Frequency (mass) errors and phase dependence in magnitude-mode apodized Fourier transform—ion cyclotron resonance spectra

~n~~~~tj~n~ Journal of Mass Spectrometry and Zon Processes, 89 (1989) 187-203 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

187

FREQUENCY (MASS) ERRORS AND PHASE DEPENDENCE: IN OGRE-LODE APODIZED FOURIER TRANSFORM-ION CYCLOTRON FONDUE SPECTRA

KIM H. CHOW and MELVIN B. COMISAROW Department of Chemistry, University of British Columbia, Vancouver, British Columbia V6T 1 Y6 (Canada) (Received 11 August 1988)

ABSTRACT Closely spaced Fourier transform-ion cyclotron resonance (FT-ICR) magnitude peaks give a composite lineshape in which the apparent peak locations are shifted from the true peak positions. The shift in peak location is a function of the actual peak separation, the windowing function used for apodizing the lineshape and the damping ratio, T/T, the ratio of the acquisition time to the relaxation time of the Fourier transform-ion cyclotron resonance time domain signal. For example, two equal intensity Blackman-Harris-apodized peaks where T/T =l.O, which should be separated by 1.0 unit of the non-zero-filled channel spacing, i.e., l/T, are actually separated by 2.32 units. Two similar peaks which are actually separated by 2.0 units appear as one merged peak. In addition, the apparent peak locations of nearby peaks are a function of the initial phases of the time signals, which when transformed give the spectrum. The amplitude of the anomalous frequency shift is greater for windows of greater dynamic range. The occurrence of the shift phenomenon can be expected whenever two magnitude peaks are closer than 3/T and can be confirmed by variation of the window function. Phase-corrected absorption spectra provide a means of obviating the problem.

INTRODUCTION

Fourier transform-ion cyclotron resonance (FT-ICR) [l-9] mass spectrometry differs from other forms of mass spectrometry in that the signal from the spectrometer is not the mass spectrum but rather a time signal which is a composite signal from all of the excited ion cyclotron resonance (ICR) motion of the sample. Numerical Fourier transformation of a stored version of this signal gives the ICR frequency spectrum, which is equivalent to the mass spectrum of the sample. The most fundamental advantage of the FT-ICR instrument is its speed, which enables the instrument to give the whole ICR spectrum at once. Processing of the ICR signal/spectrum is intrinsic to the technique and gives the operator great control over the 0168-1174/89/$03.50

0 1989 Elsevier Science Publishers B.V.

188

appearance of individual peaks in the spectrum. This control is effected by instrumental parameters in operation at the time of signal acquisition but also as part of the “off-line” data manipulations needed to produce the final spectrum. In this paper we discuss certain aspects of these data manipulations and note that closely spaced FT-ICR rna~tud~rn~e peaks are typically shifted from their true positions. The origin of this shift, which is closely related to a recently recognized distortion of FT-ICR magnitude intensities [lo] is discussed. Methods for predicting and confirming the occurrence of the “position-shifting” effect are given in addition to methods for obviating the effect. THEORY

In FT-ICR spectrometers the time domain signal consists of a sum of signals, each of the form

f(r) = [cos(w,t+ S)] exp( - f

)

OK&CT

where w, is the cyclotron frequency, 8 is the phase of the time signal, 7 is the relaxation time of the time signal and t is time. This time signal is sampled N times at a rate S for a time period T, the a~uisition time, given by

The amount of damping in the signal is characterized by the damping ratio T/r. The time signal may be augmented by having n sets of zeros added to the end of the sampled time signal, in which each zero-filling doubles the length of the sampled time signal [11,12]. Discrete, complex Fourier transformation of this zero-filled discrete time signal gives a complex frequency spectrum, consisting of a discrete real and a discrete imaginary spectrum. The real spectrum is the discrete counterpart to the continuous absorption spectrum from a scanning ICR spectrometer. The imaginary spectrum is the counterpart to the continuous dispersion spectrum. Most often, the FT-ICR spectrum is presented as the magnitude spectrum, which is the modulus of the complex spectrum (the point-by-point square root of the sum of the squares of the real and imaginary spectra). Each of the above spectra is defined at only M (which is equal to 2”-IN) discrete frequencies given by

fm= T

m=o, 1,2, . . . 2”-lN-l

where m is the index of the discrete spectrum.

