Crystal anisotropy energy

Crystal anisotropy energy is associated with the work required to move the polar direction away from an easy crystal direction this is really the variation of the interaction energy with orientation relative to the crystal lattice. 

The direction of the alignment of magnetic moments within a magnetic domain is related to the axes of the crystal lattice by crystalline electric fields and spin-orbit interaction of transition-metal t5 -ions (24). The dependency is given by the magnetocrystalline anisotropy energy expression for a cubic lattice (33) ... 

Magnetic Anisotropy Energy. There are several kinds of magnetic anisotropy energy and perhaps the most weU known is the magnetocrystaUine anisotropy. Only a crystalline soHd has this property because the energy is dictated by the symmetry of the crystal lattice. For example, in bcc Fe, the easy axis is in a (100) direction and in fee Ni, it is in a (111) direction. 

For agglomerated structures, the dipolar interaction between two neighboring crystals contributes to the anisotropy energy. This contribution increases when the inter-crystal distance decreases. 

For large super-paramagnetic crystals or for crystals with a very high anisotropy constant (12), the anisotropy energy is larger than the thermal energy, which maintains the direction of the crystal magnetic moment very... 

When the anisotropy energy is large enough it prevents any precession of the magnetic moment of super-paramagnetic crystals. The magnetic fluctuations then arise from the jumps of the moment between different easy directions. The precession prohibition is introduced into the Freed equations in order to meet that requirement every time the electron Larmor precession frequency appears in the equations, it is set to zero 12). 

A quantitative evaluation of the relaxivities as a function of the magnetic field Bo requires extensive numerical calculations because of the presence of two different axes (the anisotropy and the external field axis), resulting in non-zero off-diagonal elements in the Hamiltonian matrix (15). Furthermore, the anisotropy energy has to be included in the thermal equilibrium density matrix. Figures 7 and 8 show the attenuation of the low field dispersion of the calculated NMRD profile when either the crystal size or the anisotropy field increases. 

For USPIO particles containing only one nanomagnet per particle, the main parameters determining the relaxivity are the crystal radius, the specific magnetization and the anisotropy energy. Indeed, the high field dispersion is determined by the translational correlation time t. ... 

The maximum of relaxivity is proportional to the squared saturation magnetization of the crystal. The low field relaxivity depends on the anisotropy, and the presence of a low field dispersion indicates a low anisotropy energy. 

The first type is the crystal anisotropy which is responsible for the dependence of the magnetization curves on the orientation of the crystal in the magnetic field. The energy E, which is necessary to deflect the magnetic moment in a single crystal from the easy direction into less easy ones is given for cubic crystals by ... 

The approximate expression for K (Eq. (9.28)) leaves out all but two of the terms of an infinite power series, and even the term involving K2 can often be safely neglected. If it is assumed that K2 is negligible and Aj is positive, then K has a minimum value of zero if any two of the direction cosines are zero, i.e. the anisotropy energy is a minimum along all three crystal axes and these are therefore the easy directions. If Kx is negative, the minimum occurs for ai = a2 = a3 = 1/ /3, i.e. the body diagonal is the easy direction. 

Figure 5. Physical origin of magnetic anisotropy (a) compass-needle analogy of shape anisotropy and (b-c) magnetocrystalline anisotropy. In (b) and (c), the anisotropy energy is given by the electrostatic repulsion between the tripositive rare-earth ions and the negative crystal-field charges.

In 3d atoms, the spin-orbit coupling is much smaller than the crystal-field energy, and the magnetic anisotropy is a perturbative effect [7, 8, 16]. Typical second- and fourth-order transition-metal anisotropies are of the orders of 1 MJ/m3 and 0.01 MJ/m3, respectively. A manifestation of magnetocrystalline anisotropy is magnetoelastic anisotropy, where the crystal field is changed by mechanical strain [5, 16]. 

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