Contr. Mineral. and Petrol. 40, 259--274 (1973)
9 by Springer-Verlag 1973
Closure Temperature in Cooling Geochronological
and Petrological Systems
M a r t i n I{. D o d s o n
Department of Earth Sciences, The University, Leeds
Received March 5, 1973
Abstract. Closure temperature (Tc) of a geochronological system may be defined as its
temperature at the time corresponding to its apparent age. For thermally activated diffusion
(D = Doe--E/RT ) it is given by
T c = R / [ E In (A ~Do/a2)]
(i)
in which R is the gas constant, E the activation energy, z the time constant with which the
diffusion coefficient D diminishes, a is a characteristic diffusion size, and A a numerical
constant depending on geometry and decay constant of parent. The time constant z is related
to cooling rate by
= R / ( E d T-1/dt) = -- R T~/(Ed T/dt).
(ii)
Eq. (i) is exact only if T -1 increases linearly with time, but in practice a good approximation
is obtained by relating T to the slope of the cooling curve at T c.
If the decay of parent is very slow, compared with the cooling time constant, A is 55, 27,
or 8.7 for volume diffusion from a sphere, cylinder or plane sheet respectively. Where the
decay of parent is relatively fast, A takes lower values. Closure temperatures of 280-300 ~ C
are calculated for R b - - S r dates on Alpine biotites from measured diffusion parameters,
assuming a grain size of the order 0.5 mm.
The temperature recorded by a "frozen" chemical system, in which a solid phase in contact
with a large reservoir has cooled slowly from high temperatures, is formally identical with
geochronological closure temperature.
1. Definition of Closure T e m p e r a t u r e
W h e n t h e " a g e " of a r o c k or m i n e r a l is c a l c u l a t e d f r o m its a c c u m u l a t e d p r o d u c t s
of r a d i o a c t i v e decay, w h e t h e r those p r o d u c t s be c r y s t a l s t r u c t u r a l changes caused
b y fission fragments, or radiogenic isotopes such as sTSr or a~
t h e result i d e a l l y
r e p r e s e n t s a p o i n t in t i m e a t which a c o m p l e t e l y mobile d a u g h t e r p r o d u c t b e c a m e
c o m p l e t e l y immobile. " M o b i l i t y " , in this c o n t e x t , m e a n s either r a p i d diffusion
from t h e l a t t i c e site a t which a r a d i o g e n i c isotope is formed, or v e r y fast annealing of a d i s t u r b e d c r y s t a l lattice. A t one t i m e i t was believed t h a t t h e change
in m o b i l i t y of a radiogenic isotope could a l w a y s b e i d e n t i f i e d w i t h either t h e
c r y s t a l l i s a t i o n of a n igneous r o c k f r o m a melt, or r e c r y s t a l l i s a t i o n during m e t a m o r p h i s m . I n r e c e n t years, however, i t has b e c o m e i n c r e a s i n g l y clear t h a t for
some m e t h o d s of age d e t e r m i n a t i o n , n o t a b l y t h e d a t i n g of s e p a r a t e d minerals
b y t h e R b - - S r a n d K - - A r m e t h o d s , such a simple i n t e r p r e t a t i o n is i n a d e q u a t e ;
radiogenic a r g o n a n d s t r o n t i u m e v i d e n t l y a r e m o b i l e in some m i n e r a l s a t t e m p e r a t u r e s well below t h a t of crystallisation. T h e b e s t evidence for this view
comes f r o m t h e R b - - S r a n d K - - A r age p a t t e r n on micas f r o m t h e c e n t r a l Alps
(J/tger, 1965; J/iger et al., 1967; A r m s t r o n g et al., 1966), for which t h e s i m p l e s t
i n t e r p r e t a t i o n is t h a t closure of t h e R b - - S r a n d K - - A r s y s t e m s occurred d u r i n g
260
M.H. Dodson:
post-metamorphic cooling. A comparable interpretation of age patterns in the
British Caledonides has been presented by Harper (1967), while Armstrong (1966)
developed similar concepts in a review of K - - A r dating of orogenic belts. Workers
on fission track-dating have been led to similar conclusions (Fleischer et al., 1968;
Wagner and Rcimer, 1972), because track loss occurs by annealing of the crystal
structure at rather low temperatures in some minerals.
Fig. 1 shows how calculated ages are related to the real situation in a cooling
radiogenic system. At high temperatures the daughter product escapes as fast
as it is formed, and so cannot accumulate. At low temperatures its rate of escape
is negligible, so that it can accumulate undisturbed. There is a continuous transition from one extreme to the other. Calculation of the apparent age corresponds
to extrapolating the low-temperature portion of the accumulation curve back
to the time axis. The effective closure temperature, Tc, can therefore be defined
as the temperature of the system at the time represented by its apparent age.
