the same ( $[M_z \neq 0]$ ) both above and below [T_c] . From the point of view of symmetry, no transition occurs in this case.

References

Kouvel, J. S. & Fisher, M. E. (1964). Detailed magnetic behavior of nickel near its Curie point. Phys. Rev. A, 136, 1626–1632.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 1.5, p. 123