This is to accompany Slide #9 in the file: Radiometric dating for SCI 15.
This slide is meant to illustrate the physics underlying the Isochrone
Method. This method would typically be applied to a chunk of meteorite or
rock, which is large enough to suggest that the entire body might not have
had uniform distributions of the elements when it formed. That is, some
parts of the rock or meteorite would have more or less of the parent and/or
daughter atoms. So the Isochronal method will use samples from various
parts of the original body.
When you animate the slide, this is what the different stages are meant to
convey:
1. First you see a coordinate system where:
a. Horizontal axis is amount of parent atoms (e.g. Rb-87)
b. Verical axis is amount of daughter atoms (Sr-87)
c. Both of these are NORMALIZED—i.e. dividing by the number
of atoms of the inert=non-radioactive isotope of the daughter
(Sr-86). This does two things: first, instead of plotting zillions
of atoms vs zillions of atoms, we plat ratios, and the numbers
are much more manageable; and second, the initial (t = 0) value
of the ration Sr-87/Sr-86 is constant, that is, the same
everywhere in the rock when it forms. This is so because these
two isotopes are essentially chemically identical and so occur in
the same ratio throughout.
2. Next a straight line will appear. A point on this line corresponds to the
pair of ratios of (Rb87/Sr86,Sr87/Sr86) at a given time. Time starts at
zero on the lower right end of the line and increases up and to the
left. Two times are called out for illustration—t = t1/2 and t = 2 x t1/2.
We say this line gives the ‘locus’ of values of parent and daughter vs.
time. So, if we assume for this part of the rock/meteorite that the
ratio of Rb87 to Sr86 was = 8 when it formed, and that the initial value
of the ratio of daughter Sr87 to Sr86 was as given (1.5) then the line
tells us how the parent declines and the daughter increases with time.
[Those interested in the math my note that this straight line comes
from combining the two equations on slide 3 to get the following
expression:
3.
4.
5.
6.
7.
8.
9.
Sr87/Sr86(t) = Sr87/Sr86(0) + Rb87/Sr86(0) – Rb87/Sr86(t) x [1 –
(1/2t/t(1/2))]
The next line that appears is the same relationship, this time plotted
for a sample of the rock/meteorite where the initial ratio Rb87/Sr86 =
6. It looks the same except it’s shifted over to the left.
Animate one more time and two more lines appear, corresponding to
Rb87/Sr86 = 4 and 2. So now there is a family of lines plotted for four
possible values of these initial values.
Animate again. This time another line appears, labeled “t1/2 isochrone”.
This line was constructed by connecting all the points on the first 4
lines where t = t1/2 . Notice that the isochrone (= time) line is a
STRAIGHT line and that it intercepts the vertical axis at the initial
(or original) value of Sr87/Sr86(0). What does this mean? This means
that if we now get ACTUAL DATA and plot it on the same kind of a
graph (parent on the horizontal, daughter on the vertical, and both
normalized by the inert isotope of the daughter), we should get a
STRAIGHT LINE whose SLOPE tells us the age of the whole
rock/meteorite, and whose VERTICAL INTERCEPT tells us how much
of the daughter we started with!
Animate again. Now we connect the 2 x t1/ points on the 4 lines.
Again a straight line and intersecting the vertical axis at the initial
value of daughter (normalized to inert isotope). It’s a steeper slope
because it corresponds to an older rock.
Note a couple of neat things about this method.
a. If we get a straight line, we can be assured that the system
stayed closed—no parent/daughter contamination flowed into or
out of the system.
b. We don’t have to guess at how much daughter we started with,
it’s given by the vertical intercept.
c. If we don’t get a straight line, then we better throw that
sample out because something has contaminated it.
Slides 10-12 illustrate what contamination would look like.
Finally, slides 13-15 show actual data, and the fit to a straight line is
pretty phenomenal!