All?gre Claude J. Isotope Geology

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6.3.1 The chronology of extraction of continental crust

We saw when examining the results of geochronology that the continents are made up of

provinces ofvariable but well-de¢ned ages (see Plate 5).Thisgeological distribution seems

to show thatcontinental crusthasbeen progressively extracted fromthe mantle throughout

geological time. But if this is so, what is the exact rate ofextraction ofthe material making it

up? Isitauniform rate? Is it modulated?

In opposition to this idea of continuous extraction, it can be assume d that continental

cru st was extracted from the mantle right atthebeg inni ng, in early times, when di¡erentia-

tionoftheEarthoc curred,andthatsincethen ithasbeen recycledcontinuously throughout

geological time (erosion, sedimentation, orogeny) but without any new material being

created from the mantle, just ‘‘old material reworked into new.’’ Age provinces would

then be evidence ofthese recycling processes or internal rearrangementsofthe continental

cru st (reworking).

One way of addressing this issue is to calculate what is called the m ean age of

conti nental cru st materials. Let us suppose the continental crust is made up of

segments of ages t

, t

, ..., t

of respective mass es m

, m

, ..., m

.The fractions by mass of

these segments are thereforex

¼m

/M, x

¼m

/M, ..., x

¼m

/M,wherethemagnitude

M ¼m

þm

...

þm

isthe mass ofthe continentalcrust.

The m ean age ofthematerialsis equalto

hTi¼x

þ x

þþx

;

in other words itis an ageweightedby thefractionbymass:

i¼1

(see Jacobsenand Wasserburg,1979).

Wec an de¢neanisotopicageofthematerialsassumingthatthedi¡erentiationofthecon-

tinental cr ustoccurredall atonetimeatT

di¡

. L etus note :





isthe

Sr/

Sror

143

Nd/

144

Nd isotope ratioofthe continentalcrust;





isthe

Sr/

Sror

143

Nd/

144

Nd isotope ratioofthe primitive mantle;





is the

Rb/

Sr or

147

Sm/

144

Nd parent^daughter isotope ratio of the continental

cru st;





is the

Rb/

Sr or

147

Sm/

144

Nd parent^daughter isotope ratio of the primitive

mantle.

Then, 

(present-day) ¼

(T) þ

di¡

, but s ince the primitive mantle follows the evo-

lution

(present-day) ¼

(T) þ

di¡

, fromwhich, eliminating

(T) gives:



 

ðpresent-dayÞ¼lð

 

ÞT:

The model age is therefore written:

diff



 



 



249 The continental crust–mantle system

Moreover, the model age of th e continental material is equal to the model age of mantle

depletion, which is written (subscriptd denotesthe depletedm antle):

T ¼



 



 



We shall now demonstrate that this age is indeed the mean age de¢ned above. To do

th is, we are going to exploit mathematically a geochemi cal property of the continental

cru st, which is that its average chemical composition is al most constant whatever its

age. Let us calcul ate, then, the mean age of the continental crust using the constancy

of 

(and of course of 

hTi¼

lð

 

f½

ðt

Þ

x

þ½

ðt

Þ

x

þþ½

ðt

Þ

x

where 

)isthepresent-dayisotoperatioofasegmentofcontinentalcrustofaget

and

isthe present-day Bulk Earthvalue.



ðt

Þx

þ 

ðt

Þx

þþ

ðt

Þx

¼ 

present day:

Sincex

þx

...

þx

¼1,

T ¼



 



 



This is themodel age.Thereforewe dohavethefundamentalrelationship of:

mean age ¼ model age

hTi¼

As seen, we know 

and 

for Rb^Sm and Sm^Nd isotope systematics.We can estimate



quite well because the upper mantle corresponding tothe MORB source is fairlyhomo-

geneous. Estimating 

is more di⁄cult sin ce, when basalt is form ed, chemical fractiona-

tion occurs andthe Rb/Sror Sm/Nd ratios measured in MORB arenotthoseofthe mantle.

