Isotope Geochemistry (3)

8/8/2019 Isotope Geochemistry (3)

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LU 6 ISOTOPE GEOCHEMISTRY

ISOTOPES

DEFINITIONTwo or more nuclides having the same atomic number , thus constituting the same element, but differing in the mass number. Isotopes of a givenelement have the same number of nuclear protons but differing numbers of neutrons .

Atoms: Atomic number number of protonsDifferent atomic numbers elements

Isotopes: Nuclide - protons and neutronsMass numbers - total number of protons and neutronsDifferent number of neutrons in nuclei create varieties of an element

- isotopesDifferent mass numbers due to different number of neutrons

Nuclide: Nucleus of an isotope is called a nuclideStable nuclides - maintain atomic configuration over long periods.Unstable nuclides - spontaneously change into new atoms.

TYPES:STABLE ISOTOPES

The atomic nuclei of these elements do not change to nuclei of other elements.

RADIOACTIVE ISOTOPESThe atomic nuclei of these elements give out radiation spontaneously and thereby change to nuclei of other elements.

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http://ie.lbl.gov/education/glossary/#AtomicNumberhttp://ie.lbl.gov/education/glossary/#Protonhttp://ie.lbl.gov/education/glossary/#Neutronhttp://ie.lbl.gov/education/glossary/#AtomicNumberhttp://ie.lbl.gov/education/glossary/#Protonhttp://ie.lbl.gov/education/glossary/#Neutron


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RADIOACTIVE ISOTOPES

NUCLIDE

Nucleus of an isotope is called a nuclide

Stable nuclides - maintain their atomic configuration over long periods of time.Unstable nuclides - spontaneously change of an unstable nuclide into another

nuclide.

RADIOACTIVE DECAY

Radioactive Isotopes

Unstable nuclides - spontaneously change of an unstable nuclide into another nuclide.

This phenomenon is called decay

The process is called radioactivity

The isotope is called a radioactive isotope with a radioactivity nuclide.

Parent nuclide (unstable) before decay the atom containing the radioactivenuclide

Daughter nuclide (stable) after decay to new configuration

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Modes of Radioactive Decay

Radioactive decay occurs by one of three processes.

1. Alpha decay

Alpha emission results in releasing an alpha particle . An alpha particle has twoprotons and two neutrons, so it has a positive charge. (Since it has two protons itis a helium nucleus.) It is written in equations like this:

2. Beta decay Beta emission is when a high speed electron (negative charge) leaves the nucleus .Beta emission occurs in elements with more neutons than protons, so a neutron splitsinto a proton and an electron. The proton stays in the nucleus and the electron isemitted. Negative electrons are represented as follows:

3. Gamma Emission Gamma Emission is when an excited nucleus gives off a ray in the gammapart of the spectrum . A gamma ray has no mass and no charge. This oftenoccurs in radioactive elements because the other types of emission can result inan excited nucleus. Gamma rays are represented with the following symbol.

The two types of artificial radiation are positron emission and electron capture.Positron emission Positron emission involves a particle that has the same mass as an electron but apositive charge. The particle is released from the nucleus.

Electron capture Electron capture is when an unstable nucleus grabs an electron from its inner shell tohelp stabilize the nucleus. The electrons combine with a proton to form a neutron whichstays in the nucleus.

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Principle of Radioactive Decay

A key principle of radioactive decay is that there is a constant probability per unit of time (e.g. 1 year) of a decay event from parent atom to a new daughter atom.

This probability is expressed as the decay constant . Here we explore the decay process graphically.1. Imagine a batch of 36 parent atoms .

These spontaneously decay to daughter atoms (in green).2. The probability of such a decay for each parent atom is 1/6 per unit of time .

So after 1 unit of time the most probable outcome is that 5/6 of the original batchof parents remain (i.e. 30).

3. During the next interval of time 1/6 of the remaining parents will decay (leaving 5/6of 30 =25 parents).

4. And so it continues.Because the number of parents reduces for each new time interval , the

number of events per unit of time reduces (although the probability of eachparent decaying is constant ). This gives the graph a characteristic shape -exponential decay .

The graph also shows the half-life concept. The half-life is the amount of timenecessary to reduce the number of parent atoms by 50% from the originalnumber .

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The Basic Equation of Radioactive Decay

In any large number of atoms of a radioactive isotope, the decay follows astatistical rule:

During any fixed time interval , a definite proportion of the parent atoms

change to the daughter product .The number of decays you will measure each second from a sample dependson the number of atoms in the sample , N.

