All?gre Claude J. Isotope Geology
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Remark
The cases most studied by these methods are of thermal variations during orogeny (mountain
building) where estimations are made of how the crustal block cooled as it rose to the surface;
at what speed and over what temperature range, etc.
T he studyof con tact metamorphism
The decisive step in this type of semi-quantitative approach was the study by Stanley
Hart (1964) (Figure 3.6), then a student at the Massachusetts Institute of Technology,
con¢r m e d by Gil Hanson and Paul Gast (1967), then at the University of Minnesota,
who studied the ages obtained by the di¡erent methods for an instance of contact
metamorphism.
Contact metamorphism occur s when granite is intruded into a geological series.
Physically, such metamor phism corresponds to sudden heating by a body of de¢ned geo-
metry.The thermal evolution of such a problem can be readily processed mathematically.
The isotherms and their variations over time are easily obtained. Hart chose to study the
result of a Terti ary intrusion some 60 Ma old (Eldora) in the Precambrian terrain of
Colorado. There is a b ig age di¡erence between the intrusion and the surrounding rock.
He studied the variation in Rb^Sr, K^Ar, an d U^P b ages on di¡erent rich minerals of
the Precambrian formation with distance from the point of contact. The ‘‘apparent’’ age
of all the minerals varied w ith distance from the contact, which was what he expected. At
the contact point, it is the same as the age of the intrusion and then it progressively con-
verges towards the age of the surrounding rock at a great distance from the contact.What
is interesting is that at any given distance from the contact, the order of apparent ages
determined by the various m ethods on the various min erals obeys a coherent log ical
schema.
The reliabilityof ‘‘rich’’chronometershasbeen estimated onthe basis ofsuch a sequence
and a few laboratory experiences.They are ranked here in decreasing order of reliability of
the ages obtained.
T (°C)
K/Ar ages (Ma)
Feldspar
Biotite
Hornblende
400
1000
900 800 700 600 500
200
600
800
Figure 3.5 Cooling curve of an orogenic segment of Grenville Province, Canada, based on K–Ar apparent
ages and on the retentivity of the different minerals. After Berger and York (1981).
67 Rich systems
206
Pb
207
Pb zircon
4
206
Pb
238
U zircon
4
40
Ar
40
K amphibole
4
87
Rb
87
Sr muscovite
4
87
Rb
87
Sr biotite
4
40
Ar
40
K biotite
4
40
Ar
40
K feldspar
Although there are many exceptions to this ranking, it allows a series ofapparent ages to be
gauged rapidly.Thus when, say,
207
Pb^
206
Pb on zircon gives the same age as
40
Ar^
40
Kon
biotite there is agood chance the age is correct.When the di¡erence between the
40
Ar^
40
K
on amphibole and
87
Rb^
87
Sr on muscovite ages is slight, we consider we are close to the
true age, etc. While this empirical approach is useful, it does not, alas, provide a
1400
1200
1000
800
600
400
200
10
500 15001000 2000 m
100
Distance from contact
Age
of the
intrusion
Apparent age (Ma)
100
0
200
300
400
500
600
100 1 000 10 00010
C
A
Max. temperature (°C)
1000
Biotite
K-Ar
Feldspar
208
Pb/
204
Pb
Feldspar
206
Pb/
207
Pb
Hornblende
K-A
Zircon
207
Pb/
206
Pb
Zircon
206
Pb/
238
U
10 000 20 000
1600
Biotite
Rb-Sr
60 Ma
intrusion
Metamorphic
zone
Basement dated
1450 to 1600 Ma
a
b
Figure 3.6 Contact metamorphism of the Eldora stock, Colorado, studied by Stanley Hart. This figure is
famous. Top: the geological section. The scale of distances in (a) and (b) is logarithmic. (a) The thermal
profile, that is, the maximum temperatures reached at each point versus distance. Curves A and C are
from two values of thermal diffusivity. (b) Apparent ages versus distance.
