Isotope Geochemistry (2)

8/8/2019 Isotope Geochemistry (2)

1/25

LU 6 ISOTOPE GEOCHEMISTRY

ISOTOPES

DEFINITION

Two or more nuclides having the same atomic number, thus constitutingthe same element, but differing in the mass number. Isotopes of a givenelement have the same number of nuclearprotons but differing numbers ofneutrons.

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 otherelements.

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

1
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


2/25

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 anothernuclide.

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

2


3/25

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 decayBeta 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 EmissionGamma 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 emissionPositron emission involves a particle that has the same mass as an electron but apositive charge. The particle is released from the nucleus.

Electron captureElectron 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.

3


4/25

4


5/25

Principle of Radioactive Decay

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

This probability is expressed as the decay constant.

Here we explore the decay process graphically.1. Imagine a batch of36 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 after1 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 foreach 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.

5
http://earth.leeds.ac.uk/dynamicearth/dating/principles1.htmhttp://earth.leeds.ac.uk/dynamicearth/dating/principles1.htm


6/25

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 otherbut 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 ofhow quickly or slowly a material willdecay.

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

6

2 kg 1 kg


7/25

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

IfN is the number of atoms remaining, then

- dN/dt = N (1)where is the proportionality constant known as the decay constantNis 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 forNas a function of time

Rearrangement

- dN/N = dt (2)

Integrate both sides

N t

-dN/N= dt (3)No 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 eCis the constant of integration

The integration constantCmay be expressed in terms of the original number of

parent atoms whent=0

Whent=0,

No= number of nuclidesatt=0

-InNo= (0) + C

C = - InNo

Therefore the integrated form of the equation is

In N = t - InNo (5)

Rearrangement

In N - InNo = -t (6)

7


8/25

Present Original

In (N/No ) = -t (7)

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

N/No = 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 timetfrom the original number of atoms(No)present at timet=0.

Rearrangement

Present Original

N = Noe-t (9)

N number of parent atoms currently presentNo 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.

8
http://en.wikipedia.org/wiki/Exponential_decayhttp://en.wikibooks.org/wiki/Image:NM4_11a.gifhttp://en.wikipedia.org/wiki/Exponential_decay


9/25

All three curves here are exponential in nature, only the Decay Constant isdifferent.

When the Decay Constant has a low value the curve decreases relativelyslowly

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

The equation can be rearranged

Original Present

No = Net (10)

Nnumber of parent atoms currently presentNonumber of parent atoms originally present when mineral was formedEach radioactive isotope/ radionuclide has a characteristic decay constant that

must be determined experimentally.

Note:

N = N0e-kt (exponential decay)

[ N = N0ekt (exponential growth) ]

where

N0is the initial quantity tis time N(t) is the quantity after time t kis the decay constant and exis the exponential function (e is the base of the natural logarithm)

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

9
http://en.wikibooks.org/wiki/Image:NM4_12a.gif


10/25

Decay ofparent produces daughteror radiogenic nuclides.

Number ofdaughters produced is simply the differencebetween initialnumberofparents and number remaining after time t.

original presentD = No N (11)

Substituting (10) into (11) we obtain (forNo)

D = Net N = N (et 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. whent= 0, a more general expression is:

D = Do + N (et 1) (13)

Where Dois the number of daughters originally present.

10


11/25

Rearrangement

T = 1/ In (D/Do+ 1) (14)

P

This is the time during which an amount of the daughterrepresented byDhasaccumulated, leaving undecayed an amount of the parent represented byP.

Values ofDand Pare found by analyzing the rock or mineral in which theradioactive isotope occurs.

If we can also find values forand Dothe equation will give us the age of therock or mineral in years.

The decay constantis found by laboratory measurement of decay rate.

11


12/25

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

Afterone half-life there will be 5 grams of the radioactive element left.Afteranother half-life, there will be 2.5 g of the original element left.Afteranother 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.

12


13/25

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 ofdaughter isotope to amount ofparent

originally there

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

What is the age of the mineral?Given: Half-life of235U is 0.7 billion years (b. y.)

13


14/25

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 numberby 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 neveruse 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 life

Potassium 40 Argon 40 1.25 billion yrs

Rubidium 87 Strontium 87 48.8 billion yrs

Thorium 232 Lead 208 14.0 billion years

Uranium 235 Lead 207 704 million years

Uranium 238 Lead 206 4.47 billion years

Carbon 14 Nitrogen 14 5730 years

XXXXXXXXXXHalf-life Equations:

14


15/25

1) Suppose the amount of time for the number of parent atoms to decrease tohalf the original number i.e. twhen N/No =1/2 is required to be determined.

Take equation (4) below

In (N/No ) = -t (4)

and setting N/No to 1/2 rearrange it to get

In 1/2 = -t1/2 or In 2= t1/2 (5)

to finally gett1/2= 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

15
http://www.eas.asu.edu/~holbert/eee460/decay.htmlhttp://www.eas.asu.edu/~holbert/eee460/decay.html


16/25

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 elementand the amount of stable daughter elements.

RADIOACTIVE DECAY SYSTEMS OF GEOCHRONOLOGICAL INTEREST

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

16


17/25

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 87Sr(Z=38)

The decay constant is

= 1.42x10-11 y-1.

17


18/25

Background

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

Strontium: Divalent Occurs as four stable

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

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

87Rb 27.8346 86.90918

85Rb 72.1654 84.91171

Strontium Isotopes

88Sr 82.53 87.9056

87Sr 7.04 86.9089

86Sr 9.87 85.9094

84Sr 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:

18


19/25

i) In this equation, tis the time since the system was formed(Note that the system is assumed to have remained closed to theexchange of Rb and Sr since its formation date)

The (87Sr/86Sr)ois the isotopic composition of Sralready in the system at thetime of its formation (the initial ratio) and 87Rb/86Sris 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 original

daughter concentration.This is done by normalising against a stable reference isotope that is notitself radioactive or produced by radioactive decay of another isotope.Thereferenceisotope is 86Sr(86Sr)..

iii) The abundance of87Sr (daughter) is measured relative to a referenceisotope. Thus, the Sr isotopic composition of a sample is reported as theratio of87Sr to 86Sr i.e. 87Sr/86Sr

iv) Of these terms, (87Sr/86Sr)t , which is the total87Sr/86Sr, is measured in the

laboratory; 87Rb/86Sris calculated from the measured Rb and Sr

concentrations in the sample; and (87Sr/86Sr)o andtare 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 ifthe 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/Srratios

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

19


20/25


21/25

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 for87Rb, 87Sr, and 86Sr

21


22/25

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 of87Sr that you measure is equal to the original amountPLUSwhat has been generated by radioactive decay of rubidium.

Samples with varying Rb fall on a straight line in a plot of 87Sr/86Srvs87Rb/86Sras the axes.

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

22


23/25

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 initial87Sr/86Sr ratio (shown by theblack dashed line) but different 87Rbcontent so different 87Rb/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.

23


24/25

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 et-1.

The Isochron Method thus consists ofplotting measured 87Sr/86Sr valuesversus calculated 87Rb/86Srvalues 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 = et- 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

24


25/25

Date post:	09-Apr-2018
Category:	Documents
Upload:	gengkapak
View:	225 times
Download:	0 times

Isotope Geochemistry (2)

Documents