How Do You Determine the Best Value of β in L-S Coupling?

[tasos]

Homework Statement

[/B]
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.

 

[blue_leaf77]

You shouldn't have removed the other sections of the templates. You are required to follow the template as it's in the guideline of Homework section and also because no one is supposed to provide help before the poster posts his own effort.

 

[blue_leaf77]

So we see the splitting:

$$E_{^3F_3} - E_{^3F_2} = 2\beta$$
$$E_{^3F_4} - E_{^3F_3} = \beta$$

You seemed to be making mistakes when calculating the energy difference in terms of $\beta$. For instance, in calculating $E_{^3F_3} - E_{^3F_2}$, the terms containing $L$ and $S$ in each energy expressions will cancel because the two states have the same values for those quantum numbers. Only the terms containing $J$ will contribute, the first term has $J=3$ and second term has $J=2$.

Also take care of which system of unit you are using. If you use atomic unit in which $\hbar=1$, you should also convert the given energy difference (in terms of wave number) into atomic unit of energy.

 

[tasos]

Ok sorry. So here the energy interval between adjacent J levels is

$$ΔE_{FS}= E_J -E_{J-1}= \beta J$$So for J=4 we have $$ ΔE_{FS}= E_4 -E_{3} = 4 \beta $$ so here $$ \beta =42,75 cm^−1$$
and for J=3 we have $$ΔE_{FS}= E_3 -E_{2} = 3 \beta$$ so here $$ \beta =72 cm^−1$$ANd if I am correctly how i decide the best value of β

 

[blue_leaf77]

Ignoring any common constant factors in energies and the units, for $E_4-E_3$,
$$
\beta = \frac{E_4-E_3}{4} = \frac{171}{4} = ?
$$
while for $E_3-E_2$,
$$
\beta = \frac{E_3-E_2}{3} = \frac{387-171}{3} = ?
$$
The two $?$'s obtained from the two energy differences are alike, as they should be. It appears really strange to me that you keep getting very different values of $\beta$.