Quantum numbers | Principle, azimuthal, Magnetic Quantum Number

An electron in an atom can be described using four quantum numbers, each giving us a unique detail about its position and behavior. They are denoted by the letters n, ℓ, m, and s, called the principal quantum numbers, azimuthal quantum number, magnetic quantum number, and spin quantum number, respectively.

Principal Quantum Number (n)

This quantum number tells us how an electron moves around the nucleus and indicates which shell or orbit it belongs to.. In other words, it is a rough measure of the size of the orbit. The larger the value of n, the greater is the volume of the bulk of the electronic density. n may have any integral value, i.e., 1, 2, 3, 4, 5, … It may take the values up to 7 corresponding roughly to seven horizontal rows of the Periodic Table. The value of n describes the binding force and distance between the nucleus and the orbit in which an electron moves. The lower value of n corresponds to lower energy, while the higher value of n corresponds to higher energy. The energy levels K, L, M, N, O, P, and Q correspond to n = 1, 2, 3, 4, 5, 6, and 7, respectively.
The value of n also determines the number of breaks and discontinuities in the electron cloud. These breaks are also called nodes. The number of breaks is
given by n – 1 and is denoted by ℓ called the azimuthal quantum number.

Azimuthal Quantum Number (l):

This quantum number describes the shape of an orbit. Its value corresponds to the value of n and is given by n – 1. It is also called a secondary quantum number. The value of ‘ℓ’ tells whether the orbit is spherical, is like a dumb bell, sausage shaped or even more complicated. It means that it is associated with a certain value of the angular momentum. The larger the value of ℓ, the more complicated the shape of the electronic cloud. The angular momentum is given by

This quantum number explains the fine structure of the spectral lines in the hydrogen spectrum, thus lending support to the Sommerfeld’s assumption that the orbits are somewhat elliptical rather than being circular. The values of l and the number of orbitals are given in Table.

The resolution of single line into two or more than two lines indicates that there are various sublevels in an energy level, in which an electron revolves. The
energy is also quantized in various sublevels. The main energy shell thus can be considered as being made up of one or more energy sublevels. According to Sommerfeld modification the electrons in any particular cnergy level could either

have a circular orbit or a number of elliptical orbits about the nucleus. The number of sublevels is always equal to the value of n. For example, when n = 1, ℓ = 0, this means that the main energy level and sublevel coincide with each other. When n = 2, ℓ = 0 or 1, which means that there are two sublevels in the second energy level, one having an elliptical shape and the other a circular shape. Similarly, when n = 3, ℓ = 0, 1, 2. This means that there are three sublevels, one having circular shape and the two other with elliptical shapes.

When n = 4, ℓ can have four values (0, 1, 2, 3) corresponding to 4 sublevels, one with circular and three with elliptical shapes. Thus, it is seen that the number of sublevels for a given principal quantum number is equal to the value of that quantum number. So if l = 0, 1, 2, or 3, the electrons are said to be in the s, p, d, or f sublevels after the spectral names called sharp, principal, diffused, or fundamental, respectively.

The Magnetic Quantum Number (m):

This quantum number explains the magnetic properties of an electron. The motion of an electron around a nucleus produces a magnetic field, which can be presented as a vector in the direction of an externally applied field. This vector is zero for an s orbital because of its spherical symmetry. When there is more than one orbital of a given type (ℓ value), these cannot possibly all line up equally with an external magnetic field. This difference in orientation is represented by magnetic quantum number m, which may have values from zero to ± ℓ .

In the above figure, the splitting of the spectral lines is shown for ℓ = 1 (p-orbitals), ℓ= 2 (d-orbitals), and ℓ = 3 (f-orbitals) with the values of 3, 5, and  7 with different orientations in space. The arrows indicate an appropriate quantum number, i.e., the value of m. The angular momentum in the direction of the applied field is given by

The values of m are given in Table 1.5. It is clear from the table that when n = 1, ℓ = 0, m = 0. Hence, the number of orbitals is one, which is circular and is
called the s orbital.

When n = 2, ℓ = 1, m= + 1, 0, -1, the number of orbitals is 3 corresponding to three p orbitals with their orientations parallel to three co-ordinate axes x, y, and z and called p_x p_y and p_z, respectively, Similarly, for ℓ = 2 and ℓ = 3, the number of orbitals will be 5 and 7 orbitals, respectively corresponding to 5 type of d orbitals and 7 type of f orbitals. Each of these orbitals can accommodate two electrons at the most. Thus, s, p, d, and f orbitals accommodate 2, 6, 10, and 14 electrons, respectively, as shown in Table 1.5.

The Spin Quantum Number (s):

This quantum number is associated with the spin of an electron in the atom. All the electrons spin either in a clockwise direction or in an anticlockwise direction of the motion. The direction of the motion can be found by the application of an external magnetic field. Since the probability of motion in each case is 50 % clockwise, the motion is described by the spin quantum дumber having a value of + 1/2. Similarly, the motion in an anticlockwise direction has a value of 1/2. This quantum number is denoted by the letter ‘s’. It is also a measure of the number of units of magnetic moment associated with a given electron due to its interaction with a magnetic field externally applied. The value of the spin momentum is given by

The positive value has a lower energy and the negative one has a higher energy.

Table 1 Quantum Numbers for the first four levels of Orbitals in the Hydrogen atom

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