The Smoothness and Shapes of Chondrite-normalized Rare Earth Element Patterns in Basalts
Authors: O’Neill Hugh St. C. 
Published: February 2017

Abstract
The customary practice for displaying REE abundances is to normalize them to chondritic abundances and then to plot these normalized abundances in order of atomic number, Z , although the 3 + ionic radius, r REE , is proposed here as a preferable independent variable. In basalts, the resulting CI-normalized REE patterns usually appear smooth (excepting Eu), such that they may be fitted to polynomials in r REE with three to five terms, depending on analytical precision. The polynomials can be rearranged into an orthogonal form:
ln([REE]/[REE] CI )=λ 0 +λ 1 f1orth +λ 2 f2orth +…
where f1orth , f2orth , etc. are themselves polynomials of r REE , chosen such that the coefficients λ 0 , λ 1 , λ 2 , etc. are not correlated with each other. The terms have a simple, intuitive meaning: λ 0 is the average of the logarithms of the CI-normalized REE abundances; the term in f1orth describes the linear slope of the pattern; that in f2orth describes the quadratic curvature, etc. For most basalts, fits using only three terms (λ 0 , λ 1 , and λ 2 ) capture REE patterns to better than ±5%. The λ n , called the ‘shape coefficients’, can be used to compare the shapes of CI-normalized REE patterns quantitatively, allowing large numbers of data to be assessed, revealing trends not evident from studies of single localities. Especially instructive are λ 2 vs λ 1 diagrams. The usefulness of this approach is demonstrated using the REE patterns of common types of basalts from (mainly) oceanic settings: ocean floor basalts (OFB), ocean island basalts (OIB), and some convergent margin basalts. It is shown that the global population of OFB is characterized by a narrow dispersion of λ 0 at a given MgO content, but with large variations of λ 1 and λ 2 . Convergent margin basalts have much greater variation of λ 0 at a given [MgO], but most plot in the same area of the λ 2 vs λ 1 diagram. OIB are well separated from the OFB global array on this diagram, with Hawaiian shield basalts occupying a unique area. Because REE mineral/melt partition coefficients are also smooth functions of r REE , many mass-balance equations for petrogenetic processes that relate observed concentrations to initial concentrations, [REE] o , such as batch or fractional melting, or crystallization, may be fitted to the same orthogonal polynomials:
ln([REE]/[REE] o )=ψ 0 +ψ 1 f1orth +ψ 2 f2orth +… .
The orthogonality ensures that all λ n and ψ n terms of the same order n sum independently of the terms of the other orders, such that λ n = λn0 + ψ n , where λn0 is the shape coefficient of the source or parent magma. On λ 2 vs λ 1 diagrams, this approach can be used to relate the shapes of patterns in parental basalts to the shapes of the patterns of their sources, or differentiated basalts to their parental melts, by means of ‘petrogenetic process vectors’ consisting of the ψ 1 and ψ 2 terms, which plot as vectors on the λ 2 vs λ 1 diagrams. For example, the difference between OIB and the global array of OFB can be shown to be due to garnet in the sources of OIB. The global array of OFB requires a remarkably constant degree of partial melting ( F ) of a source with constant λ 0 to produce their parental magmas, or a compensating correlation between F and source λ 0 . Assuming a constant source, with previously suggested depleted mantle compositions, F is ∼19%, with the standard deviation of the population being only 2%. Hawaiian shield tholeiites may be products of 1–2% melting at substantially higher pressures, perhaps straddling the garnet-to-spinel transition, of a source with REE patterns near the median of the REE patterns of OFB sources. Other OIB are the result of lower degrees of melting, usually of more light REE-enriched sources.