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×22nd Jan 2020 @ 4 min read

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.

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.

If you have notice above, the number of orbitals for a given value of *l* is equal to 2*l* + 1. The number 2*l* + 1 is the number of possible values of the magnetic quantum number *m _{l}* ranging from −

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

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

The table down gives a summary of the article.

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 | - |

… | … | … | … | … | - |

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