All?gre Claude J. Isotope Geology

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resolving power RP ¼

M

where M

isthe mass.M is de¢ned as M

¼M

þM,whereM

isthe closest mass to M

thatdoes notoverlapbymorethan 50% inthe collector.

We canalso de¢ne a resolving powerat1%.

The distance xbetween twobeams inthefocal plane is written:

x ¼ K

m

Dependingonthe angle oftheincidentbeam tothe m agnet, K ¼R foranangleof incidence

of908;K ¼2R for an angle of 278.

From the formula above:

RP ¼ C

x

;

R being the radius of curvature and x the distance between two beams of M and

M þM.

This is just to show that when one wants to separate two masses more e⁄ciently, the

radius has to be increased and then the voltage adjusted accordingly. Suppose we want to

separate

Rb and

Sr by the di¡erence in mass of neutrons and protons alone. A‘‘mon-

ster’’ mass spectrometer would be required. However, interfe rences between two masses

can be avoided when separating isotopes of an element from contaminating molecules.

(Methane

has the same mass as

OandbenzeneC

interferes at mass 78 with

krypton.)

Abundance sensitivity

Another importantcharacteristicisthe xdistance(in millimeters)betweenthebeams.We

have to come back to the slits in the collectors.The problem is easy enough to understand.

At ¢rst, the bea m is rectangular. Collisions with residual air molecules means that, when it

reaches the collectorslit, thebeam is wider andtrapezoid-shaped with long tails. Collector

slits areopen so thatthey can receive one massbut no contribution from the adjacent mass.

When the abundances oftwoadjacent isotopes areverydi¡erent, the tail of the moreabun-

dant isotope forms background noise for th e less abundant one. Measuring the less abun-

dant isotope involves reconstructing the tail of the more abundant one mathematically.

This is possible only if the tail is not too big. Narrowing the collector slit brings about a

rapid decline in sensitivity.

Abundance sensitivity is the measurement ofthe contributi on of the tai l of o ne isotope

to the signal of the neighboring isotope. It is given as a signal/noise ratio multiplied by

the mass ratios. Special instruments have been developed for measuring abundance

sensitivity in extreme cases, such as measuring

C close to the massively more abundant

C. Abundanc e sensitivity is related to resolving power but also to the quality of the

ion opti c s.

7 The mass spectrometer

Exercise

The isotopic composition of the element rubidium (Rb) is measured, giving a current

¼10

11

A for the mass of

Rb. How many ions per second is that? If the measurement

lasts 1 hour how much Rb has been used if the ionization yield is 1%?

Answer

The intensity of an electrical current is deﬁned by

¼d

,whered

is the quantity of electrical

charge and d

the unit of time. Electrical current is therefore the quantity of electrical charge

ﬂowing per unit time. The ampere, the unit of electrical current, is deﬁned as being 1 coulomb

per second, the coulomb being the unit of electrical charge. The charge of an electron is

–1.6  10

19

coulombs. The positive charge is identical but with the opposite sign. An

intensity of 10

11

amps therefore corresponds to 10

11

coulombs per second / 1.6  10

19

coulombs ¼62.5  10

ions per second.

If this intensity is maintained for 1 hour: 6.25  10

3600 ¼2.2464  10

ions of

.As

the ionization is 1%, this corresponds to 2.2464  10

atoms of 87

placed on the emitter

ﬁlament. As

Rb/

Rb ¼2.5933, Rb

total

(in atoms) ¼

Rb (in atoms) (1 þ2.5933).

So a total number of 8.0719  10

atoms of Rb is placed on the ﬁlament. As the atomic

mass of Rb is 85.468 g, the total weight of Rb is 11 ng.

Exercise

How much rock is needed to determine the isotopic composition of Rb by measuring a sample

for 20 minutes at 10

11

A if its concentration in Rb is 10 ppm (parts per million)?

Answer

If 11 ng of Rb are needed for 1 hour, for 20 minutes we need (11 20)/60 ¼3.66 ng, that is

3.66  10

9

/10

5

¼0.36 mg of rock or mineral.

