Angular distribution of 4f electrons in the presence of a crystal field

The angular distribution of 4f electrons in the presence of a crystal field has been derived by applying the Stevens operators formalism. For axial symmetry the progressive localization of the 4f electrons from the z-axis to the plane perpendicular may be mainly correlated with the variation of MJ (Mj = J, J 1, ...) according to the sign of the Stevens factor 03B1j. For cubic symmetry, due to the mixing of the |JMj> states in the 4f wavefunctions, the angular distribution consists in several lobes depending on their actual composition. J. Physique 47 (1986) 677-681 AVRIL 1986,., Classification Physics Abstracts 71.50 71.70 75.10 The properties of rare-earth 4f-electrons are generally well described in the Russel-Saunders scheme by considering the set of 2 J + 1 functions I LSJMJ > (or J, Mj )) representing the basis of the ground multiplet with angular momenta L, S and J, satisfying the Hund’s rule. In a crystal this 2 J + 1 degeneracy of the ground multiplet is removed by the Crystalline Electric Field (CEF) due to the neighbouring charges, and split into several CEF levels according to the symmetry of the crystal [1]. The Stevens’ « operator equivalent » method [2] constitutes a very convenient technique to calculate quantitatively this effect of the crystal field, and to obtain the relative energy position of the various CEF levels as well as their nature, in particular with regard to the CEF ground state. Owing to the mixing of the one-electron wavefunctions in the J, Mj > functions, the angular distribution of the 4f-electrons is no more described by the spherical harmonics Y3 (e, 0). A representation in terms of « cigars >> or « pancakes >> has been often used for the Ij, MJ"’" ) and J, MJmax) states, but it is too simplistic. The aim of this paper is to show that the Stevens formalism allows a true description of the spatial distribution of the 4f-shell, in addition to the use mentioned above. This method will be applied to axial and cubic symmetries. 1. Axial symmetry. Let us first consider the simple case where there is no mixing in the CEF wavefunctions, which are thus pure I J, Mj ) states. This is roughly realized in a crystal when the second-order CEF term B2 is preponderant [3]. The corresponding total n-electron wavefunction ’P J,MJ(i , ..., f, a... (In) in real space includes the angular fi and spin ai coordinates of each of the n 4f-electrons. The radial part of the wavefunctions will not be considered here since this paper is concerned only with their angular distribution. For a given Mj, the probability P,,M J(6, ø) of finding a 4f-electron at the angular position (0, q6) (in spherical coordinates) may be expanded in spherical harmonics :