Here are two blocks of exactly the same radioisotope. The chance of an atomdecaying from one is exactly the same as in the other but there are twice asmany atoms in the 2 kg block so there will be twice as many decays per secondin the 2 kg block.

Thus the rate of decay , or the number of atoms of decay , is simplyproportional to the total number of parent atoms present:

where = the constant of proportionality, called the Decay Constant .

The decay constant is the proportion of atoms that decay in an interval of timeThe decay constant gives you an idea of how quickly or slowly a material willdecay.

A large value means that the sample will decay more quickly.

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Rate of decay of a radioactive nuclide is proportional to the number of atomsof that nuclide remaining at any time .

If N is the number of atoms remaining, then

- dN/dt = N (1) where is the proportionality constant known as the decay constantN is the number of atoms remaining/present

and the minus sign indicates that the rate of decay decreases with time .

This is a first order differential equation

Solve for N as a function of time

Rearrangement

- dN/N = dt (2) Integrate both sides

N t - dN/N = dt (3) N o 0

By integrating and expressing as natural logarithm (logarithm to base e) we obtain

-In N = t + C (4)

Where In is the logarithm to the base e C is the constant of integration

The integration constant C may be expressed in terms of the original number of parent atoms when t=0

When t=0,N o = number of nuclides at t=0

-In N o = (0) + C C = - In N o

Therefore the integrated form of the equation is

In N = t - In N o (5)

RearrangementIn N - In N o = -t (6)

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Present Original In (N/N o ) = -t (7)

Switching to an exponential format (In y = x so y = e x )

N/N o = e -t (8)

The equation above is the basic relationship that describes all radioactive decayprocesses .

With it, we can calculate the number of parent atoms (N) that remain at any time t from the original number of atoms (N o ) present at time t=0 .

Rearrangement

Present Original N = N o e -t (9)

N number of parent atoms currently presentN o number of parent atoms originally present when mineral was formed Each radioactive isotope/ radionuclide has a characteristic decay constant that

must be determined experimentally.

This expression above is known as the Radioactive Decay Law.It tells us that the number of radioactive nuclei will decrease in anexponential fashion with time with the rate of decrease being controlled by

the Decay Constant .

The Law is shown in graphical form in the figure below:

The graph plots the number of radioactive nuclei at any time, Nt, against time ,t. We can see that the number of radioactive nuclei decreases from N0 that isthe number at t = 0 in a rapid fashion initially and then more slowly in the classicexponential manner.

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All three curves here are exponential in nature, only the Decay Constant isdifferent. When the Decay Constant has a low value the curve decreases relatively

slowly When the Decay Constant is large the curve decreases very quickly .

The equation can be rearranged

Original Present N o = N e t (10)

N number of parent atoms currently presentN o number of parent atoms originally present when mineral was formed Each radioactive isotope/ radionuclide has a characteristic decay constant that

must be determined experimentally.

Note:N = N 0e -kt (exponential decay)

[ N = N 0e kt (exponential growth) ]

where N0 is the initial quantity t is time N(t) is the quantity after time t k is the decay constant and e x is the exponential function ( e is the base of the natural logarithm)

www.earth.northwestern.edu/people/seth/202/DECAY/decay.pennies.slow.html

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Decay of parent produces daughter or radiogenic nuclides .

Number of daughters produced is simply the difference between initialnumber of parents and number remaining after time t .

original present D = N o N (11)

Substituting (10) into (11) we obtain (for N o )

D = Ne t N = N (e t 1) (12)

This tells us that the number of daughters produced is a function of thenumber of parents present and the time .

Since in general there will be some atoms of the daughter nuclide around to

begin with , i.e. when t = 0 , a more general expression is:

D = D o + N (e t 1) (13)

Where D o is the number of daughters originally present .

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Rearrangement

T = 1/ In (D/D o + 1) (14) P

This is the time during which an amount of the daughter represented by D hasaccumulated , leaving undecayed an amount of the parent represented by P .

Values of D and P are found by analyzing the rock or mineral in which theradioactive isotope occurs.

If we can also find values for and D o the equation will give us the age of therock or mineral in years.

The decay constant is found by laboratory measurement of decay rate.

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Half-Life

A half-life is the time it takes for half of the parent radioactive element todecay to a daughter product.So if you have 10 grams of a radioactive element

After one half-life there will be 5 grams of the radioactive element left.After another half-life , there will be 2.5 g of the original element left.After another half-life , 1.25 g will be left.

Radioactive decay occurs at a constant exponential or geometric rate.The rate of decay is proportional to the number of parent atoms present.

