**The electron in the outer shell of the sodium atom can absorb energy and transition to a higher energetic level (energy shift). However, it has two different
choices as it can jump not to one but to two different energetic levels; and in the presence of a magnetic field it has even more energetic jump choices!**

These phenomena are explained by the spin-orbit interaction and the Zeeman effect.

At school we first learned about the gross structure of the atom with the electrons found in orbits around the nucleus and then we were taught about the orbitals corresponding to different energetic
levels. For instance, in the case of sodium, which has 11 electrons, we first learned to fill the 1st orbit or shell with 2 electrons, then the next with 8 and then have one single electron at the
outer shell. Later, we were taught about the subshells s,p,d and how to write the electron configuration of the sodium atom as 1s2,2s2,2p6, 3s1.

How do we explain the spectral lines of sodium mentioned at the section "**Spectroscopy**"(http://www.information-book.com/physics/spectroscopy)? The electron at the outer subshell, the 3s subshell, absorbs energy and transitions to the next energetic level. That would
be a p subshell. We would expect to obtain a spectral line corresponding to this energetic transition from the s shell to the p shell (absorption or emission). However, it is demonstrated that we
obtain not only one but two spectral lines! That means that are two different possible energetic jshifts or jumps, or in other words, two energetic transitions that can be made by the electron in the
outer shell. This is explained by the theory of the spin-orbit interaction which gives rise to the fine structure of the atom.

What happens upon the effect of a magnetic field?

In that case, there are even more possible energetic shifts or jumps that can be made by the electron! More energetic level options are produced because of the interaction between the atom and the
external magnetic field. This is explained by the Zeeman interaction which gives rise to the hyperfine structure of the atom. Please refer to Figure 1 and the relevant sections.

The energy shifts of the atomic electrons are determined by the influence of an orbital parameter, the orbit angular momentum L and the spin angular momentum S or simply spin.

Let us examine the system consisting of a nucleus and an electron orbiting around the nucleus.

What frame of reference shall we use? Consider the frame of reference of the electron, for instance by imagining being on the electron! In that case, it is like being "the center of the world" and you see the nucleus rotating around you (around the electron).

https://en.wikipedia.org/wiki/Fine_structure#Spin-orbit_coupling

"The spin-orbit correction can be understood by shifting from the standard frame of reference (where the electron orbits the nucleus) into one where the electron is stationary and the nucleus instead orbits it. In this case the orbiting nucleus functions as an effective current loop, which in turn will generate a magnetic field."

The movement of the nucleus, which is similar to that of a charged particle in a closed loop, is equivalent to a current I running in the loop as shown in the figure at the link "magnetic field in the electron frame". It is demonstrated at the figure, that this current I generates a magnetic field B. We are referring to the latter as an effective magnetic field in the (rest) frame of reference of the electron.

Due to this, the nucleus exerts a magnetic force on the electron (the electron "sees" or senses this). We can calculate the intensity (strength) of the magnetic field at the distance of the electron (r). As this is due to a circular or orbital motion, an expression including the orbital angular momentum L of the electron is deduced for this magnetic field.

The magnetic field will exert a torque that will produce a change in the angular momentum which is perpendicular to that angular momentum. The vector of the angular momentum L had the same direction as the magnetic field, but as a result of the exerted torque, it will "tilt" and will be precessing around the direction of the magnetic field with a certain angle as shown at this link http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/larmor.html#c1. This results in Larmor precession. We can calculate the angular velocity or Larmor frequency associated with this precession movement.

As mentioned at the link http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/vecmod.html#c2, the orbital angular momentum for an atomic electron can be represented with a vector model where the angular momentum vector is shown to be precessing about a direction in space.

Figure 1: From http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/vecmod.html#c2.

Please note the the L vector is a "special kind of vector" as its projection along a specific direction (e.g. usually z) can take only certain values; in other
words it is quantized as mentioned at this link http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/vecmod.html#c2. Specifically, it is quantized to values of one unit of angular momentum apart. Its values are dependent on the
"magnetic quantum number m_{l"} . For instance for l=2 it can take the values ml=-2, -1, 0, 1, 2.

Figure 2: From http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/vecmod.html#c2.

This effective magnetic field in the electron frame due to the orbital motion (linked to the angular momentum) interacts with the spin magnetic moment of the electron (the magnetic energy of the electron spin) and affects the energy levels of the atomic electrons. In order to calculate these energetic levels, we need to find the total energy as represented by the total angular momentum and determine how is that total energy distrubuted in different levels. To this purpose we need to add the vectors of orbital angulal momentum L and the spin angular momentum S in order to determine the total angular momentum J.

*Figure 3: From Wikipedia (selection) - By Maschen - Own
work, CC0, included at https://en.wikipedia.org/wiki/Angular_momentum*

The above figure (Figure 3) from Wikipedia shows on the left an object that rotates upon itself or "spins" and which has spin angular momentum S. On the right there is an object that performs a rotation around an origin at distance r (it is in orbit around an origin) and has orbital angular momentum L. We can calculate the total angulal momentum J by adding the two vectors L and S following the principle shown at the figure.