The value of Tc will depend on the exact cooling history of a particular system,
but it should be independent of the starting temperature if the latter is sufficiently high.
Thermodynamically we can consider a crystal in which radiogenic isotopes
are accumulating to be out of equilibrium with its environment. The equilibrium
concentration of the radiogenic daughter product may for many purposes be
considered to be zero. Thus, at high temperatures, the loss of isotopes by rapid
diffusion can be regarded as the maintenance of an equilibrium state, and there
is a close resemblance to other petrological situations. For example, oxygen
isotopes in a cooling mineral assemblage commonly record a temperature well
below that at which erystallisation occurred, because at higher temperatures
isotopic equilibrium was continuously maintained. A formal mathematical
similarity between this kind of situation and geochronological closure temperature
is established in Section 5.
Little work has been published on the mathematics of diffusion in a cooling
solid. Gentner et al. (1954) presented a theoretical analysis of argon and helium
diffusion in a slowly cooling cubic crystal, and used the results to interpret
observed relationships between apparent age and grain size in sylvite. Their
primary differential equation was criticised by Amirkhanoff et al. (1961), but
their principal solution appears to be correct. Wood (1964) and Goldstein and
Short (1967) determined cooling rates of iron meteorites from the nickel concentration distributions in neighbouring lamellae of kamaeite and taenite, using finite
difference methods. Damon (1970) discussed the relative importance of radiogenic
production of argon-40 and its loss by volume diffusion in various geological
environments, but did not attempt to analyse the consequences of steady cooling.
In the present paper some simple equations which relate closure temperature
to cooling rates and diffusion parameters are derived, and their application to
palaeo-thermometry is discussed.
2. Foundations of the Theory
Solid diffusion processes arc thermally activated, and often follow the simple
Arrhenins equation.
D ----D Oexp (-- E / R T )
(1)
Closure Temperature in Cooling Systems
261
Fig. 1. Definition of closure temperature. The time tc corresponds to the
apparent age
t
temperature
Tc
t
D/P
tc
time
--~
where D is diffusion coefficient at absolute temperature T, R is the gas constant,
and E is the activation energy for the diffusion process. D o represents the diffusion
coefficient at infinitely high temperature. Because of the very strong temperature
dependence of the diffusion coefficient the transitional temperature range (Fig. 1)
can be expected to be fairly short. Over a limited range of temperature the
cooling history of a geochronological system can conveniently be approximated
by a linear increase in l I T . The change in the loss coefficient with time then
takes the form of exponential decay, whence the mathematical analysis is made
tractable. The exponential decrease in D is conveniently described in terms of
a time constant 3, which is the time taken for D to diminish b y a factor e-z or,
in other words, for E / R T to increase b y 1. Hence we can write, for the time
dependence of D:
D ----D o exp (-- E / R T o - - t/z)
= D (0) e-t/~
(2)
where D (0) and To are the values of diffusion coefficient and temperature at
t-----0. Also, from the above definition of 3, we have
d ( E / R T ) / d t =-- 1/3
---- R / ( E dT-Z/dt)
(3)
= -- RT~/(E dT/dt).
For E = 30 kcal/mole, T = 600 K, and d T / d t = 5 ~ C/My, T is about half a million
years.
Cooling processes in orogenic belts are likely, in general, to be short compared
with the half lives of the geochronologically important radionuclidcs ( > 700 My),
262
M.H. Dodson:
so insignificant errors will be produced by assuming a constant rate of production
of radiogenic daughter isotopes. Studies of meteorite history using xenon-129,
however, may require a more general treatment, since the half life of the parent
1~9I is only 16 My.
I t will be seen that the derived relationships are of the form,
E/R T c ----In (A
T Do/a ~)
where A is a numerical constant depending on geometry and the ratio half-life/%
and a is a characteristic dimension of the system. In Appendix A it is shown
by functional analysis that the above relationship follows directly from the
Arrhenius equation, provided that one accepts the intuitive assumption that
closure temperature is independent of initial temperature. The determination of
A by analytical methods is a laborious process, even for the simple geometries
considered in this paper, and it is likely that numerical methods would be preferable for more complicated situations.
I t is conceivable that loss of radiogenic isotope in certain circumstances may
follow a first-order reaction law, rather than volume diffusion (see, for example,
Hanson and Gast, 1967). Because of its relative simplicity this type of loss is
considered first, and the results are adapted to solve the corresponding volume
diffusion problem. First-order equations are not adequate to describe the process
of fission-track annealing, which will be considered in a separate paper.
Several attempts have been made, in the course of the work, to integrate the
various differential equations by means of the Laplace transformation, but no
way of doing so has yet been found.
