 #1
Ted55
 3
 0
 Homework Statement

Find the spinorbit Coupling constants of a 4p4d 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!
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!