Disordered magnetics

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, pp. 106-107

Section 1.5.1.1. Disordered magnetics

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

1.5.1.1. Disordered magnetics

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A crystal placed in a magnetic field $[\bf H]$ is magnetized. The magnetized state is characterized by two vectors, the magnetization $[\bf M]$ (the magnetic moment per unit volume) and the magnetic induction $[\bf B]$ . The Gaussian system of units is used in this chapter (see Table 1.5.10.1 at the end of the chapter for a list of conversions from Gaussian to SI units). The magnetic induction is given by $[{\bf B} = {\bf H} +4\pi{\bf M}. \eqno(1.5.1.1)]$ This equation shows that the dimensions of B, H and M are the same in the Gaussian system. The unit for B, the gauss (G), and for H, the oersted (Oe), also coincide in magnitude, whereas the unit for M, usually called emu cm⁻³, is 4π times larger. These units are related to the corresponding SI units as follows: 1 G = 10⁻⁴ tesla (T), 1 Oe = 10³/(4π) A m⁻¹, 1 emu cm⁻³ = 10³ A m⁻¹.

In disordered magnetics, the vectors B and M are linear in the magnetic field. Using a Cartesian coordinate system, this can be expressed as $[M_{i} = \chi _{ij} H_{j}\; \hbox{ and } \;B_{i} = \mu _{ij}H_{j}, \eqno(1.5.1.2)]$ where $[\chi _{ij}]$ is the dimensionless magnetic susceptibility per unit volume and $[\mu _{ij}]$ is the magnetic permeability. The susceptibility is frequently referred to 1 g or to one mole of substance. The mass susceptibility is written as $[\chi _{{\rm g}}]$ , the molar susceptibility as $[\chi_{\rm {mol}}]$ .

All three vectors H, M and B are axial vectors (see Section 1.1.4.5.3 ), the symmetry of which is $[\infty / m]$ . Accordingly, the components of these vectors perpendicular to a mirror plane do not change sign on being reflected by this plane, whereas the components parallel to the plane do change sign. Consequently, these three vectors are invariant with respect to inversion. The quantities $[\chi _{ij}]$ and $[\mu _{ij}]$ are components of second-rank polar tensors. In principal axes, the tensors become diagonal and both the magnetic susceptibility and permeability of a crystal are characterized by the three values of the principal susceptibilities and principal permeabilities, respectively.

All disordered magnetics are divided into two types: diamagnets (χ < 0) and paramagnets (χ > 0).

Diamagnetism is a universal property of all materials. It is associated with the tendency of all the electrons to screen the applied external field according to the Lenz law. For materials in which the electron orbits are spherically symmetric, the relation for the diamagnetic susceptibility was calculated by Langevin. For monoatomic substances he obtained $[\chi = -{Ne^{2}\over{6mc^{2}}} \left(\sum \limits_{i=1}^{i=Z} \overline {r_{i}^{2}}\right), \eqno(1.5.1.3)]$ where N is the number of atoms per unit volume, Z is the number of electrons per atom, e and m are the charge and the mass of the electron, respectively, and $[\overline {r_{i}^{2}}]$ are the mean squares of the radii of the electron orbits. In polyatomic substances, the summation must be done over all types of atoms. In most chemical compounds, the orbits are not spherical and the calculation of the diamagnetic susceptibility becomes more complicated. In metals, the conduction electrons contribute significantly to the diamagnetic susceptibility. The diamagnetic susceptibility of most substances is very small ( $[\chi \sim 10^{-6}]$ ) and isotropic. Rare exceptions are bismuth and some organic compounds, in which the diamagnetism is strongly anisotropic.

Most paramagnetic materials contain ions (or free atoms) with a partly filled inner electronic shell. Examples are the transition metals and the rare-earth and actinide elements. Atoms, molecules and point defects possessing an odd number of electrons are also paramagnetic. Ions with a partly filled inner electronic shell possess orbital L and spin S angular momenta, which determine the total angular momentum J if the spin–orbit interaction is strong compared with the crystal field.

