2006 lecture 12

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Lecture 12: Radioactivity

Questions

How and why do nuclei decay?

How do we use nuclear decay to tell time? What is the evidence for presence of now extinct

radionuclides in the early solar system?

How much do you really need to know about secularequilibrium and the U-series?

Tools

First-order ordinary differential equations

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Modes of decay

A nucleus will be radioactive if by decaying it can lower

the overall mass, leading to larger (negative) nuclearbinding energy Yet another manifestation of the 2nd Law of thermodynamics

Nuclei can spontaneously transform to lower mass nucleiby one of five processes

-decay

-decay positron emission

electron capture

spontaneous fission

Each process transforms a radioactive parent nucleus into

one or more daughter nuclei.

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-decayEmission of an -particle or 4He nucleus (2 neutrons, 2 protons)

The parent decreases its massnumber by 4, atomic number by 2.

Example: 238U -> 234Th + 4He

Mass-energy budget:238U 238.0508 amu234Th 234.04364He 4.00260

mass defect 0.0046 amu

= 6.86x10-13

J/decay= 1.74x1012 J/kg 238U= 7.3 kilotons/kg

This is the preferred decay mode of nuclei heavier than 209Biwith a proton/neutron ratio along the valley of stability

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-decayEmission of an electron (and an antineutrino) during

conversion of a neutron into a proton

The mass number does not change,the atomic number increases by 1.

Example: 87Rb -> 87Sr + e + Mass-energy budget:87Rb 86.909186 amu87Sr 86.908882

mass defect 0.0003 amu= 4.5x10-14 J/decay= 3.0x1011 J/kg 87Rb= 1.3 kilotons/kg

This is the preferred decay mode of nuclei with excessneutrons compared to the valley of stability

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-decay and electron captureEmission of a positron (and a neutrino) orcapture of an inner-

shell electron during conversion of a proton into a neutron

The mass number does not change,the atomic number decreases by 1.

Examples: 40K -> 40Ar + e+ + 50V+ e -> 50Ti +

+

In positron emission, most energy isliberated by remote matter-antimatterannihilation. In electron capture, a gamma

ray carries off the excess energy.

These are the preferred decay modes of nuclei with excess

protons compared to the valley of stability

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Spontaneous Fission

Certain very heavy nuclei, particular those with even mass

numbers (e.g., 238U and 244Pu) can spontaneously fission.Odd-mass heavy nuclei typically only fission in response to

neutron capture (e.g., 235U, 239Pu)

There is no fixed daughter product but rather astatistical distribution of fission products withtwo peaks (most fissions are asymmetric).

Because of the curvature of the valley of

stability, most fission daughters have excessneutrons and tend to be radioactive (-decays).

You can see why some of the isotopes peopleworry about in nuclear fallout are 91Sr and 137Cs.

Recoil of daughter products leavefission tracksof damage in crystals about 10 m long, whichonly heal above ~300C and are therefore

useful for low-temperature thermochronometry.

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Fundamental law of radioactive decay

Each nucleus has a fixed probability of decaying per unit

time. Nothing affects this probability (e.g., temperature,pressure, bonding environment, etc.)

[exception: very high pressure promotes electron capture slightly]

This is equivalent to saying that averaged over a largeenough number of atoms the number of decays per unit time

is proportional to the number of atoms present.

Therefore in a closed system:

dN

dt N (Equation 3.1)

N= number of parent nuclei at time t

= decay constant = probability of decay per unit time (units: s1)

To get time history of number of parent nuclei, integrate 3.1:

N t Noet

(3.2) No = initial number of parent nuclei at time t= 0.

