Transcript Lecture 3: Radioactivity - asimow.com forwarding page

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 secular
equilibrium and the U-series?
• Tools
– First-order ordinary differential equations
1
Modes of decay
• A nucleus will be radioactive if by decaying it can lower
the overall mass, leading to larger (negative) nuclear
binding energy
– Yet another manifestation of the 2nd Law of thermodynamics
• Nuclei can spontaneously transform to lower mass nuclei
by one of five processes
–
–
–
–
–
a-decay
b-decay
positron emission
electron capture
spontaneous fission
• Each process transforms a radioactive parent nucleus into
one or more daughter nuclei.
2
a-decay
Emission of an a-particle or 4He nucleus (2 neutrons, 2 protons)
The parent decreases its mass
number by 4, atomic number by 2.
Example: 238U -> 234Th + 4He
Mass-energy budget:
238U
238.0508 amu
234Th
–234.0436
4He
–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 209Bi
with a proton/neutron ratio along the valley of stability 3
b-decay
Emission 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– + n
Mass-energy budget:
87Rb
86.909186 amu
87Sr
–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 excess
neutrons compared to the valley of stability
4
b+-decay and electron capture
Emission of a positron (and a neutrino) or capture of an innershell 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+ + n
50V+ e– -> 50Ti + n + g
In positron emission, most energy is
liberated by remote matter-antimatter
annihilation. 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
5
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 a
statistical distribution of fission products with
two peaks (most fissions are asymmetric).
Because of the curvature of the valley of
stability, most fission daughters have excess
neutrons and tend to be radioactive (b-decays).
You can see why some of the isotopes people
worry about in nuclear fallout are 91Sr and 137Cs.
Recoil of daughter products leave fission tracks
of damage in crystals about 10 mm long, which
only heal above ~300°C and are therefore
useful for low-temperature thermochronometry.
6
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 large
enough number of atoms the number of decays per unit time
is proportional to the number of atoms present.
• Therefore in a closed system:
dN
(Equation 3.1)
= - lN
dt
– N = number of parent nuclei at time t
– l = decay constant = probability of decay per unit time (units: s–1)
• To get time history of number of parent nuclei, integrate 3.1:
N (t ) = No e- lt
– No = initial number of parent nuclei at time t = 0.
(3.2)
7
Definitions
• For an exponential process*, the mean life t of a parent
nuclide is given by the number present divided by the
removal rate (recall this later when we talk about residence time):
N 1
* For a linear process,
t=
=
this is off by a factor of 2
lN l
– This is also the “e-folding” time of the decay:
No
N(t ) = No e
= Noe =
e
• The half life t1/2 of a nucleus is the time after which half the
parent remains:
No
ln 2 .693
- lt1/2
Þ
l
t
=
ln
2
(3.3)
N(t1/2 ) =
= Noe
Þ t1/2 =
»
1/2
2
l
l
• The activity is decays per unit time, denoted by parentheses:
( N ) = lN
(3.4)
- lt
-1
8
Activity
ln(lN)–ln(lNo)
Decay of parent
Some dating schemes only consider measurement of parent nuclei
because 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 death
this 14C decays with half life 5730 years. Hence even through the
daughter 14N is not retained or measured, age is calculated using:
14
t=
( C) / C
ln 14
l14 ( C) / C
1
[
]o
9
Radiocarbon dating in practice
10
Radiocarbon dating in practice
11
Evolution of daughter isotopes
• Consider the daughter isotope D resulting from decays of
parent isotope N. There may be some D in the system at time
zero, so we distinguish initial Do and radiogenic D*.
D(t ) = Do + D* (t )
• 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 + N o 1 - e - lt
)
• Under most circumstances, No is unknown, so substitute
No = Ne lt
(
)
D(t) = Do + N (t ) elt - 1
(3.5)
12
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:
D(t) Do N (t ) lt
=
+
e -1
S(t) S(t) S(t)
æ Dö = æ Dö + æ N ö e lt - 1
(3.6)
è S øt è Sø o è S øt
(
)
Concentration ratios
(
)
*
13
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)t
and (N/S)t on the same sample.
• Example: 40K-40Ar dating
– Ar diffusivity is very high, so it is lost by minerals above some blocking
temperature (~350 °C for biotite). We assume 40Aro = 0 and measure
time since sample cooled through its blocking temperature.
– If 36Ar is used as the stable denominator isotope, an alternative to
assuming 40Aro = 0 is to assume initial Ar of atmospheric composition.
