Dipole-Dipole Coupling and Other Interaction Effects in Polar Dielectrics and Magnetic Relaxation of Single Domain Ferromagnetic Nanoparticles

Abstract

The main purpose of this thesis is to provide the precise details of the very onerous calculations underlying our two published papers [1] and [2]. Thus, the latter can be better understood and utilised later for future research as the details form an invaluable archive which for space reasons could not be published in the journals. These details which are essential for understanding the effect of two body interactions are provided in the thesis via my comprehensive appendices written in Chapters 5 and 7. In addition, through the use of Wolfram Mathematica, the results are corroborated by calculating the complex electrical susceptibility in Figures 2-4 of [1] as well as the integral relaxation time and complex magnetic susceptibility in Figures 2-4 of [2]. The published papers may be summarised as follows: In the Budó paper [1] a fractional Smoluchowski equation for the orientational distribution of dipoles incorporating two body interactions with continuous time random walk Ansatz for the collision term is obtained for polar molecules. This equation is written via the non-inertial Langevin equations for the evolution of the relevant Eulerian angles and their associated Smoluchowski equation for the orientational probability distribution function. These equations govern the normal rotational diffusion of an assembly of noninteracting dipolar molecules with similar internal interacting polar groups hindering their rotation owing to their mutual potential energy. The resulting fractional Smoluchowski equation is then explicitly solved in the frequency domain using scalar continued fractions yielding the linear dielectric response as a function of the fractional parameter for extensive ranges of the interaction parameter and friction ratios. Thus, the main result is that Budó s treatment can possibly be extended to disordered materials. In the magnetic paper [2], the magnetisation response including dipole-dipole interactions of a pair of macrospins (single-domain ferromagnetic particles) following the sudden alteration of a dc magnetic field is calculated from the stochastic Landau-Lifshitz-Gilbert equation for the magnetisation by reducing the overall task to an infinite hierarchy of differential-recurrence relations in the time domain for the statistical moments (averaged products of spherical harmonics in this case). This is exactly solved in the frequency domain by matrix continued fractions. The greatest relaxation time and dynamic susceptibility are then compared with the corresponding results for two exchange-coupled spins using the same exact method. I believe that this is effectively the first exact treatment of dipole-dipole effects in the relaxation of macrospins. Generally, both the dielectric and magnetic calculations are essential as a starting point for the understanding of the effects of two body interactions on relaxation processes.

References [1] S. V. Titov, W. T. Coffey, M. Zarifakis, Y. P. Kalmykov, M. H. Al Bayyari, and W. J. Dowling, "Generalization to anomalous diffusion of Budó s treatment of polar molecules containing interacting rotating groups," J. Chem. Phys., 153, 4, 044128, 2020. [2] Y. P. Kalmykov, S. V. Titov, D. J. Byrne, W. T. Coffey, M. Zarifakis, and M. H. Al Bayyari, "Dipole-dipole and exchange interaction effects on the magnetization relaxation of two macrospins: Compared," J. Magn. Magn. Mater., 507, 166814, 2020.