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ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979
Theoretical Signal-to-Noise Ratio and Mass Resolution in Fourier Transform Ion Cyclotron Resonance Mass Spectrometry Alan G. Marshall Deparfment of Chemistry, University of British Columbia, Vancouver, British Columbia V6T 7 W5 Canada
Theoretical expressions are derived for relative single-to-noise ratio in Fourier transform ion cyclotron resonance (FT-ICR) mass spectroscopy, for arbitrary sample pressure and arbitrary fraction of the total duty cycle spent in observing the timedomain FT-ICR signal. Plots of signal-to-noise ratio and resolution as functions of acquisltion period at fixed sample pressure show that for optimal duty cycle, signal-to-noise ratio is maxlmized at a short acquisition period, while resolution is maximized at a long acquisition period; a suggested experimentally optimal acquisition period is 2-3 times as long as the FT-ICR time-domain signal damping constant. The present calculations and graphs confirm that the high-resolution capability inherent to fixed magnetic field FT-ICR operation can be realized while retaining most of the signal-to-noise enhancement devolving from the Fellgett multichannel advantage of Fourier data reduction. FT-ICR provides particularly convenient tradeoff of signal-to-noise ratio vs. resolution, simply by varying the acquisition period at a given sample pressure. Flnally, the common features of signal detection in ICR and in nuclear magnetic resonance are tabulated and discussed.
The theory and practice of Fourier transform ion cyclotron resonance (FT-ICR) spectroscopy have been presented in several previous papers. Beginning from the first FT-ICR spectrum obtained by discrete Fourier transformation of the digitized time-domain transient excited by a short radiofrequency pulse ( I ) , broad-band excitation ( 2 ) and phaseselected frequency response ( 3 ) were quickly demonstrated. Experimental examples of high-resolution spectra ( 4 , 5 ) were then produced. Detailed theoretical analyses of the FT-ICR line shapes (absorption, dispersion, and magnitude- or absolute-value-mode spectra) were provided, first in the limit of very low sample pressure (6, 7), and more recently for arbitrary pressure (8). The Fellgett multichannel advantage of FT-ICR over conventional ICR has been discussed (9). I t has been shown (8) that for any fixed sample operating pressure, FT-ICR mass (or frequency) resolution is maximized by using a longer data acquisition period. However, since the time-domain FT-ICR signal decays during the acquisition period, the time-domain time-averaged signal-to-noise ratio will be optimized by using a shorter acquisition period. I t is the purpose of this paper to establish quantitatively the dependence of FT-ICR signal-to-noise ratio upon the data acquisition period, FT-ICR time-domain damping constant, number of accumulated transients, and fraction of the total duty cycle (ion formation, time-delay for chemical reaction, ICR excitation, data acquisition, ion removal) spent in actual data acquisition, for a given total available period for the experiment. These results may then be combined with prior analysis of FT-ICR resolution (8) to give theoretical expressions and graphs for optimization of the product of signal-to-noise ratio and resolution a t arbitrary sample pressure. 0003-2700/79/035 1-1710$01 .OO/O
Finally, since the above analysis cannot be obtained simply by correspondence to the superficially similar problem in nuclear magnetic resonance (NMR) spectroscopy, a comparison between FT-ICR and FT-NMR is provided and the results are tabulated for these two types of radiofrequency spectroscopy.
