All?gre Claude J. Isotope Geology

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inside the starswhichgive o¡energy. Electromagnetic radiation,and light radiation in parti-

cular, is given o¡ from the star’s surface. Abalance is established between the two processes

anditishowlongthisbalanceis maintainedthatdeterminesthestar’slifetime.Whenthe equi-

libriu m is disrupted,thatis,whenthefuel is exhausted, thestarchangesnature.

The characteristic feature of nuclear reactions is that they need a certain amount of

energy to‘‘ignite.’’ Once triggered, mostofthe m produce considerable quantities ofenergy

intheirturn,becausetheyconvert matter into energy, inaccordancewithEinstein’sfamous

formula E ¼mc

What is the source of energy triggering these nuclear reactions? Gravitational en er gy.A

star is born when an interstellar cloud ofgas and dustcontract under the in£ uence ofgravity,

that is the mutual attraction exerted by all components of matter. The contraction of the

cloudgeneratesenergyby the impacts producedbetween‘‘portions’’of matter.Whenthetem-

perature reaches about10 million degrees, hydrogen fusion occurs and a star is born.This is

whathappenedforour Sun some 4.57 Ga ago.Thebirthofordinarystars isthe mosteveryday,

the most commonplace phenomenon in the Universe. Most stars are of this type. Like the

Suntheycan make onlyhelium.Th e starsare classi¢ edbytheHertzsprung ^Russell (H^R)

diagram on which we plot brightn ess against color, that is, the star’s temperature

(Figure 4.23).White is veryhot and red is cooler. Most starslie on the leading diagonal of the

diagram,which istherefore calledthe main sequence.

OXYGEN

MAGNESIUM

SILICON

IRON

Constitutive

elements

inner

planets

–2

20 30 40 50 60 70 80 90 10010

Proton number (Z)

Abundance relative to

silicon normed to 10

{

HYDROGEN

HELIUM

CARBON

NITROGEN

{

Constituent

elements

giant

planets

Trace elements in planets and meteorites

elements

Iron

peak

Neutron addition

Figure 4.22 Abundance charts of elements in the cosmos. The scale is logarithmic. Silicon (Si) is taken as

the reference. After Broecker (1986).

147 Cosmic irradiation: from nucleosynthesis to stellar and galactic radiation

Remark

The Sun was born from gravitational contraction of a cloud of gas and dust already containing all

of the chemical elements we now ﬁnd in the Solar System (in the planets and in the Sun itself). The

origin of these elements therefore pre-dates that of the Sun itself.

For starswith low masses (likethatofour Sun orlower), when the nuclear fuel is exhausted

the starburns out, expands slightlyandthen contracts, itshrivels up togivebirthto atinystar

knownasawhitedwarf(bottom ofFigure 4.23).This iswhatwillbecomeofthe Sun in 5 Ga!

For more massive ‘‘ordinary’’ stars, the stellar adventure w ill continue more gloriously.

The core of the star, made of helium, will contract again and its temperature rise while at

the same time the outer envelope of the star will expand (and so its temperature fall). In the

20 000

15 000

10 000

9000

8000

7000

6000

Surface temperature

Brightness

violet blue white yellow yellow orange red

–1

–2

–3

–4

–5

Color

White

dwarfs

Sun

Planetary

nebulae

Red

giants

Super

red

giants

Horizontal branch

SUPERNOVAE

Figure 4.23 Hertzsprung–Russell (H–R) diagram of stars. Note that the temperature scale on the

-axis

increases from right to left.

148 Cosmogenic isotopes

core,fusion reactionswillbetriggeredproducingconsiderable energy.Littlebylittlethestar

will come tobe madeup ofa series oflayers like an onion, with each layer corresponding to

a temperature and therefore to a type of nuclear fusion reaction (Figure 4.24).These stars

arealreadyredgiants(like Betelgeuse). Some havethree or fourlayers, othersixor seven.

Atthe centeroftheseredgiantsreigntemperaturesof1to6billiondegrees,whiletheouter

envelopes are atonly100million degrees. Itis inthese redgiants, throughthe course oftheir

evolution,thattheheavyelementsaremanufacture dandthatthefusion reactionsofcarbon,

oxygen, etc. occurandthestatisticalequilibriumofthe iron peak is reached.

Some ofthesegiantstars will evolve furtherand reach an extremely rare stage whe re they

explodein afractionofasecond.Thesearesupernovae.In ourgalaxythereisonesupernova

explosion per century. Such events generate the en ormous neutron £u xes (themselves the

outcome of nuclear reactions) requ ired by the r-process. The s-process s eems to occur as

redgiants evolvetowards the explosivezone, alongwhatastronomersknowasthe asympto-

tic giantbranch(AGB star).