3. Closure Temperature for First-Order Loss
Let the concentration of radiogenic decay product be x, and the corresponding
concentration of parent element cp. We consider first the case of c~ constant
(i.e. very slow decay of parent). The net rate of increase of x is given by the
rate of production minus the rate of loss, so that we can write
d x / d t -----)~ c~ - - k (t) x
(4)
where 2~ is the production constant and k(t) the loss coefficient. Assuming
k (t) ----K e-E/~T =/c (0) e-t/~ we obtain
d x / d t ~- k (0) e-tl~ x = ~x %
(5)
which is conveniently integrated by substituting the dimensionless variable v,
namely
v(t) = 7: k(t) = T k(O) e - t / L
(6)
Eq. (5) thus becomes
d x / d v - - x -~ )~ % T/v.
(7)
The solution is obtained by multiplying though by an integrating factor e -y,
and is :
x : 2x % T e v I f (e-V/v) d v + constantS.
(8)
Closure Temperature in Cooling Systems
263
The exponential integral is available as tables of the definite form
Ei(--v)-=--
(9)
~-f e ~ c l t
t
v
so that, given x = 0 at t = 0, (8) becomes
x = ~ % T e v(t){E i [ - v (0)] - E i [ - - v (t)]}.
For v e r y large v, E i ( - - v )
(10)
tends to zero and for v e r y small v, we have
E i ( - - v) ~- - - l n v - -
C=ln(yv)
where C is Euler's constant (0.5772), so 7 = 1.78. Noting from Eq. (6) t h a t v(t)
becomes v e r y small for large values of t, and t h a t for sufficiently high initial
t e m p e r a t u r e s and sufficiently slow cooling (large /c(0) and ~) v(0) becomes
v e r y large, Eq. (10) becomes at large times
x = 2, % ~ In [7 v (t) ].
(11 )
Substituting for v (t) using Eq. (6) and taking logarithms we obtain finally
x = X~ % { t - ~ i n [~ ~ ~ ( 0 ) ] }
(12)
= ~ % {t -- v [ln (7 ~ K ) - - E / R To]}
where T Ois the t e m p e r a t u r e at t = 0.
I n Eq. (12) the t e r m in T is the t o of Fig. 1 ; it could be regarded as a correction
to be added to a calculated age, if T 0 were, say, the t e m p e r a t u r e of m e t a m o r p h i c
recrystallisation. To find the closure t e m p e r a t u r e , however, we note t h a t from
the definition of ~ (see Eq. (2))
E / R T o - - E / R T O= to/r
(13)
= in (y ~ K ) - - E / R T O
so t h a t
E / R To = In (7 T K )
(14:)
is the desired relationship between To, cooling rate, and the loss p a r a m e t e r s
E and K.
If the parent decays rapidly we replace Cp by coe -zt and change the variable t
as before. Then in place of (8) we obtain
v(o)
x=(~/~)Co~eV(t)[Tlc(O)]
~ f v zT l e - V d v .
(143)
v(t)
If v (0) is sufficiently large the integral as t tends to infinity tends a p p r o x i m a t e l y
to the g a m m a function F ( ~ 3). Thus x tends to the limit x~ given b y
zoo = (~,~/,~) c o F ( ~ ~ + 1)/[~ k(0)] z~.
(14b)
F r o m the definition of closure time this limit must be related to the a m o u n t
of p a r e n t then remaining, i.e. :
x~----(~x/~)CoeXp[
18
Contr. ~IineraI. and Petrol., Vo]. 40
~ T ( E / R T e - E / R T o ) j.
(14c)
264
M.H. Dodson:
Equating right-hand sides, taking logarithms, and substituting for k(0) as before
we obtain:
E/RT~ = l n { [ F ( ~ T if- 1)] 1/~rK}.
(14d)
The constant in (14d) tends to y as 2 T tends to 0 (see Appendix B). Its value
for several values of 2 ~ is given in Table 1. For positive 2 v it is always less than y :
t h a t is to be expected, since the rate of production of daughter product is greater
when the temperature is higher, so the system should appear to close at a higher
temperature than it would with a constant production rate.
4. Closure Temperature Ior Volume Diffusion
There are several ways in which known solutions of the diffusion--heat conduction equation can be adapted to this problem. The most rigorous, given in
Appendix ]3, is based on a change of variables to transform the problem to
one of heat conduction with constant thermal diffusivity and time-varying
boundary conditions, for which general expressions have been given by Carslaw
and Jaeger (1959). (Besides confirming the results derived below, this approach
yields convergent expressions, which have not been obtained otherwise, for the
terminal concentration profile in a cooling mineral grain.)
An alternative method utflises the fact that, for variable diffusion coefficient,
the product Dt which appears in constant coefficient solutions of the diffusion
t
equation can be replaced b y fD(t')dt' (Crank, 1956). This approach results in
0
series solutions rather similar to t h a t obtained by Gentner et al. (1954) for cubic
geometry.
The method presented here is not rigorous but is mathematically the simplest.
