\nextTopic{gd_07.latex}
\previousTopic{gd_05.latex}
\chapter{Isotope Geodynamics}
The crust, mantle lithosphere and asthenosphere represent distinct chemical reservoirs.
The evolution of these reservoirs through time can be traced by analysis of their
radiogenic isotopic signatures. In this section we provide the essential concepts needed
to utilize radiogenic isotopes as a constraint on the evolution of these
reservoirs. Three decay schemes are used extensively in the study of geodynamics: the
{Rb-Sr} scheme, the {Nd-Sm} scheme and the {U-Pb} scheme.
\section{Geochemical reservoirs}
Why are there distinctive geochemical reservoirs? The differentiated state of the present
Earth is largely due to the formation and segregation of {\it partial melts} in the
interior of the planet. Because magmas are less dense than the solid rocks in which they
form, they naturally migrate towards regions of lower pressure. The key
process in the evolution of distinct geochemical reservoirs formed due to the segregation
of partial melts is the way in which elements are partitioned between the melt and the
residual solid. A simple model for the variation in concentration of elements in magmas
during partial melting based on batch melting in which the melt stays in equilibrium with
the solid until the specified fusion proportion is: \begin{equation}
\label{eq:shaw} \frac{C_{l}}{C_{o}} = \frac{1}{F+D \,(1-F)} \end{equation} where
{$F$} is the fraction of fusion, {$C_{o}$} is the concentration of the element in the
initial solid, {$C_{l}$} is the concentration of the element in the melt, {$ D$} is the
bulk distribution coefficient:
\begin{equation} \label{eq:distr} D = \sum_{i=1}^{n} w_{i} K_{D_{i}} \end{equation}
where \[ K_{Di} = \frac{X_{i}^{a}}{X_{melt}^{a}} \] and where $n$ is the number of
mineral phases, $w$ is the weight fraction of mineral $ i$, and $X_{i}^{a} $ is the
concentration of element $a$ in phase $i$. The distribution coefficient of trace elements
with respect to specific minerals is a function of ionic charge and radius. Different
minerals have different affinities for individual trace elements and therefore different
$K_D$ values. The $D$ values represent the effective {\em bulk distribution coefficient}
for the whole mineral assemblage coexisting with the melt. Trace elements that have high
$D$ values are referred to as {\it compatible}, while those with $D < 1 $ are referred to
as {\it incompatible}. Note that what constitutes a compatible or incompatible element
depends on the specific mineral assemblage. Thus, Sr is an incompatible element for
mantle-melting involving olivine and pyroxene assemblages whereas it is a compatible
element during crustal melting where feldspars are involved. The crust is the repository
of trace elements which are incompatible with respect to the mineralogy of the Earth's
mantle.
Equation \ref{eq:shaw} has a number of important special solutions (Figure
\ref{shawplot}). If $D = 1$ then $C_{l}/C_{o} = 1$ and there is no fractionation between
source and magma. If {$D >> 1$} then $C_{l}/C_{o} < 1$ and the element is preferentially
retained in the unmelted residue. If $D = 0 $ then $C_{l}/C_{o}$ tends to $1/F$ and
the trace element is strongly partitioned into the melt phase and its enrichment is simply
a function of the amount of melting. Figure \ref{shawplot} also illustrates the effect of
extraction of partial melts on the relative concentration of trace elements in the
unmelted solid residue. Note that highly incompatible elements are very quickly and
efficiently extracted from the melting solid and the residues of such melting are highly
depleted in these elements.
\begin{figure}[h,t,b] \label{shawplot} \vspace{3.5in} \hspace{0.5 in}
\special{epsf=:gdfigs:chap5:Fig5_1.eps
scale=0.75}
\caption{ \protect\scriptsize{Solutions to Eqn 4.1 are shown as a
function of the proportion of fusion the degree of enrichment or depletion relative to
their source ({$C_{o}$}), of incompatible and compatible trace elements whose bulk
distribution coefficients range from zero to ten. }} \end{figure}
The solution to Eqn \ref{eq:shaw} for varying amounts of melting of two trace elements
both of which have {$D < 1$} but for which $ D_{E_{1}} (= 0.01) < D_{E_{2}} (= 0.8) $
where both elements have $C_o = 5$ppm is shown in Figure \ref{frac},
Figure \ref{comp}
illustrates the variation of the ratio {$(E{_1}/E_{2})_{liquid}$} as a function of F.