The main lines of this theory of nucleosynthesis are well understood. It unites in

a single coherentwholeastronomic observations, the abundance ofchemical elements and

isotopes,andthe parameters measured in nuclear physic s in reactorsoraccelerators.

Forourownp urposes,in isotopegeology,we candrawafewpracticalconclu sionsfrom it.

The Sun burns hydrogen only. The heavy elements in it therefore pre-date the Sun. They

Hydrogen

fusion

Unburnt

hydrogen

Hydrogen

fusion

Unburnt

hydrogen

Burning hydrogen H He 1–60 x 10

Process Fuel Product Temperature

Burning helium He C, O 200 x 10

Burning carbon C O, Ne, Na, Mg 800 x 10

Burning neon Ne O, Mg 1500 x 10

Burning oxygen O Mg to S 2000 x 10

Silicon fusion Mg to S Elements close

to iron

3000 x 10

Helium

fusion

Unburnt

helium

H fusion

He fusion

C fusion

Ne fusion

O fusion

Si fusion

Figure 4.24 Stages in the evolution of a massive star.

149 Cosmic irradiation: from nucleosynthesis to stellar and galactic radiation

were form ed in red giants or ancient supernovae. This simple remark allows us to under-

stand thatthehistoryofthe Universe is a historyofstellar cycles. Stars areborn, synthesize

elements, and die, and their m aterial is scattered into interstellar space.There the various

clouds mix and mingle the material synthesized previously. They wander around the cos -

mos until th e day n ew gravitational contraction occurs.This g ives rise to a new star, wh ich

in turn makes matterand disperses itin the Universe and so on. One supe rnovaperc entury

is not a lot, but that still makes 10 thousand every million years, or 40 million s ince the

Earth ¢rstformed (an d inourgalaxyalone).

These pre-solarheavyelementsweretherefore made in redgiants and supernovae dating

from solar prehistoryand then extensively mixed in interstellarspace.

For us earthlings,asparts ofthe Solar System,thereare two majordatesin this cosmichis-

tory: the Big Bang and the formation of the Solar System.The Big Bang was the event with

which our universe began. Astronomers now date itto13 billionyears ago. For them, it is the

instant at which matteraswe know it came into being in a phenomenal process ofexpansion

and heating. In the course ofthis process hydrogen, helium, and a little lithiumwere formed.

Then everyth ing arranged itself into galaxies, stars,andplanetary system s. It was in the

courseofthisstellarhistorythatall theotherelementsheavierthanlith iumweresynthesized.

The formation of the Solar System occurred 4.6^4.5 billionyears ago.We shall go into this

datemorecloselybutatthispointanorderofmagnitudeisenough.TheSolarSystemgathered

upmattermadeinpreviousstellarprocessesincludingexplosionsofsupernovaeorredgiants.

All this matter was mixed and arranged itself into a central star with orbiting planetary

bodies. Butto account for th e existence at the time the Solar System formed ofwhat are now

extinct forms of radioactivity we must accept that red giants and supernovae existed an d

explodedjustbeforeits formation andthattheyarethesource oftheseformsofradioactivity.

Thisscenario oflate explos ive nu cl e o sy nt h es is (redgiantand/orsupernovae) tosynthe-

size heavy elements in the Solar System has recentlybeen challenged (or supplemented) by

asecondscenario,thatofp rimitive ir radiation.Inthis modelitis accepted thatas itformed

the young Sun emitted extremely high-energy particles which in turn produced spallation

reactions on th e solar material and that these reaction s in particular produced the extinct

forms ofradioactivity.This scenariois thought necessary toproduce the

Be discovered by

ateam fromNancy(France) (McKeegan etal.,20 00)andthe

Cldiscovered morerecently

bya Chineseteam (Lin etal., 20 05). However, itdoes not seemtob e abletomake

Mn.The

two processes probablyoccurred in succession, but in what proportions? Th is is a ver y hot

subject atpresent.

All these events havelefttraces i n meteorites.These objects, which date fromthe time the

Earth, plan ets, and Sun were formed, contain extraordinary information. Near th eir sur-

faces are cosmogenic isotopes which inform us about cosmic radiation, but also aboutthe

timetheyhave spentin the Universe as small pieces ofrock. However, some ofthem contain

grainsof interstellardust whose isotopic composition in certain elementsinforms us about

the r- and s- processes of nucleosynthesis. These process es, as described, occurred well

before the beginnings ofour Solar System.This is true for most, save one category: extinct

radioactivity, whichwehavealreadycome across.Meteoritesarean invaluablelinkbetween

cosm ic and terrestrial processes. For physics, there is a continuum between stellar nucleo-

synthesis and nuclear reactions produced by particle £uxes from the Sun or from cosmic

radiation.

