closure temperature in cooling geochronological and ...€¦ · constant depending on geometry and...

Contr. Mineral. and Petrol. 40, 259--274 (1973) �9 by Springer-Verlag 1973

Closure Temperature in Cooling Geochronological and Petrological Systems Mart in I{. Dodson

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 Rb- -Sr 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 Tempera ture

W h e n the " a g e " of a rock or minera l is ca lcu la ted f rom i ts accumula ted produc ts of r ad ioac t ive decay, whe the r those p roduc t s be crys ta l s t ruc tu ra l changes caused b y fission f ragments , or radiogenic isotopes such as sTSr or a~ the resul t idea l ly represents a po in t in t ime a t which a comple t e ly mobile daugh te r p roduc t became comple te ly immobi le . " M o b i l i t y " , in th is context , means e i ther r a p i d diffusion f rom the l a t t i ce si te a t which a radiogenic i sotope is formed, or ve ry fast an- neal ing of a d i s tu rbed c rys ta l la t t ice . A t one t ime i t was bel ieved t h a t the change in mob i l i t y of a radiogenic i sotope could a lways be ident i f ied wi th e i ther the c rys ta l l i sa t ion of an igneous rock f rom a mel t , or recrys ta l l i sa t ion dur ing metamorphism. I n recen t years , however , i t has become increas ingly clear t h a t for some me thods of age de te rmina t ion , n o t a b l y the da t ing of s epa ra t ed minerals b y the R b - - S r and K - - A r methods , such a s imple 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 rgon a n d s t ron t i um ev iden t ly are mobi le in some minera ls a t t empe ra tu res well below t h a t of crys ta l l i sa t ion. The bes t evidence for this view comes f rom the R b - - S r and K - - A r age p a t t e r n on micas f rom the cent ra l Alps (J/tger, 1965; J / iger et al . , 1967; Arms t rong et al . , 1966), for which the s imples t i n t e r p r e t a t i o n is t h a t closure of the R b - - S r and K - - A r sys tems occurred dur ing

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- -Ar 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 transi- tion 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 T c 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 at tempt 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 O exp (-- E / R T ) (1)

Closure Temperature in Cooling Systems 261

t

temperature

Tc

Fig. 1. Definition of closure temperature. The time t c corresponds to the apparent age

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 t ime 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 t ime constant 3, which is the t ime taken for D to diminish by a factor e -z or, in other words, for E / R T to increase by 1. Hence we can write, for the t ime dependence of D:

D ---- D o exp ( - - E / R T o - - t /z)

= D (0) e-t/~ (2)

where D (0) and T o 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)

= - - R T ~ / ( E dT /d t ) .

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 prefer- able 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

dx /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

dx /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

dx /dv - - 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) dv + constantS. (8)


The exponent ia l integral is available as tables of the definite form

E i ( - - v ) - = - - ~-f e ~ c l t (9) t

v

so tha t , given x = 0 a t t = 0, (8) becomes

x = ~ % T e v(t){E i [ - v (0)] - E i [ - - v (t)]}. (10)

For ve ry large v, E i ( - - v ) tends to zero and for ve ry small v, we have

E i ( - - v) ~- - - l n v - - C = l n ( y v )

where C is Euler ' s constant (0.5772), so 7 = 1.78. Noting f rom Eq. (6) t h a t v(t) becomes ve ry small for large values of t, and t h a t for sufficiently high initial t empera tu res and sufficiently slow cooling (large /c(0) and ~) v(0) becomes ve ry large, Eq. (10) becomes a t large t imes

x = 2, % ~ In [7 v (t) ]. (11 )

Subst i tu t ing for v (t) using Eq. (6) and taking logari thms we obtain finally

x = X~ % { t - ~ i n [~ ~ ~ ( 0 ) ] } (12)

= ~ % {t - - v [ln (7 ~ K ) - - E / R To] }

where T O is the t empe ra tu r e a t t = 0.

I n Eq. (12) the t e rm 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 empera tu re of me tamorph ic recrystall isation. To find the closure t empera tu re , however, we note t h a t f rom 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 relat ionship between To, cooling rate, and the loss pa ramete 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 = ( ~ / ~ ) C o ~ e V ( t ) [ T l c ( O ) ] ~ f v zT l e - V d v . (143)

v(t)

I f v (0) is sufficiently large the integral as t tends to infinity tends approx ima te ly to the g a m m a funct ion F (~ 3). Thus x tends to the limit x~ given by

zoo = (~,~/,~) coF(~ ~ + 1)/[~ k(0)] z~. (14b)

F r o m the definition of closure t ime th i s limit mus t be related to the amoun t of pa ren t then remaining, i.e. :

x~----(~x/~)CoeXp[ ~ T ( E / R T e - E /RTo) j. (14c)

18 Contr. ~IineraI. and Petrol., Vo]. 40

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). I ts value for several values of 2 ~ is given in Table 1. For positive 2 v it is always less than y : tha 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 t ime-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 by fD(t ')dt ' (Crank, 1956). This approach results in 0

series solutions rather similar to tha t obtained by Gentner et al. (1954) for cubic geometry.

