Consider the transition $5$s $3^$S$->4$p $3^$P only in calcium $($Ca$).$

$($a$)$ Explain why the transition splits into three separate lines when fine structure is taken into account. Give the initial and final J quantum number for each transition.

$($b$)$ The L$-$S coupling constant for the $4$p $3^$P state of Ca is $\backslash$beta $\_$LS $=6\mathrm{.}53$ meV. In the absence of L$-$S coupling, the transition wavelength is $614\mathrm{.}219$ nm$.$ Calculate the wavelength of the lines including the effect of the fine str