Concepto de derecho - Apuntes - Filosofia del Derecho, Apuntes de Filosofía del Derecho

Filosofía del Derecho

Descripción: Apuntes de Filosofía del Derecho sobre el Concepto de derecho, Problemática de la definición de derecho, Formas de vivir el derecho, Vías para llegar a un concepto científico del derecho, Tipos de definiciones del concepto derecho - UAB Universidad Autònoma de Barcelona

Geol. 656 Isotope Geochemistry MANTLE MODELS THE TWO-RESERVOIR MANTLE MODEL The initial interpretation of isotopic variations in the mantle (ca. 1975) was a two-reservoir model: an upper depleted mantle overlying a lower mantle that was ‘primitive’, or possibly enriched in incompatible elements. The idea that the lower mantle was primitive gained favor with the acquisition of Nd isotope data. The first Nd data obtained showed that Nd and Sr isotope ratios in oceanic basalts were well correlated and that Nd isotope ratios fell between typical MORB values of about e Nd = +10 and the primitive mantle value of eNd = 0. Mixing between these two reservoirs could explain most of the isotopic variation seen in mantle-derived rocks. This sort of model is illustrated in Figure 18.1. There were, and are, good arguments why the depleted reservoir should overlie the primitive one. First, it is generally thought the depleted reservoir acquires its characteristics through loss of a partial melt to form the crust. Obviously this reservoir should then be nearer the continental crust. In addition, depleted peridotite is less dense than undepleted peridotite. Second, the depleted reservoir seems to be sampled wherever rifting occurs, not only at major mid-ocean ridges, but also at smaller rifts. For example, the Cayman Trough, or Fracture Zone, is a transform fault in the Caribbean separating the South and North American Plates. Because of the nature of plate motion, there is a very small amount of spreading occurring within the Trough. B a salts erupted within the Trough are indistinguishable from those at the Mid-Atlantic Ridge. If the primitive reservoir overlay Figure 18.1 The two reservoir the depleted one and the depleted one were sampled only where model of the mantle. The demajor mantle convection currents carried it upward, we certainly pleted mantle is the source of would not expect to find it sampled in a place like the Cayman MORB and has e Nd = +10, t h e Trough. On t

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he other hand, the deeper reservoir seems to be sam- lower mantle is primitive and pled exclusively, or nearly so, where there is independent evi- has bulk Earth characteristics, dence for major mantle upwelling in the form of mantle plumes. e.g., eNd = 0. The geophysical evidence for this includes both gravity and elevation anomalies. In a simple three reservoir model such as that pictured in Figure 18.1, it is possible to compute t h e relative masses of the depleted and primitive mantle if several parameters are known. The basic equations are simple mass-balance ones. For example, for the Nd isotopic system we may write t h e following mass balance equations: Since we assume that the bulk Earth has eNd = 0, we can write: ÂM j j j C jeNd = 0 18.1 j where Mj is the mass of the jth reservoir, Cj is the concentration of Nd in that reservoir, and eNd is t h e value of eNd in that reservoir. We also assume the Sm/Nd is chondritic. We’ll use f Sm/Nd to denote the relative deviation of the Sm/Nd ratio from the chondritic value, i.e.: † f Sm / Nd = 147 Sm /144 Nd -147 Sm /144 NdCHUR 147 Sm /144 NdCHUR 117 18.2 Then we may write a similar mass balance for the Sm/Nd ratio for the Earth: † Geol. 656 Isotope Geochemistry ÂM j j j C j f Sm / Nd = 0 18.3 The mass balance for the Nd concentration is: ÂM † j j j o CNd = M oCNd 18.4 where Mo is the mass of the silicate Earth and CNd in the concentration of Nd in the silicate Earth. Finally, the masses of our three reservoirs must sum to the mass of the silicate Earth: o † ÂM j j = Mo 18.5 The first equation simply states that the bulk-earth eNd = 0, the second that the Sm/Nd ratio is equal to the chondritic one, the third is the mass balance equation for Nd concentration (CNd), the fourth states that the masses of the three reservoirs must equal the total mass of the silicate earth (denoted by the superscript 0). We have implicitly assumed there is no Nd or Sm in the core. Ass

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uming t h a t † the crust has grown from primitive mantle, then* c c eNd = f Sm / Nd QT c 18.6 where TC is the average age of the crust. If the Earth consists of only three reservoirs for Nd, namely the continental crust, depleted mantle, and primitive mantle, and if the depleted mantle and crust evolved from a reservoir initially identical to ‘primitive mantle’ then the mass balance equations † 18.1, 18.3, and 18.4 must hold for crust and depleted mantle alone. In this case, equations 18.1, 18.4 and 18.6 can be combined to derived a relationship between the mass of the crust and the mass of t h e depleted mantle: c c c M dm Ê CNd ˆ Ê CNd ˆ f Sm / Nd QT c = Á o - 1˜ - Á o ˜ dm M c Ë CNd ¯ Ë CNd ¯ eNd 18.7 Thus the mass ratio of depleted mantle to crust can be calculated if we know the Sm/Nd ratio of t h e crust, the eNd of the depleted mantle, and the concentration of Nd in the crust and in primitive mantle. Figure 18.2 shows a plot that shows the solutions of 18.7 as a function of TC for various values of m e d obtained by† DePaolo (1980). Most estimates of the average age of the crust are between 2 and 2.5 m Ga, and ed is about +10. Possible solutions for the ratio of depleted mantle to whole mantle are in the range of 0.3 to 0.5. A number of such mass balance calculations that included other isotopic systems as well were published between 1979 and 1980, all of which obtained rather similar results. Interestingly, the fraction of the mantle above the 650 km seismic discontinuity is roughly 0.33. The mass balance calculations suggested the seismic discontinuity was the chemical boundary between upper and lower mantle. * A note on notation: The growth equation for 1 4 3Nd/1 4 4Nd is: 143 147 Nd/144Nd = 143Nd/144Nd0 + 147 Sm/144Nd(elt – 1) Since the half-live of Sm is long compared to the age of the Earth, we may use the approximation: elt ≈ l t + 1 and hence: Nd/ Nd = 1 4 3Nd/1 4 4Ndi + 1 4 7Sm/1 4 4Nd lt The equation may be tra

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