I t uses the fact t h a t the fractional approach by a diffusing system towards
equilibrium can be expressed as an infinite series of negative exponentials. With
a constant diffusion coefficient, D, and zero surface concentration, the fraction ~b
of diffnsant, initially at uniform concentration, which remains in a system after
time t is given b y
r -= ~. (B/a~) e x p ( - - a~ nt/a ~)
(15)
where a is a characteristic dimension of the system, while the constant B and
the terms an depend on geometry. From Crank (1956, Eqs. (6.20), (5.23) and
(4.18)) we have: B----6, an=-n7~ for a sphere of radius a; B ----4, an is the n t h root
of Jo(x), and is approximately ( n - ~ ) ~ for a cylinder of radius a; and B = 2 ,
an----(n--~)~ for a plane sheet of thickness 2a. Cylindrical or plane geometry
m a y be appropriate for highly anisotropic minerals such as micas. B and an
are related by ~ B/a~ ~ 1.
We interpret Eq. (15) as follows. Each t e r m in the infinite series corresponds
to a sub-system with " w e i g h t " B / ~ and first-order loss coefficient ~z~nD/a2. The
latter corresponds to the coefficient ]c of the preceding section, and the "weight"
can be thought of as the fraction of radiogenic daughter product which is associated with that particular sub-system. The effect of cooling will be to change
each coefficient a]D/a 2 with the same time constant T as D, so that we can
Closure Temperature in Cooling Systems
265
calculate in these circumstances a closure temperature Ten for the n th sub-system
using Eq. (14), namely
E/RT~n = In (y v ~ Do~a2).
(16)
I t simply remains to determine the appropriate average of the infinite set of
values of Ten. Because E / R T varies linearly with time, the quantity of radiogenic
isotope remaining in the n th subsystem after a long period will be linearly related
to E/RTe n ; thus, for the whole system, it is reasonable to determine a weighted
arithmetic mean of E/RTe n using weights B/~.~. Hence we obtain for Te;
E/RTe = ~,, (B/~) In (y v ~n2 D o / a 2)
n=~
= In (Y ~ Do~a2) -~ 2 B ~ In u~/~.
1
(17)
The three series which are represented by the last t e r m in Eq. (17) have been
summed by a method given by Hartree (1958, p. 266). After removing the factor
from c~n we obtain:
sphere:
~
In n/n 2
=
0.938
ln(~n/~)
(%/~)2
--
0.529
1
cylinder:
1
plane sheet:
~
ln(~-- 1/2)
*
(~-
~/~)~
-- 1.748.
(For the cylinder the first ten terms of the series were obtained from a table
of the roots of Jo(x), and the remainder was summed using the approximation
(n -- ~) ~).
Thus the result can be expressed in the form:
E/RTe = In (A T D0/a2 )
(19)
where A = 5 5 for the sphere, 27 for the cylinder, and 8.7 for the plane sheet.
The variation of A with geometry is intuitively sensible, since the configuration
with greatest surface-to-volume ratio of the three, namely the sphere, would
be expected to be associated with more rapid diffusion loss and therefore lower
closure temperature. I n fact the relative values of surface to volume ratio, 3 : 2 : 1,
are of the same order as the relative values of A, 6:3 : 1.
The case of a rapidly decaying parent is covered by Appendix B. Alternatively
a similar t r e a t m e n t to the above m a y be used. Starting from Eq. (14d) an
expression for E/RT~ is obtained, comparable to Eq. (16). In forming the sum,
however, one must take account of the decay of the parent by including in the
weight allotted to each t e r m a factor proportional to exp (-- 2 tc~,).
5. Closure of Petrological Systems
Consider an oxygen isotopic exchange reaction between a siheate mineral grain
and a large quantity of well-mixed pore fluid (or adjacent minerals in which
18.
266
M.H. Dodson :
To 1
Tc 1
t e m p -1
..~
.,~,o
c ~ . ~ ~~
I
~/
/ ~O
m
mean
Fig. 2. Relationship between
geochronologicalclosure (A) and
"frozen" equilibrium (B) for
identical diffusion parameters,
assuming 1/T increases linearly
with time
conc'n
C,
0
tc
time
~
diffusion occurs rapidly). If a state of equilibrium is reached at a certain temperature, and then a stepwise temperature change occurs, the new equilibrium
abundance of oxygen --18 will be attained immediately on the surface of the
grain, and thereafter isotopic mixing by volume diffusion will tend to produce
a uniform concentration throughout its volume. The approach to equilibrium is
quantitatively described by Eq. (15). Now if the temperature changes continuously with time, the equilibrium concentration at the surface will also change
continuously. Assuming, as before, that 1 / T increases linearly; the problem takes
the mathematical form
~c
D(O)
~ta s e-tl~[72c
(20)
where c represents the isotopic abundance of oxygen--18, with the boundary
conditions that c = c o everywhere at t = 0, and c = c s (t) on the surface for t > 0.
