Approximate estimation of the degree of lanthanide tetrad effect from the data potentially involving all lanthanides

Abstract

Here is presented a mathematical method which enables us to evaluate the degrees of lanthanide tetrad effects from the data involving all of REE. Each of the four spans corresponding to four subgroups, La-Nd, Pm-Gd, Gd-Ho, and Er-La, is approximately fitted to a quadratic function and the resultant quadratic coefficient is employed as a numerical indicator for the tetrad effect. For the purpose of approximation by a quadratic function, the least-squares method it used under the constraint that the first and second quadratic curves intersect at the middle point between Pm and Sm, and that the third and fourth ones intersect at the middle Point between Ho and Er. Since, mathematically, the subject is a problem on the conditional extremum, the method of Lagrange undetermined multipliers should be used. It is noted that the degrees of four tetrad effects can be evaluated simultancously. This method can lead us to the mathematically well-defined estimation of the tetrad effect.