In case of two valence electron system, there will be four individual magnetic moments associated with mechanical moment S_1*, S_2*, l_1* and l_2*. The vector sum of mechanical moment will depends on the coupling scheme.

In L- S coupling, the two orbital vectors l_1* and l_2* are coupled together and are quantised to form a resultant L* and two spin vectors S_1* and S_2* coupled together and are quantized in form a resultant S* with total spin and orbital magnetic moments as

Then L* and S* are coupled together to form resultant J* with total magnetic moment as

Then L* and S* are coupled together to form resultant J* with total magnetic moment as

is Lande g factor in L-S coupling

In j-j coupling the orbital and spin vector l_1* ans s_1* of one electron system coupled together to form a resultant j_1* and l_2* AND s_2* of second electron system are coupled together to form a resultant j_2*. Finally j_1* and j_2* are coupled together to form a resultant J* with total magnetic moments.

When a weak magnetic field is applied to an atom total angular momentum J* start to precess around the field direction with an angular velocity given by

w_L = eH/2m ....4

Thus precession of J* will change the energy of the system. To calculate the change in energy, let's assume the magnetic field is so weak that

1. S_1* and S_2* precess rapidly around S*

2. l_1* and l_2* precess rapidly around L*

3. L* and S* precess slowly around J*

4 J* precess more slowly around field direction

Then with these assumptions, the normal components of individual moments will cancel out and only parallel component contribute the change in energy i.e magnetic interaction energy is given by

where M_J = J* cos (J*H) is total magnetic quantum number number for given value of J, M_J takes values from +J to -J i.e 2J+1 values.

The interaction energy depends upon the product gM_J. Thus the given single state split into (2J+1) sub level.

The Zeeman effect in a two-valence electron system, such as in atoms like sodium (Na), is significant for several reasons:

Quantum Mechanics Validation: The Zeeman effect provides experimental confirmation of the quantum mechanical principles governing atomic systems. It demonstrates how the energy levels of electrons split under the influence of an external magnetic field, validating theoretical predictions such as the Landé g-factor and the magnetic quantum number.
Understanding Atomic Structure: By observing the Zeeman effect, researchers can infer details about the internal structure of atoms, particularly how electrons interact with each other and with external fields. For two-valence electron systems, this interaction can reveal information about the orbital and spin angular momentum coupling within the atom.
Precision Spectroscopy: The splitting of spectral lines due to the Zeeman effect allows for high-precision measurements of magnetic field strengths. This is crucial in fields like astrophysics, where spectral lines from distant stars and galaxies can provide insights into the magnetic fields present in these celestial bodies.
Applications in Technology: Understanding the Zeeman effect has practical applications in various technologies. For example, it is used in the design and calibration of sensitive instruments such as magnetometers, which are essential in fields ranging from geophysics to medical diagnostics (e.g., MRI machines).
Investigation of Chemical Bonds: In molecules containing transition metals or lanthanides, where multiple valence electrons are involved, the Zeeman effect helps in studying the magnetic properties and electronic structure of chemical bonds. This is particularly important in fields like catalysis and materials science.
Astrophysical Observations: The Zeeman effect is crucial in astrophysics for studying magnetic fields in celestial objects. Observing the splitting of spectral lines in stars and galaxies provides valuable data about magnetic field strengths and their effects on stellar evolution and galactic dynamics.

In summary, the Zeeman effect in a two-valence electron system like sodium not only deepens our understanding of fundamental atomic physics but also finds practical applications in diverse scientific and technological domains, making it a significant area of study and application.

This note is a part of the Physics Repository

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