A refined solution to Earth's hidden niobium: implications for evolution of continental crust and mode of core formation

New high-precision niobium (Nb) and tantalum (Ta) concentration data are presented for early Archaean metabasalts, metabasaltic komatiites and their erosion products (mafic metapelites) from SW Greenland and the Acasta gneiss complex, Canada. Individual datasets consistently show sub-chondritic Nb/Ta ratios averaging 15.1+/-11.6. This finding is discussed with regard to two competing models for the solution of the Nb-deficit that characterises the accessible Earth. Firstly, we test whether Nb could have sequestered into the core due to its slightly siderophile (or chalcophile) character under very reducing conditions, as recently proposed from experimental evidence. We demonstrate that troilite inclusions of the Canyon Diablo iron meteorite have Nb and V concentrations in excess of typical chondrites but that the metal phase of the Grant, Toluca and Canyon Diablo iron meteorites do not have significant concentrations of these lithophile elements. We find that if the entire accessible Earth Nb-deficit were explained by Nb in the core, only ca. 17% of the mantle could be depleted and that by 3.7 Ga, continental crust would have already achieved ca. 50% of its present mass. Nb/Ta systematics of late Archaean metabasalts compiled from the literature would further require that by 2.5 Ga, 90% of the present mass of continental crust was already in existence. As an alternative to this explanation, we propose that the average Nb/Ta ratio (15.1+/-11.6) of Earth's oldest mafic rocks is a valid approximation for bulk silicate Earth. This would require that ca. 13% of the terrestrial Nb resided in the Ta-free core. Since the partitioning of Nb between silicate and metal melts depends largely on oxygen fugacity and pressure, this finding could mean that metal/silicate segregation did not occur at the base of a deep magma ocean or that the early mantle was slightly less reducing than generally assumed. A bulk silicate Earth Nb/Ta ratio of 15.1 allows for depletion of up to 40% of the total mantle. This could indicate that in addition to the upper mantle, a portion of the lower mantle is depleted also, or if only the upper mantle were depleted, an additional hidden high Nb/Ta reservoir must exist. Comparison of Nb/Ta systematics between early and late Archaean metabasalts supports the latter idea and indicates deeply subducted high Nb/Ta eclogite slabs could reside in the mantle transition zone or the lower mantle. Accumulation of such slabs appears to have commenced between 2.5 and 2.0 Ga. Regardless of these complexities of terrestrial Nb/Ta systematics, it is shown that the depleted mantle Nb/Th ratio is a very robust proxy for the amount of extracted continental crust, because the temporal evolution of this ratio is dominated by Th-loss to the continents and not Nb-retention in the mantle. We present a new parameterisation of the continental crust volume versus age curve that specifically explores the possibility of lithophile element loss to the core and storage of eclogite slabs in the transition zone. (C) 2003 Elsevier Science B.V. All rights reserved.

An Analytical Solution for the Elastic Lateral-Distortional Buckling of I-section Beams

Lateral-distortional buckling may occur in I-section beams with slender webs and stocky flanges. A computationally efficient method is presented in this paper to study this phenomenon. Previous studies on distortional buckling have been on the use of 3(rd) and 5(th) order polynomials to model the displacements. The present study provides an alternative way, using Fourier Series, to model the behaviour. Beams of different cross-sectional dimensions, load cases and restraint conditions are examined and compared. The accuracy and versatility of the method are verified by calibrating against the results of other published studies. The present method is believed to be a simple and efficient way of determining the buckling load and mode shapes of I-section beams that are susceptible to lateral-distortional buckling modes.

Estimating variable returns to scale production frontiers with alternative stochastic assumptions

Two stochastic production frontier models are formulated within the generalized production function framework popularized by Zellner and Revankar (Rev. Econ. Stud. 36 (1969) 241) and Zellner and Ryu (J. Appl. Econometrics 13 (1998) 101). This framework is convenient for parsimonious modeling of a production function with returns to scale specified as a function of output. Two alternatives for introducing the stochastic inefficiency term and the stochastic error are considered. In the first the errors are added to an equation of the form h(log y, theta) = log f (x, beta) where y denotes output, x is a vector of inputs and (theta, beta) are parameters. In the second the equation h(log y,theta) = log f(x, beta) is solved for log y to yield a solution of the form log y = g[theta, log f(x, beta)] and the errors are added to this equation. The latter alternative is novel, but it is needed to preserve the usual definition of firm efficiency. The two alternative stochastic assumptions are considered in conjunction with two returns to scale functions, making a total of four models that are considered. A Bayesian framework for estimating all four models is described. The techniques are applied to USDA state-level data on agricultural output and four inputs. Posterior distributions for all parameters, for firm efficiencies and for the efficiency rankings of firms are obtained. The sensitivity of the results to the returns to scale specification and to the stochastic specification is examined. (c) 2004 Elsevier B.V. All rights reserved.