Simple Decay: Radioactive Parent  Stable Daughter
ingrowth of daughter
1000
900
D* = N0 (1 - e-lt)
800
700
600
500
400
300
200
100
0
N = N0 e-lt
D0
0
1
decay of parent
2
3
4
5
6
7
Half-lives
1
8
9
10
Decay Series: Radioactive Parent  Radioactive Daughter
200
N1 (T
100
80
1/
2
= 10
hr)
60
Decay of parent
40
N
20
N2 (T1/ = 1 h
r)
2
10
8
6
Ingrowth from parent
and decay of daugther
4
2
1
0
1
2
3
4
5
6 7
8
9 10 11 12
Time in hours
Image by MIT OpenCourseWare. After Figure 4.4 in Faure.
2
Consider the decay series N1  N2  N3
Remember:
1)
-dN1/dt = l1 N1
Now we consider N2, that is produced by decay of N1 and itself decays to N3:
2)
dN2/dt = l1 N1 – l2 N2
Remember:
3)
N1 = N1,0 e –l1t
Substitute 3) into 2):
4)
dN2/dt = l1 N1,0 e –l1t – l2 N2
Rearrange:
5)
dN2/dt + l2 N2 – l1 N1,0 e –l1t = 0
Solving this first order differential equation for N2 yields:
6)
N2 = l1/(l2 – l1) N1,0 (e –l1t - e –l2t) + N2,0 e –l2t
The solution, as well as equivalent solutions for three nuclides and the general case, are known as Bateman
(1910) equations/solutions.
The first term in equation 6) is the number of N2 atoms decayed from N1 not yet decayed to N3
The second term in equation 6) is number of N2 atoms that remain from the initial N2,0
3
The classic Bateman
paper on the famous
“Bateman equations”
Bateman, Harry. "The solution of a
system of differential equations occurring
in the theory of radioactive
transformations." In Proc. Cambridge
Philos. Soc, vol. 15, no. pt V, pp. 423427. 1910.
4
continue…
If there are no atoms of the original daughter N2,0, then 6) simplifies to:
7)
N2 = l1/(l2 – l1) N1,0 (e –l1t – e –l2t)
Example: 238U  234U  230Th  …  206Pb
8)
(230Th) = (230Th)x + (230Th)s
x = excess, not supported, s = supported from 238U
Let’s first consider the excess activity only, at some time, t:
9)
(230Th)x = (230Th)x,0 e –l230 t
Normalize by a “stable” isotope. Relative to the short-lived daughters, 232Th is “stable”
10)
(230Th/232Th)x = (230Th/232Th)x,0 e –l230 t
Now let’s consider the 238U-supported (230Th) – see equation 7)
11)
(230Th)s = l234/(l230 – l234)
At secular equilibrium (234U) = (238U)
Also
l230 - l234 ≅ l230
234U
0
then
(e –l234t – e –l230t)
234U
l234 = (238U)
as l234 is small: e –l234 t ≅ 1
5
Application!
238U
22.3 y
Peucker-Ehrenbrink, 2012
Decay Series
19.9 m
6
238U
Decay Series
± constant supply
from decay of 238U
Size of spout
equals λ
Fill level in each
tank equals N
Flux equals (A)
Courtesy of Mineralogical Society of America. Used with permission.
7
Application in determining chronologies of sediments
Basic decay equation: N = N0 e -l t
assuming no U-supported activity
Replace time (t) with depth in sediment column (d) divided by sedimentation rate (sr)
Decay equation:
t = d / sr
N = N0 e -l d/sr
ln N = ln N0 - ld/sr
t = d / sr
In a diagram of ln N (y axis) and d (x axis)
the slope (m) is
Figure of log(230Th/232Th)A versus
Core Depth, cm, has been removed
due to copyright restrictions.
m = - l / sr
and
sr = - l / m
Faure, Gunter. “Principles of
Isotope Geology.” (John Wiley &
Sons, 1986): 367.
This method is also known as the ionium
(230Th) method of dating.
8
Complications…
b)
c)
d)
log (230Th/232Th)
a)
Depth in the core
9
Complications…
b)
log (230Th/232Th)
a)
a) Constant sedimentation rate
b) Change in sedimentation rate
c) Mixing at the top
d)
c)
d)
Depth in the core
10
238U-supported 230Th
dominates
continue…
If there are no atoms of the original daughter N2,0, then 6) simplifies to:
7)
N2 = l1/(l1 – l2) N1,0 (e –l1t – e –l2t)
Example:
8)
238U
 234U  230Th … 
(230Th) = (230Th)x + (230Th)s
206Pb
x = excess, not supported, s = supported from 238U
Let’s first consider the excess activity only, at some time, t:
9)
(230Th)x = (230Th)x,0 e –l230 t
Normalize by a “stable” isotope. Relative to the short-lived daughters, 232Th is “stable”
10)
Application!
