Brown s equation

The use of Brown s equation (logiQ kjkf, = p+cr+) with electrophilic substitutions in general has been fully discussed and reference will be made later to its treatment of particular substituents in nitrations. 

III. Solution of Brown s Equation Using Spherical Harmonics... 

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 III we expand the density of orientations in Brown s equation in spherical harmonics to write that equation as a set of differential-difference equations. The following problems are reduced to the solution of a set of differential-difference equations. 

Section VII is concerned with the theory of dispersion of the magnetic susceptibility of fine ferromagnetic particles. Brown s equation, including... 

The appendices contain an account of those parts of the theory of Brownian motion and linear response theory which are essential for the reader in order to achieve an understanding of relaxational phenomena in magnetic domains and in ferrofluid particles. The analogy with dielectric relaxation is emphasized throughout these appendices. Appendix D contains the rigorous derivation of Brown s equation. 

III. SOLUTION OF BROWN S EQUATION USING SPHERICAL HARMONICS... 

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. 

This leads to an infinite set of differential-difference equations, on substitution into Brown s equation. The calculation of the relaxation times is then achieved by selecting the relevant indices n and m, from which we obtain a differential-difference equation describing the time evolution of the average value of the desired spherical harmonic. Those spherical harmonics which are of interest to us here are P,(cos ), with n = l and m = 0, which describes the evolution of the alignment with the 2 axis and Pj(cos i )cos , with n = 1 and m = l which describes alignment perpendicular to the z axis. 

We first outline the approach of the following sections. Starting with Brown s equation we expand the distribution W(r, t) of orientations of M in spherical harmonics as in the previous chapter whence we obtain an infinite set of differential-difference equations. We then select the spherical harmonic of interest P for the longitudinal relaxation and P e" for the transverse relaxation. The relaxation times can then be expressed as in the previous section in terms of a continued fraction. Since we consider only the response with respect to a small applied field H, such that 1, it is only necessary to evaluate quantities linear in Furthermore we assume that equilibrium has been attained for the ratio R s) in the continued fraction and that this ratio is expressible in terms of the equilibrium values of the relevant spherical harmonics. This expression is achieved as follows. The average value of any spherical harmonic is... 

Our starting point in this section is again Brown s equation,... 

The first analytic expression we will obtain is for a small longitudinal DC field H, which is suddenly applied to a system which is in equilibrium in a larger DC field H. We begin with Brown s equation, which as we have seen when there is no azimuthal or dependence, reduces to... 

The spherical harmonic analysis so far presented for uniaxial anisotropy is mainly concerned with the relaxation in a direction parallel to the easy axis of the uniaxial anisotropy. We have not considered in detail the behavior resulting from the transverse application of an external field and the relaxation in that direction for uniaxial anisotropy. Thus we have only considered potentials of the form V(r, t) = V(i, t) where the azimuthal or dependence in Brown s equation is irrelevant to the calculation of the relaxation times. This has simplified the reduction of that equation to a set of differential-difference equations. In this section we consider the reduction when the azimuthal dependence is included. This is of importance in the transition of the system from magnetic relaxation to ferromagnetic resonance. The original study [17] was made using the method of separation of variables on Brown s equation which reduced the solution to an eigenvalue problem. We reconsider the solution by casting... 

W = W(i7, t) as before. We assume a solution of Brown s equation of the form given in Section III, namely... 

C. The Longitudinal Susceptibility t (") Brown s equation for a weak field superimposed on a large field applied along the z or easy anisotropy axis is ... 

We reiterate that the analysis given above (quite apart from the question of identification of the largest relaxation time of Brown s equation with that of the magnetization, in the Neel limit, a justification for which is provided by the numerical calculations presented in [17]), makes the assumption that the Debye and Neel processes may be treated independently. 

The multiplicative noise problem in several dimensions will presently arise in the rigorous derivation of Brown s equation. It has already arisen in the main text in connection with the calculation of the relaxation times. 

This however will not affect Brown s equation which by writing... 

This is Brown s equation in the form it appeared in his 1963 paper [8],... 

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