# Electron shell

In atomic spectroscopy, an **electron shell** is set of spatial orbitals with the same principal quantum number *n*. There are *n*^{2} spatial orbitals in a shell; see hydrogen-like atoms. For instance, the *n* = 3 shell contains nine orbitals: one 3*s*, three 3*p*, and five 3*d* orbitals. A shell is **closed** if all orbitals in it are doubly occupied, once with spin up (α) and once with spin down (β). For example, the closed *n* = 1, 2, and 3 shells contain 2, 8, and 18 electrons, respectively.

An **subshell** is a set of 2*l* + 1 spatial orbitals with a given principal quantum number *n* and a given orbital angular momentum quantum number *l*. A subshell is *closed*, if there are 2(2*l* + 1) electrons in the subshell. For instance, the 2*p* subshell in the neon atom contains 6 electrons and, hence, it is closed. Likewise in the copper atom the 3*d* subshell is closed (contains 10 electrons).
A subshell *l* containing a number of electrons *N*, with 1 ≤ *N* < 2(2*l* + 1), is called *open*. The fluorine 2*p* subshell, with electronic configuration 2*p*^{5}, is open.

A closed subshell is an eigenstate of total orbital angular momentum operator squared **L**^{2} with quantum number *L* = 0. That is, the eigenvalue of **L**^{2}, which has the general form *L*(*L* + 1), is zero. A closed subshell is also an eigenstate of total spin angular momentum operator squared **S**^{2} with quantum number *S* = 0. That is, the eigenvalue of **S**^{2}, which has the general form *S*(*S* + 1), is zero. The proof of these two statements will be omitted. Briefly, they rest on the fact that closed (sub)shells have wavefunctions that are Slater determinants and that closed-shell Slater determinants are invariant under the orbit and spin rotation groups, SO(3) and SU(2), respectively.

In the case of hydrogen-like—one-electron—atoms all orbitals within one shell are degenerate, i.e., have the same orbital energy. In the case of more-electron atoms this degeneracy is lifted to a large extent. Provided the orbitals of more-electron atoms are solutions of rotationally invariant effective one-electron Hamiltonians, the orbitals of a *subshell* are still degenerate. This degeneracy of a subshell means that *l* is a "good" quantum number, that is, the one-particle angular momentum operator **l**^{2} commutes with the effective one-electron Hamiltonian. This commutation occurs if, and only if, the effective one-electron Hamiltonian is rotationally invariant.

*See also Hund's rules and Russell-Saunders coupling*.