For convenience, we can change the variable to q = c - - c o, so the boundary
conditions are q = 0 at t = 0, and q = q8(t) on the surface: for a small temperature
range q8 (t) would be approximately given by q~= k t where k is a constant. Eq. (20)
is unchanged except that c is replaced by q.
Eq. (20) and associated boundary conditions can be compared with the equation
for diffusion of a radiogenic isotope produced at a constant rate, namely
~c
~t --
D(O) e_t/~V2c_J_ L
a2
(21)
where c is the concentration of daughter product, and L is its constant rate of
production. The boundary conditions are C~= 0 for t > 0 and c = 0 everywhere
for t = 0 . The above equation, however, becomes formally identical with (20)
if the variable c is replaced by the cumulative de]icit of radiogenic isotope. If
we denote this by q, it is given by
q = L t -- c.
(22)
Closure Temperature in Cooling Systems
267
We then obtain b y substitution in (21),
~q
St - -
1)(0)
as
e-tl~V2q
and the boundary conditions are q = 0 at t = 0, and at the surface qs = L t for t > 0.
From this formal identity it m a y be inferred t h a t the apparent temperature
recorded by such an oxygen isotope system after it has cooled completely is
equal to Tc in Eq. (19), and m a y therefore be referred to, without risk of confusion,
as its closure temperature. An obvious potential application of palaeotemperature
observations of this kind is the estimation of approximate cooling rates in orogenic
belts.
Diffusional exchange with a finite reservoir requires a more elaborate analysis
which has not so far been attempted.
6. Dependence of Closure Temperature upon Cooling Rate
To use Eq. (19) to determine the closure temperature for a particular cooling
rate, we must take into account the dependence of the time constant T upon
temperature and activation energy [Eq. (3)]. The equation m a y then be written
in the form
E
In ( - A R Tc~Do/a 2 ]
(23)
Eq. (23) can be used directly for iterative determination of To, for a given value
of d T / d t . A rough estimate T~ on the right hand side gives a reasonably precise
value Tc on the left, and two or at most three iterations would normally be
sufficient. Faster convergence is obtained by writing the iteration in the form,
E I R + 2m~
Tc = ln[(A ttTcX~Do/a~)/(EdT/dt)] "
(24)
When diffusion coefficients D are quoted at a particular temperature T, computational effort m a y be saved by substituting in the above equation l n D 0 = l n D
-f- E I R T .
A simple example will illustrate the insensitivity of Tc to choice of cooling rate.
H o f m a n and Giletti (1970) studying diffusion in biotite, obtained 21 kcal/mole
for E and 2 x 10-1% -1 for D / a 2 at 600 ~ C, assuming cylindrical geometry (i.e.
diffusion only along basal cleavage planes). If we suppose the cooling rate to be
100 ~ C per million years, and guess 600 K for the trial closure temperature Tc1,
we obtain a value of 1.1 X i01s8 or 0.35 million years for the time constant 7.
Substituting in (23) this gives on a first iteration
E/RT c =In
(5.8 X 104) ~- 10.8 =21.8
whence T c =485 K =212 ~ C. Further iterations give 496 K and 494 K (221 ~ C).
Eq. (24) gives 222 ~ C immediately. Because of the large value of the logarithmic
term, a change of cooling rate by a factor of 10 only changes T c by 11% or 55 ~ C.
7. Transitional Interval
I t is only possible to define a transitional temperature range in terms of some
finite approximation to the high temperature total-loss (continuous equilibrium)
268
M.H. Dodson:
and the low temperature zero-loss (frozen equilibrium) situations. The 5 % range,
for example, would be the interval over which the rate of accumulation of
radiogenic isotope increased from 5 to 95% of its ideal value.
The calculation for first-order loss is not difficult, and involves substitution
of Eqs. (10) and (2) in Eq. (5) to give an expression for dx/dt, together with the
approximation
E i (--v) = e-V (1Iv -- 1Iv 2)
for large v (small times), and E i ( - - v ) = - - l n ( ~ v ) for small v. The 5% transitional
interval comes out to be 7.2 T and the closure time and temperature lie roughly
in the middle of this interval.
For volume diffusion the 5 % interval appears to be about 10 v, but this figure
has not been derived rigorously.
The 5 % transitional intervals obtained above are surprisingly large, and m a y
give a misleading impression of the magnitude of possible errors arising from
the approximations made in the theory. Considering the approximation made in
deriving Eq. (14) it is found from inspection of tables of E i ( - - x ) (Jahnke-Emde,
1933) that these approximations amount to only 0.03 for v (t) = 0.01 and v (0) = 2.3,
with the additional approximation e0"01~ 1 . The latter contributes the main
error, which in other words is about 1%, in E / R T e. The above range in v(t)
corresponds to less than four time constants.
