Carbon 14 – Willard’s World

Willard Libby believed he could model the entire world of Carbon 14 by using a grid system.

This grid system is based upon a cell size of one square centimetre.

In this grid system every cell is identical and every cell behaves in exactly the same way.

Therefore, Willard Libby’s world model contains only one cell of one square centimetre.

This one square centimetre cell encapsulates the entire planet in miniature.

This one square centimetre cell encapsulates the entire Carbon Cycle in miniature.

This one square centimetre cell encapsulates the entire Carbon 14 Cycle in miniature.

Willard Libby’s single cell model was devised by playing the Carbon 14 numbers game.

Libby’s initial numerical challenge was the half-life decay process.

Half-life (t½) is the amount of time required for a quantity to fall to half its value as measured at the beginning of the time period.

While the term “half-life” can be used to describe any quantity which follows an exponential decay, it is most often used within the context of nuclear physics and nuclear chemistry – that is, the time required, probabilistically, for half of the unstable, radioactive atoms in a sample to undergo radioactive decay.

The original term, dating to Ernest Rutherford’s discovery of the principle in 1907, was “half-life period”, which was shortened to “half-life” in the early 1950s.

Rutherford applied the principle of a radioactive elements’ half-life to studies of age determination of rocks by measuring the decay period of radium to lead-206.
…
A half-life usually describes the decay of discrete entities, such as radioactive atoms.

In that case, it does not work to use the definition “half-life is the time required for exactly half of the entities to decay”.

For example, if there are 3 radioactive atoms with a half-life of one second, there will not be “1.5 atoms” left after one second.

Instead, the half-life is defined in terms of probability: “Half-life is the time required for exactly half of the entities to decay on average“.

In other words, the probability of a radioactive atom decaying within its half-life is 50%.

http://en.wikipedia.org/wiki/Half-life

One strange statistical quirk of the half-life probabilities is that the period of time needed to complete the radioactive decay process depends upon the size of the original population.

For example:
1,024 radioactive atoms require 11 half-lives before they are completely decayed.
8,192 radioactive atoms require 14 half-lives before they are completely decayed.

This statistical quirk meant that Willard Libby had to know the size of the original Carbon 14 population before he could date a specimen containing radioactive Carbon 14.

However, Willard Libby had a statistical card up his sleeve.

A second strange statistical quirk of the half-life probabilities is that the average age of the radioactive atoms [when they decay] stabilises at 1.5 times the half-life when the second half-life cycle has been completed.

This second statistical quirk led Willard Libby to the inspirational realisation that IF a population of Carbon 14 atoms could [somehow] remain constant THEN their average decay rate would remain constant and their average age [when they decayed] would remain constant at 1.5 times their half-life after the second half-life decay cycle was completed.

For example:
A constant population of 45,208,000,000 Carbon 14 atoms [with a half-life of 5,730 years] would have an average decay rate of 7.5 atoms per minute and the average age of the atoms would always be 8,595 years after the second half-life decay cycle was complete.

However, before Willard Libby could develop a Radiocarbon Dating model he needed to nail down the half-life of Carbon 14.

This was no easy task.

Origins: 1940-1945
Libby at Columbia: World War II Years

Requested 14C half-life measurement at Argonne National Laboratory:
26,000 ± 13,000 and 21,000 ±4000 years

Radiocarbon Dating: A History
2009 Radiocarbon in Ecology and Earth System Science
University of California, Irvine

Click to access UCI_C14HISTORY_2009.pdf

carbon-14 has a half-life of about 5,600 years…
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The half-life itself was measured in 1949 in collaboration with A. G. Engelkemeir, W. H. Hamill, and M. G. Inghram to be 5,580 ± 45 years, a value which when combined with independent values of 5,589 ± 75 by W. M. Jones, and 5,513 ± 165 by W.W. Miller, R. Ballentine, W. Bernstein, L. Friedman, A. O. Nier, and R. D. Evans, gave 5,568 ± 30 by weighting according to the inverse square of the errors quoted.

Radiocarbon Dating – Willard Libby
Nobel Lecture, December 12, 1960
http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1960/libby-lecture.pdf

By bombarding graphite with a strong deuteron beam from a cyclotron, the (d, p) reaction on the 13C present in the natural mixture gave enough radiocarbon for them to detect.

On the basis of this they gave the tentative value of 25 000 yr as the lifetime.

