Author: Chris Stassen Subject: FAQ: Isochron Dating Updated: 04/22/94 (added Section 6) CO

```

========================================================================
Author:		Chris Stassen
Subject:	FAQ: Isochron Dating
========================================================================

CONTENTS:

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

3. Generic Isochron Dating

4. What's NOT wrong with isochron dating methods?

5. Some talk.origins questions on isochron methods

6. A little math

7. 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
at once.

--------------------------------------------------------------------------

(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 these measurements be taken from
several different objects which all formed at the same time 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:

---------	--------	--------	--------
Sample No.	Parent		Daughter	Isotope
---------	--------	--------	--------
1             4 ppm           1 ppm           2 ppm
2             2 ppm           4 ppm           8 ppm
3             6 ppm           2 ppm           4 ppm
---------	--------	--------	--------

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):

---------	--------	--------	--------
Sample No.	Parent		Daughter	Isotope
---------	--------	--------	--------
1		 2 ppm		 3 ppm		 2 ppm
2		 1 ppm		 5 ppm		 8 ppm
3		 3 ppm		 5 ppm		 4 ppm
---------	--------	--------	--------

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/Di vs. P/Di ("Di" is
the other isotope of the same element as the daughter product).
The first set of measurements results in:

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

P/Di

Note that all of the samples lie on a straight, flat line.  This
is what we expect: they all have the same D/Di ratio,  and hence
the same Y-value.

(Note: 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/Di   1 -		 ...
|	       ..
|	     ..
|	  (2)
-
|
|
|
+-------------|-------------|-------------|-------------|
0		  0.5		 1	      1.5	     2

P/Di

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) Some talk.origins questions on isochron methods

The following are interesting questions that were asked in talk.origins
about isochron dating.  The names of the "questioners" have not been
included because permission to use their names has not been obtained.

Q1:
# Sr-87?

For the Rb/Sr isochron method, the ratio of Sr-87 to Sr-86 (at time of
formation) is not needed as an input to the equation; is given by the
Y-intercept of the isochron line.  It falls out of the computation of
the method -- provided that all of the points fall on an isochron line.

Q2:
# What is isochron dating?  a method, an equation, a graph ...?

An "isochron" is a set of data points in a plot which all fall on a
line representing a single age ("isochron" comes from:  "isos" equal +
"chronos" time).  The line itself is also sometimes called an "isochron."
The plot on which these data points appear is sometimes called an
"isochron diagram" or "isochron plot."

A dating method which uses such a plot to determine age is called an
"isochron dating method."  When "isochron dating" is mentioned in
this FAQ, the intent is to cover the methodology which is common to
all "isochron dating methods."

Q3:
# How is the half life of an element determined?  For something that
# takes 60 billion years to partially decay, how is an exact measure
# of the decay rate determined in a few hours?

These experiments don't necessarily take only "a few hours."  Davis
et al. (See Faure p. 119) measured the decay rate of Rb-87 (48.9
+/- 0.4 billion years) by counting the accumulation of Sr-87 over a
period of nineteen years.

While it may take "60 billion years" for half of the atoms to decay,
in a large enough sample there will be many decays over a shorter span
of time.  If the sample's size can be measured accurately and is large
enough (in terms of number of atoms) to be statistically significant,
and the number of decays can be counted accurately, then the half-life
can be computed accurately.  That's the basis for the "direct counting
experiments" from which half-lives are calculated.

Q4-a:
# The line is telling us that no matter what size sample we take we
# always have the same ratio of parent to daughter.
[...]
# So let's say that when the rocks were formed, certain amounts of
# both the parent and daughter were present.  But in the process of
# forming, everything got evenly distributed.  You would get your
# nice straight isochron line, but still not know the age of your
# sample.

This statement would be correct if the isochron plot were quantity of
parent (P) versus quantity of daughter (D).  But, it's not.  Three
measurements are taken, where the third is the quantity of a different
isotope of the daughter product (Di).  The graph isn't P vs D, it's
P/Di vs D/Di.  That's not the same as P vs D; Di will vary over
different minerals.  A plot of P/Di vs D/Di might form a line when
P vs D does not.

It's easy to understand how different minerals in a rock could get
different P/Di ratios.  P and Di have different chemical properties,
and P will fit better into some minerals than Di (and vice versa).
This explains why data plots don't all hit the same X-value.