Clearly, for small to moderate amounts of melting, there is a very significant
fractionation of the two trace elements; and the ultimate destination of the melts when
segregated (i.e., the outer regions of the Earth) will become preferentially enriched in
{$ E_{1}$} compared to {$E_{2}$}. By contrast, the residue of melting always has a lower
ratio {$E_{1}/E_{2}$} than either the original source or the derived melt.
\begin{figure}[h,t,b] \label{frac} \vspace{2.2in} \hspace{0.4in}
\special{epsf=:gdfigs:chap5:Fig5_2.eps
scale=1.0} \caption{ \protect\scriptsize{Concentration in the melt produced by
progressive partial fusion of trace elements whose bulk distribution coefficients are 0.01
and 0.8. Initial concentration in the source was 5 ppm. Also shown is the concentration in
the residual solid of the element whose {$D = 0.01$}. } } \end{figure}
It is important to note that Eqn \ref{eq:shaw} describes equilibrium batch partial
melting which assumes that the liquid remains in contact and equilibrium with the residue
up to the specified degree of fusion, at which stage it is totally extracted. This is a
rather simplified model of the processes likely to operate in the earth and we will
discuss some more physically realistic mechanisms later.
\begin{figure}[h,t,b] \label{comp} \vspace{2.1in} \hspace{0.7in}
\special{epsf=:gdfigs:chap5:Fig5_3.eps
scale=1.0}
\caption{\protect\scriptsize{ The concentration ratio of a highly
incompatible element ({$C_{l} (1)$}) whose {$D = 0.01$} and a moderately incompatible
element ({$C_{l} (2)$}) whose {$D = 0.8$} in the melt formed by progressive fusion. }}
\end{figure}
\section{Radioactive isotopic decay}
We can use the models of the type described above to infer something about the
compositions of the source regions of magmatic rocks. However, as discussed in the
previous section, partial melts of rocks only reflect the composition of those source
rocks indirectly. By contrast, it is assumed that at the time of melting, the isotopic
compositions of the liquid and it's source rocks are the same and thus the initial
isotopic compositions of magmas are the same as those of their source regions at the time
of melting.
Naturally occurring radioactive isotopes decay spontaneously to yield stable daughter
isotopes. The rate of decay is a physical constant unique to each specific decay scheme.
We refer both to the decay constant, {$\lambda$}, and to the half-life, $t_{1/2} = (ln \,
2) /\lambda $, of an isotope. The half life being the time taken for half the initial
number of parent isotope nucleiides to decay to daughter products.
The equation describing radioactive decay as a function of time is: \begin{equation}
\label{eq:radio} N = N_{0} \, exp\left(-\lambda t \right) \end{equation}
where {$N_{0}$} is the initial number of radioactive parent atoms and {$N$} the number
left after time {$t$}. The numbers of new daughter atoms, {$d^{*}$}, produced in time $t$,
is given by: \begin{equation} \label{eq:radt} d^{*} = N_{0} \left(1
-exp\left(-\lambda t \right) \right) \end{equation} In most geological materials, there
will already be some of the daughter element present at the start of the time interval of
interest which must be added to by the products of radioactive decay. Thus the total
radiogenic isotope concentration, {$d$}, of a rock or mineral is given by:
\begin{equation} d = d_{0} + N(exp\left(-\lambda t\right) -1) \end{equation} and is
clearly dependent on: \begin{enumerate} \item the amount of the parent isotope present at
the start, \item the amount of the daughter isotope present at the start, {$d_{0}$} and,
\item the time interval during which this system was closed, {$t$} and at this stage the
amount of the parent isotope left is {$N$}. \end{enumerate}
Isotopes in geological samples are analyzed by use a mass spectrometer as a ratio
relative to a reference non-radiogenic isotope of the same element. In the Sr-system, the
reference isotope is $^{86}$Sr and in the Nd-system it is $^{144}$Nd. The so-called
isochron equation shows that after the passage of $t$ years (in 10$^9$ units) the present
day isotope ratio of a closed system depends on its initial ratio and the ratio which is
purely a function of the concentration ratio of the parent and daughter elements. For the
Rb-Sr scheme the isochron equation is \begin{equation} \label{eq:sr} \left(\frac{
^{87}Sr}{^{86}Sr}\right) = \left(\frac{ ^{87}Sr}{^{86}Sr}\right)_{i} + \left(\frac{
^{87}Rb}{^{86}Sr}\right) \left(exp\left(\lambda\, t\right) - 1\right) \end{equation}
(note that in this case $^{87}$Sr is the radiogenic isotope and $^{86}$Sr the reference
and $^{87}$Rb is the radioactive parent however we could equally apply the same equation
to the other decay schemes with appropriate decay constants).
