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Electronic Structure and Properties
Published in Alan Cottrell, An Introduction to Metallurgy, 2019
In 4 per cent silicon-iron the domains are typically about 10−4 m across. A lower limit to the domain size is set by the surface energy of the Bloch walls. This energy is positive and so favours a structure of coarse domains. Its origin is explained by reference to Fig. 24.25 which shows how the direction of electron spins varies across a −180° Bloch wall’. The exchange interaction energy does not reach its lowest value in the Bloch wall, because the electron spins there are not in parallel. They become more parallel as the boundary thickens, but opposing this tendency is the effect of magnetic anisotropy. Magnetization occurs more readily in some crystal directions than others. The spontaneous magnetization within the domains occurs along these directions of easy magnetization, e.g. 〈100〉 in iron and 〈111〉 in nickel. Across a domain boundary the magnetic axis is forced to rotate away from the direction of easy magnetization and the thicker the boundary the more atoms there are with unfavourable directions. This effect opposes the tendency of the exchange interaction to widen the boundary and a balance is struck at an equilibrium thickness which is about 300 atoms in iron. The surface energy of a 180° wall parallel to {100} in iron is 0.0018 J m−2.
Magnetic Anisotropy of Nanocomposites Made of Magnetic Nanoparticles Dispersed in Solid Matrices
Published in Mahmood Aliofkhazraei, Advances in Nanostructured Composites, 2019
Magnetizing a single crystal/nanocrystal ferromagnetic (or ferrimagnetic), which may even be a nanoparticle or a composite containing a magnetic nanoparticle, with an external magnetic field applied by different directions in relation to the main crystallographic axes, it obtains a different magnetization. Thus, it was found that there are preferential directions of spontaneous magnetization in a crystal, compared to the crystallographic axes, corresponding to a minimum energy (equilibrium energy). This shows that there is a magnetic anisotropy, called in this case magnetocrystalline anisotropy. The direction (axis) of magnetization based on which the magnetization is the easiest to have is called easy magnetization axis (e.m.a.). The direction (axis) of magnetization based on which the magnetization makes the most difficult, is hard magnetization axis (h.m.a.). In this respect, the classic experimental results obtained as a result of the magnetization of the ferromagnetic iron, nickel and cobalt single crystals (Honda and Kaya 1926, Kaya 1928a,b) are highly suggestive.
Fluid in a Nanoslit Symmetry Breaking
Published in Eli Ruckenstein, Gersh Berim, Wetting Theory, 2018
A fluid in a slit with identical walls is subjected to an external potential due to the fluid—solid interactions which possess a reflectional symmetry about the middle of the slit. The same symmetry is present in the basic Euler—Lagrange equation,1 which describes the equilibrium density profile across the slit and provides a symmetrical solution for the profile.1–7 However, it is known that in some cases the symmetry of the stable state of a system can be lower than the symmetry of the equations describing the system. This symmetry breaking (SB) is well-known in physics, chemistry, biology, etc. A classical example from the physics of fluids is the freezing transition8 in which the continuous translational symmetry of a fluid (in the absence of an external potential) is broken. The discrete symmetry, e.g., reflectional, or the left—right one, can also be broken. A well-known example is the occurrence of spontaneous magnetization in ferromagnetic materials in the absence of an external magnetic field at temperatures lower than the Curie temperature. The possibility of a reflectional SB was demonstrated in ref 9 for an open system consisting of two parallel identical layers of a two-dimensional liquid coupled by an interlayer potential.
Exact solution of the magnetic properties for ternary ferrimagnetic mixed-spin Ising nanoislands
Published in Phase Transitions, 2022
We know that when the particle size is smaller than the critical size, the ferromagnetic material with single-domain structure shows paramagnetism at temperature below Curie temperature and above Block temperature, and hysteresis does not appear under the action of external magnetic field. Due to the up-down symmetry of spin direction in spin space, the Ising nanoisland system will not show spontaneous magnetization and macroscopically will show superparamagnetism. The spontaneous magnetization and hysteresis behavior in nanoisland system found by mean-field theory or correlated effective field theory are due to the assumption of the order parameters existing in the system at finite temperature in these theories. The single spin flipping in Monte Carlo simulation may be the reason that the nanoisland shows spontaneous magnetization and hysteresis loop. In this way, only the local environment of a spin is considered but the symmetry of the whole for spins is ignored. In this paper, we will solve exactly the ternary ferrimagnetic mixed-spin Ising nanoisland model based on the finite number of spins and geometric symmetry by using the transfer matrix method. In the second section, we introduce the application of transfer matrix method to the nanoisland model. The temperature behavior of susceptibility and magnetization behavior of the system are given in the third section. The fourth part is a summary of the paper.