THEORY FT-ICR offers two major improvements over conventional ICR spectroscopy. First, experimental FT-ICR mass resolution is approximately 100-fold better than for magnetic field-swept conventional ICR detection. One possible explanation for this improvement is that the FT-ICR experiment is conducted a t constant magnetic field in which the excited ions orbit a t a fixed radius during detection, thus effectively “spinning the sample” (7,101 to reduce magnetic field static spatial inhomogeneity. For ex‘ample, if the magnetic field inhomogeneity consisted of a radial gradient from the center of the ICR orbital motion, then excited ions in an FT-ICR experiment would orbit about a constant magnetic field, while excited ions in a conventional ICR experiment would encounter a range of magnetic field values during the detection period. Second, Fourier data reduction effectively provides the “multichannel” (“Fellgett”) advantage (9) of detecting the entire spectrum in the time that would normally be required to detect just a single peak using conventional field- or frequency-scanning ( I I ) operation. This multichannel timesaving advantage for a wide mass range FT-ICR spectrum translates into a signal-to-noise ratio improvement of a factor of about 100 compared to conventional ICR (9). In experimental FT-ICR, it is usual to acquire many separate digitized time-domain transients, sum their amplitudes point-by-point, and then perform a discrete Fourier transformation to obtain a frequency-domain spectrum. I t has been shown that in such data reduction, the time-domain time-averaged signal-to-noise ratio (integrated time-domain amplitude divided by integrated time-domain noise) is directly proportional to the frequency-domain signal-to-noise ratio. It therefore suffices to consider the time-averaged time-domain signal and noise only. Three representative FT-ICR time-domain signals as functions of time are plotted in Figure 1, based on the generalized time-domain FT-ICR signal, 0 it iT f ( t ) = K exp(-t/T) cos ( a t ) f(t)=O
tT
(1)
in which K is independent of time and is proportional to the number of ions of a given mass-to-charge ratio, T is the duration of the data acquisition period, o = qR/m ( q is ionic charge in Coulombs, B is applied magnetic field strength in tesla, and m is ionic mass in kg) is the ICR frequency in the absence of (minor) correction for the effect of the trapping electric field (12),and T is the time constant for (exponential) decrease of the time-domain excited FT-ICR signal, f ( t ) . The form and frequency-dependence of K are discussed in Ref.
D 1979 American Chemical Society
~ N A L Y T I C A LCHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979
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On summing many individual time-domain transients, the accumulated signal amplitude increases proportional to the number of transients, n, while the (random, detector-limited) noise that accompanies the time-domain FT-ICR signal accumulates as the square root of n, as is readily justified by treating the noise as a random walk about an average noise level of zero (9). Thus, at any sample pressure, @ / N ) = (S/N),
Iz = ( S / N 0 f i
6
(6)
Furthermore, the longer the time between successive acquisitions (Tduty),the smaller is the maximum number of accumulated transients (n): (number of transients)(duration per transient) = (total experimental observation period) nTduty = Ttotal (7) 2
3
4
1
6
8
1
IO
Figure 1. FT-ICR time-domain theoretical normalized signals. (a) zero-pressure limit ( T > 7 ) . T = data acquisition period and T is the lifetime for (exponential)decay of the FT-ICR timedomain signal. The bottom plot is intended to describe a signal that is observed for many decay lifetimes
13. Finally, the limits, T > T , known as the "zero-pressure" and "high-pressure" limits, are both encountered often experimentally. Since the following analysis depends on a time-aueraged signal-to-noise ratio, we need examine only the (time-dependent) normalized amplitude, exp(-t/T), of f ( t ) (14). Consider then the following definitions: (S/N), = Timeaveraged signal-to-noise ratio for one time-domain transient; (SIN) = Signal-to-noise ratio for a s u m of many time-averaged transients; n = number of transients to be summed; T = acquisition period for a single time-domain transient; To = time period in duty cycle for all steps except data acquisition; and Tduty= T To = total time between successive acquisitions. We begin by computing the average signal-to-noise ratio for a single transient. Suppose that K = 1 in Equation 1 for simplicity. Then for a single transient, the instantaneous time-domain amplitude is exp(-t/.r), and the total time-integrttted time-domain signal accumulated during one acquisition period, T, is given by
+
So= JTexp(-t/T)dt
= r ( l - exp(-T/r))
a
fi
( S I N ) = (S/N), Finally, we may combine the result for a sum of many transients (Equation 8) with the expression for signal-to-noise ratio for a single transient (Equation 4) to give an expression for the signal-to-noise ratio of the sum of many time-averaged transients, valid at arbitrary pressure:
a t arbitrary pressure (9) The limiting behavior of Equation 9 at very low or very high pressure is readily computed: lim (S/N)
(zero-pressure limit) (9a)
0:
TI
(2)
The time-integrated noise in a single transient accumulates according to (15),
No
Solving Equation 7 for n, and substituting for n in Equation 6, we obtain a fundamental expression for signal-to-noise ratio of a sum of time-domain transients, valid at any pressure:
(3)
Equations 2 and 3 may be combined to give the time-averaged time-domain signal-to-noise ratio for a single transient,
For the relatively long data acquisition periods that are experimentally desirable to achieve high resolution (see Discussion),most of the duty cycle is devoted to ICR detection, so it is of interest to examine Equations 9a and b in the further limit that (T/Tduty) 1:
-
lim (S/N) (T/Tduty) 1
0 :
+
6(;)
(1 - exp(-T/T))
(arbitrary pressure) (10)
Equation 4 exhibits two physically reasonable limiting forms (see Discussion): lim
T95 'YO). The application of the two-stage Au amalgamation method for the measurement of Hg in water and other natural materials is described.
