Azimuthal Quantum Number

The azimuthal quantum number is a set of non-negative integers which define the shape of an orbital. It is denoted by l. l = 0, 1, 2, 3, 4… It is also called as the orbital angular momentum quantum number or orbital quantum number because it determines the orbital angular momentum, which is analogous to the classical angular momentum.

It is one of the four quantum numbers that identify an electron in an atom. The others are the principal quantum number, magnetic quantum number, and spin quantum number.

The azimuthal quantum number is a very important since it governs the geometry of an orbital and the orbital angular momentum. It was put forward by Arnold Sommerfeld, a German theoretical physicist.

The orbital angular momentum is conserved and quantised. Its magnitude is an integral multiple of l(l + 1); in other words, l quantises the momentum.

Here, ℏ is the reduced Planck constant and equal to , and l is the azimuthal quantum number, l ∈ {0, 1, 2, 3…}. The value of the orbital angular momentum increases with l and it is zero when l = 0, which is the s subshell described below.

Each value of the azimuthal quantum number corresponds a subshell, for example l = 0 represents an s subshell. The s subshell is the simplest of all and has a spherical shape. It has only one orbital, s orbital.

l = 1 represents a p subshell. It has three dumbbell-shaped orbitals: p_x, p_y, and p_z. Each orbital is aligned with one of the three axes as shown in the figure below.

l = 2 represents a d subshell. It is complicated than p; it has 4 dumbbell-shaped orbitals and 1 doughnut-shaped orbital: d_xy, d_xz, d_yz, d_x²−y², and d_z².

l = 3 is an f subshell, which is more complex than d, and has 7 orbitals. As the value of the azimuthal quantum increases, the subshell becomes more convoluted.

The letter s, p, d, or f assigned by early spectroscopists from the first letter of the description of spectral lines in certain alkali metals. The letter s stands for sharp, p for principal, d for diffuse, and f for fundamental.

From l = 4 onwards, the naming follows the alphabetical order except in the case of j, which is omitted. Thus, l = 4 is g, l = 5 is h, l = 6 is i, l = 7 is k, and so on.

Relation with other quantities

Principal quantum number

For a given value of the principal quantum number n, the restricted values of l are 0, 1, 2, 3 … n − 1. l cannot be greater than or equal to n, i.e. l < n.

When n = 1, l = 0, when n = 2, l = {0, 1}, when n = 3, l = {0, 1, 2}, when n = 4, l = {0, 1, 2, 3}, and so forth.

Magnetic quantum number

If you have notice above, the number of orbitals for a given value of l is equal to 2l + 1. The number 2l + 1 is the number of possible values of the magnetic quantum number m_l ranging from −l, −(l − 1) … −1, 0, 1 … (l − 1), l.

For example, l = 4 has 2 × 4 + 1 = 9 orbitals and m_l = {−4, −3, −2, −1, 0, 1, 2, 3, 4}.

No. of electrons

Each orbital can have up to 2 electrons. Thus, the maximum number of electrons in a subshell is 2(2l + 1). For example, l = 4 (g subshell) is 18.

The table down gives a summary of the article.

Azimuthal quantum number

Azimuthal quantum number, l	Subshell	Name	No. of orbitals (2l + 1)	Max no. of electrons (2(2l + 1))	Shape
0	s	sharp	1	2	spherical
1	p	principal	3	6	dumbbell
2	d	diffuse	5	10	dumbbell and doughnut
3	f	fundamental	7	14	-
4	g	-	9	18	-
5	h	-	11	22	-
6	i	-	13	26	-
7	k	-	15	30	-
8	l	-	17	34	-
…	…	…	…	…	-

Associated articles

• Principal quantum number
• Bohr's atomic model