(3)

189

Typically, the stored version of the time signal, Eq. 1, is multiplied by a special function called a window, prior to any zero-filling and Fourier transformation. The process is called windowing in the time domain or apodization in the frequency domain. Apodization reduces the intensity of auxiliary maxima and narrows the peak width near the peak base. The purpose of windowing or apodization is to minimize the interference of a given peak with nearby peaks. Recently, we have examined the properties of a large number of windows and have suggested that the dynamic range of the FT-ICR spectrum is the spectral parameter to be used for the selection of the specific window which is appropriate for a particular FT-ICR spectrum [13,14]. Our recommended windows for the magnitude mode were the rectangle window R(t) (corresponding to no windowing) R(t) = 1.0,

Oct
(4)

the Hamming window H(t) H(t) = 0.54 - 0.46 cos

(5)

O
the three-term Blackman-Harris

window BH( t)

BH( t) = 0.42423 - 0.49755 cos

O
(6) and the Kaiser-Bessel window, KB( t) KB(t)=I,,

( 35. r{ lo.

[

(2t;T)]2jOJ/1,(3.5s)

O
(7)

where I,, is the zero-order modified Bessel function. The latter three windows were recommended [13,14] for use with spectra having dynamic ranges of 100 : 1,lOOO : 1 and 10 000 : 1 respectively. Since FT-ICR spectra often have a dynamic range of a few hundred, the Blackman-Harris window was suggested as the “standard” FT-ICR window [14]. As stated above, windowing or apodization is done to minimize the mutual interference between separate peaks. It should be noted that properly chosen window functions accomplish this minimization both for undamped transients, which have many auxihary maxima in their Iineshapes, and for heavily damped transients, whose Iineshapes have a single maximum but a very wide skirt. The undamped case is well documented in the engineering literature. The heavily damped case is illustrated in the Figures in refs. 13 and 14.

190 RESULTS

Frequency dependence of the frequency shift Figure 1 shows the FT magnitude-mode spectrum of a two-frequency time signal of the form

f(t) = [cos(o,t) + cos(w,t)] exp + (

O
1

where for simplicity,the two frequenciesare set to have the same damping ratio, T/7 = 1.0. The signal was windowed with the Blackman-Harris window(Eq. 6) prior to Fourier ~~sfo~ation. The frequency scale in Fig. 1 is in dimensionlessunits of l/T. The true separation Af between the two peaks is given by

The scale is defined in this way so that Fig. 1 applies for any value of T. It should be noted that Af is 2.0, whereas the two peak maxima have merged

BLACKMAN-HARRIS T/T =‘l,

3 TERM

hf = 2.0

100 go80-

-5

-4

-3

-2

-1

0

1

2

3

4

s

FREQUENCY

Fig. 1. FT ma~tude-rn~e spectra of two overlapping peaks. The frequency scale is in dimensionless units of l/T. The spectra were calculated by magnitude-mode Fourier transformation of a Elac~~-~~-~dow~ (Eq. 6) time domain signal of the form shown in Eq. 8. The heavy black Iines indicate the positions of the individual frequencies, u1/2n and ws/27r (Fq. 8). The frequency separation, Af = 2.0 is given by IQ. 9. S is the shift of the apparent peak position from the true peak position. It should be noted that the peaks have merged and their apparent positions have moved towards each other by 1.0 frequency unit. This type of shift, S, is defined as being a negative shift in Figs. 3-6.