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Definitions

The mean life

of a parent nuclide is given by the number

present divided by the removal rate (recall this later when we talkabout residence time):

N

N

1

(3.3)

The half life t1/2 of a nucleus is the time after which half theparent remains:

This is also the e-folding time of the decay:

N() Noe

Noe1

No

e

N(t1/2) No

2 Noe

t1/2 t1/2 ln2 t1/2 ln 2

.693

The activity is decays per unit time, denoted by parentheses:

N N (3.4)

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Decay of parent

0 2 3 4 5time

No

No

2

t1/2

No

e

0 2 3 4 5time

t1/2

0

-1

-2

-3

-4

-5

slope = -1

Activity

ln(N)ln(N

o)

Some dating schemes only consider measurement of parent nucleibecause initial abundance is somehow known.

14C-14N: cosmic rays create a roughly constant atmospheric 14C inventory,so that living matter has a roughly constant 14C/C ratio while it exchanges

CO2 with the environment through photosynthesis or diet. After deaththis 14C decays with half life 5730 years. Hence even through thedaughter 14N is not retained or measured, age is calculated using:

t

1

14 ln

(14

C) /C

(14C) /C o

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Radiocarbon dating in practice

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Radiocarbon dating in practice

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Evolution of daughter isotopes

Consider the daughter isotopeD resulting from decays of

parent isotopeN. There may be someD in the system at timezero, so we distinguish initialDo and radiogenicD*.

D t Do D* t

(3.5)

Under most circumstances,No is unknown, so substitute

Each decay of one parent yields one daughter (an extension

is needed for branching decays and spontaneous fission),

so in a closed system

D(t) Do No N t Do No 1 et

No Net

D(t) Do N t et1

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Evolution of daughter isotopes

Parent and daughter isotopes are frequently measured with

mass spectrometers, which only measure ratios accurately,

so we choose a third stable, nonradiogenic nuclide S such

that in a closed system S(t) = So:

(3.6)

D(t)

S(t) Do

S(t) N t

S(t)et1

t

D

S

o

D

S

t

N

S

e

t

1

0 2 3 4 5

No/S

o

t1/2

0

Daughter D/S

Parent N/S

time

C

oncentrationra

tios *

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Evolution of daughter isotopes When the initial concentration of daughter isotope can be taken as

zero, a date can be obtained using a single measurement of (D/S)tand (N/S)ton the same sample.

Example: 40K-40Ar dating Ar diffusivity is very high, so it is lost by minerals above some blocking

temperature (~350C for biotite). We assume 40Aro = 0 and measuretime since sample cooled through its blocking temperature.

If36Ar is used as the stable denominator isotope, an alternative toassuming 40Aro = 0 is to assume initial Ar of atmospheric composition.

40K/36Ar ratios are hard to measure well, so 40Ar-39Ar method is moreaccurate. The sample is irradiated with neutrons along with a neutronfluence standard of known age, converting 39K into 39Ar. 39K/40K isconstant in nature, so one gets the 40K content of the sample by step-

heating and measuring39

Ar/40

Ar ratios, which can be done very precisely. 40K has a branching decay; it can either electron capture to yield 40Ar or

-decay to 40Ca. The relevant decay constant is therefore (ec/40) Another example is U,Th-4He thermochronometry, which dates the passage of

apatite through the blocking temperature for4

He retention, ~80

C (!). This isuseful for dating the uplift of mountain ranges.

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K-Ar dating vs. Ar-Ar dating Here is an example of the relative precision of K-Ar and Ar-Ar

methods. The top point below is an Ar-Ar measurement, theothers are K-Ar.

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Isochron method Most often the initial concentration of neither parent nor daughter

is known, and more than one measurement is required to extract ameaningful date and also solve for the initial (D/S) ratio.

Ideally we need multiple samples ofequal age with equal initialratio (D/S)o but different ratios (N/S). In this case equation 3.6

defines a line on an isochron plot:

N/S

Do/So

slope = et-1

D/S

t

D

S

o

D

S

t

N

S

e

t1

y = intercept + x * slope

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Isochron method The best way to guarantee that all samples have the same initial

(D/S) ratio is to use different isotopes of the same element asDand S so that at high temperature diffusion will equalize this ratiothroughout a system.