It is di⁄cultto evaluate 

(average contin ental crust) since the continents are made up of

p rovinces ofdi¡erent ages and these must be averaged.This is a di⁄cultoperation because

di¡erent segments have di¡erent means, di¡erent isotopic compositions, and so on. It is

much easier to estimate 

as the chemical composition ofcontin ental crustis almost con-

stant and can be measured directlyby sampli ng and analyzing surface rocks.To overcome

these di⁄culties, welooktouse afew tricksofthe trade(Figure 6.20).

FortheRb^Srsystem,itcanbeobservedthat

¼0sincetheRb/SrratioofMORBisvery

low (0.001) and the ratio measured in basalts is a maximumvalue, as wehave said, be cause

the Rb/Sr ratio increases during basalt formation.We can therefore calculate

T with 

, 

and 

. Using the ratios 

¼0.7022, 

¼0. 705 , and 

¼0.0091 , we get T ¼ 2:16 Ga.

Graphically, this amounts to drawing a horizontal line from the value 0.7022 and observing

theageofintersectionwith thestraightlineofevolutionoftheprimitive mantle(Figure 6.20).

The second trick concerns the Sm^Nd system, in which the average isotope ratio of the

continentalcrustcanbe evaluatedbecauserareearthsareinsolubleelementsand¢neoceanic

250 Radiogenic isotope geochemistry

sedimentsprovideanaturalaverageofthe continentalcrustsubjecttoerosion.Withthevalu es



¼0.5 1 16, 

¼0.5 12 638, 

¼0. 1 1 , and 

¼0. 196, we ¢ndT  2Ga.

Both methods therefore yield an age close to 2 Ga. One concerns the mantle and

the Rb^Sr system, the other the continental crust and the Sm^Nd system. We can

therefore consider this age of 2 Ga as a good approximation of the average age of

cru s t^ mantl e di ¡erentiation.

Continental crust did not therefore form by primordial

di¡erentiation, otherwise

T would b e somewhere around 4.5 or 4 Ga, but probably

formed throughout geological ti me. However, the idea of an average age allows great

simpli¢cation becaus e it reduces the problem to a two-stage story, which simpli¢es

all quantitative reasoning but, on the other hand, conceals the real processes

(Figure 6.21).

Exercise

The MORB isotope composition used in the previous calculation is

Sr/

Sr ¼0.7022. This

value implies that the lowest value 0.7022 measured on the mid-ocean ridges is chosen.

Suppose the most representative value is 0.7025 or even, less probably, 0.7028. What would

be the model age of differentiation in this case? How can you express the uncertainty on

diff

Sr/

2.0 Ga

0.7022

0.7050

Time

Bulk Earth

MORB

143

Nd/

144

2.0 Ga

C.C.

0.512 638

Time

Bulk Earth

Continental

crust

= sediments,

shales

Figure 6.20 Calculating the model age (average age) of crust–mantle differentiation. (a) The

Rb–

system and determination of the model age of MORB using direct analysis of basalt. (b) The

147

Sm–

143

Nd system and use of shales of mostly continental origin.

A more reﬁned calculation yields an age of 2.3 Ga, which is not fundamentally diﬀerent.

251 The continental crust–mantle system

Answer

(

Sr/

Sr)

MORB

¼0.7025 and

¼1.93 Ga.

We ﬁnd (

Sr/

Sr)

MORB

¼0.7028 and

¼1.70 Ga.

As can be seen, the ﬁrst result barely changes the reasoning. Taking the ﬁrst value, the result,

allowing for uncertainty, may be written

¼ 2

þ0:1

0:3

Ga.

