Carbon-14 Dating Calculations

If we represent the number of C14 atoms at any given time after the death of a plant or animal by N, and original number of C14 atoms at the time of origin of the organism by No,  then the decay rate of C14 atoms  is given mathematically as follows:

N/No = e-kt                                                                                                                                       [1]

where t is the elapsed time since the original living organism’s death in years, k is the decay constant which for carbon-14 has the value  0.0001209, and e is the number 2.71828…, the base number of natural logarithms. This mathematical relationship graphs as follows:

We see that the carbon-14 concentration N/No decays to 0.5 ( i.e. 50%) after about 5730 years; this is called the ‘half-life’ of carbon 14.

Example 1: No contamination

If we assume, for example, that the true known age of a hypothetical linen test material is 1 A.D., i.e. 1999 years before, say, 2000 A.D., then the value of N/No = e-.0001209 x 1999 = 0.785.  That is to say, 78.5% of the original carbon 14 atoms present at 1 A.D. still remain, and  21.5% have decayed, over the 1999 years since the flax plants from which the linen was woven were harvested.

Therefore, in running a radiocarbon dating test on the sample, we would expect to get an instrumental reading corresponding  to a ‘radiocarbon age ‘t’ of 1999 years [1].

This would yield an historical age for the sample of 2000 – ‘t’ = 2000 – 1999 = 1 A.D. in agreement with what we would already have known for the example.

Example 2: 100% modern contamination

Weight of clean linen sample: 10 milligrams.   Origin of linen sample taken as being 1 A.D.,

Assume a weight of olive oil contamination of  10 milligrams  (added, say, at the time of the radiocarbon test by mistake).  Known age of olive oil contamination:  1 year.

Total sample weight is now 20 mg.

Value of N/No for clean linen sample ( t = 1999 yrs.)   = e-.0001209 x 1999 = o.785 as before.

Value of N/No for olive oil ( t =1 yr.) = e-.0001209 x 1  = 0.999879

Combined N/No for contaminated sample = 10/20 x 0785 + 10/20 x 0.999879 = 0.892

Inserting this value for N/N0 in Eqn. 1 and solving  we get  the experimental radiocarbon age  ‘t’ to be expected as:

t = ln[0.892 ]/ -.0001209 =  945 ‘years’

The calculated ‘historical age’ for the sample then (again assuming we are in the year 2000 A.D.) is 2000 – ‘t’ = 2000 -945 = 1055 A.D.

So we see that the effect of 100% contamination with modern carbon ( age 1 year) would be to artificially advance the  historical date for the linen forward from a true age of 1 A.D. to  1055 A.D., that is by over a thousand years.

Exampe 3: 100% contamination of average date  1000 A.D.

Weight of clean linen sample 10 mg. Known age of sample 1 A.D..

Weight of contamination 10 mg. Assumed average or mean age of the olive oil contamination  is now taken as 1000 A.D.

Value of N/No for clean sample:  0.785

Value of N/No for olive oil of age 1000 A.D.  = e -0.0001209 x 1000 = 0.886

Combined N/No for contaminated sample = 10/20 x 0.785 + 10/20 x 0.886 = 0.8355

Radiocarbon age ‘t’ to be expected = ln [0.8355.] /-.0001209  = 1487 yrs B.P.

Historical age of linen sample = 2000 – 1487 = 513 A.D.

Therefore the experimentally  obtained  radiocarbon result for a test sample of known origin of 1 A.D. which has been contaminated with 100%  of carbon of average age of 1000 A.D. will give an apparent computed historical date of origin of 513A.D. instead of the correct age of 1 A.D.

From these three examples we see that when radiocarbon tests are run  two questions must be asked and answered.

First, Is the sample clean? If so then the historical age is simply  [Year of test] minus[the number of radiocarbon  years “t” before the year of the test].

Second. If the sample is contaminated, can it be cleaned? If it cannot be cleaned then the non-removable contamination must be taken into account in the calculation of the historical age as in example 3 above..

Were the Shroud test samples contaminated?  How to correct for contamination

The unit area weight (specific weight, weight per square centimeter) of the 1988 test samples cut from the corner of the Shroud is simply the weight of each sample divided by its area. If this unit weight is found to be around 23 mg/sq cm, which is the known  average unit area weight for the Shroud as a whole, then the sample is clean and representative.  If it is more than 23 mg/sq cm,  then the sample is contaminated and non-representative.

