### Abstract:

In this dissertation, I develop some theoretical tools to interpret measurements of magnetization in rocks, sediments and soils. I show that the magnetization curve for an ensemble of superparamagnetic particles depends only on odd moments of the volume distribution $\rm(\langle V\rangle,\langle V\sp3\rangle,\...).$ As long as the ensemble is isotropic, the magnetic anisotropies of individual particles do not affect the curve. I derive analytical expressions for acquisition and loss of isothermal remanent magnetization in single-domain (SD) particles with uniaxial anisotropy. These curves depend only on the volume-average anisotropy. Plots of acquisition against loss of remanence can be used to distinguish uniaxial anisotropy from cubic anisotropy. I show that existing multi-domain (MD) hysteresis models, including the theory of Neel (1955) for thermoremanent magnetization, are internally inconsistent. I develop a simple self-consistent two-domain model and show that the slope of the hysteresis curve is always $1/N,$ where N is the demagnetizing factor for a two-domain particle.Using micromagnetic theory, I derive analytical expressions for the critical sizes $L\rm\sb{sw},$ the upper limit for SD hysteresis, and $L\rm\sb{n},$ the upper limit for stability of the SD remanent state. $L\rm\sb{sw}$ depends weakly on elongation and not at all on magnetocrystalline anisotropy, but$$L\sb{\rm n} = L\sb{\rm sw}\ \left({N\sb{\rm b}\over N\sb{\rm a}}\right)\sp{\sp{1/2}}\left(1-{2\kappa\over\mu\sb0M{\sbsp{\rm s}{2}}N\sb{\rm a}}\right)\sp{\sp{-1/2}}$$where $N\sb{\rm a} > N\sb{\rm b}$ are demagnetizing factors and $\kappa$ depends on the combined magnetocrystalline and magnetoelastic anisotropy. Mainly because of the the difference in $Ms,\ L\rm\sb{n}$ is orders of magnitude larger for a particle of $\rm Fe\sb{2.4}Ti\sb{0.6}O\sb4$ than for a particle of magnetite with the same aspect ratio.I develop a technique for eliminating unstable solutions of three-dimensional numerical micromagnetic models. I show that nucleation theory can be extended to non-ellipsoidal particles. The nucleation field $H\sb{\rm n}$ for a cuboid can be precisely located by a change in slope dM/dH and the appearance of curl in the magnetization. For a cube with $K\sb1 = 0,$ the plot of $H\sb{\rm n}$ against $1/L\sp2$ has the same slope as for a sphere, but the intercept is lower, reflecting a smaller average demagnetizing field. $H\sb{\rm n}$ is not affected by the demagnetizing field in the corners of the particle.