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The numerous phenomenological equations used in the study of the behaviour of single-domain magnetic nanoparticles are described and some issues clarified by means of qualitative comparison. To enable a quantitative

Enduring interest in colloidal dispersions of ferro- and ferrimagnetic nanoparticles is fueled by their applications. Colloids of small particles of iron oxide are used in magnetic resonance imaging (MRI) [

The LLG equation describes the movement of a magnetic dipole in the presence of a time-dependent magnetic field. Measurements on ferrofluids provide data on the magnetization, that is, the average over a large number of single-domain nanoparticles. Solutions of the LLG equation for various initial states are not sufficient to calculate the magnetization; the question of averaging is still left open. In this paper we report the solution of an equation of motion of the magnetization, which contains explicitly the torque driving the dipoles to the direction of the field. In itself, this torque leads to an exponential approach to the direction of a static external field, the Debye relaxation. Debye has studied the movement of electric dipoles carried by molecules [

In the next section, we give an overview of the various equations of movement used for the description of the simultaneous effect of external torques and relaxation, showing the place of the modified Bloch equation among them. En route, we give the shortest derivation of the Landau-Lifshitz-Gilbert [

The behaviour of single-domain ferro- or ferrimagnetic nanoparticles in an external magnetic field ^{2}/Js.

The vector product in (

A comparison of the Landau-Lifshitz and Gilbert equations reveals that

Shliomis [

The Langevin function,

Clearly, the parameter

Recently, Cantillon-Murphy et al. [

Equation (

(colour online) The characteristic change of the equilibrium magnetization within a periodic driving cycle as given by the Langevin function (

The dependence of the discrepancy between the two curves on the amplitude of the AC field is shown in Figure

(colour online) Equilibrium magnetization

In Sections

In this section we give the analytical solution of (

Here

The basic functional dependence of

(colour online) The dependence of

The basic functional dependence of

The specific absorption rate (SAR) defined as energy loss per second and per kg of the colloid is the gauge of energy losses relevant to applications:

This is the figure of merit for colloids to be used in hyperthermia. In this application the weak magnetic fields are quite weak (

(colour online) The dependence of

To find the equation of motion for the magnetization subjected to a magnetic field oscillating along the

The last equation in (

Keeping only the

The effective magnetization of (

In practice, the summation in (

(colour online) Convergence of the Taylor series expansion of

We have calculated the energy loss per cycle, using the formula

(colour online) Energy loss as a function of frequency

(colour online) Specific absorption rate as a function of frequency

At high driving field amplitudes

In Section

The analytic solutions of the modified Bloch-Bloembergen equation for circular and linear polarization, presented in Sections

The frequency dependence of the energy dissipation is shown in Figure

To illustrate the difference between circular and linear susceptibilities mentioned in Section

The dependence of

For linear polarization, ^{2}/Js.

As to the relative merits of linear and circular polarization, we come to the conclusion followed from our work in the Landau-Lifshitz-Gilbert frame [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors wish to express special thanks to I. Nándori and to colleagues in the lab for their valuable critical advices and continuous support. The authors acknowledge support from the Hungarian Scientific Research Fund (OTKA) no. 101329. This work was supported by the TAMOP 4.2.2.A-11/1/KONV-2012-0036 project, which is cofinanced by the European Union and European Social Fund.