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-Lu, 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 is 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 simultaneously. This method can lead us to the mathematically well-defined estimation of the tetrad effect.

Y-Ho fractionation and lanthanide tetrad effect observed in cherts.

“…Tb and Dy (Masuda et al, 1994;Minami and Masuda, 1997). The values after consideration of mathematical bias for the third tetrad effect due to the above geochemical effect also indicate M type tetrad effect.…”

“…The degrees of four tetrad effects are obtained from the first (La Pr-Nd), second (Sm-Gd), third (Gd-Th-Dy-Ho) and fourth quadratic curves (Er-Tm-Yb-Lu) approxi mately fitted so that the first and second curves intersect each other at the middle point between Nd and Pin, and that the third and fourth ones (Er-Tm-Yb-Lu) intersect each other at the middle point between Ho and Er. Detailed discussion on the approximate calculation is given by Minami and Masuda (1997). The calculated result is given in Sugitani (1992); PHC(S): Pillow Hill chert by Sugitani (1992).…”

The tetrad effect and geochemistry of apatite from the Altay Koktokay No. 3 pegmatite, Xinjiang, China: implications for pegmatite petrogenesis

“…However, with more and more data obtained using ICP-MS where all REE are determined with high accuracy and sensitivity, tetrad effects are increasingly documented in geosciences, such as in highly evolved igneous rocks (Lee et al 1994;Bau 1996;Irber 1999;Jahn et al 2001;Zhao et al 2002;Wu et al 2004;Monecke et al 2007;Yasnygina and Rasskazov 2008;Lee et al 2010;Peretyazhko and Savina 2010b;Zhao et al 2010), pegmatite (Bau 1996Irber 1999;Monecke et al 2002), chert (Minami et al 1998), clastic sediments (Liu et al 1993), meteorites (Inoue et al 2009), pegmatite minerals (Liu and Zhang 2005), fluorite (Monecke et al 2002;Wu et al 2011), zircon (Wu et al 2011), garnet (Wu et al 2011) and monazite (Wu et al 2011), uraninite (Hidaka et al 1992), kimuraite (Akagi et al 1993), scheelite (Liu et al 2007) and xenotime (Masau et al 2000). Masuda et al (1994) and Minami and Masuda (1997) presented a mathematical method to evaluate the degrees of lanthanide tetrad effects. The method of quantification requires that all elements of four tetrads are approximately fitted to a quadratic function and that the resultant quadratic coefficient is employed as a numerical indicator for the degree of the tetrad effect.…”

A depth coincidence between the lanthanide tetrad effect maximum and the oxic/anoxic interface at the CariacoTrench off the Venezuelan coast.

“…However, Ce and Eu are omitted except for rather special conditions. Mathematical Minami and Masuda (1997) When the intra-tetrad fractionation is approxi mated by a quadratic function y = axe + bx + c, one can use either of two parameters to quantita tively indicate the effect under consideration. For one of them, a quadratic coefficient a can be used .…”