191 BLACKMM-HARRIS 3 To?” l/r= l,Af = l.0 100

-7

90-l

so708

M)-

1

504030 1 20lo

0,. -5

, -4

I,

*, -3

I

-2

-1

0

1

2

I 3

181 4

5

FREQUENCY

Fig. 2. F+Tmagnitude-mode spectra of two overlapping peaks. The frequency scale is in dimensionlessunits of l/T. The spectra were calculated by magnitude-mode Fourier transformation of a Blackman-Harris-windowed(Eq. 6) time domain signal of the form shown in Eq. 8. The heavy black hnes indicate the positions of the individualfrequencies o,/2n and ws/2rr (Eq. 8). ‘Ihe frequency separation, Af=l.O is given by Eq. 9. S is the shift of the apparent peak position from the true peak position. It should be noted that the apparent peak positions have moved away from each other by 0.56 frequency units. This type of shift, S, is defined as being a positive shift in Figs. 3-6.

into one. Each of the peak maxima is shifted inward from its true position by 1.0 frequency unit. Figure 2 shows a spectrum like Fig. 1 but for peaks separated by 1.0 frequency unit. It should be noted that, although Af is smaller in Fig. 2 than in Fig. 1, each apparent peak position is shifted outward by 0.56 frequency units. Figures 1 and 2 illustrate the differences between the composite peak maxima locations and the true locations for a particular case of damping ratio, true peak separation and windowing function. Since the frequency shift is a function of all of these parameters we have carried out a systematic examination of the frequency shift as a function of these variables. The results are summarized in Figs. 3-6. Figure 3 shows the frequency shift as a function of Af (Eq. 9) and of T/T for the rectangle window. The damping ratio range is from T/r = 0.0, which describes an undamped time signal to T/7= 3.0, which describes a time signal which has decayed to 5% of its initial amplitude and is essentially completely damped. Our results then cover all cases from und~p~ to essentially completely damped time signals. It should be noted that the apparent peak separation oscillates as Af increases. Some sort of oscillation is unsurprising since the rect~~e-apod~~d lineshape itself has prominent

192

RECTANGLE m 0-l

Fig. 3. Frequency shift as a function of spacing Af between two equal intensity peaks and the damping ratio T/r for rectanglaapodized (EQ. 4) magnitude-mode FT spectra. Both the shift scale and the frequency separation scale Af are in dimensionless units of l/T, where T is the acquisition time of the time domain signal. The frequency shift displayed in the Figure is labeled S in Figs. 1 and 2. A positive shift “to the outside” is illustrated in Fig. 2. A negative shift “to the inside” is illustrated in Fig. 1. The largest shifts are -0.25 (at a peak separation of 0.5) and +0.25 (at a peak separation of 0.75). For the data in this Figure the shift is 0.0625 where Af = 4.0 but is negligible for the shifts in Figs. 4-6.

auxiliary maxima. What is surprising, and is demonstrated here for the first time, is the oscillation in peak separation for two peaks which have been apodized by the H amming (Fig. 4), Blackman-Harris (Fig. 5) and Kaiser-Bessel (Fig. 6) windows. The most positive shifts and the most negative shifts for the cases in each of Figs. 3-6 are indicated in the appropriate Figure legend. The preceding paragraph emphasized the “anomalous frequency shift” for small (less than 3.0) values of Af. It should be noted, however, that the anomalous frequency shift S is greater at large (greater than 3.0) values of Af for the rectangle (unapodized) case illustrated in Fig. 3 than for any of the apodized cases illustrated in Figs. 4-6. For example, two heavily damped (T/r = 3.0) transients for which Af = 3.0 have a spectrum in which S is 0.125 for the rectangle window but is less than 0.031 for any of the apodizing windows. The frequency shifts at large values of the peak separation for unapodized spectra and the corresponding lower shifts for apodized spectra illustrate the importance of apodization [13,14] as a general FT-ICR data handling procedure.

193

HAMMING

Fig. 4. Frequency shift as a function of spacing Af between two equal intensity peaks and the damping ratio T/r for Hamming-apodized(Eq. 5), magnitude-modeFT spectra. Both the shift scale and the frequency separation scale Af are in dimensionlessunits of l/T, where T is the acquisitiontime of the time domain signal. The frequency shift displayedin the Figure is labeled S in Figs. 1 and 2. A positive shift “to the outside” is illustrated in Fig. 2. A negative shift “to the inside” is ihustrated in Fig. 1. The largest shifts are -0.25 (at a peak separation of 0.5) and + 0.38 (at a peak separation of 1.0). The shift is negligibleif Af > 3.0.

