Isochron dating is a common technique of radiometric dating and is applied to date certain events, such as crystallization, metamorphism, shock events, and differentiation of precursor melts, in the history of rocks. Isochron dating can be further separated into mineral isochron dating and whole rock isochron dating; both techniques are applied frequently to date terrestrial and extraterrestrial rocks (meteorites and Moon rocks). The advantage of isochron dating as compared to simple radiometric dating techniques is that no assumptions are needed about the initial amount of the daughter nuclide in the radioactive decay sequence. Indeed, the initial amount of the daughter product can be determined using isochron dating. This technique can be applied if the daughter element has at least one stable isotope other than the daughter isotope into which the parent nuclide decays.
Basis for method All forms of isochron dating assume that the source of the rock or rocks contained unknown amounts of both radiogenic and non-radiogenic isotopes of the daughter element, along with some amount of the parent nuclide. Thus, at the moment of crystallization, the ratio of the concentration of the radiogenic isotope of the daughter element to that of the non-radiogenic isotope is some value independent of the concentration of the parent. As time goes on, some amount of the parent decays into the radiogenic isotope of the daughter, increasing the ratio of the concentration of the radiogenic isotope to that of the non-radiogenic isotope of the daughter element. The greater the initial concentration of the parent, the greater the concentration of the radiogenic daughter isotope will be at some particular time. Thus, the ratio of the radiogenic to non-radiogenic isotopes of the daughter element will become larger with time, while the ratio of parent to daughter will become smaller. For rocks that start out with a small concentration of the parent, the radiogenic/non-radiogenic ratio of the daughter element will not change as quickly as it will with rocks that start out with a large concentration of the parent.
Assumptions An isochron diagram will only give a valid age if all samples are cogenetic, which means they have the same initial isotopic composition (that is, the rocks are from the same unit, the minerals are from the same rock, etc.), all samples have the same initial isotopic composition (at t0), and the system has remained closed.
Isochron plots The mathematical expression from which the isochron is derived is
D ∗ = D 0 + n ⋅ ( e λ t − 1 ) , {\displaystyle {\mathrm {D*} }={\mathrm {D} }_{\mathrm {0} }+\mathrm {n} \cdot (e^{\lambda t}-1),}
where
t is age of the sample, D* is number of atoms of the radiogenic daughter isotope in the sample, D0 is number of atoms of the daughter isotope in the original or initial composition, n is number of atoms of the parent isotope in the sample at the present, λ is the decay constant of the parent isotope, equal to the inverse of the radioactive half-life of the parent isotope times the natural logarithm of 2, and (eλt-1) is the slope of the isochron which defines the age of the system.
Because the isotopes are measured by mass spectrometry, ratios are used instead of absolute concentrations since mass spectrometers usually measure the former rather than the latter. (See the section on isotope ratio mass spectrometry.) As such, isochrons are typically defined by the following equation, which normalizes the concentration of parent and radiogenic daughter isotopes to the concentration of a non-radiogenic isotope of the daughter element that is assumed to be constant:
( D ∗ D r e f ) p r e s e n t = ( D 0 D r e f ) i n i t i a l + ( P t D r e f ) ⋅ ( e λ t − 1 ) , {\displaystyle \left({\frac {\mathrm {D*} }{\mathrm {D} _{ref}}}\right)_{\mathrm {present} }=\left({\frac {\mathrm {D_{0}} }{\mathrm {D} _{ref}}}\right)_{\mathrm {initial} }+\left({\frac {\mathrm {P_{t}} }{\mathrm {D} _{ref}}}\right)\cdot (e^{\lambda t}-1),}
where
D r e f {\displaystyle D_{ref}} is the concentration of the non-radiogenic isotope of the daughter element (assumed constant),
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