Electron shell: Difference between revisions
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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''<sup>5</sup>, is open. | 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''<sup>5</sup>, is open. | ||
A closed subshell is an eigenstate of total [[Angular momentum (quantum)#Orbital angular momentum|orbital angular momentum operator]] squared '''L'''<sup>2</sup> with quantum number ''L'' = 0. That is, the eigenvalue of '''L'''<sup>2</sup>, which has the general form ''L''(''L''+1), is zero. A closed subshell is also an eigenstate of total [[Angular momentum (quantum)#Spin angular momentum|spin angular momentum operator]] squared '''S'''<sup>2</sup> with quantum number ''S'' = 0. That is, the eigenvalue of '''S'''<sup>2</sup>, 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 determinant]]s | A closed subshell is an eigenstate of total [[Angular momentum (quantum)#Orbital angular momentum|orbital angular momentum operator]] squared '''L'''<sup>2</sup> with quantum number ''L'' = 0. That is, the eigenvalue of '''L'''<sup>2</sup>, which has the general form ''L''(''L''+1), is zero. A closed subshell is also an eigenstate of total [[Angular momentum (quantum)#Spin angular momentum|spin angular momentum operator]] squared '''S'''<sup>2</sup> with quantum number ''S'' = 0. That is, the eigenvalue of '''S'''<sup>2</sup>, 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 determinant#Atomic closed shells|Slater determinant]]s 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 atom|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'''<sup>2</sup> commutes with the effective one-electron Hamiltonian. This commutation occurs if, and only if, the effective one-electron Hamiltonian is rotationally invariant. | In the case of [[hydrogen-like atom|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'''<sup>2</sup> commutes with the effective one-electron Hamiltonian. This commutation occurs if, and only if, the effective one-electron Hamiltonian is rotationally invariant. | ||
Revision as of 09:38, 12 January 2008
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 cupper 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.