Phase dependence of the frequency shifi Equation 8 describes the tw~fr~uency signal which was used to generate Figs. 1-6. A more general signal is the following equation, which allows for a difference in phase between the signals

f(t) =

[cost w,t + 8,) + cos(w,t

The phase factor 8/T

-t S,)] exp( $)

O
between the signals, given by

also affects the appearance of two closely spaced magnitude peaks [15,16]. This effect of phase for moderately damped (T/T = LO), unapodized and Blacks-adds-apo~ed overlapping magnitude lineshapes is illustrated in Figs. 7-10. Figure 7 shows the ET magnitude spectrum of two overlapping peaks which were calculated from the magnitude FT of Eq. 10 with Af of 2.0 (Eq. 9) and 8/T equal to 0.0 (Eq. 11). The broken curves gives the rectangle-apodized (unapodized) spectrum; the full curve is the Blackman-

194

BLACKMAN-HARRIS

3 TERM

Fig. 5. Frequency shift as a function of spacing Aj between two equal intensity peaks and the damping ratio T/r, for Blackman-Harris-apodized (Eq.6) ma~tude-mode FT spectra. Both the shift scale and the frequency separation scale Aj are in dimensionless units of l/T, where T is the acquisition time of the time domain signal. The frequency shift displayed in the Figure is labeled S in Figs. 1 and 2. A positive shift “to the outside” is illustrated in Fig. 2. A negative shift “to the inside” is illustrated in Fig. 1. The largest shifts are - 1.0 (at a peak separation of 2.0) and +0.56 (at a peak separation of 1.0). The shift is negligible if Aj > 3.0.

Harris-apodized spectrum. Except for the logarithmic intensity scale in Fig. 7 and the linear intensity scale in Fig. 1, the Blackman-Harris spectra are identical in the two figures. The logarithmic scale in Fig. 7 was chosen to display the rectangle lineshape together with the wide dynamic range of the Blackman-Harris lineshape. The heavy lines in Fig. 7 indicate the positions of the cyclotron frequencies (Eq. 10) of the individual components of the magnitude spectrum. Also displayed in Fig. 7 are the frequency shifts, S, and %l for the rectangle and Blackman-Harris lineshapes respectively. Figures 8-10 show spectra which are like those in Fig. 7 but arise from different values of B/T (Eq. 11). B/T is n/2 for Fig. 8, v for Fig. 9 and 3?r/2 for Fig. 10. It should be noted that although Af, at a value of 2.0, is constant for all spectra in Figs. 7-10, the positions of the apparent peak maxima for both the rectangle lineshape and the Blackman-Harris lineshape shift as 8/T changes. DISCUSSION

In any spectroscopic experiment it is desirable to measure and to record spectral positions as accurately as possible. Normally, there is no special

195

KAISER-BESSEL

Fig. 6. Frequency shift as a function of spacing Af between two equal intensity peaks and the damping ratio T/T for Kaiser-Bessel-apodized (E@. 7) magnitude-modeFT spectra. Both the shift scale and the frequencyseparation scale Af are in dimensionlessunits of l/T, where T is the acquisition time of the time domain signal. The frequency shift displayed in the Figure is labeled S’ in Figs. 1 and 2. A positive shift “to the outside” is ilh&rated in Fig. 2. A negative shift “to the inside” is ilhrstrated in Fig. 1. ‘Ihe largest shifts are - 1.0 (at a peak separation of 2.0) and + 0.69 (at a peak separation of 1.0). The shift is negligibleif Af > 3.0.

meaning to be inferred from a precise and accurate measure of the spectrum. In mass spectroscopy, where line positions are directly related to molecular masses, spectral position is meaningful in an absolute sense which is usually absent in other forms of spectroscopy. In ICR mass spectroscopy the experimental measurement is one of frequency, which can be related to mass via the cyclotron equation