It can be seen, then, that mass spectrometry is a very sensitive technique.

1.2.3 Various developments in mass spectrometers

Mass spectrometers have come a long way since the ¢rst instruments developed by J. J.

Thomson and F. Aston.To givesome ideaofthe advances made, when Al Nier was measur-

ing lead isotopes as apostdoctoral fellow at Harvardin1939 (moreaboutthis later), heused

agalvano meter projectingabeam oflightontothewallandmeasured thepeakwith aruler!

Nowadayseverything takestheform ofa computeroutput.

Ionizat ion

The ¢rst technique was to use the element to be measured as a gaseous compound.When

b ombarded byelectrons, atoms ofthe gas lose electrons and so become ioniz ed (Nier,1935,

1938,1939). Later came the thermal-ionization technique (TIMS) (Inghram an d Chupka,

1953). In the so-called solid-source mass spectrometer, a saltofthe element is depositedon

ametal ¢lament (Ta,W, Pt).The¢lamentisheatedbytheJoule e¡ectofanincreasingelectric

cur rent. At a certain temperature, the element ionizes (generally as positive ions [Sr, Rb,

Sm, Nd, U, Pb ] but alsoas negativeion s [ Os]). Ionizationbecame afundamental character-

istic ofmass spectrometry.

8 Isotopes and radioactivity

Nowadays, asanalternative,plasmaisused foroptim al ionization in instruments named

ICP-MS.

Ion optics

Substantial e¡ort has been put into optics combin ing various geometries and assemblies.

Bai nb ri dge and Jordan (1936) used a magnetic ¢eld to turn the beam through 1808.

Mattauch and Herzo g (1934) combined electric an d magnetic ¢elds to separate ions and

focusbeams. Magnetshapesweremodi¢edtoimprovetransmission e⁄ciency.

Computerized numerical simulation has allowed tremendous advances in ion optics

des ign. All of the techniques u sed tended to maximize ionization and transmission , to

increase resolution power and abun dance sensitivity, and to minimize the high voltage

requirement and the si ze of the magnet, which areboth big factors in c ost. However, when

the ionization process created a wide dispersion in ion energies, more sophisticated ion

optics were require d to refocus the ion beam in a narrow band on the collectors. So ICP-

MS, ionprobe, and AMSinstrumentshavebecomelarge and more expensive.

Collectors areanother importantissue.Early massspectrometershadasinglecollector. By

scanningthemagnetic¢eld,theionbeampassedinsequencethroughthecollectorandaspec-

trum of ion abundance was recorded (Figure 1.1). Nowadays most mass spe ctrometers use

simultaneous ion collectionwithan arrayofcollectorssidebyside, each collectorcorrespond-

ing to a distinct mass. This se ems an obvious technique to use as it eliminates £uctuations

betweentherecordingsofonemass(peak)toanother.However,itistechnicallyextremelydi⁄-

culttoachieve,both mechanically,accuratelyinstalling severalcollectors in asmallspace,and

electronically, controlling drifting of the electronic circuits with time. These problems have

now been virtually eradicated. It is worth noting that, un like in most areas of science, all

advancessince 1980havebeenmadebyindustr ialengineersratherthanbyacademicscientists.

However, because ofelectronic‘‘noise’’and electrical instabilities, all isotopic measurements

are statistical. On each run, thousands of spectra are recorded and statistically processed.

Onlysince microcomputershavebeenavailablehave suchtechniquesbecomefeasible.

1.2.4 Preparatory chemistry and ﬁnal accuracy

In most mass spectrometry techniques (except for ion probes) chemic al separation is used

before m easurement to puri fy the element whose isotopic composition is under study.

Since the elements tobe measured are present as traces, th ey have to be separated from the

majorelements wh ichwould otherwiseprevent any ionization asthe majorelementsrather

than the trace elements would give out their electrons. For example, an excess of K inhibits

any Rb ionization. Chemical separation also prevents isobaric interference between, say,

Rb and

Sror

187

Re and

187

Os.

Chemicalseparationcanbedoneingaseousform inpuri¢c ationlines,asforraregasesor

for oxygen or hydrogen measurement, or in liquid solution for most elements. The basic

technique in the latter case is the ion-exchange column as introduced by Aldrich et al.

