Crystal-Field splitting

In summary, the crystal-field splitting for f-orbitals is not explained in the same way as it is for d-orbitals, as f-orbitals are treated using complex-orbitals. The lack of degeneracy for 4f-orbitals in Gadolinium(III) ion's 8S-state is due to the strong LS coupling, where L, S, and J are good quantum numbers. The resulting orbitals can also be chosen to be real. Mathematically, there are four distinct CF Eigenstates for J=7/2, forming four Kramers doublets with a very small splitting due to the spherical symmetry of the original atomic orbits.
  • #1
HAYAO
Science Advisor
Gold Member
376
235
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.
 
Physics news on Phys.org
  • #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
http://www.sciencedirect.com/science/article/pii/0022369762901920

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.
 

What is crystal-field splitting?

Crystal-field splitting is a phenomenon that occurs when a metal ion is surrounded by ligands, causing the degenerate energy levels of the ion's d orbitals to split into different energy levels. This splitting is due to the electrostatic interaction between the metal ion and the ligands, and it affects the optical and magnetic properties of the metal ion.

What causes crystal-field splitting?

Crystal-field splitting is caused by the presence of ligands around a metal ion. These ligands are typically negatively charged ions or molecules that are attracted to the positively charged metal ion. The electrostatic interaction between the metal ion and the ligands causes the degenerate energy levels of the metal ion's d orbitals to split into different energy levels.

How does crystal-field splitting affect the properties of a metal ion?

The splitting of the energy levels due to crystal-field splitting affects the optical and magnetic properties of a metal ion. This is because the energy levels determine the color and magnetic behavior of the ion. The number and relative energies of the split energy levels can also provide information about the geometry and bonding in a complex.

What is the difference between high spin and low spin configurations in crystal-field splitting?

In high spin configurations, the energy difference between the split energy levels is small, so the electrons occupy the higher energy levels first before pairing up. This results in a larger number of unpaired electrons and a higher magnetic moment. In low spin configurations, the energy difference between the split energy levels is large, so the electrons prefer to pair up in the lower energy levels. This results in a smaller number of unpaired electrons and a lower magnetic moment.

How is crystal-field splitting relevant in chemistry and materials science?

Crystal-field splitting is relevant in chemistry and materials science because it helps to explain the optical and magnetic properties of metal complexes. It also provides information about the geometry and bonding in these complexes, which is important in understanding their reactivity and potential applications in areas such as catalysis and materials science.

Similar threads

  • Advanced Physics Homework Help
Replies
5
Views
1K
  • Atomic and Condensed Matter
Replies
3
Views
2K
Replies
2
Views
2K
  • Quantum Physics
Replies
18
Views
1K
  • Atomic and Condensed Matter
Replies
11
Views
9K
  • Chemistry
Replies
2
Views
2K
  • Biology and Chemistry Homework Help
Replies
1
Views
3K
  • Atomic and Condensed Matter
Replies
3
Views
1K
  • Quantum Physics
Replies
5
Views
736
Replies
10
Views
911
Back
Top