– 40K/36Ar ratios are hard to measure well, so 40Ar-39Ar method is more
accurate. The sample is irradiated with neutrons along with a neutron
fluence standard of known age, converting 39K into 39Ar. 39K/40K is
constant in nature, so one gets the 40K content of the sample by stepheating and measuring 39Ar/40Ar ratios, which can be done very precisely.
– 40K has a branching decay; it can either electron capture to yield 40Ar or
b-decay to 40Ca. The relevant decay constant is therefore (lec/l40)
• Another example is U,Th-4He thermochronometry, which dates the passage of
apatite through the blocking temperature for 4He retention, ~80°C (!). This is
useful for dating the uplift of mountain ranges.
14
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, the
others are K-Ar.
15
Isochron method
D/S
• Most often the initial concentration of neither parent nor daughter
is known, and more than one measurement is required to extract a
meaningful date and also solve for the initial (D/S) ratio.
• Ideally we need multiple samples of equal age with equal initial
ratio (D/S)o but different ratios (N/S). In this case equation 3.6
defines a line on an isochron plot:
(
)
æ Dö = æ Dö + æ N ö e lt - 1
è S øt è Sø o è S øt
y = intercept + x * slope
16
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 as D
and S so that at high temperature diffusion will equalize this ratio
throughout a system.
• The best way to guarantee that all samples have the same age is to
use different minerals from the same rock, which chemically
fractionate N from D when they crystallize. The whole rock can
also 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.
(
)
æ 87 Sr ö æ 87 Sr ö æ 87 Rb ö l87 t
-1
ç 86 ÷ = ç 86 ÷ + ç 86 ÷ e
è Sr ø t è Sr ø o è Sr ø t
17
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 during
melting, with geochemical affinity similar to Ca.
• Igneous processes like melting and crystallization therefore
readily separate Rb from Sr and generate a wide separation of
parent-daughter ratios ideal for quality isochron measurements.
Age of the Chondritic meteorites
from Rb-Sr isochron: A
compilation of analyses of many
mineral phases from many
chondrites define a high precision
isochron with an age of 4.56 Ga
and an initial 87Sr/86Sr of 0.698
• implies solar nebula in chondrite
formation region was well-mixed
for Sr isotope ratio and all
chondrites formed in a short time.
18
Example 2: Sm-Nd systematics
• Parent isotope is 147Sm, alpha decay half-life 106 Ga.
• Daughter isotope is 143Nd, 12% of natural Nd.
• Stable nonradiogenic reference isotope is 144Nd.
(
)
æ 143 Nd ö æ 143 Nd ö æ 147Sm ö l147 t
-1
ç 144 ÷ = ç 144 ÷ + ç 144 ÷ e
è Nd ø t è Nd ø o è Nd ø t
• Nd isotopes are useful not only for dating but as tracers of largescale geochemical differentiation. For these purposes, Nd isotope
ratios are given in the more convenient form eNd:
é 143 Nd 144 Nd
ù
sample
ê
ú ´ 10 4
(3.7)
e Nd (t ) = 143
1
144
ê
ú
Nd
Nd
CHUR
ë
û
(
(
é
e Nd (0 ) = êê
êë
)
)
( 143 Nd 144 Nd) sample - ( 147Sm 144 Nd )sample (e l147t - 1) ùú 4
- 1 ´ 10
l147t
143
144
147
144
Nd
Nd
Sm
Nd
e
1
(
)CHUR (
)CHUR (
) úúû
where CHUR is the chondritic uniform reservoir, the evolution of a reservoir19with
bulk earth or bulk solar system Sm/Nd ratio and initial 143Nd/144Nd.
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.
Sample
CI chondrite
• 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 partial
melting 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/Nd
ratio than the source.
• Normalizing concentration of
each element to CI chondrite
serves two purposes…it makes
primitive (aka chondritic)
compositions a flat line and it
takes out the sawtooth pattern
from the odd-even effect in the
solar abundances.
20
Example 2: Sm-Nd systematics
One-stage Nd evolution
• Since the rock crystallized from
the extracted melt phase has a
lower Sm/Nd ratio than the
source, it evolves with time to a
less radiogenic isotope ratio.
• Since the residual solids have a
higher Sm/Nd ratio than the
source they evolve with time to a
more radiogenic isotope ratio.
Initial Nd isotope ratios are reported by extrapolating back to the measured or
inferred age of the sample and comparing to CHUR at that time.
