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I Crystal-Field splitting

  1. Jun 28, 2016 #1
    There is several things I don't quite understand about crystal-field splitting.

    Q1: Why is crystal-field effect is not explained for f-orbitals like in d-orbitals in terms of real-orbitals?
    Correct me if I am wrong, but typically for d-orbitals, crystal-field splitting is explained classically in terms of alignment of d-orbitals in real-orbitals form (xy, yz, zx, z2, x2-y2) to the surrounding electric field potential (such as oxygen). Why is the same type of explanation not done for f-orbitals? Crystal-field splitting for f-orbitals (Stark-splitting) is explained instead through use of complex-orbitals (so magnetic quantum number is a good quantum number) and their use in eventual derivation of term symbols.

    Q2: I don't quite understand the lack of degeneracy of 4f-orbitals for 8S-state Gadolinium(III) ion.
    I read that because L = 0 (hence the 8S-state), the crystal-field splitting (to the first order) is nonexistent. That much I can understand since L = 0, which means the magnitude of orbital angular momentum is 0. But it also explain that this is because the orbitals are nondegenerate. That sounds contradicting. There are seven 4f-orbitals and term 8S means that this state has one electron in each of those orbitals which should be degenerate.
  2. jcsd
  3. Jun 30, 2016 #2
    Q1: For (3d) transition element d-orbitals, the spin-orbit coupling is in general much weaker than the crystal field splitting. Therefore one starts with the CF levels and then fills them with electrons. The orbitals (xy, ...) are essentially the same for all 3d elements. For 5d transition metals the situation becomes a bit more complex, as the LS coupling becomes stronger.

    For 4f elements, the situation is the opposite: LS coupling is strong, and the CF is just a perturbation. L, S and J are assumed to be good quantum numbers. You therefore first find the LS ground state (which is different for each element) and then see how the 2J+1 levels are split by the crystal field, i.e. m_J is no longer a good quantum number. Unless I am mistaken, the resulting orbitals can in general also be chosen to be real. See e.g. Lea, Leask and Wolf

    Q2: Mathematically you will find well-defined and distinct CF Eigenstates for J=7/2. From a symmetry point of view they form 4 Kramers doublets (half-integer J are always at least doubly degenerate). The CF analysis takes only the crystal field and the J-state into account, therefore the 4 doublets are assumed to be split. Symmetry alone, however, cannot make any statements about the magnitude of the splitting - except when it vanishes because the orbitals are symmetry related. The actual splitting between them, however, will be very very small, as the original atomic orbits were L=0 and thus have spherical symmetry.
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