Chapter 2
K-Ar, Rb-Sr, Sm-Nd, Lu-Hf and Re-Os
Figure 2.1
Decay Systems
 Geochronologists have a variety of decay systems to work
with. Each is in some ways unique. Success in dating
depends on choosing the right tool. Factors to consider:
 Half-life: shorter half-live better for younger samples
 Chemical behavior of parent and daughter: specifically
 abundance
 fractionation of parent and daughter in different minerals
 mobility of parent and daughter
 ‘closure temperature’ of system: at what temperature is the
clock reset?
Basic Equations
 Basic Equation
dN
= -l N
dt
 Rearrange and integrate:
t
dN
=
òN0 N ò0 -l dt
N
ln
N
= lt
N0
N = N 0 e- l t
 Half-life:
1
ln 2
ln = lt1/2 t1/2 =
2
l
Case where some daughter
already present
 D, daughter; D* =N0 – N, radiogenically produced
daughter
D* = N0 - N = Nelt - N = N(elt -1)
 Since some daughter is initially present (D0):
D = D0 + N(elt -1)
 Its more convenient to work with ratios than absolute
numbers, so we divide by the amount of another, nonradioactive, non-radiogenic isotope. For Sr:
87
Sr / 86 Sr = ( 87 Sr / 86 Sr)0 + 87 Rb / 86 Sr(elt -1)
Isochron Dating
 In the general case:
R = R0 + RP/D (elt -1)
 we have two unknowns, R0 and t (we measure R and R0).
 We need two equations. If we have two samples with
(presumably) identical R0 and t, subtracting one from the
other, we have:
 Rearranging:
∆R
= elt-1
∆ RP/D
∆ R = ∆ RP/D (elt -1)
or
t=
ln(∆ R /∆ RP/D +1)
l
Isochrons
 From equ. 2.20, we see that t is
proportional to the slope on a
plot of R vs. RP/D.
 Looking at equation 2.19 with
this in mind:
R = R0 + RP/D (elt -1)
 we see that it has the form y=a
+xb, where a is the intercept and
b the slope.
 So we can determine the age by
determining the slope through a
set of cogenetic samples.
 This line is called an isochron.
Assumptions
 We are assuming each sample analyzed has the same
value of R0 and t.
 In other words, they are cogenetic: they formed at the same
time with the same isotope ratio at the time.
 The latter requires isotopic homogeneity (isotope equilibrium).
Typically achieved by diffusion (± convective mixing), requiring
elevated temperature. Hence we are usually dating thermal
events.
 We also assume the only change in R and RP/D is due to
radioactive decay. In other words, we assume the system
has remained closed, no migration of parent or daughter in
or out (we’ll see some ‘work-arounds’ for this).
Additional Considerations
 In addition to a closed system that was initially
isotopically homogenized, an accurate date requires:
 Large amount of radiogenic isotope to have been
produced (essentially requiring a large parent/daughter
ratio)
 For isochron dating, we want a large range of parentdaughter ratios, which minimizes the uncertainly on the
isochron.
Determining the slope
 Relations between observations are commonly determined using
regression (commonly included in calculators, Excel, etc.).
 Classical regression assumes x values are known absolutely – not the
case with analytical data.
 We should take errors in both x and y into account in computing our
slope – this is done by weighting each point inversely by its
associated analytical error.
 Known as two-error regression, mathematically a bit more complicated and
requires iteration.
 In practice, many geochronologists use the Isoplot Excel add-in from
the Berkeley Geochronology Laboratory (unfortunately, latest version
runs only on Excel for Windows).
 http://bgc.org/isoplot_etc/isoplot.html.
The K-Ca-Ar System

40K

The ratio of electron captures to beta decays is called
the branching ratio and is defined as:
is can decay to either 40Ca (by β– decay) or 40Ar
by electron capture (or more rarely β+ decay).
R=
lec
lb

Total decay constant, sum of these two, is 5.5 x 10-10
yr-1 corresponding to a half-life of ~1.28 Ga.

Most (~90%) decays to 40Ca, but 40Ca is doubly
magic and a very abundant nuclide. Thus the
radiogenic fraction is small. 40Ar, on the other hand is
a rare gas and fairly rare (on the Earth anyway). Thus
mostly we are interested in the decay to 40Ar.

The usually large K/Ar ratio in rocks and the relatively
short half-life makes this a good choice for many
dating applications, particularly young events.