(1953).Alltheseoperationshavetobeperformedinverycleanconditions,otherwisesample

contaminationwill ruin measurement.The greater the accuracyofmass spectrometry, the

cleaner the chemistry required. The chemistry is carried out in a clean room with special

9 The mass spectrometer

equipmentusing specially preparedultra-clean reagents,thatarefarcleaner than anycom-

mercialversions.

Whenjudging measurementreliability, investigatorshavetostatetheleveloftheirblan ks.

Theblankistheamountofthetargetelements measuredin achemi calprocess donewithout

anysample.Theblankhastobeneglig ibleor verysmallcomparedwiththeamountofmater-

ial tobe measured. Soincreases in accuracyarelinked notonly withthe imp rovementofthe

mas s spe ctrometerbut also oftheblanks.

Althoughthis isnotthe placetogive full technical details about conditions for preparing

and measuring samples, as these can only be learned in the laboratory and not from text-

b ooks,afewgeneral remarks maybemade.

Moderntechniquesallow isotope ratiostobe measuredwithadegree ofprecision of10

5

or10

6

(a few ppm!) on samples weighing justa few nanograms (10

9

g) or even a few pic o-

grams (10

12

g). For example, if a rock contains10 ppm of strontium, its isotope composi-

tion can be measured on10

9

g with a degree of pre cision of 30 ppm.Therefore just10

4

that is, 0.1mg would be needed to m ake the measurement. This method can be used for

studying precious rocks, such as s amples of moon rock or meteorites, or minor or rare

minerals,that is, minerals that are di⁄culttoseparate and concentrate.What do such levels

ofprecision mean? They meanwe can readily tell apart twoisotope ratios ofstrontium, say

0.702 21and0.702 23, thatis,towithin 0.000 03, evenwherelowconcentrations areinvolved.

To achieve such precision the measurement mustbe‘‘internally calibrated.’’ When measur-

ing the abundance ratio (A

) of two isotopes, the electrical current ratio (I

) detected

is slightlydi¡erentfrom (A

).The di¡erenceis engendered by the measurementits elf.This

istermed mass discrim ination.

Eitheroftwo methods isusedforcalibrating measurements.

The ¢rst is the internal standard method. If the element has three or more isotopes one

particular ratiois chosen asthe reference ratioandc orrection is made for mass discrimina-

tion. So if the abundances are A

, A

,wetake(A

) ¼R.The measurement (I

)is

written R(1 þm), where m is the di¡erence in massbetween A

and A

.The fractiona-

tion coe⁄ci ent is calculatedand then appliedtothe measurementofthe ratio (A

The second method is to measure a standard sample periodically and to express the

values measuredin termsofthatstandard.

The extraordinary prec ision the mass spectro meter can achieve must notbe jeopardized

byaccidental contamination when preparing samples.To this end ultra-clean preparatory

chemistry isdevelopedusing ultra-pure chemical reag ents in cleanrooms(Plate 3bottom).

1.2.5 Ionization techniques and the corresponding spectrometers

Four major ionization techniques are used depending on the characteristics of the various

chemicalelements(ioni zationpotential).

Th e r mal-io nization ma ss spe ctromet r y (T I M S)

The element to be analyzed is ¢rst puri¢ed chemically (especially to separate any isobars)

and depositedon arefractory ¢lament. Heating the ¢lamentin avacuumby the Joule e¡ect

Such discrimination depends on the type of mass spectrometer used. It decreases with mass for any given

type.

In high-precision mass spectrometry an exponential law rather than a linear one is used to correct mass

fractionation.

10 Isotopes and radioactivity

ion izes the elements, whi ch eitherlose an electronbecoming positive ions, as in the cases of

,Sr

,andPb

, or gain an electron b ecoming negative ions as with OsO



and WO



Instrumental mass fractionation is of the order of1% by mass deviation for light elements

(Li)and 0.1%bymass deviation forheavyele ments(Pb, U).

