All?gre Claude J. Isotope Geology

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organic beam of any molecules and other impurities by passing them either through

th in sheets of gold or through gas streams. The carbon ions pass through these whi le

any molecules are‘‘d eactivated’’andstopped.Inthiswaythe method’s speedofanalys is,sen-

sitivity, and precision have been increased compared with counting. A measurement that

oncetookmorethanaweekcan nowbe made in an hour. But aboveall, the method requires

samples of one-tenth of th e size and achi eves levels of precision 100 times greater (see

Figure4. 7).

Exercise

If the maximum that can be measured by AMS for the

C ratio in 1 mg of carbon is 1.2 

17

, what is the maximum age that can be measured with

Answer

10 000 years.

Exercise

If the limit of detection by counting methods is 10 disintegrations per hour, what quantity of

carbon is required to measure the same age?

Answer

2.213 kg!

Pre-selection electrode Stripping canal

Stripping gas

pump

Voltage

multiplier

Gas

ionization

detector

Extraction electrode

Sample holder

Microscope

Cesium

gun

Figure 4.7 Accelerator mass spectrometer. This device is much larger and more complex than an

ordinary mass spectrometer as it operates at much higher ion acceleration energies, but the principle

is unchanged. The ions are extracted on the left by being bombarded with a beam of cesium ions and

are then deﬂected ﬁrst by an electrostatic ﬁeld E1 and then by a magnetic ﬁeld M1. In the center of the

ﬁgure, a microchannel injects gas which collides with the beams and destroys any molecules (this is

known as ‘‘stripping’’). The puriﬁed ions are then deﬂected by two electromagnets M2 and M3 and

collected by a detector similar to an ionization chamber used in particle physics.

117 Carbon-14 dating

4.2.3 Conditions for performing

C dating

Applying the formula

C ¼

Cð0Þ e

l

to ¢nd an age entails meeting a number of

extremelystringentconditions.

Thesystem musthave remained cl o s e d sincethe time thatis tobe dated.This means ifwe

wish to date a parchment, it must have neither gained nor lost any

Csincethetimethe

lamborsheepwhosehidewasusedtomaketheparchmentdied.Weneedtoknow

C(0)pre-

cisely, that is the init ia l

C content, that is the proportion of

C/C inthe animal atthe time

of its death.Theremustremain enough

Cforittobemeasu rable.

Lastly, and this is not th e least of the problems, the event whose age is to be determined

mustbe de¢ n ed.

Canwebe certain aboutall fourconditionsin practice?

T he closed system

The greatest danger from an open system is contamination ofthe sample, whether naturalor

arti¢cial.Take the case ofatree trunkwewishto date. Ifthe trun k hasbeen lying in humusfor

some time it will be impregnated, contaminated, by the carbon in the humus. Now, this car-

bon is‘‘older’’thanthe carbonofthetreethathasjustdiedandsodistortsthemeasurement.

Letus takethe example ofatree trun kwhose speci¢cactivity, that is, the

C radioactivity

per gram ofcarbon, is r

. Itlies for some ti me in humus whose radioactivity, in

Cpergram

ofcarbon, is r

.W hatis thelevelofradioactivityofthe mixturesupposing that10%ofthe car-

bon comesfromthehu mus?



tree

; r



humus

total

tree

humus

tree

þ C

humus



tree

þ C

humus



humus

tree

þ C

humus

total



tree

w þ



humus

ð1  wÞ:

Thevalue ofw ¼C

tree

total

varies between 0 and1. (This is the same mixing formula as in

the section on isotopic dilution.)

Assuming that (

C/C)

tree

¼12.5 dpm g

1

and that (

C/C)

humus

¼4dpmg

1

,then

total

¼12.5 0.9 þ4 0. 1 ¼11.65 dp m g

1

, w ¼0.9.

