Finding the spin-orbit coupling constants of an Alkai transition?

[Ted55]

Homework Statement

Find the spin-orbit Coupling constants of a 4p-4d transition of potassium given the following information:

There are three spectral lines in the transitions fine structure corresponding to wavelengths 693.9,696.4,696.5nm respectively.

The coupling constant for the 4d level is << 4p level. I.e] C_(4p) >> C_(4d)

Relevant Equations

Delta(E_j) = C/2 [ j(j+1) -l(l+1) -s(s+1)]
E = hc/x where x = wavelength

The set of quantum numbers for the 4p orbital is: 4, 1, {-1,1}, +-1/2 (n,l,m,s)
The set of quantum numbers for the 4d orbital: 4,2,{-2,2},+-1/2
Hence we can calculate DeltaE for the 4p sub levels for j=1+- 1/2
And for the 4d sub levels as j=2+-1/2.
Giving four total values for Delta E as:
C_4p /2, -C_4p , C_4d, -3/2 C_4d
Now given that E= hc/x where x is wavelength we can say that dE= hc dX/ X^2 for small energy splittings.
Now I get stuck, my thinking is to use the fact that the smaller E is the larger the lambda transition is and vice versa, however the 3 values for wavelength is throwing me off. This would give a value for C_4p as 2.6x10^-4 eV if this was the right thinking.

As the lambda split for the 4d level would be roughly 3nm, I’m not sure I could use the differential form of the equation here. So I am doubting my method.
Am I missing a trick here? Any pointers would be much appreciated! Thank you!

 

[DrDu]

1. combining L=2 and S=1/2 gives which values of J?
2. Which selection rules apply?

 

[DrClaude]

Now I get stuck, my thinking is to use the fact that the smaller E is the larger the lambda transition is and vice versa, however the 3 values for wavelength is throwing me off. This would give a value for C_4p as 2.6x10^-4 eV if this was the right thinking.

I don't understand what you are saying here. Maybe you should start by assigning the different wavelengths to the possible transitions.

Edit: @DrDu beat me to it.

As the lambda split for the 4d level would be roughly 3nm, I’m not sure I could use the differential form of the equation here. So I am doubting my method.

I think there is less possibility for errors if you first convert the wavelengths to energies.