The method presented here is not rigorous but is mathematical ly the simplest. I t uses the fact tha 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 t ime t is given by

r -= ~. (B/a~) exp ( - - 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 th 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 may 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 te rm in the infinite series corresponds to a sub-system with "we igh t" B / ~ and first-order loss coefficient ~z~nD/a 2. 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 t ime constant T as D, so that we can


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/RT varies linearly with time, the quanti ty 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 2 2 , v ~n D o / a ) n=~ (17)

= In (Y ~ Do~ a2) -~ 2 B ~ In u~/~. 1

The three series which are represented by the last te rm in Eq. (17) have been summed by a method given by Hart ree (1958, p. 266). After removing the factor

from c~ n we obtain:

sphere: ~ In n/n 2 = 0.938 1

ln(~n/~) cylinder: (%/~)2 -- 0.529 1

~ ln(~-- 1/2) plane sheet: -- 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. In 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 rea tment to the above may 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 te rm 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 quanti ty of well-mixed pore fluid (or adjacent minerals in which

1 8 .

266 M.H. Dodson :

conc 'n

C,

To 1 Tc 1 t e m p -1 ..~

. , ~ , o c ~ . ~ ~ ~

I / ~ O m ~ / mean

0 tc t i m e ~

Fig. 2. Relationship between geochronological closure (A) and "frozen" equilibrium (B) for identical diffusion parameters, assuming 1/T increases linearly with 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) ~ t - a 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 D(O) e_t/~V2c_J_ L (21) ~t -- a 2

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)


We then obtain by substitution in (21),

~q 1)(0) St - - a s e - t l ~ V 2 q

and the boundary conditions are q = 0 at t = 0, and at the surface qs = L t for t > 0. From this formal identi ty it may be inferred tha t the apparent temperature

recorded by such an oxygen isotope system after it has cooled completely is equal to T c in Eq. (19), and may 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 at tempted.

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 t ime constant T upon temperature and activation energy [Eq. (3)]. The equation may 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 t tTcX~Do/a~)/(EdT/dt)] " (24)

When diffusion coefficients D are quoted at a particular temperature T, computa- tional effort may 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 T c to choice of cooling rate. Hofman 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). I f we suppose the cooling rate to be 100 ~ C per million years, and guess 600 K for the trial closure temperature Tc 1, 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 /d t , 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 t ime 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 may 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 e 0"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 t ime 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 may be supposed tha t the diffusion coefficient D• is measured at a temperature Tin, and tha 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/T m 0.5. Thus T c 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 tha t the biotite dates correspond to cooling through a critical tern- perature, 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.


From the data of Hofman 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 tha 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 Tc~600K , Tp=300 K, we obtain 3=0 .057 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 T c ~- 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 tha 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 mm for all materials. An alternative explanation of the observed data is tha 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 tha t it requires extrapolation of labo- ra tory 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 by 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 t rack 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 T c to changes in any factor of the argument of the log- arithm, 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 determina- tions.

Table 1. Vahles of A in In [AT (Do/a 2, K)]

Type of loss ]~

0 1 2

First order reaction 1.78 1 Diffusion perpendicular to plane sheet 8.65 3.0 Radial diffusion in cylinder 27 8.0 Radial diffusion in sphere 55 15

0.71 1.9 4.9 8.5


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 / R T c

For a temperature change of the form d T 1/dt = constant, we can write the general diffusion- cooling-accumulation equation in the dimensionless form.

where 0 = dimensionless time t/T, c = concentration of daughter product, c o = 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). (A.2) 0---~r162

To find the closure time 0 c we equate / to zero, and write

0 c = g (3D (O)/a 2, 4 3). (A.3)

From the definition of 3 and 0 we have

0 c = 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 o e -E/~To/a2, 4 3). (A.5)

Writing for brevity X = E / R To, u = xDoe-X/aS , the condition that T c is independent of T o can be writ ten

(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 = l n u + b ( 4 3 )

= 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

S o l u t i o n o / A c c u m u l a t i o n - D i / [ u s i o n - C o o l i n g E q u a t i o n i n T e r m s

o / H e a t C o n d u c t i o n w i t h Var iable B o u n d a r y Cond i t i ons

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/c o relative to what would be produced with no diffusion losses. Substituting M = ~D(O)/a ~, we thus obtain from (A.1):

~ q / ~ O = M e ~ (B.2)

with the new boundary conditions (qs = value of q at surface)

q s = l - - e - ~ ~ (B.3)

and the same initial condition, q = 0 at 0 = 0.

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) whence (B.2) and (B.3) become

~q/au=V~q

q, = 1 - ( 1 - u/M)~*.