\section{Isotopic fractionation during melting} For elements of high atomic mass
melting processes do not affect the ratios of two isotopes of the same element because the
fractionation is the same for each. However, melting frequently does produce large
fractionations between parent and daughter isotopes. The relationship between the
isotopic ratio, $\alpha$, the parent to daughter ratio, $\mu$, which is a compositionally
dependent term and the time of differentiation, $t$, in a closed system is given by
\begin{equation} \label{eq:rmelt} \frac{d\:\alpha}{d\:t} = \lambda \: \mu
\end{equation} where $\lambda$ is the decay constant. The importance of this is
considerable not only for the determination of specific rock ages, but also because it
provides a connection between the chemistry of specific regions of the earth and their
radiogenic isotopic compositions. As discussed in Section 4.1 the changes in
concentration of trace elements in both extracted melts and in residues during fusion
depends on the $D$ value. Because parent and daughter elements will have different {$D$}
values partial melting will produce melts and residues whose parent to daughter ratios are
different from the source. For instance in general {$D_{Rb} < D_{Sr}$ and $D_{Sm}
>D_{Nd}$. Therefore melts have higher Rb/Sr and lower Sm/Nd ratios than their sources.
Figure \ref{rbehav}a illustrates the general process of the production of a new separate
reservoir, {$M$}, by melting and melt extraction of a source, {$S$}, at time, {$TD$},
leaving a slightly modified source region, {$R$}. This specific example is true for a
situation in which the parent isotope is more incompatible than the daughter (as is the
case for Rb - Sr). If the opposite were true (as in the case of Sm-Nd), the extracted melt
trajectory would be less steep than that of the residue. Figures \ref{rbehav}b and c
illustrate two other petrogenetic processes which are also relevant. Of particular
importance is isotopic homogenisation which will occur by diffusion during geologically
meaningful timescales at temperatures above the so-called blocking temperature. It is an
important observation that even at temperatures above the blocking temperature, the rate
of diffusional equilibration between adjacent solids or non-convective, non-mixing liquids
is only of the order of a meter in {$10^{7}$} years. For this reason the establishment
of large separate geochemical reservoirs like the crust or asthenospheric upper mantle,
requires both large scale melt extraction and convective heat and mass transfer.
\begin{figure}[h,t,b] \label{rbehav} \vspace{3.1in} \hspace{0.3in}
\special{epsf=:gdfigs:chap5:Fig5_4.eps
scale=0.75}
\caption{\protect\scriptsize{ Generalized model for three broad types of
temporal control on rock radiogenic isotopic composition. a) differentiation by partial
melting, S = source, m = melt, TD = time of melting. b) mixing of two compositionally and
isotopically distinct melts at time TM to form a homogeneous hybrid. c) Solid-state
isotopic homogenisation of two adjacent rocks masses or minerals, A and B, due to
metamorphism. } } \end{figure}
\section{The chondritic earth}
An important process in the chemical differentiation of the earth from an initial
compositionally homogeneous globe has been the generation and segregation of partial
melts to form several new separated reservoirs (crust, asthenosphere etc). A comprehensive
geodynamic model of the earth requires knowledge of the age, life-span and source of these
separate regions.
Because the earth has undergone continuous differentiation since its formation at about
4.55 x 10$^{9}$ years ago it is not a simple problem to determine what the bulk
composition of the whole, undifferentiated planet was. We only have easy access to the
outer crustal portions of the earth and the entire crust only represents a small
proportion of the earth's total mass. As we have already seen, this part of the earth has
an unrepresentative enrichment of those elements whose distribution coefficients with
respect to the mantle minerals are very low. For instance approximately 70\% of the
earth's entire supply of K, Rb and Cs all now reside in the crust.
By contrast, the Moon which is a considerably smaller planetary body, has undergone much
less differentiation than has the earth. It's crust is much more like what we suppose it's
whole composition to be and the oldest dates obtained from Moon rocks are much closer to
4.55 Ga. than the oldest terrestrial dates (Amitsoq Gneiss from S.W. Greenland dated at
3.9 Ga.). This illustrates that some form of crustal re-cycling on earth has destroyed
most of the original crust, activity absent from the Moon since the earliest Proterozoic.
