For single valence electron system, the spin orbit interaction energy in terms of wavenumber is given by

In case of two valence electrons, there are four angular momenta l_1*, s_1*, l_2*, S_2* which give rise to six possible interaction. Therefore for those six possible interaction will be six energy relations in the form of equation 1. Thus,

For ideal LS coupling, the value of Γ_5 and Γ_6 are negligible. Therefore we have to calculate Γ_1, Γ_2, Γ_3, Γ_4.

Since S_1* and S_2* precess around their resultant S* with fixed angle of inclination. We have from cosine law

S*² = S_1*² + S_2*² + 2S_1*S_2*cos (S_1*S_2*)

S_1*S_2*cos (S_1*S_2*) = S*² - S_1*² - S_2*² / 2

Γ_1 = a_1/ 2 ( S*² - S_1*² - S_2*² ) .....3

Similarly l_1* and l_2* precess around their resultant L* with fixed angle of inclination. We have

l_1* l_2* cos( l_1*l_2*) = L*² - l_1*² - l_2*² / 2

Γ_2 = a_2/ 2 ( L*² - l_1*² - l_2*² ) .... 4

Now L* and S* also precess around their resultant J* and interaction energy due to their resultant J* and interaction energy due to their precession is attributed to the coupling between l_1*² and S_1*, l_2*² and S_2* i.e Γ_1 and Γ_2

Since angle between the vectors ( l_1*, S_2*) and (l_2*² ,S_2*) are continuously change. Thus the cosine of angle between those vectors are averaged which are given as,

Using cosine law we can write

For any given state, the values of l_1*² , S_1*, l_2*² and S_2*, L* and S* are fixed. Therefore a_3, a_4, 𝛼_3 and 𝛼_4 are taken constant.

Now the term value for nay fine structure level is given by

where To is term value for hypothetical centre of gravity. Equation 11 is the required expression.

In practice, the quantum numbers L, S, and J are determined based on the particular atomic configuration and can be found in atomic spectroscopy tables or calculated theoretically using quantum mechanical methods.

This note is a part of the Physics Repository.

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