\[l\]
and \[m\]
and respectively will be similarly opposed, so that spin and angular momentum will sum to zero for electrons in full shells.The spin and angular momentum of the atoms are due solely (for atoms in the ground state) to electrons in the outermost shell.
If electron in the outermost shell have spins
\[s_1, \, s_2,...,s_k \]
the total spin in\[s=s_1+s_2+...+s_k \]
and the magnitude of the total spin is \[S=\sqrt{s(s+1)} \hbar\]
.If electron in the outermost shell have orbital angular momentum numbers
\[l_1, \, l_2,...,s_k \]
then the total orbital angular momentum is \[l=l_1+l_2+...+l_k \]
and the magnitude of the total orbital angular momentum is \[L=\sqrt{l(l+1)} \hbar\]
.The total angular momentum number is
\[j=s+l\]
and the magnitude of the total angular momentum is \[\sqrt{j(j+1)} \hbar\]
.