8. Propagation of Errors in Loss Parameters
The theoretical relationships can be used to estimate the magnitude of error
propagation from E and D to the calculated closure temperature. For simplicity
it m a y be supposed t h a t the diffusion coefficient D• is measured at a temperature
Tin, and t h a t E can be determined independently of the absolute value of D~/a 2.
Writing Eq. (19) in abbreviated form, substituting for D O by (1) and rearranging
we obtain:
T: 1 - T,g 1 = (R/E) In (A ~:D~/a2).
~m can be assumed to be exactly known, so we have from the usual relationships
for combining independent normally distributed errors:
Typically the measuring temperature will be 800 to 1100 K, so if T~ is 500-600 K
we have 1--Tc/Tm 0.5. Thus Tc can be determined within about 5%, or 25-30 ~ C,
if E is known to 7% and D~/r 2 within a factor of 2, supposing l n ( A z D , Ja 2) to
be about 10 as in the calculated example.
9. An Alpine Example
J~ger et al. (1967) presented Rb/Sr data on miens from the central Alps and
suggested t h a t the biotite dates correspond to cooling through a critical ternperature, probably about 300 ~ C from mineralogical data. Using the Hofman and
Giletti (1970) data on diffusion in biotites, we can make an independent estimate
of the closure temperature.
Closure Temperature in Cooling Systems
269
From the data of H o f m a n and Giletti we have D / a 2 = 10-1~ -1 at 600 ~ C for
diffusion of Rb in biotite flakes radius a = 0.0034 cm, using a cylindrical model.
They suggest t h a t their data will be approximately correct for strontium. The
grain size of the Alpine biotites is typically ten times greater than that of the
experimental material. Thus we must use D/a 2 =10-12s -1 = 3 0 My -1, together
with the previously quoted activation energy E = 2 1 kcal/mole.
The problem can be completely solved if we assume a uniform increase of 1 / T
up to the present day, since the present day temperature T~ can be considered
Go be known. For a biotite Rb/Sr age t we have from Eq. (3)
3 -= R t / [ E (T• 1 -- T/Z)]
in which T c represents a first guess at the closure temperature. Taking T c ~ 6 0 0 K ,
T p = 3 0 0 K, we obtain 3 = 0 . 0 5 7 t. Thus for a 10My biotite 3 = 0 . 5 7 My, and
E / R T c =1n(27 x 0.57 x 30) + 12.0
=18.3.
Hence Tc ~- 570 K = 300 ~ C.
For a 20 My biotite the result is slightly lower, namely 280 ~ C, because of
the implied slower rate of cooling.
Clearly the similarity of this result to the geologically estimated closure
temperatures should be regarded as encouraging rather than conclusive. An
important objection is t h a t the apparent ages of biotites in the zone of cooling
ages is independent of grain size over a very wide range, up to tens of centimetres
(J/igor, personal communication). Acceptance of the theory then requires that
the effective diffusion radius is of the order of 0.5 m m for all materials. An
alternative explanation of the observed data is t h a t they are related to the
cessation of migration of pore fluid acting as the sink for radiogenic strontium.
A further objection to the calculation is t h a t it requires extrapolation of laboratory diffusion data over a large temperature range. Data obtained for diffusion
in minerals in contact aureoles (Hart, 1964; Hanson and Gast, 1967) are free
from this objection, but are subject to other uncertainties related to the validity
of the models used to calculate the thermal history, and the effective diffusion
size in an ancient rock undergoing thermal metamorphism.
Given other geochronological measurements, using different minerals or
methods, it would be possible, in principle, to make a more detailed analysis of
Alpine cooling history b y iteratively fitting a polynomial in 1 / T to the various
apparent ages. At present this cannot be done, both because there is a dearth of
convincing diffusion data on muscovites, and because the theory of track annealing
is not sufficiently developed to make full use of the Alpine fission track ages of
Wagner and Reimer (1972).
10. Discussion and Conclusions
The theory developed here obviously is not useful if the systems under consideration cool to a steady temperature at which diffusion is significant over the life
of the system. Caution would therefore be required in applying it to age data from
270
M.H. Dodson:
Mesozoic or older orogenic belts, within which rocks sampled today may have
spent long periods at depths of several kilometres. The same difficulty is unlikely
to arise in young orogenic belts or meteorites. Howeber, for certain isotopic and
chemical geothermometers diffusion may be insignificant at temperatures below,
say, 200 ~ C, so it may be feasible to use them to determine approximate cooling
rates in ancient orogenic belts.
A further important condition for validity of the theory is that the cooling
should be "slow". Quantitatively this means that TD(O)/a 2 must be much
greater than unity, or, in physical terms, the cooling time constant ~ must be
much greater than the initial value of the characteristic diffusion time a2/D (0).
