A direct algebraic parametrization of the high‐symmetry crystal‐field Hamiltonians

An algebraic, calculational, non‐fitting method of parametrization of the cubic and hexagonal crystal‐field (CF) Hamiltonians ( H CF ) with only two independent CF parameters B 40 and B 60 is proposed. It is based on the linear relation σ 2 ( J ) = ∑ k A k 2 ( J ) S k 2 between σ 2 ( J ) the square of the second moment of the electron state | J 〉 CF splitting and the squares of the multipolar CF strengths S k 2 = 1 2 k + 1∑ q | B kq | 2 , where A kJ = 〈 J | | C ( k ) | | J 〉 . For a pair of | J i 〉 and | J j 〉 electron states an algebraic solution for S 4 2 and S 6 2 , and hence for the B 40 and B 60 CF parameters can be gained. The necessary conditions for physical correctness of the solutions ( S k 2 ≥ 0 ) impose limitations on the observed σ 2 ( J i ) / σ 2 ( J j ) ratios depending on the respective ratios A k 2 ( J i ) / A k 2 ( J j ) . These restrictions can be used to verify postulated CF splitting diagrams. In the high‐symmetry crystal fields, the electron states with J = 2 or 5 / 2 undergo splitting only by the k = 4H CF multipole, and from their CF splitting second moments the B 40 parameter can be directly obtained. For J = 2 , | B 40 | =63 10Δ ( J ) | A 4 ( J ) | , whereas for J = 5 / 2 , | B 40 | = 7Δ ( J ) | A 4 ( J ) | , where Δ is the energy gap between the two Stark's levels of the | J 〉 state. The proposed method is demonstrated and discussed for several systems of RE3 +ions in some high‐symmetry crystal matrices.