All?gre Claude J. Isotope Geology

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With

¼4  10

years, the result for the mantle is: 8.68  10

16

þ10.56  10

16

þ6. 59  10

16

18.4  10

16

¼44.23  10

16

cal g

1

¼1.84  10

14

1

For granite of the continental crust: 4.96  10

14

þ6.03  10

14

þ3.76  10

14

þ8.537  10

14

23.28  10

14

cal g

1

or 97.3  10

14

1

Notice that, at the present time, radioactive heat is supplied above all by the disintegration

238

U and to a lesser extent

K and

232

Th. Four billion years ago heat was supplied mainly by

K and

235

U (Figure 1.8). It will be observed, above all, that 4 billion years ago the mantle

produced 3.5 times as much heat as it does today. So it may be thought that the Earth was 3.5

times more ‘‘active’’ than today.

Problems

1 Which molecules of simple hydrocarbons may interfere after ionization with the masses of

oxygen

O, and

O when measured with a mass spectrometer? How can we make sure

they are absent?

2 The lithium content of a rock is to be measured. A sample of 0.1 g of rock is collected. It is

dissolved and 2 cm

of lithium spike added with a lithium concentration of 5  10

3

moles per

liter and whose isotope composition is

Li/

Li ¼100. The isotope composition of the mixture is

measured as

Li/

Li ¼10.

Given that the isotopic composition of natural lithium is

Li/

Li ¼0.081, what is the total

lithium content of the rock?

3 A sample contains 1 mg of strontium. What must be the maximum acceptable chemical blank,

that is, the quantity with which the sample is accidentally contaminated, if precision of

measurement with the mass spectrometer of

Sr/

Sr 0.7030 bears on 0.0001?

The

Sr/

Sr ratio of the blank is 0.7090. What must the blank be if precision is 10 times

better?

4 We are to construct a mass spectrometer for separating

Rb from

Sr. What should its

radius be?

1000 02000300040004600

100

total

Time (Ma)

Heat production (10

–12

W kg

–1

)

235

232

238

Figure 1.8 Heat production by various forms of radioactivity in the Earth versus geological age.

27 Problems

5 Suppose that 1 mg of puriﬁed uranium has been isolated. It contains two (main) isotopes

238

and

235

U in the current proportions of

238

235

U ¼137.8. What was the total activity of this

milligram of uranium 4.5 billion years ago and what is its activity today if

238

0.155 125  10

9

1

and

235

¼0.9875  10

9

1

6 The Urey ratio is the ratio of heat from radioactivity to total heat which includes heat from the

accretion of the Earth and the formation of its core. The average heat ﬂow measured at the

Earth’s surface is 4.2  10

W, which is 42 terawatts.

(i) In a ﬁrst hypothesis the mantle composition is assumed to be uniform, with:

U ¼21 ppb (Th/U)

mass

¼4 and K ¼210 ppm.

Calculate the Urey ratio today.

(ii) In a second hypothesis, it is assumed that the entire mantle is similar to the upper mantle.

The upper mantle has a uranium content of 5 ppb, and the Th/U ratio is 2. Calculate the

Urey ratio.

28 Isotopes and radioactivity

CHAPTER TWO

The principles of radioactive dating

It can never be repeated enough that radioactive dating was th e greatest revolution in the

geological sciences. Geology is an historical science which cannot readily be practised

withoutaprecisewayofmeasuringtime.Itis safetosaythatno moderndiscover yingeology

could havebeen made without radioactive dating: reversalsofthe magnetic ¢eld, platetec -

tonics, the puzzle ofthe extinction ofthe dinosaurs, lunar exploration, the evolution oflife,

humanancestry, notto mentiontheageofthe Earthorofthe Universe!

The ages i nvolved in the earth sciences are very varied.Th ey are measured in years (yr),

thousands ofyears(ka), millionsofyears(Ma), andbillions ofyears(Ga). Geological clocks

mustthe reforebevariedtoo, with mean livesranging from ayear toabillionyears.

2.1 Dating by parent isotopes

Imagine we have a radioactive isotope R and N



is the number of atoms of this isotope.

Supposethatgeological circumstances (crystallization ofarockor mineral, say) enclos e an

initial quantity of R, i.e., the number ofatoms of R at time zero, written N



ð0Þ, in a‘‘box. ’’ If

thebox hasremained closedfromwhen it¢rstformeduntil today, the numberofatomsof R

remaining is N



ðtÞ¼N



ð0Þe

lt

,wheret is the time elapsed since the boxwas closed. Ifwe

know the quantity N



ð0Þand the decay constant l, by measuring N



ðtÞ we can calculate

th e age t at whichthebox closedby using the radioactivity formula‘‘upside down’’:

t ¼



ð0Þ



ðtÞ



Methods where the initial quantities of radioactive isotopes are well enough known are

above all those where the radioactive isotope is produced by irradiation by cosmic rays.