The proportion of parent to daughter tells us the number of half-livesFor example,If there are equal amounts of parent and daughter , then one half-life has passed.If there is three times as much daughter as parent , then two half-lives have passed.

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We can use the number of half-lives to find the age in years.o Age is usually the time of crystallization or formation

Approacho Compare amount of daughter isotope to amount of parent

originally there

Example:Problem: The 235 U: 207 Pb ratio in a mineral is 1:7 .

What is the age of the mineral?Given: Half-life of 235 U is 0.7 billion years (b. y.)

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The half-life of an isotope equals the number of years it takes for an initialnumber of parent atoms to be reduced to half that number by radioactive decay.

The half-life figure enables us to relatively quickly understand the useful agerange of a particular isotopic system.

For instance, the half life of the C-14 system is 5,730 years - you would never use C-14 to determine the age of material older than 40 000 years which is thepractical upper limit; all of the radioactivity would be gone.

Each radioactive isotope has its own unique half-life .

Radioactive Parent Stable Daughter Half lifePotassium 40 Argon 40 1.25 billion yrs

Rubidium 87 Strontium 87 48.8 billion yrsThorium 232 Lead 208 14.0 billion years

Uranium 235 Lead 207 704 million yearsUranium 238 Lead 206 4.47 billion years

Carbon 14 Nitrogen 14 5730 years

XXXXXXXXXXHalf-life Equations:

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1) Suppose the amount of time for the number of parent atoms to decrease tohalf the original number i.e. t when N/ N o =1/2 is required to be determined.

Take equation (4) below

In (N/N o ) = -t (4)and setting N/ N o to 1/2 rearrange it to get

In 1/2 = -t 1/2 or In 2= t 1/2 (5)

to finally gett 1/2 = In2/ (6)

which gives the half-life.

2) Another equation for half-life calculations is as follows:

AE is the amount of substance left A0 is the original amount of substance t is the elapsed time t1/2 is the half-life of the substance

3) Another variations of the half-life equation are as follows:

An example problem is if you originally had 157 grams of carbon-14 and the half-life of carbon-14 is 5730 years, how much would there be after 2000 years?

There would be 123 grams left.http://www.eas.asu.edu/~holbert/eee460/decay.html

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http://www.eas.asu.edu/~holbert/eee460/decay.htmlhttp://www.eas.asu.edu/~holbert/eee460/decay.html


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http://www.earthsci.org/fossils/geotime/radate/radate.html

USE FOR RADIOMETRIC DATING: GEOCHRONOLOGY

Natural radioactive decay provides a variety of clocks that allow the

determination of geological time .Many radioactive elements can be used as geologic clocks.

PRINCIPLE OF RADIOMETRIC DATING

Naturally-occurring radioactive materials break down into other materials atknown rates.

Each radioactive element decays at its own nearly constant rate.

Once this rate is known, the length of time over which decay has been occurringcan be estimated by measuring the amount of radioactive parent element and the amount of stable daughter elements .

RADIOACTIVE DECAY SYSTEMS OF GEOCHRONOLOGICAL INTEREST

The course examines K-Ar, U-Th-Pb, Rb-Sr decay systems and Carbon-14 .

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RUBIDIUM-STRONTIUM

Rubidium decays to Strontium via a one step beta decay process with a half-life of 4.7 Ga.

(This method is good for minerals like micas, k-spar, pyroxene, olivine and whole metamorphic rocks)

The Rb-Sr system exists because

87

Rb (Z=37) decays by beta (-)decay to 87 Sr (Z=38)

The decay constant is

= 1.42x10 -11 y -1.

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Background

Rubidium: Univalent Not very common in the Earth's crust

Strontium: Divalent Occurs as four stable

isotopes(88Sr, 87Sr, 86Sr and 84 Sr).

The table below lists the naturally occurring isotopes of both Rb and Sr alongwith their isotopic abundances (in atom %) and their nuclide weights in atomicmass units (a. m. u.).

Isotope Atom% abundance Nuclide mass (amu)

Rubidium Isotopes 87 Rb 27.8346 86.90918 85 Rb 72.1654 84.91171

Strontium Isotopes 88 Sr 82.53 87.9056 87 Sr 7.04 86.9089 86 Sr 9.87 85.9094 84 Sr 0.56 83.9134

Systematics of the Rb-Sr system

The isotopic composition of Sr in a sample that contains both Sr and Rb is givenby:

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i) In this equation, t is the time since the system was formed (Note that the system is assumed to have remained closed to the

exchange of Rb and Sr since its formation date)The (87 Sr/ 86Sr) o is the isotopic composition of Sr already in the system at thetime of its formation (the initial ratio) and 87Rb/ 86 Sr is the ratio of Rb to Sr inthe system .

