Absolute value of the total magnetic moment of an electron in an atom – physics.stackexchange.com
cipper
The magnetic moment operator being
$$\mu = -\frac{\mu_\text{B}}{\hbar}(L+2S),$$
a direct evaluation of its absolute value would be
$$|\mu| = (\mu^2)^{1/2} = \frac{\mu_\text{B}}{\hbar}(L^2 + 4S^2 + ...