However, it's less easy to understand how different minerals in a rock
could different D/Di ratios.  The chemical properties of D and Di are
the same.  If the pool of matter were homogeneously distributed, then
all minerals should get the same D/Di ratio.  How do the data points end
up with different Y-values?

What the isochron plot shows is that there is a perfect correlation
between minerals enriched in D/Di -- exactly proportional to how they
are enriched in P/Di.  This pattern is easily explained if the enriched
D is a result of the enriched P decaying over time in a closed system.
(If the matter didn't start out homogeneously distributed, then there
is no reason for those minerals which got extra D/Di to get such a
precise extra amount of P/Di in proportion.)

Q4-b:
# Let's say we have a bunch of red marble and blue marbles.  If we put
# the marbles in a jar with blue first and then red then whenever
# you grab a handful, depending on how deep you go in the jar, you
# will probably get different ratios of blue and red.  However, if
# you stir the marbles so that they are evenly distributed, then you
# will always grab basically the same ratio now matter how large or
# line.

Let's modify this analogy to make it more like isochron dating (per the
response to Q4-a, above).  It gets a lot more complicated, unfortunately.

Suppose that you have a barrel of three kinds of marbles.  Some are
made of metal (P).  The other two kinds are lighter (but of density
equal to each other), made of wood.  Some are ivory colored (Di) and
others are black (D).

Now, you mix up the barrel thoroughly.  Since the metal marbles are
slightly heavier, they will be present in higher proportion near the
bottom and in lower proportion near the top.  The wooden marbles will
vary relative to the metal marbles by depth.  But since the properties
of the two kinds of wooden marbles are identical, they are always
present in the same ratio to each other.  If you divide the barrel
into quarters and count the marbles in each quarter, you will find
something like this:

--------        --      --      --      ----    ----    ----
location        P       D       Di      P/Di    D/Di    P/D
--------        --      --      --      ----    ----    ----
top             30      40      80      0.375   0.5     0.75
2nd qtr         60      30      60      1       0.5     2
3rd qtr         90      20      40      2.25    0.5     4.5
bottom         120      10      20      6       0.5     12
--------        --      --      --      ----    ----    ----

Note that these do *not* have the same P/D ratio, but they *do* have
the same D/Di ratio.  For this reason, when you make an isochron plot
(P/Di vs D/Di), they all have the same Y-value and the result is a
*horizontal* line.  The equation for the line is:

D/Di = 0.5

Now suppose that you keep the quarters of the barrel separate, as if
the "atoms" (marbles) were now pretty much stuck with their individual
"minerals" (quarters of the barrel).  And in each quarter you take half
of the metal marbles (P) and substutite in their place an equal number
of black wood marbles (D).  (This simulates radioactive decay.)  You get:

--------        --      --      --      ----    ----    ----
location        P       D       Di      P/Di    D/Di    P/D
--------        --      --      --      ----    ----    ----
top             15      55      80      0.1875  0.6875  0.2727
2nd qtr         30      60      60      0.5     1.0     0.5
3rd qtr         45      65      40      1.125   1.625   0.6923
bottom          60      70      20      3       3.5     0.8571
--------        --      --      --      ----    ----    ----

When you make an isochron plot now, you will still get a line, but this
time it  has a slope of one.  The equation for the line is:

D/Di = 1.0 * P/Di + 0.5

In fact, regardless of the fraction you select for replacing P with D
(as long as you replace the same fraction in each quarter), the data
points will remain on a line.  The slope of the line will vary, however,
depending on the size of fraction which you substitute.  The equation
for slope (m) as a function of that fraction (f) is:

m = -1 + ( 1 / (1-f) )

The computation works in the other direction, too.  From the slope of
the line, you can compute what fraction of P marbles have been replaced
with D marbles.  The equation is:

f = 1 - ( 1 / (m+1) )

Or, to turn this analogy back to the world of dating methods:  from the
slope of the line you can figure out what fraction of parent element has
been replaced with daughter element, from which you can calculate the
amount of time required for that to happen via radioactive decay.

Q-5:
# If an area is homogeneously mixed, then you will always get the same
# ratio of everything you grab.  And they will all be equally related to
# each other.
[...]
# In a few thousand years the decay is insignificant, so the isochron
# line would just represent uniform mixing during formation.

The situation you describe here won't result in an isochron line.  If
there is no chemical separation of P vs (D and Di), then all data
points will have the same P/Di and D/Di ratios.

