Models for interacting superparamagnetic particles

The main mechanism for acquiring a record of the Earth's magnetic field is called thermoremanent magnetization (TRM). This refers to the magnetization that occurs when a rock containing magnetic minerals cools in a magnetic field. In the standard picture of TRM, a magnetic mineral cools below the Curie temperature, it first becomes superparamagnetic (SP). Its magnetic moment constantly fluctuates, and these thermal fluctuations occasionally drive the moment over energy barriers between minimum-energy states. Eventually it cools to a "blocking temperature" below which transitions no longer occur between minima, so the magnetic moment is frozen in to a particular state. The standard picture, known as the Néel model, strictly applies only to an isolated single-domain particle with at most two stable states to choose from. The acquisition of TRM becomes much more complex if the particles interact magnetostatically with each other. Calculation of the thermal relaxation rates is far more difficult than it is in the Néel model. Two widely used models that have been developed to simplify the calculations for weak interactions are the Dormann-Bessais-Fiorani (DBF) model and the Mørup and Tronc (MT) model. The DBF model assumes that each moment can be treated as slowly varying under the influence of a Boltzmann average of the fluctuations of the other moments. The MT model makes the complementary assumption that each moment can be treated as rapidly varying while the others are fixed. Both models are attempts to solve the same physical equations. A new method, using algorithms based on algebraic geometry, makes it possible to obtain a complete and accurate solution of these equations. It is shown that both approximations are incorrect, and a new first-order approximation is developed using matrix perturbation theory.