The test samples in the 1989  report in Nature [2] each weighed approximately  50 milligrams, which agrees with the official figure given by Testore who weighed the samples in 1988  in Turin. The area of each samples was about 1.166 sq. cm each, as given  from the data of  Moretto [3], former Secretary to the  International Centre of Sindonology of Turin and Secretary of the Journal Sindon :

“The sample (was) reduced to 7 x 1 cm after removal of the frayed bits round the edges. This was then divided into two roughly equal parts, one of which was retained, and the other subdivided into three . Each of the three laboratories ...was allocated a  little more than a square centimeter of the Shroud textile, for this to undergo dating by the carbon 14 method.”

Thus the 7 x 1 cm trimmed  strip was first cut in half ,  and then three equal samples were cut from this one-half piece. This makes each of the three  samples  1/3 of 3.5 sq. cm or 1.166 sq. cm. in area  ( i.e. “ a little more than a square centimeter” as stated by Moretto.).

The  unit area weight of each sample is then calculated as  its weight, 50 mg,  divided by its area of 1.166 sq.cm for  42.9 mg.per sq. cm. ( 50/1.166 = 42.9).  Since the average weight for the Shroud as a whole, away from any heavily handled corner, is only  23 mg per sq., we have a ratio  of contamination  42.9/23 or 1.87 i.e.  87%.  This is not at all surprising, since the samples were cut from a heavily handled corner which even shows up visually as a slightly stained or darker area on photographs of the Shroud.

(Experimental tests have also been  run to determine the amount of extra weight that is taken up from human fingerprints in handling  an object.  The results easily reproduce  the 87% level of contamination on the samples cut from the corner of the Shroud)

How to correct for the 87% contamination to get the valid  estimate of the historical age of the Shroud

Given the 87% contamination  of the samples,  the task for the radiocarbon laboratory is then to determine if it is removable or not.  Here the Nature  Report is  ambiguous. The authors say only that the samples were vigorously cleaned and that the Zurich sample showed “no evidence of contamination” after the cleaning.  This could, of course, mean either that the contamination could not be removed by cleaning or, alternatively, that there was none. But, since  we already know that  it was 87% contaminated because of the very high  unit area weight of 42.9 mg. per sq. cm.,  the only conclusion is that the contamination was totally non-removable.

Later, however,  Prof. E. T. Hall of the Oxford radiocarbon team and himself one of the Nature authors, increased the ambiguity by stating in conversation that all samples lost about 20% of their weight in the cleaning!  This weight loss  could, of course, be partly a loss of contaminating carbon and partly a loss of some linen fibres from abrasion. However, let us conservatively assume that the 20% weight loss was all due to contamination,  and reduce the non-removable carbon contamination ( on all samples, including Zurich)  by Hall’s full 20%  from 87% to 50% ( 0.8 x 1.87 = 1.496 = 1.50) The radiocarbon age must then be adjusted, to account for this 50% contamination by more recent carbon picked up from human handling at the corner of the Shroud from which the samples were cut, in order to estimate  the true historical date for the origin of the linen.

If we estimate that the average date of origin of the 50% carbon contamination is, say, 1000 A.D , then the necessary adjustment to the  ‘radiocarbon age’ t of 691 years  reported in  Nature is as follows:

Specific weight of each sample                       42.9 milligram per sq.cm.

Less 20% loss on cleaning                            - 8.6 mg

Weight of clean linen plus non-removable carbon contamination 34.3 mg

Specific weight of clean linen  of Shroud                                   23.0

Net contamination                                                                     11.3

N/No for the clean linen ( t = 1988-33=1955 years) = e-0.0001209 x 1955 = 0.789

N/No for contamination (t = 1988-1000 = 988) = e-.0001209 x 988  = 0.887

Correction to t of  ‘691 years’  for contamination  ( N/No = 0.887 ) is:

N/No (comb.) = { fa x 0.789}  +  { fb x 0.887}

=  { 23/34.3  x 0.789} + { 11.3/34.3 x 0.887}

=            0.529             +         0.292  = 0.821

t (combined)   = loge 0.821/-0.0001209 =  1631 years ( instead of  691 yrs).