&)_!i!! m

@Iunits)

w-4

or equivalently f= 1.535611 x lo7 $

Wb)

Equation 12b gives f, the ICR frequency in hertz as a function of q, the ion charge in units of the electronic charge; B, the magnetic field in tesla and m, the ion mass in daltons. Equation 12a is the corresponding equation in SI units for w, the (angular) cyclotron frequency. It follows, therefore, that a thorough understanding of the factors which could shift the measured

196

0/T

= on, 27T

too

BLACKMAN-HARRIS

FREQUENCY

Fig. 7. Phase dependence of overlapping magnitude peaks. The broken curve is the FT magnitude spectrum of Eq. 10, which was unapodized (E@. 4) prior to Fourier transformation. The fuIl curve is the Blacks-His-ap~ed mag~tude spectrum of Eq. 10. The damping ratio T/r= 1.0 for both curves. The heavy lines indicate the center frequencies w,/2rr and u2/2r (Eq. 10) of the two components which make up the two-peak spectra. The phase factor B/T (Eiq. 11) = 0. S, (+0.13) is the shift in frequency from the true value of a peak of a component of the rectangle-apodized Iineshape. Sa, (- 1.0) is the corresponding shift for the Blacks-His spectrum. The phase factor B/T has different values in Figs. 8-10. It should be noted that SR f S,, and that Sa and Snn in this Figure differ from their values in Figs. S-10.

cyclotron frequencies from those expected from Eq. 12 is of the utmost importance in FT-ICR mass spectroscopy.As is true in any form of spectroscopy, random FT-ICR amplitude noise will introduce a random error in FT-ICR frequency [17]. One well-understood cause of systematic deviation from E!q 12 is the ICR trapping-field shift [18,19]. In FT-ICR spectroscopy an important source of systematic inaccuracy in both position and intensity comes from the discrete nature of the experimental FT-ICR spectrum [11,12,20,21]. The spectrum is not continuous but rather is defined only at the discrete frequencies f, given by Eq. 3. Frequency and intensity errors wili occur if the true frequency falls between two discrete frequencies, as it always does. This error can be minimized by zero-filling [11,12,21] and/or interpolation [12,20,21] and is also well documented in the literature. A separate and previously unrecognized source of systematic frequency inaccuracy is described in this work. The examples illustrated in Figs. l-10 indicate that care should be taken when frequency (mass) data from magnitude l&reshapes are to be interpreted if the peaks are less than about 3/T frequency units apart. The spectral frequencies wiIl be shifted by the

197

e/T

= n/2

-

0.01

I,, -10

I.,

-5

BLACKMAN-HARRIS

I....,.

0

+5

FREQUENCY

+10

I

Fig. 8. Phase dependence of overlapping magnitude peaks. The heavy lines which are at the same positions as in Figs. 7,9 and 10 indicate the center frequencies w1/2s and wJ2a (Eq. 10) of the two components which make up the two-peak spectra. The phase factor e/T (Eq. 11) = r/2. The broken and full curves are as in Fig. 7. S, (-0.03) is the shift in frequency from the true value to a peak of a component of rectangle-apodized Iineshape. San (+ 0.06) is the corresponding shift for the Blackman-Harris spectrum. It should be noted that S, # San and that Sa and San in this Figure differ from their values in Figs. 7, 9 and 10.

8/T =

7T

Fig. 9. Phase dependence of overlapping magnitude peaks. The heavy lines which are at the same positions as in Figs. 7, 8 and 10 indicate the center frequencies o,/2n and w2/2rr (Eq. lo), of the two components which make up the two-peak spectra. The phase factor B/T (Eq. 11) = s. The broken and full curves are as in Fig. 7. Sa (-0.17) is the shift in frequency from the true value to a peak of a component of the rectangle-apodized Iineshape. San (+0.17) is the corresponding shift for the Blackman-Harris spectrum. It should be noted that S, # S,, and that Sa and S,, in this Figure differ from their values in Figs. 7, 8 and 10.

