# What is the difference between shell, subshell, and orbital?

Dec 18, 2014

The Schroedinger equation for an electron bound to a spherically symmetric coulomb potential of a hydrogen like nuclei shows that the wave function of the electron forms standing waves called stationary states. These states are characterized by three quantum numbers.

1. Principal Quantum Number ($n$) : $n = 1 , 2 , 3 , \setminus \cdots \setminus \infty$
For single electron systems the allowed energy values (energy levels) are determined purely by the principle quantum number. This quantum number can only take integer values starting with $1$ with no upper bound. All the electronic states with the same principle quantum number are said to belong to the same shell.
The $n = 1$ states are labeled K-shell , $n = 2$ states are labeled L-shell , $n = 3$ states are labeled M-shell , $n = 4$ states are labeled N-shell and so on.

2. Angular Momentum Quantum Number ($l$): $l = 0 , 1 , 2 , \ldots , n - 1$.
This quantum number determines the magnitude of the orbital angular momentum of the electron. It can take only integer values starting from 0 but has an upper bound. It can only go up to a number that is one less than the principal quantum number. While the shells are a bigger group of quantum states, this quantum number breaks them into smaller groups of quantum states called sub-shells. All quantum states with the same orbital angular momentum quantum number are said to belong to the same sub-shell. The $l = 0$ states are labeled s-subshell, $l = 1$ states are labeled p-subshell, $l = 2$ states are labeled d-subshell, $l = 3$ states are labeled f-subshell and so on. Thus the quantum states belonging to a shell with principal quantum number $n$ are divided into $n$ subshells.

3. Magnetic Quantum Number (${m}_{l}$): ${m}_{l} = - l , - \left(l - 1\right) \setminus \cdots , 0 , \setminus \cdots , + \left(l - 1\right) , + l$
This quantum number determines the magnitude of the component of the orbital angular momentum vector of the electron along a reference direction (usually the direction of an applied external magnetic field). This quantum number breaks the sub-shells into further small groups called orbitals . All the electronic states with the same Magnetic Quantum Number (${m}_{l}$) belong to the same orbital. A sub-shell characterized by a angular momentum quantum number $l$ has $2 l + 1$ orbitals. Thus the s-subshells have 1 orbital, p-subshells have 3 orbitals, d-subshells have 5 orbitals and so on.