8.1.1 $\mu$mag standard problem #3

This problem has been chosen to verify the equilibrium states obtained by the finite difference program. A lattice of $11 \times 11 \times 11$ magnetic moments and the fixpoint iteration method (section ) have been used to calculate the equilibrium states.

The http://www.ctcms.nist.gov/ rdm/mumag.html$\mu$mag standard problem #3 [48] consists in calculating the single domain limit of a cubic magnetic particle. This is the size of equal energy for the so-called flower state (fig. ) on the one hand, and the vortex or curling state (fig. ) on the other hand. The easy axis of magnetocrystalline anisotropy is parallel to a principal axis of the cube. The uniaxial anisotropy constant is given by $K_\mathrm{u}=0.1\, K_\mathrm{m}$, where $K_\mathrm{m}=\frac{1}{2}\mu_0 M_\mathrm{s}^2$ is a magnetostatic energy density.

The results are given in the column marked ``FD'' in tables , , and and compared with those published by Rave et al. [49], Ribeiro et al. and Hertel et al. on the web page of NIST [48]. The partial energies are given in units of $K_\mathrm{m}$, the magnetization in units of $M_\mathrm{s}$, and the single domain limit $L$ in units of $l_\mathrm{ex}=\sqrt{A/K_\mathrm{m}}$, where $A$ is the exchange constant.

Figure: Magnetization configurations in the $\mu$mag standard problem #3

[flower state] [vortex state]

Figure: Energy of flower ($\times$) and vortex ($\circ$) state

Table: Flower state, partial energy densities and average magnetization

	Rave	Martins	Hertel	FD
$E_\mathrm{demag}$	0.2794	0.2792	0.2839	0.2807
$E_\mathrm{exch}$	0.0177	0.0177	0.0158	0.0175
$E_\mathrm{ani}$	0.0056	0.0056	0.0052	0.0054
$\langle {M_x} \rangle$	0.000
$\langle {M_y} \rangle$	0.000
$\langle {M_z} \rangle$	0.971	0.9710	0.973	0.972

Table: Vortex state, partial energy densities and average magnetization

	Rave	Martins	Hertel	FD
$E_\mathrm{demag}$	0.0783	0.0780	0.0830	0.0756
$E_\mathrm{exch}$	0.1723	0.1724	0.1696	0.1761
$E_\mathrm{ani}$	0.0521	0.0521	0.0522	0.0519
$\langle {M_x} \rangle$	0.000
$\langle {M_y} \rangle$	0.352	0.3516	0.351	0.348
$\langle {M_z} \rangle$	0.000

Table: Single domain limit

	Rave	Martins	Hertel	FD
$L$	8.47	8.4687	8.52	8.32