Because Ar is a gas and mobile (K also readily
mobilized), the system is readily reset. This can be a
good thing for dating low temperature events.
K-Ar Dating
 Our relevant decay equation is:
40
Ar* =
le 40
K(elt -1)
l
 where λis the total decay constant.
 Most rocks have little Ar; lavas, for example, almost
completely degas. What little Ar is present is generally
adsorped atmospheric Ar, whose isotopic composition
is well known. Thus our equation becomes:
Ar
l
= 295.5 + e
36
Ar
l
40
K lt
(e -1)
Ar
40
36
Diffusion, Cooling and
Closure Temperature
 We mentioned radiogenic chronometers are generally reset
by thermal events.
 This occurs when diffusion is sufficiently rapid to isotopically
homogenize our system.
 Or, the case of K-Ar, Ar is able to diffuse out of the rock.
 The temperature at which the chronometer is reset is known
as the closure temperature. It differs for each decay system,
mineral, and, as we’ll see, cooling rate.
 Let’s first consider diffusion.
Temperature Dependence of
Diffusion

The diffusio flux is given by Fick’s
first law:
æ ¶C ö
J = -D ç
è ¶x ÷ø


Diffusion coefficient in solids
depends on temperature according
to:


where D is the diffusion coefficient
and ∂C/∂x the concentration gradient.
- E A / RT
D
=
D
e
0
is the activation
energy and D is
EA
the frequency factor.
We can determine these by making
measurements at multiple
temperatures, taking the log of the
above equation, then plotting up the
results.
Ar in biotite
Figure 2.2
Closure Temperature
 Using the data in the previous figure, we would find there is
no significant loss of Ar at 300˚C even on geologic time
scales. At 600˚C, loss would be small but significant. At
700˚C, about 1/3 of Ar would be lost from a 100 µ biotite in 2
to 3 weeks!
 If the rock cools rapidly from 700˚C, it will quickly close. If it
cools slowly, closure will come much later.
 Think about Ar in a cooling intrusive igneous or metamorphic
rock. Unlike a lava, cooling will occur on geologic time
scales. At first, most Ar is lost, but as the rock cools, loss
slows.
 What is the closure temperature?
Diffusion Calculations
 To determine the distribution of a diffusion species with
time c(x,t), we use Fick’s Second Law:
æ ¶2 c ö
æ ¶c ö
=
D
çè ÷ø
çè ¶x 2 ÷ø
¶t x
 Solutions depend on circumstances. Easy way to solve
it is to look in Crank (1975) who gives this equation for
diffusion out of a cyclinder of radius a
4 æ Dt ö
f @ 1/2 ç 2 ÷
p èa ø
1/2
Dt
1 æ Dt ö
- 2 - 1/2 ç 2 ÷
a
3p è a ø
 where ƒ is the fraction lost
3/2
Ar loss from biotite
Figure 2.3. Fraction of Ar lost from a 150 µ cylindrical crystal as a
function of temperature for various heating times. All Ar is lost in 10
Ma at 340°C, or in 1 Ma at 380° C.
Dodson’s Closure
Temperature
 Dodson (1973) derived an equation for ‘closure temperature’ (also
sometimes called blocking temperature) as a function of diffusion
parameters, grain size and shape, and cooling rate:
Tc =
EA
æ ARTc2 D0 ö
R ln ç - 2
÷
è a E At ø
 where Tc is the closure temperature, τ is the cooling rate, dT/dt
(for cooling, this term will be negative), a is the characteristic
diffusion dimension (e.g., radius of a spherical grain), and A is a
geometric factor (equal to 55 for a sphere, 27 for a cylinder, and 9
for a sheet) and temperatures are in Kelvins.
 Unfortunately, this is not directly solvable since Tc occurs both in
and out of the log, but it can be solved by indirect methods
(MatLab, Solver in Excel).
40Ar–39Ar
 You might wonder what this is all
about. 39Ar has a 269 yr half-life
and does not occur naturally.

