For several decades the formation of different kinds of superstructures in solids
has been a topical issue in condensed matter physics. The superstructures (or spatially
modulated structures) may be of a different nature: magnetic patterns like spin-density
waves, inhomogeneous charge distributions in charge-ordered compounds, dipolar and
quadrupolar ordering in ferroelectrics or ferroelastics, regular lattice distortions and
related orbital structures, stripe-like arrangements of dopants in alloys, etc. The phase
diagrams of such compounds can be rather complicated involving a large number of
phases with non-trivial types of ordering. Fortunately, all this wealth of seemingly
unrelated phenomena can be often described by rather simple models with a due
account taken of a competitive character of the most important interactions.
In this thesis I will investigate the Ising model in 2-D with nearest and next-nearest
neighbour interactions using several methods including exact diagonalisation
of small clusters, transfer matrix technique and Monte Carlo simulation of large
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of the degree of Doctor of Philosophy at Loughborough University.