All?gre Claude J. Isotope Geology

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When apolycrystalline assemblage is in conditionsoftemperatureandpressureatwhich

itmelts, melti ng is par t ial mel t i ng pretty well obeying thethermodynamic ru les ofeutectic

equilibrium. Melting is characterize d by a de g ree of part ial mel t i ng F and a chemical

compositionofthe liquidwhich is di¡erentfromthatofthe solid.

Trace ele mentsarepartitionedbetween liquidand solidstates.Thispartition is character-

ized by the partition coe⁄cient D

¼concentration in solid/concentration in liquid of

elementi.

Figure 6.63 The types of heterogeneity the mantle may conceal. (a) Different domains scattered in a

whole. Sampling a small part brings out the heterogeneities, while more extensive sampling makes them

disappear. (b) Large-scale heterogeneities with several domains, each of which is homogeneous.

Heterogeneity is observed geographically. (c) Types (a) and (b) combined. (Of course, other types may exist.)

319 Chemical geodynamics

Itobeys a simplebalance equation:

¼ C

F þ C

1 FðÞ

where C

is the initial concentration of the solid for element i, C

is the concentration of

element i in the magma, C

the c oncentration of element i in the res idual solid, and F the

degree of partial melting. If the overall partial melting coe⁄cient of element i is de¢ned in

the pro c ess considered by C

¼D

, without worrying whether the process is in equili-

brium or not, whether it is complex or simple, but such that D

is about constant, then the

partial meltingequation canbewritten:

F þ D

ð1  FÞ

The c oncentration inthe residualsolidis therefore:

¼ D

F þ D

ð1  FÞ

Afundamentalresultoftheseformulaefor thegeodyn amic cycleisTreuil’s approximation.

Michel Treuil of the Institut de Physique du Globe in Paris noticed in 1973 (Treuil, 1973;

Treuil and Joron, 1975) that for some elements with low D, D

(1 F) becomes negligible

compared with1; therefore he nam ed those elements‘‘magmaphile’’ but some others have

na medthem‘‘incompatible.’’

’

For these elements the concentration depends directly on F (the smaller F is the more

concentrated the liquid) and, yet more importantly, if the ratio ofthe two elements is taken,

F is eliminated:

;

that is, the ratios are conserved across partial melting, which is true for isotope ratios of

h eavy ele ments and for ratios of a number of trace elements whose partition coe⁄cients D

arevery, verysmall and so negligible compared w ith1, such asTh, U, Rb, Nd, and Pb under

certain circumstances. Th is is fundamental because not only can we obtain the isotope

ratios of the mantle such as

Sr/

Sr,

143

Nd/

144

Nd , etc. but we can also obtain the Rb/Sr

and Sm/Nd chemical ratiosofthe mantle (orat any rategood estimatesofthem).

Exercise

Is Treuil’s approximation by which (for low values of

) we can write:

320 Radiogenic isotope geochemistry

still valid in the two instances below?



¼0.01 and

¼0.02. Study the Th/U ratio as a function of the degree of partial melting

, taking the values of 0.11, 0.05, and 0.01.



¼0.02 and

¼0.1.

In both cases, plot the curve (Th/U)

liquid

/(Th/U)

source

(

) ¼

Answer

We write:



þ Fð1 

¼ R:

In the ﬁrst case, we ﬁnd

¼1.08 (0.1), 1.159 (0.05), 1.4897 (0.01). In the second case, we

ﬁnd

¼1.61 (0.1), 2.10 (0.05), 3.65 (0.01). This shows the limits of Treuil’s approximation.

Exercise

At the ocean ridges, U/Pb fractionation is such that  ¼

238

204

Pb shifts from 5 in the upper

mantle to 10 and more in the oceanic crust. If

ðRÞ

¼ 0:03 and if

¼0.08, what is the value

ðRÞ

This oceanic crust is melted in the S-process. In the andesite lava  ¼8.5. Supposing that

ðSÞ

¼ 0:03 and

¼0.10, what is the value of

ðSÞ

Answer

First we ﬁnd

ðRÞ

¼ 0:135 and then

ðSÞ

¼ 0:01. Lead is less compatible than uranium during

subduction.

We have the answer to an earlier question: whydoes lead distinguish the mantle from the

continental crust so poorly? Because the U/Pb R and S chemical fractionation processes

partially o¡set one another in th is case. There are two results fro m these investigations.

