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The difficulties with single-stage systems can be circumnavigated with multi-stage systems.
Though multi-stage lead samples cannot be used for generating isochrons, they can be used to produce valuable information through concordia-discordia plots.
Another situation in which single-stage systems give unreliable information is the extraction of lead from uranium to form lead ore.
It is possible that a system could undergo a geological process that extracts lead, leaving the new system without any uranium.
Since both decay series contain a unique set of intermediate radioactive isotopes, and because each has its own half-life, independent age calculations can be made from each (Dalrymple 20).
The presence of a stable lead isotope that is not the product of any decay series (204Pb) allows lead isotopes to be normalized, allowing for the use of isochrons and concordia-discordia diagrams as dating tools.
An example isochron from Dalrymple (2004) is shown in Figure 4.8 below. Unlike other isochrons, the slope of the Pb-Pb isochron decreases with increasing age.
As stated above, the Gerling-Holmes-Houtermans model requires that assumptions about the genetic relationship between the Earth and meteorites be made.
Lead isotopes are commonly used in dating rocks and provide some of the best evidence for the Earth's age.
In order to be used as a natural clock to calculate the age of the earth, the processes generating lead isotopes must meet the four conditions of a natural clock: an irreversible process, a uniform rate, an initial condition, and a final condition.
Second, using two isotopes of the same element makes the sample immune to chemical fractionation during a post-crystallization disturbance (Dalrymple 208). This model ultimately led to the development of isochrons, in which two isotopes are plotted against each other to calculate an age for the mineral or rock.
The commonly accepted 4.5 billion year age of the earth is derived from radiometric dating of lunar rocks and meteorites in addition to dating methods based on the Gerling-Holmes-Houtermans model. Those who developed the method utilized Pb, lead isotopes that are the product of radioactive decay, normalized to 204Pb.
If that system were dated at that point in time, it would fall on the isochron and give the correct age of the mineral.