68 Radiometric dating methods
thoroughgoing solutiontothe issue ofreliability. Inparticular, howcan an agebe calculated
whenthehypothesesofthe simpl e modelarenotsatis¢ed?
Exercise
Analysis of a granite yields: U–Pb on zircon, 520 Ma; K–Ar on amphibole, 480 Ma; K–Ar on
biotite, 400 Ma; Rb–Sr on biotite, 460 Ma; Rb–Sr on muscovite, 470 Ma. What seems the
likely age of intrusion of this granite to you? How reliable is the result?
Answer
The true age is probably around 550–540 Ma. The reason being that some disruptive
phenomenon has affected the region as there is clearly a discordance in ages. The most
robust of the chronometers is U–Pb on zircons, but it is not perfect and this apparent age
may be made a little older. But this reasoning is very rough and ready as can be seen!
3.2.2 The concordia method
This method has been devised for calculating a system’s true age even though the system is
an open one.This is done by exploiting pairings of chemical behavior which may be found
withvariousradiochronometers.
Uranium ^lead systems
Uranium^lead systems using
238
U !
206
Pb and
235
U !
207
Pb decay have an interesting
feature: both parents and both daughters are of the same che mical nature but have very
di¡erent decay constants. This pairing is exploite d for determin ing geological ages even
whenthesystem isan open one.
Let us consid er a uranium-rich material. It may be zircon (ZrSiO
4
), a common mineral
in granite rocks, sphene, uraninite, or apatite, minerals containing little ‘‘common’’ lead
(detected by measuring the non-radiogenic isotope
204
Pb).The initial lead canthereforebe
neglected in chronometric equations (rich systems) andwewrite :
206
Pb
238
U
¼ e
l
238
t
1
207
Pb
235
U
¼ e
l
235
t
1
:
Twoage s canthereforebe calculated bymeasuringthe
206
Pb,
207
Pb,
238
U, a n d
235
Ucontents
of minerals. If everything complied with th e assumptions (closed system, rich system, etc.)
these two ages should be identical, that is, concordant.The common age would therefore
indicate the time the zircon or apatite crystallized in the granite magma.This is sometimes
the case, but generally ages are found to be di¡erent and so di sc ordant.Tobringoutthese
age discrepancies we consid eraplot:
x ¼
207
Pb
235
U
; y ¼
206
Pb
238
U
:
69 Rich systems
Weplotthe curveoftheparametric equation:
y ¼ e
l
238
t
1
; x ¼ e
l
235
t
1
:
This curve, which canbe graduated for time, is kn own as the con cordia curve. It is the geo-
metric locus of concordant ages. Any concordant age li es on the curve and anydiscordant
agelies o¡the curve.
The South African Louis Ah rens notic ed in 1955 that when the
206
Pb*/
238
Uand
207
Pb*/
235
Uratios (asbefore * indicates the isotopes are radiogenic) measured on suites of
cogenetic m inerals are plotted, th ey tend tobe aligned.Thesealignments cutthe concordia
cur ve at two points, corresponding to two ages (t
0
and t
1
). In 1956 George Wetherill of the
Carnegie Institution inWashington showed that the two ‘‘ages’’obtained by Ahrens’con-
structions couldbe interpreted astheageofuranium orecrystallization and theageofadis-
ruption that a¡ected the minerals and caused uranium and lead exchanges with the
environment. Hereis asimpli¢ed demonstration of Wetherill’s model.
The uranium decays in a closed system from the time at which the mineral is crystallized
attimet ¼0untiltimet
1
.
