# Author: Chris Stassen Subject: FAQ: Isochron Dating Updated: 02/27/92 Outline: 1. Generic

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Author:               Chris Stassen
Subject:      FAQ: Isochron Dating
Updated:      02/27/92
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Outline:

2. What's wrong with non-isochron dating methods?
3. Generic Isochron Dating
4. What's NOT wrong with isochron dating methods?
5. For further information (some things to read)

Generally, radiometric dating is done by performing a simple
calculation on a sample, involving three measurements:

a)        The first "measurement" is actually a "known quantity" -- the
half-life of the radioactive element used by the method.  This
value can be experimentally measured in a lab -- but since many
experiments have failed to effect a noticeable change in the
rates relevant to radiometric dating, it is usually taken from a
table.

b)        The second measurement is the amount of "parent" element (the
radioactive element used by the method).

c)        The third measurement is the amount of "daughter" element (the
element that the radioactive one decays into).

Since each atom of the parent element decays into one atom of the
daughter element, we calculate that the original quantity of the
parent element is the sum of the current amounts of parent and
daughter elements.  We then apply the following (frosh physics)
equation (the infamous "radioactive decay" equation):

P =      P0 / (2 ^ (T / H) )

or   P = (P + D) / (2 ^ (T / H) )

Where:
P is the current amount of parent element
P0 is the original amount of parent element (= P + D)
T is time that has passed ("age" of the sample)
H is the half-life of the element

Solving for T, we calculate the sample's age as:

T = H * log2 ( (P + D) / P)

(2)  What's wrong with non-isochron dating methods?

Obviously, there are a few assumptions above which have been
made for the sake of a simple expanation, but which will not
always work in the real world.  These include:

a)        The amount of daughter element at the time of formation of the
sample is zero.  Possible ways to avoid this problem include:
work on a mineral that can't incorporate any of the daughter
compound when it forms; somehow calculate the amount of initial
daughter product and subtract it out.

b)        No parent element or daughter element has entered or left the
sample since its time of formation.  Possible ways to avoid this
problem include: only date samples whose geological history does
not appear to include events which might cause this problem; date
several different parts of the same sample and only accept the
result if they all agree because contamination is not likely to
affect all parts of a large sample in the same way.

The invention of isochron methods solves both of these problems

(3) Generic Isochron Dating

Isochron dating requires a fourth measurement to be taken, which
is the amount of a _different_ isotope of the daughter element.
In addition, it requires that the second through fourth
measurements be taken from several different objects which all
formed from a common pool of materials.  (Rocks which include
several different minerals are perfect for this.)

When any rock forms, minerals "choose" atoms for inclusion by
their _chemical_ properties.  Since the two isotopes of the daughter
element have identical chemical properties, they will be mixed
evenly when the sample forms.
However, the parent element, with different properties,
will not be mixed evenly relative to the daughter elements.  So,
at formation time, a sample would contain measurements like the
following:

Mineral No.     Parent          Daughter        Isotope
---------       --------        --------        --------
1            4 gm            1 gm            2 gm
2            2 gm            4 gm            8 gm
3            6 gm            2 gm            4 gm

Note that (for this example) there is always twice as much of the
"isotope" as there is of the "daughter" in every mineral.  Also
note that the ratio of "parent" element to either one of the
others varies (as the parent element has different chemical
properties).  After one half-life's worth of time has passed, the
values will have changed (as half of the parent atoms in each
mineral will have decayed into daughter compounds):

Mineral No.     Parent          Daughter        Isotope
---------       --------        --------        --------
1            2 gm            3 gm            2 gm
2            1 gm            5 gm            8 gm
3            3 gm            5 gm            4 gm

Note that half of the amount in the Parent column has been
taken away and added to the Daughter column for each mineral.
Also note that the Isotope column, since it doesn't decay and
isn't a decay product, doesn't change at all.

I can do some math here, but it's easier to see it on a graph.
The isochron graph is drawn by graphing D/I vs. P/I.  The
first set of measurements results in:

D/I    1 -
|
|
|
-     (2)................................(3)...........(1)
|
|
|
+-------------|-------------|-------------|-------------|
0             0.5            1            1.5            2

P/I

Note that all of the samples lie on a straight, flat line.  This
is what we expect: they all have the same D/I ratio,  and hence
the same Y-value.
Note that, if the sample were homogeneously distributed
with respect to parent and daughter, then all of the data points
fall on the same point and no line can be derived.

The graph for the second set of measurements is:

2 -
|
|
|
-                        .(1)
|                      ..
|                 ..(3)
|              ...
D/I    1 -                 ...
|         ..
|       ..
|    (2)
-
|
|
|
+-------------|-------------|-------------|-------------|
0             0.5            1            1.5            2

P/I

Once again, all the points lie on a straight line.  And the slope
of the line is 1.  (I know it doesn't look like it on the screen,
but that's because I used different units for X and Y -- you can
calculate it for yourself from the table above.)

We can make a simple table of slope of line versus age:

Slope                   Age
--------                -------------------
0                     0
1                     1 half-life
3                     2 half-lives
7                     3 half-lives
...                    ...
N                     log2( N + 1 ) half-lives

(4) What's NOT wrong with isochron dating methods?

Now that the mechanics of plotting an isochron have been
mentioned before and describe why isochron methods don't fall
prey to them.

a)        Initial daughter compound.

Any amount of initial daughter compound is compensated by the
isochron method.  If one of the minerals happened to have none
of the parent element (the Y-intercept of the line), then its
amount of daughter compound wouldn't change over time -- because
it has no parent atoms to produce daughter atoms.
Regardless of whether there's a data point there or not,
the Y-intercept of the line doesn't change as the slope of the
line does.  (You can verify this for yourself; the Y-intercept
of both lines above is 0.5.)  The Y-intercept of the isochron
line actually gives the ratio of daughter to the other isotope
at the time of formation.
For each mineral, we can then measure the amount of the
other isotope and calculate the amount of daughter product that
was present when the sample formed.  If we then subtract it out,
we could derive a "traditional" age for each mineral by the
equations described in the first section.  Each such age would
match the result given by the isochron.

b)        Random contamination (parent or daughter entering or leaving the
system)

For the sake of brevity, our example only included three data
points.  While isochrons are performed with that few data points,
their value is not treated as seriously as those which have tens
of points.
Any non-systematic contamination is _extremely_ unlikely
to leave all of the data points on the line.  Even in our little
example, any contamination of one of the minerals would require
a specific contamination of one of the other two in order to keep
all three points on the same line.

When we get to an isochron with tens of data points, the suggestion
that contamination "just happened to place the points on a (fake)
isochron line" can be discarded out of hand.  It's too unlikely.
[Now, there is a form of isochron contamination, known as "mixing,"
which basically amounts to a _partial_ resetting of the isochron
clock.  However, there are tests to detect it.]

c)        General dating assumptions

All radiometric dating methods must assume certain initial
conditions and lack of contamination over time.  The wonderful
property of isochron methods is that *if one of these assumptions
is violated*, it is nearly *certain* that the data will show that
by the points not plotting on a line.

(5) For further information, see:

G. Faure, _Principles of Isotope Geology_  (a textbook/handbook;
very technical, but very good.)

G. B. Dalrymple, _Radiometric Dating, Geologic Time, and the Age of
the Earth_  (Email Chris Stassen if you want a copy.)

A. N. Strahler, _Science and Earth History_, pp. 130-135

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