This isthe case ofcarbon-14 (

C) and beryllium-10 (

Be).

Exercise

The half-life of

C is 5730 years. The

C content of the atmosphere is 13.2 disintegrations per

minute and per gram (dpm g

1

) of carbon (initial activity

). We wish to date an Egyptian

artefact dating from approximately 2000 BC. What is the approximate activity (

) of this

artefact? If our method can measure 1 dpm, what mass of the (probably precious) sample will

have to be destroyed?

Answer

If the half-life

¼5730 years, the decay constant is l ¼ ln2=T ¼ 1:209  10

4

1

. 2000 BC

corresponds to a time 4000 years before the present, therefore since

¼13.2 dpm g

1

and



¼7 dpm g

1

Making the measurement means using at least 1/7 g, or 142 mg of the sample.

As seen in the examples, the abundance of a radioactive isotope is estimated relative to a

reference. For

C we use total carbon. The dating formula is therefore:

C=CÞ



where (

C) and (C) represent the concentrations of

C and total carbon, respectively.

In other cases, a neighboring stable isotope that is not subject to radioactive decay is taken

as the reference. So for

C, we use stable

C and we write:

CÞ

This formulation has the advantage of bringing out isotopic ratios, that is, the ratios mea-

sured directly by mass spectrometry.

2.2 Dating by parent–daughter isotopes

2.2.1 Principle and general equations

The di⁄culty with datingbytheparentisotopeisofcourseknowing N



ð0Þ,thatis,knowing

exactlyhowmanyradioactiveatomsweretrappedintheboxatthebeginning.This di⁄culty

is overcome by involving the stable daughter isotope produced by the disintegration noted

(D) (the asterisk being a reminder ofthe radioactive origin of the isotope R*).The parent^

daughter relation iswritten:

ðRÞ



!ðDÞ

Fromthe Curie^Rutherford^Soddylaw, we canwrite:



ðtÞ

¼l N



ðtÞ

¼



ðtÞ

¼ l N



ðtÞ:

Integrating the ¢rst equ ation yields the decay l aw, N



ðtÞ¼N



ð0Þ e

lt

.Thesecondis

therefore written:

ðtÞ

¼ l N



ð0Þe

lt

;

30 The principles of radioactive dating

which integrates to:

ðtÞ¼N



ð0Þe

lt

þ C:

The integration constant C is determined by writing in t ¼0, N

¼N

(0), hence:

C ¼ N

ð0ÞþN



ð0Þ.Thisgives:

ðtÞ¼N

ð0ÞþN



ð0Þð1  e

lt

Þ:

But th is expression leaves the troublesome unknown N



ð0Þ. This is advantageously

replaced by:



ð0Þ¼N



ðtÞ e

Thisgives:

ðtÞ¼N

ð0ÞþN



ðtÞðe

 1Þ:

If the box remains closed for both the radioactive isotope and the radiogenic isotope, by

measuring the present values N

(t)andN

(t), we can calculate t, providedwe know N

(0).

This canbe plotted as (N

(t), t).The slopeofthe curveateachpoint equals:

ðtÞ

¼



ðtÞ

¼ lN



ðtÞ¼lN



ð0Þe

lt

It therefore equals lN

(t), at l times the parentisotope content. Asthis contentdecays con-

stantly, theslope ofthetangentdoeslikewise, andthe curveis concavedownwards.To calcu-

latean age, wewritethe datingequation:

t ¼

ðtÞN

ð0Þ



ðtÞ



þ 1



Thevalues of N

(t)andN



ðtÞcanbe measured, butt can onlybe calculated if N

(0) canbe

estimated or ignored and, of course, ifwe know the decay constant l. Figure 2. 1illustrates

all these points.

EXAMPLE

Rubidium–strontium dating

Let us take rubidium–strontium dating by way of an example. As seen,

Rb decays to

Sr with a decay constant

¼1.42  10

11

1

. The parent–daughter dating equation is

written:

Srð

Þ

Srð0Þ

Rbð

þ 1



;

where

Sr(0) is the quantity of

Sr present at time

¼0, and

Rb(

) and

Sr(

) the quantities

Rb and

Sr present at time

. (The term

quantity

must be understood here as the number

of atoms or atom–grams.) Notice that time can be reversed and the present time considered

as the starting point, which is more practical. The initial time is then in the past, age (

) such

that

31 Dating by parent–daughter isotopes

The equation is then written:

SrðpÞ

Srð

RbðpÞ

þ 1



;

where

Sr(p) and

Rb(p) relate to the present-day values (p ¼present), and

Sr(

) relates to

the initial values at time (

). When dealing with minerals that are very rich in Rb such as

biotite and muscovite, the initial

Sr is negligible relative to the

Sr produced by decay of

Rb. For such systems, which are said to be radiogenically rich (or just rich for short), the

dating formula is written:

SrðpÞ

RbðpÞ

þ 1



This formula can be extended to rich systems in general:

ðpÞ

þ 1



As seen, if

is known, the age can be calculated directly from measurements of the present-

day abundances of

(p) and

(p). The only assumption made, but which is crucial, is

that the box to be dated, that is, the mineral or rock, has remained closed ever since the

time it formed and that closure was short compared with the age to be measured. This

is indeed the case when a mineral crystallizes or a magma solidiﬁes as with volcanic or

plutonic rock.

