Cubic dynamic susceptibilities

III. Low-Frequency Nonlinear Susceptibilities of Superparamagnetic Particles in Solid Matrices A. Linear and Cubic Dynamic Susceptibilities Numerical Solutions... 

A. Linear and Cubic Dynamic Susceptibilities Numerical Solutions... 

In the case of isotropic magnetic particles, that is, TJ = -p(e H), both linear and cubic dynamic susceptibilities may be obtained analytically. To show this, we first transform Eq. (4.90) into an infinite set of differential recurrence relations ... 

To summarize this part of the chapter, we have constructed a consistent theory of linear and cubic dynamic susceptibilities of a noninteracting superparamagnetic system with uniaxial particle anisotropy. The scheme developed was specified for consideration of the assemblies with random axis distribution but may be easily extended for any other type of the orientational order imposed on the particle anisotropy axes. A proposed simple approximation is shown to be capable of successful replacement of the results of numerical calculations. 

The theory was tested with the aid of an ample data array on low-frequency magnetic spectra of solid Co-Cu nanoparticle systems. In doing so, we combined it with the two most popular volume distribution functions. When the linear and cubic dynamic susceptibilities are taken into account simultaneously, the fitting procedure yields a unique set of magnetic and statistical parameters and enables us to conclude the best appropriate form of the model distribution function (histogram). For the case under study it is the lognormal distribution. 

In Section IV.B a procedure of numerical solution for Eq. (4.329) is described and enables us to obtain the linear and cubic dynamic susceptibilities for a solid system of uniaxial fine particles. Then, with allowance for the polydispersity of real samples, the model is applied for interpreting the magnetodynamic measurements done on Co-Cu composites [64], and a fairly good agreement is demonstrated. In our work we have proposed for the low-frequency cubic susceptibility of a randomly oriented particle assembly an interpolation (appropriate in the whole temperature range) formula... 

Raikher YL, Stepanov VI (1997) Linear and cubic dynamic susceptibilities of superparamagnetic fine particles. Phys Rev B 55 15005-15017... 

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