Geoscience Reference
In-Depth Information
rock. The time required for a spherical mineral of radius
a
to lose a fraction
F
of an element
it initially contained complies fairly well with the equation:
6
F
=
1
−
τ/π
+
3
τ
(5.11)
a
2
is a scalar proportional to time known as dimensionless time. This
equation is widely utilized for calculating diffusion coefficients; for example, for rare
gases, by evaluating the rate of degassing of a particular mineral at a certain temperature
as a function of time, or for soluble compounds, by immersing them in a liquid.
When advection and diffusion are both operative at the same time, their relative impor-
tance can be evaluated with the Peclet number Pe, with Pe
D
i
t
τ
=
/
where
D
i
, where
d
is a
characteristic dimension of the system (e.g. grain size or conduit diameter). Diffusion
predominates when Pe
=
v
d
/
<
1, and advection for Pe
>
1.
5.2.1 Closure temperature: chronometers, thermometers, and barometers
Diffusion theory can be applied to calculate the resistance of radioactive chronometers to
thermal disturbances or for evaluating the equilibration temperature of a mineral assem-
blage (thermobarometry). A number of important applications in geochronology rely on
high activation energies of diffusion in solids
(Eq. (5.8)
)
cause diffusion coefficients to
vary closely with temperature. The transition from open system to closed system therefore
therefore for a diffusion coefficient greater than
D
c
, where:
D
i
0
exp
E
i
RT
c
D
c
=
−
(5.12)
the system can be considered open. The radiogenic isotope will only begin to accumulate
below
T
c
, the temperature at which the stop-watch is started, with the transition occurring
within a few tens of degrees. Let us see how to evaluate
T
c
for a given isotope in a given
process is dimensionless time
D
i
t
a
2
. Let us decide that the system is open for
τ
=
/
τ>τ
c
and closed for
τ
c
is a constant factor dependent on geometry (0.02 for a
sphere and 0.12 for a sheet). If the significance of the radius
a
of the mineral is clearly
perceived, what is the time characteristic to be introduced in
τ<τ
c
, where
τ
? Dodson suggests that a
characteristic time is the time
t
=
θ
required for the diffusion coefficient to reduce by a
factor
e
, and is equal to:
dln
D
i
d
t
E
i
R
E
i
RT
2
1
θ
d(1
/
T
)
d
T
d
t
=−
=
=−
(5.13)
d
t
This time can be estimated from experimental data on diffusion coefficients and even
approximate knowledge of the local rate d
T
d
t
of cooling after formation of the minerals
under analysis. Let us define the dimensionless time
/
τ
0
as:
D
0
θ
a
2
τ
0
=
(5.14)