This corresponds to an er ror of 7% by default. The age measured is older than the real

age. Conversely, ifwesuppose thetree trun k found in thesedimenthadbeen in soilwhichwas

still involvedin atmospheric exchanges, thenthe contamination,that is,the mixturebetween

Cinthedeadtree andtheg roundwould continuallymakethewood‘‘younger.’’

Determining the initial

Ccontent

Libbyhadassumedthatthevalueof13.5dpmg

1

ofcarbonwasthe currentvalueandvalidas

the initial value throughout time. However, several complications have arisen since then.

Not all plants ab sorb carbon isotopes in an identical manner. If (

C/C)

present

¼r

,the

118 Cosmogenic isotopes

value absorbed by living organisms is (

C/C)

organism

¼Rr

,whereR is a partition coe⁄-

cient.This coe⁄cient has been determined and to check it we measure the

C ratio of

the objecttobe dated (Figure 4.8) (this is morefullyexplained in Chapter 7).

Onthe left is the 

C scale,suchthat:

C ¼



sample





standard



standard

 10

The standard is a carbonate. In practice, this fractionation is converted into an age di¡er-

encebytaking areferencestandard.

Weassumethatfractionation is directlyproportionaltothe di¡erencein mass.Therefore,

if R

3/2

is the partition coe⁄c ient between some plant and the atmosphere, for the

ratio de¢nedby:



plant



atmosphere

¼ R

3=2

the partition coe⁄cientfor the (

C) ratio



plant



atmosphere

¼ R

4=2

is suchthatR

4/2

¼2 R

3/2

It ispossible, then, tocor r ect fo r natura l f rac tionation of

C and express this correction

in age, which is done onthe rightof Figure4.8.

It was realized that the

C content of the atmospherehad varied over the c ourse of time.

This e¡ect was shown by measuring the

C of tree rings.Tree rings are evidence of annual

C (‰)

Age correction (

Δt), years

400

–10

300

200

100

–100

–20

–30

Marine

carbonates

Soils

carbonates

Freshwater

carbonates

Bicarbonates in

subsurface water

plants

(grass)

plants

(wood)

Atmospheric CO

Animal

bone

Figure 4.8 Carbon isotope fractionation relative to the atmosphere depending on the nature of the

substrate.

119 Carbon-14 dating

growth, with each ring representing one year. By counting the rings, we can work back

throughtime, yearby year.This is known as de n d roch ron olog y. Itwasnoticedthatthe ages

ofrings measured by

C andtheagesobtainedbycounting thesame ringsfailedto match.

Systematic investigation has shown that multiple phenomena are involved in this. The

burning of coal and oil, that is, materials whose

C is‘‘dead,’’ has diluted the‘‘natural’’

contentofthe atmosphere.The present-day content is toolow c ompared withwhatit would

be without this e¡ect. Likewise, for recent periods, nuclear testing by explosions in the

atmosphere has injected a large quantity of

C, which has largely disrupted the natural

cycle(Figure4.9) and,ofcourse, increasedthe

C contentofth e atmosphere.

The £uxofcosmic particles and therefore the production of

C varies over the course of

time. This is related to solar activity, to £uctuations in the Earth’s magnetic ¢eld, etc.We

therefore need a calibration curve describing variation ofthe

C/C ratio over time. At ¢rst

sight, this looks something like a chicken-and-egg situation because we wish to determine

an age fro m the

C/C measure ment but at the s ame ti me we wish to calculate the

C/C

ratio by using age measurement.The curve of

C/C variation in the atmosphere was con-

structed by dating samples using a di¡erent method. For recent per iods, tree rings were

used as the benchmark, and for earlier periods dates were ascertained by radioactive dise-

quilibrium methods.