The best way to guarantee that all samples have the same age is to

use different minerals from the same rock, which chemicallyfractionateNfromD when they crystallize. The whole rock canalso form a data point.

Example 1: 87Rb-87Sr The parent is 87Rb, half-life = 48.8 Ga

The daughter is 87Sr, which forms only 7% of natural Sr.

The stable, nonradiogenic reference isotope is 86Sr.

t

87Sr

86Sr

o

87Sr

86Sr

t

87Rb

86Sr

e

87t1

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Example 1: Rb-Sr systematics Rb is an alkali metal, very incompatible during melting,

with geochemical affinity similar to K.

Sr is an alkaline earth, moderately incompatible duringmelting, with geochemical affinity similar to Ca.

Age of the Chondritic meteoritesfrom Rb-Sr isochron: Acompilation of analyses of manymineral phases from manychondrites define a high precision

isochron with an age of 4.56 Gaand an initial 87Sr/86Sr of 0.698

implies solar nebula in chondriteformation region was well-mixedfor Sr isotope ratio and all

chondrites formed in a short time.

Rb+

Sr2+

Igneous processes like melting and crystallization thereforereadily separate Rb from Sr and generate a wide separation ofparent-daughter ratios ideal for quality isochron measurements.

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Example 2: Sm-Nd systematics Parent isotope is 147Sm, alpha decay half-life 106 Ga.

Daughter isotope is143

Nd, 12% of natural Nd. Stable nonradiogenic reference isotope is 144Nd.

t

143Nd

144

Nd

o

143Nd

144

Nd

t

147Sm

144

Nd

e

147t1

Nd isotopes are useful not only for dating but as tracers of large-scale geochemical differentiation. For these purposes, Nd isotope

ratios are given in the more convenient form Nd:

Nd t sample

143Nd

144Nd

CHUR

143Nd

144Nd

1

104

Nd 0 sample

143Nd

144Nd

sample

147Sm

144Nd e147t 1

CHUR

143Nd

144Nd

CHUR

147Sm

144Nd e147t 1

1

104

where CHUR is the chondritic uniform reservoir, the evolution of a reservoir withbulk earth or bulk solar system Sm/Nd ratio and initial 143Nd/144Nd.

(3.7)

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Example 2: Sm-Nd systematics Both Nd and Sm are Rare-Earth elements (REE or lanthanides), a coherent

geochemical sequence of ions of equal charge (+3), smoothly decreasing ionic

radius from La to Lu, and hence smooth variations in partition coefficients.

In most minerals, Nd is more incompatible than Sm (opposite of Rb-Sr system,where daughter Sr is more compatible than parent Rb). Hence after a partialmelting event, the rock crystallized from the extracted melt phase has a lower

Sm/Nd ratio than the source whereas the residual solids have a higher Sm/Ndratio than the source.

residue

crust

1

10

0.1

La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb L u

primitive source

Sample

CI

chondrite

Normalizing concentration of

each element to CI chondriteserves two purposesit makesprimitive (aka chondritic)compositions a flat line and ittakes out the sawtooth patternfrom the odd-even effect in thesolar abundances.

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Example 2: Sm-Nd systematics

Since the rock crystallized from

the extracted melt phase has alower Sm/Nd ratio than thesource, it evolves with time to aless radiogenic isotope ratio.

Since the residual solids have ahigher Sm/Nd ratio than thesource they evolve with time to amore radiogenic isotope ratio.

143 Nd144 Nd

time

age

0 (today) 4.5 Ga

.511847

CHUR

meltingevent

residue

crust

Initial Nd isotope ratios are reported by extrapolating back to the measured orinferred age of the sample and comparing to CHUR at that time.

Thus, Nd(t)=0 in an igneous rock implies that the source was chondritic(or primitive) at the time of melting.Typical continental crust has Nd=-15 (requires remelting enriched source!) Typical oceanic crust has Nd=+10 (requires remelting depleted source!).

