This journal is © The Royal Society of Chemistry 2003
Electronic supplementary information
In this ESI, a model to calculate propagated errors for isotopic compositions of
unknown samples (ui), a method of data reduction for calculating internal precision of
each run, optimization method of the
205
Pb-204Pb spike composition, and calibration
procedures of the compositions of the 207Pb-204Pb and 205Pb-204Pb spikes are described.
Error model
The propagated errors for isotopic ratios of samples are calculated using the model
modified from that of Hamelin et al. (ref. 1 in the text). Equations (5) and (6) in the text
can be combined to express as follows
AB  D,
(1)
where
 M 1  s1
M  s
2
 2
 M 3  s3

A 0
 0

 0
 0

B   p q rm
D   M 1
M 1  1
M 2  2
M 3  3
0
0
0
0
0
0
0
N 0  t0
N1  t1
N 2  t2
N 3  t3
rn
 M2
u1 u 2
 M3
In equation (3), the variables rm and rn are
1
0
0
0
N0  0
N1   1
N2  2
N3  3
1 0 0 
0  1 0 
0 0  1

0 0 0  (2)
1 0 0 

0 1 0 
0 0  1
u3  T
 N0
(3)
 N1
 N2
 N3  T .
(4)
rm  ( 1  p )   m
rn  ( 1  q )   n .
The solution of equation (1) is given by
B  A 1  D ,
(5)
from which the corrected isotope ratios of the sample (ui) can be directly obtained.
The errors for the corrected isotopic ratios of the sample are obtained through
the variance-covariance matrix of B:
VB  dB  dB T .
(6)
By following the method of Hamelin et al. (ref. 1 in the text), the matrix VB is obtained
as
VB  ( A 1 )  [ p 2  VS '  q 2  VT '  K  VM '  K  L  VN '  L ]  ( A 1 ) T , (7)
where K and L are the diagonal matrices with elements 1 + p + i · rm and 1 + q + i ·
rn, respectively,
0
0
1  p  1  rm

0
1  p   2  rm
0


0
0
1  p   3  rm

K
0
0
0

0
0
0

0
0
0


0
0
0

2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0


0

0
0

0
0
(8)
0
0

0

L  0
0

0
0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1  q   0  rn
0
0
0
0
0
1  q  1  rn
0
0
0
0
0
1  q   2  rn
0
0
0
0
0


0


0

0
.

0

0

1  q   3  rn 
0
(9)
In equation (7), VS’, VT’, VM’, and VN’ are the variance-covariance matrices for the
isotopic ratios of the
207
Pb-204Pb spike,
205
Pb-204Pb spike,
207
Pb-204Pb spike–sample
mixture, and 205Pb-204Pb spike–sample mixture, respectively. For example,

 M2 1
cov( M 1 , M 2 ) cov( M 1 , M 3 )

 M2 2
cov( M 2 , M 3 )
cov( M 1 , M 2 )
cov( M 1 , M 3 ) cov( M 2 , M 3 )
 M2 3

VM '  
0
0
0

0
0
0

0
0
0


0
0
0

0
0

0

VN '  0
0

0
0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0


(10)




0
0
0
0


0
0
0
0

2
 N0
cov( N 0 , N1 ) cov( N 0 , N 2 ) cov( N 0 , N 3 ) .
cov( N 0 , N1 )
 N2 1
cov( N1 , N 2 ) cov( N1 , N 3 ) 

cov( N 0 , N 2 ) cov( N1 , N 2 )
 N2 2
cov( N 2 , N 3 )

cov( N 0 , N 3 ) cov( N1 , N 3 ) cov( N 2 , N 3 )
 N2 3

0
0
0
0
(11)
In the matrix VB, the elements VB
(k,l)
(5 ≤ k ≤7; 5 ≤ l ≤7) give the variance and
3
covariance of the corrected isotope ratios of the sample (ui): VB (5,5) =  u2 1 and VB (5,6) =
VB (6,5) = cov(u1, u2), for example.
Data reduction for calculating internal precision of each run
In order to calculate the propagated error for isotopic ratios of unknown samples, the
analytical errors (internal precisions) for the two spiked runs (matrices VM and VN), in
addition to the uncertainty of the spike compositions (matrices VS and VT), are
necessary. A scatter of the raw scan data cannot be used to obtain the internal precisions
of individual runs, because it also includes the variations resulting from the mass
fractionation during mass spectrometry (ref. 19 in the text). In this study, internal
precisions for both runs are obtained through the following “average-normalization
method”.
All raw scan data are normalized using their average (corresponding to the
uncorrected raw measured data), in the same way as the 86Sr/88Sr ratio is used for the Sr
isotope measurements. For each of the scan data, the fractionation factor is calculated
using its
206
208
Pb/204Pb ratio and the ratio of the average data. Then,
Pb/204Pb, and
205
207
Pb/204Pb,
Pb/204Pb ratios of the scan data are normalized using the calculated
fractionation factor. The
208
factor calculated from the
Pb/204Pb ratio itself is normalized using the fractionation
207
Pb/204Pb ratio of each scan datum and that of the average
data. The internal precision of the normalized scan data is independent of mass
fractionation effects, and thus it gives a realistic estimate of the internal precision of
both runs.
Optimization of parameters
Optimization of the 205 spike composition was performed using the error model
described above. In order to optimize the spike composition, as well as other parameters,
such as p, q, and R (the ratio of the amount of Pb sample in the 205 spiked run to that in
the 207 spiked run), internal precision for the measured ratios of the spiked runs
(matrices VM and VN) is modeled as follows. Assuming that the individual ion currents
4
of lead isotopes are not correlated with each other, the uncertainty of the isotope ratio
20X
Pb/204Pb (X = 5, 6, 7, 8) can be expressed as
 20X Pb 
 ( 20X Pb 204Pb)   204 
 Pb 
  20X Pb    204Pb 
 20X    204  ,
Ι
  I