To c a l i b r a t e t h e

C/C ratio, we ¢rst seek to calculate (

C/C)

initial

¼(

C/C)

measured

where t is the independently determined age. In this way a correction can be applied to the

age measured by the conventional method (Figure 4. 1 0). It is then p ossible to draw a curve

conne cting

C dates tohistorical calendar ages.To do this,1950 was taken as the reference

date for

C. Suppose we calculate a

C date of 3000 yearsbefore1950 (ifwe made the mea-

surementintheyear2000 wewouldremove 50 years!).Toobtainthe BCdate,wemustthere-

fore subtract1950: 3000 ^ 1950 ¼1050 BC(Figure 4. 11).

Years

Percentage excess

100

1963 1964 1965

41°

1966 1967 1967

36° N

21°

9° N

18° N

71° N

38° N

Figure 4.9 Changes in excess

C injected into the atmosphere by U.S. atom-bomb testing in the

Paciﬁc. Each symbol corresponds to a latitude. The various, ever weaker, injection dates can be seen:

1963, 1964, 1965, 1966. It took several decades to return to equilibrium. After Libby (1970).

120 Cosmogenic isotopes

Exercise

We use the curve of

C/C variation with time (Figure 4.11). Suppose we ﬁnd an age of 200

years for a piece of fabric using the standard

C formula. What date does this correspond to?

Answer

As can be seen, this ‘‘age’’ corresponds to two possible dates: 1630 or 1800. How can we

decide between the two? The answer is obtained by using other independent information

such as the nature of the fabric. This illustrates the limits of precision of

Δt

atm

130

–1000

–800

–600

–400

–200

+200

–30

8000 6000

from –10

000 to 2000 years

Time (ka)

Time (years)

from –30

000 years to today

4000 2000 20000

500

400

100

200

300

25 20 15

15 10 5 0202530

10 5 0

C age (calendar year)

C (‰)

Figure 4.10 Calibration curves for

C. (a) For the older period, we use methods based on radioactive

disequilibrium as did Edouard Bard (Bard

., 1990); (b) for the recent period we can use tree rings

(Stuiver, 1965).

8000

6000

4000

2000

6000

4000 2000

Calendar age (years)

Radiocarbon age

0 2000

200

1600

1800 2000

400

Figure 4.11 Correspondence between calendar ages and

C ages. The

C age is the time between the

death of the organism and 1950. The calendar date begins at the birth of Jesus Christ. To this difference,

we must add the ﬂuctuations caused by calibrations because of historical variations of (

C/C)

initial

121 Carbon-14 dating

Synthe t i c for m ula forage cal cu lation

When making calculations, we can convert everythingtothe additionofmagnitudeswhich

themselves have age dimensions.When applying the uncorrected conventional formulawe

obtain:

apparent



where A

is the initial conventional activityand A

is the activity as measured.The true age

mustbewritten:

true

0;real

2;4



where A

0,real

isthe real,initialactivityofthe atmosphere andP

2,4

isthe isotopefractionation

factor.We can also introduce in the parenthesis the reference value A

for the initial activity.

Dividing thetopand bottom gives:

true

0;real



 P

2;4



Taking advantageofthe additivepropertyoflogarithms, wehave:

true



ln P

2;4



0;real



Fromwhich, bytransform ingeverything intotimeunits:

true

¼ t

apparent

þ Dt

fractionation

þ Dt

atmospheric correction

where:

fractionation

lnðP

2;4

Þ and Dt

atmospheric correction

0;real



;

withalittleextracalculationtrickbecause A

0;real



canbeevaluatedasafunctionoftime

only.

We the refore calculate t

¼t

apparent

þt

fractionation

,thenwecanuset

to calculate

t

atmospheric correction

usingthe £uctuation curves of Figure 4. 11 ,an dwethen obtaint

true

Exercise

We measure the activity of a camel bone found in the tomb of the vizier Hemaka as 6.68 dpm g

1

What is the real date of the vizier’s death? (Use the existing calculations and graphs.)