(B.4)

(B.5)

(B.6)

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

(M--u) ~xn

exp(~nM) e -v [1-- v ~ I n -- ~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) [i-- {r(~+1) m~ ~ , - ~ ] " (]3.10) In, oo=olim I n -

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:

q = 2 ~ , (--1) n + l cos[(n--1/2)zcx]{ F( ,~ 'c+l ) ~ n = l (~t__l /2) 7~ 1-- (B.11) [(n--1/2)2 ~2MJ~J

3 = 1 r ( ~ T + l ) ~ , 2 (B.12) M.~'~ [ ('y& - - 1/2 ) 2 ;i~:2 ] ).T + 1

n~l �9

cylinder:

q=2n--~_l ungl(~n) (~nM) ~ J (B.13)

where ~n is the n th root of J0 (x);

r(i~+ I) ~ 4 3= I M ~ ~ ~ ~2n(1 + ~) (B.14) n = l

sphere:

q = - - 2 ~ (--1) n s i n ( n ~ x ) [ 1 F ( ~ T + I ) ] (B.15) n=l n ~ x (n2~2 M)~ �9

plane sheet:


F ( ~ z + 1) co 6

l~eturning to the notation of Section 4, and introducing the dimensionless closure time 0 c : tc/~, we may now write for all three geometries:

q = l -- exp (-- ~ 0 c )

/ '(TT + 1) ~ B (B.17) 1

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 0 c (which is exp(--2~0c) after infinite time). Equating terms on the right, taking logarithms, and substituting for 0 c 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

- - lny , 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 at tempt 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.

Re fe rences

Amirkhanov, K. I., Brandt, S. B., Bartnitsky, E. N.: Radiogenic argon in minerals and its migration. Ann. N.Y. Acad. Sci. 91 (2), 235-275 (1961).

Armstrong, R. L. : K-Ar dating of plutonic and volcanic rocks in orogenic belts. In: Potassium- argon dating, O. A. Schaeffer, J. Z~ihringer, eds., p. 117-133. Berlin-Heidelberg-NewYork: Springer ]966.

274 M.H. Dodson: Closure Temperature in Cooling Systems

Armstrong, R. L., J~ger, E., Eberhardt, P.: A comparison of K-Ar and Rb-Sr ages on Alpine biotites. Earth Planet. Sci. Left. l , 13-19 (1966).

Carslaw, H. S., Jaeger, J. C.: Conduction of heat in solids. Oxford: Clarendon Press 1959. Crank, J. : Mathematics of diffusion. Oxford: Clarendon Press 1956. Damon, P. E.: A theory of " r ea l " K-Ar clocks. Eclogue Geol. Helv. 58, 69-76 (1970). Fleischer, R. L., Price, P, B., Walker, R .M. : Identification of Pu 24~ fission tracks and the

cooling of the parent body of the Toluca meteorite. Geochim. Cosmochim. Acta. 32, 21-31 (1968).

Gentner, W., Goebel, K., PrEg, R.: Argonbestimmungen an Kalium-Mineralien. I II . Ver- gleichende Messungen nach der Kalium-Argon- und Uran-Helium-Methode. Geochim. Cosmochim. Acta. 5, 124-134 (1954).

Goldstein, J. I., Short, J. M. : Cooling rates of 27 iron and stony-iron meteorites. Geochim. Cosmochim. Acta. 81, 1001-1023 (1967).

Hanson, G. 1~., Gast, P . W . : Kinetic studies in contact metamorphic zones. Geochim. Cosmochim. Acta 31, 1119-1153 (1967).

Harper, C. T. : The geological intepretation of potassium-argon ages of metamorphic rocks from the Scottish Caledonides. Scot. J. Geol. 8, 46-66 (1967).

Hart, S. R. : The petrology and isotopic-mineral age relations of a contact zone in the Front Range, Colorado. J. Geol. 72, 493-525 (1964).

Hartree, D. R. : Numerical analysis. Oxford: Clarendon Press 1958. Hofmann, A.W. , Giletti, B. J. : Diffusion of geochronologically important nuclides under

hydrothermal conditions. Eclogue Geol. Helv. 63, 141-150 (1970). JEger, E.: Rb-Sr age determination on minerals and rocks from the Alps. Sci. Terre 1O,

395-406 (1965). JEger, E., 1Niggli, E., Wenk, E. : Rb-Sr Alters-Bestimmungen an Glimmern der Zentralalpen.

Beitr. Geol. Karte Schweiz, N.F. 134 Lieferung (1967). Jahnke-Emde: Funktionentafeln. Leipzig: Teubner 1933. Wagner, G. A., Reimcr, G. M. : Fission track tectonics: the tectonic interpretation of fission

track apatite ages. Earth Planet. Sci. Lett. 14, 263-268 (1972). Wood, J. A. : The cooling rate and parent planets of several iron meteorites. Icarus 3, 429459

(1964).

Dr. M. H. Dodson Dept. of Earth Sciences The University Leeds LS2 9JT, England

closure temperature in cooling geochronological and ...€¦ · constant depending on geometry and...

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