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Discussion of shot noise limit with decreasing beam intensity:
Shot noise was nicely defined by Merritt & Haye [1] and the shot noise limit (),
or the smallest standard error that can be reached based on counting statistics, can be
calculated with the following equation:
2 = 2 x 106 * (R/R)2
(S1)
This can be rewritten as: 2 = 2 x 106 * (1+R)/R * (44*qe)/(t*V44)
(S2)
Figure S15: Mass of reacted
carbonate vs calculated (line) and
observed (points) shot noise limit.
0.005
shot noise limit (permil)
0.010
0.015
0.020
where R is the ratio of the number of ions collected per time unit (current) in the mass of
interest (47) relative to the most common mass (44) (=i47/i44 = 4.8x10-5), 44 is the
resistor on m/z 44 (=3x107 ohms), and qe is the charge on each ion (=1.6x10-19 C).
In the traditional dual-bellows measurement configuration, V44 is the target
voltage for the m/z 44 beam at which every cycle is measured. The quantity V44*t/qe,
where t is the total number of seconds of integration time over all cycles and acquisitions,
gives an estimate of the total number of ions collected over the whole measurement
period.
In our dual-reservoir measurement configuration, the beam intensity (and
therefore V44) is constantly changing throughout the run. In order to quantify the total
number of ions collected over the measurement period, we “integrate” the beam strength
over time. We take V44 for each cycle, multiply by 26 seconds, the integration time of a
single cycle, and then sum all the cycles. This is computationally equivalent to taking the
average V44 value over the whole run and inputting that in place of the target voltage V44
in the equation above.
We observe a linear relationship between the average V44 and the initial V47,
which we can relate to sample size by the following equations:
V44-average = 0.4197* V47-initial + 50.2
V47-initial = 2231*(Mass of CaCO3) -1714
These equations are defined based on the sensitivity levels observed during this study
(Smaller U-trap, tuning 2), and would change with changes in sensitivity. We can use
Sample Size vs. Shot noise limit (for 7acquisitions x 14 cycles)
these equations to plot the shot
noise limit against sample size,
assuming that all samples were run
Calculated from equations
for the same amount of time (7
Measured 1sigma SE on D47
acquisitions x 14 cycles x 26
second integration time). We see
our measured standard error
increasing in line with the
predicted shot noise limit at small
sample sizes.
1.0
17
1.5
2.0
mass of reacted carbonate (mg)
2.5