Azimuthal quantum number

The Azimuthal quantum number (or orbital angular momentum quantum number) l is a quantum number for an atomic orbital which determines its orbital angular momentum. Their wavefunctions take the form of spherical harmonics, and so are described by Legendre polynomials. The various orbitals relating to different values of l are sometimes called sub-shells, and (mainly for historical reasons) are referred to by letter, as follows:

Each of the different angular momentum states can take 2(2l+1) electrons. This is because the third quantum number m_l (which can be thought of [somewhat inaccurately] as the [quantised] projection of the angular momentum vector on the z-axis) runs from −l to l in integer units, and so there are 2l+1 possible states. Each distinct nlm_l orbital can be occupied by two electrons with opposing spins (given by the quantum number m_s), giving 2(2l+1) electrons overall. Orbitals with higher l than given in the table are perfectly permissible, but these values cover all atoms so far discovered.

For a given value of the Principal quantum number, n, the possible values of l range from 0 to n−1; therefore, the n=1 shell only possesses an s subshell and can only take 2 electrons, the n=2 shell possesses an s and a p subshell and can take 8 electrons overall, the n=3 shell possesses s, p and d subshells and has a maximum of 18 electrons, and so on (generally speaking, the maximum number of electrons in the nth energy level is 2n²).