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tasos
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Homework Statement
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We have that the three lowest energy states of a system are $$ ^3F_2, ^3F_3, ^3F_4 $$ (these are the Term symbols) with relative energy gap $$0,\ 171,\ 387 \ cm^{-1}$$
Now using the perturbation $$H_{LS}=\beta \ \vec{L}\cdot \vec{S}$$ i have to find the best value of the parameter β that fits best with the energy gaps.
Homework Equations
The Attempt at a Solution
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As i read fine structure is responsible for splitting $$ ^3F$$ into the 3 degenerate states $$ ^3F_2, ^3F_3, ^3F_4 $$
I calculate the term $$\langle\ \vec{L}\cdot \vec{S} \rangle = \ \vec{J}(\ \vec{J} + 1) -\ \vec{L}(\ \vec{L}+1) -\ \vec{S}(\ \vec{S}+1) $$
So now we have that $$E_{LS} = \frac{\beta}{2}( \ \vec{J}(\ \vec{J} + 1) -\ \vec{L}(\ \vec{L}+1) -\ \vec{S}(\ \vec{S}+1) )$$
So we see the splitting:
$$E_{^3F_3} - E_{^3F_2} = 2\beta$$
$$E_{^3F_4} - E_{^3F_3} = \beta$$
The first abstraction give the value $$\beta =85.5$$
The second abstraction gives the valut $$\beta = 108$$
So if all the above are correct what's the best value of the parameter β that fits best with the energy gaps?Thnx.
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