Linear dynamic susceptibilities

The other components, namely, b f and b j, may be constructed straightforwardly using their relations with the given ones [see Eqs. (4.192)]. For a random system, that is, for an assembly of noninteracting particles with a chaotic distribution of the anisotropy axes, the average of any Legendre polynomial is zero, so that b[1> = b, and the linear dynamic susceptibility reduces to... 

Figure 4.21. Real (a) and imaginary (b) parts of the linear dynamic susceptibility at fho = 10-4 for the bias field = 0.1 (1), 0.2 (2), 0.3 (3), 0.5 (4). For figure (b) the curve with = 0.5 does not resolve. Circles show the result of the effective time approximation.

Let us first analyze the effect of the bias field on the magnetic SR in the framework of the linear response theory formulated for SR [96,97,99]. The main idea of the linear response treatment is a direct use of the fluctuation-dissipation theorem, which expresses the thermal (fluctuational) power spectrum Qn(a>) of the magnetic moment of the system, namely, magnetic noise, through the imaginary component of its linear dynamic susceptibility ImX = to a weak probing ac field of an arbitrary frequency co as... 

Because of their high sensitivity, fluorescence detectors are particularly useful in trace analysis when either tire sample size is small or the analyte concentration is extremely low. Although fluorescence detectors can become markedly nonlinear at concentrations where absorption detectors are still linear in response, their linear dynamic range is more than adequate for most trace analysis applications. Unfortunately, fluorometric detectors are often susceptible to background fluorescence and quenching effects that can plague all fluorescence measurements. 

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.4 we have shown that the quadratic dynamic susceptibilities of a superparamagnetic system display temperature maxima that are sharper than those of the linear ones. If the maximum occurs as well at the temperature dependence of the signal-to-noise ratio, this should be called the nonlinear stochastic resonance. However, before discussing this phenomenon, one has to define what should be taken as the signal-to-noise ratio in a nonlinear case. 

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... 

The equation for Qfv is derived in Section V.C together with the scheme of its solution down to the third order in From the solutions obtained, the terms yielding the linear and cubic low-frequency responses to the probing magnetic field H(t) = H cos cot are extracted. In terms of linear and cubic susceptibilities those quantities evaluated numerically are compared in Figures 4.31 and 4.32 primes and double primes there denote, as usual, the in-phase and out-of-phase components of the dynamic susceptibilities. 

Figure 4.32. Linear dynamic magnetic susceptibilities of a randomly oriented super-paramagnetic assembly (a, b) and of a magnetic fluid of the same particles (c, d) for all the graphs the ratio TdAb = 10-4. Vertical axes for % are scaled in the units of cp2/r so that at to —> 0 both % tend to 1/3.

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