# To: DAVID BUMP Re: RADIOMETRIC DATING It's been a while since I posted an explanation of r

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From: HENRY SHAW
To: DAVID BUMP

It's been a while since I posted an explanation of radiometric dating here,
and it looks like we have enough new readers that it's time to go through
the details again. I apologize in advance for the length of this message,
but I wanted to cover quite a few points.

My training is in isotope geochemistry, a field that includes, among other
things, the age-dating of rocks using naturally occurring radioactive
elements.  Although it has never been the major part of my research, I have
dated a number of rocks in the past.

decay at a constant characteristic rate according to the law:

dP(t)
decay rate =   -----  = -k*P(t)
dt

where P(t) is the number of radioactive (or "parent") atoms present at time
t. The constant k is called the decay constant, and is related to the
halflife (T1/2= ln(2)/k).  We can easily integrate this equation to get:

P(t) = P(0)*exp(-k*t)

where P(0) is the number of parent atoms present initially (at some
arbitrary time that we can call t=0).

This law has been found to hold to extremely high accuracy, and there is no
reason to suspect that it has not held for all times in the past (at least
as long as there have been atoms).  There is no reason to suggest, as is
often done by Biblical creationists, that the values of the decay constants
have changed with time; there is simply no evidence for this, and there is
quite a bit of evidence against it.  I will return to this point later.
The assumption that the above decay law has always held is, however, the
first and most fundamental assumption involved in radiometric dating.  It's
not much of an assumption, because if the decay constants varied over time,
one would have to throw out much of the rest of physics, too.

Every radioactive atom that decays turns into a "daughter atom" (I know
it's sexist, but no one calls them "son atoms".)  The transformation is
exactly 1-for-1, so that the increase in the number of daughter atoms is
exactly equal to the decrease in the number of parent atoms:

dD(t)/dt = -dP(t)/dt
=  k*P(t)
=  k*P(0)*exp(-kt)

This equation can be integrated to give:

D(t) = D(0)+ P(0)*[1-exp(-k*t)]

where D(0) is the number of daughter atoms at time t=0 (chosen arbitrarily,
but the same time we chose for P(0)).

The factor P(0) can be eliminated by substituting in the equation for P(t)
given above.  If we do this, we get:

D(t) = D(0) + P(t)*[1-exp(-k*t)]/exp(-k*t)
D(t) = D(0) + P(t)*[exp(k*t)-1]

This equation is the basis for nearly all radiometric dating techniques.
(C-14 is one exception, but C-14 is not used for dating old rocks;  it can
only be used to date things a few tens of thousands of years old because
its halflife in only about 5700 years.)

There are quite a few naturally occurring radioactive elements that can be
used to date rocks.  The following are the most commonly used parent-
daughter pairs:

Parent        Daughter       Halflife     Decay
Isotope        Isotope       (x10^9yrs)     Mode
(P)            (D)

K-40           Ar-40         11.97  electron capture &
positron emission
Rb-87          Sr-87         48.9        beta decay
Sm-147         Nd-143       106.         alpha decay
Lu-176         Hf-176        37.         beta decay
Re-187         Os-187        45.         beta decay
U-235          Pb-207        0.704 long chain of alpha
and beta decays
U-238          Pb-206         4.47            "
Th-232         Pb-208        14.0             "

Some rarely used pairs include:

K-40           Ca-40          1.43        beta decay
La-138         Ce-138       349.          beta decay
La-138         Ba-138       165.    electron capture

So how is any of this useful?

Let's say we have a box, and we put some radioactive "P" atoms into the
box. For the sake of argument right now, let's say there are no daughter
atoms in the box to start with.  We seal the box very tightly so that none
of the parent or daughter atoms can escape.  Let the moment we seal the box
be called time t=0.  Now, let the box sit for a while, then open it and
count the number of both the parent and daughter atoms.  That is, we now
know P(t) and D(t) in the above equation, and we assumed that D(0) = 0.  We
can look up the value of k for our radioactive nucleus in a handbook.  That
means the only unknown variable in the above equation is t, the time since
we sealed the box, and we can therefore solve for its value:

1
t = --- * [ln(D(t)/P(t))+1]
k

The extension to the case of a rock or mineral should be obvious.  This
simplest case corresponds to a rock or mineral that forms and incorporates
a radioactive element while completely excluding the daughter element.  By
measuring the amount of parent and daughter isotope present now, we can
calculate the age of the sample if:

1)  None of the daughter was present initially
2)  The sample has neither gained nor lost parent or daughter since it
formed.