To go back to your marble analogy, you'll get something like this
(allowing for samples to be different sizes, which I didn't include
the last time for sake of simplifying the data):

--------        --      --      --      ----    ----
location        P       D       Di      P/Di    D/Di
--------        --      --      --      ----    ----
top             30      15      30      1       0.5
2nd qtr         60      30      60      1       0.5
3rd qtr         50      25      50      1       0.5
bottom          60      30      60      1       0.5
--------        --      --      --      ----    ----

... every data point plots at ( 1 , 0.5 ), and no isochron line results.

One depends on some chemical differentiation between P and (D and Di)
in order to get a spread of data along the X-axis.

Q-6:
# But when scientists get data for something that appears contaminated,
# what do they do with it?  If data does not conform to the isochron
# method and fall along a line it is interpretted as contamination, I
# presume, as your FAQ also says. Why keep around bad samples?

It sounds as if you are suggesting that geologists might keep trying
isochron plots on a single item until they get one where the data points
line up, which probably isn't representative of its "real" age, and
only that one gets published.  (This is about one pace away from some
pretty heavy-duty "conspiracy-theorizing.")  Four reasons why I doubt
that this is done:

First, that is recognized as being dishonest.  If a geologist were to
plot 30 data points, and then throw away the ten which fell furthest
from the least-squares-fit isochron line, s/he would be in deep doo-doo
when anyone else found out.  (And the next person to attempt to replicate
the work *would* find out.)  The same would be true of someone who
buried evidence of many bad plots in favor of one good one.

Second, that would be outlandishly expensive -- in terms of both time
and money.  Performing just one isochron age requires quite a few
measurements that are with rather expensive equipment.  The ICR has
spent over \$30,000 in getting *one* Rb/Sr isochron "age" for their
"Grand Canyon Dating Project."  (While that is a little extreme, it's
not cheap in any case.)  If attempt number one showed the object to be
unsuitable, chances are that attempts number two through N would merely
give the same result.  Who has that sort of time or money to waste?

Third, negative results often get published.  Even when the plot does
not yield the age directly, it is often possible to ascertain useful
information about the history of the object from the data.  (For
example, see Faure p. 126.)

Fourth -- and most importantly -- if it were the case that isochron ages
were essentially random, fictitious numbers, then we would not expect
different researchers, etc.  For example, assuming you have Dalrymple's
"Open-File Report 86-100", see page 44.  Several different investigators
using several different dating methods all get about the same age for
the amitsoq gneiss.
This is easily explained (indeed, required) if these methods
yield the actual age of the formation.  How is it explained if the
"ages" are essentially random numbers?  Suppose that the first researcher
publishes an age of X years for the amitsoq gneiss.  Do you think that
the next guy is going to keep repeating the isochron method until he
gets a result that both plots as a line and agrees with the first guy's
data?  Heck, it's no skin off *his* nose if the first researcher's "age"
is a half billion years off.

--------------------------------------------------------------------------

(6) A little math

The author of this FAQ prefers to deal mainly in "intuitive" arguments,
but there have been some requests for something more rigorous.  Here
is a mathematical derivation of isochron methodology, written by Dave
Thomas (P.O. Box 1017 / Peralta, NM 87042):

Assume a variety of minerals (k=1,2,3,...N) with different initial
rubidium-87/strontium 86 ratios P0(k) (parent, horizontal axis),
but all with the same initial strontium-87/strontium-86 ratios D0(k)
(daughter, vertical axis).  It is important to normalize the ratios to
an isotope of strontium (namely Sr-86) that is *not* a decay product
of rubidium-87.  For one of these minerals (say, k=5),
the amount of the parent isotope would decrease from the initial

P(T) = P0*exp(-lambda*T)          (Equation 1)

where exp is the exponential function, lambda is the Rb decay rate
(for rubidium-87, decay rate lambda = 1.39E-11 / year ), and T
is the elapsed time since rock crystallization.  For the same
mineral (k=5), the amount of the daughter would increase from the
initial ratio:

D(T) = D0 + P0*(1-exp(-lambda*T)) (Equation 2)

Note that P(T) + D(T) = D0 + P0 = constant for all ages T.

At age T=0, P(T) = P0, and D(T) = D0+P0*(1-exp(0)) = D0+P0*(1-1) = D0.

At age T > 0, P(T) < P0, and D(T) > D0.

At age T approaching infinity, P(T) goes to zero, and D(T) to P0+D0.