Electronicbombardment

Lightelementssuchashydrogen (H), carbon (C), nitrogen (N),andoxygen (O) or raregases

are analyzed as gases (H

,CO

, or ato ms of He, Ne, At, or Xe) bombarded in a

vacuum by an electron beam. Positive ions are thus formed by stripping an electron from

such molecules or atoms. The ions are then accelerated and sorted magnetically as in

TIMS. Substances are prepared for analysis in gaseous form by extracting the gas from the

sample under vacuum either by fusion or by chemical reaction.The gas is then puri¢ed in

vacuum lines where other gases are captured either byadsorption orby manipulating their

liquefactiontemperatures.

In duct ivelycoupled plasma mas s spec trometry (ICPMS) inanargon plasma

The sample is ionized in an argon plasma induced by a high-frequency electrical ¢eld

(plasma torch).The high temperature of the plasma, about 10 000 K, means elements like

hafnium or thorium, which are di⁄cult to ionize by thermal emission, can be completely

ion ized. The ele ment to be analyzed is atomized and th en ionized. It is sprayed into the

plasma from a solution as a liquid aerosol. Or it may be released from a solid sample by

laserablation.Thesolid aerosolsoformed isinjected into theplasmatorch.Mass fractiona-

tion is between a twentieth of1% for a light element like boron and 1% for heavy elements.

Fractionation is corrected for by using the isotope ratios ofother similar elements as inter-

nal standards,because,atthe te mperature ofthe plasma, fractionation is duetomass alone

and is nota¡ectedbythe element’s chemical characteristics.

Ionic bombardment in secondary-ion mass spectrometry (SIMS)

The solid sample (rock, mineral) containing the chemic al element for analysis is cut,

polishe d, and put into the vacuum chamber where itis bombarded by a‘‘primary’’ beam of

ions (argon, oxygen, or c esium). This bombardment creates a very-high- te mperature

plasma at about 40 000 K in whi ch the element is atomized and ionized.The development

ofhigh resolution secondary-ion mass spectrometers (ion m icroprobe s) means in-situ iso-

tope measurements can be made on very small samples and, above all, on tinygrains.T his

is essential for studying, say, thefewgrainsof interstellar materialcontainedin meteorites.

Remark

All of the big fundamental advances in isotope geology have been the result of improved

sensitivity or precision in mass spectrometry or of improved chemical separation reducing con-

tamination (chromatographic separation using highly selective resins, use of high-purity materials

such as teﬂon). These techniques have recently become automated and automation will be more

systematic in the future.

1.3 Isotopy

Assaid,eachchemicalelementis de¢nedbythenumberofprotonsZ in itsatomicstructure.

It is the numberofprotons Z thatde¢nes the element’s position in the periodic table. But in

11 Isotopy

each position there are several isotopes which di¡er by the numberof neutrons N they con-

tain, that is, by their mass.These isotopes are created during nuclear processes which are

collectively referred to as nucle osy nthe sis and which have been tak ing place in the stars

throughoutthehistoryofthe Universe (see Chapter 4).

The isotopic composition of a chemical element is expressed either as a percentage or

more convenientlyas aratio. A referenceisotope is chosen relativeto whichthequantitiesof

other isotopes are expressed. Isotop e rati o s are expressed in terms of numbers of atoms

and notofmass. Forexample,tostudy variations intheisotopic composition ofthe element

strontium brought about by the radioactive decay of the isotope

Rb, we choose the

Sr/

Sr isotope ratio. To study the isotopic variations of lead, we consid er the

206

Pb/

204

Pb,

207

Pb/

204

Pb,and

208

Pb/

204

Pb ratios.

1.3.1 The chart of the nuclides

The isotopic composition of all the naturallyoccurring chemical elements has been deter-

mined, that is, the numberof isotopes and their proportions havebeen identi¢ed.The ¢nd-

ings havebeen plotted as a (Z, N) graph, that is, the number ofprotons against the number

ofneutrons. Figure1.3, details ofwhich aregiven inthe Appendix, prompts a few remarks.