150 Cosmogenic isotopes

This scenario is the one that has occurred tirelessly in th e Universe over 10 or 14 billion

years. But in the beginning, what was there? What matter? The beginning was the Big

Bang, which synthesized the elementary particles, hydrogen, helium, and lithium an d

enacted an essential event ...which lies outside the scope of this book, although it impreg-

nates eve ry partof it!

Problems

1 Carbon-14 is produced by nuclear reactions on

N, but it also decays to

N. We propose to use

this decay scheme for dating purposes as is done with

Rb–

Sr or

K–

Ar, for example.

We consider that to make a measurement,

N must differ from the normal value (276)

by 1  10

5

. What should the C/N ratio be to apply this method to objects that are, say, 15 000

years old? Do you think this would be feasible?

2 The Holy Shroud of Turin long posed a problem for believers. Was it really the shroud in which

Christ had been wrapped? It ﬁrst appeared in 1350 and the monks asserted it was a genuine

relic. It could not be dated by the traditional

C method as there was not enough material, but

with AMS it could be dated with just 150 mg of cloth. The

C age was between 600 and 800

years old. Given the correspondence curve in Figure 4.25, what is the age of the Holy Shroud?

3 We measure the isotopic composition of K in an iron meteorite having undergone cosmic

irradiation for 500 Ma. The mass spectrum of K measured is a mixture between the cosmo-

genic mass spectrum and the normal mass spectrum. Estimating that (

normal

is negligible,

establish the equation for determining (

cosmogenic

. (Suppose the normal K isotope

ratios are known.)

500

300

1100

850

Years BP

650 450

1300 1500

700

900

1100

Calendar age

Radiocarbon age

Figure 4.25 Correspondence curve of the radiocarbon age and calendar age of the Turin Shroud.

151 Problems

4 Chlorine-36 is created in the atmosphere by spallation reactions on

Ar, and

1/2

¼3  10

years. From the time of its creation, the chlorine is incorporated in water masses and in the

hydrological cycle. The

Cl/Cl

total

ratio is 100 20  10

5

in rainwater at the Earth’s surface.

We analyze the water sampled from a deep aquifer and ﬁnd 5  10

15

for the

Cl/Cl

total

ratio.

How old is the groundwater?

5 Here is a series of

Be/

Be measurements on a manganese crust.

(i) What is the rate of accretion?

(ii) What is the error?

Depth (mm)

Be/

Be 2



Surface 3.94 0.32  10

8

0–1.0 3.23 0.12  10

8

1.5–2.0 2.19 0.25  10

8

2.5–3.0 1.79 0.08  10

8

3.5–4.0 1.41 0.06  10

8

4.0–5.0 1.04 0.07  10

8

5.0–5.7 9.00 0.52  10

9

5.7–6.0 5.98 0.88  10

9

6.0–7.0 5.20 0.44  10

9

8.0–8.5 3.94 0.26  10

9

8.5–9.0 2.69 0.62  10

9

9.0–10.0 2.39 0.40  10

9

10.7–11.7 1.85 0.44  10

9

152 Cosmogenic isotopes

CHAPTER FIVE

Uncertainties and results of

radiometric dating

5.1 Introduction

Thepurposeofthis chapteristoreviewradiometricdatingmethodsasappliedbothtoterres-

trialandtocosm icproblems.Weshallspeakofthemajorresultsthatnow¢xthechronologi cal

frameworkofthe Earth’shistory, notforgettingthat whatthey mean is inseparable from how

reliabletheyare.The central questionwe shall be dealing with in this chapter is: how reliable

arethegeologicalageswedetermine? In other words,whatisthelevelofuncertaintya¡ecting

anage determination?

Whatguaranteehavewethatthis age isgeologically meaningful?

Ascertainingtheuncertainty, whenweobtainavalueT, allowingustowritethattheage

is T , is a problem that breaks down into various sub-problems, some of which are

uncommon even in books on uncertainty. Let u s get our ideas straight with a simple

example.

