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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.
The magnetization is the negative derivative of the free energy with respect to the applied field, and so the magnetization per unit volume is = , where n is the number density of magnetic moments. [1]: 117 The formula above is known as the Langevin paramagnetic equation.
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 .
The electric field was formally defined as the force exerted per unit charge, but the concept of potential allows for a more useful and equivalent definition: the electric field is the local gradient of the electric potential. Usually expressed in volts per metre, the vector direction of the field is the line of greatest slope of potential, and ...
Electric motors and generators: Some electric motors rely upon a combination of an electromagnet and a permanent magnet, and, much like loudspeakers, they convert electric energy into mechanical energy. A generator is the reverse: it converts mechanical energy into electric energy by moving a conductor through a magnetic field.