Answer

In an exercise in Section 4.2.1 we dated the death of vizier Hemaka (4880 years ago) using an

ordinary wooden beam.

apparent

þ D

fractionation

122 Cosmogenic isotopes

Using Figure 4.8 and the previous result, we get

¼4800 þ100 ¼4980 years, or 3030 BC.

By using the graph in Figure 4.10, we ﬁnd for a date of 3000 BC a correction of –250 years:

real

¼ 4980  250 ¼ 4730 years:

4.2.4 Other forms of cosmogenic radioactivity

Carbon-14 is far more important than any other spallation product created in the atmo-

sphere by interaction of cosmic rays, but some oth ers exist as well. These can be divided

intothreegroups(see Figure4.3):



The rare gases, which are free gases. These are incorporated into solid material by

trapping partof the atmosphe re.These are

Ar (T

1/2

¼269 years) and

Kr (T

1/2

¼2.1 

years);



Gaseous products, which, similarly to

C, are incorporate d into reactive atmospheric

molecules. Forexample

H(tritium;T

1/2

¼12.43 years)or

Cl (T

1/2

¼3.08 10

years);



Productswhicharenotgaseousbutadhereimmediatelyafter their formationtodustpar-

ticles and follow the history of the dust rather than the gaseou s atmosphere. Examples

are

Be(T

1/2

¼1.51 10

years) or

Al(T

1/2

¼7.16 10

years).

Beryllium-10

Beryllium-10 (

Be)isproducedbyspallationofnitrogenandoxygen intheatmosphere.As

soon as it forms, it is adsorbed onto particles in the atmosphere an d incorporated in rain.

Accordingly, it is more or less well mixed. Its lifetime in the atmosphere is short and as its

production varies with latitude (as with

C), its distribution may be extremely heteroge-

neous an d evenerratic (Figure 4.1 2).

A further complication arises b e cause

Be, common beryllium (which has just this one

isotope), has a completely di¡erent geochemical history to

Be. It is incorporated in rocks

andsoin dustderivedfrom erosionandscatteredbythewind.The

Be/

Beratioistherefore

just as erratic as the absolute

Be content.We cannotuse

B e like

C, assuming auniform

Latitude, N or S

Be flux

(10

atoms cm

–2

–1

)

30°

60° 90°

Figure 4.12 Variations in the ﬂux of

Be with latitude display substantial dispersion.

123 Carbon-14 dating

value for the entire Earth at anygiven time. One m ethod is to assume that, at a given place,

£uctuations in the

Be/

Be ratio are low and at any rate much lower than those arising

from the radioactive decayof

Be. Itisthenpossibletouse

Beas achronometer.

Exercise

We wish to measure the rate of accretion of manganese nodules found in the ocean. How can

we set about this, knowing that the variation in the

Be/

Be ratio with depth is

Be/

Be ¼(

Be/

Be)



Answer

thicknessðÞ

a accretion rateðÞ

from which:





Switching to logarithms gives:



¼ ln





The logarithm of the ratio

Be/

Be plotted as a function of thickness is a straight line of slope

aÞð , which gives the rate of accretion,

O’Nions et al.(1998) measured the

Be/

Be ratios in manganese crusts of the North

Atlantic. Figure 4. 13 shows the results theyobtained.This methodyielded an accretion rate

of 2.37 0. 15 mm Ma

1

–8

–9

–10

–7

5 10 15 20 25

Thickness (mm)

Be/

a = 2.37 ± 0.15 mm/ Ma

–1

Figure 4.13 Variation in the

Be/

Be ratios in a manganese concretion from the Atlantic Ocean.

Magnitude

indicates the rate of accretion of the nodule. After O’Nions

.(1998).

124 Cosmogenic isotopes

4.2.5 Dating oceanic cycles with

This is a rather unusual m ethod of dating, but one that is very important in oceanography.

The residence time  of a chemi cal element in a reservoir is the average time the element

spends in the reservoir (Figure 4. 1 4) (s ee further discussion in Chapter 4). If W is the £ux

and V the reservoir volume, we positthat  ¼(V/W).

