ESS 480/580 Clumped Isotope Geochemistry – ASSIGNMENT II
The goal of this exercise is for you to continue working through some basic concepts,
definitions and calculations in preparation for next week when we will reduce raw mass
spectrometer measurements to interpretable clumped isotope (47) data. It may be
helpful to read Huntington et al., 2009, J Mass Spec before responding.
Carbonate Clumped Isotope Thermometry Concepts
(1) Explain the difference between  and the  used in clumped isotope geochemistry.
(2 points)
(2) Both conventional stable isotope and clumped isotope measurements are made vs. a
reference gas, or working gas standard. Concisely explain why this is, and why clumped
isotope measurements also must be normalized to measurements of heated gases.
(2 points)
Measurement Precision
Isotope ratios are determined from ion-current ratio measurements, and the precision of
these measurements is limited by Poisson or shot noise (Merrit and Hayes, 1994). Shot
noise in a mass spectrometer consists of random fluctuations of the electrical beam
current caused by the fact that the current is carried by discrete ions that hit a collector in
a series of independent, discrete arrival events. A property of Poisson processes is that the
number of observed occurrences—in this case ions counted (N)—fluctuates about its
mean with a standard deviation of s = N . Therefore at the shot-noise limit (i.e., the
theoretical absolute standard of precision that would be obtained of any noise in the
measurement resulting from analytical artefacts), the precision will scale as N ; that is,
precision improves by integrating more ions (increasing N). As a consequence, the limit
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of precision (i.e., the best precision that could possibly be achieved) in isotope-ratio
measurements is set by the number of ions counted, which depends on the product of the
integration time (the time interval over which ions are counted) and the beam signal
intensity (the ion beam current or measured voltage drop across the 1012 ohm resistor).
We express the precision of an isotope-ratio measurement in per mil units on a ‘’ scale
(i.e., following common convention). The  scale is defined by the relation:
[
]
d º ( Rsample /Rsta ndard ) -1 ×1000
(Equation 7)
Following the work of Merritt and Hayes (1994), we can come up with the following
simplified expression for variance ( s 2) of  at the shot-noise limit of performance is
derived, assuming that Rsample and Rstandard are essentially equal
sd 2 = 2 ´106 (s R /R)2 = 2 ´106 (1+ R)2 /44 NR
(Equation 8)
For clumped-isotope thermometry the precision-controlling ratio in temperature
calculations, R, is the ratio of mass-47 ion current, 47i, to the mass-44 ion current, 44i.
All else being equal (e.g., stable magnet current and other instrument settings), increasing
the number of ions counted by either lengthening the time over which ion currents are
integrated or increasing the beam intensity improves precision. The optimal standard of
performance can be calculated using equation 7 for the characteristic ion current ratio for
analysis of sample CO2 gas (in this case R47), and 44N, the number of mass-44 ions
collected. The abundance of mass-44 ions collected is equal to 44it/qe, where 44i is the
mass-44 ion current, t is the integration time, and qe is the electronic charge. The current
44
i is given by 44V/Rf, where 44V is the mass-44 signal and Rf is the feedback resistance of
the mass-44 electrometer. The value of the resistance on the mass-44 electrometer, Rf is
3x108 Ω, and the abundances of the mass-44 and mass-47 isotopologues in CO2 with the
stochastic distribution are 98.40% (16O12C16O) and 46.5 ppm (given by 45 ppm
18 13 16
O C O+ 1.5 ppm 17O12C18O + 1.5 ppb 17O13C17O), respectively, for CO2 assuming
bulk 13C/12C ratios equal to PDB, and bulk 18O/17O/16O ratios equal to SMOW. Using
these abundances, we compute R = 47i/44i = 4.7x10-5 (equation 7).
(3) Using Excel or MATLAB, make two plots:
(a) Assume integration time t =320 s, and plot the standard deviation of 47 (as a
function of mass 44 voltage (44V = 4 to 32 V) using these R and Rf values (Equation 8).
You can check your work by comparing the result to Huntington et al. (2009) online
supplementary information figure S2 (part e). Label and attach plot. (5 points)
(b) Assume 44V=16V, and plot the standard deviation of 47 (as a function of
integration time t in seconds. Label and attach plot. (5 points)
(c) How many seconds would you have to integrate to get precision of 0.03‰ or better?
Note: useful measurements for paleothermometry require on the order of 0.01‰
precision! (1 point)
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(4) What did you think of this assignment? How long did it take you to complete it? If you
don’t want to answer here, please put anonymous response under Kate’s office door
(JHN 353) or in her mailbox in JHN 070.
For your reference only: The shot noise limit calculations presented here are after
those of Merritt and Hayes (1994). For clumped-isotope thermometry, the
precision-controlling ratio in temperature calculations, R, is the ratio of mass-47 ion
current, 47i, to mass-44 ion current, 44i. The current 44i is given by 44V/Rf, where 44V is
the mass-44 signal and Rf is the feedback resistance of the mass-44 electrometer.
The observed ion current, i, is a function of the number of ions, N, collected over
integration time, t, and the electronic charge, qe (1.6x10-19 C/ion)
i=
Nqc
t
(Equation B1)
Since i and N are proportional for a fixed value of t,
R=47i/44 i=47N/44 N
(B2)
and from the standard treatment of sampling statistics we can therefore write the variance
of R as a function of either i or N. As a function of N, the variance is
s R2 = (¶R/¶ 47 N) s 47 2 + (¶R/¶ 44 N) s 44 2
2
2
(B3)
where 47 and44 are the standard deviations of 47N and 44N, respectively. Evaluating the
derivatives in (A3), we have
[
s R2 = R2 (s 47 /47 N) + (s 44 /44 N)
2
2
]
(B4)
At the shot-noise limit of precision, the variance of a measured variable is equal to that
variable, i.e., the variance of N is N. From this it follows that (N/N)2 = 1/N, and (B4) can
be recast as
(s R /R)2 = (1/47 N) + (1/44 N)
(B5)
Recalling that R=47N/44N from (B2), we can write (B5) as a function of only the majorbeam ion current 44N and R
(s R /R)2 = (1/44 N)[(1+ R) /R], or
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(s R /R)2 = (1+ R)2 /44 NR
(B6)
The quantity of interest in isotopic measurements is  , which is defined as
[
]
d º10 3 ( Rsample /Rsta ndard ) -1
(B7)
and the variance of  from the standard treatment of sampling statistics is
sd 2 = (sd /sRsample)2s Rsample2 + (sd /sRsta ndard )2 s Rsta ndard 2
(B8)
where  is the characteristic standard deviation of the observed  value. If Rsample and
Rstandard are essentially equal and the derivatives in (B8) are evaluated, we can write a
simplified expression for shot-noise limited performance using (B8) and (B6):
sd 2 = 2 ´106 (s R /R)2 = 2 ´106 (1+ R)2 / EmNA R
(B9)
The optimal standard of performance can be calculated using Equation B9 for R, the
characteristic ion current ratio for analysis of sample CO2 gas (in this case R47), and
EmNA, the number of mass-44 ions collected (denoted as 44N). The abundance of mass-44
ions collected is equal to 44it/qe, where 44i is the mass-44 ion current, t is the integration
time, and qe is the electronic charge. The current 44i is given by 44V/Rf, where 44V is the
mass-44 signal and Rf is the feedback resistance of the mass-44 electrometer.
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