Quantum theory of Paramagnetism

The quantum theory of paramagnetism describes the magnetic properties of materials in terms of the behavior of their constituent particles at the quantum level. Paramagnetism is a phenomenon that occurs when the magnetic moments of unpaired electrons in an atom or molecule are not cancelled out by opposing magnetic moments, resulting in a net magnetic moment.

At the quantum level, the magnetic moment of an electron can be quantized, meaning that it can only take on certain discrete values. This is due to the intrinsic angular momentum, or spin, of the electron. In a magnetic field, the spins of electrons can be oriented in different directions, which results in a net magnetic moment.

The quantum theory of paramagnetism also takes into account the effects of temperature on the magnetic properties of a material. At higher temperatures, thermal energy can cause electrons to become paired, reducing the material's overall magnetic moment. This is why the paramagnetic behavior of materials typically decreases as the temperature is increased.

Overall, the quantum theory of paramagnetism provides a framework for understanding the magnetic properties of materials at the quantum level, which is important for a variety of applications in materials science, electronics, and other fields.

According to Quantum theory, magnetic moments of substances are not freely rotating but are restricted to finite set of orientation relative to the external field.

Here, Magnetic moment of an atom or ion in free space is given by

μ = γ ℏJ

γ is gyromagnetic ratio and g is Lande 'g' factor.

In presence of magnetic field, the energy levels of system are

where mJ = Jcos θ is azimuthal quantum number. Its values are -J to +J through 0. There are (2J +1) ways in which atom can align itself in a magnetic field. For single electron system in ground state,

This shows the system has two energy levels.

Let N1 and N2 be number of atoms in lower and upper levels respectively.

Total population is given by

N = N1 + N2

According to Boltzmann statistics,

The resultant magnetisation for N atoms per unit volume is

hence paramagnetic susceptibility for two level system is

χ = M/ B = C/ T

where C is Curie constant and is given by

The paramagnetic susceptibility for two level system according to quantum theory is similar as classical theory.