Supposewe measurea series ofrocksby the

Rb^

Sr method and the analytical results

arejust about alignedonthe (

Sr/

Sr,

Rb/

Sr)diagram. Howcanwe measurethe slop e

of the straight line isochron to c alculate the age of the rocks? Any experimental physicist

will tell you that you ¢rst need to know the uncertainty a¡ecting each individual physical

measurement: the strontium isotope ratios and the absolute values in

Rb an d

Sr. An d

thenthatyouneedtoestimatetheslopeofthestraightlineby usingtheleast-squaresmethod

where the ‘‘weight’’ of each data point is measured by its individual u ncertainty.We then

obtain an age and an unc ertainty value re£ecting the dispersion of data points relative to

the straight line we calculated.This approach, which has been tried and tested by years of

statisticalpractice,seemsunassailable onthefaceofit.Andyet, as soon asitisputintoprac-

tice, it raises questions that are not to do w ith mathematical statistics but with what we

might call geological understandi ng.

The ¢rst question is: should we attribute the analytical uncertainty measured in the

laboratory to all the datapoints? D espite the reassu ri ng lookofthis question, which invites

us to answer ‘‘yes,’’ we need to realize that such‘‘automatic’’attribution is not as safe as we

mightthink. In fact, the rockswiththehighest

Rb/

Sr values andtherefore

Sr* values,

are probably those that, in the course of geological history, have been the most likely to

To avoid any confusion with the word ‘‘error’’ used in ordinary language it is now recommended to use the

term uncertainty in scientiﬁc work. The word error is used when we know the difference between the true

value and the measured value; uncertainty is an estimate of this error. I shall try to abide by this logical rule,

although it goes against decades of habit. Indeed, it has often been used in the ﬁrst three chapters.

lose

Sr* through exchanges with adjacent rocks. Logically, we should therefore attribute

greater uncertainty totheir ages than tothose ofother rocks and m easure the de¢citin ge o-

logical robustness. Moreover, suchuncertainty wouldbe asymmetricalbecausethe loss of

Sr* leads to a reduction i n the

Sr/

Sr ratio (Figure 5.1). Should we therefore increase

the analyticaluncertaintyofageologicaluncertainty?

Another question arises. Given that we are working with a natural system and that the

conditionsofthebasic model (perfectinitialisotopichomogenization, boxclosed sincethe

origin)haveprobablynotbeen stringentlyobserved , thedatapointsaredistributed‘‘statisti-

cally’’(thatis,moreorl essaccurately)aroundth ebeststraight-line¢t.Howmuchdispersion

is acceptable for it to be considered that we are ind eed working within the frameworkof a

theoretical model and therefore thatthe age determination is legitimate? This is a di⁄cult

question and even more so ifyou thinkthat, as we goback i n time, thevagaries ofdisruptive

geological phenomena become ever more probable and that the alignment must be less

good for Preca mbrian rocks than for rocks of the Secondary era, say.Therefore ge olo g ical

tol eranc e for non-alignmentmustlogicallyincreasewithage. Byhow much?Towhatextent

canwe accept non-alignment, even for veryold rocks? We must remember that, in addition

to measurementunc ertainty, an alytical uncertai nty toois a fu nction ofage.There is, then, a

series ofpriorqu estionsto ourcentralquestion:is age determination reliable?

There is a second aspect to this issue of the reliability of measurements, that of the geo-

logical sign i¢cance ofthe age obtaine d. Imagine we have an age in thousands, millions, or

billions ofyears, together with itsun certainty.What does th e number obtained correspond

to geolog ically? Letus return toour series ofro cks whose ages havebeen determined by the

Rb^

Sr method. Isthe

Rb^

Sr straightline a‘‘mixing line’’or an isochron? Chapte r 3

(S ection 3.4) gave us a test for deciding between these two hypotheses. Suppose we thin k it

Geological

uncertainties

Sr/

Rb/

Excluded

Figure 5.1 A series of Rb–Sr measurements plotted on the isochron diagram. After examining the

dispersion of data points, there seem to be three options for obtaining the ‘‘best age.’’ (a) We can

calculate the best straight-line ﬁt using all of the data points to which analytical uncertainty has been

attributed and obtain an age determination. The uncertainty of the pattern suggests that this age is

very much inﬂuenced by samples 6 and 7. (b) We add a geological uncertainty to samples 6 and 7

because they are the most radiogenic and so liable to have lost

Sr. We then calculate the best straight-

line ﬁt. (c) We eliminate samples 6 and 7 and use samples 1–5 alone to calculate the best straight-line ﬁt.

154 Uncertainties and results of radiometric dating

plausible we have an isochron.What does this mean? Our basic model is of a closed box in

whi chwe enclose radioactive and radiogenic isotopes at awell-de¢ ned‘‘instant’’ T

. But to

what does this formal theoretical model correspond in the world of nature? Suppose that,

to getour ideas straight, the series ofrocks in question is a series ofgranite ro cks from a sin-

gle massif. Isthe age obtainedthe agewhen thegranitewas emplaced as a magmaor the age

of the metamorphic basement whose partial melting gave rise to the granite mass?

