... Gibbs^1.1

In Ref. [20] this free energy is called Landau free energy $G_L$ to distinguish from the Gibbs free energy $G$. Here we perform an abuse of notation.

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... stationary^1.2

It can be shown that metastable equilibria are minima of the free energy (1.17). In this sense the minimization of the free energy (1.17) generalizes the minimization of the Gibbs free energy that holds in equilibrium thermodynamics.

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... equilibrium^1.3

although the whole body is not in equilibrium, one assumes that each elementary volume is in equilibrium

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... magnitude^1.4

We will discuss this aspect in section 1.3.5

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... meanings^2.1

The term shape anisotropy recalls the fact that magnetostatic field depends on the geometry of the body, whereas the crystalline anisotropy depends on the lattice structure of the material.

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... components^2.2

The external field is supposed to be $\textbf{h}_$a$=-h_{ax}\mathbf{e}_x+h_{ay}\mathbf{e}_y$, with $h_{ax}>0$.

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... values^2.3

We recall that the applied field is expressed here as $\textbf{h}_$a$=-h_{ax}\mathbf{e}_x+h_{ay}\mathbf{e}_y$.

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... (see^2.4

Notice that here the role of easy axis $z$ is played by the $x$ axis.

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... functions^2.5

See Appendix B.

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... energy^2.6

We notice that a fixed point is a degenerate periodic trajectory.

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... charges^3.1

Recalling the Coulomb approach to magnetic materials, which is in term of equivalent volume charges $\rho_m=-\nabla\cdot\textbf{M}$ and surface charges $\sigma_m=\textbf{M}\cdot\textbf{n}$. See Ref. [13] for details.

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... thin-films^3.2

The S-state was obtained by first saturating the thin-film along the $[1,1,1]$ direction and then by switching the external field off.

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... elliptic^3.3

See Appendix B

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... media^3.4

Since the medium ``sees'' the external field as applied at 135$^\circ$ off the easy axis, this can be understood by looking at the Stoner-Wohlfarth astroid (see Fig. 3.16) in the direction at 135$^\circ$ off the $x$ axis, where the critical field is about $h_{SW}/2$.

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... field^3.5

This can be inferred by observing that directions at about 135$^\circ$ off the $x$ axis intersect the SW astroid at almost the same distance from the origin. This leads to very close values of critical fields.

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... dynamics^3.6

We have already seen an example of switching below SW limit, namely the precessional switching analyzed in section 2.4.3.

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... conservative^4.1

We have seen examples of this in chapter 2, for uniformly magnetized particles.

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