There are a few natural cases that are very good approximations to the
first condition.  For example, one dating technique is based on the branch
of K-40 decay that results in Ar-40.  Now, Argon is a gas, a so-called
"noble gas" for that matter, and it simply does not fit or bond properly
into the crystal structure of any mineral.  It is thus strongly excluded
from a crystal growing in, say, a body of molten rock (magma).  On the
other hand, there are many minerals that include potassium as an intrinsic
part of their structures (examples include K-feldspar, biotite, muscovite,
amphiboles).  In conventional K-Ar dating, one assumes that the mineral
being dated contained no Ar when it formed, and that any Ar-40 present is
due to the decay of K-40 in the mineral.  There are more sophisticated
techniques that allow one to check on the assumption of no initial Ar, and
in general, it has been found that unless one is dealing with rocks that
formed under high pressures (deep in the earth), "initial" or "inherited"
Ar is not a problem.

About the only other system that conforms to this simplest case is the U-Pb
system in the mineral zircon (and a few other less suitable minerals).
Zircons readily incorporate uranium into their crystal structure, but
exclude lead, which is the last daughter product in a long chain of
intermediate daughters produced by the decay of uranium.  In the case of
Pb, we can check to see if any Pb was present initially.  Pb has 4
isotopes, Pb-208 (produced by the decay of Th-232), Pb-207 (produced by the
decay of U-235), Pb-206 (produced by the decay of U-238), and Pb-204, which
has no radioactive parent.  If there is any Pb-204 present in the zircon
then it is an indication that there is some "common Pb", (Pb not due to the
in situ decay of U or Th) in the sample.  One can then either argue that
the amount of common Pb is too little to affect the age, or make the
appropriate corrections to account for the initial presence of Pb.

In all cases, it is important that the minerals have remained "closed
systems" since they formed, and that parent and daughter elements have
neither been gained nor lost.  I will return to this point in a while.
First, I want to explain how one can date things that contain both some
parent and daughter element initially.

Let's go back to the equation I wrote before:

D(t) = D(0) + P(t)*[exp(k*t)-1]

There are five variables in this equation (D(t), D(0), P(t), k, and t).  We
can measure the quantities of parent and daughter elements pesent in a
sample today (D(t), and P(t)) and we can look up the value of k.  That
means that we know, or can measure 3 out of the 5 unknowns.  We need to
find a way to find out what D(0) was as well as what t is (t is the age of
the rock).  We could do this if we had *two* equations in the two unknowns.
Let's make a small change in the above equation.  Let's divide both sides
of the equation by the number of atoms of a stable isotope of the daughter
element that has no radioactive parent.  Call this quantity "S".  Since S
has no radioactive parent, the number of atoms of S does not change with
time.  The equation now looks like:

D(t)   D(0) + P(t)
---- = ----   ---- *[exp(k*t)-1]
S      S       S

Now, instead of measuring the number of atoms in the sample, we will
measure the RATIOS D(t)/S and P(t)/S, which is actually a much easier thing
to do anyway.  To make this specific, let's take the Rb-Sr system as an
example.  Sr, the daughter element has a number of isotopes, only one of
which, Sr-87, is produced by the decay of a radioactive parent, Rb-87.  We
normally use Sr-86 as the stable isotope to do the ratioing:

Sr87       Sr87         Rb87
----(t) =  -----(0)  +  ----(t) * [exp(k*t)-1]
Sr86       Sr86         Sr87

The same idea holds for any of the parent-daughter pairs in the table I
gave above.

Let's say we have a rock that formed by crystallizing from a magma (an
igneous rock).  A rock is made up of several different types of minerals.
Each mineral has a distinct chemical composition and crystal structure.
Some of these minerals will "like" Rb more than Sr and will incorporate
more of it into their structures, so in general, each mineral will have a
different Rb/Sr ratio when it forms.  On the other hand, for elements
heavier than about mass 32 (sulphur), different isotopes of the same
element behave essentially identically chemically, so each mineral will not
distinguish between Sr-86 and Sr-87 when it is forming and it will
incorporate Sr with the same isotopic composition (Sr87/Sr86 ratio) as the
magma bulk magma.  The magma cools, and eventually the temperature falls
enough so that the diffusion of atoms into and out of the crystals
(minerals) essentially stops (this is called the closure temperature).  As
long as the rock remains undisturbed from then on, the mineral grains act
as closed systems.  This point in time is what we will get as the "age" of
the rock.  Different minerals have different closure temperatures, but for
a rock that is hundreds of millions of years old, the difference in time
between closure for the different minerals is not normally significant,
except, perhaps, for the K-Ar system.