Firstofall,thestableisotopesfallintoaclearlyde¢ned zone,knownasthevalleyofstability

b ecause it corresponds to the minimum energy levels of nuclides. Initially this energy valley

followsthe diagonal Z ¼N.Then,afterN ¼20, the valley fallsaway from the diagonal onthe

side of a surplus of neutrons. It is as if, as Z increases, an even g reater number of neutrons is

neededtopreventthe electrical lychargedprotonsfrom repelling eachotherandbreaking the

nucleusapart.(Things areactually morecomplicatedthanthissimplisticimagesuggests!)

20 40 60 80

Neutron number (N)

Proton number (Z)

100 120

Natural stable isotopes

Figure 1.3 The distribution of natural stable isotopes in the neutron–proton diagram. The diagram is

stippled because natural or artiﬁcial radioactive isotopes lie between the stable isotopes. After

¼20,

the zone of stable nuclei moves away from the diagonal for which the number of neutrons equals the

number of protons. For

> 20, the number of neutrons then exceeds the number of protons. This zone

is called the valley of stability as it corresponds to a minimum energy level of the nuclides.

12 Isotopes and radioactivity

A second remark relates to parity. Elementsfor which Z is an even numb erhave far more

isotopes than elements for which Z is an odd number. Fluorine (Z ¼9), sodium (Z ¼11),

phosphorus (Z ¼15), andscandium (Z ¼21)havejusta single isotope.

Lastly, and notl east importantly, theheaviest ele mentwithstable isotopes islead.

1.3.2 Isotopic homogenization and isotopic exchange

As the isotopes ofanygiven chemical element all have the same electron suite, theyall have

pretty much the same chemical properties. But in all chemical, physical, or biological pro-

cesses, isotopes ofanygiven elementbehaveslightlydi¡erently from each oth er, thus giving

rise to isotopic f ract ionatio n. Such fractionation is very weak and is apparent above all in

lightelements. Itis also exploited in isotopegeologyas shallbe seen in Chapter7.

Initially we shall ignore such fractionation, except where allowance has tobe made for it

as with

C or when making measurements with a mass spectrometer where, as has been

seen, correction mustbe madefor mass discrimination.This virtuallyidenticalbehaviorof

chemical isotopes entails afundamental consequenceinthetendencyfor isotopic homogen-

ization to occur.Where two or more geochemical objects (minerals within the same rock,

ro cks in solution, etc.) are in thermodynamic equilibrium, the isotope ratios ofthe chemical

elements present are generally equal. If theyare unequal initially, they exchange some atoms

until they equalize them. It is important to understand that is otopic homogen ization occurs

through isotopic exchange without chem ical homogeni zat ion. Each chemi cal component

retains its chemicalidentity, ofcourse.Thispropertyof isotopichomogenization‘‘across’’che-

mical diversityis one ofthefundamentalsof isotope geochemistry. A simplewayofobserving

this phenomenon is to put calcium carbonate powder in the presence of a solution ofcarbo-

nate in water in proportions corresponding to thermodynamic equ ilibrium. Therefore no

chemical reaction occurs. Repeat the experiment but with radioactive

C in solution in the

form ofcarbonate. Ifafter10 daysorsothesolidcalcium carbonateis isolated, it willbefound

to have become radioactive. It will have exchanged some of its carbon-14 with the carbonate

ofmass12and13whichwereinthesolution.

Exercise

A liter of water saturated in CaCO

whose Ca

2 þ

content is 1  10

2

moles per liter is put in the

presence of 1 g of CaCO

in solid form. The isotopic ratio of the solid CaCO

Ca/

Ca ¼151.

The isotopic ratio of the dissolved Ca

2þ

has been artiﬁcially enriched in

Ca such that

Ca/

Ca ¼50. What is the common isotopic composition of the calcium when isotopic

equilibrium is achieved?

Answer

Ca/

Ca ¼121.2.

As said, when two or more g eoche mical objects with di¡erent isotopic ratios are in

each other’s presence, atom exchange (whi ch occurs in all chemical reactions, including at

Until recently it was believed to be bismuth (Z ¼83), whose only isotope is

209

Bi. In 2003 it was

discovered to be radioactive with a half-life of 1.9  10

years!