This is evidence that the upper mantle (from which oceanic crust recently

came) is depleted, and that the complementary enrichedreservoir is thecontinents. The mean age of depletion of the upper mantle is ~2.5 Ga.

One-stage Nd evolution

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Example 3: Extinct nuclides

Since the parent is extinct, we cannot use equation 3.6 to measure an isochron

We can show that certain nuclei with half-lives between ~1 and 100 Ma werepresent in the early solar system even though they are extinctnow.

Chronometry based on these short-lived systems gives superior time resolutionfor studies of early solar system processes.

Example: 26Al-26Mg

half-life of26Al is 0.7 Ma. It is present in supernova debris.

t

D

S

o

D

S

t

N

S

e

t1N 0,t

1

t

D

S

o

D

S

0 1 ?

Instead, to interpret measured (D/S) ratios we need another, stable isotope S2of the same element as short-lived parentN, so that we can expect (N/S2)owas constant. This gives a new equation for a line:

tDS

t o

DS o

NS o

DS o

NS2

S2S

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Example 3: Extinct nuclides

Wasserburg used stable 27Al as thesecond, stable isotope of Al to provethat 26Al was present when the Ca,Al-

rich inclusions in chondrites formed.

He demonstrated a correlation between26Mg/24Mg and Al/Mg amongcoexisting mineral phases.

The correlation proves the presence oflive 26Al when the inclusion formed,and the slope is the initial 26Al/Al ratio,~5 x 10-5 in the oldest objects.

Given estimates of26Al production insupernovae, this places a maximum ofa few million years betweennucleosynthesis and condensation of

solids in the solar system!

Example: 26Al-26Mg

half-life of26Al is 0.7 Ma. It is present in supernova debris.

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Joys of the U,Th-Pb system 238U decays to 206Pb through an elaborate chain of 8 -decays and 6 -decays,each with its own decay constant. To understand U-Pb (or Th-Pb)

geochronology, we need to understand decay chains.

238 U

234 Th

92

91

90

144 145 146

#

protons

# neutrons

#nucleo

ns

234

235

236

2

37

238

235 U

231 Th 232 Th

234 U

234 Pa231 Pa

230 Th228 Th227 Th

228 Ac227 Ac

223 Ra224 Ra 226 Ra 228 Ra

223 Fr

219 Rn220 Rn 222 Rn

219 At218 At215 At

218 Po216 Po215 Po214 Po210 Po211 Po 212 Po

215 Bi214 Bi210 Bi211 Bi 212 Bi

214 Pb212 Pb211 Pb210 Pb206 Pb207 Pb 208 Pb

210 Tl206 Tl207 Tl 208 Tl

206 Hg

89

88

87

86

85

84

83

82

81

80

141 142 143138 139 140135 136 137132 133 134129 130 131126 127 128124 125

229

230

231

232

233

224

225

226

227

228

219

220

221

222

223

214

215

216

217

218

212

213

210

211

209

207

208

4.5Ga0.7Ga

14Ga

247k a

7 h33 ka

24 d26 h80 ka2 a18 d

6 h22 a

6 a1.6ka4 d11 d

22 m

4 d55 s4 s

1 m2 s0.1m s

3 m0.15 s2 m s0.2m s0.3 s0.5 s138 d

7 m20 m61 m2 m5 d

27 m11 h36 m21 a

1 m3 m5 m4 m

7.5 m

Decay series of 238 U, 235 U, and 232 Th

-decay -decay

232 Th chain

s = 10 -6 seconds

ms = 10-3

secconds

s = seconds

m = minutes

h = hours

d = days

a = years

ka = 103

years

Ga = 109

years

half-life unit abbreviations:

235 U chain

238 U chain

(mas

sn

umb

ers

modul

o4

)

0

3

2

(length of

chain)

-deca

ys

-decay

s

6

7

8

4

4

6

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Decay chain systematics

Consider a model system of three isotopes:

N11 N2

2 N3

ParentN1 decays toN2. Intermediate daughterN2 decaystoN

3. Terminal daughterN

3is stable.