Our rock sits around for a bunch of years, until eventually it is exposed
at the surface and some geologist whacks a chunk of it off with his hammer,
takes it back to the lab, and carefully separates the different minerals.
He/she then analyzes the minerals for their Rb87/Sr86 and Sr87/Sr86 ratios.
The minerals that had higher Rb/Sr ratios when they formed will have higher
Sr87/Sr86 ratios today because there was more radioactive Rb87 relative to
Sr86 in those minerals.  If we have just two minerals with different
parent/daughter (Rb/Sr, in this case) ratios, then we can solve the
resulting two equations for BOTH the age, t, and the initial Sr87/Sr86
ratio.  In practice, one analyzes as many minerals as possible from a
sample, and does a least-squares fit to the data to obtain the values of
the two unknown.  This method is known as the isochron method.

One can portray this situation graphically, by plotting the measured
Sr87/Sr86 on the y-axis, and the measured Rb87/Sr86 ratio on the x axis.
If all the samples incorporated Sr with the same Sr87/Sr86 ratio initially,
AND each mineral has behaved as a closed system since it formed, then the
analytical data will lie on a straight line in this diagram (see figure),
with intercept equal to the initial Sr87/Sr86 ratio, and a slope that is
related to the age of the sample by:

t = 1/k*ln(1+slope)

Isochron Diagram
/
|                    X
|                  /
|                /
|              /
87Sr  |            /
----  |          X   <-- measured compositions at some
86Sr  |        /         later time
|      /
|    /
|  /X
initial ->|/____X__________X__________X____ <-initial
Sr    |   min1       min2        min3     compositions
|
|
|--------------------------------
87Rb/86Sr

If the data do NOT lie on a straight line, then we know that one of the two
assumptions was not satisfied (common initial Sr composition and closed
system behavior).  This provides a very powerful internal check on the
validity of the assumptions, and it is quite common to find that the
assumptions were not satisfied, in which case, one does not (or should not)
claim to know the age of the rock.

Instead of individual mineral grains, this same technique can be applied to
samples of "whole rocks" that are "related".  For instance, when a magma
chamber cools, some minerals crystallize at lower temperatures than others,
so the chemical composition of the solid rock that is forming will change over
time
as different minerals solidify.  These different composition rocks will
have different Rb/Sr ratios.  Since they all crystallized from the same
parent magma, they all started with the same Sr87/Sr86 ratio, and we can
treat them just the same as in the case of individual minerals I discussed
above.  This method has some advantages, which I'll discuss below, but it

It often is not clear that a suite of rocks *did* crystallize from the same
parent, and even if they did, the Sr isotopic composition of the magma can
change over time, for instance by mixing in of new magma formed by melting
the rocks enclosing the magma chamber.  In addition, if one has two
*different*, completely unrelated magmas, the different rocks formed by
mixing the magmas in different proportions will lie on a straight line in
the isochron diagram.  This line has absolutely no age significance,
however.  One must be careful, then, in interpreting "whole rock" isochrons
in terms of ages.  Creationists like to take examples of such "pseudo-
isochrons" and use them to try to invalidate the entire idea of radiometric
dating, which is, to my mind, intellectually dishonest behavior.

A few days ago someone on the echo asked several *excellent* questions

> 1) Can ALL rocks be dated with this method? i.e. what
> percentage of rocks do actually contain Uranium, Argon or
> whatever...

For the K-Ar method, one needs minerals that contain large amounts of K, of
which there are several.  Not all rocks contain these minerals so not all
rocks can be dated with K-Ar.

For the U-Pb method using zircons, obviously, the rock must contain
zircons.  Not all rocks do.  Granites and related rocks are the most
commonly dated using this technique.

To obtain an accurate age using the isochron method, which can use any of a
number of parent-daughter pairs (including U-Pb), what is important is that
the sample have a reasonably large spread in parent-to-daughter ratio.  One
can see that this is important by noting that the age of a sample is
determined by the slope in an isochron diagram, a large spread will define
the slope of the "isochron" better. In most cases, the absolute
concentrations are not that important because we have *very* sensitive
methods for measuring isotopic compositions and concentrations.  For
example, I can measure the 143Nd/144Nd ratio to better than 3 parts in
10,000 on as little as 5 nanograms of Nd (about 2 x 10^13 atoms).  Nature
is a sloppy chemist and there is a little bit of every element in
everything.  Techniques have been developed over the years to work with
trace levels of these elements in geological samples.

More limiting on the types of rocks that can be dated are some of the other
assumptions.  For instance, sedimentary rocks, which are formed by the
accumulation of debris from the weathering and erosion of other rocks
cannot usually be dated radiometrically.  They are mixtures of random
pieces of other rock, which, in general, will not be related to one
another.  Igneous rocks, which crystallize from a melt (magma) are usually
suitable, though not always.  Metamorphic rocks, which form by heating and
pressurizing other rocks (either igneous, sedimentary, or metatamorphic)
until chemical reactions occur that transform preexisting minerals into new
ones, are sometimes suitable, depending on whether the reactions go to
completion.  Sedimentary rocks are dated "indirectly", usually by
bracketing their age with ages determined on igneous rocks that pre- and
post-date the deposition of the sediments.