Assuming it is the emplacement of the granite that is dated, what is the exact time of that

emplacement? When th e magma ¢rst intruded? When it cr ystallized? Or is it when hydro-

thermal circulation occurred, which seems to end the emplacement of granite massifs as

attestedbyquartz veinssometimescontaining metallicm inerals?Onceagain,this question

is not unconnected to th e age measured. And if we measure the minerals, have they not

been subjectedtosome subsequentevent?Didtheyall crystallizeatthesame instant?

The duration of emplacement of granite being estimated at 1Ma in all (perhaps 2 Ma),

when we date a granite that is 2 billion years old this question is irrelevant as our power of

resolution is insu⁄cient to distinguish the various episodes in its emplacement. But when

we datethegranitesoftheislandofElba (Italy) as 7 or 8 Maold, thequestionbecomesacru-

cialone.We are left to‘‘date’’something w ithoutreally kn owing whatitis we are dating! The

agewill be an‘‘average age’’ofthe emplacementofgranite on the islandofElba, but we must

not read more into it than that! T hat is, unless we have other independent information

tohand.

This is the sortofquestionwearegoing totry to answer while exploring the major results

obtained by radiometric dating methods. Eachwill be placed in the contextof how reliable

it is.We shall see that the p receding chapters have given some poi nters for some of the pro-

blems.Forothers,weshall needtogivesubstancetoour intuitionsandattempttorationalize

them ...or quali fy them. But the ultimate test of any age determ ination by radiometric

methodsis howcoherentitiswithwhat we alreadyknow.

Thisknowledgeformsthe chronologicalsetting withinwh ichthe newdeterminationisto

be¢tted.The overal l framework is formedbythe geologic al and cosmic timescale.

The age ofthe Earth orofthe Sun is a ¢xed point and our ultimate reference point in any

‘‘cosmic’’ chronology. At the same time, these general references are only known with a

degree ofu ncertainty because of all the sources we have referred to and are going to study.

There is, then, constantfeedbackbetween advances on uncertainties and the major results.

The major results are to be looked on as numbers that are liable to vary somewhat or to see

their underlying meaning change (we shall see this is the case for the age of the Earth).We

shall see there is a broad range ofuncertainty as regards the chron ology offormation of the

elements.Itisto emphasizethefundamentalconnectionbetweentheuncertaintiesinherent

to the methods of radiochronology and its main ¢ndings that they have been brought

together here in a single chap ter.

5.2 Some statistical reminders relative to the

calculation of uncertainties

When we make N measurements of x, they are represented in a (N, x) plot by groupi ng the

values of x into classes.This is a histogram.Wh en N is large and the results are distributed

155 The calculation of uncertainties

ran do mly, we consider that the best estimation of the tru e value sought is th e mean of th e

measuredvalues(note d



x).



x ¼

þ x

þþx

To estimatetheuncertaintyassociatedwiththis,we consider the dispersion ofthemeasure -

ments obtaine d byc alculating the deviations from the mean foreach measurementandtak-

ing their average.Inp ractice, we calculatethe sum ofthesquared deviations:

ðx





xÞ

andwe dividebythe numberofmeasurements N minus1togetthe mean variance:

ðx





xÞ

ðN  1Þ

The s tandard dev iation is the squarerootofthevarianc e:



ﬃﬃﬃﬃﬃﬃ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðx





xÞ

ðN  1Þ

The drawbackofusing 

is that it is expressed in units ofx.Sowecannotcompare,say,the

dispersion for the isotopic measurement of lead and that of strontium.To make such com-

parisons we de¢nethe reduced dispersion bydividing 



 ¼





This dispersion or reduc ed deviat io n is express ed as percent, per mil, etc.

Whenuncertainty is considered tobe a random £uctuation due tofactors thatare poorly

determined but which a¡ect each measurement and make it deviate around its true value,

then results of theoreti calstatistics canb e applied. Itcanbe shown that if there were an in¢-

n ite nu mberof random measurements, thevalues measured would be distributed around a

meanvalueinaccordancewithwhatisknownasthe Laplace^Gaussor normaldistribution

(Figure5.2).Th e curveofthenormal distribution is written:

y ¼



ﬃﬃﬃﬃﬃﬃ

exp 

x 







where y isth e measurementfrequencyand is a functionofth e variablex.

We use the square so that positive and negative deviations are added and do not cancel out by

subtraction.

Suppose we have just one measurement. If we divide by N, the uncertainty would be zero whereas if we

divide by N 1, that is by zero, it is indeterminate, which is indeed the case!

156 Uncertainties and results of radiometric dating