First, if the melting rates are high, the ratios between the elements that inte rest us (Rb/Sr,

Sm/Nd, U/Th, and U/Pb) arejust about preserved.

Accordingly,themi xing line sinthe isotope correlationdiagrams(seethebeginni ngof

this chapter) aremore or les s st raight.This is why roughly correct correlations were found

between many isotope ratios. Becaus e everything is al mo s t li n ear: fractionation, mixing,

and radioactive decay. (For lead isotope ratios, where growth of radioactive origin is non-

linear, the parent^daughter identity preserveslinearity; when such linearity is lost, the iso-

tope diag ramsbecome more co mplex.)

Bycontrast, when dealing withelementsthatfractionatedduringdi¡erentp rocesses,lin-

earity is not preserved.This is th e case, for example, of isotopes of lithophile elements (Sr,

Nd , Hf) and of the raregases, because theyobey two di¡erentrationales. Fractionation for

one is related to extraction from the continental crust, and for others to mantle outgassing.

The former are sensitive to processes of reinjection of oceanic or continental crust, the

others are not reinjected. An example of this is the (Sr, He) non-correlation established by

Mark Kurz of the Woods Hole Oceanographic Institution ( Kurz et al., 1982). Another

321 Chemical geodynamics

example is the case thatimplies osmium, because itsgeochemicalbehavior is atypical com-

paredwiththatofotherelements (Lambe rt etal.,1989).

Forelementswhoseonlyisotopefractionsare createdbyextractionofcontinentalcrust,a

double chemical fractionationprocess occurs: oneatthe mid-ocean ridges, theother inthe

subduction zon es.We have just seen with the example of lead thatthe two processes are not

identical (in the case of Pb/U, fractionation is reversed).We shall look more closely at this

questionusingother isotopetracers, thosefrom radioactive disequilibria.

6.6.5 Consequences for the Earth’s energy regime

Uranium, thorium, andpotassium, whichare all ele mentswhose radioactivity is an impor-

tantsourceofheatforth e internalactivityofour planet,aregreatlyenrichedinthe continen-

tal crust and therefore depleted in the upper mantle. The lower mantle is much richer in

these elements.It i s th e refo ret h e lowe r mantl e that su ppl ies t he upp e r mantl e’s convect ive

heat.Theupper mantleisheatedfrombelowand cooled fromabovebytheocean. Itisthe re-

fore in an almost ideal situation for convecting, by an admitte dly complex process but

which is akintowhatisknown inphysics as Rayleigh ^Be

nard convection (seeTurcotteand

S chubert, 2002), which can be observed when a p an of water is put on the cooker to boil.

The lower mantle too c onvects, be cause it is very hot and is of large dimensions, but it con-

vects in a di¡erent manner, by internal heating particularly by the radioactive elements it

contains (it mightbe worth lookingback atthe end of Chapter1, where the calculations set

outwerepreparation forcurrentreasoning).

Exercise

We wish to test the two-layer convective model in respect of heat ﬂow (see Chapter 1). The

measurements on MORB give the results: U ¼50 ppb, Th/U ¼2.2, and K/U ¼10

. Accepting

that the three elements are wholly incompatible, that is, that

liquid

solid

where

is the

degree of partial melting estimated at 10%, calculate the heat ﬂow of radioactive origin

produced by the upper mantle. Calculate the heat ﬂow produced by the lower mantle if we

assume that for the mantle U ¼21 ppb, K/U ¼10

, and Th/U ¼4.2. Calculate the heat ﬂow of

the continental crust given that K ¼1.2%, K/U ¼10

, and Th/U ¼5. Given that the total ﬂow is

41  10

W, what proportion is produced by radioactivity? What makes up the difference?

Answer

The heat ﬂows from radioactivity are: upper mantle: 0.95  10

W (terawatts), lower mantle

15.3  10

W, continental crust: 6.25  10

W, making a total of 22.5  10

W. The proportion

created by radioactivity (Urey ratio) is 0.5. The difference comes from energy stored when

the Earth formed and from differentiation of the core. It is stored in the lower mantle and

core (gravitational energy transformed into heat). The lower mantle therefore transmits

36.8  10

W whereas the upper mantle produces just 1  10

6.6.6 Radioactive disequilibria and the difference between

R- and S-processes

Wehavejusttouch edupon itwithregardtoU/Pbfractionation,butweshallseeindetailwith

the exampleof

230

Th/

238

Usystems,whichwehavealreadydealtwith intermsofchronology,

322 Radiogenic isotope geochemistry

there is a considerable di¡erence between the S- and R-processes, as Michel Condomines

and the present author established in developing this method in 19 82 (Alle

gre and

Condomines,1976; 1982). It was extremely well explored and explained later on by a large

amountofoutstanding work.