where I20X is the ion beam intensity of

204
Pb
2
2
(12)
Pb (background-corrected),  2 0X Pb and
20X
are the uncertainties of the measurements of I20X and I204, respectively.19 The
uncertainty  ( 20X Pb 204Pb) is used as an analytical error of each isotopic ratio,
corresponding to the internal precision through the “averaged-normalization method” in
actual spectrometry analyses. For simplicity, covariance among the different isotope
ratios is ignored. The uncertainty  2 0Y Pb (Y = 4, 5, 6, 7, 8) is calculated according to
the model of ref. 18 and 19 in the text.
To calculate the uncertainty  ( 20X Pb 204Pb) , ion beam intensities of individual
isotopes must be given. In the following optimizations, the total ion intensity (Itot) is
expressed as a function of the amount of Pb sample, by assuming that the total ion
intensity solely depends on the amount of Pb, irrespective of isotopic composition. For
analyses of small Pb samples processed with column chromatography in our laboratory,
ion beam intensities for
208
Pb are about 300, 1200, and 2500 mV for 1, 5, and 10 ng of
Pb, respectively. Using the isotopic abundance of
208
Pb in NBS981, the total ion
intensity is approximated as Itot = 480 × x, in which x indicates the amount of Pb (ng).
Because the 205Pb/204Pb ratio is important in the compositions of the 205 spike,
this ratio was treated as a variable. In the optimization, the parameter p and R are fixed
to p = 200 and R = 5 (see text). This treatment does not significantly affect the result of
the optimization of the spike composition. The amount of Pb sample, x, is 1 ng, and the
isotopic compositions of NBS981 is used. Fig. S1 shows the variation of the relative
error for the corrected
205
206
Pb/204Pb ratio of the sample with the parameter q. As the
Pb/204Pb ratio of the 205 spike increases, the propagated error tends to be reduced.
However, the error is not significantly improved for
5
205
Pb/204Pb ratios higher than ~5.
From this result, we selected a 205Pb/204Pb ratio of ~5 for the 205 spike.
Relative error
0.010
0.008
205
204
Pb/
0.006
Pb
0.5
1.0
2.0
5.0
10.0
0.004
0
2
4
6
8
10
q
Fig. S1 Comparison of the propagated errors for the corrected isotopic ratio
(206Pb/204Pb) of the unknown sample, for various compositions of the 205Pb-204Pb double
spike as a function of q. The composition of the double spike with a 205Pb/204Pb ratio of
> ~5 is desirable.
6
Calibration method of the spike compositions
The compositions of each double spike were calibrated by the “normal double spike
method”: the true composition of NBS982 (given in the text) was the known “spike
composition”, the measurement of a pure double spike represented an “unspiked run”,
and the measurement of a spike–NBS982 mixture constituted a “spiked run”, according
to ref. 1 in the text. In the case of the calibration of the 207 spike composition, the
mixing relationships between the 207 spike (si) and NBS982 (wi) is written as
mi ( 1 + p ) = si + wi · p,
(13)
where mi indicates the true isotopic composition of the 207 spike–NBS982 mixture. The
linear fractionation law for the “unspiked run” and “spiked run” can be expressed as
si = Si ( 1 + i · s )
(14)
mi = Mi ( 1 + i · m )
(15)
and
respectively, where Si and s denote the measured composition of the pure 207 spike and
the mass fractionation factor for the “unspiked run”, and Mi and m are the measured
composition of the 207 spike–NBS982 mixture and the mass fractionation factor for the
“spiked run”. Combining (14) and (15) gives:
( Mi – wi ) · p + Mi · i · ( 1 + p ) · m – Si · i ·s = Si – Mi
(i: 1,2,3) (16)
Three unknown variables, p, m, and s, can be obtained from the above three
independent equations. The true isotopic composition of the 207 spike is then obtained
7
from equation (14), by substituting s and the measured composition Si.
The calibration procedures of the 205Pb-204Pb double spike is essentially similar
to those described above. Note, however, that we have four independent equations,
although there are only three unknown variables, q, n, and t:
( Ni – wi ) · q + Ni · i · ( 1 + q ) · n – Ti · i ·t = Ti – Ni
(i: 0,1,2,3) (17)
in which Ti is the measured composition of the pure 205 spike, Ni is the measured
composition of the 205 spike–NBS982 mixture, and t and n are the mass fractionation
factors for the “unspiked run” and “spiked run”, respectively. Therefore, these unknown
variables can be obtained from any three equations from the four equations. The
calibrated composition of the 205 spike is mostly independent of the combinations of
three equations from the four equations.
8