198

e/T

= 31i/2

%

Fig. 10. Phase dependence of overlapping magnitude peaks. The heavy lines, which are at the same positions as in Figs. 7,8 and 9, indicate the center frequencies w1/2n and oJ2r (Eq. 10) of the two components which make up the two-peak spectra. The phase factor B/T (I$. 11) = 3~/2. The broken and fuII curves are as in Fig. 7. Sa (+0.03) is the shift in frequency from the true value to a peak of a component of the rectangleapodized Iineshape. San ( - 0.14) is the corresponding shift for the Blackman-Harris spectrum. It should be noted that S, # San and that Sa and San in this Figure differ from their values in Figs. 7, 8 and 9.

phenomenon of this work and conversion to masses could lead to incorrect molecular formulae. The origin of the shifted frequencies derives from the non-additive nature of magnitude spectra [15,16]. While absorption spectra and dispersion spectra are each separately additive, the magnitude spectrum M of two overlapping magnitude lines is given by [16] M = {M; +

+ M; + 2[ (44,

(A,& -A,D,)

+ D,D,)

sin(B, -

cos( @I- e,)

e2)]}1’2

(1%

the individual magnitude, absorption and dispersion spectra respectively are each centered on We. Only when the individual lines are far apart do the cross terms become negligible and the magnitude spectrum (Eq. 13) reduces to

where Wc2j, Alc2)ad 42~

M= (M,z+M;)1’2

(14)

which is the magnitude spectrum of two separate magnitude lines [16]. As mentioned in the introduction, FT-ICR spectra are usually presented in the magnitude mode. This is because in FT-ICR spectroscopy, the phase

199

corrections required for producing the absorption spectrum are more difficult than in for example, FT-nuclear magnetic resonance spectroscopy. The magnitude spectrum has a slightly lower resolution than the absorption spectrum [22,23] but this is a minor disadvantage in FT-ICR spectroscopy as FT-ICR is an ultrahigh resolution technique [18,1-g]. A more serious disadvantage of FT-ICR magnitude presentation is illustrated in Figs. l-10 of the present work. Apodization of magnitude lineshapes is done to minimize the interference of one spectral peak with nearby peaks. It follows, therefore, that apodization will be most needed when peaks are closely spaced. Unfortunately, as illustrated in Figs. l-10, this is exactly the situation in which anomalous magnitude frequencies are most likely to occur. Furthermore, the anomalous frequency shift phenomenon increases with increasing dynamic range of the window (Figs. 3-6), for small ( Af< 3.0) peak separations. Initially, it might appear that “correction nomographs” could be developed from which the true spectral positions could be inferred from experimental magnitude data. These nomographs could, for example, be easily developed from the data in Figs. 3-6. However, all of the figures of this work were developed for the single case of two peaks of equal intensity. The phase dependence of the spectra (Figs. 7-10) adds a further level of parametric complexity. The large number of nomographs needed to describe more than two overlapping peaks or even two peaks of unequal intensity renders the nomograph approach unworkable. Figures 3-10 show that the magnitude and sign of the frequency shift of this work are functions of the apodizing window and this fact provides a clue as to how the presence of the anomalous shift of this work may be confirmed whenever it is suspected, for example by noting two magnitude peaks which are within three units of l/T of each other. Changing the windowing function will cause a shift in peak location of the anomalous shift of this work is operative but not if the peaks are far enough apart for the situation in Eq. 14 to apply. The FT-ICR spectrometer is inherently a “frequency meter” which can be used as a “mass meter” only through Eq. 12 and its related corrections. Differentiating Eq. 12 with respect to mass and equating infinitesimals with small changes in mass or frequency gives, after omitting the minus sign arising from the inverse relationship between w and m [22], Am

_

Awm_ 2;;p2 w

(SI units)

(15)

Equation 15 gives, in SI units, the relationship between a mass scale increment Am and a frequency increment Ao in rad s-i or the dimension-

200

less frequency Af. Assigning the dimensionless frequency l-10) to the frequency increment Af (Eq. 15) gives Am=-

2&?z= @T

(SI units)

shift 5’ (Figs.

