The general expression for potential energy of a magnetic dipole placed in magnetic field B is given by,
E = -m.B
Where m=magnetic dipole moment
It is given in a book that due to spin orbit interaction, the potential energy possessed by an electron can be written as,
E(s) = K(L.S)

They argue that m is proportional to S(spin quantum number) and B is proportional to L(orbital quantum number).What is K in the above expression? They have calculated the value of K in one of the problems and I found that it has different values for different atoms.

 

Without seeing the text in context, might I suggest K is a constant of proportionality which serves to allow the expression to be equated?

i.e:

E(s) α L.S

With constant of proportionality:

E(s) = K(L.S)

K will vary according to the proportionality of S and B as described in your OP.

Dave