6.3.2 The mass of the depleted mantle and the mantle structure

Letus consider the system made up of three reservoirs: mantle, continental crust, and pri-

mitivemantle.Extractionofthe continentalcrustandthe correspondingdepletionofafrac-

tion ofthemantle is written likea massbalance relation:

primitive mantle ¼ continental crust þ depleted mantle:

This is the opposite of a mixture, although arithmetically it is the same thing. Returni ng to

the formulafor isotope dilution or mixing, we canwrite:



¼ 

W þ 

ð1WÞ

withW ¼ðm

Þ=ðm

CÞ:

Here,W is the mass fraction of continental crust, m

is the mass of the continental

cru st, which we know from seismology: m

¼2.2 10

kg, m

is the mass of the primitive

mantle that has been depleted and whichwe are trying to estimate, C

is the concentration

Continental

crust

Depleted

mantle

670 km

Depth

2 Ga (model age)

Today

Time

Primitive

mantle

Figure 6.21 Diagram summarizing the approach and results obtained with the simple model of

evolution of the crust–mantle system, assumed to be a two-stage process separated by a sudden

period of differentiation.

252 Radiogenic isotope geochemistry

ofthe elementin the continental cr ust,while C

isthe initi al concentration ofthe elementin

the primitiv emantle.

Theid ea is toc alculateW from the¢rstequation, andthento calculatem

, themassofthe

mantle which hasbeen depleted. Is this the total mass of the mantle? Is it merelya fraction?

Tobri ng thisout,we canwrite:

W ¼





where M

isthe total mass ofthe mantle:M

¼4. 10  10

kg. Ifwe notem

¼th e frac-

tionofthe depleted mantle, weget:

f ¼



withm

2:2  10

4:10  10

¼ 0:0054.

The di⁄culty lies in estimating the  paramete rs.We k now 

and we can estimate 

quitewell from MORB measurements. Hereagainthe di⁄culty is in estimati ng 

because

of the segmentation of the continents. How do you take the average value of a mosaic? For

Sr, it is di⁄cult to do. A statistical estimate of 0.720 by Patrick Hurley et al.(1962), then at

the Massachusetts Institute of Technology, is a likely value but subject to considerable

error. For Nd, we have seen that ¢ne marine sediments seem to be a good approximation

with"

¼þ10.ThenW

¼0.32. Ifwetake for the absolute concentrations C

¼28 ppm,

¼1.1ppmwe calculatef ¼0.43.The depleted mantle therefore makes up about 43% of

thetotal mantle.

Seismology has shown there is a marked seismic discontinu ity at a depth of 670 km,

which is inte rpreted as the transition from a silicate with tetrahedral structures (SiO

)to

perovskite-type octahedral structures (SiO

) and that this discontinuity is a strong barrier

against movement of matter.This boundary is taken as the lower limitof the upper mantle

in a convectivemantle.

The upper mantle above 670 km represents 25% of the mantle. So the mass of the

depleted m antle is greater than the upper mantle alone.While the continental crust has

di¡erentiated from the upper mantle, the upper mantle has exchanged matter with the

lower mantle.

Exercise

Write the equation giving

in the " notation. If the extreme values of "

are 16 and  25,

and those of "

are þ15 and þ10, what are the extreme values of the estimate of

? What

extreme values of  do they correspond to?

Answer

Results are given by the matrix. The ﬁrst number is

and the second in brackets is

þ10 þ15

25 0.285 (0.4822) 0.375 (0.366)

16 0.3846 (0.3617) 0.483 (0.286)

253 The continental crust–mantle system

The extreme values of

are 0.28 and 0.48, which correspond to -values of 0.482 and

0.286. As can be seen, these results differ with the extreme values but the reasoning does not

change qualitatively, namely a part only of the mantle is depleted and that part is greater

than the upper mantle alone. This implies that exchanges have occurred between the upper

and lower mantle.

