On the determination of magnetic grain-size distributions of superparamagnetic particle ensembles using the frequency dependence of susceptibility at different temperatures

Abstract

S U M M A R YMagnetic grain-size and coercivity distributions of a superparamagnetic (SP) particle ensemble together determine its frequency dependence of susceptibility (FDS). Investigating the mathematical theory of this dependence leads to a general dispersion relation between real and imaginary parts of the complex susceptibility for SP particle ensembles, which extends the previous treatment by Néel. Using the new theory, it is demonstrated that the inverse problem of determining the combined grain-size and coercivity distribution from FDS measurements is not uniquely solvable. The inversion of the FDS at one temperature can be described by a deconvolution integral, the kernel of which is analytically calculated. The deconvolved FDS corresponds to an energy barrier distribution. Only using a priori assumptions about the relation between particle volume and coercivity it can be interpreted in terms of a volume or grain-size distribution. In order to deconvolve natural rock measurements, a semi-analytical parametric deconvolution method has been developed, which allows to reconstruct the SP grain-size distribution even from relatively noisy data. Dense measurements of FDS at several temperatures can be used to check for the applicability of the above theory. Observed deviations can be interpreted in terms of magnetostatic particle interaction. A quantitative estimate is presented, which allows to determine the average interaction field together with the volume distribution.

Characterizing the superparamagnetic grain distribution f(V, Hk) by thermal fluctuation tomography

“…In both cases the strong nonlinearity at all but the lowest temperatures undermines the analysis. (Table 2) in previous studies [Schlinger et al, 1991;Worm and Jackson, 1999;Egli and Lowrie, 2002;Shcherbakov and Fabian, 2005]. The disparity is undoubtedly due in part to the diminished slope at higher temperatures, related to a separate population of larger grains.…”

“…For example, Dunlop [1965] calculated f(V, H K0 ) from AF demagnetization of a set of weak field pTRMs, and then used it to predict the thermal demagnetization spectra of pTRMs acquired in different DC fields. Here we use the grain distributions obtained from thermal fluctuation tomography to model the frequency and temperature dependence of susceptibility, k(f, T), using Néel theory [Néel, 1949;see also Worm and Jackson, 1999;Shcherbakov and Fabian, 2005].…”

“…Thermomagnetic analyses indicate that these grains are lowtitanium titanomagnetites with a composition of Fe 3-x Ti x O 4 (x $ 0.10 or TM10). Careful transmission electron microscopy (TEM) studies [Schlinger et al, 1991] and numerous magnetic investigations [Schlinger et al, 1991;Rosenbaum, 1993;Worm and Jackson, 1999;Egli and Lowrie, 2002;Shcherbakov and Fabian, 2005] have shown that at each stratigraphic level the distribution of grain sizes and shapes is very narrow, and that it closely approximates a StonerWohlfarth population of identical, randomly oriented elongate grains. Sizes increase from approximately 5 Â 5 Â 15 nm (4.5 Â 10À25 m 3 ) to 25 Â 25 Â 250 nm (1.6 Â 10 À22 m 3 ) over a 3-m span near the base of the flow (Table 2) [Schlinger et al, 1991].…”

“…This requires us to make the assumption that the distribution of coercivities is much narrower than that of grain size (as pointed out by Dunlop [1965]), since the unblocking temperature (the SSD-SP transition in zero field) depends on the product of grain volume and coercivity (see section 2 for details). Similarly Shcherbakov and Fabian [2005] have shown that temperature-and frequencydependent low-field susceptibilities can be mapped into a volume distribution only when some relationship between grain volume and coercivity is specified a priori.…”

Inferred time- and temperature-dependent cation ordering in natural titanomagnetites

“…It is a basic assumption of this theory that T C is a material constant, depending only on chemical composition and crystal structure of the remanence-carrying minerals, and that the function M S (T) is a single-valued and invariant material property, governing not only magnetization intensities but also the scaling of temperature-dependent anisotropies and energy-barrier distributions [44][45][46][47][48][49][50][51][52][53] . When the T C is itself a function of thermal history, and when M S (T) depends on the time-and temperaturedependent degree of cation ordering, additional complexity is required in quantitative modelling of remanence blocking and unblocking.…”