Figure 4: From Wikipedia - By Maschen - Own work, Public Domain, - https://commons.wikimedia.org/wiki/File:Angular_momentum_conservation.svg

The above figure (Figure 4) from Wikipedia also shows the principle of the addition of the vectors of the two angular momenta to calculate the total angular momentum.

In order to calculate the energy levels of the atomic electrons, we will use the vector model for total angular momentum described at the link http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/vecmod.html#c2 and shown in Figure 5 which illustrates a single electron case for which l=1 and s=1/2. We need to add the vectors of orbital angulal momentum L and the spin angular momentum S in order to determine the total angular momentum J.

Figure 5 (not shown):

From http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/vecmod.html#c2.

A specific procedure described here https://en.wikipedia.org/wiki/Spin–orbit_interaction#Evaluating_the_energy_shift which includes the evaluation of the quantum numbers is followed. For reference, the five quantum numbers are: n (the "principal quantum number"), j (the "total angular momentum quantum number"), l (the "orbital angular momentum quantum number"), s (the "spin quantum number"), and j_{z}(the "z component of total angular momentum").

There is also mj which is known as "projection of the total angular momentum along a specified axis".

In the case of sodium (1s22s22p63s1), due to the spin-orbit interaction, as mentioned at the link http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/sodzee.html, "the 3p level is split into states with total angular momentum j=3/2 and j=1/2 by the magnetic energy of the electron spin in the presence of the internal magnetic field caused by the orbital motion".

Please refer to the figure of the above link for an image showing a ΔE of 0.0021 eV between the two 3p levels and also to Figure 6 below.

Wikipedia mentions: "note that the spin-orbit effect is due to the electrostatic field of the electron". Other say that in this case we have an internal magnetic field caused by the orbital motion of the electron and this is the reason that this effect is known as an "internal Zeeman effect" (more explanations at the relevant section).

An interesting discussion on "What Causes Electron Energies to Depend Upon the Orbital Quantum Number" is found at the link http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/orbdep.html#c1 (note: the relative position of the orbitals and their overlap determine their shielding from the effect of the nucleus).

We can continue with the Zeeman interaction, where we place the above vector model configuration in a magnetic field (as mentioned at the relevant figure as well).

It is of note that the spin-orbit interaction mechanism couples the electron spin with an electric field as mentioned here https://en.wikipedia.org/wiki/Electric_dipole_spin_resonance. This is relevant to EDSR.

Ref. for this section: https://bit.ly/2GYlWtK from physics.stackexchange.com

*Figure 6: From the gross structure of the atom to the fine and hyperfine structure. The spin-orbit interaction is linked to the fine structure and the Zeeman
interaction to the hyperfine structure.*

The spin-orbit interaction can be considered as a case of an "internal magnetic field" affecting atomic energy levels. In this section, the influence of external magnetic fields is examined.

For this purpose, we will transfer the above vector model for total angular momentum in a magnetic field as shown at this section:

When a magnetic dipole is placed in a magnetic field it will experience a torque.

As mentioned at https://en.wikipedia.org/wiki/Magnetic_moment, the magnetic (dipole) moment of a magnet is a quantity that determines the torque it will experience in an external magnetic field.

A magnetic source in general is inherently a dipole source that can be visualized as a coil, or let us say a loop with current I and area A as shown at the picture below from link http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html#c1.

*Figure 2 (not shown): Image from hyperphysics.phy-astr.gsu.edu *

*http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html#c1.*

The magnetic (dipole) moment can be considered to be a vector μ with a value equal to I * A and a direction perpendicular to the current loop in the right-hand-rule direction. The torque is given by τ = μ * B.

Due to this, a "capacity for doing work" is generated as a result of the magnetic field. The "capacity for doing work" or "potential for doing work" (cf. common expression "this candidate for the job has got potential!") is translated to the notion of "potential energy" as mentioned at http://hyperphysics.phy-astr.gsu.edu/hbase/pegrav.html#pe. In particular, due to the fact that this potential is linked to a magnetic field, this is termed "magnetic potential energy" (cf. http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magpot.html#c1).

The magnetic potential energy U of a magnetic (dipole) moment μ due to the presence of a magnetic field B is given by:

U = - μ * B

The magnetic potential energy is expressed as a scalar product (μ and B are vectors) and it implies that the torque that is exerted tends to line up/to align the magnetic moment with the magnetic field B, so this represents its lowest energy configuration.

For simplicity purposes, we will refer to a perturbation V_{M} due to the magnetic field as mentioned at https://en.wikipedia.org/wiki/Zeeman_effect#Theoretical_presentation.

where μ is the magnetic moment of the atom given by the following expression (please refer to the orbital and spin magnetic moments).

You can consult the vector model at this reference http://bcs.whfreeman.com/webpub/Ektron/Tipler%20Modern%20Physics%206e/More%20Sections/More_Chapter_7_2-The_Zeeman_Effect.pdf at Figure 7.29. It is mentioned that this model
is quite complicated and therefore only the results are discussed. In conclusion, the model allows to calculate the splitting of the energy levels, with each level being split into 2j+1 levels,
corresponding to the possible values of m_{j}.