Exercise

The estimation of the Nd concentration of the primitive mantle is highly constrained:

its error is estimated at 0.1 ppm at most; that of the continental crust is estimated

at 2 ppm. Do possible errors with the absolute Nd concentrations in the continental crust

and in the primitive mantle greatly increase the estimated limits of ?Whatarethetwo

extreme values we obtain for  by combining the preceding uncertainties and uncertainties

on concentrations?

Answer

The extreme values for  are 0.567 and 0.231. Once again, quantitatively the difference is

great, but qualitatively the result is not overturned in geochemical terms and so yields

quantitative limits.

Howcanwe calculateW

and the parameters for the Rb^Sr system fordi¡erentiation of

the crustand mantle?

Theidea isto comebacktothe Nd systemandtoexploitthe reliable cross-referenced data

we have between the Sm^Nd and Rb^Sr systems. The way to evaluate W

is to return to

n eodymium.

We wr ite :

Nd NdSr

SrðÞ

Nd NdSr

SrðÞ

;

with Nd

¼28 ppm, Sr

¼230 ppm, Nd

¼1.1, and Sr

¼20. This gives W

¼0. 1 44.

ThenW

¼2.21 2.

Exercise

Roughly speaking, the continental crust is made up of a deep layer (lower crust) and a surface

layer (upper crust). Studies of heat ﬂow have shown that the deep crust is depleted in certain

soluble elements (uranium, potassium, and probably rubidium too) which have been trans-

ferred to the upper crust. Accordingly, the Rb/Sr ratio measured in the upper parts of the crust

is not representative of the whole crust. Given that

is 0.144 and that the

Rb/

Sr ratio of

surface continental rocks is 0.8, what is the maximum thickness of the deep crust? The Sr

content is assumed to be identical for the whole of the continental crust, which is certainly an

oversimpliﬁcation.

254 Radiogenic isotope geochemistry

Answer

The value of

is the same for budget equations of  and  values. Since 

0, we have:



0:09

0:144

¼ 0:625:

Let us write the equation giving the average composition of the continental crust.



¼ 

þ 

ð1 

Þ:

where 

and 

are the  parameters for the upper and lower crust.

If we note

¼ mass of upper crust=mass of whole crust

thickness is maximum when 

¼0, corresponding to all Rb being transferred to the upper

continental crust. In this case

¼0.625/0.8 ¼0.78.

Maximum thickness of the upper crust is about 78% of that of the whole crust. Here is a

new structural constraint derived from isotope data. If we consider the upper crust is enriched

in Rb by a factor of 2 compared with the lower crust, then the lower crust represents 56% of

the continental crust.

Remark

In practice, all of the unknowns such as ages,

, , , etc. are calculated globally by what is termed the

generalized inverse model in mathematics, by which all the equations can be solved simultaneously.

All the equations, including the  balances,  balances, and the model ages as well as the various

relations among the elements, for example on the Sr/Nd ratio measured in the crust or the mantle

samples, are calculated with a margin of uncertainty for each, which is essential. We then calculate all

the values in one go. To do this, we introduce values estimated on an a-priori basis, optimizing the

result with their estimated uncertainties and we solve a gigantic least-squares problem. It is difﬁcult

to solve as it is non-linear, with some unknowns multiplying each other. We shall see the spirit of it in

one of the problems proposed at the end of the chapter (see Alle

gre

., 1983b).

6.3.3 Changes in the Rb–Sr and Sm–Nd systems

over geological time

In the foregoing, we have u sed present-day results and have proje cted intothe past. But this

techniquehas alim ited powerofresolution.

Weknowthattheaverageageofthe continentalcrustisabout2 Ga.Butwewouldalsolike

toknowexactlyhowthisgrowth cameabout: wasthe rateofextraction fromthe mantleuni-

form, modulated, etc.?

Exercise

We consider the following eight models of continental growth (A–H). Each is represented by a

distribution of continental orogenic segments for which the table shows the contribution

from continents and the average age of each segment.

