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The crystal field splitting third moment in crystal field analysis: The tetragonal symmetry case
Author(s) -
Mulak Maciej,
Mulak Jacek
Publication year - 2017
Publication title -
physica status solidi (b)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.51
H-Index - 109
eISSN - 1521-3951
pISSN - 0370-1972
DOI - 10.1002/pssb.201700109
Subject(s) - tetragonal crystal system , crystal (programming language) , parametrization (atmospheric modeling) , hamiltonian (control theory) , field (mathematics) , symmetry (geometry) , crystal field theory , physics , ion , condensed matter physics , mathematics , chemistry , quantum mechanics , crystal structure , geometry , crystallography , mathematical optimization , pure mathematics , computer science , programming language , radiative transfer
The second‐ and third‐order crystal field splitting moments are applied to the analysis of the crystal field Hamiltonian H CF = ∑ k ∑ q B kq C q ( k )parametrization in the tetragonal systems. To this end, a useful expression for the crystal field splitting third moment σ 3 3 ( J ) of the state | J 〉 has been formulated. It consists of the sum of 18 terms where each contains a product of three crystal field parameters B k 1 q 1B k 2 q 2B k 3 q 3, for k = 2 , 4 and 6, and q = 0 , ± 4 . The derived σ 3 3 ( J ) formula allows us to investigate the interplay between the B kq parameters in forming the total σ 3 3 ( J ) . For the Eu 3+ :YOCl system (of the C 4 vcentral ion point symmetry), taking advantage of the unique setting of the Eu 3+ ion lowest states with J = 1 , 2 , and 3, all the five H CF parameters B 20 , B 40 , B 44 , B 60 , and B 64 (including their signs) have been calculated successively for all the three multipoles ( k = 2 , 4 , 6 ) by means of the relevant second and the third crystal field splitting moments of the states.A good qualitative agreement with the conventional fitted parametrization has been gained while the quantitative differences are instructive. Within this approach, four various H CF parametrizations for the Nd 3+ :CaWO 4 system (of the S 4 central ion point symmetry) has been verified comparing the second and third crystal field splitting moments of the 4 I 9/2 and 4 I 11/2 states. They have been found by the two independent methods: directly from the splitting diagrams, and from the analytical calculations. The difference between the proper parametrization (positively verified) and the remaining incorrect ones occurs due to the underestimation of the B 44 parameter.