Azimuthal quantum number
The Azimuthal quantum number l corresponds to the angular momentum of the state. These states take the form of spherical harmonics, and so are described by Legendre polynomials. The various states relating to different values of l are sometimes called sub-shells, and (mainly for historical reasons) are referred to by letter, as follows:
| l value | Letter | Maximum number of electrons in shell |
|---|---|---|
| 0 | s | 2 |
| 1 | p | 6 |
| 2 | d | 10 |
| 3 | f | 14 |
| 4 | g | 18 |
Each of the different angular momentum states can take 2(2l+1) electrons. This is because the third quantum number ml (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 nlml state can be occupied by two electrons with opposing spins (given by the quantum number ms), giving 2(2l+1) electrons overall. States with higher l than given in the table are perfectly permissible in theory, but these values cover all atoms so far discovered.
For a given value of 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).