13 Isotopy

equilibrium) tends to homogenize the whole in terms of isotopes.This is known as isotopic

exchan ge. Itis akineticphenomenon,depending thereforeonthetemperatureandphysical

state of the phases present. Simplifying, isotope exchange is fast at high temperatures and

slow at low temperatures like all chemical reactions which are accelerated by temperature

increase. In liqu ids an d gases, di¡us ion is fast and so isotope exchange is fast too. In solids,

di¡usionis slowandsoisotopeexchangeisslowtoo.In magmas (high-temperatureliquids),

then, both trends are compounded and isotope homogenization occurs quickly.The same

istrueofsupercritical£uids,thatis,£uidsdeepwithintheEarth’scrust.Conversely,insolids

at ordinary temperatures, exchange occurs very slowlyan d isotope heterogeneities persist.

Two important consequences follow from these two properties. The ¢rst is that a mag ma

has the sam e isotope composition asthe solid sourcefromwhich ithas issuedbyfusion, but

notthe same chemical composition.The second is that, conversely, a solid atordinary tem-

peratures retains its isotopic composition over time without becomi ng homogeneous with

itssurroundings.Thisiswhyro cks are reli ab l e isotopere c ords.Thispropertyisadi rectcon-

sequenceofthe di¡usionproperties ofnaturalisotopes in liquids andsolids.

The theory of di¡usion, that is, the spontaneous motion of atoms in£uenced by di¡er-

encesi n con centration, provides anapproximatebut adequateformula:

x 

ﬃﬃﬃﬃﬃﬃ

wherex isthe distancetraveledbythe element,t istime inseconds, andD the di¡usion coef-

¢cient(cm

1

Exercise

In a liquid silicate at 1200 K the diffusion coefﬁcient for elements like Rb, Sr, or K is

¼10

6

1

. In solid silicates heated to 1200 K,

¼10

11

1

How long does it take for two adjacent domains of 1 cm diameter to become homogeneous:

(1) within a silicate magma?

(2) between a silicate magma and a solid, which occurs during partial melting when 10% of

the magma coexists with the residual solid?

Answer

(1) In a silicate magma if

¼10

6

1



¼10

s, or about 11 days.

If it takes 11 days for the magma to homogenize on a scale of 1 cm, on a 1-km scale

(¼10

cm), it will take



¼10

s, or close to 3  10

years, given that

1 year 3  10

In fact, homogenization at this scale would not occur by diffusion but by advection or

convection, that is, a general motion of matter, and so would be much faster.

(2) In the case of a magma impregnating a residual solid with crystals of the same dimen-

sions (1 cm),



¼10

s or about 3  10

years, or 300 000 years, which is rather fast

in geological terms. For 1-mm crystals, which is more realistic,

10

2

¼3  10

years,

or 3000 years. So isotope equilibrium is established quite quickly where a magma is in the

presence of mineral phases.

A second important question is whether rocks at ordinary temperatures can retain their

isotope compositions without being modi¢ed and without being re-homogenized. To

14 Isotopes and radioactivity

answer this, it must be remembered that the di¡usion coe⁄cient varies with temperature

bythe Arrheniuslaw:

D ¼ D

exp

DH



where H is the activation energy, which is about 40 kcal per mole, R is the ideal gas con-

stant(1.987 calper K per mole), and T theabsolutetemperature(K). If D ¼10

11

1

solids at1300 8C, what is the di¡usion coe⁄cientD

atordinarytempe ratures?

1300

¼ exp

DH



1300



¼hBi

from which D

¼ B

1300

Calculate Bhito¢ndthat itgives1.86 10

25

, thereforeD

¼2 10

36

1

Tohomogenizea1-mmgrainatordinary temperaturestakes

t ¼

2

2  10

36

¼ 0:5  10

s  1:5  10

years;

whi ch is in¢nitelylongto allintentsandpurposes.

Important remark

Rocks therefore retain the memory of their history acquired at high temperatures. This is the prime

reason isotope geology is so incredibly successful and is the physical and chemical basis of isotope

memory. The phenomenon might be termed isotopic quenching, by analogy with metal which, if it is

immersed when hot in cold water, permanently retains the properties it acquired at high temperature.