> 3)Is it possible to know the original ratio of, e.g.
>  Uranium to Lead in the sample.

The initial parent/daughter ratio is not important, but yes, it is possible
to determine the important initial ratios.  See above

> 4) Is it possible for some of the parent or daughter
> atoms to be removed from the sample by natural means,
> e.g. water percolating through rocks...

ABSOLUTELY.  Radiometric dating assumes that the systems being dated are
closed.  It is quite common for rocks and minerals to have been open
systems, in which case, an attempt to construct an isochron will fail.  One
can therefore tell when this has happened.  Generally, this makes it
impossible to obtain an age on the sample.  The U-Pb method using zircons
is special, because there are *two* isotopes of U involved, which decay at
different rates to different isotopes of Pb.  Using a diagram known as a
concordia, one can deal with the situation in which a zircon undergoes a
single episode of open system behavior.  In this case, one can determine
both the initial time of formation of the zircon, as well as the time at
which the open system behavior occurred.

There are two basic ways rocks and minerals can behave as open systems;
diffusive loss/gain of material; and chemical reactions caused by a change
in temperature, pressure, or composition (e.g., the introduction of a fluid
like water).  The second case is simply a form of metamorphism.  If the
reactions go to completion (i.e. equilibrium), one ends up with a whole new
set of minerals, and the radiometric "clock" gets reset.  An age
determination will give you the time at which the disturbance occurred.  If
the reacions don't go to completion, then one has a mixed situation, and
because the minerals don't all react at the same rate, you won't get
anything that looks like an isochron and you can't determine the age.

Diffusive loss of an element, usually the daughter, can occur if a rock is
heated up so that the rates of diffusion of elements in the solid are
significant.  In most cases, a mineral structure is not very "happy" about
having daughter element present in the structure where once there was an
atom of the parent because usually the parent and daughter have different
chemical characteristics.  For example, Rb is a large, monovalent cation
while its daughter, Sr is a much smaller divalent cation which really
shouldn't be where it finds itself in the structure.  The crystal would
like to rearrange itself to get rid of this intruder, which it can do via
diffusion.  Again, this behavior would show up in an isochron diagram.  The
Sm-Nd isotopic system is rather resistant to this sort of open system
behavior because both Sm and Nd are rare earth elements, and behave very
similarly geochemically.

> 2) Can the rate of radioactive decay be influenced by external events,
> e.g.  sunspots, fluctuations in C etc...

The decay constant for electron-capture, in which a neutron-deficient
nucleus "captures" one of the orbital electrons, can be affected by the
chemical and physical environment of the atom (these effects have been
measured), but, so far as we know, other decay modes are not affected by
external events, and have remained constant over time.  Dave Knapp recently
posted a message that explained how the decay of emissions from the
Supernova 1987a prove that the decay constant for an isotope of Ni (?) has
not changed in the past several thousand years.

Creationists often raise this point in an attempt to invalidate radiometric
dates, but there is simply no basis for proposing this.  On the other hand,
let's say the Earth really formed in 4004 BC, making it ~6000 years old
instead of ~4,570,000,000 years old.  Now, when a nucleus decays, it
releases energy (which is why it decays in the first place).  This energy
is converted to heat (radioactive decay is why the interior of the earth is
hot).  If we have to compress all the decays that have occurred in the
4.57b.y. history of the earth into 6000 years, then on average, the heat
generation rate would have been 4,57x10^9/6x10^3 = 7.6x10^5 times what it
is today.  There is simply no way that this much heat could have been lost
from the planet in that short a time; the earth would still be largely
molten underneath a thin solid crust....unless, of course, you change the
fundamental physics of heat transfer, which a die-hard Biblical creationist
would probably have no qualms about proposing, because increasing the
decay rate requires changing the fundamental physics of the electro-weak
interactions, strong interactions...in short, most of modern physics.

Finally, there is no counterargument to the creationist argument that god
created everything in just such a way as to make it *appear* old.  God
could then have created the universe at the moment I typed the next letter
in this sentence, with everyone's memories of a false past, light on its
way to us from distant stars, just the right parent-daughter relationships
in rocks, and so on.  This, however, leads to an intellectually bankrupt,
know-nothing philosophy that I find repugnant.

That's all, class.  There will be a short quiz on this material next
echo-period.  ;-)

-!- TBBS v2.1/NM
! Origin: Diablo Valley PCUG-BBS, Walnut Creek, CA 415/943-6238 (1:161/55)

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