Uðt
1
Þ¼U
0
e
l
238
t
1
:
Suppose that at t
1
some of the uranium was lost. Let us term the lost prop ortion .There
remains (1^ ). At time t
1
þt, theuraniumbecomes:
Uðt
1
þ tÞ¼ð1 ÞU
0
e
lt
1
:
Thisuraniumd ecaysin aclosedsystemuntil timet whenthe analysisis conducted:
UðtÞ¼Uðt þ tÞe
lðtt
1
Þ
¼ð1 ÞU
0
e
lt
1
e
lðtt
1
Þ
UðtÞ¼ð1 ÞU
0
e
lt
:
Letus see whatbecomes ofthe corresponding lead isotope. From t
0
to t
1
the mineral system
is closed.We have Pbðt
1
Þ as ðt
1
Þ¼U
0
ð1 e
lt
1
Þ, as we assume there is no initial l ead. At
time t
1
the system loses a proportion () of lead so at (t
1
þt) the lead is therefore:
Pbðt
1
tÞ¼ð1 ÞU
0
ð1 e
lt
1
Þ:
Betweent
1
and t, leadisproducedbydecayofthe u ranium remainingatthattime.
PbðtÞ¼Pbðt
1
þ tÞþUðt
1
þ tÞ 1 e
lðtt
1
Þ
hi
:
Replacingby their valuesPb(t
1
þt) andU (t
1
þt) gives:
PbðtÞ¼ð1 ÞU
0
1 e
lt
1
þð1 ÞU
0
e
lt
1
1 e
lðtt
1
Þ
hi
:
We replace U
0
by U(t) toreturntothe traditionalexpression:
70 Radiometric dating methods
U
0
¼
UðtÞe
lt
ð1 Þ
:
We have to change variablesbecause wehave decided to take the present time as the origin.
Inthe new notation (in capitals),T ¼t,T
1
¼t ^ t
1
.Thisgives:
Pb
U
¼
1
1
e
lT
e
lT
1
þ e
lT
1
1
:
But for the decay constant the expressions are identical for both
206
Pb^
238
Uand
207
Pb^
235
U pairs, because the two lead isotopes and the two uranium isotopes must react
chemicallyin an identical mannertothe‘‘crisi s’’which a¡ectedthesystems at T
1
.We write:
206
Pb
238
U
¼ r
1
207
Pb
235
U
¼ r
2
r
1
¼
1
1
e
l
8
T
e
l
8
T
1
þ e
l
8
T
1
1
r
2
¼
1
1
e
l
5
T
e
l
5
T
1
þ e
l
5
T
1
1
:
We can eliminate ð1 Þ=ð1 Þ
½
betweenthetwoequations, so giving:
r
1
e
l
8
T
1
1
r
2
e
l
5
T
1
1½
¼
e
l
8
T
e
l
8
T
1
e
l
5
T
e
l
5
T
1
:
For ¢xed Tand T
1
, this expression takes theform:
r
1
y
0
r
2
x
0
¼ constant:
This is the equation ofa straightline in an (r
1
, r
2
)plot.When ¼ ¼0, that is, when the sys-
tem has remained closed, r
1
¼ e
l
8
T
1
and r
2
¼ e
l
5
T
1
.The upper intercept with
the concordiatherefore corresponds toT, which isth e initial ageofthe mineral population.
When ¼1, that is, when the minerals have lost all their lead at T
1
,
r
1
¼ e
l
8
T
1
1
and r
2
¼ e
l
5
T
1
1
:
The lower intercept with the c oncordia correspon ds toT
1
.When and are between 0
and 1, the data points are on the straight line joining the two points of the concordia Tand
T
1
.When more lead than uranium is lost the points are betweenTand T
1
.When more u ra-
nium than lead is lost the points are ‘‘above’’ T and therefore above the c oncordia (see
Figure3.7).
We can then address the inverse problem and say thatwhen the datapoints obtained on a
series ofuranium-rich cogenetic m inerals are aligned in the (
206
Pb/
238
U,
207
Pb/
235
U) plot,
the intersections of the straight line they de¢ne with the concordia curve determine the
initial age ofthefam ilyofmin erals andthe age ofthe disruption experienced.Wegive a sim-
ple historical example, that of the Morton gneiss of Minnesota.The U^Pb measurements
being madeon zircons,weobtaintheageofthe perturbation (Figure 3.8).