The cooling time constants are of the order of 106 years for strontium diffusion
in Alpine biotites, for which a2/D (0) is of the order of 30000 years at temperatures
of 600 ~ C, according to the data used here, so the condition of slow cooling is fully
satisfied in that example.
Under slow cooling conditions no significant errors will be introduced by
neglecting the approximate steady-state daughter concentration which can be
expected to exist at the commencement of cooling (Damon, 1970). This neglected
component corresponds to the approximation made at the upper limit of integration (large v (0) at zero time) in Eqs. (8) to (10). Another component discussed by
Damon, the "environmental" argon-40 introduced at (or after) the time of
crystallisation of the minerals, has been ignored : by assuming a zero concentration
of radiogenic daughter product at the surface of the mineral grains throughout
cooling, consideration of such a component is effectively excluded from the theory.
While fairly easy to justify for radiogenic argon, this simplification carries obvious
risks in relation to strontium-87.
Values of the constant in the formulae for closure temperature [Eqs. (14), (19),
and (B. 18)], are summarised in Table 1. The logarithmic relationship implies
marked insensitivity of Tc to changes in any factor of the argument of the logarithm, including A, and is strong justification for using the above formula,
rather than a complicated numerical computation, in attempting to derive a first
approximation to cooling history from mineral ages and petrochemical data.
In conclusion, it must again be emphasised that closure temperatures may
not necessarily be always determined by thermally activated diffusion processes.
However, the mathematical relationships presented in this paper offer a relatively
simple method of testing whether solid diffusion models are adequate to explain
observed patterns of mineral ages and anomalous palaeotemperature determinations.
Table 1. Vahles of A in In [AT (Do/a2, K)]
Type of loss
First order reaction
Diffusion perpendicular to plane sheet
Radial diffusion in cylinder
Radial diffusion in sphere
]~
0
1.78
8.65
27
55
1
1
3.0
8.0
15
2
0.71
1.9
4.9
8.5
Closure Temperature in Cooling Systems
271
Appendix A
P r o o / o/ the General L o g a r i t h m i c E x p r e s s i o n / o r
E/RT c
For a temperature change of the form d T 1/dt = constant, we can write the general diffusioncooling-accumulation equation in the dimensionless form.
where 0 = dimensionless time t/T, c = concentration of daughter product, co = initial concentration of parent, and D ( 0 ) = D o e x p ( - - E / R T o ) is the initial value of diffusion constant.
Thus c/c o is a function of 3D(O)/a 2, O, and ),3, whose form will depend on the geometry of
the system. At very large times c/c o tends to the no-loss accumulation curve (the first term
on the right hand side of the equation tends to zero), and we can write
lira c/c o = / (0, v D (O)/a 2, 4 3).
0---~r162
(A.2)
To find the closure time 0c we equate / to zero, and write
0c = g (3D (O)/a 2, 4 3).
(A.3)
From the definition of 3 and 0 we have
0c = E / R T c -- E / R T o
(A.4)
where T o is initial temperature. When we substitute in (A.3) for D(0) and Oc, we obtain
E / R T e = E / R T o + g (vD oe -E/~To/a2, 4 3).
(A.5)
Writing for brevity X = E / R To, u = xDoe-X/aS, the condition t h a t T c is independent of T o
can be written
( E / R Tc)/OX = O.
Differentiating, therefore, the r.h.s, of (A.5) we obtain
0 = 1 + (~g/Ou)~. d u / d X
= 1-- u ( S g / ~ u ) ~
which gives on integration
g=lnu+b(43)
= In @Do/a 2) -- E / R T O+ b (43)
b being an integration constant which is a function of 4v. Substitution in (A.5) confirms that,
if g has the above form, we obtain Eq. (19).
Appendix B
Solution o/Accumulation-Di/[usion-Cooling
Equation in Terms
o / H e a t C o n d u c t i o n w i t h Variable B o u n d a r y C o n d i t i o n s
We start from Eq. (A.1), and rewrite it in terms of the new variable q, defined by
q = 1-- e - ~ ~
C/Co
(B.1)
which is a measure of the deficiency in c/co relative to what would be produced with no
diffusion losses. Substituting M = ~D(O)/a ~, we thus obtain from (A.1):
~q/~O=Me
~
(B.2)
with the new boundary conditions (qs = value of q at surface)
qs=l--e -~~
and the same initial condition, q = 0 at 0 = 0.
(B.3)
272
M.H. Dodson:
When the diffusion coefficient varies with time t, one can simplify the problem by changing
t
the time variable to u = f D (t')dt" (Crank, 1956). Applying this to (B.2) we obtain for the new
variable
0
u = M ( 1 - - e -o)
(B.4)
whence (B.2) and (B.3) become
~q/au=V~q
(B.5)
(B.6)
q, = 1 - ( 1 - u/M)~*.
At 0 = 0 , u = 0 , but as 0-+oo, u-->M.