ii) As in practice there are commonly daughter atoms already present in amaterial. So in this case we must make a correction, estimating the originaldaughter concentration.This is done by normalising against a stable reference isotope that is notitself radioactive or produced by radioactive decay of another isotope . Thereference isotope is 86Sr (86Sr)..

iii) The abundance of 87Sr (daughter) is measured relative to a referenceisotope . Thus, the Sr isotopic composition of a sample is reported as theratio of 87Sr to 86 Sr i.e. 87Sr/ 86Sr

iv) Of these terms, (87 Sr/ 86Sr) t , which is the total 87 Sr/ 86 Sr, is measured in thelaboratory ; 87Rb/ 86Sr is calculated from the measured Rb and Sr concentrations in the sample ; and (87Sr/ 86 Sr) o and t are unknowns.

v) The initial ratio and age.For an individual sample,

the initial ratio can be calculated from the measured isotopic compositionof the sample if the age of the sample is known or

the age of the sample can be calculated if the initial ratio is known.

However, if neither the initial ratio nor the age of the sample is known,then neither can be computed using the equation above.

This limitation can be overcome by studying rocks with different Rb/Sr ratios

If the body of rock under study contains rocks with different Rb/Sr ratios and the rocks are known, based on geological observations, to

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have formed at the same time from the same source , then an equationlike the one above can be written for each sample .

If only two samples are available, the two equations may be solvedsimultaneously to give both the initial ratio and the age of the samples.

If more than two samples are available, t hen all of the equations are

solved simultaneously using least squares methods to give best fitvalues for the initial ratio and the age of the samples. The latter approach ispreferred and is called the Isochron Method .

Methods

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Method #1: Direct comparison

Analyze 87Rb - free sample to find non radiogenic 87Sr/ 86Sr ratio

(Since no87

Rb in this sample all87

Sr must have been present to start with-- itis not radiogenic). Analyze 87Rb rich sample for 87 Rb, 87 Sr, and 86Sr

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Method #2: The Isochron Method

In this method minerals with varying amounts of Rb are analyzed that are thesame age .

At time of crystallization87

Sr/86

Sr ratio is the same for all minerals of the samerock.

The amount of 87Sr that you measure is equal to the original amount PLUSwhat has been generated by radioactive decay of rubidium .

Samples with varying Rb fall on a straight line in a plot of 87Sr/ 86Sr vs87 Rb/ 86Sr as the axes.

Radioactive decay equation used as the equation for a line (y = mx + b),where the slope is proportional to the age .

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How does this work?

The diagram below illustrates the isochron method.

Consider the four samples shown as black dots in the diagram.

All four of these samples have the same initial 87 Sr/ 86Sr ratio (shown by theblack dashed line ) but different 87Rb content so different 87 Rb/ 86Sr ratios .

With time , some of the87

Rb in the samples decays to87

Sr . The red arrowsshow how the locations of the samples move as a function of time (note thatone Sr is produced by each Rb that decays).

The 87Rb decreases while the 87Sr increases . As Rb decays to form Sr andthe samples evolve , they remain colinear .

You can think of the horizontal line originally defined by the initial ratio of thesamples rotating with its fixed point located at the initial 87Sr/ 86Sr ratio andan 87Rb/ 86Sr value of zero .

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Now consider the form of the Sr isotope evolution equation from above:

( = decay constant) For the variables in the diagram above, this equation is the equation of

a straight line (y = mx + b) , where y = ( 87Sr/ 86Sr) t, x = ( 87Rb/ 86Sr) , b =(87Sr/ 86Sr) o and the slope of the line (m) is e t -1 .

The Isochron Method thus consists of plotting measured 87Sr/ 86Sr valuesversus calculated 87Rb/ 86Sr values for the samples.

A straight line is then fit to the data using linear regression (most spreadsheets and hand calculators have linear regression functions).

The slope of the straight line (m) is then equal to:m = e t - 1

Thus, the age of the sample suite is given by:t = ln (m + 1)/

The intercept of the best fit line gives the initial ratio [(87Sr/ 86Sr) o] for thesample suite.

The use of this method is based on the validity of the following assumptions :

1. All of the samples are of the same age2. All of the samples came from the same source and had the same initial

ratio3. The samples were closed to Rb and Sr exchange during their complete

histories

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Recap

Recall the equation (9) above D = D o + N (e t 1) (9)

So! If we substitute in theappropriate players in the Rb-Sr system:

Recall that there is some 87Sr inthe rock to start with, so what youmeasure is equal to the originalamount PLUS what has beengenerated by radioactive decay of rubidium.