Meteorites are pieces of a fragmented planetary body (probably more than one) which
probably occupied an orbit between earth and Mars. Like the Moon, this appears to have
had only a short history of geological activity and was then broken-up to form the
asteroid belt. Meteorites therefore represent a unique opportunity to directly examine
and analyze samples from the interior of a terrestrial planet like earth. They fall into
three groups: \begin{itemize} \item Metallic Iron-Nickel alloy meteorites; fragments of
the core of an earth-like planet. \item Stoney meteorites; these are mainly silicate-rich
composed of olivine and pyroxenes and are derived from the equivalent of the earth's
mantle. \item Chondritic meteorites; a unique class of meteorites which have had no high
temperature history. They are carbon-rich and often contain significant concentrations of
volatile compounds. \end{itemize} The stoney meteorites provide mineral isochrons in
both the Rb-Sr and Sm-Nd systems (O'Nions et al, 1979) that indicate the age of the
terrestrial planets to be 4.5 -4.55 Ga. and indicate initial $^{87}$Rb / $^{86}$Sr and
$^{143}$Nd / $^{144}$Nd ratios to be 0.69898 and 0.50682, respectively.
The chondritic meteorites have not been part of a planet that has differentiated core and
mantle and their bulk composition is thought to be very similar to the inter-stellar dust from which the
planets accreted. The first approximation of the whole-earth's composition is therefore
the composition of the chondritic class of meteorites and it is informative to present
data on terrestrial samples normalized relative to this as illustrated by Figure
\ref{chond}.
\begin{figure}[h,t,b] \label{chond} \vspace{2.5in}
\special{epsf=:gdfigs:chap5:Fig5_5.eps
scale=0.95}
\caption{ \protect\scriptsize{The dashed line represents the normalized rare
earth content of of a chondritic source and the normalized compositions of melts due to 1
and 15\% partial fusion are shown. The residue in equilibrium with the 15\% melt is the
lower most curve.}} \end{figure}
Given that we are able to define the initial Nd and Sr isotopic composition and age of
the Earth and we can use the composition of chondritic meteorites to define its Rb/Sr and
Sm/Nd ratios, we are now in a position to predict the variation in the $^{87}$Rb /
$^{86}$Sr and $^{143}$Nd / $^{144}$Nd ratios for this chondritic uniform reservoir (CHUR)
as a function of time (McCulloch and Wasserburg, 1978). The "mantle" lines in the the two
lower diagrams in Figure \ref{res} are a representation of this reservoir.
It is very common that the Sr and Nd isotopic compositions of rocks are not quoted
directly as ratios but are compared to the expected values of the chondritic reservoir.
This is referred to as epsilon, {$\epsilon$}, notation. \begin{equation}
\label{eq:eps} \epsilon_{Nd}\left(0\right) =
\left(\frac{\left(\frac{^{143}Nd}{^{144}Nd}\right)\left(0\right) }
{I_{chur}\left(0\right) } - 1\right) \; {\rm x} \;10^{4} \end{equation} where
\begin{displaymath} I_{chur}\left(0\right) = I_{chur}\left(T\right) +
\left(\frac{^{147}Sm}{^{144}Nd}_{Chur}\right) \,\left(e^{\lambda\,t} - 1\right) \
\end{displaymath} \begin{displaymath} \lambda = 6.54 \; {\rm x} \;10^{12}
\end{displaymath} \begin{displaymath} I_{chur}\left(0\right) = 0.512638 \; \; \;
{\rm and}\; \; \;
\frac{^{147}SM}{^{144}Nd_{chur}} = 0.1967 \end{displaymath} $\epsilon_{Nd}\left(0\right)$
(or $\epsilon_{Sr}\left(0\right)$) refer to the comparison between the present day value
of the chondritic reservoir and the present day isotope value of the sample (ie the
measured value). If this rock is not of zero age and if the age is known, then its initial
isotopic composition at its age, {$T$}, can be calculated and compared to the
chondritic reservoir at that time, {$\epsilon_{Nd}\left(T\right)$} and
{$\epsilon_{Sr}\left(T\right)$}. In the upper diagram of Figure \ref{res}, the axes could
equally be presented as isotope ratio values or $\epsilon$ values. The "bulk earth"
position in this diagram is the intersection point of $\epsilon_{Nd}\left(0\right)$ = 0
and {$\epsilon_{Sr}\left(0\right)$} = 0. This figure shows the field modern day ocean
island basalts (OIB) and mid ocean ridge basalt (MORB). These reflect the composition of