For a different mineral (say, k=13), let's use the prime (') symbol:

P'(T) = P0'*exp(-lambda*T)          (Equation 1')

D'(T) = D0' + P0'*(1-exp(-lambda*T)) (Equation 2')

Because of mixing prior to crystallization, the initial daughter
ratios for both minerals are identical: D0 = D0' (*important*).

Now, the slope of the line connecting point (P,D) to point (P',D') is:

D'(T) - D(T)
slope m = ------------         (Equation 3)
P'(T) - P(T)

Putting equations 1,2,1', and 2' into equation 3, we obtain:

(D0'+P0'*(1-exp(-lambda*T))) - (D0+P0*(1-exp(-lambda*T)))
slope m = ---------------------------------------------------
P0'*exp(-lambda*T) - P0*exp(-lambda*T)

which can be simplified by noting that initial strontium ratios
D0' = D0 (*important*), and multiplying top and bottom by exp(+lambda*T) :

P0'*(exp(+lambda*T)-1) - P0*(exp(+lambda*T)-1)
slope m = --------------------------------------------------
P0' - P0

which further simplifies to

(P0' - P0) * (exp(+lambda*T)-1)
slope m = --------------------------------
( P0' - P0 )

or just

slope m = (exp(+lambda*T)-1)       (Equation 4)

Equation 4 is the basic connection between age T and isochron
slope m.  It can be inverted to give the age T as a function of
the measured slope on an isochron plot:

T = log_e( m+1 ) / lambda           (Equation 5)

where log_e is the natural logarithm.  We can also express
decay rate lambda in terms of half-life, H. From Equation 1,

P(H) = P0/2 = P0*exp(-lambda*H)  (for age T = Halflife H)

which reduces to log_e(2) = lambda*H, or

lambda =  log_e(2) / H            (Equation 6)

Putting Equation 6 into Equation 5, we get:

T = H * log_e( m+1 ) / log_e(2)     (Equation 7)

Multiplying Eqtn. 7 by log_e(2), and using log_e(x^n) = n*log_e(x),
we get

T*log_e(2) = log_e(2^T) = H*log_e( m+1 ) = log_e( (m+1)^H ), or

2^T  =  (m+1)^H     , or just

2^(T/H)  =  (m+1)                  (Equation 8).

For m = 1, m+1=2 = 2^1 so that T = H = 1 half-life.

For m = 3, m+1=4 = 2^2 so that T = 2H = 2 half-lives.

For m = 7, m+1=8 = 2^3 so that T = 3H = 3 half-lives.

The method requires minerals with a spread of initial rubidium-87
ratios (P0 not equal to P0'), else the denominator of Equation 3
(and subsequent equations leading to Equation 4) would be zero,
indicating an undefined slope.

--------------------------------------------------------------------------

(7) For further information, see:

G. Faure.  1986.   _Principles of Isotope Geology_  Second Edition.
John Wiley and Sons.  ISBN 0-471-86412-9.  589 pp.

Faure is a textbook/handbook on isotope dating.  It is relatively
technical and has many references to the scientific literature.
Isochron methods are first detailed in Chapter 8: "The Rb-Sr
Method of Dating."

G. B. Dalrymple.  1991.  _The Age of the Earth_.  Stanford
University Press.  ISBN 0-8047-1569-6.  474 pp.

Dalrymple is an excellent work on the age of the earth, which
does not discuss creationist arguments but details the evidence
and mainstream science's position.  It is written for the
layman, but is well-referenced.  Isochron methods are introduced
in a section titled "Age-Diagnostic Diagrams" (pp. 102-124).

G. B.  Dalrymple.  1982.  _Radiometric Dating, Geologic Time, and
the Age of the Earth:  A Reply to "Scientific" Creationism_  (U.S.
Geological Survey Open-File Report 86-110).  76 pp.  Available from
Wesley Elsberry (welsberr@orca.tamu.edu) for the cost of postage.

Brent Dalrymple has given permission for this paper to be copied
and distributed.  The paper details radiometric dating methods,
creationist criticisms of radiometric dating methods, and
critiques creationists' own means for deriving an age for the
earth.  Isochron methods are introduced with the Rb/Sr isochron
method (pp. 31-33).

A. N. Strahler.  1987.  _Science and Earth History_.  Prometheus
Books.  ISBN 0-87975-414-1.  552 pp.

Strahler is aimed at discussing creationism and creationist