文档介绍：Chapter 12
Pb Isotope Modeling
Pb isotope evolution plicated. Mathematical analysis of the Pb
isotope diagrams can help understand challenging problems related to Pb
isotope evolution. Here we conduct new theoretical analyses of the
fundamental equations related to four useful diagrams in Pb isotope
geochemistry: the Tera-Wasserburg diagram, the conventional U-Pb
Concordia diagram, the Holmes-mon Pb evolution
model and the two-stage Pb evolution model. The U-Th-Pa Concordia
diagram will also be treated in a similar manner. You may find that these
quantitative treatments are quite stimulating. At the end of this chapter,
we talk about the error analyses of U-Pb isotope data.
. Why is the Tera-Wasserburg Concordia Diagram Concave
Upward?
In the Tera-Wasserburg Concordia (Tera and Wasserburg, 1972),
238U/2MPbratios are plotted directly against the 207Pb/2MPbratio. The
equations for 238U/2MPband 207Pb/2MPbare given by:
()
pb 235u e'hiSf - 1
207 - (1 )
2381/ - 1
206 pb e'h7st '
where /$38 = 125 X lo-'' and h3,= X lo-'' y-' are decay
constants for 238Uand 235U,respectively.
248
Lead Isotope Modeling 249
Because the second derivative ( d2y/dX2) of a function ( y = f(x) )
determines whether the curve of the function is concave upward or down,
to answer the question of this section, we need to derive the second
derivatives of the Tera-Wasserburg Concordia plot.
For the Tera-Wasserburg Concordia diagram (Fig. ), we have
x= 23sU/206Pb,
y = 207Pb/2MPb.
Differentiation of x in Eq. () with respect to t gives
(1 )
( )
From Eqs. () and () we obtain the first derivative of y with
respect to x by”=”/($) using chain rule
ax at
( )
Differentiation of the above $/ax in () with respect to t gives
250 Quantitative Geochemistry
()
The key to obtain a2y/ax2 is to use the following chain rule:
--aZy - a(ay/ax) - a(ay/ax) -at
ax2 ax