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Radiometric Dating: General Theory 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. 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. 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. λ, the decay constant, is the probability of an atom decaying in unit time. It is different for each isotope.

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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: log e N = -λ t + c If at t=0 there are N 0 atoms present, then c = log e N 0 N = N 0 e -λt

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The half-life, T ½, is the length of time required for half of the original atoms to decay. N 0 /2 = N 0 e -λT½ or T ½ = (log e 2) / λ Consider the case of a radioactive Parent atom decaying to an atom called the Daughter. After time t, N = N 0 – D parent atoms remain and N 0 – D = N 0 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 = N 0 (1 – e -λt )

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However, it is not possible to measure N 0, but only N Use the previous equation and N = N 0 e –λt yields D = N (e λt – 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 = log e (1 + D/N) / λ In practice, measurements of D/N are made using a mass spectrometer.

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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 238 U 206 Pb 4.5 billion 10 million – 4.6 billion ZirconApatite 235 U 207 Pb 0.7 billion 10 million – 4.6 billion ZirconApatite 40 K 40 A 1.3 billion 50,000 – 4.6 billion MuscoviteBiotiteHornblende 87 Rb 87 Sr 47 billion 10 million – 4.6 billion MuscoviteBiotite Potassium Feldspar 14 C 14 N ,000 Wood, charcoal, peat, bone and tissue, shell and other calcium carbonate, groundwater, ocean water, and glacier ice containing dissolved CO 2

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Radiometric dating is not always that simple! There may have been an initial concentration of the daughter in the sample There may have been an initial concentration of the daughter in the sample Not all systems are closed. There may have been exchange of parent and/or daughter with surrounding material. Not all systems are closed. There may have been exchange of parent and/or daughter with surrounding material. 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. 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. 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. 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. 87 Rb to 87 Sr is a simple one step decay. The two U to Pb series have a number of intermediate daughter products. Radioactive decay schemes are not all as simple as a parent and exactly one daughter. 87 Rb to 87 Sr is a simple one step decay. The two U to Pb series have a number of intermediate daughter products.

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Fission Track Dating As well as decaying to 206 Pb as described before, 238 U is also subject to spontaneous fission. As well as decaying to 206 Pb as described before, 238 U is also subject to spontaneous fission. It disintegrates into two large pieces and several neutrons. This is a very rare event, occurring just once per 2 million α decays. 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. 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. These fission tracks can be observed by etching the polished surface of certain minerals. The tracks become visible under a microscope.

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Spontaneous Fission Tracks

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Consider a small polished sample of a mineral. Assume that it has [ 238 U] now atoms of 238 U distributed throughout its volume The number of decays of 238 U, D r, during time t is: The number of decays of 238 U by spontaneous fission, D s, which occur in time t is: Where s is the decay constant for spontaneous fission of 238 U. To determine an age, we must count the visible fission tracks, estimate the proportion of the tracks visible (crossing) the surface, and measure [ 238 U] now.

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Fortunately, we do not need to do this in an absolute manner, because another isotope of Uranium, 235 U, 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 N s and N I are the numbers of spontaneous and induced fission tracks counted in an area.

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The equation can be rearranged and the known present ratio of the two isotopes of Uranium, [ 238 U] now /[ 235 U] now =137.88,can be inserted to give: In practice, after the number of spontaneous fission tracks N s 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 N I are counted and the age calculated. Spontaneous Fission Tracks Induced Fission Tracks

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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.

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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. 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).

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