(a) Explain the two sources of magnetic moments for electrons. (b) Do all electrons have a net magnetic moment? Why or why not? (c) Do all atoms have a net magnetic

Imagine electrons as tiny planets orbiting a sun, which in this case is the nucleus of an atom. This motion around the nucleus is not just responsible for creating the orbital pattern but also contributes a fundamental property known as the orbital magnetic moment.

The electron carries a charge, and when it moves, it creates a magnetic field, similar to how a wire carrying electrical current generates a magnetic field around it. This phenomenon is grounded in electromagnetism, where moving charges produce magnetic fields. The formula \(\mu_L=\frac{e}{2m_e}L\) captures this relationship mathematically, linking the magnetic moment (\(\mu_L\)) to the charge of the electron (\(e\)), mass of the electron (\(m_e\)), and the orbital angular momentum (\(L\)).

Angular momentum here is a measure of the 'quantity' of rotation of the electron, and when you consider this motion in a quantum mechanical perspective, specific quantized values come into play, further affecting the magnetic moment's magnitude. The complexity of these interactions within atoms results in the orbital magnetic moments that play a pivotal role in phenomena such as magnetism in materials.

Another intrinsic quality of an electron, aside from its orbital motion, is its spin. This property isn't something you can visualize like an object spinning in space, because in the quantum realm, 'spin' doesn't quite correspond to a literal rotation. However, for the sake of understanding, you can think of it as a type of internal gyroscope that creates its own magnetic moment, aptly named the spin magnetic moment.

Unlike the orbital motion, which depends on the electron's path around the nucleus, spin is an inherent property of the electron. Quantum mechanics tells us that every electron possesses this intrinsic angular momentum that contributes to the overall magnetic behavior of the atom. The official formula \(\mu_S=\frac{e}{m_e}S\) shows the direct proportionality between the spin magnetic moment (\(\mu_S\)) and the spin angular momentum (\(S\)).

This intrinsic spin is crucial, especially when we talk about magnetic materials or elements in the periodic table. The difference in behavior between materials that are magnetic and those that are not often comes down to the spin properties of their electrons.

Bringing together the concepts of orbital motion and intrinsic spin gives rise to the net magnetic moment of an electron. This net moment is essentially the vector sum of an electron's orbital and spin magnetic moments.

Why don't all electrons showcase a net magnetic moment then? The key lies in electron pairing. Within an atom, electrons tend to pair up in orbitals with their spins pointing in opposite directions. This pairing causes their magnetic moments to cancel out—no more different than two equally strong but opposing magnets nullifying each other's effect.

However, when you have unpaired electrons, typically found in atoms with partially filled orbitals, these lone rangers maintain their individual magnetic moments. Therefore, atoms containing unpaired electrons often exhibit a net magnetic moment—a fundamental reason behind the magnetic properties of certain substances. The balance or imbalance of these minuscule magnets at the atomic level explains why some materials behave as magnets while others do not, which is a cornerstone in the field of magnetism and materials science.