Isochrons isochron equations

Ca = 0.000031, and Ca = 0.001824. The high relative abundance of °Ca (a result of its mass number A, which is a multiple of 4 and thus exceptionally stable cf section 11.3) is one of the two main problems encountered in this sort of dating (in Ca-rich samples, the relative enrichment in °Ca resulting from °K decay is low with respect to bulk abundance). The other problem is isotopic fractionation of calcium during petrogenesis (and also during analysis see for this purpose Russell et al., 1978). These two problems prevent extensive application of the K-Ca method, which requires extreme analytical precision. The isochron equation involves normalization to the Ca abundance... 

The low decay energy prevents accurate determinations of half-life by direct counting. Reported half-lives range from 3 to 6.6 X 10 ° a. The most recent direct determination (Lindner et al., 1989) assigns a half-life of (4.23 0.13)Xl0 °a to the decay process of equation 11.117, which is fairly consistent with the indirect estimates of Hirt et al. (1963) [(4.3 0.5) X 10 ° a]. The isochron equation is normalized to the °Os abundance ... 

To use Equation l-47c for dating, one has to overcome the difficulty that there are two unknowns, the initial amount of Nd and the age. With this in mind, the most powerful method in dating, the isochron method, is derived. To obtain the isochron equation, one divides Equation l-47c by the stable isotope of the product (such as " Nd) ... 

Therefore, to kineticists and informed geochronologists, the age obtain from an isochron equation or from Example 1-5 is an apparent age, and is called the closure age (Dodson, 1973) because it means the age since the closure of the mineral, not necessarily since the formation of the mineral. The closure age may differ from the true age or formation age because of diffusive loss (or exchange) of the daughter nuclide. For the closure age to be the same as the formation age, the mineral must have cooled down rapidly (for volcanic rocks) or formed at not-so-high a temperature (for metamorphic rocks) so that diffusive loss from the mineral is negligible. 

After transformation into the isochron equation, both the age and the initial isotopic ratio ( Nd/ Nd)o can be obtained. That is, in a given mineral, the initial concentration of the daughter nuclide Nd as well as the initial concentration of the parent nuclide in each mineral can be found. Hence, with the isochron method, we determine not only the age, but also the initial amount of the daughter and parent nuclides. 

Using Pb as the normalizing isotope of Pb, the isochron equation may be written as... 

Converting the above equation into a linear equation similar to the isochron equation requires a couple of steps. First, we normalize the above equation to a stable and nonradiogenic Mg isotope ( " Mg), yielding... 

The only unknown on the right hand side is a value for modulus E. For the plastic this is time-dependent but a suitable value may be obtained by reference to the creep curves in Fig. 2.5. A section across these curves at the service life of 1 year gives the isochronous graph shown in Fig. 2.13. The maximum strain is recommended as 1.5% so a secant modulus may be taken at this value and is found to be 347 MN/m. This is then used in the above equation. 

The resulting corrected age of 59.8 3.8 ka can be calculated either from the two-point isochron defined by the data, or by application of Equations (16)-(18) of this chapter. 

A method is described for fitting the Cole-Cole phenomenological equation to isochronal mechanical relaxation scans. The basic parameters in the equation are the unrelaxed and relaxed moduli, a width parameter and the central relaxation time. The first three are given linear temperature coefficients and the latter can have WLF or Arrhenius behavior. A set of these parameters is determined for each relaxation in the specimen by means of nonlinear least squares optimization of the fit of the equation to the data. An interactive front-end is present in the fitting routine to aid in initial parameter estimation for the iterative fitting process. The use of the determined parameters in assisting in the interpretation of relaxation processes is discussed. 

The term within the square brackets in equation 11.79 is the normalized (equilibrium) distribution coefficient between the minerals a and )3 (cf section 10.8) at the closure condition of the mineral isochron. Ganguly and Ruitz (1986) have shown it to be essentially equal to the observed (disequilibrium) distribution coefficient between the two minerals as measured at the present time. can be assumed to be 1, within reasonable approximation. Equation 11.79 can be calibrated by opportunely expanding AG° over P and T ... 

Based on equations 11.79 and 11.80, the measured P b/sr value for each mineral of a given isochron must furnish concordant indications in terms of the T and P of equihbrium. Moreover, the deduced parameters can be assumed to correspond to the condition of closure of exchanges. 

Whenever it is possible to analyze in a given rock at least two minerals that crystallized at the same initial time t = 0, equation 11.86 can be solved in t. On a Cartesian diagram with coordinates Sr/ Sr and Rb/ Sr, equation 11.86 appears as a straight line ( isochron ) with slope exp(At) — 1 and intercept ( Sr/ Sr)o. As shown in figure 11.14A, all minerals crystallized at the same t from the same initial system of composition ( Sr/ Sr)o rest on the same isochron, whose slope exp(At) — 1 increases progressively with t. 

The intercept term C Ar/ Ar)o, which accounts for igneous, metamorphic, or atmospheric sources, is regarded as the excess contribution present at time = 0, whereas the second term is the radiogenic component accumulating in the various minerals of the isochron by decay of If all the minerals used to construct the isochron underwent the same geologic history and the same sort of contamination by excess " Ar, the slope of equation 11.100 would have a precise chronological... 

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