40Ar–39Ar
dating is simply a
specific analytical technique for
40K–40Ar dating.
 The sample is irradiated with
neutrons in a reactor and 39Ar is
created from 39K by: 39K(n,p)39Ar.
 Since the amount of 39Ar is
proportional to the amount of 39K
and that is in turn proportional to
the amount of 40K, the 39Ar/40Ar
ratio is a proxy for the 40K/40Ar
ratio.
Dating
40Ar–39Ar
Technique
 The amount of 39Ar produced is a function of the amount of
39K
present, the reaction cross-section (analogous to the
neutron capture cross-section), the neutron flux, neutron
energies, and the irradiation time:
39
 The
40Ar*/39Ar
Ar = 39 Kt ò fes e de
Ar * le
=
39
Ar
l
40
is then:
40
39
K(elt -1)
Kt ò fes e de
 This is way too much nuclear physics for simple geochemists.
The trick is to combine several of these terms in a single term,
C:
C=
le
1
l t f s de
ò e e
 then determine C by irradiating a ‘standard’ of known age and
solving this equation for C:
Ar *
=C
39
Ar
40
40
K ( elt -1)
39
K
Advantages
 The parent-daughter ratio can be determined simply by
determining the isotopic composition of Ar in the
irradiated sample (rather than having to separately
measure K).
 Ar can be extracted from a sample simply by heating it
in vacuum. This can be done in temperature steps,
allowing for multiple cases and multiple isotope ratios.
 In fact, this is what is typically done.
 Even newer techniques involve spot heating with a
laser, allowing for high spatial resolution.
A Textbook Plateau
Figure 2.4. Here, there is been some diffusional loss from
the edges of the biotite, giving a younger age, but
subsequent temperature steps all give the same age.
Partial reseting in contact
metamorphic aereole
Figure 2.5
Figure 2.6. Ar release spectrum of a hornblende in a Paleozoic gabbro
reheated in the Cretaceous by the intrusion of a granite. Anomalously old
apparent ages in the lowest temperature release fraction results from
diffusion of radiogenic Ar into the hornblende during the Cretaceous
reheating.
Figure 2.7. Ar release spectrum from a calcic plagioclase from Broken Hill,
Australia. Low temperature and high temperature fractions both show
erroneously old ages. This peculiar saddle shaped pattern, which is
common in samples containing excess Ar, results from the excess Ar being
held in two different lattice sites.
40Ar-39Ar
Isochrons
 The data from various
temperature release steps are
essentially independent
observations of Ar isotopic
composition. Because of this,
they can be treated much the
same as in conventional
isochron treatment.
 Since for all release fractions
of a sample the efficiency of
production of 39Ar from 39K is
the same and 40K/39K ratios
are constant, we may
substitute 39Ar × C for 40K:
Ar æ 40 Ar ö
=
+
36
Ar çè 36 Ar ÷ø 0
40
Ar
C(elt – 1)
36
Ar
39
Figure 2.8
Inverse Isochrons
 The problem with that
approach is 36Ar will not be
very abundance and there will
be a relatively large error in
measurement - not something
we want when it occurs as
both denominators.
 We can invert that ratio and
plot it vs 39Ar/40Ar.
 The x intercept is then the age
(case of no trapped Ar) and
the y-intercept gives the
isotopic composition of
trapped Ar.
Figure 2.9
Two Trapped Components
after correction for inherited Ar
Figure 2.10
Rb-Sr System

Rb: alkali; soluble, mobile, highly incompatible, substitutes for K

Sr: alkaline Earth; soluble, somewhat mobile, incompatible,
substitutes for Ca

Both concentrated in the Earth’s crust; particularly Rb. High Rb/Sr in
granitic rocks and their derivatives.
Sr chronostratigraphy

Sr present in relatively high
concentration is seawater.

Also concentrated in carbonates
(abundant marine bio-sediment).

Long residence time; therefore
 Sr isotopic composition of open
ocean water is uniform (in space).

Sr isotope ratio varies in time,
mainly due to changes in the relative
fluxes from the continents (erosion)
and mantle (ridge-crest
hydrothermal activity).

Particularly in the Tertiary, Sr isotope
ratio of marine sediment can be
used to date the horizon..
Figure 2.12
Sm-Nd System
Figure 2.13
Sm-Nd characteristics
 Long half-life (~100 Ga).
 Sm and Nd both rare earths elements (REE), similar
behavior, typically small variation in Sm/Nd
 generally more variation in mafic igneous rocks that granitic
ones and their derivatives
 garnet has quite high Sm/Nd.
 REE behavior well understood.
 Both form 3+ ions and are quite insoluble and immobile.
 High closure temperature.
Cosmic Rare Earth
Abundances
Figure 2.14
Normalized abundances
Figure 2.15
Ionic Radii and Partition
Coefficients
The Epsilon Notation
 Because Sm and Nd are refractory lithophile elements
(condensed at high T in solar nebula and partition into
silicate part of the planet) and because the Sm/Nd
differs little in nebular materials (i.e., chondritic
meteorites), it was assumed the Earth had a chondritic
Sm/Nd ratio and therefore that the evolution of
143Nd/144Nd in the Earth should follow that of
chondrites. Furthermore, variations in 143Nd/144Nd are
small. Thus the ε notation was introduced by DePaolo
and Wasserburg (1976):
e Nd
é 143 Nd / 144 Ndsample - 143 Nd / 144 Ndchondrites ù
=ê
ú ´10, 000
143
Nd / 144 Ndchondrites
ë
û
Nd isotopic evolution of the Earth
 CHUR: “Chondritic Uniform
Reservoir” – (silicate Earth if
it is chondritic)
 Mantle is Nd depleted
relative to Sm, has high
Sm/Nd, evolves to high εNd.
 Continental crust is Nd
enriched relative to Sm, has
high Sm/Nd, evolves to
lowεNd.
Figure 2.16
Figure 2.17. Garnet bearing granulite from Dabie UHP metamorphic
belt in China.
Crustal Residence Times

Basic idea is that there is a relatively large fractionation between Sm and Nd during
melting to form new additions to crust. Subsequent crustal processing produces little
change in Sm/Nd.

Consider our isochron equation:
143

Nd /144 Ndsam =143 Nd /144 Nd0 +147 Sm /144 Ndsam (elt -1)
Since λt << 1, we can use the approximation that for x<<1, e x ≈ x+1 and linearize this
equation:
143
Nd /144 Ndsam @143 Nd /144 Nd0 +147 Sm /144 Ndsam lt

On a plot of 143Nd/144Nd vs. t, the slope = 147Sm/144Ndλ

We want to know t assuming 143Nd/144Nd0 is the mantle value at the time the material
was added to crust.

We project back along the slope defined by
the mantle evolution curve.
147Sm/144Nd
to the point of intersection on
Sm-Nd Model Ages (aka
Crustal Residence Times)
Figure 2.18
Lu-Hf System
Lu-Hf in Chondrites
Figure 2.19
Lu-Hf System

176Lu

Decays to 176Hf with half-life of 36 Ga
(possible it might also decay to 176Yb, but
extremely infrequently – not
demonstrated.

Lu and Hf both refractory lithophile
elements -implies silicate Earth has
chondritic Lu/Hf (?).

Lu slightly incompatible, Hf moderately
incompatible.

Lu has 3+ valance state, Hf 4+ valance
state, both quite insoluble and immobile.

Strong chemical similarity of Hf to Zr;
therefore Hf strongly partitioned into
zircon (ZrSiO4) – a highly resistance
accessory mineral in many crustal rocks.
is another odd-odd nuclide.
epsilon Hf notation

ratio commonly represented as εHf – exactly
analogous to εNd:
176Hf/177Hf
e Hf
é 176 Hf / 177 Hfsample - 176 Hf / 177 Hfchondrites ù
=ê
ú ´10, 000
176
177
Hf / Hfchondrites
ë
û
Hf in the crust
 Unlike Sm/Nd, the Lu/Hf ratio does change significantly during weathering and
other crustal processes – mainly related to zircon.
 Zircon is extremely resistant chemically and physically and when weathering
occurs will go into the sand fraction (taking Hf with it).
 Lu will mainly go in the clay fraction.
 Thus sedimentary processes act to fractionate Lu/Hf.

Thus sediments sometimes deviate from the otherwise strong correlation between
εHf and εNd.
 So there is no analogous Lu-Hf crustal residence time.
 On the other hand, zircons have very low Lu/Hf, so preserve, or nearly so,
their initial εHf. This together with U-Pb ages of zircons provides analogous
information on provinance of sediments.
Figure 2.20
Lu-Hf in dating
 Lu and Hf are immobile - good for dating older rocks.
 Lu half-life relatively short.
 Lu/Hf ratio more variable that Sm/Nd.
 Consequently, larger variation in Hf isotope ratios than
Nd isotope ratios.
 Lu very strongly concentrated in garnet, so this system
again very useful for dating garnet-bearing rocks.
Figure 2.21
Re-Os

187Re

decays to 187Os with a half-live of 42 Ga.
Also a decay of 190Pt to 186Os with very long half-life (450 Ga), so that resulting
variation is usually not detectable
 Unlike most elements we’ve considered, Re and Os are siderophile (and
also chalcophile) meaning they are concentrated in the Earth’s core.
 Consequently, Re and Os concentrations in the crust and mantle are very low
(ppb and usually lower).
 Within the silicate part of the Earth, Os behaves as a very compatible
element (hence remains in the mantle with very low concentrations in the
crust), while Re is moderately incompatible and concentrates in the crust.
 Large variation in Re/Os leads to quite large variations in 186Os/188Os.
Gamma notation
 In a manner somewhat analogous to the epsilon
notation, Os isotope ratios are often reported as γOs :
percent deviations from Primitive Upper Mantle:
g Os
æ 186Os / 188Ossam - 186Os / 188OsPUM ö
=ç
186
÷ø ´100
Os / 188OsPUM
è
Figure 2.22
Figure 2.23: Re/Os isochron for a komattite from
Munro Township, Ontario.
Re-Os dating of diamond
inclusions
Figure 2.24: sulfide inclusion in diamond
Tracing Os isotope evolution of seawater
from hydrogenous sediments and black
shales.
Figure 2.26
Dating Hydrocarbon
Generation (?)
Figure 2.27