Let us plot the results on the graph (of activity)

230

Th=

232

Th;

238

232



(Figure6.64), inwhich(a) is areminderofhow itisread.

Let us pick up on what was developed in Chapter 3 about dating based on

238

U

230

Th radioactive disequilibrium using the isochron construction. We take the

(

230

Th/

232

Th,

238

232

Th) diagram in the form of activities. Any system in secular

equilibrium has its data point on the equiline. This can be assumed to be the case of

mantle rock.

1.2

1.4

1.0

0.8

[

230

Th/

232

Th]

[

230

Th/

232

Th]

radioactive

decline

equiline

[

238

232

Th]

MORB

source

(upper

mantle)

Old

MORB

Bulk Earth

OIB

Enrichment

in U

radioactive

growth

1.2

1.4

1.0

0.8

[

238

232

Th]

Old

MORB

Sediments

Island arc

Depleted

mantle

Enrichment

in Th

Figure 6.64 General principles of the reasoning in this section. (a) The two types of fractionation are

shown in the (

230

Th/

232

Th,

238

232

Th) diagram. Left: fractionation where Th is enriched in the ﬂuid

compared with U; right: enrichment of U compared with Th. (b) The two types of observations and

explanations. Left: genesis of MORB and OIB; right: genesis of island arc volcanoes. After Alle

gre and

Condomines (1982).

See the excellent book by B. Bourdon, G. M. Henderson, C. C. Lundstrom, and S. P. Turner (2003)

Introduction to U-Series Geochemistry, Reviews in Mineralogy and Geochemistry Vol. 52. Washington,

DC: Mineralogical Society of America.

See Chapter 3. Remember that the activity of a radioactive element R* is written [R] ¼l

R*, where R*

is the number of atoms and l

the decay constant.

323 Chemical geodynamics



If the partial melting process involves enrichment of the liquid in (Th) compared

with (U), the point representing the magma will shift horizontally to the left. The sys-

tem will return to equilibrium through decay of its excess

230

Th (excess relative to

the

238

U value). The

230

Th/

232

Th isotope ratio of the magma will decrease over time

with th e data point shifting vertically downwards. When it returns to equilibrium, it

will li e on the equiline again but in a lower position (with a lower

230

Th/

232

Th ratio

in U/Th).



If, on th e contrary, the partial melting process involves enrichment of the melt in (U)

compared with (Th), the data p oint will shift horizontally to the right. The system will

return to equilibrium by recreating

230

Th through

238

U decay. The point will therefore

rise vertically.When it reaches equilibrium, the

230

Th/

232

Th ratio will have increase d.

The U/Th ratiowillthenbefargreater thanthatofthe initialsolid.

The evolution equation is, it will be re called (with ratios in square brackets in the form of

activities):

230

232



230

232



l

230

238

232



1  e

l

230



If we measure the composition of a magma, the age since it und erwent partial melting is

written:

t ¼

230

232





238

232



230

232





238

232



Calculating an age like this involves measuring the isotopic composition ofthorium and

the

238

232

Th (chemical) ratio. But one datum is missing: the initial isotope ratio of the

thorium. One can also reason backwards and calculate the initial isotope ratio of the thor-

iumiftheageisknown.Letusleaveasidethis issueofcalculatingtheageandtheinitialratios

for th e moment andturnto the resultsofanalyses on present-daylavasfromvarioustypes of

volcanic rocks.

Ofcourse,therocksmustbefromthepresentdaysosigni¢cantcomparisons canbemade.

Otherwise radioactive decay since eruption will have shifted the position of the data point

verticallyinthe(

230

Th/

232

Th,

238

232

Th)diagram.Wemustbewaryofcontaminationpro-

cesses, too. As for raregases, seawater is a powerful contaminantfor the

230

Th/

232

Th ratio

because it has extremely high

230

Th/

232

Th ratios (values of 50 or 100 are commonplace

while the values of basalt lavas are close to unity). Even slight contamination by sea water

shiftsthe datapointupwards.

Havingtakenthe requisiteprecautions, whatdoweobserve? MORB and OIB arelocated

to the left of the equiline.They therefore correspond to cases of magma being en riched in

Th relativeto Usinc e the U/Th ratiohas fallen.

