The purpose of static micromagnetics is to solve for the spatial distribution of the magnetization \mathbf{M} at equilibrium. In most cases, as the temperature is much lower than the
Curie temperature of the material considered, the modulus |\mathbf{M}| of the magnetization is assumed to be everywhere equal to the
saturation magnetization M_s. The problem then consists in finding the spatial orientation of the magnetization, which is given by the
magnetization direction vector \mathbf{m}=\mathbf{M}/M_s, also called
reduced magnetization. The static equilibria are found by minimizing the magnetic energy, :E_\text{exch} = A \int_V \left((\nabla m_x)^2 + (\nabla m_y)^2 + (\nabla m_z)^2\right) \mathrm{d}V where A is the
exchange constant; m_{x}, m_{y} and m_{z} are the components of \mathbf{m}; and the integral is performed over the volume of the sample. The exchange energy tends to favor configurations where the magnetization varies slowly across the sample. This energy is minimized when the magnetization is perfectly uniform. The energy of the demagnetizing field favors magnetic configurations that minimize magnetic charges. In particular, on the edges of the sample, the magnetization tends to run parallel to the surface. In most cases it is not possible to minimize this energy term at the same time as the others. The static equilibrium then is a compromise that minimizes the total magnetic energy, although it may not minimize individually any particular term.
Dzyaloshinskii–Moriya Interaction Energy This interaction arises when a crystal lacks inversion symmetry, encouraging the magnetization to be perpendicular to its neighbours. It directly competes with the exchange energy. It is modelled with the energy contribution E_\text{DMI} = \int_{V}\mathbf{D}:(\nabla \mathbf{m}\times \mathbf{m}) where \mathbf{D} is the spiralization tensor, that depends upon the crystal class. For bulk DMI, :E_\text{DMI} = \int_{V}D \mathbf{m}\cdot(\nabla \times \mathbf{m}), and for a
thin film in the x-y plane interfacial DMI takes the form :E_\text{DMI} = \int_{V}D(\mathbf{m}\cdot\nabla m_{z} - m_{z}\nabla\cdot\mathbf{m}), and for materials with symmetry class D_{2d} the energy contribution is :E_\text{DMI} = \int_{V}D \mathbf{m}\cdot\left(\frac{\partial \mathbf{m}}{\partial x}\times \hat{x} - \frac{\partial \mathbf{m}}{\partial y}\times \hat{y}\right). This term is important for the formation of
magnetic skyrmions.
Magnetoelastic Energy The magnetoelastic energy describes the energy storage due to elastic lattice distortions. It may be neglected if magnetoelastic coupled effects are neglected. There exists a preferred local distortion of the crystalline solid associated with the magnetization director \mathbf{m}. For a simple
small-strain model, one can assume this strain to be isochoric and fully isotropic in the lateral direction, yielding the deviatoric
ansatz \mathbf{\varepsilon}_0(\mathbf{m}) = \frac{3}{2} \lambda_{\text{s}}\, \left[\mathbf{m}\otimes \mathbf{m} - \frac{1}{3}\mathbf{1}\right] where the material parameter \lambda_{\text{s}} is the isotropic magnetostrictive constant. The elastic energy density is assumed to be a function of the elastic, stress-producing strains \mathbf{\varepsilon}_e := \mathbf{\varepsilon} -\mathbf{\varepsilon}_0. A quadratic form for the magnetoelastic energy is E_\text{m-e} = \frac{1}{2} \int_{V}[\mathbf{\varepsilon} -\mathbf{\varepsilon}_0(\mathbf{m})] : \mathbb{C} : [\mathbf{\varepsilon} -\mathbf{\varepsilon}_0(\mathbf{m})] where \mathbb{C} :=\lambda \mathbf{1}\otimes \mathbf{1} + 2\mu \mathbb{I} is the fourth-order
elasticity tensor. Here the elastic response is assumed to be isotropic (based on the two Lamé constants \lambda and \mu). Taking into account the constant length of \mathbf{m}, we obtain the invariant-based representation E_\text{m-e} = \int_{V} \frac{\lambda}{2} \mbox{tr}^2[\mathbf{\varepsilon}] + \mu \, \mbox{tr}[\mathbf{\varepsilon}^2] - 3\mu E \big\{ \mbox{tr}[\mathbf{\varepsilon}(\mathbf{m}\otimes\mathbf{m})] - \frac{1}{3}\mbox{tr}[\mathbf{\varepsilon}] \big\} . This energy term contributes to
magnetostriction. == Dynamic micromagnetics ==