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 ^{206}Pb and red balls being ^{207}Pb, and the right hand side box represents the U fraction of the sample, with blue balls being ^{238}U and pink balls being ^{235}U. 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 ^{238}U/^{206}Pb 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 ^{206}Pb and the gold balls represent ^{207}Pb. There are 100 red balls and 60 gold balls therefore the ^{206}Pb/^{207}Pb ratio = 100/60 = 1.667. The right hand side box represent the U and has pink blue balls. The pink balls represent ^{235}U and the blue balls ^{238}U. There are 40 pink balls and 120 blue balls therefore the ^{238}U/^{235}U 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 ^{205}Pb and for U we could use ^{233}U and/or ^{236}U. 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 ^{205}Pb tracer (teal balls) and a ^{233}U 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 ^{205}Pb/^{206}Pb 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 ^{205}Pb to our sample and measured a ^{205}Pb/^{206}Pb ratio of 0.15 then we would have 6,666.67 moles of ^{206}Pb in our sample. Moving to the right hand side container we have 16:36:108 green:pink:blue balls. The ^{233}U/^{238}U 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 ^{238}U. Completing this simple calculation we can estimate that the ^{238}U/^{206}Pb 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 (^{205}Pb) to green balls (^{233}U) = 1/20, then we could add a unknown amount of mixed tracer balls and still determine the ^{238}U/^{206}Pb 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 ^{205}Pb and ^{233}U) 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., ^{238}U/^{206}Pb) 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.