Radiometric Dating: General Radiometric Dating: General TheoryTheory

The radioactive decay of any radioactive The radioactive decay of any radioactive atom is an entirely random event, atom is an entirely random event, independent of neighboring atoms, physical independent of neighboring atoms, physical conditions, and the chemical state of the conditions, and the chemical state of the atom.atom.

It depends only on the structure of the It depends only on the structure of the nucleus.nucleus.

λλ, the decay constant, is the probability of an , the decay constant, is the probability of an atom decaying in unit time. It is different for atom decaying in unit time. It is different for each isotope.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 Major radioactive elements used in radiometric datingdating

Parent Parent IsotopeIsotope

Daughter Daughter IsotopeIsotope

Half Life of Half Life of Parent (years)Parent (years)

Effective dating Effective dating range (years)range (years)

Materials that Materials that can be datedcan be dated

238238UU 206206PbPb 4.5 billion4.5 billion 10 million – 10 million – 4.6 billion4.6 billion

ZirconZircon

ApatiteApatite

235235UU 207207PbPb 0.7 billion0.7 billion 10 million – 10 million – 4.6 billion4.6 billion

ZirconZircon

ApatiteApatite

4040KK 4040AA 1.3 billion1.3 billion 50,000 – 4.6 50,000 – 4.6 billionbillion

MuscoviteMuscovite

BiotiteBiotite

HornblendeHornblende

8787RbRb 8787SrSr 47 billion47 billion 10 million – 10 million – 4.6 billion4.6 billion

MuscoviteMuscovite

BiotiteBiotite

Potassium Potassium FeldsparFeldspar

1414CC 1414NN 57305730 100 - 70,000100 - 70,000

Wood, charcoal, peat, Wood, charcoal, peat, bone and tissue, shell bone and tissue, shell

and other calcium and other calcium carbonate, carbonate,

groundwater, ocean groundwater, ocean water, and glacier ice water, and glacier ice containing dissolved containing dissolved

COCO22

Radiometric dating is not always that Radiometric dating is not always that simple!simple!

There may have been an initial concentration of the daughter in the There may have been an initial concentration of the daughter in the samplesample

Not all systems are closed. There may have been exchange of parent Not all systems are closed. There may have been exchange of parent and/or daughter with surrounding material.and/or daughter with surrounding material.

If dates from different isotope systems match within analytical error, 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 we say the ages are concordant. If they are not, then we say they are discordant.discordant.

When discordant, we suspect problems like those above with one or When discordant, we suspect problems like those above with one or all of the systems.all of the systems.

The date The date tt obtained is not always the date of formation of the rock. It 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 may be the date the rock crystallized, or the date of a metamorphic event which heated the rock to the degree that chemical changes event which heated the rock to the degree that chemical changes took place.took place.

Radioactive decay schemes are not all as simple as a parent and Radioactive decay schemes are not all as simple as a parent and exactly one daughter. exactly one daughter. 8787Rb to Rb to 8787Sr is a simple one step decay. The two Sr is a simple one step decay. The two U to Pb series have a number of intermediate daughter products.U to Pb series have a number of intermediate daughter products.

Fission Track DatingFission Track Dating As well as decaying to As well as decaying to 206206Pb as described before, Pb as described before,

238238U is also subject to spontaneous fission.U is also subject to spontaneous fission.

It disintegrates into two large pieces and several It disintegrates into two large pieces and several neutrons. This is a very rare event, occurring just neutrons. This is a very rare event, occurring just once per 2 million once per 2 million αα decays. decays.

Each event is recorded as a trail of destruction Each event is recorded as a trail of destruction about 10 about 10 m long through the mineral structure.m long through the mineral structure.

These “fission tracks” can be observed by etching These “fission tracks” can be observed by etching the polished surface of certain minerals. The the polished surface of certain minerals. The tracks become visible under a microscope.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:

The number of decays of 238U by spontaneous fission, Ds, which occur in time t is:

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.

)1(][238 tnowr eUD

)1(][238 tnow

ss eUD

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:

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:

Where Ns and NI are the numbers of spontaneous and induced fission tracks counted in an area.

nUD nowI ][235

I

s

I

st

now

nows

N

N

D

D

n

e

U

U

)1(

][

][235

238

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:

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.

88.1371log

1 n

N

Nt

sI

se

Spontaneous

Fission Tracks

Induced

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.

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

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).