The two boxes represent looking down to the top of a containers of unknown, and likely, different, depth. In the context of U-Pb isotope dilution the left hand side box represents the Pb fraction of the sample, with gold balls being 206Pb and red balls being 207Pb, and the right hand side box represents the U fraction of the sample, with blue balls being 238U and pink balls being 235U. Although we do not know the depth of the container we can easily count the number of balls of each colour in a given container and determine their relative abundances. This is what mass spectrometers do, very well, we can readily determine the ratio of isotopes of the same element. The challenge in U-Pb geochronology is to accurately determine the ratio of U balls to Pb isotopes without directly measurement. In the case of 238U/206Pb we would want to know the ratio of blue to gold balls. However it is not possible to determine the ratio of ball between boxes of unknown volume. In the case of isotopes of different elements, different elements have different ionisation potentials therefore the measured elemental ratio in a mass spectrometer does not reflect the true ratio in the sample.
In the top panel there are two boxes. The left hand side box represents Pb and has red and gold balls. The red balls represent 206Pb and the gold balls represent 207Pb. There are 100 red balls and 60 gold balls therefore the 206Pb/207Pb ratio = 100/60 = 1.667. The right hand side box represent the U and has pink blue balls. The pink balls represent 235U and the blue balls 238U. There are 40 pink balls and 120 blue balls therefore the 238U/235U ratio = 120/40 = 3. However, given that we do not know the depth of either container it is not possible to determine the ratio of balls from different boxes.
The bottom panel represents the same two boxes but we have now added some 'tracer' balls. In the example of U-Pb geochronology, these 'tracers' would be isotopes of the same element that do not occur in the natural samples. For Pb this could be 205Pb and for U we could use 233U and/or 236U. By adding a known amount of tracer balls/isotopes to our sample and measuring the tracer isotope/sample isotope ratio we can quantify amount of sample isotope present.
In the bottom panel we have added two tracers to our sample, a 205Pb tracer (teal balls) and a 233U tracer (green balls), and we have added 1,000 teal balls and 20,000 green balls. Following the simple 'balls in a container' analogy, in the left had side container we now have 12:48:80 teal:gold:red balls, therefore the 205Pb/206Pb ratio = 12/80 = 0.15. If there were 1,000 teal balls added and the ratio of teal/red is 0.15 there must be 6,666.67 red balls in the container. If we think about Pb isotopes and say we added 1,000 moles of 205Pb to our sample and measured a 205Pb/206Pb ratio of 0.15 then we would have 6,666.67 moles of 206Pb in our sample. Moving to the right hand side container we have 16:36:108 green:pink:blue balls. The 233U/238U ratio = 16/108 = 0.1484, therefore with 20,000 moles of 233U added to the container we can quantify that the container contains 135,000 moles of 238U. Completing this simple calculation we can estimate that the 238U/206Pb ratio = 135,000/6,666.67 = 20.25.
This is the basic principle that underpins U-Pb isotope dilution geochronology. Yet, in practical terms this is exercise is not so straightforward for U-Pb geochronology, largely because the amounts of materials we are dealing with would require very accurate weighing of small volumes of materials which is simply not practicable at the level of precision required. To circumvent the need to know the quantity of tracer added to the sample to a very high-precision we use mixed tracers. These mixed tracers allow us to determine the relative concentration of sample isotopes from different elements to a very high level of precision. Completing the 'balls' in a container' analogy, if we had a mixed tracer of teal and green balls, with a known ratio of teal balls (205Pb) to green balls (233U) = 1/20, then we could add a unknown amount of mixed tracer balls and still determine the 238U/206Pb ratio = 20.25. Try this yourself by changing the number of teal and green balls added to the boxes but remember to keep the ratio of teal to green constant at 1:20.
In U-Pb U-Pb isotope dilution geochronology this is how we are able to determine sample U/Pb ratios to a high-precision. Each lab will have a bottle of mixed U-Pb tracer (commonly 205Pb and 233U) which is added to each sample prior to dissolution. The isotopic ratio for each element can be determined via mass spectrometry and the elemental ratio (e.g., 238U/206Pb) can be quantified using the isotope dilution approach outlined above. What is critical to the accuracy of the elemental ratio is the elemental ratio of the mixed tracer. The method for calibrating a mixed elemental tracer back to the S.I units is outlined elsewhere in this book.
Links to the EARTHTIME website for additional instruction.