064

or equivalently Sm2 Ana = 6.512064 x 1O-8 4BT

(16b)

Equation 16b gives Am, the anomalous mass shift of this work in dahons as a function of S (Figs. l-lo), m, B, q and T. Equation 16a is the corresponding equation in SI units. For example, consider the case where 4 = 1 electronic charge, B = 1 T and T = 1 s. For a frequency shift S (Figs. l-10) of 0.5, the mass shift Am (Eq. 16b) would be 0.0003256 daltons at 100 daltons, 0.03256 daltons at 1000 daltons and 3.256 daltons at 10000 daltons. Mass accuracy to about a ~d~ton is usually needed for molecular formula determination. It follows from the above examples that the anomalous frequency shift described in this paper could obviate this determination for closely spaced peaks, particularly above mass 200. For a wide mass range FT-ICR spectrum, the acquisition time, T = 1.0 s, of the preceding paragraph is about the maximum value permitted by the current generation of FT-ICR computers. The sampling rate (Eq. 2), which must satisfy the Nyquist criterion, is proportional to B. The maximum acquisition time T is limited by the amount of available computer memory (Eq. 2) and is thus inversely proportional to B. AU is proportional to B. For an FT-ICR experiment with a lower mass limit of 46 daltons, and a 3 T magnet, the sampling rate (Eq. 2) must be at least 2 MHz. The acquisition time will then be limited to 0.125 s for a 256 K word data table or 0.5 s for a 1 M word data table. It follows then that for the current gen~ation of FT-ICR instruments, which have memories of this size, at masses of a few hundred or above the anomalous mass shift of this work could prevent accurate mass determination for closely spaced FT-ICR peaks. It is worth discussing methods to obviate (not merely detect and confirm) the anomalous shift of this paper. Larger computer memories will help, as discussed in the preceding paragraph. The spectral segment extraction (mixer or heterodyne) technique [18] allows longer acquisition times but with a restricted mass range. The latter technique is feasible today; the former will occur with the next generation (32 bit) FT-ICR computers. The anomalous shift used in this work is characteristic of magnitude spectra but not of absorption spectra. For pulse excitation [24], where ICR motion is simultaneously excited throughout the excitation bandwidth, we

201

could expect the real part of the complex FT to be identical with the absorption spectrum and the imaginary part of the complex FT to be identical with the dispersion spectrum. However, because of (a) finite delays between the end of FT-ICR excitation and the start of FT-ICR detection and (b) frequency-d~end~t phase shifts t~ou~out the detection electronics, the experimental real spectrum, Fa and the experimental imaginary spectrum Fr are both weighted sums of the true absorption and dispersion spectra [20,25]. The desired pure absorption spectrum (A(w)) is then, in experimental practice, an admixture of the experimental real and imaginary spectra A(w)=cos

@F,+sinBF,.

07)

If 6 in Eq. 17 is not zero, FR and _FIare said to be “phase distorted” [20]. If it is assumed that phase shift 8 at each frequency o can be given by a sum of zero and first-order terms

e(~~=e~+e~#

081

then the spectra can be phased manually by the operator in a straightforward manner [20]. For a multipeak spectrum which was pulse excited, Eq. 18 should suffice to give a properly phased spectrum, at least over a moderate mass range. Pulse excitation may be implemented either directly [24] or by the stored waveform inverse Fourier transform (SWIFT) [26] technique. For a two-peak spectrum excited by any method Eq. 18 will always suffice to give a pure absorption spectrum [27], provided that the higher order phase variation across a single peak can be neglected. It is apparent from Eq. 16 that the anomalous mass shift will be more serious at higher masses. The pulse power required to excite a given bandwidth is proportional to that bandwidth [28] and at higher masses a larger mass range can be excited with a given pulse power level. This follows from Eq. 15 by noting that the mass range Am, which corresponds to a given frequency range Aw, is directly proportional to the square of the ion mass. Therefore, at high masses, where the anomalous frequency shift of this work would be most serious, pulse excitation followed by zero and first-order phase correction of the absorption spectrum using Eq. 18 may provide a suitable means of obviating the anomalous frequency shift of this work. Finally, it should be noted that all of the equations in this work have neglected the “negative half’ of the Fourier spectrum. For an extremely high mass, the negative Fourier ICR frequency will overlap the positive Fourier ICR frequency and each frequency will be anomalously shifted by the effect described herein [29]. For this case of negative-positive overlap Af (Eq. 9) and Figs. l-6) = w (Eq. 12)/r.

