in free atoms

How does that tiny bar magnet at the nucleus make itself felt? We will examine this first for a single atom, with a nucleus that is more than just a point charge: the nucleus has a magnetic moment. What does this mean for the energy levels of the atom?

You will first review the (probably/perhaps) known topic of the coupling of spin and orbital angular momenta. Having understood this, the coupling of the nuclear spin to the total electronic angular momentum is just copy-paste. You will understand how perturbation theory allows to get the energy levels of a free atom with a nuclear spin.

There are three tasks related to this video, two to submit via a post first forum, and one to submit as a pdf file: 1. The first task is meant to train you in making the connection between the sometimes abstract angular momentum algebra and a visual picture about what is going on. Consider the three energy levels of a free atom, drawn in the picture. At the right-hand side of the picture, three descriptions of the orientation of L and S are given. Which J-value corresponds to which orientation? Describe your reasoning in the forum.

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2. In the second task, we ponder how we can exploit this magnetic hyperfine interaction to measure useful information. Imagine you are given an isotope with a known nuclear spin I1 and known nuclear magnetic moment μ1. You have at your disposal an experimental setup that can measure the hyperfine splitting in free atoms (i.e. you can measure the hyperfine coupling constant ‘a’ that was defined in the video). You get now a different isotope of the same element, with known nuclear spin I2yet unknown magnetic moment μ2. Describe a procedure how you could determine experimentally the value of μ2. Post your reasoning in this forum:

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3. In order to familiarize yourself with the expressions that were derived in the video: Explicitly calculate the value of C11 in the expression <1|Hjj|1>, for a nucleus with spin 3/2 and an electron cloud with total angular momentum 3/2. Write this down in a handwritten or typed pdf file, and upload:

C11 calculation

Expected time: 60 minutes (report)

A04-01