R(0)

D(0)

Time graduated in periods

D(0)

24 60

Time graduated in periods

Number of atoms of R or D

24 6

R(0)

= N

D(0)

+ N

R(0)

(1–e

–

λt

)

= N

R(0)

–

λt

+ (e

–1)

Figure 2.1 Left: the decrease in the radioactive isotope and the increase in the radiogenic isotope. Right:

the increase in the radiogenic/radioactive isotope ratio.

number of atoms of the radioactive isotope (R)

number of atoms of the radiogenic isotope (D)

(0) number of atoms of R at time

¼0

(0) number of atoms of D at time

¼0

radioactive decay constant

32 The principles of radioactive dating

Remark

Where

is very small relative to the age, the exponential can be approximated by e

1 þ

. This

is the case for the constant

of rubidium (

¼1.42  10

11

1

) and for many others. The dating

formula is then:



SrðpÞ

RbðpÞ



Exercise

Suppose we have a specimen of biotite from Que

rigut granite (Pyre

es Orientales, France)

whose age is to be determined from

Rb !

Sr decay. We measure the total content of

Rb ¼500 ppm and of Sr ¼0.6 ppm. Knowing that Rb is composed of

Rb and

Rb in the

proportions

Rb/

Rb ¼2.5, that the Sr is ‘‘pure’’ radiogenic

Sr, and that the decay constant

¼1.42  10

11

1

, calculate the age of the biotite in Que

rigut granite.

Answer

The Rb content is written Rb

total

Rb þ

Rb ¼(2.5 þ1)

Rb, therefore:

Rb ¼

total

3:5

 142:8 ppm:

The

Sr content is 0.6 ppm. (There is no need to come back to atom–grams since

Rb and

have virtually the same atomic mass.) We can therefore write directly:



1:42  10

11

0:6

142:8



 298  10

yr ¼ 298 million years:

If we had not adopted the linear approximation, we would have obtained:



1:42  10

11

0:6

142:8

þ 1



 297:36  10

yr:

As can be seen, the linear approximation is valid for the

Rb/

Sr system when the age is not

too great.

2.2.2 Special case of multiple decay

Letus considerthe caseofthelong-period

K radioactiveisotope which decays in two dif-

ferent ways, either by electron capture giving

Arorby 



decaygiving

Ca, eachwith its

own decayconstantl

and l





, respectively:

–

cap

–

Whatis the dating formula? Letusgetbacktobasics:

d=dt½N

ðtÞ ¼ ðl

þ l



ÞN

ðtÞ

33 Dating by parent–daughter isotopes

where N

isthe numbe rof

K nuclides andN

the numberof

Ar nuclides.

d=dt½N

ðtÞ ¼ l

ð0Þe

ðl

þl



Þt

Integrating using theusual notationwith

Kand

gives:

ArðtÞ¼

KðtÞ

þ l



ðl

þl



Þt

 1



withl

¼0.581 10

10

1

and l



¼4.962 10

10

1

l ¼ l

þ l



¼ 5:543  10

10

1

The initial

Ar is usually negligible.We are therefore generally dealing with r ich systems

but not withvery young systems where whatis known as excess argon raises di⁄culties for

accurateage calculations.

Exercise

We analyze 1 g of biotite extracted from Que

rigut granite by the

K–

Ar method. The biotite

contains 4% K and

K ¼1.16  10

4

of K

total

The quantity of argon measured at standard temperature and pressure is 4.598  10

5

Ar. What is the radiometric age of this biotite?

Answer

The dating formula to calculate the age is written:

þ l



þ l



þ 1



We must therefore calculate the

Ar/

K ratio in atoms.

As there are 22 400 cm

in a mole at standard temperature and pressure, the value of

in number of moles is:

4:598  10

5

22 400



¼ 2:053  10

9

moles of

Ar:

The value of

K is:

0:04  1:16  10

4

¼ 1:16  10

7

moles of

Therefore:

0:5543

2:053  10

9

1:16  10

7

 0:1048



þ 1



¼ 280 million years:

Comparing this with the result obtained previously using the

Rb–

Sr method, we ﬁnd

about the same age but slightly younger.

There is another branched decay used in geology, that of

138

La which decays into

138

and

138

Ce(Naka

ı etal.,1986).

