2.1 The uniform mode approximation

In many technological applications, where the size of the magnetic media has reached the nanometric scale, it is reasonable to assume that the exchange interaction is prevalent with respect to the others and, therefore, that the particle tends to be uniformly magnetized. In other words, the uniform mode is energy-favored with respect to disuniformities as soon as the characteristic dimension of the body is comparable or even smaller than the exchange length. In this framework, it does make sense to neglect non-uniform modes and consider the particle as uniformly magnetized. This has considerable simplifications as far as the mathematical model is concerned, but nevertheless the uniform mode analysis can give, in certain applications, very interesting analytical indications and, in some cases, the predictions are also very accurate with respect to non-uniform micromagnetic simulations, as we will see in the following chapter. Last but not least, the uniform mode analysis has been used for long time in the design of magnetic recording devices. In our analysis we will use quite extensively the tools provided by dynamical systems theory [43], since in the case of single domain particle we deal with low dimensional systems (2D and 3D).