Gilbert’s equation

Derivation of the Explicit Form of Gilbert s Equation Ferrofluids... 

V. Gilbert s Equation as Modified to Include the Inertia of a Ferrofluid Particle... 

VI. The Calculation of Relaxation Times for a Single-Domain Ferromagnetic Particle from Gilbert s Equation... 

We may summarize the contents of this chapter in more detail as follows. In Section I we demonstrate how the explicit form of Gilbert s equation describing Neel relaxation may be written down from the gyromagnetic equation and how, in the limit of low damping, this becomes the Landau-Lifshitz equation. Next the application of this equation to ferrofluid relaxation is discussed together with the analogy to dielectric relaxation. 

In Section II it is shown how the effects of thermal agitation may be included in Gilbert s equation and how the Fokker-Planck equation for the density of orientations of the magnetic moments on the unit sphere may be written down in an intuitive manner from Gilbert s equation. (The rigorous derivation of the Fokker-Planck equation from Gilbert s equation is given in Appendix D). We coin the term Brown s equation for this particular form of the Fokker-Planck equation. 

In Section V we are concerned with Gilbert s equation as applied to the Debye relaxation of a ferrofluid particle with the inertia of the particle included. It is shown, by averaging Gilbert s equation for Debye relaxation corrected for inertia and proceeding to the noninertial limit, how analytic expressions for the transverse and longitudinal relaxation times for Debye relaxation may be obtained directly from that equation thus bypassing the Fokker-Planck equation entirely. These expressions coincide with the previous results of the group of Shliomis [16]. 

This is Gilbert s equation. It is implicit in dMIdt. It is possible, for small values of the dimensionless damping factor, a, to express it explicitly in M by iterating, as follows ... 

This explicit form of Gilbert s equation for the case of low damping is of the same form as the previous Landau-Lifshitz equation. The neglect of the terms 0(17 ) and higher corresponds to the assumption of Landau and Lifshitz that that is, of small damping. This correspondence... 

This is the form of Gilbert s equation used, though not stated, by Brown in [8]. It is essentially of the same form as the Landau-Lifshitz equation except that both prefixes g and h depend on the damping level. 

In Section II, Gilbert s equation describing the Neel relaxation is augmented by a random field term, representing thermal fluctuations. The underlying Fokker-Planck equation is then constructed from this augmented equation. The time constant in this equation is the Neel relaxation time... 

The last term in Gilbert s equation above is the aligning term and consequently the term of interest in relaxation of the magnetization with respect to a magnetic field. For low damping (a< l), we see that precessional motion is the dominant motion and that the relaxation time is approximately given by... 

In this section we summarize the approach used by previous authors [8, 16-19] to find expressions for the relaxation times of single domain ferromagnetic and ferrofluid particles. We begin with the Fokker-Planck equation obtained from Gilbert s equation, in spherical polar coordinates, augmented by a random field term, that is, with Brown s equation. We then expand the probability density of orientations of M, that is. 

Here we note that the complex quantity resulting from dWld describes the oscillation of the x and y components of the magnetization on the x-y plane that is, the gyromagnetic precession. The characteristic frequency of this is apparent from the explicit form of Gilbert s equation and is obtainable from the above by writing... 

V. GILBERT S EQUATION AS MODIFIED TO INCLUDE THE INERTIA OF A FERROFLUID PARTICLE... 

In order to discuss the Langevin equation for a single domain ferrofluid particle we first consider Gilbert s equation for the dynamic behavior of the particle s magnetization vector M in the presence of thermal agitation, which is Eq. (1.12),... 

VI. THE CALCULATION OF RELAXATION TIMES FOR A SINGLE-DOMAIN FERROMAGNETIC PARTICLE FROM GILBERT S EQUATION... 

The method we describe is analogous to the one used by us [52] in order to study dielectric relaxation of polar fluids in the presence of a DC electric field using the Langevin equation. Our calculations are carried out by interpreting the Cartesian components of Gilbert s equation as a set of stochastic nonlinear differential equations of Stratonovich type [12]. 

As we already know Gilbert s equation, (6.1) can be transformed to the equivalent Landau-Lifshitz form... 

We have based our analysis throughout on Gilbert s equation. Raikher and Shliomis [17] based their analysis of this problem on the Landau-Lifshitz equation which has a different direction of precession convention. This results in a dimensionless damping factor a of opposite sign. If we bear this in mind and make use of the low cr approximations of Eq. (6.78) we see that the above equations coincide precisely with those of Raikher and Shliomis s equation (33) [17] for small cr, when we neglect all terms of order and higher. 

We have shown how to obtain the drift and diffusion coefficients for simple additive noise. A difference arises in the case of multiplicative noise, which we encounter in Gilbert s equation as augmented by a random noise field. In that case the system is governed by a Langevin equation of the form (we take the one-dimensional case for simplicity)... 

Gilbert s equation when augmented by a random field term as the Gilbert-Langevin equation is... 

Equations (D.6) and (D.7) are Gilbert s equation in spherical polar coordinates. To obtain the Gilbert-Langevin equation in such coordinates we augment the field components //, and with random field terms and h. By graphical comparison of the Cartesian and spherical, polar coordinate systems, we find these to be... 

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