71 Rich systems
T
1
T
0
T
1
T
0
206
Pb*/
238
U
206
Pb*/
238
U
206
Pb*/
235
U
WETHERILL'S MODEL
Uranium loss
at T
1
Lead
loss
at T
1
TILTON'S MODEL
207
Pb*/
235
U
C
o
n
c
o
r
d
i
a
C
o
n
c
o
r
d
i
a
Continuous
lead
loss
a
b
Figure 3.7 Theoretical diagrams of concordia construction for
238
U–
206
Pb and
235
U–
207
Pb systems. (a)
The episodic loss model (Wetherill’s model) at
T
1
for zircons of age
T
0
; (b) the constant loss model
(Tilton’s model) by diffusion for the same zircons of age
T
0
. The parametric curves are plotted on each
diagram. The ratios are given in number of atoms.
C
o
n
c
o
r
d
i
a
C
o
n
c
o
r
d
i
a
207
Pb*/
235
U
1.0
0.8
T = 1850 Ma
1
2
3
4
T
0
= 3550 Ma
MORTON GNEISS CONTINENTS
0.6
0.4
0.2
8
207
Pb*/
235
U
206
Pb*/
238
U
0.6
2 500
2 000
1 500
1 000
100
50
10
25
x 10
–12
yr
–1
= D/a
2
Europe
Africa
North America
Asia
Australia
0.5
0.4
0.3
0.1
0.2
5 10 15
16 24 32 40 48
500
a b
Figure 3.8 Concordia plots. (a) The Morton gneiss of Minnesota; (b) different continental zircons of
common age 2.7 Ga. These plots can be interpreted as constant losses although they have different
geological histories.
72 Radiometric dating methods
Wet her i l l ’s model has been used abundantly. Generally it is borne out but it has often
been observed that the lower intercept is di⁄cult to interp ret in geologic al terms because
the age d etermined corresponds to neither a known tectonic nor a m etamorphic episode.
Thus, taki ng zircons of several continents of the same age of 2.7 Ga, and plotting the
U^Pb results on the concordia diagram, they seem to de¢ne the same lower intercept,
while the geological histories of the various parts of the continents and in particular the
tectono-metamorphic episodes later than 2.7 Ga are very di¡erent.To acc ou nt for this, in
the 1960s Ge orge Ti lton, at the Carnegie Institution, developed another model (Tilton,
196 0). He assumed a continuous loss of l ead throughout the rock’s history. In such a case,
the ¢rst part of the curve of evolution in the concordia diagram is a straight line starting
from the initial age. Itthen curves towards the origin. From this mo del, the lower intercept
isgeologically meaningless and depends solelyon the initial age.This canbe demonstrated
mathematicallyby using kinetic equations. Ifwe consider a continuous leadloss, the equa-
tions are written:
dPb
dt
¼ lU G Pb
dU
dt
¼lU
(
where G is a coe⁄cient of lead loss which is assumed constant to simplify matters (we
assume thereis nouranium loss).Therefore:
d
Pb
U
dt
¼ l þðl GÞ
Pb
U
:
BypositingPb*/U ¼randbyintegrating r
8
and r
5
, we obtain:
r
8
¼
l
8
l
8
G
e
ðl
8
GÞt
1
; r
5
¼
l
5
l
5
G
e
ðl
5
GÞt
1
:
Nu m er i cally, it can be s een that in an (r
8
, r
5
) plot supposing G is identical for both
lead isotopes, the curve is a straight line in its ¢rst part and only curves towards the ori-
gin. It therefore has no signi¢cant lower intercept. Only th e initial age is signi¢cant. If
a straight line is drawn through the data points, the upper interse ction with the concor-
dia indeed gives the initial age but the lower intersection with the concordia is
meaningless.
Alle
'
gre, A lbare
'
de, Gr u« nenfelder,andKo
«
ppel (1974) showed that we could switch from
one modeltotheother. Ifseveral tectono-metamorphic crises aresuperimposed, they may
also generate a discordiawhose lower intersection with the concordia is meaningless.This
is the case ofold inherited zircon s in the Alps. A complexpolymetamorphic h istorygener-
atesregularities similar to acontinuous loss.