the present day mantle and show that though some ocean island magmatism is sampling mantle
whose composition is close to the predicted chondritic Earth, the modern mantle shows some
very strong deviation from this model. This is particularly so for those portions of the
mantle sampled by the voluminous magmatic production at the mid ocean ridges. On this same
diagram from Figure \ref{res}, the field of continental crust is also shown. This has high
$^{87}$Rb / $^{86}$Sr (or positive $\epsilon_{Sr}$) values and low $^{143}$Nd / $^{144}$Nd
(or negative $\epsilon_{Nd}$) values. As the range of Sm/Nd ratios of continental rocks
is relatively limited, Eqn \ref{eq:eps} reduces to express $^{143}$Nd / $^{144}$Nd as a
function of time and this is illustrated as the age contours in the continental crustal
quadrant in Figure \ref{res}. Note that this is not so for $^{87}$Rb / $^{86}$Sr as there
is a very large range of Rb/Sr values for common crustal rocks.
\begin{figure}[h,t,b]
\label{res} \vspace{6in}
\special{epsf=:gdfigs:chap5:Fig5_6.eps
scale=0.9}
\caption{\protect\scriptsize{
Upper: Present day Nd - Sr isotopic covariations for important terrestrial reservoirs.
MORB is mid ocean ridge basalts, OIB is ocean island basalts. Lower: Sr- and Nd-time
evolutionary diagrams. $T_{ model}$ is a model age. Data for the ocean island basalts
suggests that in addition to the depleted domain there still exists a mantle reservoir
that has not suffered relative Rb and Nd depletion. There is some suggestion that this is
a lower mantle or lower upper mantle reservoir which still has a chondrite-like chemistry.
This region is also a probable source of rare gas emissions including the now extinct
$^{129}$Xe suggestive of long term isolation without melt extraction. If
this were true it would have important implications in constraining the extent and
dimensions of mantle convective cells.}} \end{figure}
\section{The depleted mantle}
Figure \ref{chond} shows that for the partial fusion of a chondritic source like that
assumed to represent the undifferentiated earth, the degree of total rare earth element
enrichment in the melt diminishes with larger amounts of melting and that the residual
source is depleted in these incompatible elements. More importantly, the melts have lower
Sm/Nd ratios than the original pristine source and the residue has a higher Sm/Nd ratio.
To a simplified first approximation (with respect to the rare earth elements), Figure
\ref{chond} illustrates the evolution of the two important terrestrial geochemical
reservoirs: \begin{itemize} \item a melt repository (= crust) and, \item a source region
depleted in the crust forming elements by crustal extraction (= some part of the present
mantle). \end{itemize} The lower two diagrams in Figure \ref{res} show the present day
range of Nd and Sr isotopic compositions of the continental crust with respect to a mantle
reservoir. These represent simplifications of the true situation, because (Figure
\ref{chond}) the extraction of melts from the mantle bring about complimentary changes to
the composition of that mantle. A mantle region that has suffered partial melting and melt
extraction will have higher Sm/Nd and lower Rb/Sr and U/Pb ratios than the original
chondritic unmelted mantle.
Mid-ocean ridge basalts are melts of the contemporary decompressed asthenospheric mantle
beneath the oceanic ridges. Even though these are melts they have rare earth patterns
which are like the solid residue in Figure \ref{chond} and are strongly LREE-depleted.
This suggests they are the products of remelting of mantle that has already lost a melt
component before. In Figure \ref{res}, the MORB field is at high $\epsilon_{Nd}$ and low
$\epsilon_{Sr}$ confirming that their source mantle had achieved its depleted chemistry
over a long period of geological time (perhaps 2 Ga). We then view the continents as being
complimentary to the depleted mantle now being sampled by mid ocean ridge magmatism. The
average age of the continental crust is given by: \begin{equation} = \int_{0}^{T}\,
\tau_{m}\, \left(\tau\right) \, dt \end{equation} where $\tau_{m}$ is the mass of
increment of crustal addition aged $\tau$.