ItisobservedthatstatisticallytheOIBliesbelowthe MORB.An enrichmentcoe⁄cientis

de¢ne d k ¼(U/Th)

magma

/(U/Th)

source

. From the exercises i n the previous section,the

lower the degree of partial melting, the smal ler k is. Petrologists tell us that the extent of

324 Radiogenic isotope geochemistry

partial melting of OIB is much less than that of MORB (1% versus10%). Now, the relative

position on the (

230

Th/

232

Th,

238

232

Th) diagram is notso clear.This can be explained by

acc epting that the genesis of OIB is often complex, i nvolving phenomena of mixing, con-

tamination by ancient o ceanic crust, and multiple transfer times (Figure 6.65).When the

data points for island arcvolcanoes are plotted on the same diagram, theyarefound, on the

contrary, to lie to the right of the equiline.Their genesis involved processes which enriched

the magmain Umorethan inTh.Itisalsoobservedthat many pointsrepresentingthesevol-

canoes have much greater thorium isotope ratios than are found for MORB and OIB.This

is interprete d i n the frameworkofmodelswhich havebeen developedbythe genesisofmag-

mas in subduction zones. It is accepted that the downgoing plate begins to melt, but that it

passes on to the overlying mantle wedge a silica-rich £uid with a very high water content.

This £uid has probably also derived some of its material from the subducted sediments.

The £uid induces melting ofthe material in the mantle wedge overlying the plunging plate.

The initi al material is therefore a mixture ofupper mantle similar to that whi ch gave rise to

the MORB, ofa subducted sediment whose

238

232

Th ratios maybe as lowas 0.4, and of a

water-rich£uidwhich is enriche d morein Uthani nTh as Uis moresolubleinwater thanTh.

There is abig di¡erence, then, between magmas which result fro m par tial melting of the

mantle, like MORB, and island arc magmas whose genetic processes are more complex.

The genesis of OIB belongsto the family where m elting processes dominate, but in detail it

isobservedthattheTh isotope ratios ofone andthesame islandarevariable.

Detailed studies have shown complex phenomena to be involved. First of all, there are

variable types of source, as seen with lead isotopes.Then there are processes of interaction

with the oceanic cr ust through which the OIB makes its way. So there are three types of

magma-producing processes.Simplepartial melting (MORB), partial meltingfollowed by

mixing(OIB),andpartialmeltinginthepresenceofwaterandamixtureofmultiple compo-

nents (island arcs). But the diagrams show a continuous spread of data points among these

di¡erenttypes, indicatingthe existenceofcer tain a⁄nities.

Exercise

A domain of the mantle is considered whose [

230

Th/

232

Th] ratio in terms of activity is 1.4.

Suppose this domain undergoes 8% partial melting to form oceanic crust. This oceanic crust

returns to radioactive equilibrium. Then it undergoes 5% partial melting in a subduction zone.

The transfer time of the magma to the surface is 40 000 years. What are the [

230

Th/

232

Th] and

[

238

232

Th] ratios of magma in terms of activity? It is assumed the solid/melt global

fractionation coefﬁcients are

¼0.015,

¼0.0034.

Answer

[

230

Th/

232

Th] ¼1.15. [

238

232

Th] ¼1.

Letus moveon now totransittimesbetween the source and the surface. Despite complex

theoretical models that predicted a quite long and complex transfer from th e melting zone

to the surface (McKenz ie, 1989) the transfe r rate for MORB is quite high. For MORB, the

issue hasbeen addressed intwoways.The¢rstis to calculateth e initialvalue of [

238

232

Th]

of the source mantle by estimating the d egree of partial melting using the fractionation

325 Chemical geodynamics

238

232

230

Th/

232

equiline

1.00

Tonga

Kermadec

Sunda

Marianas

Kamchatka

Aleutians

Nicaragua

2.00

3.00

1.000 2.00 3.00 4.00

150

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

0.7

0.8

0.9

1.1

1.2

1.3

1.4

1.5

1.6

230

Th/

232

238

232

MORB

Azores

4.1

3.1

2.8

2.6

2.4

2.25

= 0.5

1% 2% 5%

Degree of

partial melting

Th/U

10%

= 0.6 k = 0.7 k = 0.8

= 0.9

Bulk Earth

equiline

Pitcairn

Comoroes

Faial

Tahiti

Hawaii

Réunion

Pico

Ardoukola

MAR 37

Iceland

Macdonald

Figure 6.65 Actual cases of disequilibrium data. (a) The MORB zone in blue, the OIB zone in white with

the main OIBs shown. Note the two domains are continuous. (b) Subduction zones with various

isochrons showing a very different representation of MORB and OIB. After Condomines

.(1988);

Condomines and Sigmarsson (1993).