34 The principles of radioactive dating

–

cap

138

Cap e

– = 4.44

–12

–1

= 2.29

–12

–1

138

–

138

There is also a case where even more intense multiple decay occurs, in the spontaneous

¢ssion of

238

U, which yields a whole series of isotopes. Luckily, ¢ssion decay is negligible

comparedwith decay. In the dating equationwe can consider the constant l



alone asthe

decay constant, but allowance mustbe made, of course, for the ¢ssion products. For exam-

ple, for

136

Xewewritethe dating equation:

136

radio

238

¼ Y

fission



ðe



 1Þ

whereY is th e yieldof

136

Xeproducedduring ¢ssion 0.067 3, with l

¢ssion

¼8.47 10

17

1

and l



¼1.55 10

10

1

Wesawwhen lookingatdatingbyparentisotopesthatitwasconvenienttoexpressthe dat-

ing equation by introducing isotope ratios rather than moles of radioactive and radiogenic

isotopes.Thisis callednormalization.Thusfor

Rb^

Sr weuseastablestrontium isotope,

Sr.The dating equation isthenwritten:

t ¼

ðtÞ

ð0Þ

þ1

This is the form in which dating equations will be expressed fro m now on. A system will be

richwhen:

ð0Þ

ðtÞ:

2.2.3 Main geochronometers based on simple

parent–daughter ratios



Rubidium ^ Strontiu m

Rb 

87

Sr ; l ¼1.42 10

11

1

.The normalization isotope is

Sr. Developed in its modern formbyAldrichet al.(1953).



Potassium ^Argon

Ar, with the constants alreadygiven.The reference isotope is

Ar.DevelopedinitsmodernformbyAldrichandNier(1948a).



Rhenium^ Osmium

187

Re 

 187

Os with l ¼1.5 10

11

1

. The reference isotope is

186

Os and more recently

188

Os. Developed by Luck, Bi rck,andAlle

gre in 1980 after a

¢rstattemptbyHirtetal.in1963.

These are the three simple clocks that are commonly found as rich systems in nature.We

shallseethatother formsofdecay c anbeusedbutunder moredi⁄cultcircumstances.



Sam a ri um ^Neodymi um

147

Sm 

143

Nd ; l ¼6.54 10

12

1

. Normalization by

14 4

Nd.

Develop edbyLug mai rand Mart i in1977 after an at temptbyNotsu etal.in1973.

35 Dating by parent–daughter isotopes



Lutetium ^ Hafnium

176

Lu 

176

Hf ; l ¼2 10

11

1

. Norm alization with

177

Hf.

DevelopedbyPatchett andTa t s u m ot o (1980a,1980b).

These methods are supplemented by others related to radioactive chains which are to be

examined next, by the extinct radioactive methods covered in Section 2.4,andbythe

inducedradioactivemethods examined inChapter 4.

2.3 Radioactive chains

2.3.1 Principle and general equations

When

238

235

U, a n d

232

Th decay, theygive risetothreeother radioactiveisotopeswhich,in

turn, decay into new radioactive elements, and so on.The process stops when the last iso-

topes produced are the three lead isotopes

206

Pb,

207

Pb, and

208

Pb, which are stable. It was

radioactive chains which allowed both Pierre and Mar i e Cu r ie and Rutherford and

Soddy to discover the mechanisms ofradioactivity. A radioactive chain can be represented

by writing:

ð1Þ



!ð2Þ



!ð3Þ



!ð4Þ



! ...!ðnÞ stable:

Decayinvolves and radioactivity. Alpharadioactivitygives o¡helium nuclei.Their path,

inthe (Z, N)plot,bringsthe end product into thevalleyofstability.

Figure 2.2 shows the prec ise structure ofthe three chains. Mathematically, as studied by

Bateman (1910), radioactive chains can be described by the Curie^Ruth erford^Soddy

lawswritten one after the other:

ðtÞ

¼l

ðtÞ

¼ l

ðtÞl

ðtÞ

¼ l

ðtÞl

ðtÞ

¼ l

ðtÞl

ðtÞ

¼ lN

n1

ðtÞ

where N

is the number of nuclides of elements i and N

the number of nuclides of the ¢nal

stable isotope. Successive integration of these equation s presents no real di⁄culty and is

even agood revision exercise for integrating ¢rst-order di¡e rential equations with constant

coe⁄cients. Letus leave thatfor nowand concentrate on a fewsimple and importantlimit-

ing cases.

2.3.2 Secular equilibrium: uranium–lead methods

We suppose that, in view of the length of geological time, the radioactive chain reaches a

stationarystatewhere the contentofall the intermediate radioactive isotopes remains con-

stant(this is known assecularequilibrium):

¼ ¼

n1

¼ 0:

36 The principles of radioactive dating