Isochronous method

The isochronous method is a widely used method for radiometric dating of rocks . The advantage over conventional radiometric dating is that no assumptions have to be made about the initial concentration of the decay product in the rock in order to reliably date a rock. In addition, the isochronous method can also be used to detect disturbances in the isotope system used for dating , which would falsify dating if they remained undetected. The isochronous method is therefore a very powerful tool for radiometric dating.

Usability

The isochronous method can be used with isotope systems in which the element into which the mother nuclide breaks down has at least one other non-radiogenic stable isotope as a reference isotope in addition to the daughter isotope . An example is the Rb - Sr system ( rubidium-strontium dating ). In addition to ⁸⁷ Sr, which is the decay product of the radionuclide ⁸⁷ Rb, the stable isotope ⁸⁶ Sr also occurs in nature , which is non-radiogenic, i.e. not itself the decay product of a radionuclide occurring in the sample. Other examples are Sm-Nd and U-Pb. ${\displaystyle M}$ ${\displaystyle D}$ ${\displaystyle R}$

Process of the analyzes

For dating, the corresponding isotope concentrations are determined either in different minerals of an individual rock sample (mineral isochrones , English mineral-isochronous ) or in different types of rock of cogenetic origin ( whole rock isochronous ), which come from a rock melt, for example .

In the case of a mineral isochrone, various mineral fractions must first be separated from the rock to be dated. This mineral separation is done by various methods, such as density separation , magnetic separation, chemical separation, manually with tweezers and microscope, etc. The aim is to obtain mineral fractions with the greatest possible difference in the frequency ratio of parent isotope to reference isotope, which ultimately increases the accuracy of the dating.

The different fractions are then chemically resolved and the elements used for dating are extracted by chromatographic methods . The samples thus obtained are then prepared for measuring the isotope ratios with a mass spectrometer and the element abundances , for example with an atomic emission spectrometer . The results obtained during the subsequent measurement are then drawn in a so-called isochronous plot.

Isochronous plot

Diagram 1: Example of a hypothetical isochronous plot for four mineral fractions from a rock sample. The isotope ratios of the individual mineral fractions migrate from their initial position at the time ( time of crystallization of the rock) to the new position at the time . The more time has passed, the steeper the straight line; however, the point of intersection with the ordinate remains constant. ${\displaystyle t_{0}}$ ${\displaystyle t_{1}}$

The isochronous plot is a diagram in which the ratio of the daughter isotope to the reference isotope ( ) is plotted against the ratio of the parent isotope to the reference isotope ( ). If the data in the isochronous plot lie on a straight line, this straight line is referred to as an isochronous. The slope of the isochrones is then a measure of the age of the sample. The point of intersection with the ordinate of the coordinate system shows the relationship between the child and reference isotopes at the dated point in time . ${\displaystyle D/R}$ ${\displaystyle M/R}$ ${\displaystyle t_{0}}$

It can be shown that the following relationship applies to the slope m and the age t (see below):

${\displaystyle t={\frac {1}{\lambda }}\cdot \ln \left(m+1\right),\quad \lambda ={\text{Zerfallskonstante}}}$

It is noteworthy that in this formula for determining the age only the slope and not the initial ratio of the daughter isotope to the reference isotope is included. This initial ratio is obtained as a secondary result of the isochronous method, but it is not needed to determine the age.

Immediately after the formation of a rock, the ratio of daughter isotope to reference isotope is the same in all mineral fractions, provided that sufficient homogenization has taken place. The isochronous is therefore a horizontal straight line at the beginning. The relationship between age and slope of the isochrones can be interpreted in such a way that the greater the frequency of the parent isotope in the respective fraction, the further a mineral fraction is on the right in the isochronous plot, the more decays into the daughter isotope take place. A fraction on the right in the isochronous diagram will therefore move faster upwards and at the same time to the left than one on the left. Since this migration is proportional to the abscissa value, the values ​​of all fractions always lie on a straight line, provided that the isotope system is not disturbed by environmental influences. The extrapolation of the isochrones to the point of intersection with the ordinate can be interpreted as an extrapolation to a hypothetical mineral fraction in which there is no parent isotope, i.e. no decays take place and the initial ratio of daughter to reference isotope therefore remains constant.

In principle, two points in the isochronous diagram are sufficient to determine the isochronous slope and thus to determine the age. As a rule, however, at least three or more fractions are separated, measured and entered in the isochronous diagram. The reason for this is that a straight line can always be drawn through two points; only three or more points can be used to check whether it is really a straight line and whether consistency is guaranteed. If, for example, the initial homogeneity in the ratio of daughter isotope to reference isotope is not guaranteed when a rock is formed, or if the isotope system was disturbed after the rock was formed, for example by diffusion, the fractions concerned deviate from the straight line. This would not be seen with only two measuring points. If, however, more measuring points were determined and they lie on a straight line in the isochronous plot, it is ensured that this straight line can be interpreted as an actual isochronous, interferences are excluded, and the initial homogeneity was guaranteed. The age determination is then considered to be very reliable.