255 The continental crust–mantle system

Segments

Model A (1) (2) (3) (4)

3.5–2.5 Ga 2.5–1.5 Ga 1.5–0.5 Ga 0.5–0 Ga

Average ages 3 Ga 2 Ga 1 Ga 0.25 Ga

Proportions 39% 32% 18% 11%

Segments

Model B (1) (2) (3) (4)

3.5–2.5 Ga 2.5–1.5 Ga 1.5–0.5 Ga 0.5–0 Ga

Average ages 3 Ga 2 Ga 1 Ga 0.25 Ga

Proportions 21% 59% 19% 1%

Segments

Model C (1) (2) (3) (4)

3.5–2.5 Ga 2.5–1.5 Ga 1.5–0.5 Ga 0.5–0 Ga

Average ages 3 Ga 2 Ga 1 Ga 0.25 Ga

Proportions 40% 10% 10% 40%

Segments

Model D (1) (2) (3) (4) (5)

4.7–3.8 Ga 3.8–2.5 Ga 2.5–1.5 Ga 1.5–0.5 Ga 0.5–0 Ga

Average ages 4 Ga 3 Ga 2 Ga 1 Ga 0.25 Ga

Proportions 20% 20% 20% 20% 20%

Segments

Model E (1) (2) (3) (4)

3.5–2.5 Ga 2.5–1.5 Ga 1.5–0.5 Ga 0.5–0 Ga

Average ages 3 Ga 2 Ga 1 Ga 0.25 Ga

Proportions 10% 20% 30% 40%

Segments

Model F (1) (2) (3) (4) (5)

4.2–3.8 Ga 3.8–2.5 Ga 2.5–1.8 Ga 1.8–0.8 Ga 0.8–0 Ga

Average ages 4 Ga 3 Ga 2 Ga 1 Ga 0.5 Ga

Proportions 40% 15% 15% 15% 15%

Segments

Model G (1) (2) (3) (4) (5)

4.2–3.8 Ga 3.8–2.8 Ga 2.5–1.8 Ga 1.8–0.8 Ga 0.8–0 Ga

Average ages 4 Ga 3 Ga 2 Ga 1 Ga 0.5 Ga

Proportions 3% 3% 85% 5% 4%

Segments

Model H (1) (2) (3) (4) (5)

4.2–3.8 Ga 3.8–2.5 Ga 2.5–1.8 Ga 1.8–0.8 Ga 0.8–0 Ga

Average ages 4 Ga 3 Ga 2 Ga 1 Ga 0.5 Ga

Proportions 10% 50% 30% 10% 0%

256 Radiogenic isotope geochemistry

In each case, calculate the average age of the continent and draw the cumulative growth

curve of the continents and variations in the rate of growth (see Figure 6.22). Which

models do you think are compatible with the results for the crust–mantle system with

current data?

Answer

Four models are acceptable with an age of 2 1 Ga: A, B, D, and G.

Comments

As the exercise has just demonstrated, theaverage age found for the continental crust using

the calculation of th e previous paragraph can indeed be used to eliminate some scenarios,

but still leaves scope for a range of possible scenarios. For example, one might equally well

envisage eithe r steady continuousgrowth (model D) or suddengrowth (model G).

Toget arealpictureofthekineticsofcontinentalextraction(examination oftheshapes of

the curves ofgrowth rates is extremelydemonstrative) outsidethe mantle, wehadtheideaof

trying to obtain isotopic information for Sr and Nd for di¡erent geological epo chs so as to

calculateWas a function oftimeW(t).

These studies, whi ch looked as if they should be straightforward, have proved very di⁄-

cult. The ¢rst problem was the relevance of the evidence of ancient massifs. For recent

rocks, weknow thatbasaltrocks collectedfro m ocean ridges are evidence ofdepleted man-

tle. But in the past, when we no longe r have mid-ocean ridges, how canwe recognize wh ich

basalts can be used as evidence of ancient depleted mantle like the present-day MORB?