As seen, interpreting the lower intercept is neither straightforward nor unequivocal!
Each case must be carefully examined, that is, one must have sound geologic al knowledge
of the region before reaching any conclusion. However, the upper intercept, when well
de¢ned mathemati cally(space d datapoints), seems more robust.
73 Rich systems
Of course, an ‘‘apparent age’’ can be determined for each discordant mineral by the
207
Pb/
206
Pb method alone.These apparent
207
P/
206
P ages vary from the age of the lower
intersectiontotheinitial age.Wethereforehaveawholeseries ofapparent ages.
2
And yet the chronometric equation is misleading a priori since if we c onsider only the
207
Pb^
206
Pb dating equation and suppose that lead islost withoutisotopic fractionation, it
is easy to believe the system has remained closed for lead! This is what was thought some
50 years agowhen some workers asserted thatsince the same granite had
207
Pb/
206
Pb ages
ranging from s ay 200 million to1billionyears, thatwas evidence ithad formed bydi¡usion
in solid state over hundreds ofmillions ofyears! In fact, the dati ng equation contains a hid-
denvariable ^ uranium! Thetrue equationofthe
207
Pb/
206
Pb ratio,i n the case ofaepisodic
loss,iswritten:
207
Pb
206
Pb
¼
1
137:8
1
1
e
l
5
T
e
l
5
T
1
þ e
l
5
T
1
1
1
1
e
l
8
T
e
l
8
T
1
þ e
l
8
T
1
1
2
6
6
4
3
7
7
5
:
This equation shows thatthe ratiodepends on the coe⁄cients ofloss and (unlessT
1
¼0,
that is, iftheloss is atthe present day).This also illustrates that apparent and true ages must
not be mi xed without care! Ofcourse, it is tempti ng to measure the
207
Pb/
206
Pb ratio alone
becauseitrequiresjust an isotope measurementofleadwithout measuringtheleadandura-
n ium content, but, as has be en seen, its interpretation is ambiguous. It must therefore be
used by taking certain pre cautions andbybeing familiar withthelimits.
Exercise
We assume a continuous loss with
G
¼2
l
8
. Calculate the apparent
206
Pb/
238
U,
207
Pb/
235
U,
and
207
Pb/
206
Pb* ages for a zircon whose initial age is 2.7 Ga.
Answer
206
Pb/
238
U ¼1.8 Ga,
207
Pb/
235
U ¼2.18 Ga, and
207
Pb/
206
Pb* ¼2.480 Ga.
Generalizingthe concordia method
Alle
'
gre(1964)andAlle
'
greandMichard(1964)extendedthisapproachtootherpairsof
chronometers (Figure 3.9).
Uran i u m/l ead ^tho r ium/l ead.This was anatural extension.T hebig di¡erence was that
the Th/U ratio is variable and so the ‘‘magical’’conditions of the U^Pb system do not
hold.The discordia straight line cuts the concordia at the upper intercept only.There is
nolowerinterceptasthediscordiaisactuallyacurve(SteigerandWasserburg,1966;
Alle
'
gre, 1967) (Figure3.9).
Rubidiu m /strontium^potassium /argon.Hereagainthe discordiacutsthe concordiaat
one point only.This generalization has con¢rme d the coherence of the two systems and
theirbehaviorwhendisrupted(Alle
'
greandMichard,1964).
2
Apparent ages are measured by the slope of the straight line joining the origin and the experimental point
on the (r
5
, r
8
) plots.
74 Radiometric dating methods
Uranium ^
23 0
Th,U^
231
Pa.This approach has been used for dating secondary mineral-
izations of uranium. It was then extended recently to other examples of dating corals.
These are mentioned herebutnotelaboratedon (Alle
'
gre, 1964).
