Superparamagnetic relaxation times for mixed anisotropy and high energy barriers with intermediate to high damping: 2. Uniaxial axis in a 〈111〉 direction

Abstract

[1] Superparamagnetic relaxation rates are calculated for ferromagnetic particles with mixed cubic and uniaxial anisotropy. In part 1 the uniaxial axis is in a h001i crystallographic direction, while in this article it is in a h111i crystallographic direction. When K u = 0, there are six remanent states but only one relaxation rate. As K u increases, the remanent states converge on the uniaxial axis, merging with it at K u = 0.76 K 0 1 for K 0 1 > 0 and at K u = 0.22 jK 0 1 j for K 0 1 < 0. In between the components parallel and perpendicular to the uniaxial axis relax at different rates. The rate for the perpendicular component increases with K u . If all the remanent states have the same energy, there is a single, decreasing rate for the parallel component. However, for some values of K u and K 0 1 there are two states with two different energies. There are then two rates, one decreasing and one increasing. For large K u the remanence is uniaxial. In this article and part 1 the relaxation rates have an exponential and a prefactor. The prefactors are calculated in the high energy barrier, intermediate-to high-damping approximation. For elongated particles the prefactor is smaller than predicted by the Néel-Brown theory for uniaxial particles. The predicted relaxation rates are the same order of magnitude as experimental estimates for maghemite and magnetite. Better agreement cannot be expected because order-of-magnitude uncertainties still exist in both the experimental and theoretical estimates.

Transition to superparamagnetism in chains of magnetosome crystals

“…The top curve is the critical size for a single crystal with cubic anisotropy and elongation along the crystallographic 〈111〉 axis. It is based on the relaxation calculations by Newell [2006b]. The subsequent curves are for increasing numbers of crystals.…”

“…Critical crystal lengths for a chain of six crystals and intercrystal gaps equal to 0 ( η = 0) and 0.6 L ( η = 0.6). Also shown are critical sizes for isolated crystals: a SP size based on Newell [2006b] and a MD size from Newell and Merrill [2000b]. The upper critical size is for a chain of three crystals [ Muxworthy and Williams , 2006].…”

“…For a crystal elongated in a 〈111〉 direction, Newell [2006b] showed that for sufficient elongation the energy barrier is effectively uniaxial with energy barrier ( K u ( d ) − K ′ 1 /12), where K u ( d ) represents the shape anisotropy and K ′ 1 is the cubic anisotropy parameter. The criterion for sufficient elongation is K u ( d ) > 0.2229∣ K ′ 1 ∣, which only requires a length‐to‐width ratio of 1.3.…”

“…Formally, the solution of this equation is where the exponential is interpreted as a matrix exponential [e.g., Horn and Johnson , 1994, p. 204]. Such solutions have multiple decay times, not all of which affect the moment of the chain [ Newell , 2006a, 2006b]. However, the point of Néel's theory is that, for sizes slightly larger than the critical size, the moment does not change significantly over the time scale of interest, t b , while for slightly smaller sizes the moment rapidly goes to zero.…”

Magnetic susceptibility measurements as a function of temeperature and freqeuncy I: inversion theory

“…The forward model describing the susceptibility of SP particles with non‐uniaxial anisotropy has never been solved explicitly. Superparamagnetic relaxation calculations for mixed uniaxial and cubic anisotropies show that multiple relaxation rates exist between different equilibrium states of the same particle (Newell 2006a,b). The practical consequence for susceptibility measurements is that a single particle might be characterized by multiple blocking conditions.…”