Sometimes a variation of the isochronous plot is also used, in which a stable isotope of the same element in the isotope diagram is used instead of the parent isotope. This is especially used in dating methods with "extinct" radionuclides.

Interpretation of the dating

The resulting age of the isochronous method dates, as with other radiometric dating methods, the time of "completion" of the isotopic system used, i.e. H. the point in time from which the isotopes are fixed in the corresponding minerals and rocks and are no longer exchanged with the environment. Different isotope systems react very differently to environmental conditions, so they terminate under different conditions. Depending on the isotope system used for dating, the "conclusion" can correspond to different physical events. For example, if one isotope system used for dating closes at a higher temperature than another, the former will provide a higher age for a rock crystallizing from a melt and cooling extremely slowly than the latter. The ages then indicate the times at which the respective temperature was reached, which is used in such cases to determine cooling rates.

It should also be noted that mineral isochronous and total rock isochronous date different events. While the mineral isochronous dates the crystallization of the individual rock, the total rock isochronous dates the splitting of the original melt into different partial melts, from which the different types of rock later crystallized. So it is not uncommon for the two dates to give different results.

Math

According to the law of collapse , because of

${\displaystyle M=M_{0}\cdot e^{-\lambda t}\Leftrightarrow M_{0}=M\cdot e^{\lambda t}}$

the time dependence

${\displaystyle D=D_{0}+M\cdot (e^{\lambda t}-1)}$

where = decay constant, or = frequency of the daughter isotope or of the mother isotope at the point in time , = initial frequency of the daughter isotope . Both sides of the equation can be divided by the frequency of the reference isotope : ${\displaystyle \lambda }$ ${\displaystyle D}$ ${\displaystyle M}$ ${\displaystyle t}$ ${\displaystyle D_{0}}$ ${\displaystyle R}$

${\displaystyle {\frac {D}{R}}={\frac {D_{0}}{R}}+{\frac {M}{R}}\cdot (e^{\lambda t}-1)}$

If the initial frequency of the daughter isotope is not known, then with the unknown age you have a total of two unknowns. The "conventional" radiometric age determination, in which only one value is determined for and , provides only one determination equation, which does not result in a clear solution. With the isochronous method, however, several fractions are measured, which results in a corresponding number of equations. With two fractions, and , you already have two determining equations: ${\displaystyle D_{0}}$ ${\displaystyle t}$ ${\displaystyle D}$ ${\displaystyle M}$ ${\displaystyle F1}$ ${\displaystyle F2}$

${\displaystyle \left({\frac {D}{R}}\right)_{F1}=\left({\frac {D_{0}}{R}}\right)_{F1}+\left({\frac {M}{R}}\right)_{F1}\cdot (e^{\lambda t}-1)}$

${\displaystyle \left({\frac {D}{R}}\right)_{F2}=\left({\frac {D_{0}}{R}}\right)_{F2}+\left({\frac {M}{R}}\right)_{F2}\cdot (e^{\lambda t}-1)}$

Because of the initial homogeneity:

${\displaystyle \left({\frac {D_{0}}{R}}\right)_{F1}=\left({\frac {D_{0}}{R}}\right)_{F2}}$

The system of equations thus has a unique solution and the following formula for the slope can be derived by subtracting the two equations : ${\displaystyle m}$

${\displaystyle m:={\frac {\left({\frac {D}{R}}\right)_{F2}-\left({\frac {D}{R}}\right)_{F1}}{\left({\frac {M}{R}}\right)_{F2}-\left({\frac {M}{R}}\right)_{F1}}}=(e^{\lambda t}-1)}$

Transformation according to age results in: ${\displaystyle t}$

${\displaystyle t={\frac {1}{\lambda }}\cdot \ln \left(m+1\right)}$

literature

• RW Carlson, LE Borg, AM Gaffney, M. Boyet: Rb-Sr, Sm-Nd and Lu-Hf isotope systematics of the lunar Mg-suite: the age of the lunar crust and its relation to the time of Moon formation. In: Phil. Trans. R. Soc. A. 2014, p. 372.

• GP Badgasaryan, Gukasyan, R.Kh., Veselsky, J .: The age of Male Karpaty Mts. granitoid rocks determined by Rb-Sr isochrone method. In: Geologicky Zbornik. V. 33, No. 2, 1982, pp. 131-140.

• B. Heuel-Fabianek: Natural radioisotopes: the “atomic clock” for determining the absolute age of rocks and archaeological finds. In: Radiation Protection Practice. No. 1, 2017, pp. 31–42.

• K. Suzuki, M. Adachi: Precambrian provenance and Silurian metamorphism of the Tsubonosawa paragneiss in the South Kitakami terrane, Northeast Japan, revealed by the chemical Th-U-total Pb isochronous ages of monazite, zircon and xenotime. In: Geochemical Journal. V. 25, No. 5, 1991, pp. 357-376.