Two radioisotope systems for determining the 'age' of rocks are the decay of ¹⁴⁷Sm to ¹⁴³ Nd and the decay of ⁸⁷ Rb to ⁸⁷ Sr.These elements occur in minute amounts in virtually every igneous and metamorphic rock and are invaluable aids to geophysicists in determining how long a mineral assemblage has remained an isolated system.

2) If the decay constants are

Rubidium - Strontium System
We now examine the isotope systematics in detail. If several minerals in the same rock are analysed for Rb and Sr isotopes we can determine the length of time that the rock system has been in equilibrium. If a unique figure is arrived at, then this might be termed the 'age' of the sample.In what follows we denote the total number of atoms present at the start by N_o
 

3) If decay of a parent isotope is given by N_p = N_oe^-lt and growth of a daughter product is given by N_d = N_o(1 - e^-lt )then show that in general:
 
 

4) From this relationship, show that the age of a rock would be given by
 
 

where ⁸⁷Sr and ⁸⁷Rb are the amounts of each isotope present after time 't'.Usually these would be the amounts present day sample of rock.

If a rock remains as a closed system, the minerals in it with higher Rb contents will generate more ⁸⁷Sr and a plot of ⁸⁷Sr against ⁸⁷Rb will be a straight line. It is more usual to plot the Ratio of these isotopes to the stable ⁸⁶Sr, so we plot the following (this is termed an ISOCHRON diagram:)
 
 

5) Explain why the Age of the sample is given by
 

6) What does the INTERCEPT represent ? Explain carefully.

Some of the oldest rocks in the world can be found in the Godthaab area of Greenland. (Here 'Age' means the length of time that the mineral assemblage has remained in an isolated state of equilibrium.) Here is some relevant data:
 

7) Plot an ISOCHRON diagram of this data.
8) Determine the age of the Godthaab complex and the initial ratio of ⁸⁷Sr/⁸⁶Sr.

The Earth is thought to have been formed approximately 4.6 x 10⁹ years ago . What does this tell you about the formation of certain parts of the earth's crust ? Comment in the light of your calculations.

Samarium - Neodymium System

We have seen that  some rocks from continental regions appear to have been in equilibrium for billions of years.The ocean floor is different,however.Continuous recycling of oceanic crust occurs on a time scale of a few hundred million years. An example will now illustrate this.

The Lizard complex of rocks in Cornwall is thought to be an example of ancient ocean floor that is now an outcrop of rock in Southern Britain. Here is some isotope data:
 
 

9) Plot an ISOCHRON diagram .
 

10) Which mineral is not in equilibrium with the others ?

11) Calculate the age of the Lizard complex and its initial abundance of ¹⁴³Nd / ¹⁴⁴Nd.

12) No oceanic rocks in the world give an age of more than a few hundred million years. What can you infer from this ?
 
 

13) Some Meteorites are thought to be representative of material from the early solar system that did not form into a planetary body. Consequently they have not been subject to tectonic processes in the same way that the earth and the moon have for example. Some have (¹⁴³Nd/¹⁴⁴Nd)_Initial ratios of 0.506609. Comment on this in the light of your answer to Q12.