Magnetic Quantum Number

The magnetic quantum number is a set of integers that determine the spatial orientation of an orbital. It defines the orbital and is unique to each orbital for a given value of the azimuthal quantum number. It is symbolized as m_l. m stands for magnetic and the subscript l for azimuthal. m_l = … −2, −1, 0, 1, 2…

The number along with the principal quantum number, azimuthal quantum number, and spin quantum number is used to identify each and every electron in an atom. It was introduced by Arnold Sommerfeld, a German theoretical physicist, who also proposed the azimuthal quantum number.

It determines the direction of the orbital angular momentum. The magnetic quantum number does not decide the energy of an orbital. However, it affects the energy of an orbital in the presence of an external static magnetic field. The magnetic quantum number together with the spin quantum number contributes to the magnetic moment of an electron.

The values of the magnetic quantum number are restricted by the azimuthal quantum number l. For a given value of l, we can have 2l + 1 possible values of m_l, ranging from −l, −(l − 1) … −2, −1, 0, 1, 2 … (l − 1), l.

For l = 0, m_l = 0. This represents an s orbital, which is spherical in shape. For an s orbital, the orbital angular momentum is zero and it does not point in any specific directions.

For l = 1, m_l = {−1, 0, 1}. Each value represents a p orbital. When m_l = 0, it is the p_z orbital. It is aligned along the z-direction having each lobe on either of the origin. m_l = ±1 are the p_x and p_y orbitals. p_x and p_y rest on x- and y-axes.

l = 2 is the d subshell and it has five orbitals. The magnetic quantum numbers for d subshell are −2, −1, 0, 1, and 2. m_l = 0 is the d_z² orbital, which is oriented along the z-axis. The orbitals d_xz and d_yz have m_l = ±1 and lie in the xz and yz planes. m_l = ±2 corresponds to d_xy and d_{x² − y²}; both lie in the xy plane.

l for f subshell is 3 and its has seven orbitals: f_z³, f_xz², f_yz², f_xyz, f_{z(x² − y²)}, f_{x(x² − 3y²)}, and f_{y(3x² − y²)}. And the magnetic quantum number associated with these orbitals are −3, −2, −1, 0, 1, 2, and 3.

Similarly for g orbital, m_l = −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, and 5.

Each orbital can hold utmost two electrons. Thus, every m_l can have two electrons; however, it cannot distinguish those two electrons. They can only be identified by the spin quantum number.

The magnetic quantum number m_l depends on the azimuthal quantum number l, which relies on the principal quantum number n. We can establish the relation between m_l and n. For every value of n, there can be n values of l ranging from 0, 1, 2, 3 … n − 2, n − 1.

Therefore, n² values of m_l are possible for every n.

Associated articles

• Azimuthal quantum number
• Principal quantum number
• Bohr's atomic model