part 2

The fine electronic structure, related with the atomic-like states and determined by the crystal-field and spin-orbit interactions, has been evaluated by means of different experimental techniques. The importance of the higher-order charge multipolar interactions and the local symmetry of the crystal field for the realized fine electronic structure and for the ground state are pointed out. We pointed out that significant successes of the crystal-field theory indicate on the substantial preservation of the atomic-like structure of the open-shell atoms even when they become the part of a solid. An extension of the CEF theory to a quantum atomistic solid-state theory is proposed.

The discrete electronic structure contains a number of states grouped in terms and multiplets. It is related to the incomplete 4*f (*5*f) *or 3*d* shell. The inner 3*d* shells are closed forming the core ** ^{18}Ar**. The inner 4

*f (*5

*f)*shells are closed forming the

**core. Terms, subset of states characterized by the same values of total**

^{54}Xe**L**and

**S**quantum numbers, are formed by non-central Coulomb interactions. States of the terms are grouped in the multiplets, characterized by the same L, S and J, are formed by spin-orbit interactions. The spin-orbit interactions are for rare-earth ions much stronger then the CEF interactions.

The strong spin-orbit coupling causes that J is the good quantum number and that the first excited multiplet is at least 130 meV (1500 K) above the ground state. Its thermal population in room temperature is negligible. The S, L and J of the ground multiplet are determined by Hund’s rules. The ground multiplet is 2J+1 degenerated – its degeneracy is removed by CEF interactions and magnetic interactions. CEF interactions and magnetic interactions resemble, somehow, Stark and Zeeman effect known from atomic physics. The energies and eigenfunctions of the discrete fine electronic structure are obtained by diagonalization of the (2J+1)-dimensional matrix. The fine electronic structure can be directly detected by

**I**nelastic

**N**eutron

**S**cattering (

**INS**) experiments.

The the case of strong cubic CEF (for 3*d* ions) interactions form group of levels (e.g. T2g, Eg, A2g) which are partially splited by spin-orbit interactions and (if occur) lower symmetry CEF interactions.

The energies and eigenfunctions of the discrete fine electronic structure (for the lowest term) are obtained by diagonalization of the (2L+1)(2S+1)-dimensional matrix.

In the temperature T = 0[K] (absolute zero) only the lowest state is occupied. The magnetic moment at T=0[K] is equal to the moment of the ground state. It allows evaluate the total, spin and orbital moments.

** Hund’s rules :**

**1 ^{st} Hund’s rule: **

*The ground multiplet is characterized by the maximal value of the total spin quantum number S allowed by Pauli principle.*

**2**

^{nd}Hund’s rule:*The ground multiplet is characterized by the maximal value of the total orbital quantum number L provided the 1 Hund rule is satisfied.*

**3 ^{rd} Hund’s rule **(if J is good quantum number only):

*The ground multiplet is characterized by the total angular momentum J of:*

J = |L-S| for n < 7 and J = L+S for n > 7.

J = |L-S| for n < 7 and J = L+S for n > 7.

The eigenstates and corresponding eigenfuncrions |Γ* _{n}*> can be found from direct diagonalization of CEF hamiltonian. The CEF levels are denoted by the irreducible representation |Γ

*> and the corresponding wave functions are:*

_{n}## Details of Theory

**Computable properties:**- entropy S(T)
- specific heat Cmol(T)
- magnetic moment and influence of the external magnetic field for it m(B,T)
- magnetic susceptibility along different crystal directions χi(T)
- visibility of the energy levels in spectroscopy <Γi|
**J**_|Γj>, <Γi|**J**+|Γj> , <Γi|**J**Z|Γj> - spin and orbital momenta of the magnetic ion in solid <Γi|
**L**|Γi>, <Γi|**S**|Γi>