Evolution of this system is governed by coupled equations:

dN1dt

1N1 dN2dt

1N1 2N2 dN3dt

2N2

Solution forN1 is already known (eqn. 3.2), so we have:dN2dt

1N1oe

1t 2N2dN3dt

2N2N1 t N1oe

1t

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The general solution for n isotopes in a chain was obtained

by Bateman (1910); for our 3 isotope case:

N2 (t) 1

2 1N1

oe

1t e2t N2oe2t (3.8a)N3(t) N1

o1 1

2 11e

2t 2e1t N2

o1 e2t N3

o (3.8b)

The behavior of this system depends on 1/2. Solutions fall into twoclasses. For 1/2>1, all concentrations and ratios are transient:

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25

N / N

1o

t/ 1

1/2=5N1

N2 N3

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Decay chain systematicsFor 1/2

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Consider further the case 1/2 10-12 s-1)

In this case 21 ~ 2, so 3.8a simplifies to:

N2 (t) 1

2

N1o

e1t e2t N2oe2t (3.9)

Since 2 > 1, the e2t terms decay fastest, and after about5 mean-lives ofN2, we have

N2 (t)t5/2

1

2

N1oe

1t 1

2

N1(t)

2N2 1N1 N2 N1 This is the condition of secular equilibrium: the activities

of the parent and of every intermediate daughter are equal.

The concentration ratios are fixed to the ratios of decayconstants.

(3.10)

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Applications of U-series disequilibria Violations of secular equilibrium are extremely useful for studying

phenomena on timescales comparable to the intermediate half-lives,e.g.:

230Th, t1/2 = 75000 years

226Ra, t1/2 = 1600 years

210Pb, t1/2 = 21 years Some systems incorporate lots of daughter and essentially no parent

when they form. The daughter is unsupported and acts like the parent ofan ordinary short-lived radiodecay scheme. Example: measuring

accumulation rates in pelagic sediments, where Th adsorbs on particlesbut U remains in solution.

Some systems incorporate lots of parent and essentially no daughter.Surprisingly, the daughter grows in on the time scale of its own decay,not that of the parent. Example: corals readily incorporate U and

exclude Th during CaCO3 growth. In this caseN2o = 0, e1t~1, and

230Th

234U

230

Th 238

U 1 e230t

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Applications of U-series disequilibria During partial melting, the partition coefficients of parents and

daughters may differ, producing a secular disequilibrium in melt andresidue.

For the timescales of mantle melting and melt extraction to the crust,the relevant isotopes are 230Th (75 ka), 231Pa (33 ka), and 226Ra (1.6 ka)

During melting in the mantle at pressure

2.5 GPa, the mineral garnetpreferentially retains U over Th, leading to excess (230Th) in the melt.The melt would return to secular equilibrium within ~350 ka, so thepresence of excess (230Th) in erupted basalts proves both the role ofgarnet in the source region and fast transport of melt to the crust.

(238

U)/(232

Th)

( 232 Th)

( 230 Th)

melting

d e c a

y

equilin

e

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U,Th-Pb geochronology On timescales long enough that all intermediate nuclei reach secular

equilibrium, U and Th systems can be treated as simple one-step decays to Pb.

dNndt

n1Nn1 L 1N1

Nn(t) Nio

i2

n N1o e1t1 Nn

o N1o e1t1

t

206

Pb204Pb

o

206

Pb204Pb

t

238

U204Pb

e238t1

t

207Pb

204Pb

o

207Pb

204Pb

t

235U

204Pb

e235t

1t

208Pb

204Pb

o

208Pb

204Pb

t

232Th

204Pb

e

232t1

238U, t1/2=4.5 Ga

235

U, t1/2=0.7 Ga

232Th, t1/2=14 Ga

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U,Th-Pb geochronology

*207Pb

204Pb

*206Pb204

Pb

t

207Pb

204Pb

o

207Pb

204Pb

t

206

Pb204Pb

o

206

Pb204Pb

t

235U

238

U

e235t1

e

238t1 Conveniently, 235U/238U is globally constant (except for an ancient

natural fission reactor in Gabon, and perhaps near Oak Ridge, TN) at1/138. One does not have to measure U at all for this method.

