Basis of Zeeman effect discussion paper

The reason behind the splitting of spectral line is the cessation of degeneracy of angular momentum quantum states. Degeneracy means that they are at par in terms of energy content. P orbital represents the three quantum states of angular momentum , i.e., P_xP
x
​
,P_yP
y
​
, andP_zP
z
​
. Each of these angular momentum quantum states has a specific magnetic dipole moment which is manifested in three different energy levels when they are placed in a magnetic field. Due to the magnetic field, the energy of one state is elevated and that of the other is decreased. The energy of the third state remains unchanged. The elevation in energy is equal to the decrease in energy. When electronic transition takes place between these energy levels, different spectral lines emerge. This is the simplest version of Zeeman effect.

Figure 1: L-S coupling

Zeeman effect can be explained on the basis of the classical theory developed by Lorentz. It is valid only when spin quantum number s = 0. At the time of discovery of Zeeman effect, the discovery of the spin movement of electron was not made.

Each atom behaves like a tiny magnetic dipole whose total dipole moment is the sum of the dipole moment due to orbital angular momentum and the spin angular momentum. \mu_Lμ
L
​
and\mu_Sμ
S
​
represents the magnetic moment due to orbital angular momentum L and spin angular momentum S, respectively.

{{\mu }_{L}}=\frac{-{{g}_{l}}{{\mu }_{b}}L}{\hbar }μ
L
​
=
ℏ
−g
l
​
μ
b
​
L
​

{{\mu }_{S}}=\frac{-{{g}_{S}}{{\mu }_{b}}S}{\hbar }μ
S
​
=
ℏ
−g
S
​
μ
b
​
S
​

where {{\mu }_{b}}=\frac{e\hbar }{2{{m}_{e}}}μ
b
​
=
2m
e
​

eℏ
​

In case of L-S coupling, L and S process about the resultant J

Where J = L + S

L, S, and J all lie in one plane with the whole plane processing about J. µ also lies in the same plane and it also processes about J. The frequency of precession depends on the strength of the atomic internal magnetic strength which is due to L. The internal magnetic field of an atom is about the order of 1 tesla. Therefore if the applied magnetic field is weaker than this value, the precession of µ about J is rapid and if the external magnetic field is higher than 1 tesla, then J will process about B.

image

When a magnetic dipole of value µ is placed in a magnetic field B, energy of interaction

E is created whose value is given as follows:

\Delta E={{\text{ }\!\!\mu\!\!\text{ }}_{\text{B}}}\text{B}\left( 1+\frac{j(j+1)+s(s+1)-l(l+1)}{2j(j+1)} \right){{m}_{j}}ΔE= μ
B
​
B(1+
2j(j+1)
j(j+1)+s(s+1)−l(l+1)
​
)m
j
​

\Delta E={{\mu }_{B}}Bg{{m}_{j}}ΔE=μ
B
​
Bgm
j
​

This gives the energy gap that is created in a magnetic field between energy levels that were degenerate earlier.

According to this equation, each energy level is broken into 2j +1 level, one for each value of m_jm
j
​
. Landé’s g – factor, determines the value of splitting between the levels.

If s = 0, j = 1 and g = 1

The selection rule for transition between the split levels is as follows: