Braun J., van der Beek P., Batt G. Quantitative Thermochronology: Numerical Methods for the Interpretation of Thermochronological Data

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28 Basics of thermochronology: from t–T paths to ages

The thermochronological ‘age’ of a sample is obtained from the ratio of the

concentration of the daughter element, N

, and the production rate P (which

depends on the concentration of the parent element, as described in Section

2.1) integrated over the sample volume. One can therefore write the following

evolution equation for the apparent age Ar t =N

r t/P:

Ar t

t





Ar t

r

Ar t

r



+1 (2.23)

This equation can be integrated through time for a given thermal history Tt,

which determines the value of the diffusivity DT. The initial value Ar 0 =

corresponds to the initial age of the grain, i.e. before the thermal history

determined by Tt begins. The boundary condition is A1t=0. The age of the

grain is obtained by spatially averaging Ar t over the volume of the grain (see

Appendix 6):

age =3



Ar tdr (2.24)

At low temperature, the diffusivity is small (D →0) and the evolution equation

(2.23) becomes

Ar t

t

= 1 (2.25)

which shows that, everywhere in the grain, age increases linearly with time. At

high temperature, the diffusivity is large (D →) and no daughter isotopes

accumulate within the grain (Art/t = 0). Equation (2.23) becomes



Ar t

r

Ar t

r

= 0 (2.26)

implying that no gradient in age is permitted within the grain.

The finite-difference formulation

Our purpose is to find the solution to (2.23) at a finite number of points along

the radius of the spherical grain, r

= i −1r, for i = 1n. The solution

at r

is denoted A

. Derivatives with respect to r are approximated by a centred

difference:

A

r

r

 =

i+1

−A

i−1

2 r



r

r

 =

i+1

−2A

i−1

r

(2.27)

2.5 Numerical solution 29

The time derivative is approximated by

A

t

= 

A

t+t

t

+1−

A

t

(2.28)

The factor  will determine whether we use an explicit ( = 0) or implicit

( = 1) time-marching method. As we will see below, an explicit method is

rather efficient because it does not require the solution of a large system of

algebraic equations. The equations are not coupled, i.e., each equation has only one

unknown. An implicit method leads to a system of coupled equations that require

the solution (or inversion) of a tri-diagonal matrix system. Explicit methods can

be unstable (i.e., the solution can diverge if the time step is not made sufficiently

small); implicit methods are always stable. Both require time steps shorter than

a critical length in order to provide an accurate solution. Intermediate values

of  correspond to explicit–implicit schemes that are, in general, both stable

and accurate (see Belytschko et al. (1979) for a complete description of mixed

time-integration methods).

The solution at t +t is expressed as a Taylor expansion:

t+t

= A

+t

A

t

+Ot

 +··· (2.29)

in which we neglect the terms in t

and higher orders to obtain

t+t

= A

+t





A

t+t

t

+1−

A

t



(2.30)

which, using (2.23), leads to the following finite-difference form of the evolution

equation:

t+t

= A

+t

t+t



t+t

i+1

−2A

t+t

i−1

r

t+t

i+1

−A

t+t

i−1

2 r



+1−t



i+1

−2A

i−1

r

i+1

−A

i−1

2 r



+t (2.31)

We see now that, in the case for which  = 0, i.e. the method is explicit, the

value of each A

t+t

can be directly obtained from the values of A

at i −1, i and

i +1; there is no need to find the solution of a large system of coupled algebraic

equations.

When  = 0, the solution can be obtained only by finding the solution of a

tri-diagonal system of algebraic equations. Each equation can be re-written as

t+t

i−1

t+t

i+1

= b

 for i =2n−1 (2.32)

30 Basics of thermochronology: from t–T paths to ages

where, noting that r/r

= 1/i −1,

=−

tD

t+t

r



i −1



= 1 +

2tD

t+t

r

=−

tD

t+t

r



1−

i −1



= t +

1−t D

r



i+1



i −1



−2A

i−1



1−

i −1



(2.33)

The boundary condition, Ar = a = 0, and radial symmetry,

A

r

r =a =0

lead to

=−1D

= 1 and b

= 0

= 0D

= 1 and b

= 0

(2.34)

These equations form a tri-diagonal system

LDUA = b (2.35)

which can be solved by double backsubstitution (Press et al., 1986) and marched

through time.

The age is obtained by averaging the final solution over the volume of the grain

using the trapezoidal rule:

age = 3



Ar tdr =3



i=1

r

i −1



05 for i = 1n

1 otherwise

(2.36)

2.6 Determining the diffusion parameters

The parameters entering the Arrhenius relationship (2.9), i.e. D

and E

, are

established by controlled-heating experiments in the laboratory. Because diffusion

obeys the Arrhenius relationship, experiments at high temperatures over short

times in the laboratory provide insight into diffusive behaviour at lower tempera-

tures and correspondingly longer timescales in natural settings, as long as we can

2.6 Determining the diffusion parameters 31

be confident that the same diffusional processes and pathways operate in both

cases.

During laboratory experiments, a sample of the mineral of interest is held at

various temperatures for a fixed amount of time and the amount of gas extracted

from the sample at each step is measured, from which DT/a

can be estimated.

If the Arrhenius relation (2.9) is obeyed, then a plot of lnDT/a

 against recip-

rocal temperature 1/T should produce a straight line with gradient −E

/R and

intercept lnD

 (cf. Figure 3.5). For thermochronological systems in which

the diffusion domain size a corresponds to the physical grain size (e.g., for the

(U–Th)/He system, cf. Section 3.2), the diffusivity D

of a thermochronological

system can be estimated by repeating the measurements on grains with various

radii. For systems in which the diffusion domain is smaller than the grain size

(e.g., for most Ar–Ar systems, cf. Section 3.1), D

has to be estimated jointly

for each sample. The constraints imposed by such experiments on the diffusion

parameters and closure temperatures of the most widely used low-to-medium-

temperature thermochronological systems are outlined in Chapter 3.

Because these experiments are necessarily performed at higher temperatures

and on much shorter timescales than those of natural diffusion (several minutes

to hours in the laboratory versus 10

–10

years in Nature), their results have to

be extrapolated over many orders of magnitude. Small errors in the analytical

procedure may therefore result in large uncertainties in the predicted diffusion

behaviour and closure temperature over geological timescales. Furthermore, there

is no a-priori evidence to support the hypothesis that the same diffusion pathways

and mechanisms operate on both timescales. Therefore, the results of the labora-

tory experiments need to be calibrated against natural settings in which thermal

histories can be particularly well constrained, for instance by analysing samples

from boreholes with relatively well-known temperature histories.

Tutorial 1

Use the closure temperature given in Table 1.1 to predict the (U–Th)/He apatite

ages for rocks that have experienced the following cooling histories, all starting

100 Myr ago:

(i) rapid cooling from 500 to 15



C, 40 Myr ago;

(ii) monotonic cooling from 135 to 15



C over 100 Myr;

(iii) rapid cooling from 60 to 15



C 20 Myr ago;

(iv) slow cooling from 100 to 60



C over 25 Myr, isothermal conditions at 60



C for

50 Myr, then slow cooling to 15



C over the last 25 Myr; and

(v) slow monotonic heating from 15 to 65



C during the first 95 Myr of the experiment,

followed by a rapid cooling to 15



C over the last 5 Myr.

32 Basics of thermochronology: from t–T paths to ages

Table 2.1. Diffusion parameters to be used in

Tutorial 1 to determine ages using Dodson’s

method and by numerical integration

Parameter Value

77

−1

15146kJ mol

−1

R 8.314

Use Dodson’s method to determine the ages of the same rocks (the code

Dodson.f is supplied on the accompanying CD). Compare the results with those

obtained by solving the solid-state diffusion equation numerically (use the code

MadHe.f supplied). Take the values in Table 2.1 for the diffusion parameters.

Compare the ages predicted by these three approaches. What can you deduce

from this result?

Thermochronological systems

In this chapter, we review the three main methods used for obtaining

quantitative thermochronological constraints,

Ar −

Ar, (U–Th)/He

and fission-track dating, with the aim of providing an overview of which

analytical tool may be most suited for a given tectonic or geomorphic

problem. The emphasis is not on an exhaustive review of all technical

aspects of these methods: excellent manuals already exist for each, and we

direct the reader to consult these on specific methodological questions.

Instead, we seek to provide an illuminating background to the use of these

techniques for the non-specialist, emphasising the accuracy and repro-

ducibility of data, and common problems that may affect data quality.

3.1 Ar dating methods

Potassium–argon (K–Ar) dating, and the associated

Ar/

Ar method of analy-

sis, are among the most widely used thermochronological methods (McDougall

and Harrison, 1999). Potassium is a major element that makes up about 1.5%

of the Earth’s crust and is abundant in common rock-forming minerals such as

K-feldspars, amphiboles and micas. Potassium has three naturally occurring iso-

topes:

K and

K, which are stable and together make up about 99.9% of the

natural abundance of K, and the radioactive isotope

K (Table 3.1).

The latter decays by the following reactions:

K →

Ar +

0001%

K +

−

→

Ar 103%

K →

Ca +

−

897%

(3.1)

Naturally occurring calcium, another abundant and major rock-forming ele-

ment in the Earth’s crust, is dominated by the isotope

Ca (the natural isotopic

34 Thermochronological systems

Table 3.1. Natural abundances of

K and Ar isotopes

Isotope Abundance (%)

K932581±00029

K001167 ±000004

K67302 ±00029

Ar 99.60

Ar 0.063

Ar 0.337

abundance of which is 969%). Therefore, there are commonly large quantities of

Ca bound up in mineral structures, which are indistinguishable from the radio-

genically produced

Ca. Moreover, the diffusion behaviour of Ca is complex

due to its reactivity. For these reasons, the decay reaction of

Kto

Ca is not

generally used as a dating method, although some authors have utilised this reac-

tion under particularly favourable conditions (low-Ca minerals, cf. McDougall

and Harrison (1999)). Argon, although a secondary constituent of the Earth’s

atmosphere (∼1% by weight), is sufficiently rare in rocks that the radiogenic

component can be precisely measured. In most circumstances it can be assumed

that Ar was not present in the crystal structure at its formation. Because Ar is a

noble gas, it is unreactive and its diffusion behaviour is relatively simple, making

it a particularly effective thermochronometer. On applying Equation (2.7) to the

K–Ar system, the age equation becomes

t =





∗



−1



(3.2)

where  is the total decay constant of

K(5543 ×10

−10

−1

), 

is the decay

constant of

Kto

Ar (0581×10

−10

−1

) and

∗

is radiogenically produced

Ar, that is, the amount of measured

Ar corrected for any potentially extraneous

component. The abundances of both

K and

∗

thus have to be measured in

order to obtain an age.

In the classical K–Ar approach to geochronology, the concentration of K in a

sample is measured by wet-chemical methods or flame photometry, from which

the abundance of

K in a measured aliquot can be deduced through the con-

stant abundance ratios of K isotopes in nature (Table 3.1).

Ar is measured

by fusion of a separate aliquot of the sample, purification of the noble gases

released, and measurement of their isotopic composition in a mass spectrometer.

The measurement of isotopic abundance is calibrated by adding a spike (i.e. a

known amount) of

Ar to the Ar from the sample and measuring the ratio of

3.1 Ar dating methods 35

Ar to

Ar. The presence of trace amounts of atmospheric gas, of which Ar is

the third most abundant component, adhering to the sample as well as interior

surfaces of the extraction system and mass spectrometer, can contribute signif-

icantly to the amount of

Ar measured. A correction to the measured

abundance must therefore be applied to account for this possible contamination.

This is performed through measurement of the stable, non-radiogenic isotope

relative to the

Ar released. The ratio of these two isotopes is constant in the

atmosphere (

Ar/

Ar = 2955%; cf. Table 3.1). Assuming that all

Ar present

is of atmospheric origin, the measured

Ar extracted can be corrected for this

contamination through the relationship:

∗

measured

−2955×

measured

(3.3)

If the system has cooled to temperatures such that diffusive loss of

∗

slower than the increase in

∗

caused by decay of

K, a component of the

∗

formed in the sample is retained over geological timescales, and a finite

‘age’ will begin to accumulate in the sample (cf. Section 2.2). The significance

of ages derived from K–Ar measurements depends strongly on the relative speed

of this cooling and the consequent duration of the transition from open-system to

closed-system behaviour (see below).

Diffusive transport alters the distribution of

∗

within crystals and leads to

its loss from the system.

∗

distributions within a grain population can also

be modified by alteration or recrystallisation, and the incorporation of

Ar-rich

fluid inclusions or back-diffusion into a crystal lattice due to high external partial

Ar pressures can lead to the incorporation of ‘excess’ argon (that is,

Ar from a

source other than in situ decay of

K within a sample, inheritance, or atmospheric

contamination). Neither the extent, nor indeed the occurrence, of these forms of

altered argon distribution can be directly assayed by the bulk extraction techniques

adopted in K–Ar dating, restricting the quantitative application of this approach

to the limited case of samples for which a simple thermal and petrological history

can be assumed.

These limitations are directly addressed by the

Ar−

Ar analytical technique.

As discussed below, the possibility of progressively extracting and analysing

argon with this method provides the potential to resolve the spatial distribution of

argon in a sample and, with it, the corresponding thermal and geological histories

of samples.

The

Ar −

Ar technique has the advantage that no knowledge of the absolute

K and Ar concentrations is required. Instead, the amount of

K is determined by

a proxy, in the form of

Ar produced by irradiating the sample with fast neutrons

in a nuclear reactor.

Ar is produced through the reaction

K +

n →

Ar +

p+Q (3.4)

36 Thermochronological systems

where Q represents the energy released. After irradiation, the argon content of

a sample is extracted by progressive heating under vacuum in a furnace or laser

cell, and, after correction for various potential isotopic interferences, the ratios of

Ar to

Ar released during each step of the experiment are compared to derive

an age. The

Ar/

Ar ratio is determined by



t

−1



(3.5)

where  is a proportionality factor that depends on the neutron dose and the

capture cross-section of

K for fast neutrons. Because  is constant for each

irradiation, and 

/ and

K are also constants, they can be combined into

a single parameter J (known as the irradiation factor), which can be defined as

J =





/

K

(3.6)

A value for J can be estimated by irradiating a mineral standard of known age

together with the unknown samples. In this way, no exact knowledge of the

received neutron fluence is required, since this can be assayed in comparative

terms by employing the relative gas evolution from the unknown sample and the

well-constrained age standard. The age equation then becomes

t =





1+J



(3.7)

Ar can be extracted stepwise by heating the sample over a range of temperatures

below that of fusion, which presents a powerful advantage over bulk extraction

and analysis in that it provides insight into the relative spatial distribution of

Ar.

Ar and

Ar behave essentially identically during extraction; any gas

released during a given heating step thus contains both

Ar and a direct proxy

for the

K content of the correlative area of the sample. Any departure from

uniformity in the ratio of these components during the various heating stages of

the experiment can therefore be read as a modification of the

Ar distribution

expected from the decay of

K alone. By investigating sample behaviour over

a range of temperatures in the laboratory, a pattern of argon release is derived

that can be interpreted geologically (Figure 3.1); either as partial retention of

Ar (lower ages for low-temperature steps) or as the incorporation of excess Ar

(higher ages for low-temperature steps).

Alternatively, the spatial distribution can be mapped directly: Ar can be

extracted from the sample by a spot-fusion technique using a laser coupled to

the extraction line and mass spectrometer, which allows determination of the

∗

Ar ratio in very small samples (Figure 3.2). This makes it possible to date

multiple phases of deformation by spot-dating deformation tails on metamorphic

3.1 Ar dating methods 37

Ar*

Fraction

Ar released

01.0

Apparent Age

Fraction

Ar released

0 0.2 0.4 0.6 0.8 1.0

Apparent Age (Ma)

100

Sample TK 7

K-Feldspar

Fraction

Ar released

Apparent Age (Ma)

120

Sample TK 7

White Mica

Fig. 3.1.

Ar/

Ar age spectra obtained by step-heating experiments. The upper

panel shows the three principal types of spectra that may be obtained: (a) an

ideal flat age spectrum indicating rapid cooling at time t

; (b) a monotonically

rising age spectrum indicating argon loss and either partial resetting at time t

slow cooling from t

to t

; and (c) uptake of extraneous (‘excess’) argon leads

to progressively decreasing apparent ages. Insets show schematic distributions

∗

and

Ar in the sample for these three cases. Lower panels show age

spectra from a sample from the Hohonu Range, South Island, New Zealand. Left:

the K-feldspar spectrum indicates uptake of excess Ar (high ages for the first few

extraction steps) followed by a staircase-like spectrum with ages increasing from

∼16 Myr for low-temperature steps to ∼90 Myr at fusion. Right: the white-mica

spectrum for the same sample is a relatively flat spectrum, with initial ages of

∼85 Myr over the first three heating steps, rising to a plateau at ∼100 Myr

for the remainder of the extraction steps. These data have been interpreted as

indicating rapid cooling from ≥350



C down to ∼250



C at 100–90 Myr (closure

of white mica and beginning of Ar retention in K-feldspar) followed by a second

rapid cooling phase from ∼15–20 Myr ago onwards. Modified from Batt et al.

(2004); data from Reiners et al. (2004).