The corresponding estimated historical date, corrected for the contamination, is just  1988-1631 = 357 A.D. , or 1950-1631 = 319 A.D if we take the base year as 1950 instead of the actual year of the test).

To sum up, contamination was not considered in the Nature report, whose authors  simply went ahead as though the samples were clean and published their erroneous  mediaeval date for the origin of the linen of the Shroud [1988-691= 1297   or 1950- 691  = 1259 A.D., declaring that they had 95% confidence in their calculation. Quite naturally, this assertion  has unraveled over the years, after the official Turin information on the sample sizes which showed the ignored contamination, become widely available. .

Further correction for  C-14 enhancement by neutron flux, if any:  Finally, we should account for the enhancement of  C14 by any possible neutron flux. The  exchange between Phillips and Hedges  in Nature n 1989 [4] generated widespread interest and discussion. One objection raised to the C14 enhancement by neutron flow was that, while neutrons would undoubtedly enrich the C14 content of linen, this was no proof that the new C14 atoms so produced would remain in the linen and not just simply diffuse out and evaporate into the air.

J-B. Rinaudo [5] settled this point experimentally by irradiating a piece of ancient linen of  known historical age with a neutron flow in a reactor, and then measuring the radiocarbon age. He found, as predicted, that the apparent age of the cloth had been greatly advanced by the neutrons in accordance with the predictions, thus proving that the new C14 atoms produced by the neutron flow did indeed bind to the linen and remained there to alter the radiocarbon date. Whether this result will eventually prove relevant to the Shroud mystery will depend on whether a plausible physical source for the postulated neutron flux can be established. This will be examined in further Updates to the website.

A final word of caution to the non-scientist here.  One must be careful in interpreting scientific statements about radiocarbon dates since the measurements and calculations  are not an exact historical accounting or balancing procedure .  The Nature authors for example, quite properly rounded all their date estimates only to the nearest 10 years.  Exact years here are  only misleading.  In any case, we  are only concerned with the broad question  of whether the Shroud is medieval or ancient. Therefore, the minor complications of such things as the  38 years difference between “years B P, meaning “years before 1950”  on the one hand and the  “years before 1988 the date of the test measurements ´on the other hand are of no importance to the conclusions. For another example, the Nature report gives the mean radiocarbon date of 691 BP plus or minus 31. This gives an historical age slightly different from their statistical average age of around 1325 A.D. but there is no essential contradiction. The critical analysis given here is not concerned  with minor arithmetical, statistical or calibration details, or with irrelevant slight inconsistencies.  It deals with the major non-removable contamination of the Nature  samples which was neither taken into account nor reported.

References

1. For historical reasons, it has become customary in radiocarbon work to take B.P. (  “before present” )  to mean   “before 1950 A.D.”,  in spite of the fact that tests made today are some 50-odd years later.  This small technical  correction to the calculation of dates is not made  here for reasons of simplicity and clarity of presentation to non-scientist readers. To take it into account  for the Shroud tests in 1988 would mean an adjustment to calculated dates of  only 38 years ( 1988-1950 = 38).

It should also be noted that professional Carbon-14  dating calculations are very sophisticated and specialized; the calculations given here are simplified for clarity, but with no loss of validity.

2.  P.E. Damon, et al., “Radiocarbon dating of the Shroud of Turin”, Nature, 337, 6208, pp 611-615, Sept. 11, 1989.

3. Gino Moretto,  The Shroud: A Guide ( English transl.).  Paulist Press, New York, N.Y. 1996.

4.  T.J. Phillips and R.E.M. Hedges. Correspondence in Nature, 337, 16 February, p. 594. 1989.

--------- T.J. Phillips,  Reply to Dr. R.E.M Hedges’ Nature correspondence. British Society for the Shroud of Turin Newsletter No. 22, May 1989, pp. 8-11.

5.  J. B. Rinaudo, “Image formation on the Shroud of Turin explained by a protonic model affecting radiocarbon dating”  III Congresso internazionale di studi sulla Sindone, Torino, 5-7 Giugno 1998.

---------------------,   “Theory No. 3: French Scientist Jean-Baptiste Rinaudo”.  British Society for the Shroud of Turin Newsletter. No. 38, Aug.-Sept., 1994.