Transcript Radiometric Dating: General Theory

Radiometric Dating: General Theory
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The radioactive decay of any radioactive atom is
an entirely random event, independent of
neighboring atoms, physical conditions, and the
chemical state of the atom.
It depends only on the structure of the nucleus.
λ, the decay constant, is the probability of an
atom decaying in unit time. It is different for
each isotope.
Suppose that at time t there are N atoms and that at time t+δt, δN of those
have decayed, then δN can be expressed as
δN = -λ N δt
In the limit as δN and δt go to 0, this becomes
dN/dt = -λ N
Thus, the rate of decay is proportional to the number of atoms present.
Rearrangement and integration gives:
loge N = -λ t + c
If at t=0 there are N0 atoms present, then c = loge N0
N = N0 e-λt
The half-life, T½, is the length of time required for half of the original
atoms to decay.
N0/2 = N0 e-λT½
or
T½ = (loge 2) / λ
Consider the case of a radioactive Parent atom decaying to an atom called
the Daughter. After time t, N = N0 – D parent atoms remain and
N0 – D = N0 e-λt
Where D is the number of daughter atoms (all of which have come from
decay of the parent) present at time t. Thus
D = N0 (1 – e-λt)
However, it is not possible to measure N0, but only N
Use the previous equation and
N = N0 e
–λt
yields
D = N (eλt – 1)
This equation expresses the number of daughter atoms in terms of the number
of parent atoms, both measured at time t, and it means that t can be
calculated by taking the natural log
t = loge (1 + D/N) / λ
In practice, measurements of D/N are made using a mass spectrometer.
http://www.chemguide.co.uk/analysis/masspec/howitworks.html
Major radioactive elements used in radiometric dating
Parent
Isotope
Daughter
Isotope
Half Life of
Parent (years)
Effective dating
range (years)
Materials that
can be dated
238U
206Pb
4.5 billion
10 million –
4.6 billion
Zircon
Apatite
235U
207Pb
0.7 billion
10 million –
4.6 billion
Zircon
Apatite
1.3 billion
50,000 – 4.6
billion
Muscovite
Biotite
Hornblende
10 million –
4.6 billion
Muscovite
Biotite
Potassium
Feldspar
100 - 70,000
Wood, charcoal, peat,
bone and tissue, shell
and other calcium
carbonate, groundwater,
ocean water, and glacier
ice containing dissolved
CO2
40K
87Rb
14C
40A
87Sr
14N
47 billion
5730
Radiometric dating is not always that simple!
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There may have been an initial concentration of the daughter in the
sample
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Not all systems are closed. There may have been exchange of
parent and/or daughter with surrounding material.
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If dates from different isotope systems match within analytical
error, we say the ages are concordant. If they are not, then we say
they are discordant.
When discordant, we suspect problems like those above with one or
all of the systems.
The date t obtained is not always the date of formation of the rock.
It may be the date the rock crystallized, or the date of a
metamorphic event which heated the rock to the degree that
chemical changes took place.
Radioactive decay schemes are not all as simple as a parent and
exactly one daughter. 87Rb to 87Sr is a simple one step decay. The
two U to Pb series have a number of intermediate daughter
products.
Fission Track Dating
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As well as decaying to 206Pb as described before,
also subject to spontaneous fission.
238U
is
It disintegrates into two large pieces and several
neutrons. This is a very rare event, occurring just once
per 2 million α decays.
Each event is recorded as a trail of destruction about 10
m long through the mineral structure.
These “fission tracks” can be observed by etching the
polished surface of certain minerals. The tracks become
visible under a microscope.
Spontaneous
Fission Tracks
Consider a small polished sample of a mineral. Assume that it has
[238U]now atoms of 238U distributed throughout its volume
The number of decays of 238U, Dr, during time t is:
t
Dr  [ U ]now (e 1)
238
The number of decays of 238U by spontaneous fission, Ds, which occur in
time t is:
s 238
Ds  [ U ]now (e t  1)
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Where s is the decay constant for spontaneous fission of 238U.
To determine an age, we must count the visible fission tracks, estimate
the proportion of the tracks visible (crossing) the surface, and measure
[238U]now.
Fortunately, we do not need to do this in an absolute manner, because
another isotope of Uranium, 235U, can be made to fission artificially. This is
done by putting our sample in a nuclear reactor and bombarding it with slow
neutrons for a specified time (hours). This provides us with a standard against
which to calibrate the number of tracks per unit area (track density). The
number of induced fissions is:
DI [235U ]nown
Where σ is the known neutron capture cross-section and n is the neutron
dose in the reactor.
We assume that if the two isotopes of U are equally distributed in the sample,
then the proportion of tracks that cross the surface will be the same. We can
combine equations to get:
s [ 238U ]now (et  1) Ds N s


235
[ U ]now n
DI
NI
Where Ns and NI are the numbers of spontaneous and induced fission tracks
counted in an area.
The equation can be rearranged and the known present ratio of the two
isotopes of Uranium, [238U]now/[235U]now=137.88,can be inserted to give:
 N s  n 
t  loge 1 


N

137
.
88
I
s


1
In practice, after the number of spontaneous fission tracks Ns has been
counted, the sample is placed in the reactor and then etched again. The
spontaneous tracks are enlarged and the induced tracks are exposed. The
number of induced tracks NI are counted and the age calculated.
Spontaneous
Induced
Fission Tracks
Fission Tracks
There is an additional (and very powerful) way to use
fission tracks.
Fission tracks in a mineral crystal are stable at room
temperature, but can “heal” if the temperature of the
crystal is high enough.
At very high temperature, the tracks heal completely very
quickly. This means that the “age” of a rock can be
completely “reset” by heating.
The rate at which tracks are healed varies with
temperature and mineral type. Therefore there is a
“closure” temperature that is a function of mineral type
and rate of cooling.
For example, fission track
ages determined from
sphene are always greater
than ages determined from
apatite. This is because
healing tracks in sphene
(~300C) requires much
greater temperatures than
healing tracks in apatite
(~90C).
Imagine that rocks are being uplifted and eroded during the creation of a mountain
range. The individual rocks are cooling as they are brought closer to the surface. A
progression of fission track ages in different minerals record the uplift/cooling history
of the rock.
There are newer, even more sophisticated methods, that use the rate at which
tracks heal, they actually shorten before disappearing, to determine more
complicated temperature history curves from each mineral.
http://www.geotrack.com.au/ttinterp.htm