Electron shell: Difference between revisions
imported>Paul Wormer (New page: In atomic spectroscopy, an '''electronic shell''' is set of spatial orbitals with the same principal quantum number ''n''. There are ''n''<sup>2</sup> spatial orbitals in a shell...) |
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In [[atomic spectroscopy]], an ''' | In [[atomic spectroscopy]], an '''electron shell''' is set of spatial orbitals with the same [[principal quantum number]] ''n''. There are ''n''<sup>2</sup> spatial orbitals in a shell, see [[hydrogen-like atom]]s. 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 | An '''subshell''' is a set of 2''l''+1 spatial orbitals with a given principal quantum number ''n'' and a given [[Angular momentum (quantum)#Orbital angular momentum|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 cupper atom the 3''d'' subshell is closed (contains 10 electrons). | ||
A subshell ''l'' | 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 which are invariant under the action of '''L''' and '''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]]s which are invariant under the action of '''L''' and '''S'''. | ||
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. | |||
[[Category: CZ Live]] | [[Category: CZ Live]] | ||
[[Category: Chemistry Workgroup]] | [[Category: Chemistry Workgroup]] | ||
[[Category: Physics Workgroup]] | [[Category: Physics Workgroup]] | ||
Revision as of 07:51, 11 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 which are invariant under the action of L and S.
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.