202 SUMMARY

Closely spaced FT-ICR magnitude peaks will be anomalously shifted in frequency whenever the peaks are closer than 3/T, where T is the acquisition time of the FT-ICR time signal. For closely spaced peaks above mass 200 the shift may prevent an exact mass determination. The anomalous shift phenomenon can be expected whenever FT-ICR magnitude peaks are closer than 3/T and can be confirmed by observing a dependence of peak location upon window function. Absorption-mode presentation will obviate the anomalous shift. ACKNOWLEDGMENT This research was supported by the Natural Sciences and Engineering Research Council of Canada. REFERENCES 1 M.V. Buchanan (Ed.), Fourier Transform Mass Spectrometry: Evolution, Innovation and Applications, ACS Symposium Ser. 359, American Chemical Society, Washington, DC, 1987. 2 M.B. Con&row, Anal Chim. Acta, 178 (1985) 1. 3 D.A. Laude, Jr., D.L. Johhnan, R.S. Brown, D.A. WeiI and C.L. Wilkins, Mass Spectrom. Rev., 5 (1986) 107. 4 D.H. Russell, Mass Spectrom. Rev., 5 (1986) 167. 5 A.G. Marsha& Act. Chem. Res., 18 (1985) 316. 6 B.S. Freiser in J.M. Farrar and W. Saunders, Jr. (Eds.), Techniques for the Study of Ion Molecule Reactions, Wiley, New York, 1988. 7 M.L. Gross and D.L. Rempel, Science, 226 (1984) 261. 8 K.P. Wanczek, Int. J. Mass Spectrom. Ion Processes, 60 (1984) 11. 9 N.N.M. Nibbering, Comments At. Mol. Phys., 18 (1986) 223. 10 J.P. Lee, K.H. Chow and M.B. Comisarow, Anal Chem., 60 (1988) 2212. 11 M.B. Comisarow and J. Melka, Anal. Chem., 51 (1979) 2198. 12 C. Giancaspro and M.B. Can&row, Appl. Spectrosc., 37 (1983) 153. 13 J.P. Lee and M.B. Comisarow, Appl. Spectrosc., 41 (1987) 93. 14 M. Aarstol and M.B. Comisarow, Int. J. Mass Spectrom. Ion Phys., 76 (1987) 287. 15 S.L. Marple, Jr., Rec. 1977 Int. Conf. Acous., Speech and Signal Process., Hartford, CT, May 1977, p. 74. 16 J.P. Lee and M.B. Comisarow, J. Magn. Reson., 72 (1987) 139. 17 F.R. Verdun and A.G. Marshall, Appl. Spectrosc., 42 (1987) 715. 18 M.B. Comisarow, Adv. Mass Spectrom., 8 (1980) 1698. 19 E.B. Ledford, Jr., D.L. Rempel and M.L. Gross, Anal. Chem., 56 (1984) 2744. 20 M.B. Comisarow and J.P. Lee, Anal. Chem., 57 (1985) 464. 21 A. Serreqi and M.B. Gxnisarow, Appl. Spectrosc., 41 (1987) 288. 22 M.B. Comisarow and A.G. Marshall, J. Chem. Phys., 64 (1975) 110. 23 A.G. Marsha& M.B. Comisarow and G. Parisod, J. Chem. Phys., 71 (1979) 4434.

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