Jon Fleming has added this section in 2005. The purpose is to show the derivation of Cook’s “correction” equation and how non-equal cross-sections affect the result. It is based on two major premises, both of which are almost certainly false:

- There is no
primordial lead in the samples under consideration; all lead was
created by radioactive decay and/or neutron capture.
- The neutron
flux was sufficient to create all the
^{208}Pb in the samples, since the^{208}Pb could not have come from the decay of^{232}Th (which would leave at least a little residual^{232}Th in the samples today).

Recall the neutron-capture reaction series presented in the main text:

We’ll move from simple to more complex; starting with the end of the chain, then considering the beginning of the chain, and finally looking at the middle of the chain.

Since we
“know” that all the ^{208}Pb was generated by neutron capture, using C_{207→208}
to
indicate the cross section for converting ^{207}Pb to ^{208}Pb by neutron capture, using N to denote the neutron flux and
using ^{208}Pb_{measured}
to
denote the measured amount of ^{208}Pb in the sample today:

(1) |

Or, solving for N:

(2) |

Next we move to
the beginning of the chain. Here there is one mechanism
creating ^{206}Pb (radioactive decay of ^{238}U) and another mechanism destroying ^{206}Pb (neutron capture). Therefore the measured amount of
^{206}Pb today is *less
than* the amount
created from decay of ^{238}U. Using ^{206}Pb_{Radiogenic}
as
the ^{206}Pb that was generated from radioactive decay and should be used
in the determination of age; ^{206}Pb_{Measured}
as
the ^{206}Pb that was actually measured in the sample “today”;
and ^{206}Pb_{Converted}
as
the ^{206}Pb that was generated from radioactive decay *and then was
converted to ^{207}Pb by neutron capture*:

(3) |

^{206}Pb_{Converted}
is,
of course, the amount of ^{206}Pb measured times a conversion factor that is the product of the
cross section (for converting ^{206}Pb to ^{207}Pb by neutron capture) and the total neutron flux. Since
the ^{206}Pb and ^{207}Pb were in close proximity, the total neutron flux N is the same
for both. Calling the cross section C_{206→207}:

(4) |

Since N is the same for both
^{206}Pb and ^{207}Pb, we can substitute the right-hand side of
equation (2) for the “N” term in equation
(4):

(5) |

Now we consider
the middle of the neutron capture series, the amount of
^{207}Pb generated from radioactive decay, ^{207}Pb_{Radiogenic},
which should be used in the calculation of age. Here two
mechanisms are creating ^{207}Pb (radioactive decay of ^{235}U
and neutron capture of ^{206}Pb) and one mechanism is destroying
^{207}Pb (neutron capture creating ^{208}Pb).

^{207}Pb_{Radiogenic}
is
the measured amount ^{207}Pb_{Measured}
**plus**
the
amount of ^{207}Pb that was generated by radioactive decay *but then was
converted to ^{208}Pb by neutron capture* (which in turn
is equal to the amount of

(6) |

And, finally, dividing equation (5) by equation (6):

(7) |

which, when
C_{207→208}
=
C_{206→207}
(as
Cook assumed), reduces to the equation given in the main text
except for the way that the Pb terms are grouped in the last term
of both the numerator and denominator:

(8) |

For calculating
the ^{206}Pb/^{207}Pb for the third
row of Table 5, using Dalrymple’s values for cross sections
plugged into equation (7):

I do not know why my result is slightly (0.5%) different from Dalrymple’s 16.38, but the difference is not significant.

Using more recent values for the cross sections:

which is insignificantly different from the other values.

To calculate
the age from these “corrected” values of
^{206}Pb/^{207}Pb , we use the standard equation (which cannot be solved in
closed form) for age “t” of a sample (in years) given
its ^{206}Pb/^{207}Pb ratio, the decay constant of ^{238}U
(λ_{1}
=
1.55125×10^{-10}
per
year), and the decay constant of ^{235}U (λ_{2}
=
9.8485×10^{-10}
per
year) is:

(from
Dalrymple, G. Brent, “The Age Of the Earth”, Stanford
University Press, 1991, page 101. The values
of λ_{1}
and λ_{2}
are
from the same source, page 80. Note that the equation uses
^{207}Pb/^{206}Pb, the inverse of the lead ratios used in this paper and
addendum)

The age
equation is easily solved by any of a variety of numerical
techniques. For ^{206}Pb/^{207}Pb = 16.46 the calculated age
is 630 million years, and
for ^{206}Pb/^{207}Pb = 16.47 the calculated age is 629
million years. Including the
effect of non-equal cross-sections for the neutron capture
reactions completely obviates Cook’s conclusion. Neutron
capture does not noticeably affect the measurement of ages
by ^{206}Pb/^{207}Pb ratios.

^{†} This
section copyright © 2005 Jon Fleming. Reproduction
permission granted if this copyright is noted.

I am indebted to Dr. Jamie Gilmour (Senior Lecturer, Isotope Geochemistry and Cosmochemistry, University of Manchester School of Earth, Atmospheric and Environmental Sciences) for pointing me in the right direction to understand this derivation.

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