The magnetic susceptibility of paramagnets follows the Curie–Weiss law in low magnetic fields ( $[\mu_B B \ll k_B T]$ ): $[\chi = {{Np^{2} \mu_{B}^{2}}\over{3k_{B}(T-\Delta)}}, \eqno(1.5.1.4)]$ where N is the number of magnetic ions (or atoms) per cm³, $[\mu_{B}]$ is the Bohr magneton, p is the effective number of Bohr magnetons, $[{k_{B}}]$ is the Boltzmann factor and $[\Delta ]$ is the Weiss constant. The Weiss constant is related to the interaction between the magnetic moments (mostly exchange interaction) and to the effect of the splitting of electron levels of the paramagnetic ion in the crystalline electric field . Many paramagnets that obey the Curie–Weiss law transform into ordered magnetics at a temperature $[T_{c}]$ , which is of the order of $[|\Delta|]$ . The sign of $[\Delta]$ depends on the sign of the exchange constant J [see relation (1.5.1.7)]. For the substances that at low temperatures become ferromagnets, we have $[\Delta\,\gt\, 0]$ , for antiferromagnets $[\Delta \,\lt\, 0]$ , and for ferrimagnets the temperature dependence of $[\chi]$ is more complicated (see Fig. 1.5.1.1). For those paramagnets that do not go over into an ordered state, $[\Delta]$ is close to zero and equation (1.5.1.4) changes to the Curie law.

Figure 1.5.1.1 | top | pdf |

Temperature dependence of $[1/\chi]$ at high temperatures for different types of magnetics: (1) ferromagnet; (2) antiferromagnet; (3) ferrimagnet.

The value of the effective number of Bohr magnetons p depends strongly on the type of the magnetic ions and their environment. For most rare-earth compounds at room temperature, the number p has the same value as for free ions: $[p = g [J(J+1)]^{1/2}, \eqno(1.5.1.5)]$ where g is the Landé g-factor or the spectroscopic splitting factor ( $[1 \leq {g} \leq 2]$ ). In this case, the paramagnetic susceptibility is practically isotropic. Some anisotropy can arise from the anisotropy of the Weiss constant $[\Delta ]$ .

The behaviour of the transition-metal ions is very different. In contrast to the rare-earth ions, the electrons of the partly filled shell in transition metals interact strongly with the electric field of the crystal. As a result, their energy levels are split and the orbital moments can be `quenched'. This means that relation (1.5.1.5) transforms to $[p_{ij} = (g_{\rm eff})_{ij}[S(S+1)]^{1/2}. \eqno(1.5.1.6)]$ Here the value of the effective spin S represents the degeneration of the lowest electronic energy level produced by the splitting in the crystalline field; $[(g_{\rm eff})_{ij}]$ differs from the usual Landé g-factor. The values of its components lie between 0 and 10–20. The tensor $[(g_{\rm eff})_{ij}]$ becomes diagonal in the principal axes. According to relation (1.5.1.6), the magnetic susceptibility also becomes a tensor. The anisotropy of $[(g_{\rm eff})_{ij}]$ can be studied using electron paramagnetic resonance (EPR) techniques.

The Curie–Weiss law describes the behaviour of those paramagnets in which the magnetization results from the competition of two forces. One is connected with the reduction of the magnetic energy by orientation of the magnetic moments of ions in the applied magnetic field; the other arises from thermal fluctuations, which resist the tendency of the field to orient these moments. At low temperatures and in strong magnetic fields, the linear dependence of the magnetization versus magnetic field breaks down and the magnetization can be saturated in a sufficiently strong magnetic field. Most of the paramagnetic substances that obey the Curie–Weiss law ultimately transform to an ordered magnetic as the temperature is decreased.

The conduction electrons in metals possess paramagnetism in addition to diamagnetism. The paramagnetic susceptibility of the conduction electrons is small (of the same order of magnitude as the diamagnetic susceptibility) and does not depend on temperature. This is due to the fact that the conduction electrons are governed by the laws of Fermi–Dirac statistics.

References

International Tables for Crystallography (2006). Vol. D. ch. 1.5, pp. 106-107