Geochemical mass balance estimates for the global cycling of Hg suggest that the major transfer of Hg between the continents and the ocean is through the atmosphere (1-3). Despite the apparent importance of airborne transport of natural and anthropogenic Hg to the marine environment, the concentrations and chemical species of Hg in the atmosphere and the fluxes associated with rainfall, air/sea gas exchange and dry deposition are not well known (3-5). Thus carefully conducted atmospheric studies of Hg can contribute significantly to our understanding of the global cycle of Hg and to refining our assessment of the impact of modern Hg emissions on the pre-man cycle of Hg. Here we present a description of a two-stage Au amalgamation analytical system for the accurate determination of Hg a t the subnanogram level. Since sampling inaccuracies are frequently a major source of error in environmental studies, a working outline of our field collection apparatus and sample preservation procedures for atmospheric studies of Hg is given. The major volatile Hg species found in the near ground atmosphere are collected by amalgamation on Au coated glass beads (6). Atmospheric particulate Hg species are collected on glass fiber filters. Representative data are also presented for Hg concentrations in the coastal marine atmosphere and 0003-2700/79/035 1-1714$01.OO/O
the geochemical consequences briefly discussed. Mercury analyses are conducted, using a two-stage Au amalgamation gas train with detection of the eluting Hgo by flameless atomic absorption. The Hg collected on a gilded glass bead tube in the field is transferred by controlling heating to a standardized analytical Au-coated glass bead column, using Hg free air as the carrier gas. Following this step the Hg is eluted from the analytical column by controlled heating, and the absorption of elemental Hgo determined by gas phase detection at 253.7 nm. A standard curve is prepared for the analytical column using known injections of Hg saturated air. The coefficient of variation for the determination of 0.5 ng Hg is 4% and about 0.06 ng of Hg can be confidently measured. The two-stage Au amalgamation technique has been used successfully in the laboratory (7, 8), to separate interfering substances from the mercury vapor before it is released to the detector. Potential interfering substances that may absorb at the mercury wavelength (253.7 nm) can be separated from the mercury species collected on the Au field column during air sampling and upon transfer of elemental Hgo vapor to the analytical column. The double amalgamation procedure provides an additional step toward ensuring the specific gas phase detection of Hg. Therefore, Hg can be measured accurately in air and in other materials, with good precision and a low detection limit, using a relatively inexpensive single beam atomic absorption instrument. Additional advantages of the two-stage Au amalgamation technique are operational convenience and rapid analysis of large numbers of samples. In single stage systems, for example, where many different Au columns are used both for the concentration step in the field and as the means of introducing the sample to the detector, identical mercury concentrations may yield slightly different responses. This variability results from the different characteristics each column may have with respect to the release of mercury to the detector upon heating. To acquire the same precision as a two-stage system, each column used would have to be calibrated individually. The present system avoids this additional step by using the second carefully calibrated Au column (analytical column) to introduce all collected samples to the detector. The field column used to collect and concentrate the mercury samples serves as a transfer stage, 0 1979 American Chemical Society
