In an atom, total number of electrons having quantum numbers $$n = 4$$, $$|m_l| = 1$$ and $$m_s = -\frac{1}{2}$$ is ______

We need to find the total number of electrons with quantum numbers $$n = 4$$, $$|m_l| = 1$$, and $$m_s = -\frac{1}{2}$$.

For $$n = 4$$, the possible subshells are $$4s$$ ($$l = 0$$), $$4p$$ ($$l = 1$$), $$4d$$ ($$l = 2$$), and $$4f$$ ($$l = 3$$). We require $$|m_l| = 1$$, which corresponds to $$m_l = +1$$ or $$m_l = -1$$. In each subshell the allowed $$m_l$$ values range from $$-l$$ to $$+l$$.

In the $$4s$$ subshell ($$l = 0$$) the only value of $$m_l$$ is 0, so there are no orbitals with $$|m_l| = 1$$. In the $$4p$$ subshell ($$l = 1$$) the values of $$m_l$$ are $$-1, 0, +1$$, giving two orbitals with $$m_l = +1$$ and $$m_l = -1$$. Similarly, in the $$4d$$ subshell ($$l = 2$$) with $$m_l = -2, -1, 0, +1, +2$$ there are two orbitals with $$|m_l| = 1$$, and in the $$4f$$ subshell ($$l = 3$$) with $$m_l = -3, -2, -1, 0, +1, +2, +3$$ there are also two such orbitals.

The total number of orbitals satisfying $$|m_l| = 1$$ across all subshells is $$0 + 2 + 2 + 2 = 6$$. Since each orbital can accommodate one electron with $$m_s = -\frac{1}{2}$$, the total number of electrons is $$6$$.

The answer is 6.