Electron shell

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In atomic spectroscopy, an electron shell is set of spatial orbitals with the same principal quantum number n. There are n2 spatial orbitals in a shell; see hydrogen-like atoms. For instance, the n = 3 shell contains nine orbitals: one 3s, three 3p, and five 3d 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 2l + 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(2l + 1) electrons in the subshell. For instance, the 2p subshell in the neon atom contains 6 electrons and, hence, it is closed. Likewise in the copper atom the 3d subshell is closed (contains 10 electrons). A subshell l containing a number of electrons N, with 1 ≤ N < 2(2l + 1), is called open. The fluorine 2p subshell, with electronic configuration 2p5, is open.

A closed subshell is an eigenstate of total orbital angular momentum operator squared L2 with quantum number L = 0. That is, the eigenvalue of L2, which has the general form L(L + 1), is zero. A closed subshell is also an eigenstate of total spin angular momentum operator squared S2 with quantum number S = 0. That is, the eigenvalue of S2, 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 l2 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.