• Thus, eNd(t)=0 in an igneous rock implies that the source was chondritic (or
primitive) at the time of melting.
•Typical continental crust has eNd(t)=-15 (requires remelting enriched source!)
• Typical oceanic crust has eNd(t)=+10 (requires remelting depleted source!).
This is evidence that the upper mantle (from which oceanic crust recently came)
is depleted, and that the complementary enriched reservoir is the continents. The
mean age of depletion of the upper mantle is ~2.5 Ga.
21
Example 3: Extinct nuclides
• We can show that certain nuclei with half-lives between ~1 and 100 Ma were
present in the early solar system even though they are extinct now.
Chronometry based on these short-lived systems gives superior time resolution
for studies of early solar system processes.
• Example: 26Al-26Mg
• half-life of 26Al is 0.7 Ma. It is present in supernova debris.
• Since the parent is extinct, we cannot use equation 3.6 to measure an isochron
(
)
æ Dö = æ Dö + æ N ö e lt - 1
è S øt è Sø o è S øt
Dö æ Dö
æ
N = 0, t >> Þ
=
+ 0(¥ - 1) = ?
è
ø
è
ø
l
S t
S o
1
• Instead, to interpret measured (D/S) ratios we need another, stable isotope S2
of the same element as short-lived parent N, so that we can expect (N/S2)o
was constant. This gives a new equation for a line (a fossil isochron):
Dö æ N ö
Dö æ N ö æ S2 ö
æ Dö ¾t®¥
æ
æ
¾¾® =
+
=
+
è S øt
è S ø o è S ø o è S ø o è S2 ø o è S ø
22
Example 3: Extinct nuclides
• Example: 26Al-26Mg
• half-life of 26Al is 0.7 Ma. It is present in supernova debris.
• Wasserburg used stable 27Al as the
second, stable isotope of Al to prove
that 26Al was present when the Ca,Alrich inclusions in chondrites formed.
• He demonstrated a correlation between
26Mg/24Mg and Al/Mg among
coexisting mineral phases.
• The correlation proves the presence of
live 26Al when the inclusion formed,
and the slope is the initial 26Al/Al ratio,
~5 x 10-5 in the oldest objects.
• Given estimates of 26Al production in
supernovae, this places a maximum of
a few million years between
nucleosynthesis and condensation of
solids in the solar system!
23
Joys of the U,Th-Pb system
• 238U decays to 206Pb through an elaborate chain of 8 a-decays and 6 b-decays,
each with its own decay constant. To understand U-Pb (or Th-Pb)
geochronology, we need to understand decay chains.
24
Decay chain systematics
• Consider a model system of three isotopes:
l
l
1 ® N ¾¾
2® N
N1 ¾¾
2
3
Parent N1 decays to N2. Intermediate daughter N2 decays
to N3. Terminal daughter N3 is stable.
• Evolution of this system is governed by coupled equations:
dN1
dN2
dN3
= -l1N1
= l1 N1 - l2 N2
= l2 N2
dt
dt
dt
• Solution for N1 is already known (eqn. 3.2), so we have:
dN3
o - l1t dN2 = l N o e- l1t - l N
= l2 N2
N1( t ) = N1 e
1 1
2 2
dt
dt
25
Decay chain systematics
• The general solution for n isotopes in a chain was obtained
by Bateman (1910); for our 3 isotope case:
(
l1
)
N1o e- l1t - e - l 2t + N2oe - l 2t
(3.8a)
l2 - l1
1
-l2t
- l1t ö
- l2 t
oæ
o
o
N3 (t) = N1 ç1 +
l1e
- l2e
+
N
1e
+
N
÷
2
3 (3.8b)
è l2 - l1
ø
N2 (t) =
(
)
(
)
The behavior of this system depends on l1/l2. Solutions fall into two
classes. For l1/l2>1, all concentrations and ratios are transient:
26
Decay chain systematics
• Consider the case l1/l2 << 1, which applies to all
intermediates in the U and Th decay chains (parent l are
all < 10-16 s-1; intermediates l are all >10-12 s-1)
• In this case l2–l1 ~ l2, so 3.8a simplifies to:
(
)
l1 o - l1t - l2 t
N2 (t) =
N1 e
-e
+ N2o e- l 2 t
l2
(3.9)
• Since l2 > l1, the e–l2t terms decay fastest, and after about
5 mean-lives of N2, we have
l1 o - l1t l1
N2 (t) ¾ ¾ ¾
¾® N1 e
=
N1(t)
l2
l2
l2 N2 = l1N1 ( N2 ) = ( N1 )
t>5/ l2
(3.10)
• 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 decay
27
constants.