Su ch general i z ation of t he con cord ia m etho d is ab ove al l offu n dam ental meth o dolog ical
intere st . It sh ows that th e behav ior of the var io us radioge n ic is otope s , altho ugh or ig i nal,
is not ‘‘autonomous’’ and that th ere is redun dancy which explains the regu lar itie s
obs e r ved and allows si mple mathe matical model i n g.
3.2.3 Stepwise thermal extraction
Theid eabeh ind this approach istosupposethat,when disruptivephenomenaoccur, radio-
genic isotopes migrate by di¡usion towards the mineral boundaries or to cracks and that
thereare‘‘cores’’ which have resisted andwhich canbe considered to be closed boxes.We go
0.90
231
Pa*/ U
Chalcocites
CONCORDIA
230
Io-
231
Pa
Autunites
about
100 000 years
230
Th*/U
87
Sr*/
87
Rb
207
Pb*/
235
U
Ontario and Minnesota
40
Ar*/
40
K
208
Pb*/
232
Th
CONCORDIA
0.5
C
o
n
c
o
r
d
i
a
Concordia
2600
500
1000
1500
2000
2250
2500
2750
3000
2 660
2460
2200
2000
a
b c
Figure 3.9 Generalization of the concordia method. (a) U–Th–Pb concordia; (b) Concordia applied to
methods of radioactive disequilibrium; (c) concordia applied to K–Ar, Rb–Sr methods. After Alle
`
gre
(1967), Alle
`
gre (1964), and Alle
`
gre and Michard (1964).
75 Rich systems
and g et the ‘‘information’’ by causing arti¢cial di¡usion.Two chronometers have been the
subject of such an approach: that of the K^Ar method on various minerals or rocks and
thatofthe U^Pb methodon zircon.
The
39
Ar^
40
Armethod
This method ¢rst came to light at Berkeley in the laboratory of John Reynolds (Merr i hue
and Turner, 1966) but was above all the product of work by Grenville Turner (1968)at
She¡reld University i n England. Here is what it involves. First the sample under test is irra-
diated by a stream of neutrons. A nuclear reaction transforms the
39
K into
39
Ar.Thisisa
n eutron^proton (n, p) reaction. Once this operation has been accomplished, the test sam-
ple is heated by temperature stages and the
40
Ar/
39
Ar ratio measured at each step
(Figure3.10).
Naturally, at the same time as the sample, we irradiate a control sample containing a
k nown quantityofKwhose
39
Ar contentis tobeanalyzed.Th e
40
Ar/
39
Ar ratiois equivalent
to an
40
Ar/
40
K ratio and so the dating formula can be applied to it.We therefore obtain a
rangeof (apparent) ages dependingon theoutgassing temperature.
ΔT1 ΔT2
ΔT
Closed system
Open system
MonodomainPolydomains
Computed ages
Temperature (°C)
Computed ages
Thermal
event
just before Tt just after Tt
40
K
40
K
40
K
40
Ar
40
K
40
Ar
40
Ar
40
K
40
Ar
40
K
40
Ar
40
Ar
0–d +d 0–d +d 0
100 800
Temperature (°C)
100 800
–d +d
0–d +d 0–d +d
0–d +d
ΔT
Tt
today
Figure 3.10 The
39
Ar–
40
Ar method. Each small rectangle represents horizontally the size of a mineral
(the center has coordinate zero and the edges þ
d
,–
d
). The ordinates show the
40
K and
40
Ar contents.
Top: evolution over time in a closed system. Notice that
40
K decreases and
40
Ar increases. When the
experiment is conducted we find a simple clear plateau indicating the age. Bottom: the evolution of an
open system. We suppose a thermal event occurred at
T
t
, thermal diffusion occurred, and argon escaped
from the rims, hence the bell-shaped distribution. Then the system remained closed and
40
K decayed
producing
40
Ar. When the argon is extracted stepwise, it can be seen that a certain temperature is
required to reach the plateau because the edges are degassed first. The bottom plot is analogous except
we have supposed a mineral with a complex structure with several domains separated by cracks, faults,
or grain joints. The first part of the extraction has a complex appearance.
76 Radiometric dating methods