(230Th/232Th)x = (230Th/232Th)x,0 e –l230 t
Now let’s consider the 238U-supported (230Th) – see equation 7)
11)
(230Th)s = l234/(l230 – l234)
At secular equilibrium, (234U) = (238U)
Also
then
l230 - l234 ≅ l230
234U
234U
0
(e –l234t – e –l230t)
l234 = (238U)
as l234 is small: e –l234 t ≅ 1
11
continue…
11)
230Th
s
= l234/(l230 - l234)
l230 230Ths = l234
12)
(230Th)s =
From equation 9)
238U
0
234U
0
234U
0
(e –l234t – e –l230t)
(1 – e –l230t)
(1 – e –l230t)
(230Th)x = (230Th)x,0 e –l230 t
(230Th) = (230Th)x,0 + (230Th)s
Total 230Th activity = initial excess 230Th activity + 238U-supported activity
13)
(230Th) = (230Th)x,0 e –l230t +
238U
(1 – e –l230t)
Normalize to 232Th
14)
(230Th/232Th) = (230Th/232Th)x,0 e –l230t + (238U/232Th) (1 – e –l230t)
If (230Th/232Th) is plotted against (238U/232Th), equation 14) is a linear equation, the so-called
230Th-238U
isochron diagram
12
(230Th/232Th)
230Th-238U
isochron diagram
t=0
Equipoint
(238U/232Th)
© John Wiley & Sons. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/fairuse.
13
Modified after figure 21.7 of
Faure, Gunter. “Principles of
Isotope Geology.” (John Wiley &
Sons, 1986): 379.
238U-series
238
U
234
4.47 by
235U-series
235
U
U
0.704 by
245,000 y
234
231
Pa
Th
21.4 d
Pa
32,800 y
6.69 h
234
232Th-series
230
231
Th
75,000 y
227
Th
1.06 d
232
Th
14.1 by
18.7 d
227
11.4 d
222
219
3.823 d
222
214
Po
-5
3.04 m
210
Bi
19.7 m
Pb
26.9 m
Po
138.4 d
1.6 10 s
214
214
210
Po
Pb
22.6 y
224
Ra
3.63 d
220
Rn
3.96 s
55.6 s
215
216
Po
211
Bi
211
Pb
36.1 m
stable
4.77 m
14
Pb
stable
Tl
Bi
1.01 h
207
Pb
Po
3 10-7 s
212
Bi
2.14 m
208
212
Po
0.15 s
0.0018 s
207
Peucker-Ehrenbrink, 2012
Ra
5.75 y
Rn
5.01 d
210
228
Ra
1600 y
Rn
Ac
6.15 h
223
Ra
Th
1.91 y
228
Ac
21.8 y
226
228
Th
212
208
Pb
10.6 h
Pb
stable
208
Tl
3.05 m
238U-series
238
U
234
4.47 by
235U-series
235
U
U
0.704 by
245,000 y
234
231
Pa
Th
21.4 d
Pa
32,800 y
6.69 h
234
232Th-series
230
231
Th
75,000 y
227
Th
1.06 d
232
Th
14.1 by
18.7 d
227
11.4 d
222
219
3.823 d
222
214
Po
-5
3.04 m
210
Bi
19.7 m
Pb
26.9 m
Po
138.4 d
1.6 10 s
214
214
210
Po
Pb
22.6 y
224
Ra
3.63 d
220
Rn
3.96 s
55.6 s
215
216
Po
211
Bi
211
Pb
36.1 m
stable
4.77 m
15
Pb
stable
Tl
Bi
1.01 h
207
Pb
Po
3 10-7 s
212
Bi
2.14 m
208
212
Po
0.15 s
0.0018 s
207
Peucker-Ehrenbrink, 2012
Ra
5.75 y
Rn
5.01 d
210
228
Ra
1600 y
Rn
Ac
6.15 h
223
Ra
Th
1.91 y
228
Ac
21.8 y
226
228
Th
212
208
Pb
10.6 h
Pb
stable
208
Tl
3.05 m
226Ra
- 230Th disequilibrium diagram
Figure of radioactive disequilibria
between 226Ra-230Th and 238U-230Th in
lavas from austral and southern
volcanic zones removed due to
copyright restriction.
After Figure 1 in Sigmarsson, O., J.
Chmeleff, J. Morris, and L. LopezEscobar. "Origin of 226 Ra–230Th
disequilibria in arc lavas from southern
Chile and implications for magma
transfer time." Earth and Planetary
Science Letters 196, no. 3 (2002):
189-196.