Eq. (B.5) is identical with the reduced form of the standard diffusion equation. Under the
time-varying boundary condition (B.6) and the initial condition q = 0 at u = 0, its solution
may be derived from general expressions given by Carslaw and Jaeger (1959), which are
infinite series in which each term includes a factor of the form
u
I n = f exp(c~2nu')qs(u')du"
(B.7)
0
where ~n has the same meaning as in Section 4. Substituting for qs by (B.6) and making the
further change of variable
v = ~ (M -- u')
(B.8)
we obtain
~xn (M--u)
I n --
e - v [1--
exp(~nM)
v
~
~gM
To determine the situation after a long time we need only consider what happens in the limit
as 0--->oo, i.e. u--~M. Because 3 / i s large under conditions of slow cooling we can write, to a
good approximation,
exp(~M)
m~ [i--{r(~+1)
~,-~
]"
In, oo=olim I n -
(]3.10)
This result may now be combined with the general solutions for plane, cylindrical, and spherical
geometry given by Carslaw and Jaeger (1959 : p. 104, Eq. (3) ; p. 201. Eq. (12) ; p. 233, Eq. (3)),
to yield the following expressions for the values of q and its volume average ~ after an infinite
time:
plane sheet:
q=2
~ , (--1) n + l cos[(n--1/2)zcx]{
n=l
F(,~'c+l)
1--
(~t__l/2) 7~
3=1
r(~T+l)
M.~'~
~,
n~l
~
[(n--1/2)2 ~2MJ~J
2
[ ('y& -- 1/2 ) 2 ;i~:2] ).T + 1
(B.11)
(B.12)
9
cylinder:
q=2n--~_l ungl(~n)
(~nM) ~ J
(B.13)
where ~n is the n th root of J0 (x);
3=
I
r(i~+ I) ~
M ~~
~
n=l
4
~2n(1+ ~)
(B.14)
sphere:
q=--2
~
n=l
(--1) n s i n ( n ~ x ) [ 1
n ~x
F(~T+I)]
(n2~2 M)~ 9
(B.15)
Closure Temperature in Cooling Systems
273
F ( ~ z + 1) co
6
l~eturning to the notation of Section 4, and introducing the dimensionless closure time
0c : tc/~, we may now write for all three geometries:
q = l -- exp (-- ~ 0 c )
/'(TT + 1)
1
~
B
(B.17)
since ~ is, by definition, the difference between the total quanti~y of daughter product formed
(in this case 1 after infinite time) and the quantity formed since the closure time 0c (which
is exp(--2~0c) after infinite time). Equating terms on the right, taking logarithms, and
substituting for 0c and D(0) as was done, in effect, in Section 3, we arrive at the following
general expression for closure temperature:
E/RTe=ln(TDo/a2)--(1/.~'r)ln{F(~v-]-1)
n=~i B/~x~n(l+'~7:)}=ln(AvDo/a~).
(B.18)
For the special case of a very slowly decaying parent, ()~v ~ 1) the stun in (B.18) becomes
In E (B/a~n) (1 -k 2 7 v In %)
"~ Z ( ~ 2Bin%/~).
Moreover, the limit of (1/7~)In/'(~T-[-1) as ~ tends to zero is equal to Euler's constant,
from Weierstrass' definition of the gamma function: hence, in the limit, Eq. (B. 18) becomes
identical to Eq. (17) in Section 4.
Eq. (17) can be reached independently, if we replace (B.1) by q = 2 ~ 0 - - c / c o, so that the
boundary condition on (B.5) is
q.s=ZT0
~tTln ( 1 - - u / M ) .
(B.19)
co
Forming I n as before, making the same change of variable, and using the result f In v e v d v =
0
- - l n y , we obtain for the limit of I n,
I n , ~ = 2"v exp ( ~ M ) I n (yo:~nM)/~n.
(B.20)
Expressions for q and ~ are slightly simpler than (B.11) to (B. 16), and Eq. (17) is reached by
equating the value of ~ to 7T0 c, to which it is equal by definition.
The limiting concentration distribution of the diffusing isotope is given by the above
expressions for q [Eqs. (B.11), (B.13), and (B.15)]. No attempt has yet been made to evaluate
these expressions numerically. A complete description of the concentration distribution might
be more readily obtained by numerical solution of Eq. (B.2) than by evaluation of the
explicit solutions. I t should be reasonably simple, however, to determine for plane sheetand
cylinder the value of q at x = 0, and to express this as an apparent closure temperature for
the central point in a mineral grain.
Acknowledgements. I thank E. J~ger, R. A. Cliff, and R. G. Turner for critical reading of
the manuscript.
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M . H . Dodson: Closure Temperature in Cooling Systems
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Dr. M. H. Dodson
Dept. of Earth Sciences
The University
Leeds LS2 9JT, England