This was in 1940.
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However, within a year or two, as a result of researches in the laboratory of the author’s group at the Argonne Laboratory, and at the University of Chicago, together with a number of other teams, the value of 5568 yr was established; the radiocarbon dates have always been given on this basis, even though it is now known that it is some 3% low compared to the latest value of 5730 yr.

This convention has been adhered to for the purpose of avoiding confusion.

History of Radiocarbon Dating – 1967 – Willard Libby
http://www.osti.gov/cgi-bin/rd_accomplishments/display_biblio.cgi?id=ACC0336&numPages=23&fp=N

Surprisingly, this is an issue that still generates some confusion.

Half-life 5,730 ± 40 years
http://en.wikipedia.org/wiki/Carbon-14

Carbon-14, an isotope with a half-life of 5715 years, has been widely used to date such materials as wood, archaeological specimens, etc.
Los Alamos National Laboratory
http://periodic.lanl.gov/6.shtml

Eventually, Willard Libby settled on “the value of 5568 yr” and “radiocarbon dates have always been given on this basis” to avoid confusion!

With the issue of the half-life settled Willard Libby needed to nail down a credible rate of production for Carbon 14 so that a decaying population of Carbon 14 atoms could become stable when it was continually topped-up with newly created Carbon 14 atoms.

The next step in the history of radiocarbon dating was the discovery by Korff that neutrons are produced in the atmosphere by cosmic rays.

A counter had been developed at Berkeley by Korff and the author’s group which was capable of detecting neutrons; they found, on flying this counter on a balloon, that its count rate increased with altitude to a maximum at some 50 000 ft, after which it fell off again.
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In his first reference to this work Korff pointed out how the (n, p) reaction on nitrogen would undoubtedly make carbon-14; from the data of Korff and Hammermesh it was possible to estimate that, on average, one or two atoms of carbon-14 would be produced in this way each second for each cm2 of the Earth’s surface.

History of Radiocarbon Dating – 1967 – Willard Libby
http://www.osti.gov/cgi-bin/rd_accomplishments/display_biblio.cgi?id=ACC0336&numPages=23&fp=N

Given the choice between “one or two atoms of carbon-14” Willard Libby settled on a production rate of two atoms of Carbon 14 “each second for each cm2 of the Earth’s surface”.

Willard Libby could then reverse engineer his envisaged stable population of Carbon 14 atoms.

The reverse engineering simply involved calculating the number of Carbon 14 atoms that were needed to produce an average decay rate that exactly matched his chosen production rate of two per second.

Based upon the creation rate of 2.00 atoms per second the population of atoms achieves a stable decay rate of -2.00 [rounded to two decimal places] after 10 Half-Life periods [55,680 years] and the population stabilises at precisely 527,147,333,650 atoms after 40 Half-Life periods [222,720 years].

Winding-down the stable population is simply achieved by turning off the creation process.

The population then experiences a regular Half-Life decay and the decay rate reaches 0.00 [rounded to two decimal places] after 10 Half-Life periods [55,680 years].

The cautious Willard Libby sensibly decided to limited the accuracy of his model to two decimal places and thus defined the outer limit of Radiocarbon Dating to be 55,680 years.

The final step to complete Willard’s World was to reverse engineer a carbon reservoir size that could plausibly contain 527,147,333,650 atoms of Carbon 14.

The 1960 version of Willard’s World sets the carbon reservoir size at “about 8.5 grams of carbon per cm2”

To return to radiocarbon dating – knowing that there are about 2 neutrons formed per square centimetre per second, each of which forms a carbon-14 atom, and assuming that the cosmic rays have been bombarding the atmosphere for a very long time in terms of the lifetime of carbon-14 (carbon-14 has a half-life of about 5,600 years) – we can see that a steady-state condition should have been established, in which the rate of formation of carbon-14 would be equal to the rate at which it disappears to reform nitrogen-14.

This allows us to calculate quantitatively how much carbon-14 should exist on earth (see Fig. 1); and since the 2 atoms per second per cm2 go into a mixing reservoir with about 8.5 grams of carbon per cm2, this gives an expected specific activity of living matter of 2.0/8.5 disintegrations per second per gram of carbon.

Radiocarbon Dating – Willard Libby
Nobel Lecture, December 12, 1960
http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1960/libby-lecture.pdf

Given the numerous layers of assumptions, uncertainties and probabilities that underpin Willard’s World it is hardly surprising that Willard Libby thought his “good fortune in many stages of this research was most miraculous”.

Personally, I don’t believe in miracles.