Since 207Pb-206Pb age depends only on Pb isotope ratios, not Pb or Uconcentration, it is not affected by recent alteration whether Pb-loss or

U-loss. Only addition of contaminant Pb or aging after alteration willaffect the measured age (still need to correct for common Pb).

Each of these chronometers can be used independently. If they agree,the sample is said to be concordant. However, Pb is mobile in many

environments, and samples often yield discordantages from the 238U-206Pb, 235U-207Pb, and 232Th-208Pb chronometers.

Discordance due to recentPb loss, such as during weathering, isresolved by coupling the two U-Pb systems to obtain a 207Pb-206Pb date

U Th Pb geochronology

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U,Th-Pb geochronology Any concordant group of samples plots on an isochron line in

(207Pb/204Pb)*-(206Pb/204Pb)* space; the age is calculable from its slope.

Initial Pb isotope ratios can be neglected for many materials with veryhigh U/Pb ratios (e.g., old zircons), or measured on a coexisting mineralwith very low U/Pb ratio (e.g., feldspar, troilite).

10

11

12

13

14

15

16

1718

19

9 11 13 15 17 19 21 23

( 206 Pb/204 Pb)

( 207 P

b /

204

P b )

4.56 Ga ISOCHRON

evolution curves

=5

=8

=10=12

initial Pb (Canyon Diablo FeS, Patterson 1955)

In 1955 C.C. Patterson measured initial Pb in essentially U-free troilite (FeS)grains in the Canyon Diablo meteorite and thereby determined the initial Pb

isotope composition of the solar system. It follows from measurements ofterrestrial Pb samples that the Pb-Pb age of the earth is 4.56 Ga, and that theearth has evolved with a =(238U/204Pb) ratio of about 9 (chondrite value = 0.7)


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U,Th-Pb geochronology If Pb was lost long enough in the past for continued decay of U to have

any significant effect on Pb isotopes, the 207Pb-206Pb may be impossible

to interpret correctly. In this case, we turn to the concordia diagram (G.Wetherill). Consider the family of all concordant compositions:

t

206Pb

204

Pb

o

206Pb

204

Pb

t

238U

204Pb

*206

Pb238U

e238t1

*207Pb

235U

e235t1

These equations parameterize a curve in (206Pb/238U)*(207Pb/235U)*space, the concordia.

0

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30207 Pb*/235 U

0.51.0

1.5

2.0

2.5

3.0

3.5

concordant sample after 3.0 Ga

206 Pb*238 U

concordia

age in Ga


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U,Th-Pb geochronology Imagine that a suite of samples underwent a single short-lived episode

of Pb-loss at some time. This event did not fractionate 206Pb from 207Pb,

so it moved the samples along a chord towards the origin in theconcordia plot:

If these now discordant samples age as closed systems, they remain on a line,whose intercepts with the concordia evolve along the concordia with time

0

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30( 207 Pb*/235 U)

0.5

1.01.5

2.0

2.5

3.0

3.5

206

Pb*238 U

discordant samples at time of Pb-loss,

3.0 Ga after crystallization

0

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30(207 Pb*/235 U)

0.5

1.01.5

2.0

2.5

3.03.5

discordant samples 0.5 Ga after Pb-loss,

3.5 Ga after crystallization

206 Pb*238 U


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U,Th-Pb geochronology Example: the oldest zircons on Earth (actually, the oldest anything on

Earth), from the Jack Hills conglomerate in Australia

P k l GCA 65 4215 2001

2006 lecture 12

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