50%

100%

34 2 1 0

<T> = 2.017 Ga

3 2 1 0

<T> = 2.012 Ga

50%

100%

34 2 1 0

<T> = 1.1 Ga

34 2 1 0

<T> = 2.6 Ga

34 2 1 0

<T> = 2.5 Ga

4 3 2 1 0

<T> = 1.5 Ga

4 3 2 1 0

<T> = 2 Ga

34 2 1 0

<T> = 1.97 Ga

Figure 6.22 The results for the various models calculated in the exercise. On each graph, the

-axis

shows time in Ga and the

-axis the percentage of continental crust formed. The integral curve is in

black, the derivative curve, that is the growth rate, in blue. At the top of each graph h

i indicates the

average age calculated. The models shaded in blue are those that are compatible with an average age of

about 2 Ga.

257 The continental crust–mantle system

Forcontinents,howcouldwe establishanaverage sampleofthe continents atagivenperiod

in the past? Are the current mappings of various age provinces representative i n relative

abundance ofwhatthey wereinthepast?



For ev iden c e abo ut t h e upp er mantle, we adopt a pragmatic attitude. Up to 600 Ma, we

use ophiol ite mas s i fs, which are pieces ofoceani c crust that have been obducted on the

continents. For terrai ns older than 2 Ga, we u se komatiites, which are associations of

rock put in place under the seas and which associate ultrabasic and basic lava (which is

peculiar to Archean times). But these are working hypothes es that have not really been

demonstrated (other thanthattheydonotseem absurd).



For th e contin ental crust, the only evidence of what it was like at a given time are the

shales ofthe time i n question.T heyare a sample ofeverything that had been eroded and

then mixedin theo cean atthetime.Theyarean average ofthe uppercontinental cru st.

In anyevent, the following program canbeset.

(1) Choose agroup ofrockswhose origin hasbeen identi¢ed foragiven time-span.

(2) Determin e its age and i nitial isotope ratio precisely (which takes us backto the consid-

erationson isochrons in Chap ter3).

Wethen just need to repeatthe ope ration for severalgeolog ical epo chs andso obtain Srand

Nd isotope evolution curvesforour tworeservoirs (Figure 6.23).

This program has involved some 20 or so research teams worldwide over 10 years. The

results of th e quest have been very dis appointing.Why? Not because the teams were bad,

but because, in the past, isotope systems s eldom remained strictly closed and th erefore

u ncertainties on initial ratios increase statistically with time.Without going into details

thatextendbeyondthescopeofthisbook, we can remember tworelatedideas: considerable

di⁄culties and poor and inconsistent results. However, let us summarize the all too scarce

p ositive ¢ndings.

Stronti um

Whole-rock isochrons ofancientbasic andultrabasic rocks arevery rarely reliable.To resolve

the problem, we bring in analysis of pure clinopyroxene extracte d from rocks containing

almost no Rb and providing a fairly accurate re£ection of initial

Sr/

Sr ratios.We deter-

minetheageofthebasic massifs in questionindependentlybyotherchronologicalmethods.

This method has yi eldedfewreliableresults. Evenso, the overallshapeofthe curveofevo-

lution (Figure6.24) is clear enough.It is virtuallyalignedon thestraightlineof isotope evo-

lution of the primitive (closed) medium up to 3 Ga and then is practically horizontal from

1G aonwards.

Itisimpossibletoobtainthe curveofisotope evolution oftheaveragecontinentalcrustfor

Sr from the analysis ofancient shales.There aretworeasons for this: the¢rst is thatapartof

the e rosion of Sr is soluble and was deposite d in limestone; the second is that shales them-

selves areopen systems for Sr.

Neodymium

Thesituation isfar morefavorableb ecausebothSmandNdaremuchmore chemical lyi nert

(less soluble) elements than Rb and S r. But the variations are, of course, very small: to

258 Radiogenic isotope geochemistry