1.3.3 A practical application of isotopic exchange:

isotopic dilution

Supposewe wishto measurethe rubidium contentofa rock. Rubidium has two isotopes, of

mas s 85 and 87, in the proportion

Rb/

Rb ¼2.5933. (This is the value found when mea-

suring the Rb isotope composition ofnatural rocks.) The rock is dissolvedwitha mixtureof

aci ds.To the solution obtained, we add a solution witha known Rb content which has been

arti¢cially enriched in

Rb (spike), whose

Rb/

Rb ratio in the spike is known.The two

solutionsmixandbecomeisotopicallybalanced.Onceequilibriumisreached, theabsol ute

Rb content ofthe rock canbe determined bysimply m easur ing the isotope composition of

afraction ofthemixture.

Writing





gives:



mixture

RbÞ

rock

þð

RbÞ

spike

RbÞ

rock

þð

RbÞ

spike



spike



rock

spike

1 þ

rock

spike

To p a n d b o tt o m a r e d i v i d e d b y

Rb, bringingout



A little manipulation gives:

rock

spike



spike



spike





mixture



mixture





rock

15 Isotopy

Because we knowthe

Rb contentofth espike,theRb contentoftherockcanbeobtainedby

simply measur ing the isotope ratio of the mixture, without having to recover all the Rb by

chemi calseparation.

Suppose, say, the isotopic composition of the spike of Rb is

Rb/

Rb ¼0. 12.The natu-

rally occurri ng

Rb/

Rb ratio is 2.5933.We dissolve1g of rock an d add to the solution1g

ofspi ke containing 3 ngof

Rb percubic centimeter. After thoroughlymixing thesolution

containing the dissolved rockand the spike solution, measurement of a fraction of the mix-

tureyields aratio of

Rb/

Rb ¼1.5.Th e Rb contentofthe rockcanbe calculated.We sim-

plyapply theformula:

rock

spike

0:12  1:5

1:5  2:59



¼ 1:266;

rock

¼ 1:266 

spike

¼ 1:266  3  10

9

g ¼ 3:798  10

9

As we took a sample of 1g of rock, C

¼ 3:798 ng g

1

¼ 3:798 ppb. Therefore

total

¼ C

ð1 þ 2:5933Þ¼13:42 ppb.

This method can be used for all chemical elements with several stable isotopes for which

spikeshavebeenpreparedthatarearti¢ciallyenrichedinoneormoreisotopesandforelements

with a single isotope, provided it is acceptable to use a radioactive isotope (and so potentially

dangerousfor whoeverconductstheexperiment)asaspike.Themethod hasthreeadvantages.

First,aftermixingwiththespike,chemicalseparation methodsneednotbe entirelyquanti-

tative. (The yields of the various chemical operations during analysis do not count.) Isotope

ratios alonematter, aswell as anycontamination,whichdistortsthemeasurement,ofcourse.

Then, as th e mass spectrometer makes very sensitive and very precise measurements of

isotope ratios, isotopic dilution may be used to measure the amounts of trace elements,

eventhetiniesttraces, withgreatprecision.

Isotope dilutionwas invented for the needs ofl aboratoryanalysisbut maybe extended to

natural processes. As shall be seen, variable isotope ratios occur in nature. Mixes of them

can be used to calculate proportions by mass of the geochemical elements involved just by

simple measurements of isotope ratios.

As can be seen, isotope dilution is an essential method in isotope geochemistry. But just

howpreciseisit?This exercisewillallowustospecifythe error(uncertainty)inisotopedilu-

tion measurement.

Exercise

The isotope ratios of the spike, sample, and mixture are denoted

, and

, respectively.

We wish to ﬁnd out the quantity

of the reference isotope

in the sample. To do this,

quantity

of spike has been mixed and the isotope ratios (

) ¼

of the spike, sample, and

mix have been measured. What is the uncertainty of the measurement?

Answer

Let us begin with the formula





16 Isotopes and radioactivity