Question

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

Answer 1

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, \cdots \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, ..., 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,-(l-1)\cdots,0,\cdots,+(l-1),+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 `2l+1` orbitals. Thus the s-subshells have 1 orbital, p-subshells have 3 orbitals, d-subshells have 5 orbitals and so on.