(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 moment? Why or why not?

There are two main sources of magnetic moments for electrons: 1. Orbital Magnetic Moment: This magnetic moment originates from the electron's orbital motion around the nucleus. As the electron moves around the nucleus, it creates a current loop that produces a magnetic field. The orbital magnetic moment is proportional to the angular momentum of the electron and can be expressed as: \(\mu_L = -\frac{e}{2m}L\) Where \(\mu_L\) is the orbital magnetic moment, e is the electron charge, m is the mass of the electron, and L is the orbital angular momentum. 2. Spin Magnetic Moment: The second source of the magnetic moment is the intrinsic property of the electron known as spin. Electrons behave like tiny spinning charged particles, and their spin creates a magnetic field. The spin magnetic moment is also proportional to the electron's spin angular momentum and can be expressed as: \(\mu_S = -g_s \frac{e}{2m}S\) Where \(\mu_S\) is the spin magnetic moment, \(g_s\) is the electron spin g-factor (\(g_s \approx 2\)), and S is the spin angular momentum.

Not all electrons have a net magnetic moment. The total magnetic moment of an electron is the vector sum of its orbital magnetic moment and spin magnetic moment: \(\mu = \mu_L + \mu_S\) In some atoms or atomic configurations, the total magnetic moment of electrons can be zero due to the cancelation of the orbital and spin magnetic moments. This can happen when electrons are paired in an atomic orbital, as the magnetic moments of the two electrons with opposite spins will cancel each other out. In such cases, there is no net magnetic moment for the electron pair.

Not all atoms have a net magnetic moment. Atoms that have an even number of electrons with electrons in pairs (such as noble gases) typically have no net magnetic moment since the individual magnetic moments of the electrons cancel each other out. Additionally, the total angular momentum (sum of orbital and spin angular momentum of electrons) of the atom is often zero for these atom configurations, leading to no net magnetic moment. However, atoms with unpaired electrons, such as transition metal ions and some rare earth elements, possess a net magnetic moment, making them magnetic. The magnetic properties of materials are due to the collective behavior of atoms with net magnetic moments.