Decay chain systematics
For l1/l2<<1, the system evolves to a state called secular equilibrium
in which the ratio of parent to intermediate daughter is fixed:
It takes about 5 mean-lives of N2 to reach secular equilibrium. After
this point the initial amount of N2 is the system no longer matters.
Note the N3 does not participate in secular equilibrium, it just
accumulates.
28
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 of
an ordinary short-lived radiodecay scheme. Example: measuring
accumulation rates in pelagic sediments, where Th adsorbs on particles
but 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 case N2o = 0, e–l1t~1, and
( ) = ( Th) » 1 - e-l230t
( 234U ) ( 238U )
230
Th
230
29
Applications of U-series disequilibria
• During partial melting, the partition coefficients of parents and
daughters may differ, producing a secular disequilibrium in melt and
residue.
• 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 garnet
preferentially retains U over Th, leading to excess (230Th) in the melt.
The melt would return to secular equilibrium within ~350 ka, so the
presence of excess (230Th) in erupted basalts proves both the role of
garnet in the source region and fast transport of melt to the crust.
30
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.
n
(
N n (t) = å Nio + N1o e
i=2
l1t
)
(
-1 » Nno + N1o e
)
l1t
-1
(
)
(
)
235U,
(
)
232Th,
æ 206 Pb ö æ 206 Pbö æ 238U ö l238 t
-1
ç 204 ÷ = ç 204 ÷ + ç 204 ÷ e
è Pb ø t è Pbø o è Pb ø t
æ 207 Pb ö æ 207 Pb ö æ 235U ö l235 t
-1
ç 204 ÷ = ç 204 ÷ + ç 204 ÷ e
è Pb ø t è Pbø o è Pb ø t
æ 208 Pb ö æ 208 Pb ö æ 232Th ö l232 t
-1
ç 204 ÷ = ç 204 ÷ + ç 204 ÷ e
è Pb ø t è Pbø o è Pb ø t
238U,
t1/2=4.5 Ga
t1/2=0.7 Ga
t1/2=14 Ga
31
U,Th-Pb geochronology
• 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 discordant ages from the 238U206Pb, 235U-207Pb, and 232Th-208Pb chronometers.
• Discordance due to recent Pb loss, such as during weathering, is
resolved by coupling the two U-Pb systems to obtain a 207Pb-206Pb date
*
207
207
207
æ Pb ö
ç 204 ÷
è Pb ø
æ Pb ö æ Pb ö
ç 204 ÷ - ç 204 ÷
è Pb ø t è Pbø o
(
(
)
)
l235 t
e
-1
Uö
º 206
= ç 238 ÷ l t
*
206
æ Pb ö æ Pbö
è U ø t e 238 -1
æ 206 Pb ö
ç 204 ÷ - ç 204 ÷
ç 204 ÷
è Pb ø t è Pbø o
è Pb ø
æ 235
• Conveniently, 235U/238U is globally constant (except for an ancient
natural fission reactor in Gabon, and perhaps near Oak Ridge, TN) at
1/138. 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 U
concentration, it is not affected by recent alteration whether Pb-loss or
U-loss. Only addition of contaminant Pb or aging after alteration will
affect the measured age (still need to correct for common Pb).
32
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 very
high U/Pb ratios (e.g., old zircons), or measured on a coexisting mineral
with very low U/Pb ratio (e.g., feldspar, troilite).
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 of
terrestrial Pb samples that the Pb-Pb age of the earth is 4.56 Ga, and that the 33
earth has evolved with a m=(238U/204Pb) ratio of about 9 (chondrite value = 0.7)
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:
æ 206 Pb ö æ 206 Pbö
ç 204 ÷ - ç 204 ÷
è Pb ø t è Pbø o
æ
Uö
ç 204 ÷
è Pbø t
238
æ
ºç
è
*
(
)
*
Pb ö
æ 207 Pbö
l238t
l235t
=
e
1
=
e
-1
÷
238
ç
÷
235
Uø
è
Uø
206
(
)
• These equations parameterize a curve in (206Pb/238U)*–(207Pb/235U)*
space, the concordia.
34
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 the
concordia 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
35
U,Th-Pb geochronology
• Example: the oldest zircons on Earth (actually, the oldest anything on
Earth), from the Jack Hills conglomerate in Australia…
Peck et al. GCA 65:4215, 2001
36