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At these temperatures, the available thermal energy simply overcomes the interaction energy between the spins. In general, paramagnetic effects are quite small: the magnetic susceptibility is of the order of 10 −3 to 10 −5 for most paramagnets, but may be as high as 10 −1 for synthetic paramagnets such as ferrofluids.
The energy difference between the two states is so small their populations vary significantly with temperature. In consequence the magnetic moment varies with temperature in a sigmoidal pattern. The state with spins opposed has lower energy, so the interaction can be classed as antiferromagnetic in this case. [ 14 ]
The Ising model can be solved analytically in one and two dimensions, numerically in higher dimensions, or using the mean-field approximation in any dimensionality. Additionally, the ferromagnet to paramagnet phase transition is a second-order phase transition and so can be modeled using the Landau theory of phase transitions.
The Hamiltonian for an electron in a static homogeneous magnetic field in an atom is usually composed of three terms = + (+) + where is the vacuum permeability, is the Bohr magneton, is the g-factor, is the elementary charge, is the electron mass, is the orbital angular momentum operator, the spin and is the component of the position operator orthogonal to the magnetic field.
In this case, it will be shown that (+ +), which, combined with the constant k, shows that paramagnetic materials can have energy maxima but not energy minima and diamagnetic materials can have energy minima but not energy maxima. That is, paramagnetic materials can be unstable in all directions but not stable in all directions and diamagnetic ...
Paramagnetic materials are attracted to magnetic fields, hence have a relative magnetic permeability greater than one (or, equivalently, a positive magnetic susceptibility). The magnetic moment induced by the applied field is linear in the field strength, and it is rather weak .
This energy tends to be minimized when the axis of magnetization of the domains in a crystal are all parallel. E k is magnetocrystalline anisotropy energy: Due to its magnetic anisotropy, the crystal lattice is "easy" to magnetize in one direction, and "hard" to magnetize in others. This energy is minimized when the magnetization is along the ...
This reduces the electrostatic energy of the electrons when their spins are parallel compared to their energy when the spins are antiparallel, so the parallel-spin state is more stable. This difference in energy is called the exchange energy. In simple terms, the outer electrons of adjacent atoms, which repel each other, can move further apart ...