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Condition of resonance and angular dependence of H_res

In polar coordinate, H. Suhl [H. Suhl, Phys. Rev. 97, 555 (1954)]derived the following expression for the condition of resonance:

$\omega_{\mathrm{res}}=\frac{\gamma}{M \mathrm{sin}\theta} \ \left(\frac{\partial^2 E}{\partial \theta^2} \times \frac{\partial^2 E}{\partial \phi^2} \ - \ \left(\frac{\partial^2E}{\partial \theta \partial \phi} \right)^2\right)^{1/2}$

The total magnetic energy is composed of:

The Zeeman energy $E_z=-\vec{M} \bullet \vec{H}$

The uniaxial anisotropy energy inclusing term of the first and second order.

The shape anisotropy $E_s=-\vec{M} \bullet \vec{N}$

We can simplify the expression of the energy for a polycrystalline, where all directions of the applied magnetic field in the film plane are equivalente. With $\theta_H = \frac{\pi}{2}$ , the total magnetic energy density is [Chappert, Phys. Rev. B 34, 3192 (1986)]:

$E\ = \ -M H \mathrm{sin}\theta \mathrm{cos}(\phi_H-\phi)+\frac{1}{2}$
$(4 \pi M_s^2)\mathrm{sin}^2 \theta \mathrm{sin}^2 \phi - (K_1+2K_2)\mathrm{sin}^2 \theta \mathrm{sin}^2 \phi$ $+ K_2 \mathrm{sin}^4 \theta \mathrm{sin}^4 \phi$