content_image/331492.jpg content_image/331494.jpg content_image/331496.jpg content_image/331500.jpg
<img src='content_image/331492.jpg'>

where we introduced the combinations of dipole sums Σ$_{(}$$_{±}$$_{)}$( ⃗ k ) 

<img src='content_image/331494.jpg'>

These sums are plotted in Fig. 1 as a function of ⃗ k for symmetrical directions of the reciprocal lattice. The field, H eff a = H$_{a}$ − (3 / 2) Mσ (0) is the z -projection of the effective anisotropy field, comprising an anisotropy field for the single particle and demagnetization field of the array as a whole, and σ (0) (3 m$_{0}$ /2 a $^{3}$) ∼ = 1 . 08 · (4 πM ). 

## A. Dispersion relation and stability conditions. 

<img src='content_image/331496.jpg'>

It is remarked that, the expression for the frequency can be rewritten in the universal form 

<img src='content_image/331500.jpg'>

