An Analytical and Computational Study of Magnetic Susceptibility from the Quantum Perspective

Abstract

In this study, we review the stages in the development of understanding magnetic phenomena in materials. We show how materials are classified according to the sign of their magnetic susceptibility into ferromagnetic, paramagnetic, and diamagnetic substances. We then calculate the magnetization in the classical case based on its definition as the sum of the magnetic moments of the system’s constituents, following Langevin’s theory of magnetization.

In the second part, we introduce the quantum contributions of both the orbital and intrinsic (spin) angular momenta, and we compute the expectation value of the resulting moments in a material placed in an external magnetic field. Our calculations lead to a formula involving the well-known Brillouin function, which forms the foundation for studying magnetism in both classical and quantum regimes; however, in our formulation, it incorporates the total magnetic angular momentum resulting from both orbital and spin motion.

In the final part, we present code implementations for each case and plot the resulting functions after adding a term from the Taylor expansion of the magnetization function. The obtained curves indicate that a correction to the classical results is required, related to the spin–orbit coupling term. A clear agreement appears between the examined curves within the ranges that take into account only the first and second terms.