326 Radiogenic isotope geochemistry

coe⁄cients determinedinthelaboratoryfor UandTh.T hesecond isbasedonthefollowing

remark. A radioactive pair has returned to equilibrium when the time since the disequili-

brium was created is six times the shorter half-life. As many

230

Th/

232

Th ratios of MORB

are far from equilibrium, it can be estimated that the transit times are very much less than

300 000 years.

238

232

238

Th ingrowth

over 30 ka

238

Th ingrowth

over 350 ka

U addition by fluids

230

Th/

232

equiline

0.6

0.8

1.0

1.2

1.4

0.90.5 1.3 1.7 2.1

average

Marianas

sediment

sediment melting

Figure 6.66 (a) Diagram of current understanding of the formation of island arc volcanoes; (b) the

various stages of genesis explaining the observations. The example of the Marianas Islands, studied by

Elliot

.(1997).

Volcano

Melting

Fluid

transfer

327 Chemical geodynamics

But there is a more interesting observation.The

226

Ra/

230

Th and

210

Pb/

226

Ra ratios are

not in equilibrium.The transit time is less than 6 times the half-lives of

226

Ra and

210

Pb of

1.622 10

years and 22.26 years, respectively. This corresponds to a date for

226

Ra of less

than10 000 years, but for

210

Pb of les s than135 years. If it is accep ted that mid-ocean ridge

magmasform atdepthsof30 km,this correspondstotransitspeeds of 200 myr

1

.Thecon-

straintofradium correspon ds to 3 myr

1

.Therefore, the speed is i nstantaneousfor MORB

and it should be considered that the

230

Th/

232

Th ratios m easured on present-day rocks

yield the U/Th value of their sourc e mantle di rectly and better than the directly measured

chemicalratiob ecauseitis fractionated.

Calculating transfe r times for OIB is much more complicated because of the complex

p rocesses involved, such as magma chamber pro cesses, lithosphere interaction, and so on

(see Hawkesworth,1979; Elliotetal.,1997;Turner etal.,20 03).

Island arcs have ver y long and variable transit ti mes. Some are more than 300 000 years

because the data points are very high on the equiline, far above the MORB points. In the

case of the Marianas Island arc,Tim E lliott of the University of Bristol and his colleagues

have managed to put a ¢gure on the ¢nal stages and to show that there were two stages of

input fromthe downgoing plate. A ¢rststage involvi ng sedimentshaving occurred 300 000

yearsbeforea secon d melting stage, which itselfoccurred150 000 yearsbefore the eruption

(see Figure 6.66a). All ofthis clearlyshows the di¡erenceinbehaviorofislandarcvolcanoes

comparedto MORBand OIB.

Exercise

The radioactive disequilibrium of an OIB is measured giving [

230

Th/

232

Th] ¼1.08 and

[

238

232

Th] ¼0.8. In addition, [

238

230

Th]

source

¼1.1. Calculate the fractionation coefﬁc-

ients

¼(U/Th)

source

/(U/Th)

magma

, the transit time between fractionation and reaching the

surface, as well as the rate of ascent, if it is assumed the process began 20 km beneath the

ocean ridge.

Answer

¼1.43,

¼1380 years,

¼15 m yr

1

6.6.7 The major elements

Themajorelementsarethosethatmakeupmineralsandthereforerocks.Theirabundanceis

measured in percent by mass. They are oxygen, silicon, magnesium, aluminum, calcium,

iron, and sodium.Theydetermine the physical properties of the Earth’s interior and so are

responsible for the properties measured in geophysics (velocity of seismi c waves, density,

magnetism). Now, what is importantis thatthese major elements are practically una¡ected

by the major processesgoverning chemical geodyna mics.To show this, letus takean extre-

melysimple calculation.

Let us consider th e weight analyses of peridotite (an essential constituent of the mantle)

and ofgranite(an essential constituentofthe continental crust) (Table6.9).

The proportions by mass between continental crust and upper mantle are 0.01 (for th e

upper mantle above the seismic discontinuity at 670 km). Let us suppose the peridotite

328 Radiogenic isotope geochemistry