16
230Th-234U-238U
activity ratio diagram
Courtesy of Mineralogical Society of America. Used with permission.
From: Chabaux, Riotte and Dequincey, U-Th-Ra Fractionation during weathering and river transport. In:
Bourdon et al. (Eds), Uranium-series geochemistry, Rev. Mineral. Geochem., vol. 52, chapter 13, 533-576.
17
238U-series
238
U
234
4.47 by
235U-series
235
U
U
0.704 by
245,000 y
234
231
Pa
Th
21.4 d
Pa
32,800 y
6.69 h
234
232Th-series
230
231
Th
75,000 y
227
Th
1.06 d
232
Th
14.1 by
18.7 d
227
11.4 d
222
219
3.823 d
222
214
Po
-5
3.04 m
210
Bi
19.7 m
Pb
26.9 m
Po
138.4 d
1.6 10 s
214
214
210
Po
Pb
22.6 y
224
Ra
3.63 d
220
Rn
3.96 s
55.6 s
215
216
Po
211
Bi
211
Pb
36.1 m
stable
4.77 m
18
Pb
stable
Tl
Bi
1.01 h
207
Pb
Po
3 10-7 s
212
Bi
2.14 m
208
212
Po
0.15 s
0.0018 s
207
Peucker-Ehrenbrink, 2012
Ra
5.75 y
Rn
5.01 d
210
228
Ra
1600 y
Rn
Ac
6.15 h
223
Ra
Th
1.91 y
228
Ac
21.8 y
226
228
Th
212
208
Pb
10.6 h
Pb
stable
208
Tl
3.05 m
Further Reading…
•Bourdon, B. et al. (Ed.) Uranium-Series Geochemistry, Rev. Mineral. &
Geochem., vol. 52, p. 20-21 (general solution to U-series equations).
•Bourdon, B. et al. (Ed.) Uranium-Series Geochemistry, Rev. Mineral. &
Geochem., vol. 52, p. 2-7 (U-series introduction).
•DecaySeries toy (simple Excel file).
19
Activity = (l N)
A = 1 Ci, 3.7 106 bq
1.E+18
N
1.E+16
1.E+14
1.E+12
1.E+10
1.E+08
1.E+06
1.E+04
1.E+02
1.E+00
1.E-02
1.E-04
1.E-06
l
1.E-08
1.E-10
1.E-12
1.E-04
1.E-02
1.E+00
1.E+02
1.E+04
1.E+06
Half-life (years)
20
1.E+08
1.E+10
Measurement Uncertainties
All measurements are afflicted with uncertainties. For large number of events, binomial distributions
asymptotically approach Gaussian (or normal) distributions. The spread in events (here numerical values of
isotope ratios, count rates or ion currents) is equal to N. According to Gaussian statistics about 2/3 of the
results lie within the range N ± N (one standard deviation), about 95% lie within the range N ± 2N (two
standard deviations), and ~99% lie within the range N ± 3N. The fractional uncertainty is thus N/N, or
1/N. If you measure twice as long (N*) you get twice as many events
N* = 2N
the fractional uncertainty is
(2N)/2N = 1/(2N)
= 1/2 * 1/N
i.e.
reducing the fractional uncertainty only by ~30%. The fractional uncertainty improves only as the square
root of time (or ion current, or count rate). If you attempt to improve the uncertainty by a factor of two, you
need to measure four times as long, or measure a four-times stronger ion current.
In order to evaluate if uncertainties associated with small ion beam intensities significantly affect the
measured ratios it is often helpful to assume that all uncertainties are associated with uncertainties in the
smallest ion current (least abundant isotope). By assuming an arbitrary uncertainty in the measurement of
this ion current you can plot an error trend on plots of isotope ratio versus another isotope ratio (same
isotope in the denominator, i.e. m2 = m4, if m1 and m3 are isotopes in the numerator). This trend is often
distinct from a instrumental fractionation trend and helps to assess what process dominates the uncertainty
of your analysis.
21
Measurement Uncertainty
1.E+03
1.E+02
Counting
1.E+01
f(lN)
1.E+00
1.E-01
1.E-02
1.E-03
1.E-04
Mass spectrometry
1.E-05
1.E-06
1.E-04
0.01% transmission
1.E-02
1.E+00
1.E+02
1.E+04
1.E+06
~20
Half-life (years)
22
1.E+08
1.E+10
f(N)
The Reporting of Data & Uncertainties
Text removed due to copyright restrictions.
See page 70 of Warren, Warren S. The physical
basis of chemistry. Academic Press, 2000.
23
MIT OpenCourseWare
http://ocw.mit.edu
12.744 Marine Isotope Chemistry
Fall 2012
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