What we actually measure is theratio of these elements relative tothe stable isoptope 86Sr. Theequation becomes:

This looks remarkably like thestandard equation for a line

y = mx+b

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POTASSIUM-ARGON (KAr) Dating

Principle

It is based on the fact that some of the radioactive isotope of Potassium, Potassium-40 (K-40) , decays to the gas Argon asArgon-40 (Ar-40) .

Potassium (K) is one of the most abundant elements in the Earth's crust(2.4% by mass).

One out of every 100 Potassium atoms is radioactive Potassium-40 (K-40) .

The nuclei of naturally occurring 40K are unstable so undergo

decay.

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For every 100 K-40 atoms that decay,

11 become Ar-40. The other 89% of the 40K atoms decay to 40Ca.

This aspect of this system makes it special i.e. a branched decay :

A 40K nucleus may decay to either a 40Ca by - or to a 40Ar atom byelectron capture.

It is impossible to predict how a given 40 K atom will decay, just as it isimpossible to predict when it will decay. We can predict quite accurately whatproportion of a large number of 40K atoms will decay to each, however.

The ratio of electron captures to beta decays , called the

branching ratio , is defined as:R = e /

where the two lambda's are the decay constants (probability of decay) for each mode.

The branching ratio is 0.117, e = 0.581 x 10 -10 yr, = 4.962 x 10 -10 yr.

The total decay constant for 40K is: = e + = 5.543 x 10 10 yr 1

The radioactivity of is unusual, in that two processes take place:

-decay - 89% Electron capture: 11%

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calculated as 40Ar which also occurs in the atmosphere canadhere to the sample and contaminate it making the calculateddata inaccurate.

To check for this it is necessary to analyze also for 36Ar which hasa known ratio to 40 Ar in the atmosphere ( 40Ar/ 36Ar = 296).

The formula for correcting for atmospheric argon is:

40 Ar (measured) 296 ( 36 Ar) = 40 Ar* (radiogenic)

The resulting 40Ar* and 40K can be plugged into the ageequation as follows:

How is the Atomic Clock Set?

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When the rock crystallizes it becomes impermeable to gasses. AnyAr-40 contained in the magma is already released into theatmosphere.

As the K-40 in the rock decays into Ar-40, the gas is trapped in therock.

Limitations of K-Ar Dating

The rock should not have gone through a heating-recrystallizationprocess after its initial formation. No argon should have escaped or leaked.

No extra argon should have been added.

URANIUM-THORIUM-LEAD

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Nuclear Decay of Uranium

Uranium is a common element in the earth's crust and is widelydistributed in rocks such as granite and basalts. The relevant radioactive isotopes are: 238 U, 235 U and 232 Th.

The decay schemes are very complex as below

U-Th Decay Series

Since 238 U is much more abundant than 235 U, lets look at the decaysequence for 238 U which is given below.

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(235 U and 232 Th have their unique decay sequences as well).

Lead is formed by the radioactive decay of Uranium and Thorium .

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The final daughter products are: 206 Pb, 207 Pb and 208 Pb .

One isotope of lead, 204 Pb , is of non-radiogenic origin. The leadisotope, 204 Pb, is not produced by r. a. decay.

A summary of the decay schemes and half-lives are :

Lead is formed by the radioactive decay of Uranium and Thorium, asindicated below.

One isotope of lead, 204 Pb, is of non-radiogenic origin.

Parent Daughter Half Life Relative Abundance of Parent 238 U 206 Pb 4.468 Ga 99.27%235 U 207 Pb .7038 Ga 0.72%232 U 208 Pb 14.010 Ga ~100%

The decay schemes above represent three separategeochronometers

The availability of three separate geochronometers is anadvantage.

One problem is that some minerals which were dated containPb at the outset and must be corrected for. This is calledcommon lead .

Common lead is corrected for by measurement of 204 Pb andknowledge of the ratios of r. a. Pb relative to other Pb isotopes.

As with the Rb-Sr system, we can use the stable isotope as astandard to compare the abundance of the other isotopes

With the stable isotope, 204 Pb, as a standard to compare theabundance of the other isotopes we have the three equationsbelow

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Three sets of isochron diagrams are made based on theseequations above:

The ages should agree. If they do, they are called concordant ; if not,they are discordant .

Discordant ages indicate that there has been some type of disturbance which has reset the "time clock" at some time in thepast, such that the steady decrease in U has been interruptedand reset (e.g., metamorphism).

Ideal closed system

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