Broecker and colleagues at the Lamont Observatory of Columbia University at

NewYork came up with the idea ofusing

C to determine the residence time ofwater in the

deep ocean (Broecker et al., 196 0;BroeckerandLi,1970). The ocean is divided into two

layers: a surface layer that is well mixed by currents and wh ich is constantly exchanging its

(andsoits

C)withtheatmosphere,andadeeplayerwhich exchangeswaterand matter

withthe surface layer. Let B denotethe £owofsolid carbonate particles (moles yr

1

)falling

from the surface and dissolving in the deep ocean; W and W

are the £ows of descending

and ascending water which summarize the exchange between the upper layer and the

deep ocean (m

1

):W ¼W

; C

and C

are the concentrations of carb on dissolved in

the surface and deep layers (moles m

3

); R is the

C/C ratio standardized relative to

the atmosphere, R

and R

being the ratios for the surface water and deep water.The total

carbon conservation equation is written:

ðWC

þ BÞ¼WC

The equation for

C is:

ðWC

þ BÞR

¼ WC

þ V

The residencetime in the deep water is:



;

Atmosphere

Surface water

Deep water

= 1

= 0.95

= 0.80

Water exchange Solid particle flux

Figure 4.14 The behavior of

C in the ocean. The ocean is divided into two reservoirs: the upper layer,

which is well mixed, and the deep ocean.

is the

C/C ratio.

, ﬂow of solid carbonate particles;

descending and ascending ﬂows of water.

125 Carbon-14 dating

which, giventheforegoing equations, is written:



 1



Now, l ¼1.20 9 10

4

1

, therefore

¼1550 years.

4.3 Exposure ages

4.3.1 Meteorites

Meteorites are fragments of planetarybodies resulting from collisions and whose primary

age is close to that of the Solar System (4.55 Ga, as an indication). They are p iec es of rock

‘‘£oating’’ in i nterplan etary space. Most come from the asteroid belt, between Mars and

Jupiter,which isformedofrockdebris,pie cesofwh ich maymeasureakilometerorsoacross

(Ceres). Some were ripped from Marsor the Moon.These rocks driftloose in space and are

continuallysubjectedto cosmic radiationwhose average energy is1GeV. Just as happens in

the atmosphere, when cosmic‘‘primary’’protons penetratethese rocks theygive ris e tosec-

ondary neutrons, which produce most of the nuclear reactions.

These parti cles produce

nuclear reactions at depths ranging from a few tens of centimeters up to a meter at most.

Thesurface layers ofrocks exposedto cosmic rays arethereforethe siteofnuclear reactions,

usually spallation reactions.Theygive rise to isotopes of lower mass than the target, whi ch

are known as cos mo gen ic is otope s (see Paneth et al.,1952; Honda and Arnold,1964). The

nuclei engenderedbysuch reactions include radioactive and stable isotopes.Both types are

very numerous.

Stable i sotop es

Thenumberofcosmogenicstable isotopes N

producedperunittimeiswritten :

ðtÞ

¼ 

c!s

;

where  is the particle £ux, 

is the e¡ective cross-se ction of the reaction, thatis, the prob-

abilityofa nuclear reaction o ccurring, and N

c!s

thenumb eroftarget atomsproducingthe

stable daughter isotope, s, by nuclear reaction. If th e £ux is constant in terms of intensity

and spectralenergy, then:

ðtÞ¼

c!s

Thenumberofdaughter isotopes is directly propor tionaltotheirradiationtime.

Itseemsstraightfor wardenoughto calculatetheageofirradiation,providedweknow,

and N

c!s

. It is comparatively easy to measure the parameter N

c!s

:itistheconcentrationof

As explained, protons carry a positive charge and do not penetrate readily into matter because they are

repelled electronically. Neutrons carry no charge and so penetrate much more readily.

126 Cosmogenic isotopes